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Mair, and Juan P. Garrahan +School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK and +Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, +University of Nottingham, Nottingham, NG7 2RD, UK +We study the triangular plaquette model (TPM, also known as the Newman-Moore model) in +the presence of a transverse magnetic field on a lattice with periodic boundaries in both spatial +dimensions. +We consider specifically the approach to the ground state phase transition of this +quantum TPM (QTPM, or quantum Newman-Moore model) as a function of the system size and +type of boundary conditions. Using cellular automata methods, we obtain a full characterization of +the minimum energy configurations of the TPM for arbitrary tori sizes. For the QTPM, we use these +cycle patterns to obtain the symmetries of the model, which we argue determine its quantum phase +transition: we find it to be a first-order phase transition, with the addition of spontaneous symmetry +breaking for system sizes which have degenerate classical ground states. +For sizes accessible to +numerics, we also find that this classification is consistent with exact diagonalization, Matrix Product +States and Quantum Monte Carlo simulations. +I. +INTRODUCTION +In this paper, we study the ground state phase transi- +tion of the quantum Newman-Moore model, or quantum +triangular plaquette model. The classical triangular pla- +quette model (TPM), introduced by Newman and Moore +[1], is a model of Ising spins interacting in triplets in (half +of) the plaquettes of a triangular lattice. +Despite the +absence of quenched disorder and its trivial static prop- +erties, the model has rich glassy dynamics [1–3]. +The +TPM is an important model as it realises in an interact- +ing system the paradigm of slow (super-Arrhenius) re- +laxation at low temperatures due to effective kinetic con- +straints. This phenomenon is central to the dynamic fa- +cilitation picture of the glass transition [4–6]. The physics +of the TPM can also be generalised to three dimensions, +for example in the five-spin interaction square-pyramid +model [7], or maintaining the triangular interactions in +the models of Ref. [8]. A three-dimensional generalisa- +tion of the TPM with non-commuting terms [9] actually +started what is now the field of fractons [10, 11]. +The simplest way to transform the TPM into a quan- +tum model is by adding a transverse field term to the +classical Hamiltonian. Such quantum TPM (QTPM, or +quantum Newman-Moore model) was considered in the +context of fractons in Refs. [12, 13]. Numerics in Ref. [12] +suggested that the ground state undergoes a first-order +transition. A related work studying the large deviations +of plaquette observables in the stochastic dynamics of in- +dependent spins [14] also found numerical evidence for a +first-order transition at the self-dual point of the model. +In contrast, the results from Ref. [15] indicated that the +transition is continuous, with a particular form of fractal +symmetry breaking. The classical TPM and its connec- +tion with fractals and topological order was also drawn in +∗ ksfairopoulos@gmail.com +Ref. [16]. Here we aim to resolve these discrepancies by +exploiting a general connection between D-dimensional +cellular automata (CA) [17] and the ground states of +(D + 1)-dimensional classical spin models [18]. By us- +ing this method in the specific case of the QTPM with +periodic boundary conditions, we are able to character- +ize the approach to its quantum phase transition in the +large size limit. Our key observation is that the nature of +the transition depends on the specific lattice dimensions, +and this is manifested in the finite size scaling. +For the TPM, the relevant CA is Rule 60 [17] and +not Rule 90 that might be assumed from comment [18] +in Ref. [1]. For system sizes where one dimension is a +power of two, Rule 60 has a single fixed point [19], im- +plying a single energy minimum for the classical TPM. +In such cases, we verify that the quantum phase tran- +sition in the QTPM is first-order (that is, a sequence +of such system sizes tends to a first-order transition in +the large size limit). This also holds for other sizes for +which Rule 60 has no non-trivial attractors. However, +for certain sizes there can be periodic orbits on top of +the fixed point for the CA, giving rise to classical ground +state degeneracies in the TPM. For the quantum model, +this translates into a mixed order quantum phase transi- +tion. We provide evidence for this scenario by means of +numerical simulations, namely, for small sizes using ex- +act diagonalisation, and for large sizes using both Matrix +Product State approximations of the ground state, and +continuous-time Quantum Monte Carlo [20–23]. +The paper is organised as follows. In Sec. II, we review +the classical and quantum TPM. In Sec. III, we provide +the necessary background on CA and the connection to +the ground states of the classical TPM. In Sec. IV, we dis- +cuss the ground state phase transition of the QTPM in +terms of the symmetries that follow from the properties +of the associated CA, and support our predictions with +numerical simulations. In Sec. V we give our conclusions. +In the Appendix we provide further details, including a +discussion on the case of the QTPM with open bound- +arXiv:2301.02826v1 [cond-mat.stat-mech] 7 Jan 2023 + +2 +aries. +II. +TRIANGULAR PLAQUETTE MODEL, +CLASSICAL AND QUANTUM +A. +Classical +The triangular plaquette model (or TPM or Newman- +Moore model) [1–3] is a model of Ising spins si = ±1 +on the sites i = 1, . . . , N of a triangular lattice, with +cubic interactions between the spins on the corners of +downward-pointing triangles of the lattice, Fig. 1. The +Hamiltonian of this classical model reads +ETPM = −J +� +i,j,k∈▽ +sisjsk. +(1) +In what follows, it will be convenient to consider the +equivalent model on a square lattice of size N = L × M, +with classical Hamiltonian, +ETPM = −J +L,M +� +x,y=1 +sx,ysx+1,ysx+1,y+1, +(2) +where we assume periodic boundary conditions in both +directions by identifying +x + L = x +mod L +y + L = y +mod L. +In Eq.(2), we label the spins by sx,y at site with coordi- +nates (x, y) in the square lattice. +The classical TPM has been predominantly studied in +the context of the glass transition. For lattices with at +least one dimension being a power of two, the energy +Eq.(1) reduces to that of the non-interacting plaquette +variables [1–3], +ETPM = −J +� +▽ +d▽, +(3) +where d▽ = sisjsk with i, j, k ∈ ▽ for every downward- +pointing triangle in the lattice. When at least one di- +mension is a power of two, the relation between plaque- +ttes and spin variables is one-to-one, exactly proving the +above. The thermodynamics of the TPM is therefore one +of free binary excitations and, as such, it is essentially +trivial. +In contrast to the statics, the single spin-flip dynamics +of the TPM is highly non-trivial, as flipping one spin +changes three adjacent plaquettes. +This implies that +at low temperatures, where excited plaquettes are sup- +pressed, cf. Eq.(3), the dynamics has effective kinetic con- +straints [2]. These dynamical constraints lead to an ac- +tivated relaxation similar to that of the East model [24], +with relaxation times growing as the exponential of the +inverse temperature squared [2] (a super-Arrhenius form +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +FIG. 1: Triangular plaquette model. The shaded +triangles indicate the interacting triplets, Eq.(1). +Dotted lines indicate spins that interact for periodic +boundary conditions on an N = 4 × 4 lattice. +known as the “parabolic law” [25]). Similar glassy be- +haviour is seen in generalisations of the TPM with odd +plaquette interactions [8, 26]. The TPM has also been +considered in the presence of a (longitudinal) magnetic +field [27] and in the related case of coupled replicas [28], +and the TPM with open boundary conditions was studied +in Ref. [29] through partial trace methods. The classical +TPM was studied in the context of topological order in +fractal models in Ref. [16]. Autoregressive neural net- +works were applied to the classical TPM with limited +success in [30]. +B. +Quantum: TPM in a transverse field +Taking Eq.(1) and adding a transverse field, we obtain +the Hamiltonian of the quantum TPM (QTPM), +HQTPM = −J +� +i,j,k∈▽ +ZiZjZk − h +� +i +Xi, +(4) +where Zi and Xi represent Pauli operators acting non- +trivially on site i. This quantum model was studied in +Refs. [12, 13] and its connections to models of fractons +were investigated. +The Hamiltonian (4) is expected to have a quantum +phase transition at the self-dual point J = h [14]. Nu- +merical results from [12, 14] suggested the transition to be +first-order. Ref. [14] used trajectory sampling in systems +with linear size a power of two and periodic boundaries. +In contrast, Ref. [15] found evidence for a continuous +phase transition with fractal symmetry breaking using +stochastic series expansion methods, also with periodic +boundary conditions but not restricted to power of two +sizes. The study of QTPM was connected to Rydberg +atoms in Ref. [31]. Further studies on the QTPM and its +generalisations from the viewpoint of fracton field theory +were presented in Refs. [32, 33]. + +3 +III. +CELLULAR AUTOMATA AND GROUND +STATES OF THE CLASSICAL TPM +A. +General aspects of CA +Cellular automata (CA) consist of a D-dimensional ar- +ray of sites evolving under discrete-time and synchronous +dynamics given by a CA rule. Under this evolution, the +state is updated by a deterministic rule which is local in +time (although generalisations exist) [17, 34–36]. As a +result, CA dynamics gives rise to, often rich, (D + 1)- +dimensional structures. +In what follows, we consider +D = 1, linear CA with deterministic local transition rules +with sites taking values from the finite field F2. +Discrete cellular automata are fully specified by an ini- +tial configuration of L sites and a local transition rule. +The local update rule determines the configuration of ev- +ery site at each timestep using the local neighbourhood +of size r. For D = 1, the simplest CA rules are those +defined only by r = 1. +For a neighbourhood of three +sites, r = 1, there are 8 possible configurations giving +rise to 256 possible choices for update rules. These ele- +mentary CA were classified by Wolfram [17, 34]. Figure +2(a) shows Rule 60, which will be the relevant one for +the TPM: if we identify the empty and occupied sites of +the CA with the up and down spins of the TPM, then +Rule 60 is the same as the condition that the product of +spins in Eq.(2) is one, thus maximising the local energy. +Figure 2(b) also shows the closely related Rule 90. CA +like these are called elementary [34]. +Figure 3(a,b) shows the patterns generated for Rules +60 and 90 starting from an initial single seed. Note that +these depend on the boundary conditions (for a generic +analysis on cellular automata with periodic boundaries, +see Ref. [37]). For example, Fig. 3(b) shows Rule 90 for +L = 64 and periodic boundaries: the timestep after the +last one shown will take the CA to the trivial, empty +configuration; in contrast for a length L = 63 the Sier- +pinski fractal [38] will continue. Similarly, time evolution +will bring in one timestep the pattern of Fig. 3(a) to the +trivial one for periodic boundaries with L = 32, but the +fractal shape would continue to be reproduced for open +boundaries. Focusing on Rules 60 and 90 for concrete- +ness, CA evolution can be written as polynomials (or +generating functions, see Ref. [34]), as +T60(x) = 1 + x +(5) +T90(x) = x−1 + x, +(6) +with the evolution of the whole lattice obeying [34] +A(t)(x) = TA(t−1)(x) +mod (xL − 1). +(7) +Given a configuration at timestep t for the evolution +of any given rule, the configuration at timestep t + 1 will +be its successor and the one at t − 1 its predecessor. For +Rules 60 and 90, following Ref. [34], we have: +• There are no predecessors for configurations with +an odd number of sites being equal to 1 for both +(a) +(b) +FIG. 2: (a) Rule 60 and (b) Rule 90 evolution rules. +(a) +(b) +(c) +FIG. 3: (a) Evolution from a single site for Rule 60. (b) +Same for Rule 90. (c) A stable cycle generated by Rule +60. +rules. For Rule 90, 2L−1 configurations for a system +of size L cannot be reached if L is odd and 3 × +2L−2 if L is even. For Rule 60, 2L−1 configurations +cannot be reached for any system size. This means +that rows with an odd number of ones can only +occur as initial states for any given time evolution +for Rule 60. +• Similarly, there may exist configurations that can +be reached in one timestep by more than one pre- +decessors. More specifically, given a configuration +with at least one predecessor, there are 2 predeces- +sors if the length L is odd and 4 if it is even for +Rule 90, while there are always 2 for Rule 60. +Regarding the length of cycles for Rule 90, more results +can be found in Ref. [34]. For Rule 60, which is most +relevant for the TPM, we discuss the cycle lengths and +their multiplicities next. +B. +Periodic orbits of Rule 60 +The iterated map +Dn(x) = (|x1 − x2|, |x2 − x3|, ..., |xL − x1|) +(8) +for Dn : Zn → Zn and x = (x1, x2, ..., xL) is known as the +Ducci map (also known as rule 102 in Wolfram’s nota- +tion [17, 34], the mirror image of Rule 60, e.g. Ref. [39]). +Ducci’s map (and therefore Rule 60) exhibits periodic +orbits or evolves to a fixed point depending on L being +a power of two or not. +For L = 2k, any initial state + +4 +(b) +(a) +(c) +FIG. 4: The fixed point and limit cycles of Rule 60. Filled circles indicate states of the CA and the arrows the flow +under the dynamics. (a) For L = 5 there is a cycle of length C = 15 and the fixed point (which flows into itself). (b) +For L = 6 there are two cycles of period C = 6, a cycle of period C = 3 and the fixed point. (c) For L = 8, there is +the fixed point and no cycles. +evolves to the trivial state of zeros [40], while for L ̸= 2k +it evolves to a richer attractor structure [41]. +In F2 and by using periodic boundary conditions, the +Ducci map can be brought into matrix form as follows, +Dn x = ((x1 + x2), (x2 + x3), · · · , (xL + x1)) +mod 2, +(9) +where +Dn = +� +������� +1 1 0 . . . 0 0 0 +0 1 1 . . . 0 0 0 +... +... +... ... ... +... +... +0 0 0 . . . 1 1 0 +0 0 0 . . . 0 1 1 +1 0 0 . . . 0 0 1 +� +������� +. +(10) +The matrix D can be expressed as +D = I + SL +(11) +with SL the left shift map [19, 41]. It is easy to check +that +D2k = (I + SL)2k += I + S2k +L = I + I = 0 +mod 2, (12) +with k ∈ Z and, thus, a system that has L a power of +2 ends up in the trivial configuration. For Rule 60 the +corresponding matrix is D⊺. +In order to obtain the cycles of Rule 60, we need +the following definitions and results for Rule 102 from +Refs. [19, 35, 36, 41–44]: +• For any given CA, each array of sites can be +thought of as a vector, v. For any such vector v in +F2, its order is defined through the monic polyno- +mial, which satisfies µv(D)v = 0. +• The order of the minimal annihilating polynomial, +ord µv(λ), is equal to the smallest natural number +n, such that µv(λ)|λn − 1. If µv(0) = 0, then for +µv(λ) = λk˜µv(λ) with ˜µv(0) ̸= 0 and k ∈ N, the +order of µv is ord (µv) = ord (˜µv). +• Assuming that µv(λ) = λk˜µv(λ) with k ≥ 0, then +the k-th successor of v belongs to a cycle of length +c = ord µv. This applies to a vector v of any pos- +itive integer L and any linear map (or cellular au- +tomaton rule). +The minimal annihilating polynomial for the Ducci +map was calculated in Refs. [19, 43], based on the char- +acteristic polynomial of the matrix D. It was found that +µn(λ) = pn(λ) = (1 + λ)L + 1. +(13) +We are now in a position to obtain the attractor struc- +ture of Rule 60 by following Ref. [36] (specifically “Prin- +ciple C”). We decompose the minimal annihilating poly- +nomial into the product of its irreducible polynomials, +πi(λ). We call their polynomial powers bi. Thus, +µv(λ) = +m +� +i=1 +πi(λ)bi, +(14) +and +ord µv(λ) = rpt, +(15) +where r is the least common multiple of ord πi and t the +smallest integer satisfying pt ≥ max (b1, b2, . . . , bm). +Figure 4 illustrates the different scenarios for cycles as +a function of L. We show three different sizes, L = 5, 6, 8, +small enough to be able to visualise the network of states. +Configurations of the CA are identified by blue circles, + +5 +1 +1 +0 +2 +1 +0 +3 +1 +3 +3 +4 +1 +0 +5 +1 15 +15 +6 +1 +3 +6 +15 +7 +1 +7 +63 +8 +1 +0 +9 +1 +3 63 +255 +10 +1 15 30 +255 +11 +1 341 +1023 +12 +1 +3 +6 12 +255 +13 +1 819 +4095 +14 +1 +7 14 +4095 +15 +1 +3 +5 15 +16383 +16 +1 +0 +17 +1 85 255 +65535 +18 +1 +3 +6 63 126 +65535 +19 +1 9709 +262143 +20 +1 15 30 60 +65535 +21 +1 +3 +7 21 63 +1048575 +22 +1 341 682 +1048575 +23 +1 2047 +4194303 +24 +1 +3 +6 12 24 +4095 +25 +1 15 25575 +16777215 +26 +1 819 1638 +16777215 +27 +1 +3 63 13797 +67108863 +28 +1 +7 14 28 +16777215 +29 +1 475107 +268435455 +30 +1 +3 +5 +6 10 15 30 +268435455 +31 +1 31 +1073741823 +32 +1 +0 +33 +1 +3 341 1023 +4294967295 +34 +1 85 170 255 510 +4294967295 +35 +1 +7 15 105 819 4095 +17179869183 +36 +1 +3 +6 12 63 126 252 +4294967295 +37 +1 +3233097 +68719476735 +38 +1 9709 19418 +68719476735 +39 +1 +3 455 819 1365 4095 +274877906943 +40 +1 15 30 60 120 +16777215 +TABLE I: Cycle lengths for Rule 60. The second +column indicates the cycle lengths and the third the +total number of periods for Rule 60 (each period of +length C is counted C-times), assuming that the least +common multiple of the periods for each given L divides +M, lcm C|M. The last column is constructed based on +the number of predecessors for all configurations for +Rule 60, given a length L. +and arrows indicate to which configurations they evolve +to under the CA dynamics. Figure 4(a) shows L = 5: +here there is one fixed point to which one other state +evolves to (shown at the centre of the figure), and one +limit cycle of period C = 15, to which all other configura- +tions flow. Figure 4(b) shows a more general situation of +multiple distinct cycles for the case of L = 6: there are +two cycles of period C = 6, one cycle of period C = 3 and +one fixed point. For the case of L a power of 2, as shown +in Fig. 4(c) for L = 8, there is a unique fixed point and +all states evolve towards it. +C. +Classical ground states of the TPM +From the analysis above for Rule 60, we can enumerate +all the minimum energy configurations of the TPM, cf. +Eq.(2). The classical ground states for a system of size +N = L × M are: (i) the state with all spins up, corre- +sponding to the fixed point of Rule 60, for any value of +L and M; (ii) two-dimensional spin configurations that +correspond to periodic trajectories of Rule 60, that is, +CA trajectories starting from any of the states of a limit +cycle, for all limit cycles whose period is contained an in- +teger number of times in M—this occurs only for certain +combinations of L and M (never if L is a power of two). +We show the relevant Rule 60 information in Table +I for up to L = 40. Under the column C we give the +distinct periods of the limit cycles. In the column labelled +by M we give the corresponding degeneracy of classical +ground states of the TPM, apart from the uniform spin- +up state, given that lcm C|M. For example, for L = 15, +there is one cycle of length 3, three cycles of length 5, +and 1091 cycles of length 15, which means 1 + M = +1+3+3×5+1091×15 = 16384 distinct two-dimensional +spin configurations that minimise the energy in a TPM +of size N = 15 × M as long as 15|M. In contrast, for +L = 15, if 5|M but 3 ∤ M and 15 ∤ M, then there are +1+15 different ground states. The symmetries of a TPM +of size N = L × M can be similarly constructed from +the number of ground states and their multiplicities, as +determined by Rule 60. +IV. +PHASE TRANSITION IN THE QUANTUM +TPM +As we will now show, the set of minimum energy con- +figurations and the associated symmetries of the classical +TPM, as obtained from the Rule 60 CA, determine the +properties of the ground state phase transition in the +quantum TPM, Eq.(4). +A. +Symmetries of the QTPM +Like its classical counterpart, the QTPM with periodic +boundaries has full translational invariance. The symme- + +6 +(b) +(a) +FIG. 5: (a) Symmetry operators for the 3 × 3 QTPM +with periodic boundaries. (b) One of the symmetry +operators of the 9 × 9 QTPM. +tries of the QTPM will then be deduced by the results of +Sec. III. +For systems with dimensions N = L × M, which can +accommodate non-trivial cycles of Rule 60, the symme- +tries of the corresponding QTPM easily follow. Consider +as a simple example the case of N = 3 × 3 with periodic +boundaries in both dimensions. +From Table I, we see +that L = 3 has one cycle of period 3. This means that +for M = 3 there are three non-trivial symmetries, given +by the operators +G1 = X1,2X1,3X2,1X2,2X3,1X3,3 +(16) +G2 = X1,1X1,3X2,2X2,3X3,1X3,2 +(17) +G3 = X1,1X1,2X2,1X2,3X3,2X3,3 +(18) +see Fig. 5(a). Note that translational invariance plays a +crucial role in the determination of the symmetries: in +the example above G2 = TxG1T −1 +x +and G3 = TxG2T −1 +x +, +where Tx is the translation operator in the x direction, +and these symmetries alongside the identity form the +Klein group K4, which, in turn, is isomorphic to Z2 ⊗Z2. +This approach generalises to other system sizes, and +the symmetries are products of K4. +For example, the +symmetries of the N = 5×15 and the N = 6×6 systems +form the group K4 ⊗ K4. Since the symmetry operators +are products of the X Pauli matrices, cf. Eqs.(16-18), +they commute with the transverse field term in Eq.(4), +and, therefore, are symmetries of the QTPM for all values +of J and h. +B. +The character of the quantum phase transition +The character of the phase transition for the QTPM +is now easy to predict based on the information of its +symmetries for a given system size. By this we mean: +given a sequence of increasing system sizes with periodic +boundaries, all having the same number of symmetries, +the progressively sharper finite size crossovers will be in- +dicative of an eventual phase transition in the large size +limit whose character—first-order or continuous—will be +determined by the underlying symmetries of the given +system sizes in the sequence. As these symmetries in a +system of size N = L × M depend on the precise values +of both L and M, this analysis has to be done carefully. +In the next section we show numerical evidence for the +general considerations we give here. +First consider the system sizes N = L × M such that +the underlying Rule 60 has only one fixed point, as de- +scribed previously. Then, the QTPM has no non-trivial +symmetries. In the limit J ≫ h, there is a single ground +state corresponding to the all-up state, and a vanishing +number of excited plaquettes, as the classical TPM has +a single minimum. +In the opposite limit, J ≪ h, the +ground state is paramagnetic, with spins aligned in the +x direction, with an explicit Z2 symmetry for its ground +state, corresponding to a high density of excited plaque- +ttes. +In the limit of large N, we expect the quantum +phase transition at the self-dual point J = h to be first- +order. +A second scenario is that of system sizes where the +underlying Rule 60 cycles give rise to non-trivial sym- +metries in the QTPM. In this case, in the limit J ≪ h, +the ground state is the same paramagnetic one as be- +fore, invariant under a global Z2 symmetry. +However, +for J ≫ h there exists spontaneous breaking of the sym- +metries emerging from Rule 60. +This case, therefore, +has characteristics of a first-order phase transition (the +explicit symmetry breaking of the global symmetry of +the paramagnetic ground state) with those of a contin- +uous transition (spontaneous symmetry breaking). This +is similar to the first-order phase transitions observed in +kinetically constrained models [45] and suggests that, for +local observables, the phase transition will appear first- +order; for example, in a discontinuous jump in the excited +plaquette density, +Mzzz = 1 +N +� +i,j,k∈▽ +ZiZjZk. +(19) +Appropriate operators will quantify the symmetry break- +ing of the degenerate classical ground states. +The choice of these operators depends on the size and +specific lattice dimensions of the system. Consider, for +example, the case of L = 3 and M = 3k with k ∈ N, +where we know from Rule 60 that there is a single fixed +point and the three non-trivial ground states. For the +trivial ground state this operator is just the magnetisa- +tion Mz = +1 +N +�N +i Zi. To detect the symmetry breaking +into the other three states, we can define the three op- +erators ˜ +M m +z = +1 +N +�N +i (−1)niZi, where ni = 1 if the spin +i is flipped for a state of the cycle m of Rule 60, and +ni = 0 otherwise. For example, for the state associated + +7 +to Eq.(16), we have +˜ +Mz = 1 +N (Z1,1 − Z1,2 − Z1,3 − Z2,1 − Z2,2 + Z2,3 +−Z3,1 + Z3,2 − Z3,3) . +(20) +Note that when there are multiple non-trivial ground +states connected by translations, there will be no sin- +gle operator taking the form of a sum of local terms for +which it will be possible to discern between them. For +example, in the N = 3k case, Mz will only distinguish be- +tween the trivial ground state and the 3-fold degenerate +ones, while the operators ˜ +M m +z +will be able to distinguish +between one of the non-trivial ground states. +C. +Numerics +We now provide evidence for the general observations +above from numerical simulations. +For small systems +we use exact diagonalization (ED) [46–48], allowing the +study of system sizes up to 28 sites. For larger systems +we estimate the properties of the ground states using +two different approaches. +The first of these is Matrix +Product States (MPS) [49–52], which we “snake” around +the 2D lattice, and optimize with the 2D Density-Matrix +Renormalization Group (DMRG) [53, 54]. By employing +a bond dimension up to D = 1000, we are able to reli- +ably estimate the ground state properties for system sizes +on the square lattice for up to N = 16 × 16. Time and +memory constraints hinder progressively the convergence +in the paramagnetic phase, where J ∼ h. As discussed +below, we are also able to apply MPS to cylindrical sys- +tems, which can be considered to be quasi-1D, allowing +us to reach much larger sizes than in the case of square +geometries. +To confirm the results of 2D DMRG, we also employ +Quantum Monte Carlo (QMC) methods. In particular, +we use the continuous-time expansion (ctQMC) [20], with +local spin updates which re-draw the entire trajectory +of a single spin, subject to a time-dependent environ- +ment, where the trajectories of unmodified spins are con- +sidered to act as a “heat bath”, e.g., see Refs. [21, 22]. +We run our simulations with an inverse temperature of +β = 128, which we find to be large enough to converge +to the ground state [23]. +1. +First-order transition +As explained above, when the underlying Rule 60 has a +single fixed point, and the classical TPM a single energy +minimum with all spins up, we thus expect the phase +transition of the QTPM to be first-order. +Figure 6 shows results for N = L×L with L = 4, 8, 16. +We show that our MPS and ctQMC results coincide with +the large deviations results of Ref. [14], obtained via tran- +sition path sampling (TPS). What is plotted is the pla- +quette density Mzzz as a function of the coupling J for +□□□□□□□□□□□□□□□□ +▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇ +■ +▼ +▼ +◆ +◆ +0.95 +1.00 +1.05 +1.10 +0.4 +0.6 +0.8 +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +□ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +□ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +▼ +▼ +▼ +▼ ▼ +▼ +▼ ▼▼ +▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ +◆ +◆ +◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ctQMC +□ +▽ +◇ +MPS +■ +▼ +◆ +TPS +△ +○ +FIG. 6: Normalized three-spin correlator, Eq.(19), in +the QTPM as a function of J for fixed h = 1, with +N = L × L with L a power of two. We compare results +from MPS and ctQMC obtained here with results from +Ref. [14]. The numerical data indicates a first-order +transition at J = h. +fixed transverse field h = 1. The data indicates a first- +order transition at J = h, as expected. For J > 1.0, we +see small deviations in the TPS results, due to the extra +field used for the acquisition of this data in Ref. [14]. Sim- +ilar issues are observed for ctQMC close to the J = 1.0 +point, where the single spin updates do not allow for the +collective effects necessary to move between phases. +In Fig. 7, we show the results for several system sizes +in a square geometry, N = L × L. Figure +7(a) shows +the ground state energy as a function of J (at h = 1) +for L = 5 to 16, obtained from MPS numerics. For the +smallest size we also show ED results, which coincide +with the MPS ones. The kink near J = 1 indicates a +quantum phase transition. Note that this behaviour is +similar in systems with a single classical ground state +(L = 5, 8, 16) or multiple ones (L = 6, 7), cf. Table I. In +Figs. 7(b,c) we show the average transverse magnetisa- +tion, Mx = 1 +N +� +i Xi, and Mzzz, respectively, for systems +with L a power of two. We get exactly the same results +for different system sizes too. +Both MPS and ctQMC +show clear indications of a first-order transition at J = 1 +in both observables. +Figure 8 shows similar results in a rectangular geome- +try, N = 3×M. For such thin stripe systems we can per- +form MPS more efficiently for larger system sizes than for +square geometries. Once again, MPS and ctQMC results +coincide, and indicate a first-order transition at J = 1 +(although weaker than in the square lattice case, in the +sense that the discontinuity in the local operators shown +is smaller). Note that these results include not only val- +ues of M which are multiples of three, for which there +are multiple classical ground state cycles, but also values +of M for which a single ground state is found. What we +see in this case is that the observables Mx and Mzzz are +unable to detect changes related to any given classical +ground states. + +8 +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +0.0 +0.5 +1.0 +1.5 +2.0 +-2.0 +-1.5 +-1.0 +(a) +△ +● +● +● +● +● +● MPS +△ ED +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +■ ■ ■ ■ ■ ■ ■ +■ +■ +■ ■ ■ ■ ■ ■ ■ +▮ ▮ ▮ ▮ ▮ +▮ +▮ ▮▮ +▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) +ctQMC +□ +▯ +△ +▽ +◇ +○ +MPS +■ +▮ +▲ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +▮ +▮ +▮ +▮ ▮ +▮ +▮ ▮▮ +▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(c) +ctQMC +□ +▯ +△ +▽ +◇ +○ +MPS +■ +▮ +▲ +FIG. 7: First-order transition in the QTPM for systems of size N = L × L. (a) The normalised by the system size +ground state energy as a function of J at fixed h = 1. Open symbols are results from ED, filled symbols from +numerical MPS. (b) Transverse magnetisation as a function of J. In this case the open symbols are from ctQMC. (c) +Average three-spin interaction as a function of J. +■ +■ +■ +■ ■ ■ ■ +■ +■ +■ +■ +■ +■ +▲ +▲ +▲ +▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ +▼ +▼ +▼ +▼ ▼ ▼ ▼ +▼ +▼ +▼ +▼ +▼ +▼ +◆ +◆ +◆ +◆ +◆ ◆ ◆ +◆ +◆ +◆ +◆ +◆ +◆ +● +● +● +● +● ● ● +● +● +● +● +● +● +■ +■ +■ +■ ■ ■ ■ +■ +■ +■ +■ +■ +■ +▲ +▲ +▲ +▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +▲ +▲ +▼ +▼ +▼ +▼ ▼ ▼ ▼ +▼ +▼ +▼ +▼ +▼ +▼ +◆ +◆ +◆ +◆ +◆ ◆ ◆ +◆ +◆ +◆ +◆ +◆ +◆ +● +● +● +● +● ● ● +● +● +● +● +● +● +0.0 +0.5 +1.0 +1.5 +2.0 +-2.0 +-1.5 +-1.0 +(a) +MPS +■ +▲ +▼ +◆ +● +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +□ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +○○○○○○○○○○○○○○○○○○○○ ○○○○○○○ +■ +■ +■ +■ ■ +■ +■ ■ +■ ■ +■ +■ ■ +▲ +▲ +▲ +▲ ▲ +▲ +▲ ▲ +▲ ▲ +▲ +▲ ▲ +▼ +▼ +▼ +▼ ▼ +▼ +▼ ▼ +▼ ▼ +▼ +▼ ▼ +◆ +◆ +◆ +◆ +◆ +◆ +◆ ◆ +◆ ◆ ◆ +◆ ◆ +● +● +● +● +● +● +● ● +● ● ● +● ● +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) +ctQMC +□ +△ +▽ +◇ +○ +MPS +■ +▲ +▼ +◆ +● +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +□ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +○○○○○○○○○○○○○○○○○○○○ ○○○○○○○ +■ +■ +■ +■ ■ +■ +■ ■ +■ ■ +■ +■ ■ +▲ +▲ +▲ +▲ ▲ +▲ +▲ ▲ +▲ ▲ +▲ +▲ ▲ +▼ +▼ +▼ +▼ ▼ +▼ +▼ ▼ +▼ ▼ +▼ +▼ ▼ +◆ +◆ +◆ +◆ +◆ +◆ +◆ ◆ +◆ ◆ ◆ +◆ ◆ +● +● +● +● +● +● +● ● +● ● ● +● ● +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(c) +ctQMC +□ +△ +▽ +◇ +○ +MPS +■ +▲ +▼ +◆ +● +FIG. 8: Same as Fig. 7 but for systems of size N = 3 × L. +2. +Symmetry breaking +In Figs. 7 and 8, we show the two terms that com- +pete in the Hamiltonian, Mx and Mzzz. For system sizes +where there is one classical ground state and no non- +trivial symmetries, the total longitudinal magnetisation +Mz can also serve as an order parameter, as it picks up +the orientation of the ground state. Figure 9(a) shows +that the transition is also clear for this observable for +square lattices. +For system sizes where degeneracies are expected for +J ≥ h, however, Mz is unable to detect the symme- +try breaking related to the extra symmetries. For these +cases, we need the staggered magnetisations, ˜ +M m +z , such +as that for N = 3 × 3 in Eq.(20). Figure 9(b) shows that +such operators are able to detect the spontaneous break- +ing of symmetry for these lattices. Note that Fig. 9(b) +was obtained through the use of a small symmetry break- +ing field. This is a standard method for the detection of +the symmetry breaking in the ground state of a degen- +erate quantum model [55]. As a result, the calculations +were performed through the use of a modified Hamilto- +nian H = HQTPM − p ˜ +Mz, where p is chosen to be small. +The detection of the spontaneous symmetry breaking can +be similarly preformed for any of the classical ground +states of the given lattice size with the appropriate oper- +ator ˜ +Mz. +In order to more clearly understand the mechanism of +the phase transition, in Figs. 10 and 11 we plot the low- +lying spectrum of the QTPM from ED as a function of +h for fixed J. +These results support our above obser- +vations: for system sizes where only a first-order phase +transition is expected, there is an avoided crossing be- +tween the ground state and the first excited state; for +system sizes with extra symmetries from the cycles of +Rule 60, we see both an avoided crossing (indicative of +first-order transitions) and a merging of eigenstates in- +dicative of spontaneous symmetry breaking. As seen in +Fig. 11 for the case of N = 3 × M, the avoided crossing +becomes apparent only with increasing system size. +We now comment on how our results compare to those +in Ref. [15]. For the numerics, Ref. [15] used a stochastic +series expansion (SSE) approach. We in turn use MPS +and ctQMC. Both SSE and ctQMC are Quantum Monte +Carlo based methods, which indicates that they, in prin- +ciple, should be able to roughly access system sizes of the +same order of magnitude. +Furthermore, while Ref. [15] also considered periodic +boundaries, there was no specific restriction on system +size, and therefore no distinction between sizes for which +there is a single classical minimum and sizes where there +are multiple ones, with the implications for symmetries of +the corresponding QTPM. Ref. [15] also used a non-local +order parameter, compared to our local ones (the stag- +gered magnetisations) that do reflect the minima of the +underlying TPM. In [15], the existence of a phase tran- +sition at J = h was confirmed through the study of the +Binder cumulant; this was done, however, with limited + +9 +accuracy on the location of the phase transition point. It +is important to note that some of the local observables +we calculate here are also studied for specific system sizes +in the Appendix of Ref. [15]. Since the temperature used +for those calculations varied for different system sizes, +it is possible that the smoothness observed in Ref. [15] +is a consequence of thermal effects. We instead used a +fixed inverse temperature β = 128 which we verified is +sufficient to make thermal effects negligible. +D. +Nature of the phase transition in the +thermodynamic limit +The discussion above and the numerical results indi- +cate the existence of a quantum phase transition in the +thermodynamic limit, N → ∞, at the self-dual point, +J = h, of the QTPM. However, the approach to the +thermodynamic limit is different across different system +size geometries. +There are three different limits to thermodynamics: (i) +across one of the two dimensions while the other one re- +mains constant (that is, infinite stripes), (ii) across both +dimensions, and (iii) on making the spins continuous. We +briefly discuss the differences between these limits and +the complications that might arise. +In the case (i), if the limit is taken for fixed L and with +M such that lcm C|M (e.g. M = 3k, with k ∈ N), the +number of classical ground states remains the same. In +our numerics we are restricted to narrow stripes to allow +convergence of the MPS algorithm. Fig. 8 suggests that +in such quasi-1D systems the transition will eventually +be slighly weaker than for square system sizes. +Case (ii) can be more involved. The simplest situation +is that of square lattices N = L × L with L a power of +two, where it is guaranteed that for all sizes there will be +a single classical ground state, and therefore the transi- +tion is certainly first-order. For other size sequences, the +number of relevant Rule 60 cycles, and therefore symme- +tries of the QTPM, may grow or decline with system size. +For some cases this growth is monotonic (as for example +for N = 3k × 3k with k → ∞), while in others it is not +(as for example when N = 3k × 3k with k → ∞), see +Table I. +In case (iii) the nature of the underlying CA is altered +[56, 57]. In this limit, Rule 60 becomes +f(p, q, r) = p + q − 2pq, +(21) +where p, q and r indicate the state of the three sites +in the neighbourhood determining the local evolution of +the CA, see Fig. 2. Basic arguments [56] indicate a single +fixed point in the evolution of this fuzzy CA. We spec- +ulate that the same behaviour will be observed in the +quantum field theory limit for QTPM; a single ground +state across different regions of the whole J − h space +and thus a first-order phase transition. However, a field +theoretic description of the QTPM might not be as obvi- +ous and straightforward to get for the above elementary +argument to hold. +V. +CONCLUSIONS +In this work, we used the cycles of the cellular automa- +ton Rule 60 to describe the symmetries of the quantum +triangular plaquette model. We found that the attrac- +tor structure of Rule 60 plays an important role in the +characterization of the degeneracies of the ground states +of the classical TPM, allowing in turn to construct the +symmetry operators of the QTPM. In this way, the exis- +tence or absence of stable cycles in Rule 60 imply whether +it is possible or not for the QTPM to display sponta- +neous symmetry breaking of the corresponding symme- +tries, which in turn impacts on the nature of the quantum +phase transition at the self-dual point. These general ob- +servations are also consistent with the finite size trends +from our numerical simulations. A full description of the +QTPM phase transition would require a field theoretical +description and a renormalization group treatment; we +leave these tasks for future works. +ACKNOWLEDGMENTS +We thank L. Vasiloiu for insightful comments. +We +acknowledge financial support from EPSRC Grant no. +EP/R04421X/1, the Leverhulme Trust Grant No. RPG- +2018-181, +and University of Nottingham grant no. +FiF1/3. LC was supported by an EPSRC Doctoral prize +from the University of Nottingham. +Simulations were +performed using the University of Nottingham Augusta +HPC cluster, and the Sulis Tier 2 HPC platform hosted +by the Scientific Computing Research Technology Plat- +form at the University of Warwick (funded by EPSRC +Grant EP/T022108/1 and the HPC Midlands+ consor- +tium). +[1] M. E. J. Newman and C. Moore, Glassy dynamics and +aging in an exactly solvable spin model, Phys. Rev. E 60, +5068 (1999). +[2] J. P. Garrahan and M. E. J. Newman, Glassiness and +constrained dynamics of a short-range nondisordered spin +model, Phys. Rev. E 62, 7670 (2000). +[3] J. P. Garrahan, Glassiness through the emergence of +effective dynamical constraints in interacting systems, +Journal of Physics: Condensed Matter 14, 1571 (2002). + +10 +□□□□□□□□□□□□□□□□□□□□□□□□□□ +□ +□ +□ +□ +□ +□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +▮ +▮ +▮ +▮ ▮ +▮ +▮ ▮▮ +▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀ +◀ +◀ +◀ +◀◀◀◀ +◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀ +◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀ +◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ +◆ +◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ +◆◆ +◆◆◆◆◆◆◆ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +▮ +▮ +▮ +▮ ▮ +▮ +▮ ▮▮ +▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ +▲ +▲ +▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +□□□□□□□□□□□□□□□□□□□□□□□□□□ +□ +□ +□ +□ +□ +□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 (a) +ED +◀ +◆ +MPS +■ +▮ +▲ +ctQMC +□ +▯ +△ +▽ +◇ +○ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 (b) +MPS +● +● +● +● +● +FIG. 9: (a) Longitudinal magnetisation for systems with no symmetries. (b) Staggered magnetisation for detecting +symmetry breaking in systems with multiple symmetries. +□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△ +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +-18 +-16 +-14 +-12 +(a) +Energy Levels +□ +1 +▯ +2 +△ 3-4 +□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△ +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +-22 +-20 +-18 +-16 +(b) +Energy Levels +□ +1 +▯ +2 +△ 3-4 +□ □ □ □ □ □ □ □ +□ +□ +□ +□ +□ +▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ +△ △ △ △ △ △ △ △ △ △ △ △ △ +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +-30 +-28 +-26 +-24 +-22 +(c) +Energy Levels +□ +1 +▯ +2 +△ 3-4 +FIG. 10: Low-lying spectrum of the QTPM as a function of h for fixed J = 1 from ED, for sizes without extra +symmetries. (a) N = 3 × 4. (b) N = 4 × 4. (c) N = 3 × 7. The avoided crossing between the ground (black squares) +and first excited (purple rectangles) states is indicative of a first-order transition. +[4] D. Chandler and J. P. 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Dongarra (Springer Berlin Heidelberg, +Berlin, Heidelberg, 2005) pp. 351–359. +Appendix A: TPM and QTPM for other boundary +conditions +In the main text, we show that ground state properties +of the TPM and, consequently, the quantum phase tran- +sition of the QTPM depends on the boundary conditions, +but we only focused on systems with periodic boundaries. +Here, we consider the cases of periodic boundaries in only +the x-dimension (PBCx) and of open boundaries (OBC), +using the same Rule 60 CA considerations as for the pe- +riodic case. +For PBCx, we use the update rule for Rule 60 as before, +with the only difference that we do not need to explic- +itly check for the periodicity across the y-direction. As +a result, for an initial array of L sites, there will be 2L +configurations and, thus, 2L ground states for the clas- +sical TPM. In this case, only the number of sites in the +x-direction matters for the number of classical ground +states. For example, a lattice with size N = 3 × 3 and +one with N = 3 × 80 will have the same number, 8, of +ground states. The identification of the classical ground +states can be worked out from the Rule 60 evolution, as +before. +For OBC, the update rule for Rule 60 is modified for +the first cell of an L-length array so that it is not updated. +This freedom on choosing two of the boundaries of the +lattice gives an increased number of ground states for +the classical TPM. Specifically, given a lattice of N spins, +N = L × M, the number of the classical ground states is +2L+M−1. +We now perform a similar numerical analysis as in +Sec. IV C, but only using MPS methods. Data is nor- +malised with the system size of the given lattice. The +system sizes accessible do not give a clear indication of +a well-formed phase transition, but only signatures of it. +The first-order transition found is weaker than in the case +of the fully PBC, which we attribute to the high number +of ground states for the classical TPM, given the system +sizes. We note that all these states for h ̸= 0 constitute +low-lying excited states which affect the convergence of +the MPS algorithm. +As seen from Figs. 12 and 13 for PBCx, the difference +between the square lattice size scaling and the quasi-1D +rectangular stripes is more pronounced, when compared +to the finite-size scaling for PBC. Extra calculations on +wider rectangular stripes verify that this difference is only +a feature of the quasi-1D geometry of the lattice and not +an inherent property of the system. Accuracy is lost with +increasing size and the MPS results for the sizes studied +are not reflective of the true thermodynamic limit. +The above considerations for PBCx are even more no- +ticeable for the case of OBC, Figs. 14 and 15. Rectangu- +lar stripes are fully continuous, while they seem to have +converged to their “thermodynamic” behaviour. +How- +ever, as seen from the square system sizes, the behaviour +of the model remains the same regardless of the bound- +ary conditions. It becomes apparent though that bigger +system sizes soon become computationally inaccessible +due to the exponential number of classical ground states. +This behaviour shows an obvious discrepancy with stan- +dard MPS methods; normally, fully periodic system sizes +are computationally harder to access. +Here, we stud- +ied a model where the exponential number of low-lying +states (ground states for h = 0) deters the convergence of +the algorithm and also significantly increases the lattice +size where the “thermodynamic limit” has been reached. +QTPM belongs to this class of models, coming from clas- +sical glasses, where the number of classical ground states +for PBCx or OBC would progressively constitute the +model numerically inaccessible. Only for PBC, the ther- +modynamic limit is evidently accessible. +Another conclusion that can be drawn from Figs. 12, +13, 14 and 15 concerns the nature of the phase transi- +tion. The possession of data for only OBC and Fig. 15, +in particular, kschangewould possibly point towards a +continuous quantum phase transition or an inconclusive +statement. Note, however, that this conclusion would be +inaccurate. Even if the “thermodynamic limit” has pos- +sibly been reached, numerics for some system sizes alone +does not provide enough evidence for the characterization +of the phase transition for these cases; further knowledge +of the ground states of the model is necessary, while also +a general understanding of the behaviour (and number) +of the low-lying states. +The significance of these arguments is further evident +from Figs. 16 and 17. +For the case of PBCx, all de- +generate ground states for the J ≫ h region are easily +found from exact diagonalization calculations and classi- +cally excited states are easily tractable too. However, the +same is not true for OBC. The number of classically de- +generate ground states increases exponentially and this +is the reason why it would be pointless to show more +ground states. +Appendix B: Gap Scaling Analysis for PBC +In this section we present a restricted and with limited +accuracy analysis on the energy difference between the +ground state and the first excited state. This analysis was +conducted based on ED and MPS methods, which limits +the validity of the conclusions that can be reached: it be- +comes quickly obvious that MPS methods are not power- +ful enough for the detection of the actual gap, especially +in regions of the parameter space with high entanglement +or with a high number of low-lying excited states, where +MPS often converge to excited states above the lowest- +lying ones. However, the analysis below still provides an + +13 +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-3.0 +-2.5 +-2.0 +-1.5 +-1.0 +(a) +□□□□□□□□□□□□□□ +□ +□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ +△ +△ +△ △ △ △ △ △ △ △ +□□□□□□□□□□□□□□ +□ +□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ +△ +△ +△ △ △ △ △ △ △ △ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ +△ +△ △ △ △ △ △ △ △ △ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ +△ +△ △ △ △ △ △ △ △ △ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 (c) +MPS +● +● +● +● +● +ED +□ +△ +FIG. 12: (a) Normalised energy, (b) Mx and (c) Mzzz of ground states from MPS and ED for square lattices with +PBCx. Data from ED are denoted as empty squares and empty triangles. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-3.0 +-2.5 +-2.0 +-1.5 +-1.0 +(a) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 (c) +MPS +● +● +● +● +● +● +FIG. 13: Same as Fig. 12 but for systems of size N = 3 × L. +indication of the behaviour of the gap with system size +when comparing systems with different symmetries. +This limited accuracy when measuring the first excited +state energy is evident in Fig. 18(a). Data is calculated +for the J = h = 1.0 point. The gap seems to decrease +with increasing the system size, but at the same time, +the power of MPS to detect it is significantly reduced. +The situation seems clearer for Fig. 18(b). However, it is +equally problematic despite the monotonically decreasing +gap. The only significance of these results are an upper +bounds of the actual gap. +For the first case, the gap +seems to decrease algebraically to zero, while for the case +of multiple classical ground states, it seems to decrease +exponentially. +This underlines the different behaviour +depending on the existence or not of multiple classical +ground states. +The same problems are encountered close to the phase +transition from the MPS results in Fig. 19(a). Both plots +are normalised by the maximum value of the gap encoun- +tered in the region of J values studied. For 0.0 < J < 2.0, +the gap appears always to be maximum at J = 2.0 for +Fig. 19(a) and at J = 0 for Fig. 19(b). In Fig. 19(b), for +J > h = 1.0 the gap approaches zero, as expected from +the existence of degenerate ground states. + +14 +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-2.5 +-2.0 +-1.5 +-1.0 +(a) +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ △ +△ +△ +△ △ △ △ △ △ △ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ △ +△ +△ +△ △ △ △ △ △ △ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 (c) +MPS +● +● +● +● +● +ED +□ +△ +FIG. 14: (a)Normalised energy, (b) Mx and (c) Mzzz of ground states from MPS and ED for square lattices with +OBC. Data from ED are denoted as empty squares and empty triangles. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-2.0 +-1.8 +-1.6 +-1.4 +-1.2 +-1.0 +(a) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 (c) +MPS +● +● +● +● +● +● +FIG. 15: Same as Fig. 14 but for systems of size N = 3 × L. +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ +◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻ +0.2 +0.4 +0.6 +0.8 +1.0 +-18 +-16 +-14 +-12 +-10 +(a) +Energy Levels +□ +1 +▯ 2-7 +△ +8 +▽ 9-12 +◇ 13-14 +○ +15 +◻ +16 +17 +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ +◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +-26 +-24 +-22 +-20 +-18 +-16 +-14 +-12 +(b) +Energy Levels +□ +1 +▯ 2-4 +△ 5-7 +▽ 8 +◇ 9 +FIG. 16: The unnormalised state diagrams for a (a) 4 × 4 and a (b) 3 × 6 lattice with PBCx. +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +0.5 +1.0 +1.5 +-30 +-25 +-20 +-15 +-10 +(a) +Energy Levels +□ +1 +▯ 2-5 +△ +6 +□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ +△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ +0.5 +1.0 +1.5 +-35 +-30 +-25 +-20 +-15 +-10 +(b) +Energy Levels +□ +1 +▯ 2-5 +△ +6 +FIG. 17: Same as Fig. 16 for OBC. + +15 +● +● +● +● +● +● +● +● +100 +200 +300 +400 +500 +0.00 +0.05 +0.10 +0.15 +0.20 +● +● +● +● +● +● +● +● +● +0 +100 +200 +300 +400 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +FIG. 18: The scaling of the gap, g, for different lattice sizes without (a) and with (b) symmetries for the QTPM for +the J = h = 1/0 point. Both ED and MPS methods are used (where appropriate) for the calculation of the given +gaps. Square and rectangular sizes are equally used. +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +◆ +◆ +◆ +◆ ◆ ◆ +◆ +◆ +◆ +◆ +◆ +◆ +● +● +● +● ● ● +● +● +● +● +● ● +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▮ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +◆ +◆ +◆ +◆ ◆ ◆ +◆ +◆ +◆ +◆ +◆ +◆ +● +● +● +● ● ● +● +● +● +● +● ● +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +MPS +■ +▮ +▲ +▼ +◆ +● +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +◆ ◆ ◆ ◆ +◆ ◆ +◆ +◆ +◆ +◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ +● ● ● ● +● +● +● +● +● +● ● ● ● ● ● ● ● ● ● ● +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +MPS +■ +▲ +▼ +◆ +● +FIG. 19: The gap, g, normalised with the maximum gap, gmax, in the given domain for different lattice sizes from +MPS without (a) and with (b) symmetries for the QTPM. For (a) gmax ≈ 3.43 − 3.50 and for (b) gmax = 2.0. + diff --git a/0dE0T4oBgHgl3EQf_gIS/content/tmp_files/load_file.txt b/0dE0T4oBgHgl3EQf_gIS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6559eebed2a4433d96c77904b67e04911f9890e7 --- /dev/null +++ b/0dE0T4oBgHgl3EQf_gIS/content/tmp_files/load_file.txt @@ -0,0 +1,1854 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf,len=1853 +page_content='Boundary conditions dependence of the phase transition in the quantum Newman-Moore model Konstantinos Sfairopoulos,∗ Luke Causer, Jamie F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Mair, and Juan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Garrahan School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK We study the triangular plaquette model (TPM, also known as the Newman-Moore model) in the presence of a transverse magnetic field on a lattice with periodic boundaries in both spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We consider specifically the approach to the ground state phase transition of this quantum TPM (QTPM, or quantum Newman-Moore model) as a function of the system size and type of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Using cellular automata methods, we obtain a full characterization of the minimum energy configurations of the TPM for arbitrary tori sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For the QTPM, we use these cycle patterns to obtain the symmetries of the model, which we argue determine its quantum phase transition: we find it to be a first-order phase transition, with the addition of spontaneous symmetry breaking for system sizes which have degenerate classical ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For sizes accessible to numerics, we also find that this classification is consistent with exact diagonalization, Matrix Product States and Quantum Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' INTRODUCTION In this paper, we study the ground state phase transi- tion of the quantum Newman-Moore model, or quantum triangular plaquette model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The classical triangular pla- quette model (TPM), introduced by Newman and Moore [1], is a model of Ising spins interacting in triplets in (half of) the plaquettes of a triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Despite the absence of quenched disorder and its trivial static prop- erties, the model has rich glassy dynamics [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The TPM is an important model as it realises in an interact- ing system the paradigm of slow (super-Arrhenius) re- laxation at low temperatures due to effective kinetic con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This phenomenon is central to the dynamic fa- cilitation picture of the glass transition [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The physics of the TPM can also be generalised to three dimensions, for example in the five-spin interaction square-pyramid model [7], or maintaining the triangular interactions in the models of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' A three-dimensional generalisa- tion of the TPM with non-commuting terms [9] actually started what is now the field of fractons [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The simplest way to transform the TPM into a quan- tum model is by adding a transverse field term to the classical Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Such quantum TPM (QTPM, or quantum Newman-Moore model) was considered in the context of fractons in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Numerics in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [12] suggested that the ground state undergoes a first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' A related work studying the large deviations of plaquette observables in the stochastic dynamics of in- dependent spins [14] also found numerical evidence for a first-order transition at the self-dual point of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In contrast, the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [15] indicated that the transition is continuous, with a particular form of fractal symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The classical TPM and its connec- tion with fractals and topological order was also drawn in ∗ ksfairopoulos@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='com Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Here we aim to resolve these discrepancies by exploiting a general connection between D-dimensional cellular automata (CA) [17] and the ground states of (D + 1)-dimensional classical spin models [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' By us- ing this method in the specific case of the QTPM with periodic boundary conditions, we are able to character- ize the approach to its quantum phase transition in the large size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Our key observation is that the nature of the transition depends on the specific lattice dimensions, and this is manifested in the finite size scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For the TPM, the relevant CA is Rule 60 [17] and not Rule 90 that might be assumed from comment [18] in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For system sizes where one dimension is a power of two, Rule 60 has a single fixed point [19], im- plying a single energy minimum for the classical TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In such cases, we verify that the quantum phase tran- sition in the QTPM is first-order (that is, a sequence of such system sizes tends to a first-order transition in the large size limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This also holds for other sizes for which Rule 60 has no non-trivial attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' However, for certain sizes there can be periodic orbits on top of the fixed point for the CA, giving rise to classical ground state degeneracies in the TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For the quantum model, this translates into a mixed order quantum phase transi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We provide evidence for this scenario by means of numerical simulations, namely, for small sizes using ex- act diagonalisation, and for large sizes using both Matrix Product State approximations of the ground state, and continuous-time Quantum Monte Carlo [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' II, we review the classical and quantum TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' III, we provide the necessary background on CA and the connection to the ground states of the classical TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' IV, we dis- cuss the ground state phase transition of the QTPM in terms of the symmetries that follow from the properties of the associated CA, and support our predictions with numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' V we give our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In the Appendix we provide further details, including a discussion on the case of the QTPM with open bound- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='02826v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='stat-mech] 7 Jan 2023 2 aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' TRIANGULAR PLAQUETTE MODEL, CLASSICAL AND QUANTUM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Classical The triangular plaquette model (or TPM or Newman- Moore model) [1–3] is a model of Ising spins si = ±1 on the sites i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' , N of a triangular lattice, with cubic interactions between the spins on the corners of downward-pointing triangles of the lattice, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The Hamiltonian of this classical model reads ETPM = −J � i,j,k∈▽ sisjsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (1) In what follows, it will be convenient to consider the equivalent model on a square lattice of size N = L × M, with classical Hamiltonian, ETPM = −J L,M � x,y=1 sx,ysx+1,ysx+1,y+1, (2) where we assume periodic boundary conditions in both directions by identifying x + L = x mod L y + L = y mod L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (2), we label the spins by sx,y at site with coordi- nates (x, y) in the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The classical TPM has been predominantly studied in the context of the glass transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For lattices with at least one dimension being a power of two, the energy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (1) reduces to that of the non-interacting plaquette variables [1–3], ETPM = −J � ▽ d▽, (3) where d▽ = sisjsk with i, j, k ∈ ▽ for every downward- pointing triangle in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' When at least one di- mension is a power of two, the relation between plaque- ttes and spin variables is one-to-one, exactly proving the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The thermodynamics of the TPM is therefore one of free binary excitations and, as such, it is essentially trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In contrast to the statics, the single spin-flip dynamics of the TPM is highly non-trivial, as flipping one spin changes three adjacent plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This implies that at low temperatures, where excited plaquettes are sup- pressed, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (3), the dynamics has effective kinetic con- straints [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' These dynamical constraints lead to an ac- tivated relaxation similar to that of the East model [24], with relaxation times growing as the exponential of the inverse temperature squared [2] (a super-Arrhenius form FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 1: Triangular plaquette model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The shaded triangles indicate the interacting triplets, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Dotted lines indicate spins that interact for periodic boundary conditions on an N = 4 × 4 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' known as the “parabolic law” [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Similar glassy be- haviour is seen in generalisations of the TPM with odd plaquette interactions [8, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The TPM has also been considered in the presence of a (longitudinal) magnetic field [27] and in the related case of coupled replicas [28], and the TPM with open boundary conditions was studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [29] through partial trace methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The classical TPM was studied in the context of topological order in fractal models in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Autoregressive neural net- works were applied to the classical TPM with limited success in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Quantum: TPM in a transverse field Taking Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (1) and adding a transverse field, we obtain the Hamiltonian of the quantum TPM (QTPM), HQTPM = −J � i,j,k∈▽ ZiZjZk − h � i Xi, (4) where Zi and Xi represent Pauli operators acting non- trivially on site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This quantum model was studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [12, 13] and its connections to models of fractons were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The Hamiltonian (4) is expected to have a quantum phase transition at the self-dual point J = h [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Nu- merical results from [12, 14] suggested the transition to be first-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [14] used trajectory sampling in systems with linear size a power of two and periodic boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In contrast, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [15] found evidence for a continuous phase transition with fractal symmetry breaking using stochastic series expansion methods, also with periodic boundary conditions but not restricted to power of two sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The study of QTPM was connected to Rydberg atoms in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Further studies on the QTPM and its generalisations from the viewpoint of fracton field theory were presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' CELLULAR AUTOMATA AND GROUND STATES OF THE CLASSICAL TPM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' General aspects of CA Cellular automata (CA) consist of a D-dimensional ar- ray of sites evolving under discrete-time and synchronous dynamics given by a CA rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Under this evolution, the state is updated by a deterministic rule which is local in time (although generalisations exist) [17, 34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' As a result, CA dynamics gives rise to, often rich, (D + 1)- dimensional structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In what follows, we consider D = 1, linear CA with deterministic local transition rules with sites taking values from the finite field F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Discrete cellular automata are fully specified by an ini- tial configuration of L sites and a local transition rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The local update rule determines the configuration of ev- ery site at each timestep using the local neighbourhood of size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For D = 1, the simplest CA rules are those defined only by r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For a neighbourhood of three sites, r = 1, there are 8 possible configurations giving rise to 256 possible choices for update rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' These ele- mentary CA were classified by Wolfram [17, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 2(a) shows Rule 60, which will be the relevant one for the TPM: if we identify the empty and occupied sites of the CA with the up and down spins of the TPM, then Rule 60 is the same as the condition that the product of spins in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (2) is one, thus maximising the local energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 2(b) also shows the closely related Rule 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' CA like these are called elementary [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 3(a,b) shows the patterns generated for Rules 60 and 90 starting from an initial single seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Note that these depend on the boundary conditions (for a generic analysis on cellular automata with periodic boundaries, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 3(b) shows Rule 90 for L = 64 and periodic boundaries: the timestep after the last one shown will take the CA to the trivial, empty configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' in contrast for a length L = 63 the Sier- pinski fractal [38] will continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Similarly, time evolution will bring in one timestep the pattern of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 3(a) to the trivial one for periodic boundaries with L = 32, but the fractal shape would continue to be reproduced for open boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Focusing on Rules 60 and 90 for concrete- ness, CA evolution can be written as polynomials (or generating functions, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [34]), as T60(x) = 1 + x (5) T90(x) = x−1 + x, (6) with the evolution of the whole lattice obeying [34] A(t)(x) = TA(t−1)(x) mod (xL − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (7) Given a configuration at timestep t for the evolution of any given rule, the configuration at timestep t + 1 will be its successor and the one at t − 1 its predecessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For Rules 60 and 90, following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [34], we have: There are no predecessors for configurations with an odd number of sites being equal to 1 for both (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 2: (a) Rule 60 and (b) Rule 90 evolution rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 3: (a) Evolution from a single site for Rule 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (b) Same for Rule 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (c) A stable cycle generated by Rule 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For Rule 90, 2L−1 configurations for a system of size L cannot be reached if L is odd and 3 × 2L−2 if L is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For Rule 60, 2L−1 configurations cannot be reached for any system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This means that rows with an odd number of ones can only occur as initial states for any given time evolution for Rule 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Similarly, there may exist configurations that can be reached in one timestep by more than one pre- decessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' More specifically, given a configuration with at least one predecessor, there are 2 predeces- sors if the length L is odd and 4 if it is even for Rule 90, while there are always 2 for Rule 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Regarding the length of cycles for Rule 90, more results can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For Rule 60, which is most relevant for the TPM, we discuss the cycle lengths and their multiplicities next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Periodic orbits of Rule 60 The iterated map Dn(x) = (|x1 − x2|, |x2 − x3|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=', |xL − x1|) (8) for Dn : Zn → Zn and x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=', xL) is known as the Ducci map (also known as rule 102 in Wolfram’s nota- tion [17, 34], the mirror image of Rule 60, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Ducci’s map (and therefore Rule 60) exhibits periodic orbits or evolves to a fixed point depending on L being a power of two or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For L = 2k, any initial state 4 (b) (a) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 4: The fixed point and limit cycles of Rule 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Filled circles indicate states of the CA and the arrows the flow under the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (a) For L = 5 there is a cycle of length C = 15 and the fixed point (which flows into itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (b) For L = 6 there are two cycles of period C = 6, a cycle of period C = 3 and the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (c) For L = 8, there is the fixed point and no cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' evolves to the trivial state of zeros [40], while for L ̸= 2k it evolves to a richer attractor structure [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In F2 and by using periodic boundary conditions, the Ducci map can be brought into matrix form as follows, Dn x = ((x1 + x2), (x2 + x3), · · · , (xL + x1)) mod 2, (9) where Dn = � ������� 1 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 0 0 0 0 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 1 1 0 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 0 1 1 1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 0 0 1 � ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (10) The matrix D can be expressed as D = I + SL (11) with SL the left shift map [19, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' It is easy to check that D2k = (I + SL)2k = I + S2k L = I + I = 0 mod 2, (12) with k ∈ Z and, thus, a system that has L a power of 2 ends up in the trivial configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For Rule 60 the corresponding matrix is D⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In order to obtain the cycles of Rule 60, we need the following definitions and results for Rule 102 from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [19, 35, 36, 41–44]: For any given CA, each array of sites can be thought of as a vector, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For any such vector v in F2, its order is defined through the monic polyno- mial, which satisfies µv(D)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The order of the minimal annihilating polynomial, ord µv(λ), is equal to the smallest natural number n, such that µv(λ)|λn − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' If µv(0) = 0, then for µv(λ) = λk˜µv(λ) with ˜µv(0) ̸= 0 and k ∈ N, the order of µv is ord (µv) = ord (˜µv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Assuming that µv(λ) = λk˜µv(λ) with k ≥ 0, then the k-th successor of v belongs to a cycle of length c = ord µv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This applies to a vector v of any pos- itive integer L and any linear map (or cellular au- tomaton rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The minimal annihilating polynomial for the Ducci map was calculated in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [19, 43], based on the char- acteristic polynomial of the matrix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' It was found that µn(λ) = pn(λ) = (1 + λ)L + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (13) We are now in a position to obtain the attractor struc- ture of Rule 60 by following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [36] (specifically “Prin- ciple C”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We decompose the minimal annihilating poly- nomial into the product of its irreducible polynomials, πi(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We call their polynomial powers bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Thus, µv(λ) = m � i=1 πi(λ)bi, (14) and ord µv(λ) = rpt, (15) where r is the least common multiple of ord πi and t the smallest integer satisfying pt ≥ max (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' , bm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 4 illustrates the different scenarios for cycles as a function of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We show three different sizes, L = 5, 6, 8, small enough to be able to visualise the network of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Configurations of the CA are identified by blue circles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 ' metadata={'source': 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='TABLE I: Cycle lengths for Rule 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The second column indicates the cycle lengths and the third the total number of periods for Rule 60 (each period of length C is counted C-times), assuming that the least common multiple of the periods for each given L divides M, lcm C|M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The last column is constructed based on the number of predecessors for all configurations for Rule 60, given a length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' and arrows indicate to which configurations they evolve to under the CA dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 4(a) shows L = 5: here there is one fixed point to which one other state evolves to (shown at the centre of the figure), and one limit cycle of period C = 15, to which all other configura- tions flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 4(b) shows a more general situation of multiple distinct cycles for the case of L = 6: there are two cycles of period C = 6, one cycle of period C = 3 and one fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For the case of L a power of 2, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 4(c) for L = 8, there is a unique fixed point and all states evolve towards it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Classical ground states of the TPM From the analysis above for Rule 60, we can enumerate all the minimum energy configurations of the TPM, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The classical ground states for a system of size N = L × M are: (i) the state with all spins up, corre- sponding to the fixed point of Rule 60, for any value of L and M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (ii) two-dimensional spin configurations that correspond to periodic trajectories of Rule 60, that is, CA trajectories starting from any of the states of a limit cycle, for all limit cycles whose period is contained an in- teger number of times in M—this occurs only for certain combinations of L and M (never if L is a power of two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We show the relevant Rule 60 information in Table I for up to L = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Under the column C we give the distinct periods of the limit cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In the column labelled by M we give the corresponding degeneracy of classical ground states of the TPM, apart from the uniform spin- up state, given that lcm C|M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For example, for L = 15, there is one cycle of length 3, three cycles of length 5, and 1091 cycles of length 15, which means 1 + M = 1+3+3×5+1091×15 = 16384 distinct two-dimensional spin configurations that minimise the energy in a TPM of size N = 15 × M as long as 15|M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In contrast, for L = 15, if 5|M but 3 ∤ M and 15 ∤ M, then there are 1+15 different ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The symmetries of a TPM of size N = L × M can be similarly constructed from the number of ground states and their multiplicities, as determined by Rule 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' PHASE TRANSITION IN THE QUANTUM TPM As we will now show, the set of minimum energy con- figurations and the associated symmetries of the classical TPM, as obtained from the Rule 60 CA, determine the properties of the ground state phase transition in the quantum TPM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Symmetries of the QTPM Like its classical counterpart, the QTPM with periodic boundaries has full translational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The symme- 6 (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 5: (a) Symmetry operators for the 3 × 3 QTPM with periodic boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (b) One of the symmetry operators of the 9 × 9 QTPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' tries of the QTPM will then be deduced by the results of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For systems with dimensions N = L × M, which can accommodate non-trivial cycles of Rule 60, the symme- tries of the corresponding QTPM easily follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Consider as a simple example the case of N = 3 × 3 with periodic boundaries in both dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' From Table I, we see that L = 3 has one cycle of period 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This means that for M = 3 there are three non-trivial symmetries, given by the operators G1 = X1,2X1,3X2,1X2,2X3,1X3,3 (16) G2 = X1,1X1,3X2,2X2,3X3,1X3,2 (17) G3 = X1,1X1,2X2,1X2,3X3,2X3,3 (18) see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Note that translational invariance plays a crucial role in the determination of the symmetries: in the example above G2 = TxG1T −1 x and G3 = TxG2T −1 x , where Tx is the translation operator in the x direction, and these symmetries alongside the identity form the Klein group K4, which, in turn, is isomorphic to Z2 ⊗Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This approach generalises to other system sizes, and the symmetries are products of K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For example, the symmetries of the N = 5×15 and the N = 6×6 systems form the group K4 ⊗ K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Since the symmetry operators are products of the X Pauli matrices, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (16-18), they commute with the transverse field term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (4), and, therefore, are symmetries of the QTPM for all values of J and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The character of the quantum phase transition The character of the phase transition for the QTPM is now easy to predict based on the information of its symmetries for a given system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' By this we mean: given a sequence of increasing system sizes with periodic boundaries, all having the same number of symmetries, the progressively sharper finite size crossovers will be in- dicative of an eventual phase transition in the large size limit whose character—first-order or continuous—will be determined by the underlying symmetries of the given system sizes in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' As these symmetries in a system of size N = L × M depend on the precise values of both L and M, this analysis has to be done carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In the next section we show numerical evidence for the general considerations we give here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' First consider the system sizes N = L × M such that the underlying Rule 60 has only one fixed point, as de- scribed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Then, the QTPM has no non-trivial symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In the limit J ≫ h, there is a single ground state corresponding to the all-up state, and a vanishing number of excited plaquettes, as the classical TPM has a single minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In the opposite limit, J ≪ h, the ground state is paramagnetic, with spins aligned in the x direction, with an explicit Z2 symmetry for its ground state, corresponding to a high density of excited plaque- ttes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In the limit of large N, we expect the quantum phase transition at the self-dual point J = h to be first- order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' A second scenario is that of system sizes where the underlying Rule 60 cycles give rise to non-trivial sym- metries in the QTPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In this case, in the limit J ≪ h, the ground state is the same paramagnetic one as be- fore, invariant under a global Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' However, for J ≫ h there exists spontaneous breaking of the sym- metries emerging from Rule 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This case, therefore, has characteristics of a first-order phase transition (the explicit symmetry breaking of the global symmetry of the paramagnetic ground state) with those of a contin- uous transition (spontaneous symmetry breaking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This is similar to the first-order phase transitions observed in kinetically constrained models [45] and suggests that, for local observables, the phase transition will appear first- order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' for example, in a discontinuous jump in the excited plaquette density, Mzzz = 1 N � i,j,k∈▽ ZiZjZk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (19) Appropriate operators will quantify the symmetry break- ing of the degenerate classical ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The choice of these operators depends on the size and specific lattice dimensions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Consider, for example, the case of L = 3 and M = 3k with k ∈ N, where we know from Rule 60 that there is a single fixed point and the three non-trivial ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For the trivial ground state this operator is just the magnetisa- tion Mz = 1 N �N i Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' To detect the symmetry breaking into the other three states, we can define the three op- erators ˜ M m z = 1 N �N i (−1)niZi, where ni = 1 if the spin i is flipped for a state of the cycle m of Rule 60, and ni = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For example, for the state associated 7 to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (16), we have ˜ Mz = 1 N (Z1,1 − Z1,2 − Z1,3 − Z2,1 − Z2,2 + Z2,3 −Z3,1 + Z3,2 − Z3,3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (20) Note that when there are multiple non-trivial ground states connected by translations, there will be no sin- gle operator taking the form of a sum of local terms for which it will be possible to discern between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For example, in the N = 3k case, Mz will only distinguish be- tween the trivial ground state and the 3-fold degenerate ones, while the operators ˜ M m z will be able to distinguish between one of the non-trivial ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Numerics We now provide evidence for the general observations above from numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For small systems we use exact diagonalization (ED) [46–48], allowing the study of system sizes up to 28 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For larger systems we estimate the properties of the ground states using two different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The first of these is Matrix Product States (MPS) [49–52], which we “snake” around the 2D lattice, and optimize with the 2D Density-Matrix Renormalization Group (DMRG) [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' By employing a bond dimension up to D = 1000, we are able to reli- ably estimate the ground state properties for system sizes on the square lattice for up to N = 16 × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Time and memory constraints hinder progressively the convergence in the paramagnetic phase, where J ∼ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' As discussed below, we are also able to apply MPS to cylindrical sys- tems, which can be considered to be quasi-1D, allowing us to reach much larger sizes than in the case of square geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' To confirm the results of 2D DMRG, we also employ Quantum Monte Carlo (QMC) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In particular, we use the continuous-time expansion (ctQMC) [20], with local spin updates which re-draw the entire trajectory of a single spin, subject to a time-dependent environ- ment, where the trajectories of unmodified spins are con- sidered to act as a “heat bath”, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=', see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We run our simulations with an inverse temperature of β = 128, which we find to be large enough to converge to the ground state [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' First-order transition As explained above, when the underlying Rule 60 has a single fixed point, and the classical TPM a single energy minimum with all spins up, we thus expect the phase transition of the QTPM to be first-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 6 shows results for N = L×L with L = 4, 8, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We show that our MPS and ctQMC results coincide with the large deviations results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [14], obtained via tran- sition path sampling (TPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' What is plotted is the pla- quette density Mzzz as a function of the coupling J for 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 ctQMC □ ▽ ◇ MPS ■ ▼ ◆ TPS △ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 6: Normalized three-spin correlator, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (19), in the QTPM as a function of J for fixed h = 1, with N = L × L with L a power of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We compare results from MPS and ctQMC obtained here with results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The numerical data indicates a first-order transition at J = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' fixed transverse field h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The data indicates a first- order transition at J = h, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For J > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0, we see small deviations in the TPS results, due to the extra field used for the acquisition of this data in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Sim- ilar issues are observed for ctQMC close to the J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 point, where the single spin updates do not allow for the collective effects necessary to move between phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 7, we show the results for several system sizes in a square geometry, N = L × L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 7(a) shows the ground state energy as a function of J (at h = 1) for L = 5 to 16, obtained from MPS numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For the smallest size we also show ED results, which coincide with the MPS ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The kink near J = 1 indicates a quantum phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Note that this behaviour is similar in systems with a single classical ground state (L = 5, 8, 16) or multiple ones (L = 6, 7), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 7(b,c) we show the average transverse magnetisa- tion, Mx = 1 N � i Xi, and Mzzz, respectively, for systems with L a power of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We get exactly the same results for different system sizes too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Both MPS and ctQMC show clear indications of a first-order transition at J = 1 in both observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 8 shows similar results in a rectangular geome- try, N = 3×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For such thin stripe systems we can per- form MPS more efficiently for larger system sizes than for square geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Once again, MPS and ctQMC results coincide, and indicate a first-order transition at J = 1 (although weaker than in the square lattice case, in the sense that the discontinuity in the local operators shown is smaller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Note that these results include not only val- ues of M which are multiples of three, for which there are multiple classical ground state cycles, but also values of M for which a single ground state is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' What we see in this case is that the observables Mx and Mzzz are unable to detect changes related to any given classical ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 8 △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='△ MPS ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (b) Transverse magnetisation as a function of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In this case the open symbols are from ctQMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (c) Average three-spin interaction as a function of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ● ● 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 (c) ctQMC □ △ ▽ ◇ MPS ■ ▲ ▼ ◆ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 8: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 7 but for systems of size N = 3 × L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Symmetry breaking In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 7 and 8, we show the two terms that com- pete in the Hamiltonian, Mx and Mzzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For system sizes where there is one classical ground state and no non- trivial symmetries, the total longitudinal magnetisation Mz can also serve as an order parameter, as it picks up the orientation of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 9(a) shows that the transition is also clear for this observable for square lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For system sizes where degeneracies are expected for J ≥ h, however, Mz is unable to detect the symme- try breaking related to the extra symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For these cases, we need the staggered magnetisations, ˜ M m z , such as that for N = 3 × 3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Figure 9(b) shows that such operators are able to detect the spontaneous break- ing of symmetry for these lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Note that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 9(b) was obtained through the use of a small symmetry break- ing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This is a standard method for the detection of the symmetry breaking in the ground state of a degen- erate quantum model [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' As a result, the calculations were performed through the use of a modified Hamilto- nian H = HQTPM − p ˜ Mz, where p is chosen to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The detection of the spontaneous symmetry breaking can be similarly preformed for any of the classical ground states of the given lattice size with the appropriate oper- ator ˜ Mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In order to more clearly understand the mechanism of the phase transition, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 10 and 11 we plot the low- lying spectrum of the QTPM from ED as a function of h for fixed J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' These results support our above obser- vations: for system sizes where only a first-order phase transition is expected, there is an avoided crossing be- tween the ground state and the first excited state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' for system sizes with extra symmetries from the cycles of Rule 60, we see both an avoided crossing (indicative of first-order transitions) and a merging of eigenstates in- dicative of spontaneous symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 11 for the case of N = 3 × M, the avoided crossing becomes apparent only with increasing system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We now comment on how our results compare to those in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For the numerics, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [15] used a stochastic series expansion (SSE) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We in turn use MPS and ctQMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Both SSE and ctQMC are Quantum Monte Carlo based methods, which indicates that they, in prin- ciple, should be able to roughly access system sizes of the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Furthermore, while Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [15] also considered periodic boundaries, there was no specific restriction on system size, and therefore no distinction between sizes for which there is a single classical minimum and sizes where there are multiple ones, with the implications for symmetries of the corresponding QTPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [15] also used a non-local order parameter, compared to our local ones (the stag- gered magnetisations) that do reflect the minima of the underlying TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In [15], the existence of a phase tran- sition at J = h was confirmed through the study of the Binder cumulant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' this was done, however, with limited 9 accuracy on the location of the phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' It is important to note that some of the local observables we calculate here are also studied for specific system sizes in the Appendix of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Since the temperature used for those calculations varied for different system sizes, it is possible that the smoothness observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [15] is a consequence of thermal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We instead used a fixed inverse temperature β = 128 which we verified is sufficient to make thermal effects negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Nature of the phase transition in the thermodynamic limit The discussion above and the numerical results indi- cate the existence of a quantum phase transition in the thermodynamic limit, N → ∞, at the self-dual point, J = h, of the QTPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' However, the approach to the thermodynamic limit is different across different system size geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' There are three different limits to thermodynamics: (i) across one of the two dimensions while the other one re- mains constant (that is, infinite stripes), (ii) across both dimensions, and (iii) on making the spins continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We briefly discuss the differences between these limits and the complications that might arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In the case (i), if the limit is taken for fixed L and with M such that lcm C|M (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' M = 3k, with k ∈ N), the number of classical ground states remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In our numerics we are restricted to narrow stripes to allow convergence of the MPS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 8 suggests that in such quasi-1D systems the transition will eventually be slighly weaker than for square system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Case (ii) can be more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The simplest situation is that of square lattices N = L × L with L a power of two, where it is guaranteed that for all sizes there will be a single classical ground state, and therefore the transi- tion is certainly first-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For other size sequences, the number of relevant Rule 60 cycles, and therefore symme- tries of the QTPM, may grow or decline with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For some cases this growth is monotonic (as for example for N = 3k × 3k with k → ∞), while in others it is not (as for example when N = 3k × 3k with k → ∞), see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In case (iii) the nature of the underlying CA is altered [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In this limit, Rule 60 becomes f(p, q, r) = p + q − 2pq, (21) where p, q and r indicate the state of the three sites in the neighbourhood determining the local evolution of the CA, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Basic arguments [56] indicate a single fixed point in the evolution of this fuzzy CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We spec- ulate that the same behaviour will be observed in the quantum field theory limit for QTPM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' a single ground state across different regions of the whole J − h space and thus a first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' However, a field theoretic description of the QTPM might not be as obvi- ous and straightforward to get for the above elementary argument to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' CONCLUSIONS In this work, we used the cycles of the cellular automa- ton Rule 60 to describe the symmetries of the quantum triangular plaquette model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We found that the attrac- tor structure of Rule 60 plays an important role in the characterization of the degeneracies of the ground states of the classical TPM, allowing in turn to construct the symmetry operators of the QTPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In this way, the exis- tence or absence of stable cycles in Rule 60 imply whether it is possible or not for the QTPM to display sponta- neous symmetry breaking of the corresponding symme- tries, which in turn impacts on the nature of the quantum phase transition at the self-dual point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' These general ob- servations are also consistent with the finite size trends from our numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' A full description of the QTPM phase transition would require a field theoretical description and a renormalization group treatment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' we leave these tasks for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Vasiloiu for insightful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We acknowledge financial support from EPSRC Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' EP/R04421X/1, the Leverhulme Trust Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' RPG- 2018-181, and University of Nottingham grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' FiF1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' LC was supported by an EPSRC Doctoral prize from the University of Nottingham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Simulations were performed using the University of Nottingham Augusta HPC cluster, and the Sulis Tier 2 HPC platform hosted by the Scientific Computing Research Technology Plat- form at the University of Warwick (funded by EPSRC Grant EP/T022108/1 and the HPC Midlands+ consor- tium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Newman and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Moore, Glassy dynamics and aging in an exactly solvable spin model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' E 60, 5068 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Garrahan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Newman, Glassiness and constrained dynamics of a short-range nondisordered spin model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' E 62, 7670 (2000).' metadata={'source': 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with no symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (b) Staggered magnetisation for detecting symmetry breaking in systems with multiple symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' □□□□□□□□□□□□□□□□□□□□□□□□□□□ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ △△△△△△△△△△△△△△△△△△△△△△△△△△△ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='9 1.' metadata={'source': 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Low-lying spectrum of the QTPM as a function of h for fixed J = 1 from ED, for sizes without extra symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (a) N = 3 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (b) N = 4 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' (c) N = 3 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The avoided crossing between the ground (black squares) and first excited (purple rectangles) states is indicative of a first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Chandler and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Garrahan, Dynamics on the Way to Forming Glass: Bubbles in Space-Time, Annual Review of Physical Chemistry 61, 191 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [5] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Speck, Dynamic facilitation theory: a statistical me- chanics approach to dynamic arrest, Journal of Statis- tical Mechanics: Theory and Experiment 2019, 084015 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Hasyim and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Mandadapu, A theory of lo- calized excitations in supercooled liquids, The Journal of Chemical Physics 155, 044504 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' [7] R.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='6 16 14 12 10 8 6 (a) Energy Levels □ 1 ▯ 2-4 △ 5 □□□□□□□□□□□□□□□□□□□□□□□□□ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ △△△△△△△△△△△△△△△△△△△△△△△△△△ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='3 26 24 22 20 18 (b) Energy Levels □ 1 ▯ 2-4 △ 5 □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='2 38 36 34 32 30 28 (c) Energy Levels □ 1 ▯ 2-4 △ 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 11: Same 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Sloot, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Dongarra (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 351–359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Appendix A: TPM and QTPM for other boundary conditions In the main text, we show that ground state properties of the TPM and, consequently, the quantum phase tran- sition of the QTPM depends on the boundary conditions, but we only focused on systems with periodic boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Here, we consider the cases of periodic boundaries in only the x-dimension (PBCx) and of open boundaries (OBC), using the same Rule 60 CA considerations as for the pe- riodic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For PBCx, we use the update rule for Rule 60 as before, with the only difference that we do not need to explic- itly check for the periodicity across the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' As a result, for an initial array of L sites, there will be 2L configurations and, thus, 2L ground states for the clas- sical TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In this case, only the number of sites in the x-direction matters for the number of classical ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For example, a lattice with size N = 3 × 3 and one with N = 3 × 80 will have the same number, 8, of ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The identification of the classical ground states can be worked out from the Rule 60 evolution, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For OBC, the update rule for Rule 60 is modified for the first cell of an L-length array so that it is not updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This freedom on choosing two of the boundaries of the lattice gives an increased number of ground states for the classical TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Specifically, given a lattice of N spins, N = L × M, the number of the classical ground states is 2L+M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We now perform a similar numerical analysis as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' IV C, but only using MPS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Data is nor- malised with the system size of the given lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The system sizes accessible do not give a clear indication of a well-formed phase transition, but only signatures of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The first-order transition found is weaker than in the case of the fully PBC, which we attribute to the high number of ground states for the classical TPM, given the system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' We note that all these states for h ̸= 0 constitute low-lying excited states which affect the convergence of the MPS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' As seen from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 12 and 13 for PBCx, the difference between the square lattice size scaling and the quasi-1D rectangular stripes is more pronounced, when compared to the finite-size scaling for PBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Extra calculations on wider rectangular stripes verify that this difference is only a feature of the quasi-1D geometry of the lattice and not an inherent property of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Accuracy is lost with increasing size and the MPS results for the sizes studied are not reflective of the true thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The above considerations for PBCx are even more no- ticeable for the case of OBC, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Rectangu- lar stripes are fully continuous, while they seem to have converged to their “thermodynamic” behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' How- ever, as seen from the square system sizes, the behaviour of the model remains the same regardless of the bound- ary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' It becomes apparent though that bigger system sizes soon become computationally inaccessible due to the exponential number of classical ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This behaviour shows an obvious discrepancy with stan- dard MPS methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' normally, fully periodic system sizes are computationally harder to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Here, we stud- ied a model where the exponential number of low-lying states (ground states for h = 0) deters the convergence of the algorithm and also significantly increases the lattice size where the “thermodynamic limit” has been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' QTPM belongs to this class of models, coming from clas- sical glasses, where the number of classical ground states for PBCx or OBC would progressively constitute the model numerically inaccessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Only for PBC, the ther- modynamic limit is evidently accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Another conclusion that can be drawn from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 12, 13, 14 and 15 concerns the nature of the phase transi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The possession of data for only OBC and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 15, in particular, kschangewould possibly point towards a continuous quantum phase transition or an inconclusive statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Note, however, that this conclusion would be inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Even if the “thermodynamic limit” has pos- sibly been reached, numerics for some system sizes alone does not provide enough evidence for the characterization of the phase transition for these cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' further knowledge of the ground states of the model is necessary, while also a general understanding of the behaviour (and number) of the low-lying states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The significance of these arguments is further evident from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 16 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For the case of PBCx, all de- generate ground states for the J ≫ h region are easily found from exact diagonalization calculations and classi- cally excited states are easily tractable too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' However, the same is not true for OBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The number of classically de- generate ground states increases exponentially and this is the reason why it would be pointless to show more ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Appendix B: Gap Scaling Analysis for PBC In this section we present a restricted and with limited accuracy analysis on the energy difference between the ground state and the first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This analysis was conducted based on ED and MPS methods, which limits the validity of the conclusions that can be reached: it be- comes quickly obvious that MPS methods are not power- ful enough for the detection of the actual gap, especially in regions of the parameter space with high entanglement or with a high number of low-lying excited states, where MPS often converge to excited states above the lowest- lying ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' However, the analysis below still provides an 13 □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 (c) MPS FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 13: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 12 but for systems of size N = 3 × L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' indication of the behaviour of the gap with system size when comparing systems with different symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This limited accuracy when measuring the first excited state energy is evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 18(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Data is calculated for the J = h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The gap seems to decrease with increasing the system size, but at the same time, the power of MPS to detect it is significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The situation seems clearer for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 18(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' However, it is equally problematic despite the monotonically decreasing gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The only significance of these results are an upper bounds of the actual gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For the first case, the gap seems to decrease algebraically to zero, while for the case of multiple classical ground states, it seems to decrease exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' This underlines the different behaviour depending on the existence or not of multiple classical ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' The same problems are encountered close to the phase transition from the MPS results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 19(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Both plots are normalised by the maximum value of the gap encoun- tered in the region of J values studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 < J < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0, the gap appears always to be maximum at J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 19(a) and at J = 0 for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 19(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 19(b), for J > h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 the gap approaches zero, as expected from the existence of degenerate ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 14 □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} 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△ △ △ △ △ □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 30 25 20 15 10 (a) Energy Levels □ 1 ▯ 2-5 △ 6 □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='30 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' 18: The scaling of the gap, g, for different lattice sizes without (a) and with (b) symmetries for the QTPM for the J = h = 1/0 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Both ED and MPS methods are used (where appropriate) for the calculation of the given gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' Square and rectangular sizes are equally used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='■ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 MPS ■ ▮ ▲ ▼ ◆ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ● ● ● ● ● ● ● ● ● ● ● ● ● 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} 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symmetries for the QTPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content=' For (a) gmax ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='43 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='50 and for (b) gmax = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQf_gIS/content/2301.02826v1.pdf'} diff --git a/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf b/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e22b36aab0b8a2e67741911a19036864bcdea6c2 --- /dev/null +++ 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b/2tFKT4oBgHgl3EQfQC0t/content/tmp_files/2301.11765v1.pdf.txt @@ -0,0 +1,2154 @@ +ExplainableFold: Understanding AlphaFold Prediction with Explainable AI +Juntao Tan 1 Yongfeng Zhang 1 +Abstract +This paper presents ExplainableFold, an explain- +able AI framework for protein structure prediction. +Despite the success of AI-based methods such as +AlphaFold in this field, the underlying reasons for +their predictions remain unclear due to the black- +box nature of deep learning models. To address +this, we propose a counterfactual learning frame- +work inspired by biological principles to generate +counterfactual explanations for protein structure +prediction, enabling a dry-lab experimentation +approach. Our experimental results demonstrate +the ability of ExplainableFold to generate high- +quality explanations for AlphaFold’s predictions, +providing near-experimental understanding of the +effects of amino acids on 3D protein structure. +This framework has the potential to facilitate a +deeper understanding of protein structures. +1. Introduction +The protein folding problem studies how a protein’s amino +acid sequence determines its tertiary structure. It is crucial +to biochemical research because a protein’s structure influ- +ences its interaction with other molecules and thus its func- +tion. Current machine learning models have gain increasing +success on 3D structure prediction (AlQuraishi, 2021; Tor- +risi et al., 2020). Among them, AlphaFold (Jumper et al., +2021) provides near-experimental accuracy on structure pre- +diction, which is considered as an importance achievement +in recent years. Nevertheless, one of the important problems +of AlphaFold, as well as other deep models, is that they can- +not provide explanations for their predictions. Essentially, +the why question still remains largely unsolved: the model +gives limited understanding of why the proteins are folded +into the structures they are, which hinders the model’s ability +to provide deeper insights for human scientists. +However, it is crucial to understand the mechanism of pro- +tein folding from both AI and scientific perspectives. From +1Department of Computer Science, Rutgers University. Corre- +spondence to: Juntao Tan , Yongfeng +Zhang . +Copyright by the author(s). +the AI perspective, explainability has long been an important +consideration. State-of-the-art protein structure prediction +models leverage complex deep and large neural networks, +which makes it difficult to explain their predictions or debug +the trained model for further improvement. From the sci- +entific perspective, scientists’ eager to conquest knowledge +is not satisfied with just knowing the prediction results, but +also knowing the why behind the results (Li et al., 2022). +In particular, structural biologists not only care about the +structure of proteins, but also need to know the underlying +relationship between protein primary sequences and tertiary +structures (Dill et al., 2008; Dill & MacCallum, 2012). +It has been established that certain amino acids play signifi- +cant roles in the protein folding process. For instance, one +single disorder in the HBB gene can significantly change +the structure of hemoglobin, the protein that carries oxygen +in blood, causing the sickle-cell anaemia (Kato et al., 2018). +Knowing the relationship between amino acids and protein +structure helps scientists to produce synthetic proteins with +precisely controlled structures (Tan et al., 2020) or mod- +ify existing proteins with desired properties (Szymkowski, +2005; Mart´ınez et al., 2020; Ackers & Smith, 1985), which +are essential for advanced research directions such as drug +design. Additionally, in certain research tasks, scientists +would like to modify the amino acids without drastically +changing the protein structure, which requires the knowl- +edge of “safe” residue substitutions (Bordo & Argos, 1991), +i.e., the knowledge of which amino acids are not the most +crucial ones in the folding process. +While currently there are few Explainable AI-based methods +to study the mechanism of protein folding, many previous +biochemical researches have been conducted for this pur- +pose. One of the best known methods is via site-directed +mutagenesis (Hutchison et al., 1978; Carter, 1986; Sarkar & +Sommer, 1990). To test the role of certain residues in pro- +tein folding, biologists either delete them from the sequence +(i.e., site-directed deletion) (Arpino et al., 2014; Gl¨uck & +Wool, 2002; Flores-Ram´ırez et al., 2007; Dominy & An- +drews, 2003) or replace them with other types of amino acids +(i.e., site-directed substitution) (Flores-Ram´ırez et al., 2007; +Bordo & Argos, 1991; Betts & Russell, 2003) and then mea- +sure their influences on the 3D structure. However, these +approaches suffer from several limitations: 1) So far, modi- +fication of such residues can be limited by methods for their +arXiv:2301.11765v1 [cs.AI] 27 Jan 2023 + +ExplainableFold: Understanding AlphaFold Prediction with Explainable AI +Figure 1. (a) Some amino acids play crucial roles in protein folding. By removing the effects of a relative small set of these residues, the +predicted structure will be different. (b) Some other residues are less important. Despite deleting a large set of these residues, the protein +still folds into a similar structure. (c) Some substitutions are radical to the protein structure and even a small number of such substitutions +can drastically change the structure. (d) Some other substitutions are conservative and have small effect on the protein structure. +installation and the chemistry available for reaction, and +the modification of some residues can be very challenging +(Spicer & Davis, 2014), 2) Wet-lab methods for determin- +ing protein structures are very difficult and time-consuming +(Ilari & Savino, 2008), and 3) The wet lab experiments de- +scribed above have many prerequisites and obstacles, and +may not be completely safe for many researchers. +Recently, AI-based dry-lab methods such as AlphaFold pro- +vide near-experimental protein structure predictions (Jumper +et al., 2021), which sheds light on the possibility to gen- +erate insightful understandings of protein folding by ex- +plaining AlphaFold’s inference process. Such (Explainable) +AI-driven dry-lab approach will largely overcome the afore- +mentioned limitations and can be very helpful for human +scientists. Furtunately, we observe that the process of testing +the effects of residues on protein structure by site-directed +mutagenesis is fundamentally similar to counterfactual rea- +soning, a commonly used technique for generating explana- +tions for machine learning models (Tan et al., 2021; 2022; +Goyal et al., 2019; Tolkachev et al., 2022; Cito et al., 2022). +Intuitively, counterfactual reasoning perturbs parts of the +input data, such as interaction records of a user (Tan et al., +2021), nodes or edges of a graph (Tan et al., 2022), pixels +of an image (Goyal et al., 2019), or words of a sentence +(Tolkachev et al., 2022), and then observes how the model +output changes accordingly. +In this paper, we propose ExplainableFold, a counterfac- +tual explanation framework that generates explanations for +protein structure prediction models. ExplainableFold mim- +ics existing biochemical experiments by manipulating the +amino acids in a protein sequence to alter the protein struc- +ture through carefully designed optimization objectives. It +provides insights about which residue(s) of a sequence is cru- +cial (or indecisive) to the protein’s structure and how certain +changes on the residue(s) will change the structure, which +helps to understand, e.g., what are the most impactful amino +acids on the structure, and what are the most radical (or safe) +substitutions when modifying a protein structure. An exam- +ple of applying our framework on CASP14 target protein +T1030 is shown in Figure 1, which shows that deletion or +substitution of a small number of residues can result in sig- +nificant changes to the protein structure, while some other +deletions or substitutions may have very small effects. We +evaluate the framework based on both standard explainable +AI metrics and biochemical heuristics. Experiments show +that the proposed method produces more faithful explana- +tions compared to previous statistical baselines. Meanwhile, +the predicted relationship between protein amino acids and +their structures are highly positively correlated with wet-lab +biochemical experimental results. +2. Related Work +The essential idea of the proposed method is to integrate +counterfactual reasoning and site-directed mutagenesis anal- +ysis in a unified machine learning framework. We discuss +the two research directions in this section. +2.1. Residue Effect Analysis by Site-directed Mutagenesis +Many studies in molecular biology, such as those involv- +ing genes and proteins, rely on the use of human-induced +mutation analysis (Stenson et al., 2017). Early metagenesis +methods were not site-specific, resulting in entirely random +and indiscriminate mutations (Egli et al., 2006). In 1978, + +a) Most Intolerant Deletion: +c) Most Radical Substitution: +TM-score: 0.47 Changed # of Residues: 80 / 273 (29%) +b) Most Tolerant Deletion: +d) Most Conservative Substitution: +TM-score: 0.66 Changed # of Residues: 142 / 273 (52%) +TM-score: 0.57 Changed # of Residues: 82 / 273 (30%)ExplainableFold: Understanding AlphaFold Prediction with Explainable AI +Hutchison et al. proposed the first method that can modify +biological sequences at desired positions with specific inten- +tions, which is known as site-directed mutagenesis. Later, +more precise and effective tools were constantly developed +(Motohashi, 2015; Doering et al., 2018). Site-directed muta- +genesis is widely utilized in biomedical research for various +applications. In this section, we focus on the use of site- +directed mutagenesis to study the impact of amino acid +mutations on protein structures (Studer et al., 2013). +Two common approaches to site-directed mutagenesis are +amino acid deletion and substitution (Choi & Chan, 2015). +The deletion approach deletes certain residues from the +sequence and observes the effects on the structure. For in- +stance, Gl¨uck & Wool (2002) identified the amino acids that +are essential to the action of the ribotoxin restrictocin by +systematic deletion of its amino acids. Flores-Ram´ırez et al. +(2007) proposed a random deletion approach to measure the +amino acids’ effects on the longest loop of GFP. Arpino et al. +(2014) conducted experiments to measure the the protein’s +tolerance to random single amino acid deletion. The substi- +tution approach, on the other hand, replaces one or multiple +residues with other types of amino acids to test their influ- +ence. For example, Clemmons (2001) substituted a small +domain of the IGF-binding protein to measure weather spe- +cific domains account for specific structures and functions. +Zhang et al. (2018) mutated a specific amino acid on the +surface of a Pin1 sub-region, known as the WW domain, and +observed significant structural change on the protein struc- +ture. Guo et al. (2004) randomly replaced amino acids to +test proteins’ tolerance to substitution at different positions. +When developing our framework, we draw insights from the +aforementioned biochemical methods, which were proven +effective in wet-lab experiments. We aim to translate the +wet-lab methods for understanding protein structures into a +dry-lab AI-driven approach. We note that there have been ex- +isting attempts which built models to understand the relation- +ship between protein structures and their residues (Masso +& Vaisman, 2008; Masso et al., 2006; Sotomayor-Vivas +et al., 2022). However, they were mostly based on statistical +analysis on wet-lab experimentation data. Our method is +the first AI-driven machine learning method developed for +understading protein structure predictions. +2.2. Counterfactual Reasoning for Explainable AI +Counterfactual explanation is a type of model-agnostic ex- +plainable AI method that tries to understand the underlying +mechanism of a model’s behavior by perturbing its input. +The basic idea is to investigate the difference of the model’s +prediction before and after changing the input data in spe- +cific ways (Wachter et al., 2017). Since counterfactual ex- +planation is well-suited for explaining black-box models, +it has been an important explainable AI method and has +been employed in various applications, including but not +limited to recommender systems (Tan et al., 2021; Ghaz- +imatin et al., 2020), computer vision (Goyal et al., 2019; +Vermeire et al., 2022), natural language processing (Yang +et al., 2020; Lampridis et al., 2020; Tolkachev et al., 2022), +molecular analysis (Tan et al., 2022; Ying et al., 2019; Lin +et al., 2021), and software engineering (Cito et al., 2022). +In this paper, we explore counterfactual explanation to ex- +plain the amino acids’ effects on protein folding. How- +ever, counterfactual explanation for protein folding exhibits +unique challenges compared with previous tasks. For exam- +ple: 1) most of the aforementioned applications are classifi- +cation tasks, for which the explanation goal is very clear – +looking for a minimal change that alter the predicted label. +However, protein structure prediction is a generation task +in a continuous space, which requires careful design of the +counterfactual reasoning objective; 2) protein structure pre- +diction models such as AlphaFold take complicated input +besides the primary sequence, e.g., the MSA and templates; +3) it is easier to evaluate the explanations for the classifi- +cation tasks, nevertheless, as a new explainable AI task, +protein structure prediction poses unique challenges on the +evaluation of explanation. We will show how to overcome +these challenges in the following parts of the paper. +3. Problem Formulation +In this section, we first provide formulation of the Explain- +ableFold problem. Since a protein tertiary (3D) structure is +uniquely determined by its primary structure (amino acid +sequences) (Dill et al., 2008; Wiltgen, 2009), according to +the key idea of counterfactual explanation, we define the ex- +planation as identifying the most crucial residues that cause +the proteins to fold into the structures they are. +Suppose a protein consists of a chain of l residues, where +the i-th residue is encoded as a 21-dimensional one-hot col- +umn vector ri. The “1” element in ri indicates the type +of the residue, which can be one of the 20 common amino +acids or an additional dimension for unknown residue. By +concatenating all the residue vectors, a protein P is de- +noted as P = [r1, r2, · · · , rl], where P ∈ {0, 1}21×l is +called the protein embedding matrix. Many state-of-the-art +protein structure prediction models predict the 3D struc- +ture not only based on the residue sequence, but also uti- +lize supplementary evolutionary information (Senior et al., +2020; Jumper et al., 2021) by extracting Multiple Sequence +Alignment (MSA) (Edgar & Batzoglou, 2006) from pro- +tein databases. Suppose m proteins are retrieved from the +evolutionary database based on their similarity with protein +P, the constructed MSAs can be encoded as another ma- +trix M(P ) ∈ {0, 1}m×21×l. A protein structure prediction +model fθ predicts the protein 3D structure S based on the + +ExplainableFold: Understanding AlphaFold Prediction with Explainable AI +residue sequence and MSA embeddings: +S = fθ +� +P , M(P ) +� +(1) +where M(P ) can be omitted if the model only takes the +residue sequence information. Though a structure predic- +tion model may predict the positions of all atoms, in many +structural biology research, only the backbone of residues +are used for comparing the similarities of protein structures +(Zhang & Skolnick, 2004; 2005; Xu & Zhang, 2010; Zemla, +2003). Therefore, we adopt the same idea in this paper, +where S ∈ R3×l only contains the predicted (x, y, z)T co- +ordinates of the α-carbon atom of each amino acid residue. +The explanation is expected to be a subset of residues ex- +tracted from the protein sequence, expressed as E. The +objective of the ExplainableFold problem is to find the mini- +mum set of E that contains the most influential information +for the prediction of the 3D structure. +4. The ExplainableFold Framework +In biochemistry, the most common methods for studying +the effects of amino acids on protein structure fall into two +categories: amino acid deletion and substitution (Choi & +Chan, 2015). We design the ExplainableFold framework +from both of the two perspectives, and we introduce them +separately in the following. +4.1. The Residue Deletion Approach +The deletion approach simulates the biochemical studies +that detect essential residues for a protein by deleting one or +more residues and measuring the protein’s tolerance to such +deletion (Arpino et al., 2014; Gl¨uck & Wool, 2002; Flores- +Ram´ırez et al., 2007). The key idea is to apply a residue +mask that removes the effect of certain residues from the se- +quence and then measure the change of the protein structure. +From the counterfactual machine learning perspective, this +can be considered from two complementary views (Guidotti +et al., 2019; Tan et al., 2022): 1) Identify the minimal dele- +tion that will alter the predicted structure and the deleted +residues will be the necessary explanation; 2) Identify the +maximal deletion that still keeps the predicted structure and +the undeleted residues will be the sufficient explanation. We +design the counterfactual explanation algorithm from these +two views accordingly. +4.1.1. NECESSARY EXPLANATION (MOST INTOLERANT +DELETION) +From the necessary perspective, we aim to find the minimal +set of residues in the original sequence which, if deleted, +will change the AI model’s (such as AlphaFold’s) predicted +structure. The deleted residues thus contain the most neces- +sary information for the model’s original prediction. +We can express the perturbation on the original sequence +as a multi-hot vector ∆ = {0, 1}1×l, where δi = 1 means +that the i-th residue will be deleted and δi = 0 means it will +be kept. Then the counterfactual protein embedding matrix +P ∆ can be expressed as: +P ∆ = P ⊙ (1 − ∆) + U ⊙ ∆ +(2) +where ⊙ is the element-wise product and U ∈ {0, 1}21×l +denotes an “unknown” matrix of the same shape with P , +but with all elements being 0 except for the last row being 1 +(i.e., unknown type amino acid). Thus, for δi = 0, the i-th +residue in the original sequence will be preserved, while for +δi = 1, the i-th residue will be treated as an unknown amino +acid without any specific chemical property. +Motivated by the Occam’s Razor Principle (Blumer et al., +1987), we aim to find simple and effective explanations. The +simpleness can be characterized by the number of residues +that need to be deleted, which should be as few as possi- +ble, while effectiveness means that the predicted protein +structure should be different before and after applying the +deletions. We can use zero-norm ∥∆∥0 to represent the +number of deletions (for simpleness), while using the TM- +score between the original and the new protein structures +TM(S, S∗) to represent the degree of change on the struc- +ture (for effectiveness). TM-score is a standard measure- +ment for comparing aligned protein structures, where TM- +score > 0.5 suggests the same folding and TM-score ≤ 0.5 +suggests different foldings (Zhang & Skolnick, 2004; Xu & +Zhang, 2010). The counterfactual expxlanation algorithm +then learns the optimal explanation by solving the following +constrained optimization problem: +minimize ∥∆∥0 +s.t. TM(S, S∗) ≤ 0.5, ∆ = {0, 1}1×l +where S∗ = fθ(P ∆, M(P ∆)) +(3) +where the objective ∥∆∥0 aims to find the minimal dele- +tion, while the constraint guarantees the effectiveness of the +deletion, i.e., the deletion will change the predicted protein +structure to be different from before. +Due to the exponential combinations of sub-sequences for +a given sequence, it is impractical to search for an optimal +solution on the discrete space. To solve the problem, we use +a continuous relaxation approach to solve the optimization +problem by relaxing the multi-hot vector ∆ to a real-valued +vector. We also relax the hard constraint in Eq.(3) and +combine them into a single trainable loss function: +L1 = max +� +0, TM(S, S∗) − 0.5 + α +� ++ λ1∥σ(∆)∥1 +s.t. ∆ ∈ R1×l, where S∗ = fθ(P σ(∆), M(P σ(∆))) +(4) +where the sigmoid function σ(·) is applied so that σ(∆) ∈ +(0, 1)1×l approximates the probability distribution between + +ExplainableFold: Understanding AlphaFold Prediction with Explainable AI +the original residues and unknown residues, and α is the +margin value of the hinge loss whose default value is 0.1. +This relaxation approach has been justified in several previ- +ous studies which also learn explanation on discrete inputs +(Ying et al., 2019; Goyal et al., 2019; Tan et al., 2022). The +1-norm regularizer assures the learned perturbation σ(∆) +to be sparse (Candes & Tao, 2005), i.e., the learned expla- +nation only contains a small set of residues. λ1 is a hyper- +parameter that controls the trade-off between the complexity +and strength of the generated explanation. Eq.(4) can be +easily optimized with gradient descent. After optimization, +we convert σ(∆) to a binary vector with the threshold 0.5. +4.1.2. SUFFICIENT EXPLANATION (MOST TOLERANT +DELETION) +Symmetrically, from the sufficiency perspective, we aim to +find the maximal set of residues in the orignal sequence +which, if deleted, will not change the AI model’s predicted +structure. The undeleted residues thus contain the most +sufficient information for the model’s original prediction. +This can be formulated as a similar but reversed optimization +process as Eq.(3), which looks for the maximal perturba- +tion ∆ while keeping the same folding (TM-score > 0.5). +Therefore, the optimization problem is formulated as: +maximize ∥∆∥0 +s.t. TM(S, S∗) > 0.5, ∆ = {0, 1}1×l +where S∗ = fθ +� +P ∆, M(P ∆) +� +(5) +Similarly, we relax Eq.(5) to a differentiable loss function: +L2 = max +� +0, 0.5 − TM(S, S∗) + α +� +− λ2∥σ(∆)∥1 +s.t. ∆ ∈ R1×l, where S∗ = fθ(P σ(∆), M(P σ(∆))) +(6) +Contrary to the necessary explanation, the sufficient explana- +tion consists the undeleted residues. Hence, after optimiza- +tion, we filter the residues according to +� +1 − σ(∆) +� +> 0.5 +and include them into the sufficient explanation. +4.2. The Residue Substitution Approach +Another popular approach in biochemistry, site-directed sub- +stitution, studies the influence of the amino acids on protein +folding by replacing certain residues with other known-type +residues (Flores-Ram´ırez et al., 2007; Bordo & Argos, 1991; +Betts & Russell, 2003). Different replacements may have +different effects on protein structures, and they can be clas- +sified into two types: conservative substitution and radical +substitution (Zhang, 2000; Dagan et al., 2002). A conser- +vative substitution is considered as a “safe” substitution for +which the amino acid replacement usually have no or minor +effects on the protein structure. A radical substitution is +considered “unsafe,” which usually causes significant struc- +tual changes. Based on the above concepts, we design the +substitution approach from these two different perspectives. +4.2.1. RADICAL SUBSTITUTION EXPLANATION +From the radical substitution perspective, we aim to find the +mimimal set of residue replacements which will lead to a +different folding, and then the learned substitutions are the +most radical substitutions for the protein. +For a target protein with binary embedding matrix P , we +learn a counterfactual binary protein embedding P ′, which +has the same shape as the original embedding matrix. The +number of substitutions is represented by ∥P −P ′∥0, which +is the 0-norm of the difference between the two matrices. +To find the minimal residue substitution that changes the +original folding, the optimization problem is defined as: +minimize ||P − P ′||0 +s.t. TM(S, S′) ≤ 0.5, P ′ ∈ {0, 1}21×l +where S′ = fθ +� +P ′, M(P ′) +� +(7) +Due to the exponential search space of the substitutions, +we use the similar continuous relaxation method as in the +deletion approach. First, we relax the binary counterfactual +embedding matrix P ′ to continuous space. We also relax +the hard constraint in Eq.(7) and define the differentiable +loss function as: +L3 = max +� +0, TM(S, S′) − 0.5 + α +� ++ λ3∥P − σ(P ′)∥1 +s.t. P ′ ∈ R21×l, where S′ = fθ +� +P ′, M(P ′) +� +(8) +After optimization, we convert the learned continuous ma- +trix σ(P ′) into binary by setting the maximum value of each +column as 1 and others as 0. Then, the changed residues +between P and P ′ are the radical substitution explanations. +4.2.2. CONSERVATIVE SUBSTITUTION EXPLANATION +From the conservative substitution perspective, we aim to +find the maximal set of residue replacements which how- +ever lead to the same folding, and then the learned substitu- +ions are the most conservative substitutions for the protein. +On the contrary to Eq.(7), we formulate an inverse optimiza- +tion problem as: +maximize ||P − P ′||0 +s.t. TM(S, S′) > 0.5, P ′ ∈ {0, 1}21×l +where S′ = fθ +� +P ′, M(P ′) +� +(9) +With the same relaxation process, the loss function is: +L4 = max +� +0, 0.5 − TM(S, S′) + α +� +− λ4∥P − σ(P ′)∥1 +s.t. P ′ ∈ R21×l, where S′ = fθ +� +P ′, M(P ′) +� +(10) + +ExplainableFold: Understanding AlphaFold Prediction with Explainable AI +After learning σ(P ′) and getting the binary matrix, again, +the changed residues between P and P ′ are the conservative +substitution explanations. +4.3. Phased MSA Re-alignment +It is impractical to re-compute MSAs in each training step. +Therefore, we propose a phased MSA re-alignment strategy. +When learning the explanations, we fix the generated MSAs +and only learn the changes on the sequence embedding for +t training steps (t = 100 by default), which is one pahse. +Then, we re-align the MSAs and start another training phase. +5. Experiments +We first introduce the datasets and implementation details. +Then, we introduce the evaluation results of the deletion +approach and substitution approach, respectively. +5.1. Datasets +We test the ExplainableFold framework on the 14th Criti- +cal Assessment of protein Structure Prediction (CASP-14) +dataset1 (Moult et al.). CASP consecutively establishes +protein data with detailed structural information as a stan- +dard evaluation benchmark for protein structure prediction. +Following Jumper et al. (2021), we remove all sequences +for which fewer than 80 amino acids had the alpha carbon +resolved and remove duplicated sequences. After filtering, +55 protein sequences are selected. +5.2. Implementation Details +Though the ExplainableFold framework can be applied on +any model that predicts protein 3D structures, we choose +Alphafold2 (Jumper et al., 2021), the state-of-the-art model, +as the base model in the experiments. More specifically, we +use the OpenFold (Ahdritz et al., 2022) implementation and +load the official pre-trained AlphaFold parameters2. +When learning the explanations, the pre-trained parame- +ters of AlphaFold are fixed, and only the perturbation vec- +tor on the input (∆ for the deletion approach and P ′ for +the substitution approach) will be optimized. However, it +still requires computing the gradient through the entire Al- +phafold network, as a result, the learning process requires +extensive memory consumption. To solve the problem, we +follow exactly the same training procedure as introduced in +the original AlphaFold paper (Jumper et al., 2021). More +specifically, we use the gradient checkpointing technique to +reduce the memory usage (Chen et al., 2016). Meanwhile, +if a protein has more than 384 residues, we cut it to differ- +ent chunks for each consecutive 384 residues, and generate +1https://predictioncenter.org/casp14/ +2https://github.com/deepmind/alphafold +explanations for each of them. +We employ the same training strategy for both deletion and +substitution explanation methods: for each training phase +between MSA re-alignments, we optimize the purturbation +vector for 100 steps with Adam optimizer (Kingma & Ba, +2014) and learning rate 0.1. After each training loop, we re- +align the MSAs with the AlphaFold HHblits/JackHMMER +pipeline. We repeat the training and alignment process for +3 phases when generating explanations for each protein. +All experiments are conducted on NVIDIA A5000 GPUs. +The entire training process (including all 3 phases) for one +protein takes approximately 5 hours. We set α = 0.1 and +λ = 0.01 in Equations (4)(6)(8)(10). To realize an incre- +mental substitution process, we initialize the counterfactual +protein embedding matrix as a duplication of the original +protein embedding matrix, i.e., we initialize ∆ with all 0’s +and initialize σ(P ′) equal to the original P . Thus, the +optimal explanations are gradually learned. +5.3. Evaluation of the Deletion Approach +Counterfactual explanations can be evaluated by their com- +plexity, sufficiency and necessity (Glymour et al., 2016; Tan +et al., 2022). First, according to the Occam’s Razor Princi- +ple (Blumer et al., 1987), we hope an explanation can be as +simple as possible so that it is cognitively easy to understand +for humans. This can be evaluated by the complexity of the +explanation, i.e., the percentage of residues that are included +in the explanation: +Complexity = |E|/l +(11) +where l is the length of the protein. +Sufficiency and necessity measure how crucial the generated +explanations are for the protein structure. We follow the def- +inition in causal inference theory (Glymour et al., 2016) and +existing explainable AI research (Tan et al., 2022) and mea- +sure the explanations with two causal metrics: Probability +of Necessity (PN) and Probability of Sufficiency (PS). +PN measures the necessity of the explanation. A set of +explanation residues is considered a necessary explanation +if, by removing their effects from the protein sequence, +the predicted structure of the protein will have a different +folding (TM-score < 0.5). Suppose there are N proteins in +the testing data, then PN is calculated as: +PN = +�N +k=1 PNk +N +, PNk = +� +1, if TM(Sk, S∗ +k) ≤ 0.5 +0, else +(12) +Intuitively, PN measures the percentage of proteins whose +explanation residues, if removed, will change the protein +structure, and thus their explanation residues are necessary. +PS measures the sufficiency of the explanation. A set of +explanation residues is considered a sufficient explanation if, + +ExplainableFold: Understanding AlphaFold Prediction with Explainable AI +Table 1. PN Evaluation. Deletion∗ is the necessity optimization. +Ave Exp. +Ave Comp. +Ave TM-score +PN +Size (|E|) ↓ +(|E|/l) ↓ +TM(S, S∗) ↓ +score↑ +Random +77.31 +0.30 +0.83 +0.07 +Evo (Masso et al., 2006) +88.42 +0.33 +0.77 +0.16 +Deletion (necessity)∗ +74.54 +0.29 +0.48 +0.44 +Table 2. PS Evaluation. Deletion∗ is the sufficiency optimization. +Ave Exp. +Ave Comp. +Ave TM-score +PS +Size (|E|) ↓ +(|E|/l) ↓ +TM(S, S∗) ↑ +score↑ +Random +102.9 +0.40 +0.38 +0.31 +Evo (Masso et al., 2006) +104.95 +0.42 +0.41 +0.40 +Deletion (sufficiency)∗ +95.20 +0.37 +0.61 +0.62 +by removing all of the other residues and only keeping the +explanation residues, the protein still has the same folding. +Similarty, PS is calculated as: +PS = +�N +k=1 PSk +N +, PSk = +� +1, if TM(Sk, S∗ +k) > 0.5 +0, else +(13) +Intuitively, PS measures the percentage of proteins whose +explanation residues alone can keep the protein structure +unchanged, and thus their explanation resides are sufficient. +5.3.1. BASELINES +We compare the model performance with a common com- +putational biology baseline (Masso et al., 2006), which +analyzes a protein’s tolerance to the change on each residue +by extracting the data from evolutionary database. More +specifically, proteins are not tolerate to the mutations at evo- +lutionary conserved positions. However, they are capable +of withstanding certain mutations at other positions. When +implementing the baseline, we refer to a protein’s MSAs +and select the evolutionary conserved residues as the expla- +nation. This is illustrated in Figure 2 using protein CASP14 +target T1030 as an example, where for each residue position, +we count the number of MSAs that conserve the residue +at this position, and show the top 30% and 40% conserved +residues. We also randomly select residues as explanation +and compute PN and PS scores as another baseline to mea- +sure the general difficulty of the evaluation task, and more +details are provided in the following subsection. +5.3.2. RESULTS +The results of PN and PS evaluation are reported in Table +1 and Table 2, respectively. The explanation complexity +of our method (29% for necessary explanation and 37% +for sufficient explanation) are automatically decided by our +optimization process. However, the baselines do not have +the ability to decide the optimal explanation complexity. For +fair comparison, we set the complexities of the baselines +to be slightly larger than our method (30% for necessary +explanation and 40% for sufficient explanation). Therefore, +the baselines will have a small advantage over our method +(a) Top 30% conserved residues (b) Top 40% conserved residues +Figure 2. Evolutionary conserved residues are considered more +important for the protein structures (the residues marked red). +because they are allowed to use more residues to achieve +the necessity or sufficiency goals. +For PN evaluation, the results of the random baseline shows +that protein structures tend to be robust to residue deletions. +For example, when randomly removing the effects of 30% +residues, only 7% of the proteins fold into different struc- +tures, which indicates that finding necessary explanations +is a challenging problem. The evolutionary baseline is able +to select more necessary residues with a PN score of 0.16, +which is 128.6% better than random selection. Compared to +them, our method shows much better performance: with a +smaller number of residues, the generated explanations are +able to cause 44% of the proteins fold into different struc- +tures, outperforming the evolutionary baseline by 175%. +For PS evaluation, the evolutionary baseline is not notice- +ably better than randomly selecting residues. The reason +may be that despite the proteins’ less tolerance to the evo- +lutionary conserved residues, there is no guarantee that the +evolutionary conserved residues alone contain sufficient in- +formation to preserve the protein structure. In comparison, +our method does generate more sufficient explanations, out- +performing the evolutionary baseline by 55% according to +the PS score with less complex explanations. Meanwhile, +our TM score is > 50%, indicating that the protein structure +is indeed preserved under our sufficient explanation. +Additionally, we show the learning curve of the optimization +for CASP14 target protein T1030 in Figure 3. For necessary +optimization, the algorithm gradually deletes the protein +residues until the TM-score is smaller than 0.4 (i.e., 0.5−α, +see Eq.(4)). Then, the explanation complexity slightly drops +back while keeping the TM-score at the same level. For +sufficient optimization, the L1-loss drastically increases at +the beginning, which suggests that the algorithm is trying +to delete as many residues as possible while keeping the +original folding structure unchanged. However, after re- +computing MSAs, the TM-score becomes too low. Thus, +the algorithm increases the number of preserved residues +to keep TM-score larger than 0.6 (i.e., 0.5 + α, see Eq.(6)). +Note that the TM-scores change sharply when re-computing +MSAs at the end of each training loop. The more frequently + +175 +150 +MSAs Num +125 +100 +75 +50 +25 +0 +20 +40 +60 +80 +0 +100 +120 +Residue Position175 +150 +iSAs Num +125 +100 +75 +50 +25 +20 +40 +60 +80 +0 +100 +120 +Residue PositionExplainableFold: Understanding AlphaFold Prediction with Explainable AI +(a) Necessary Optimization +(b) Sufficient Optimization +Figure 3. Learning Curves of the Deletion Approach +we re-align MSAs, the smoother the optimization will be. +5.4. Evaluation of the Substitution Approach +The substitution approach identifies the most radical or con- +servative amino acid substitutions, which are of particular +interest in biochemical research (Zhang, 2000). Previously, +it was impractical to conduct wet-lab experiments to investi- +gate the relative “safety” of replacing specific residues with +alternative amino acids due to their prohibitive cost (Bordo +& Argos, 1991). Alternatively, scientists infer the exchange- +ability of two types of amino acids either through the use of +heuristics based on their physical or chemical properties or +through the analysis of evolutionary data, such as: +• Epstein’s distance(Epstein, 1967): Predict the impact of +switching two amino acids based on their size and polarity. +• Miyata’s distance (Miyata et al., 1979): Predict the impact +based on their volume and polarity. +• Evolutionary indicator (Bordo & Argos, 1991): Detect +“safe” substitutions based on evolutionary data. +Note that these indicators are rather suggestions than ground- +truth. They provide general trends that are better than ran- +dom selection but cannot be expected to be precise in every +scenario (Bordo & Argos, 1991). These methods are not +perfectly consistent with each other, but are linearly related. +Therefore, we utilize the amino acid substitution data gener- +ated by our method to caculate the pair-wise exchangeability +between the amino acids, and test the correlation between +our exchangeability with the above three existing exchange- +ability indicators. The details of the pair-wise substitution +statistics and the calculation of pair-wise exchangeability +are provided in the Appendix. +In Table 3, we report the correlation of our generated pair- +wise exchangeability with the three aforementioned indica- +tors by a non-parametric method: Pearson’s correlation r. +Besides, the correlation among the three biochemical meth- +ods themselves range from 0.438 to 0.578. Additionally, +the correlation is also visualized in Figure 4, where darker +color indicates higher correlation. For Pearson’s correlation, +a value greater than 0 indicates a positive association, where +r > 0.1, r > 0.3, r > 0.5 represents small, medium, and +Table 3. Correlation between our method and each of the biochem- +ical indicators. Metrics with “*” are originally distance metrics, +for which we take the inverse to reprenst the exchangeability. The +results are significant at p < 0.001 under two-tailed test. +Epstein∗ +Miyata∗ +Evolution +Radical +0.388 +0.602 +0.382 +Conservative +0.494 +0.796 +0.405 +(a) Corr. w/ Epstein’s distance +(b) Corr. w/ Miyata’s distance +Figure 4. The correlation between the exchangeability provided by +our conservative optimization method and (a) Epstein’s distance +as well as (b) Miyata’s distance. +large correlations, accordingly (Cohen et al., 2009). Both +Table 3 and Figure 4 show that our method has clear posi- +tive correlations with all of the three biochemical methods, +indicating that ExplainableFold can provide informative +exchangeability signals (Yampolsky & Stoltzfus, 2005). Be- +sides, the results generated by ExplainableFold may further +improve when larger protein datasets are available or applied +on even better base models in the future. +6. Conclusions and Future Work +In this paper, we propose ExplainableFold—an Explain- +able AI framework that helps to understand the deep learn- +ing based protein structure prediction models such as Al- +phaFold. Technically, we develop a counterfactual explana- +tion framework and implement the framework based on two +approaches: the residue deletion approach and the residue +substitution approach. Intuitively, ExplainableFold aims to +find simple explanations that are effective enough to keep +or change the protein’s folding structure. Experiments are +conducted on CASP-14 protein datasets and results show +that our approach outperforms the results from traditional +biochemical methods. We believe Explainable AI is funda- +mentally important for AI-driven scientific research because +science not only pursues the answers for the “what” ques- +tions but also (or even more) for the “why” questions. In the +future, we will further improve our framework by consider- +ing more protein modification methods beyond deletion and +substitution. We will also generalize our framework to other +scientific problems due to the flexibility of our framework. + +1.0 +L1 Loss +TM-score +0.8 +0.6 +0.4 +0.2 +0.0 +0 +50 +100 +150 +200 +250 +300 +Training Steps1.0 +L1 Loss +TM-score +0.8 +0.6 +0.4 +0.2 +0 +50 +100 +150 +200 +250 +300 +Training StepsARN +YV +A +R +N - +D +0.8 +C- +Q: +E +G +0.6 +H - +L- +K- +0.4 +M - +F - +S - +0.2 +W- +Y +V +0.0ARN +D +C +H +M +STWYV +A +R +N - +0.8 +D +0.7 +Q +E +0.6 +G +H - +0.5 +I- +L- +K- +0.4 +M - +F +0.3 +P +0.2 +T- +W - +0.1 +Y- +V +0.0ExplainableFold: Understanding AlphaFold Prediction with Explainable AI +References +Ackers, G. K. and Smith, F. R. Effects of site-specific amino +acid modification on protein interactions and biological +function. Annual review of biochemistry, 54(1):597–629, +1985. +Ahdritz, G., Bouatta, N., Kadyan, S., Xia, Q., Gerecke, +W., O’Donnell, T. J., Berenberg, D., Fisk, I., Zanichelli, +N., Zhang, B., Nowaczynski, A., Wang, B., Stepniewska- +Dziubinska, M. 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Statistical Analysis of Amino Acid Substitutions +Table 4 shows the total count of each amino acid in the +testing proteins. In Table 5, we show how many times a +specific type of substitution happens in the generated ex- +planations learned by the conservative substitution method. +For instance, the substitution of A → R happens 19 times. +The exchangeability of X → Y can be easily calculated by +|X → Y |/|X| (Bordo & Argos, 1991; Masso et al., 2006). +The same statistics for radical substitution is provided in +Table 6. For radical substitution, the higher the number +in Table 6, the lower the exchangeability, and thus the ex- +changeability of X → Y is calculated as the reciprocal +|X|/|X → Y | (Bordo & Argos, 1991; Masso et al., 2006). +Table 4. Total number of each amino acid in testing data +A +R +N +D +C +Q +E +G +H +I +# +782 +581 +827 +776 +175 +520 +848 +853 +309 +904 +L +K +M +F +P +S +T +W +Y +V +# +1122 +911 +273 +600 +506 +916 +746 +149 +595 +780 +Table 5. Structural Conservative Statistics +A +R +N +D +C +Q +E +G +H +I +L +K +M +F +P +S +T +W +Y +V +A +0 +19 +25 +16 +50 +18 +21 +78 +23 +26 +22 +22 +19 +14 +39 +28 +15 +42 +35 +8 +R +8 +0 +15 +23 +23 +11 +15 +37 +12 +12 +19 +18 +9 +15 +23 +18 +11 +49 +18 +9 +N +15 +33 +0 +32 +51 +19 +18 +56 +16 +19 +11 +19 +50 +21 +33 +21 +16 +42 +22 +14 +D +7 +29 +22 +0 +57 +21 +25 +42 +11 +19 +23 +19 +16 +21 +44 +16 +15 +32 +16 +11 +C +2 +1 +1 +2 +0 +7 +1 +9 +8 +4 +5 +5 +2 +2 +2 +4 +0 +8 +2 +5 +Q +5 +12 +18 +12 +33 +0 +14 +42 +15 +18 +7 +15 +21 +7 +33 +5 +7 +32 +21 +18 +E +14 +14 +14 +50 +47 +30 +0 +62 +15 +35 +19 +19 +18 +29 +32 +16 +11 +79 +19 +11 +G +18 +11 +19 +19 +62 +18 +18 +0 +23 +26 +29 +9 +18 +25 +29 +22 +12 +46 +15 +9 +H +0 +15 +4 +15 +11 +14 +9 +28 +0 +5 +9 +7 +5 +12 +9 +9 +2 +21 +9 +7 +I +9 +16 +16 +14 +46 +16 +25 +58 +33 +0 +49 +19 +56 +35 +29 +14 +14 +54 +18 +32 +L +5 +28 +23 +50 +57 +30 +25 +70 +21 +53 +0 +19 +47 +40 +40 +21 +19 +51 +22 +33 +K +2 +44 +22 +56 +78 +22 +28 +51 +22 +35 +18 +0 +29 +19 +47 +7 +11 +49 +21 +15 +M +4 +5 +5 +5 +9 +8 +8 +26 +5 +12 +28 +5 +0 +7 +9 +5 +4 +16 +7 +8 +F +8 +19 +16 +25 +35 +18 +9 +36 +14 +21 +19 +7 +21 +0 +21 +9 +5 +46 +21 +2 +P +8 +11 +5 +12 +16 +9 +14 +25 +9 +28 +9 +14 +28 +16 +0 +4 +14 +30 +16 +2 +S +21 +26 +22 +33 +49 +14 +28 +53 +23 +30 +26 +21 +29 +22 +36 +0 +40 +65 +36 +19 +T +9 +19 +21 +35 +33 +22 +21 +40 +9 +33 +42 +22 +30 +28 +29 +22 +0 +47 +25 +19 +W +0 +2 +4 +4 +7 +1 +1 +12 +4 +7 +5 +0 +5 +7 +7 +4 +0 +0 +7 +4 +Y +7 +9 +16 +23 +25 +18 +15 +36 +11 +9 +5 +7 +29 +26 +15 +16 +8 +37 +0 +9 +V +8 +12 +7 +29 +44 +25 +9 +54 +25 +49 +14 +14 +43 +19 +11 +15 +11 +40 +18 +0 +Table 6. Structural Radical Statistics +A +R +N +D +C +Q +E +G +H +I +L +K +M +F +P +S +T +W +Y +V +A +0 +28 +22 +16 +39 +5 +19 +22 +30 +28 +25 +5 +25 +22 +33 +14 +14 +64 +16 +14 +R +8 +0 +11 +33 +19 +5 +28 +22 +16 +25 +8 +2 +11 +36 +25 +5 +5 +33 +11 +11 +N +11 +16 +0 +8 +47 +11 +22 +16 +14 +19 +25 +22 +19 +25 +25 +5 +8 +25 +14 +19 +D +11 +11 +11 +0 +25 +14 +22 +16 +19 +25 +11 +16 +44 +16 +16 +11 +0 +22 +14 +25 +C +2 +0 +0 +0 +0 +2 +2 +11 +2 +0 +8 +8 +2 +2 +5 +8 +5 +2 +0 +5 +Q +2 +19 +5 +11 +25 +0 +22 +16 +14 +14 +14 +16 +14 +16 +25 +5 +5 +19 +8 +8 +E +5 +11 +5 +19 +58 +5 +0 +22 +14 +19 +14 +11 +28 +36 +19 +5 +8 +39 +25 +19 +G +2 +25 +2 +16 +56 +11 +14 +0 +8 +33 +28 +22 +53 +28 +14 +14 +5 +44 +22 +8 +H +2 +5 +0 +0 +16 +14 +8 +11 +0 +11 +8 +8 +0 +0 +14 +2 +0 +5 +5 +14 +I +25 +28 +22 +36 +33 +5 +22 +44 +5 +0 +2 +8 +16 +8 +30 +11 +11 +47 +5 +11 +L +22 +28 +25 +30 +64 +28 +28 +72 +19 +25 +0 +47 +33 +11 +56 +28 +22 +36 +28 +19 +K +14 +2 +8 +25 +81 +22 +19 +30 +11 +14 +8 +0 +16 +30 +33 +5 +19 +64 +19 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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf,len=1953 +page_content='ExplainableFold: Understanding AlphaFold Prediction with Explainable AI Juntao Tan 1 Yongfeng Zhang 1 Abstract This paper presents ExplainableFold, an explain- able AI framework for protein structure prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' Despite the success of AI-based methods such as AlphaFold in this field, the underlying reasons for their predictions remain unclear due to the black- box nature of deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' To address this, we propose a counterfactual learning frame- work inspired by biological principles to generate counterfactual explanations for protein structure prediction, enabling a dry-lab experimentation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' Our experimental results demonstrate the ability of ExplainableFold to generate high- quality explanations for AlphaFold’s predictions, providing near-experimental understanding of the effects of amino acids on 3D protein structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' This framework has the potential to facilitate a deeper understanding of protein structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' Introduction The protein folding problem studies how a protein’s amino acid sequence determines its tertiary structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' It is crucial to biochemical research because a protein’s structure influ- ences its interaction with other molecules and thus its func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' Current machine learning models have gain increasing success on 3D structure prediction (AlQuraishi, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' Tor- risi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' Among them, AlphaFold (Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=', 2021) provides near-experimental accuracy on structure pre- diction, which is considered as an importance achievement in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' Nevertheless, one of the important problems of AlphaFold, as well as other deep models, is that they can- not provide explanations for their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' Essentially, the why question still remains largely unsolved: the model gives limited understanding of why the proteins are folded into the structures they are, which hinders the model’s ability to provide deeper insights for human scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' However, it is crucial to understand the mechanism of pro- tein folding from both AI and scientific perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' From 1Department of Computer Science, Rutgers University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFKT4oBgHgl3EQfQC0t/content/2301.11765v1.pdf'} +page_content=' Corre- spondence to: Juntao Tan ∆2 and ∆1 ≤ n − 2. Let Γ1 be +a zig-zag drawing of C1 with starting point vj1 and ending point vr1, and let +Γ2 be a zig-zag drawing of C2 with starting point vj2 and ending point vr2. If +∆2 +2 ≤ j2 − j1 < n−(∆1−1) +2 +, then Γ1 ∪ Γ2 has no multiple edges. +12 + +Proof. As described in the proof of Lemma 2, the edges of a zig-zag drawing of +a ∆-regular caterpillar are all drawn as segments whose slope belongs to a set +of ∆ slopes. In particular, for every spine vertex v, the edges incident to v are +drawn using all these ∆ slopes. Based on this observation, we show that the ∆1 +slopes used to represent the edges of Γ1 are distinct from the ∆2 slopes used to +represent the edges of Γ2. We use the same notation used in Theorem 2. +Consider the staring vertex vj1 of Γ1; the edges incident to vj1 are drawn +with the first ∆1 slopes of ψ(ij1). Analogously, the edges incident to the starting +vertex vj2 of Γ2 are drawn with the first ∆2 slopes of ψ(ij2). Since j2 −j1 ≥ ∆2 +2 , +the sequence ψ(ij2) is shifted clockwise by ∆2 units with respect to ψ(ij1). On +the other hand, since j2 − j1 ≤ n−(∆1−1) +2 +, the sequence of the first ∆2 slopes of +ψ(ij2) does not overlap with the first ∆1 slopes of ψ(ij1), which concludes the +proof. +Theorem 3. Let C1, C2, . . . , Ch be h caterpillars such that Ci is ∆i-regular, +for 1 ≤ i ≤ h, and ∆h ≤ ∆h−1 ≤ · · · ≤ ∆1 ≤ n − h. If �h +i=1 ∆i ≤ n − 1 and +�h +i=2 +� ∆i +2 +� +< n−(∆1−1) +2 +, then there exists a k-planar packing with k ∈ O(∆1h2). +Proof. We compute a zig-zag drawing of C1 with starting point vj1, with j1 = 1; +for each Ci, with 2 ≤ i ≤ h, we compute a zig-zag drawing Γi with starting +vertex vji where ji = ji−1 + +� ∆i +2 +� +. Notice that, each vertex v of Γ1 ∪Γ2 ∪· · ·∪Γh +has degree at most n−1; namely �h +i=1 degCi(v) ≤ �h +i=1 ∆i ≤ n−1. Moreover, +given two caterpillars Ci and Ci′ with 1 ≤ i < i′ ≤ h, we have that: (i) +ji′ − ji ≥ ji′ − ji′−1 = +� +∆i′ +2 +� +; and (ii) ji′ − ji ≤ jh − j1 = �h +i=2⌈ ∆i +2 ⌉, which +gives ji′ − ji < n−(∆1−1) +2 +< n−(∆i−1) +2 +. Putting together (i) and (ii), we obtain +∆i′ +2 +< ji′ − ji < n−(∆i−1) +2 +. Hence, Lemma 3 holds for every pair of caterpillars +and the union of all the zig-zag drawings Γ1, Γ2, . . . , Γh is a valid packing of +C1, C2, . . . , Ch. +We now prove the bound on the number of crossings along an edge. We +consider an edge of Γ1; the number of crossings along an edge of another drawing +is bounded by the same number. +Let e be an edge of the drawing Γ1. +By +Lemma 2, the number of crossings χe along e due to the edges of another +drawing Γl (with 2 ≤ l ≤ h) is at most 2(∆1 + ∆l) + 4(jl − j1). Summing over +all drawings distinct form Γ1, we obtain χe ≤ �h +l=2(2(∆1 + ∆l) + 4(jl − j1)). +Considering that jl ≥ jl−1 + +� ∆i +2 +� +, we obtain that jl − j1 = �l +i=2 +� ∆i +2 +� +. Since +∆l ≤ ∆1 for every 2 ≤ l ≤ h, we have jl − j1 ≤ (l − 1)( ∆1 +2 + 1). Therefore, we +obtain χe ≤ �h +l=2(4∆1 + 4(l − 1)( ∆1 +2 + 1)) ≤ (∆1 + 2)h2 + 4∆1(h − 1). +We now consider a special case of packing a set of h ∆1-, ∆2-, . . . , ∆h- +regular caterpillars where, for each ∆i (1 ≤ i ≤ h), we have that ∆i − 1 is a +multiple of ∆i+1 − 1. In this case, we show that the sufficient conditions of +Theorem 3 can be relaxed. For example, consider the packing of a 17-regular +caterpillar and two 9-regular caterpillars, each having 34 vertices. These three +caterpillars do not satisfy the sufficient condition of Theorem 3. However, a k- +planar packing of these caterpillars is possible, as proven in Theorem 4. We start +13 + +vj +vi +vr +vg +upper part +lower part +c = 0 +c = 1 +c = 2 +d = 0 +d = 1 +d = 2 +Figure 6: Illustration for Property 2; σ = 5. For each spine vertex, c and d are +shown. Considering adjacent spine vertices, the sum of c and d is 2 or 3. +with the following property, which immediately follows from the construction of +a zig-zag drawing (see also Fig. 6 for an illustration). +Property 2. Let Γ be a zig-zag drawing of a ∆-regular caterpillar with starting +vertex vj and ending vertex vr. If vi is a spine vertex in the upper part of Γ, +then i = j + c(∆ − 1) for some c ∈ N; if vg is a spine vertex in the lower part of +Γ, then g = r + d(∆ − 1) for some d ∈ N. Moreover, if vi and vg are adjacent +then either c + d = +� σ +2 +� +− 1 or c + d = +� σ +2 +� +, where σ is the number of spine +vertices of Γ. +Property 2 is extensively used in the proof of the following lemma. +Lemma 4. Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n- +vertex ∆2-regular caterpillar such that ∆1−1 = q(∆2−1), for some q ∈ N+ and +∆i ≤ n − 2 (for i = 1, 2). Let Γ1 be a zig-zag drawing of C1 with starting point +vj1 and ending point vr1, and let Γ2 be a zig-zag drawing of C2 with starting +point vj2 and ending point vr2. If 0 < j2 − j1 < n−(∆1−1) +2 +, then Γ1 ∪ Γ2 has no +multiple edges. +Proof. Let (vi1, vg1) be an edge of Γ1 and (vi2, vg2) be an edge of Γ2. Assume +that vi1 belongs to the upper part of Γ1 and vi2 belongs to the upper part of Γ2. +Note that this implies that vg1 belongs to the lower part of Γ1 and vg2 belongs +to the lower part of Γ2. We prove that (vi1, vg1) and (vi2, vg2) do not coincide. +We first show that it does not happen that vi1 coincides with vi2 and vg1 +coincides with vg2. We then show that it does not happen that vi1 coincides +with vg2 and vg1 coincides with vi2. In the rest of the proof we will express the +four indices i1, i2, g1 and g2 in terms of the values j1, j2, r1 and r2, according to +Property 2. Without loss of generality, we can assume that r2 ≤ n and j1 ≥ 1, +i.e., that the vertices vr2, vn, v1, and vj1 appear in this clockwise order, with +vr2 and vn possibly coincident and with v1 and vj1 possibly coincident. With +these assumptions, we have j1 < j2 < r1 < r2 and vi1 can coincide with vg2 +14 + +only if i1 = g2 − n, i.e., only if the value of g2 is greater than n and coincides +with i1 modulo n. Thus, while assuming that vi1 coincides with vi2 implies that +i1 = i2, assuming that vi1 coincides with vg2 implies that i1 = g2 − n. +Case 1: It does not happen that vi1 coincides with vi2 and vg1 coincides +with vg2. +At least one vertex per edge is a spine vertex. We distinguish four sub-cases +depending on which vertex is a spine vertex for each edge. Since all the cases +are very similar, we give here only the first case and the others can be found in +the appendix. +Case 1.a: vi1 and vi2 are spine vertices. By Property 2 we have, for some +c1, c2 ∈ N: +i1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1). +(5) +and +i2 = j2 + c2(∆2 − 1). +(6) +If vi1 coincides with vi2, we have i1 = i2; from Eq. (5) and Eq. (6) we obtain: +j2 − j1 = (qc1 − c2)(∆2 − 1). +(7) +Concerning vg1 and vg2, we have: +gm +1 ≤ g1 ≤ gM +1 ; +gm +2 ≤ g2 ≤ gM +2 . +with gm +l += rl + dl(∆l − 1), gM +l += rl + (dl + 1)(∆l − 1) for some dl ∈ N such that +cl + dl = +� σl +2 +� +− 1, where σl is the number of spine vertices of Cl, for l = 1, 2. +We prove that gM +1 +< gm +2 , which implies g1 ̸= g2. To have gM +1 +< gm +2 it must be: +r1 + (d1 + 1)(∆1 − 1) < r2 + d2(∆2 − 1) +r1 + q(d1 + 1)(∆2 − 1) < r2 + d2(∆2 − 1) +r2 − r1 > (qd1 + q − d2)(∆2 − 1). +(8) +Since rl = jl + n−(∆l−1)(σl +mod 2) +2 +, for l = 1, 2, Eq. (8) can be rewritten as: +j2 − j1 > +� +(qd1 + q − d2) + (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 +� +(∆2 − 1). +(9) +Combining Eq. (7) and Eq. (9) we obtain: +qc1 − c2 > (qd1 + q − d2) + (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 +. +Since cl + dl = +� σl +2 +� +− 1, we have dl = +σl+1(σl +mod 2) +2 +− cl − 1, for l = 1, 2; +replacing d1 and d2 in the previous equation, we obtain: +qc1 − c2 > qσ1 +2 ++ q(σ1 +mod 2) +2 +− qc1 − q + q − σ2 +2 − 1(σ2 +mod 2) +2 ++ c2+ ++ 1 + σ2 +mod 2 +2 +− q(σ1 +mod 2) +2 +15 + +which, considering that σ2 = +n−2 +∆2−1 = q(n−2) +∆1−1 = qσ1, implies: +qc1 − c2 > 1 +2 +(10) +In summary, to have gM +1 +< gm +2 Eq. (10) must hold. On the other hand, from +Eq. (7) and from the hypothesis that j2−j1 > 0 we obtain (qc1−c2)(∆2−1) > 0 +which, since (∆2 − 1) > 0, implies qc1 − c2 > 0 and, since qc1 − c2 is integer, +can be rewritten as qc1 − c2 ≥ 1. This implies that Eq. (10) holds and therefore +that gM +1 +< gm +2 and g1 ̸= g2. +Case 2: It does not happen that vi1 coincides with vg2 and vg1 coincides +with vi2. +Also in this case we distinguish four sub-cases depending on which vertex is +a spine vertex for each edge. As in Case 1, we give here only the first sub-case, +while the others can be found in the appendix. +Case 2.a: +vi1 and vi2 are spine vertices. +Since vg2 is a vertex in the +lower part of Γ2, it must be g2 = r2 + d2(∆2 − 1) + α2, for some α2 such that +0 ≤ α2 < ∆2 − 1. If vg2 coincides with vi1, as explained above, it must be +i1 = g2 − n. Combining the expression of g2 with Eq. (5) we obtain: +r2 − j1 = (qc1 − d2)(∆2 − 1) − α2 + n. +(11) +Concerning vg1, we have: +gm +1 ≤ g1 ≤ gM +1 ; +with gm +1 = r1 + d1(∆1 − 1), gM +1 += r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such +that c1 + d1 = +� σ1 +2 +� +− 1, where σ1 is the number of spine vertices of C1. +We prove that i2 < gm +1 , which implies i2 ̸= g1. To have i2 < gm +1 it must be: +j2 + c2(∆2 − 1) < r1 + d1(∆1 − 1) +j2 + c2(∆2 − 1) < r1 + qd1(∆2 − 1) +j2 − r1 < (qd1 − c2)(∆2 − 1). +(12) +Since rl = jl + n−(∆l−1)(σl +mod 2) +2 +, for l = 1, 2, Eq. (11) can be rewritten as: +j2 − j1 = (qc1 − d2)(∆2 − 1) − α2 + n +2 + (∆2 − 1)(σ2 +mod 2) +2 +, +(13) +while Eq. (12) can be rewritten as: +j2 − j1 < +� +(qd1 − c2) − q(σ1 +mod 2) +2 +� +(∆2 − 1) + n +2 . +(14) +Combining Eq. (13) and Eq. (14) we obtain: +qc1 − d2 < (qd1 − c2) − (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 ++ +α2 +∆2 − 1. +16 + +Since cl + dl = +� σl +2 +� +− 1, we have dl = +σl+1(σl +mod 2) +2 +− cl − 1, for l = 1, 2; +replacing d1 and d2 in the previous equation, we obtain: +qc1 − σ2 +2 − σ2 +mod 2 +2 ++ c2 + 1 < qσ1 +2 ++ q(σ1 +mod 2) +2 +− qc1 − q − c2− +− σ2 +mod 2 +2 +− q(σ1 +mod 2) +2 ++ +α2 +∆2 − 1 +which, considering that σ2 = +n−2 +∆2−1 = q(n−2) +∆1−1 = qσ1, implies: +qc1 − qσ1 +2 ++ c2 < −q + 1 +2 ++ +α2 +2(∆2 − 1) +(15) +In summary, to have iM +2 +< gm +1 Eq. (15) must hold. On the other hand, from +Eq. (13) and from the hypothesis that j2 − j1 < +n−(∆1−1) +2 += +n−q(∆2−1) +2 +we +obtain: +qc1 − d2 + 1 +2(σ2 +mod 2) < −q +2 + +α2 +∆2 − 1. +Replacing again d2 with σ2+1(σ2 +mod 2) +2 +− c2 − 1, we obtain: +qc1 − qσ1 +2 ++ c2 < −q + 2 +2 ++ +α2 +∆2 − 1. +(16) +We have that − q +2 − 1 + +α2 +∆2−1 < − q +2 − 1 +2 + +α2 +2(∆2−1), since +α2 +2(∆2−1) < 1 +2. +In other words, Eq. (16) implies that Eq. (15) holds and therefore that +i2 < gm +1 and i2 ̸= g1. +Theorem 4. Let C1, C2, . . . , Ch be h caterpillars such that Ci is ∆i-regular, +∆i − 1 is a multiple of ∆i+1 − 1, with 1 ≤ i < h, and ∆i ≤ n − h (for i = +1, 2, . . . , h). If n ≥ 2h + (∆1 − 1), then there exists a k-planar packing with +k ∈ O(∆1h + h2). +Proof. For each Ci, with 1 ≤ i ≤ h, we compute a zig-zag drawing Γi with +starting vertex vi. Notice that, given two caterpillars Cj1 and Cj2 with 1 ≤ +j1 < j2 ≤ h, we have that ∆j1 − 1 is a multiple of ∆j2 − 1, and the zig-zag +drawings Γj1 and Γj2 have starting vertices vj1 and vj2, respectively. Hence, +0 < j2 − j1 < h and by hypothesis h ≤ n−(∆1−1) +2 +. Hence, Lemma 4 holds for +every pair of caterpillars and the union of all zig-zag drawings Γ1, Γ2, . . . , Γh is +a valid packing of C1, C2, . . . , Ch. +The proof of the bound on the number of crossings along an edge is the same +as the one of Theorem 2, considering that ∆l ≤ ∆1 and that jl − j1 = l − 1 for +every 2 ≤ l ≤ h. +17 + +5 +Lower bounds +In this section we first give a general lower bound on the value of k for k-planar +h-packings; we then increase this lower bound for some small values of h. +Theorem 5. Every k-planar h-packing of h graphs with n vertices and m edges +is such that k ≥ +h2m2 +14.6n2 . +Proof. The number of edges of a k-planar graph with n vertices is at most +3.81 +√ +k ·n, for k ≥ 2 [1]. Since the h graphs have h·m edges in total, a k-planar +packing of these graphs can exist only if h ≤ 3.81 +√ +k n +m, i.e., if k ≥ +h2m2 +14.6n2 . +Since for a tree m = n − 1, we have the following. +Corollary 1. Every k-planar h-packing of h trees is such that k ≥ +h2 +58.4. +We now refine the lower bound above for small values of h in an h-placement +of caterpillars. Specifically we show that for values of h equal to 3, 4, and 5 the +corresponding lower bounds are 2, 3, and 5, respectively. Note that for all these +cases the lower bound implied by Corollary 1 is 1. +Theorem 6. For h = 3, 4 there exists a caterpillar C with at least h+7 vertices +for which every k-planar h-placement of C is such that k ≥ h − 1. For h = 5 +there exists a caterpillar C with at least 24 vertices for which every k-planar +5-placement of C is such that k ≥ h. +Proof. Case h = 3, 4. Let n be an integer such that n ≥ h + 7, and let Cn,h be +the n-vertex caterpillar shown in Fig. 7. Notice that the vertex of Cn,h denoted +as v in Fig. 7 has degree n − h; we call it the center of Cn,h. Consider any +h-placement of Cn,h into a graph G and denote as vi the vertex of G which the +center of Ci is mapped to (i = 1, 2, . . . , h). The vertices v1, v2, . . . , vh must be +distinct because, if two centers were mapped to the same vertex of G then this +vertex would have degree larger than n − 1. Namely, if two centers are mapped +to the same vertex, this vertex has degree 2n − 2h which is larger than n − 1 +if n > 2h − 1, i.e., if h + 7 > 2h − 1, which is true for h < 6. Since each vi +(1 ≤ i ≤ h) has degree n − h in Ci and degree 1 in each of the h − 1 other +caterpillars, its degree in G is n−1. Thus, G contains Kh,n−h. Thus, for h = 3, +G contains K3,7 (n ≥ 10 in this case), which is not 1-planar [7]; for h = 4, G +contains K4,7 (n ≥ 11 in this case), which is not 2-planar [4]. The case h = 5 is +analogous with K5,19, which is not 4-planar [3]. +6 +Concluding Remarks and Open Problems +This paper studied the placement and the packing of caterpillars into k-planar +graphs. It proved necessary and sufficient conditions for the h-placement of ∆- +regular caterpillars in a k-planar graph and sufficient conditions for the packing +of a set of ∆1-, ∆2-, . . . , ∆h-regular caterpillars with k ∈ O(∆1h2) (∆1 is the +18 + +. . . +. . . +v +h + 2 +h +n − h − 2 +Cn,h +Figure 7: +A caterpillar as described in the proof of Theorem 6. +maximum vertex degree in the set). The work in this paper contributes to the +rich literature concerning the placement and the packing problem in planar and +non-planar host graphs and it specifically relates with a recent re-visitation of +these questions in the beyond-planar context. +Many open problems naturally arise from the research in this paper. We +conclude the paper by listing some of those that, in our opinion, are among the +most interesting. +• Extend the characterization of Theorem 2 to the placement of caterpillars +that are not ∆-regular. +• Theorems 4 and 3 give sufficient conditions for the k-planar packing of +some families of caterpillars. It would be interesting to give a complete +characterization of the packability of these families into k-planar graphs. +• Theorem 6 improves the lower bound of Theorem 5 for caterpillars that +are not ∆-regular. It would be interesting to find a similar result with +∆-regular caterpillars. +Finally, we point out that one could investigate what graphs can be packed/placed +into a k-planar graph for a given value of k, instead of studying how k varies with +the number h and the vertex degree of the caterpillars that are packed/placed. +While the interested reader can refer to [3] for results with k = 1, the following +theorem gives a preliminary result for k = 2 (see the appendix for a proof). +Notice that Eq. (4) in the proof of Theorem 2 would give upper bounds in the +range [86, 137] for the caterpillars considered by the following theorem. +Theorem 7. A ∆-regular caterpillar with 4 ≤ ∆ ≤ 7 admits a 2-planar 3- +placement. +References +[1] Eyal Ackerman. On topological graphs with at most four crossings per +edge. 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Saidur Rahman. +Planar Graph Drawing, vol- +ume 12 of Lecture Notes Series on Computing. +World Scientific, 2004. +doi:10.1142/5648. +[24] Yoshiaki Oda and Katsuhiro Ota. Tight planar packings of two trees. In +22nd European Workshop on Computational Geometry, 2006. +[25] Norbert Sauer and Joel Spencer. Edge disjoint placement of graphs. J. +Comb. Theory, Ser. B, 25(3):295–302, 1978. +[26] S. K. Teo and H. P. Yap. Packing two graphs of order n having total size +at most 2n − 2. Graphs Comb., 6(2):197–205, 1990. +[27] Hong Wang and Norbert Sauer. +Packing three copies of a tree into a +complete graph. European Journal of Combinatorics, 14(2):137 – 142, 1993. +[28] Mariusz Wozniak and A. Pawel Wojda. Triple placement of graphs. Graphs +and Combinatorics, 9(1):85–91, 1993. +[29] Andrzej Zak. A note on k-placeable graphs. Discret. Math., 311(22):2634– +2636, 2011. doi:10.1016/j.disc.2011.08.002. +21 + +A +Missing cases for the proof of Lemma 4 +Case 1.b: vg1 and vg2 are spine vertices. By Property 2 we have, for some +d1, d2 ∈ N: +g1 = r1 + d1(∆1 − 1) = r1 + qd1(∆2 − 1). +(17) +and +g2 = r2 + d2(∆2 − 1). +(18) +If vg1 coincides with vg2, we have g1 = g2; from Eq. (17) and Eq. (18) we +obtain: +r2 − r1 = (qd1 − d2)(∆2 − 1). +(19) +Concerning vi1 and vi2, we have: +im +1 ≤ i1 ≤ iM +1 ; +im +2 ≤ i2 ≤ iM +2 . +with im +l = il + cl(∆l − 1), iM +l += il + (cl + 1)(∆l − 1) for some cl ∈ N such that +cl + dl = +� σl +2 +� +− 1, where σl is the number of spine vertices of Cl, for l = 1, 2. +We prove that iM +1 < im +2 , which implies i1 ̸= i2. To have iM +1 < im +2 it must be: +j1 + (c1 + 1)(∆1 − 1) < j2 + c2(∆2 − 1) +j1 + q(c1 + 1)(∆2 − 1) < j2 + c2(∆2 − 1) +j2 − j1 > (qc1 + q − c2)(∆2 − 1). +(20) +Since rl = jl + n−(∆l−1)(σl +mod 2) +2 +, for l = 1, 2, Eq. (19) can be rewritten as: +j2 − j1 = +� +(qd1 − d2) + (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 +� +(∆2 − 1). +(21) +Combining Eq. (21) and Eq. (20) we obtain: +c2 − qc1 − q > −1 +2. +(22) +In summary, to have iM +1 +< im +2 Eq. (22) must hold. On the other hand, from +Eq. (21) and from the hypothesis that j2 − j1 > 0 we obtain c2 − qc1 − q > −1 +which, since c2 − qc1 − q is integer, can be rewritten as c2 − qc1 − q ≥ 0. This +implies that Eq. (22) holds and therefore that iM +1 < im +2 and i1 ̸= i2. +Case 1.c: vi1 and vg2 are spine vertices. By Property 2 we have, for some +c1 ∈ N: +i1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1). +(23) +We also have, for some c2 ∈ N and 0 ≤ α2 < ∆2 − 1: +i2 = j2 + c2(∆2 − 1) + α2. +(24) +22 + +If vi1 coincides with vi2, we have i1 = i2; from Eq. (23) and Eq. (24) we +obtain: +j2 − j1 = (qc1 − c2)(∆2 − 1) − α2. +(25) +Concerning vg1, we have: +gm +1 ≤ g1 ≤ gM +1 ; +with gm +1 = r1 + d1(∆1 − 1), gM +1 += r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such +that c1 + d1 = +� σ1 +2 +� +− 1, where σ1 is the number of spine vertices of C1. +Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2 + d2(∆2 − 1). +We prove that gM +1 +< g2, which implies g1 ̸= g2. To have gM +1 +< g2 it must be: +r1 + (d1 + 1)(∆1 − 1) < r2 + d2(∆2 − 1) +r1 + q(d1 + 1)(∆2 − 1) < r2 + d2(∆2 − 1) +r2 − r1 > (qd1 + q − d2)(∆2 − 1). +(26) +Since rl = jl + n−(∆l−1)(σl +mod 2) +2 +, for l = 1, 2, Eq. (26) can be rewritten as: +j2 − j1 > +� +(qd1 + q − d2) + (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 +� +(∆2 − 1). +(27) +Combining Eq. (25) and Eq. (27) we obtain: +qc1 − c2 − +α2 +∆2 − 1 > (qd1 + q − d2) + (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 +. +Since cl + dl = +� σl +2 +� +− 1, we have dl = +σl+1(σl +mod 2) +2 +− cl − 1, for l = 1, 2; +replacing d1 and d2 in the previous equation, we obtain: +qc1 − c2 − +α2 +∆2 − 1 > qσ1 +2 ++ q(σ1 +mod 2) +2 +− qc1 − q + q − σ2 +2 − 1(σ2 +mod 2) +2 ++ c2+ ++ 1 + σ2 +mod 2 +2 +− q(σ1 +mod 2) +2 +which, considering that σ2 = +n−2 +∆2−1 = q(n−2) +∆1−1 = qσ1, implies: +qc1 − c2 > 1 +2 + +α2 +2(∆2 − 1) +(28) +In summary, to have gM +1 +< g2 Eq. (28) must hold. On the other hand, from +Eq. (25) and from the hypothesis that j2 −j1 > 0 we obtain (qc1 −c2)(∆2 −1)− +α2 > 0 which, since (∆2 − 1) > 0, implies qc1 − c2 > +α2 +∆2−1. Since 0 ≤ +α2 +∆2−1 < 1 +and qc1 − c2 is integer, we have qc1 − c2 ≥ 1. This implies that Eq. (28) holds +and therefore that gM +1 +< g2 and g1 ̸= g2. +Case 1.d: vg1 and vi2 are spine vertices. By Property 2 we have, for some +c2 ∈ N: +i2 = j2 + c2(∆2 − 1). +(29) +23 + +We also have, for some c1 ∈ N and 0 ≤ α1 < ∆2 − 1: +i1 = j1 + c1(∆1 − 1) + α1 = j1 + qc1(∆2 − 1) + α1. +(30) +If vi1 coincides with vi2, we have i1 = i2; from Eq. (30) and Eq. (29) we +obtain: +j2 − j1 = (qc1 − c2)(∆2 − 1) + α1. +(31) +Concerning vg2, we have: +gm +2 ≤ g2 ≤ gM +2 . +with gm +2 = r2 + d2(∆2 − 1), gM +2 += r2 + (d2 + 1)(∆2 − 1) for some d2 ∈ N such +that c2 + d2 = +� σ2 +2 +� +− 1, where σ2 is the number of spine vertices of C2. +Since vg1 is a vertex in the lower part of Γ1, it must be g1 = r1 + d1(∆1 − 1). +We prove that g1 < gm +2 , which implies g1 ̸= g2. To have g1 < gm +2 it must be: +r1 + d1(∆1 − 1) < r2 + d2(∆2 − 1) +r1 + qd1(∆2 − 1) < r2 + d2(∆2 − 1) +r2 − r1 > (qd1 − d2)(∆2 − 1). +(32) +Since rl = jl + n−(∆l−1)(σl +mod 2) +2 +, for l = 1, 2, Eq. (32) can be rewritten as: +j2 − j1 > +� +(qd1 − d2) + (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 +� +(∆2 − 1). +(33) +Combining Eq. (31) and Eq. (33) we obtain: +qc1 − c2 + +α1 +∆2 − 1 > (qd1 − d2) + (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 +. +Since cl + dl = +� σl +2 +� +− 1, we have dl = +σl+1(σl +mod 2) +2 +− cl − 1, for l = 1, 2; +replacing d1 and d2 in the previous equation, we obtain: +qc1 − c2 + +α1 +∆2 − 1 > qσ1 +2 ++ q(σ1 +mod 2) +2 +− qc1 − q − σ2 +2 − 1(σ2 +mod 2) +2 ++ c2+ ++ 1 + σ2 +mod 2 +2 +− q(σ1 +mod 2) +2 +which, considering that σ2 = +n−2 +∆2−1 = q(n−2) +∆1−1 = qσ1, implies: +qc1 − c2 > 1 − q +2 +− +α1 +2(∆2 − 1) +(34) +In summary, to have g1 < gm +2 +Eq. (34) must hold. On the other hand, from +Eq. (31) and from the hypothesis that j2 −j1 > 0 we obtain (qc1 −c2)(∆2 −1)+ +α1 > 0 which, since (∆2−1) > 0, implies qc1−c2 > − +α1 +∆2−1. Since 0 ≤ +α1 +∆2−1 < 1 +and qc1 − c2 is integer, we have qc1 − c2 > 0. Since q is a positive integer, this +implies that Eq. (34) holds and therefore that g1 < gm +2 and g1 ̸= g2. +24 + +Case 2.b: vg1 and vg2 are spine vertices. Since vg2 is a vertex in the lower +part of Γ2, it must be g2 = r2+d2(∆2−1). If vg2 coincides with vi1, as explained +above, it must be i1 = g2 − n. Combining the expression of g2 with Eq. (30) we +obtain: +r2 − j1 = (qc1 − d2)(∆2 − 1) + α1 + n. +(35) +Concerning vi2, we have: +im +2 ≤ i2 ≤ iM +2 ; +with iM +2 = j2 + (c2 + 1)(∆2 − 1) for some c2 ∈ N such that c2 + d2 = +� σ2 +2 +� +− 1, +where σ2 is the number of spine vertices of C2. +We prove that iM +2 < g1, which implies i2 ̸= g1. To have iM +2 < g1 it must be: +j2 + (c2 + 1)(∆2 − 1) < r1 + d1(∆1 − 1) +j2 + (c2 + 1)(∆2 − 1) < r1 + qd1(∆2 − 1) +j2 − r1 < (qd1 − c2 − 1)(∆2 − 1). +(36) +Since rl = jl + n−(∆l−1)(σl +mod 2) +2 +, for l = 1, 2, Eq. (35) can be rewritten as: +j2 − j1 = (qc1 − d2)(∆2 − 1)/α1 + n +2 + (∆2 − 1)(σ2 +mod 2) +2 +, +(37) +while Eq. (36) can be rewritten as: +j2 − j1 < +� +(qd1 − c2 − 1) − q(σ1 +mod 2) +2 +� +(∆2 − 1) + n +2 . +(38) +Combining Eq. (37) and Eq. (38) we obtain: +qc1 − d2 < (qd1 − c2 − 1) − (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 +− +α1 +∆2 − 1. +Since cl + dl = +� σl +2 +� +− 1, we have dl = +σl+1(σl +mod 2) +2 +− cl − 1, for l = 1, 2; +replacing d1 and d2 in the previous equation, we obtain: +qc1 − σ2 +2 − σ2 +mod 2 +2 ++ c2 + 1 + σ2 +mod 2 +2 ++ +α1 +∆2 − 1 < qσ1 +2 ++ q(σ1 +mod 2) +2 +− +− qc1 − q − c2 − q(σ1 +mod 2) +2 +− 1 +which, considering that σ2 = +n−2 +∆2−1 = q(n−2) +∆1−1 = qσ1, implies: +qc1 − qσ1 +2 ++ c2 + 1 < −q +2 − +α1 +2(∆2 − 1) +(39) +In summary, to have iM +2 +< gm +1 Eq. (39) must hold. On the other hand, from +Eq. (37) and from the hypothesis that j2 − j1 < +n−(∆1−1) +2 += +n−q(∆2−1) +2 +we +obtain: +qc1 − qσ1 +2 ++ c2 + 1 < −q +2 − +α1 +∆2 − 1. +(40) +25 + +We have that − q +2 − +α1 +∆2−1 < − q +2 − +α1 +2(∆2−1), since +α1 +2(∆2−1) < +1 +2. In other +words, Eq. (40) implies that Eq. (39) holds and therefore that iM +2 +< g1 and +i2 ̸= g1. +Case 2.c: vi1 and vg2 are spine vertices. Since vg2 is a vertex in the lower +part of Γ2, it must be g2 = r2+d2(∆2−1). If vg2 coincides with vi1, as explained +above, it must be i1 = g2 − n. Combining the expression of g2 with Eq. (23) we +obtain: +r2 − j1 = (qc1 − d2)(∆2 − 1) + n. +(41) +Concerning vg1, we have: +gm +1 ≤ g1 ≤ gM +1 ; +with gm +1 = r1 + d1(∆1 − 1), gM +1 += r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such +that c1 + d1 = +� σ1 +2 +� +− 1, where σ1 is the number of spine vertices of C1. +We prove that iM +2 < gm +1 , which implies i2 ̸= g1. To have iM +2 < gm +1 it must be: +j2 + (c2 + 1)(∆2 − 1) < r1 + d1(∆1 − 1) +j2 + (c2 + 1)(∆2 − 1) < r1 + qd1(∆2 − 1) +j2 − r1 < (qd1 − c2 − 1)(∆2 − 1). +(42) +Since rl = jl + n−(∆l−1)(σl +mod 2) +2 +, for l = 1, 2, Eq. (41) can be rewritten as: +j2 − j1 = (qc1 − d2)(∆2 − 1) + n +2 + (∆2 − 1)(σ2 +mod 2) +2 +, +(43) +while Eq. (42) can be rewritten as: +j2 − j1 < +� +(qd1 − c2 − 1) − q(σ1 +mod 2) +2 +� +(∆2 − 1) + n +2 . +(44) +Combining Eq. (43) and Eq. (44) we obtain: +qc1 − d2 < (qd1 − c2 − 1) − (σ2 +mod 2) +2 +− q(σ1 +mod 2) +2 +. +Since cl + dl = +� σl +2 +� +− 1, we have dl = +σl+1(σl +mod 2) +2 +− cl − 1, for l = 1, 2; +replacing d1 and d2 in the previous equation, we obtain: +qc1 − σ2 +2 − σ2 +mod 2 +2 ++ c2 + 1 < qσ1 +2 ++ q(σ1 +mod 2) +2 +− qc1 − q − c2 − 1− +− σ2 +mod 2 +2 +− q(σ1 +mod 2) +2 +which, considering that σ2 = +n−2 +∆2−1 = q(n−2) +∆1−1 = qσ1, implies: +qc1 − qσ1 +2 ++ c2 + 1 < −q +2 +(45) +26 + +In summary, to have iM +2 +< gm +1 Eq. (45) must hold. On the other hand, from +Eq. (43) and from the hypothesis that j2 − j1 < +n−(∆1−1) +2 += +n−q(∆2−1) +2 +we +obtain: +qc1 − d2 + 1 +2(σ2 +mod 2) < −q +2. +Replacing again d2 with σ2+1(σ2 +mod 2) +2 +− c2 − 1, we obtain: +qc1 − qσ1 +2 ++ c2 + 1 < −q +2. +(46) +Since Eq. (45) is equivalent to Eq. (46), we can conclude that Eq. (45) holds +and therefore that iM +2 < gm +1 and i2 ̸= g1. +Case 2.d: vg1 and vi2 are spine vertices. Since vg1 is a vertex in the lower +part of Γ1, it must be g1 = r1 + d1(∆1 − 1). If vg1 coincides with vi2, combining +the expression of g1 with Eq. (6) we obtain: +j2 − r1 = (qd1 − c2)(∆2 − 1). +(47) +Concerning vi1 vg2, we have: +im +1 ≤ i1 ≤ iM +1 ; +and +gm +2 ≤ g2 ≤ gM +2 . +with im +1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1), gM +2 += r2 + (d2 + 1)(∆2 − 1) − n +for some d2 ∈ N such that c2 + d2 = +� σ2 +2 +� +− 1, where σ2 is the number of spine +vertices of C2. +We prove that gM +2 +< im +1 , which implies g2 ̸= i1. To have gM +2 +< im +1 it must be: +r2 + (d2 + 1)(∆2 − 1) − n < j1 + c1(∆1 − 1) +r2 + (d2 + 1)(∆2 − 1) − n < j1 + qc1(∆2 − 1) +r2 − j1 < (qc1 − d2 − 1)(∆2 − 1) + n(∆2 − 1). +(48) +Since rl = jl + n−(∆l−1)(σl +mod 2) +2 +, for l = 1, 2, Eq. (47) can be rewritten as: +j2 − j1 = (qd1 − c2)(∆2 − 1) − q(∆2 − 1)(σ1 +mod 2) +2 ++ n +2 , +(49) +while Eq. (48) can be rewritten as: +j2 − j1 < (qc1 − d2 − 1)(∆2 − 1) + (∆2 − 1)(σ2 +mod 2) +2 ++ n +2 . +(50) +Combining Eq. (49) and Eq. (50) we obtain: +qd1 − c2 − q(σ1 +mod 2) +2 +< (qc1 − d2 − 1) + (σ2 +mod 2) +2 +. +27 + +Since cl + dl = +� σl +2 +� +− 1, we have dl = +σl+1(σl +mod 2) +2 +− cl − 1, for l = 1, 2; +replacing d1 and d2 in the previous equation, we obtain: +qσ1 +2 ++ q(σ1 +mod 2) +2 +− qc1 − q − c2 − q(σ1 +mod 2) +2 +< qc1 − σ2 +2 − σ2 +mod 2 +2 ++ ++ c2 + 1 − 1 + σ2 +mod 2 +2 +which, considering that σ2 = +n−2 +∆2−1 = q(n−2) +∆1−1 = qσ1, implies: +2 +� +qc1 − qσ1 +2 ++ c2 +� +> −q +(51) +In summary, to have gM +2 +< im +1 Eq. (51) must hold. On the other hand, from +Eq. (49) and from the hypothesis that j2 − j1 < +n−(∆1−1) +2 += +n−q(∆2−1) +2 +we +obtain: +qd1 − c2 − q +2(σ1 +mod 2) < −q +2. +Replacing again d1 with σ1+1(σ1 +mod 2) +2 +− c1 − 1, we obtain: +2 +� +qc1 − qσ1 +2 ++ c2 +� +> −q. +(52) +Since Eq. (51) is equivalent to Eq. (52), we can conclude that Eq. (51) holds +and therefore that gM +2 +< im +1 and g2 ̸= i1. +B +Proof of Theorem 7 +Theorem 7. A ∆-regular caterpillar with 4 ≤ ∆ ≤ 7 admits a 2-planar 3- +placement. +Proof. Let C1, C2, and C3 be three copies (shown in red, blue and green, respec- +tively, in Fig. 8) of a ∆-regular caterpillar C with 4 ≤ ∆ ≤ 7. We denote the +vertices of caterpillar Cj for j = 1, 2, 3 as follows; the spine vertices are denoted +as vj +0, vj +1, . . . , vj +c−1 in the order they appear along the spine; the leaves adjacent +to vertex vj +i (for i = 1, 2, . . . , c−1) are denoted as uj +i,l with l = 0, 1, . . . , d, where +d = ∆ − 2 if i = 0 or i = c − 1 and d = ∆ − 3 if 0 < i < c − 1. +Let p0, p1, . . . , pn−1 be n points on a circle in clockwise order (with indices +taken modulo n). To construct the packing, we compute a drawing for each +caterpillar such that the vertices are mapped to points p1, p2, . . . , pn and the +union of the three drawings is a 2-planar drawing. We describe the construction +for ∆ = 4, 5, 6 (see also Figs. 8(a) to 8(c)); the construction in the case ∆ = 7 +is slightly different and it is shown in Fig. 8(d). +Caterpillar C1 is drawn outside the circle so that vertex v1 +0 is mapped to +point p0, each vertex v1 +i , for i = 1, 2, . . . , c − 1 is mapped to pi(∆−1)+1, each +leaf u1 +0,l is mapped to the point pl+1, and each leaf u1 +i,l is mapped to the point +pi(∆−1)+2+l. In other words, each vertex of the spine is followed clockwise by +28 + +∆ = 4 +(a) +∆ = 5 +(b) +∆ = 6 +(c) +∆ = 7 +(d) +Figure 8: +2-planar 3-placements of ∆-regular caterpillars. +its leaves and the last of these leaves is followed by the next vertex of the spine. +Caterpillar C2 is drawn inside the circle so that vertex v2 +i is mapped to the +point immediately following clockwise the point hosting v1 +i and each leaf u2 +i,l is +mapped to the point immediately following clockwise u1 +i,l. Clearly, the drawings +of the first two caterpillars do not cross each other because they are on different +sides of the circle; also, their union has no multiple edges. Concerning C3, the +vertex v3 +i , for i = 0, 1, . . . , c−2 is mapped to the point that hosts u1 +i,d and u2 +i,d−1, +i.e., the last leaf of v1 +i and the second last leaf of v2 +i ; the vertex v3 +c−1 is mapped +to the point that hosts u1 +i,d−1 and u2 +i,d−2, i.e., the second last leaf of v1 +c−1 and +the third last leaf of v2 +i . About this mapping, observe that if we draw the edges +of the spine of C3 outside the circle, each edge of the spine of C3 intersects two +consecutive edges of the spine of C1 and each edge of the spine of C1 intersects +at most two consecutive edges of the spine of C3. To complete the drawing, we +need to draw the leaves of C3. Consider two consecutive spine vertices v3 +i and +v3 +i+1, with 0 ≤ i ≤ c − 2; between these two vertices there are ∆ − 2 points not +yet used by C3, we connect the first two of these vertices in clockwise order to +vi. Depending on the value of ∆, there remain 0, 1, or 2 points between vi and +vi+1 not yet used by C3; we connect these points to vi+1. Notice that, there +remain to map ∆ − 3 leaves adjacent to v3 +0 and 3 leaves adjacent to v3 +c−1. On +the other hand, there are ∆ points not yet used by C3 that are between v3 +c−1 +and v3 +0 clockwise; we connect the three vertices following clockwise v3 +c−1 to v3 +c−1, +and the remaining ones to v3 +0. All the edges of C3 that are incident to leaves +are drawn inside the circle. This mapping of C3 does not create multiple edges +and gives rise to at most two crossings along the edges of C2 and C3. +29 + diff --git a/3tAzT4oBgHgl3EQfR_u4/content/tmp_files/load_file.txt b/3tAzT4oBgHgl3EQfR_u4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a647b038e3a499d34ccc9c78310bc37ec52e2b6d --- /dev/null +++ b/3tAzT4oBgHgl3EQfR_u4/content/tmp_files/load_file.txt @@ -0,0 +1,1004 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf,len=1003 +page_content='k-planar Placement and Packing of ∆-regular Caterpillars Carla Binucci1, Emilio Di Giacomo1, Michael Kaufmann2, Giuseppe Liotta1, and Alessandra Tappini1 1Dipartimento di Ingegneria, Universit`a degli Studi di Perugia, via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Duranti 93, 06125, Perugia, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' {carla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='binucci, emilio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='digiacomo, giuseppe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='liotta, alessandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='tappini}@unipg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='it 2Wilhelm-Schickard Institut f¨ur Informatik, Universit¨at T¨ubingen, Sand 13, 72076, T¨ubingen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' mk@informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='uni-tuebingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='de January 4, 2023 Abstract This paper studies a packing problem in the so-called beyond-planar setting, that is when the host graph is “almost-planar” in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Pre- cisely, we consider the case that the host graph is k-planar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', it admits an embedding with at most k crossings per edge, and focus on families of ∆-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We study the dependency of k from the number h of caterpillars that are packed, both in the case that these caterpillars are all isomorphic to one another (in which case the packing is called placement) and when they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We give necessary and sufficient conditions for the placement of h ∆-regular caterpillars and sufficient conditions for the packing of a set of ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h-regular caterpillars such that the degree ∆i and the degree ∆j of the non-leaf vertices can differ from one caterpillar to another, for 1 ≤ i, j ≤ h, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1 Introduction Graph packing is a classical problem in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The original formu- lation requires to merge several smaller graphs into a larger graph, called the host graph, without creating multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' More precisely, graphs G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh with Gi = (Vi, Ei) should be combined to a new graph G = (V, E) by injec- tive mappings ηi : Vi → V so that V = V1 ∪ V2 ∪ · · · ∪ Vh and the images of 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='01226v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='CO] 3 Jan 2023 the edge sets Ei do not intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It has been often assumed that |Vi| = n for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' h, and thus the mappings ηi are bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Many combinatorial problems can be regarded as packing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For example, the Hamiltonian cycle problem for a graph G can be stated as the problem of packing an n-vertex cycle with the complement of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' When no restriction is imposed on the host graph, we say that the host graph is Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Some classical results in this setting are those by Bollob´as and Eldridge [6], Teo and Yap [26], Sauer and Spencer [25], while related famous conjectures are by Erd˝os and S´os from 1963 [10] and by Gy´arf´as from 1978 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Within this line of research, Wang and Sauer [27], and Mah´eo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [22] char- acterized triples of trees that admit a packing into Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Haler and Wang [17] extended this result to four copies of a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Further notable work on graph packing into Kn is by Hedetniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [18], Wozniak and Wojda [28] and Aich- holzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A packing problem with identical copies of a graph is also called a placement problem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', [17, 27, 29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A tighter relation to graph drawing was established when researchers did not consider Kn to be the host graph, but required that the host graph is planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The main question here is how to pack two trees of size n into a planar graph of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' After a long series of intermediate steps [11, 12, 13, 14, 24] where the class of trees that could be packed has been gradually generalized, Geyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [15] showed that any two non-star trees can be embedded into a planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Relaxing the planarity condition allows for packing of more (than two) trees, and restricting the number of crossings for each edge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', in the so-called beyond-planar setting [9, 19, 21], still keeps the host graph sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The study of the packing problem in the beyond planarity setting was started by De Luca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [8], who consider how to pack caterpillars, paths, and cycles into 1-planar graphs (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', [21] for a survey and references on 1-planarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' While two trees can always be packed into a planar graph, it may not be possible to pack three trees into a 1-planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In this work we further generalize the problem by allowing the host graph to be k-planar for any k ≥ 1, and we study the dependency of k on the number of caterpillars to be packed and on their vertex degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We consider ∆-regular caterpillars, which are caterpillars whose non-leaf vertices all have the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Our results can be briefly outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We consider the packing problem of h copies of the same ∆-regular cater- pillar into a k-planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We characterize those families of h ∆-regular caterpillars which admit a placement into a k-planar graph and show that k ∈ O(∆h + h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We extend the study from the placement problem to the packing problem by considering a set of ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h-regular caterpillars such that the degree ∆i and the degree ∆j of the non-leaf vertices can differ from one caterpillar to another, with 1 ≤ i, j ≤ h, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By extending the tech- niques of the bullet above, we give sufficient conditions for the existence of a k-planar packing of these caterpillars and show that k ∈ O(∆h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2 Finally, we prove a general lower bound on k and show that this lower bound can be increased for small values of h and for caterpillars that are not ∆-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Preliminaries are in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The placement of h ∆-regular caterpillars into a k-planar graph is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Section 4 is devoted to k-planar h-packing, while Section 5 gives lower bounds on the value of k as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Concluding remarks and open problems can be found in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2 Preliminaries We assume familiarity with basic graph drawing and graph theory terminology (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', [5, 20, 23]) and recall here only those concepts and notation that will be used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Given a graph G, we denote by degG(v) the degree of a vertex v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh be h graphs, all having n vertices, an h-packing of G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh is an n-vertex graph G that contains G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh as edge-disjoint spanning subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We also say that G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh can be packed into G and that G is the host graph of G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' An h-packing of h graphs into a host graph G such that the h graphs are all isomorphic to a graph H, is called an h-placement of H into G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We also say that G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh can be placed into G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The following property establishes a necessary condition for the existence of an h-packing into any host graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A packing of h connected n-vertex graphs exists only if n ≥ 2h and degGi(v) ≤ n − h, for each i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h} and for each vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Each Gi has at least n−1 edges (because it is connected);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' thus, if n < 2h the h graphs have more edges in total than the number of edges of any graph with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' But since graphs Gi must be edge-disjoint subgraphs of G, the number of edges of G must be at least the total number of edges of the graphs Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since degGi(v) ≥ 1 for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h} and for each v (because each Gi is connected) and since �h i=1 degGi(v) ≤ n − 1 (because G cannot have vertex-degree larger that n − 1), it holds that degGi(v) ≤ n − h, for each i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h} and for each vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A k-planar graph is a graph that admits a drawing in the plane such that each edge is crossed at most k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If the host graph of an h-packing (h-placement) is k-planar, we will talk about a k-planar h-packing (k-planar h-placement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Sometimes, we shall simply say k-planar packing or k-planar placement, when the value of h is clear from the context or not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A caterpillar is a tree such that removing all leaves we are left with a path, called spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A caterpillar T is ∆-regular, for ∆ ≥ 2, if degT (v) = ∆ for every vertex v of the spine of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of vertices of a ∆-regular caterpillar is n = σ(∆ − 1) + 2 for some positive integer σ, which is the number of vertices of the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 3 3 h-placement of ∆-regular Caterpillars into k- planar Graphs Given h copies of a same ∆-regular caterpillar, we want to study under which conditions they admit a placement into a k-planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We start by showing that the necessary condition stated in Property 1 is, in general, not sufficient to guarantee a placement even for ∆-regular caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For every h ≥ 2, let ∆ be a positive integer such that h−1 ∆−1 is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A set of h ∆-regular caterpillars with n = 2h vertices does not admit a placement into any graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since each caterpillar has n − 1 edges and the number of caterpillars is h = n 2 , the total number of edges is n(n−1) 2 and thus, if a placement exists, the host graph can only be Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove that this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Denote by C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch the h caterpillars and suppose that a packing into Kn exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let v be a vertex of Kn and let v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vh be the h vertices that are mapped to v, with vi being a vertex of Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Each vertex vi has degree in Ci that is either ∆ or 1 (because each Ci is ∆-regular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Denote by c the number of vertices among v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vh that have degree ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the degree of v in the packing is c∆ + (h − c) and since the degree of v in Kn is n − 1, it must be c∆ + (h − c) = n − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', c∆ + (h − c) = 2h − 1, which can be rewritten as c = h−1 ∆−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' But this is not possible because c is integer, while h−1 ∆−1 is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In the rest of this section we shall establish necessary and sufficient condi- tions that characterize when a set of h isomorphic ∆-regular caterpillars admit a k-planar h-placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Concerning the sufficiency, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1 we describe a constructive argument that computes a set of so-called zig-zag drawings and study the properties of such drawings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='2, we complete the charac- terization by also giving necessary conditions for an h-placement of ∆-regular caterpillars into a k-planar graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' in the same section, we give an upper bound on k as a function of h and ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We recall that a ∆-regular caterpillar has a number of vertices n that is equal to σ(∆ − 1) + 2 for some natural number σ, which is the number of vertices of the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' While ∆-regular caterpillars are defined for any value of σ ≥ 1, when we want to pack a set of h ≥ 2 caterpillars, Property 1 requires that each caterpillar has at least two spine vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', that σ ≥ 2 for each caterpillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Otherwise, the unique spine vertex would have degree n − 1 and Property 1 would not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1 Zig-zag Drawings of ∆-regular caterpillars Let C be a ∆-regular caterpillar with n vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we construct a drawing Γ of C as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of vertices of the spine of C is σ = n−2 ∆−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' consider a set of σ points on a circle γ and denote by u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , uσ these points according to the circular clockwise order they appear along γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Draw the spine 4 u1 u2 u3 u4 u5 (a) v17 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 upper part lower part hole short edge (b) v17 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 (c) Figure 1: (a) A zig-zag drawing of a 4-regular caterpillar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (b) the upper and the lower part are highlighted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' c) a 2-packing obtained by the drawing of (b) with a copy of it rotated by one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' of C by connecting, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ⌊ σ 2 ⌋, the points ui and ui+1 to the point uσ−i+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If σ is even and i = σ 2 , the points ui+1 and uσ−i+1 coincide and therefore the point u σ 2 is connected only to u σ 2 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that all points ui have two incident edges, except u1 and u⌊ σ 2 ⌋+1 which have only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We add the leaves adjacent to each vertex ui ̸∈ {u1, u⌊ σ 2 ⌋+1} by connecting uσ−i+1 to ∆−2 points between ui and ui+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we then add the leaves adjacent to u1 by connecting it to ∆−1 points between uσ and u1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we finally add the leaves adjacent to u⌊ σ 2 ⌋+1 by connecting it to ∆ − 1 points between u σ 2 and u σ 2 +1 if σ is even, or to ∆−1 points between u⌊ σ 2 ⌋+1 and u⌊ σ 2 ⌋+2 if σ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The resulting drawing is called a zig-zag drawing of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' From now on, we assume that in a zig-zag drawing the points that represent vertices are equally spaced on the circle γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let χ be the convex hull of the points representing the vertices of C in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A zig-zag drawing has exactly two sides of χ that coincide with two edges of C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we call these two edges short edges of Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' each other side of χ is called a hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Denote by v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vn the vertices of Γ according to the circular clockwise order they appear along χ with v1 ≡ u1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that (v1, vn) is a short edge and vn is the degree-1 vertex of this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider a straight line s that intersects both short edges of Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' line s inter- sects all the edges of the zig-zag drawing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Without loss of generality, assume that s is horizontal and denote by U the set of vertices that are above s and by L the set of vertices that are below s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The vertices in U form the upper part of Γ and those in L form the lower part of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Without loss of generality assume that v1 is in the upper part (and therefore vn is in the lower part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It follows that each edge has the end-vertex with lower index in the upper part, and the end-vertex with higher index in the lower part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence the short edge different from (v1, vn), which we denote as (vr−1, vr), is such that vr−1 is in the upper part and vr is in the lower part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The first vertex of the upper part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', vertex v1, is called starting point of Γ, while the first vertex of the lower part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', 5 vertex vr, is called ending point of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We observe that r = n 2 + 1 if the number of vertices of the spine σ = n−2 ∆−1 is even, while r = 1 + n−(∆−1) 2 if σ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This can be written with a single formula as r = 1+ n−(∆−1)(σ mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The two short edges separate two sets of consecutive holes, one completely contained in the upper part and one completely contained in the lower part;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' if σ is even, these two sets have the same number of holes equal to n−2 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' if σ is odd, then one of the two sets has n−∆−1 2 holes, while the other has n+∆−3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Note that the smaller set is in the upper part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let ℓ be a positive integer and let Γ′ be the drawing obtained by re-mapping vertex vi to the point1 representing vi+ℓ in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We say that Γ′ is the drawing obtained by rotating Γ by ℓ steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Note that the starting point of Γ′ is vj with j = 1 + ℓ and the ending point is vr with r = 1 + ℓ + n−(∆−1)(σ mod 2) 2 = j + n−(∆−1)(σ mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The drawing in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1(c) is the union of two zig-zag drawings Γ1 and Γ2, where Γ2 is obtained by rotating Γ1 by one step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the starting point of Γ1 is v1 while its ending point is v8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the starting point of Γ2 is v2, while its ending point is v9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ1 be a zig-zag drawing of a ∆-regular caterpillar C with starting point j1 and ending point r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' let Γ2 be a zig-zag drawing of C with starting point j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If 0 < j2 − j1 < n−(∆−1)(σ mod 2) 2 , where σ is the number of spine vertices of C, then Γ1 ∪ Γ2 has no multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We first observe that Γ2 is obtained by rotating Γ1 by ℓ steps, where ℓ = j2 − j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Suppose that a multiple edge (vi, vg), with i < g exists in Γ1 ∪ Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This implies that in the drawing Γ1 there must be an edge (vi′, vg′) that, when rotated by ℓ steps, coincides with (vi, vg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In other words, the two edges (vi, vg) and (vi′, vg′) must be such that: (i) i′ < i < r1 ≤ g < g′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (ii) g = i′ + ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (iii) the number α of vertices encountered between vi and vg when going clockwise from vi to vg is the same as the number of vertices encountered when going clockwise from vg′ to vi′ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Denote by β the number of vertices encountered when going clockwise from vi′ to vi, and by ζ the number of vertices encountered when going clockwise from vg to vg′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We have 2α + β + ζ + 4 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If σ is even, then β = ζ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2(a)), which implies α + β + 2 = n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that g = i′ + ℓ implies that ℓ = β + α + 2 (ℓ is equal to the number of vertices encountered clockwise between vi′ and vg plus one) and therefore (vi′, vg′) can coincide with (vi, vg) after a rotation of ℓ steps only if ℓ = n 2 but, when σ is even, we have ℓ = j2 − j1 < n 2 and therefore a multiple edge cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If σ is odd, then β − (∆ − 1) ≤ ζ ≤ β + (∆ − 1) (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2(b) and 2(c)) and therefore 2α + 2β − (∆ − 1) + 4 ≤ 2α + β + ζ + 4 = n ≤ 2α + 2β + (∆ − 1) + 4, which can be rewritten as n−(∆−1) 2 ≤ α + β + 2 ≤ n+(∆−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It follows that, in order to have (vi′, vg′) and (vi, vg) coincident after a rotation of ℓ steps, the value of ℓ must be such that n−(∆−1) 2 ≤ ℓ ≤ n+(∆−1) 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' but, when σ is odd, we have ℓ = j2 − j1 < n−(∆−1) 2 and therefore a multiple edge cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1In a drawing in convex position the indices of the vertices are taken modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 6 vj1 vr1 vi′ vh′ vi vh β ζ α α ℓ (a) vj1≡vi′ vr1 vh′ vi vh β ζ α α ℓ (b) vj1 vr1 vi′ vh′ vi vh β ζ α α ℓ (c) Figure 2: Illustration for the proof of Lemma 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (a) σ even;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (b)-(c) σ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We conclude this section by computing the maximum number of crossings per edge in the union of two zig-zag drawings without overlapping edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We state this lemma in general terms assuming that the two ∆-regular caterpillars can have different vertex degrees, as we are going to use the lemma to establish upper bounds on k both for k-planar h-placements and for k-planar h-packings (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ be a union of a set of zig-zag drawings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To ease the description that follows, we regard Γ as a sub-drawing of a straight-line drawing of Kn whose vertices coincide with those of Γ (and therefore are equally spaced along a circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, for each vertex vj, we denote by ej,0, ej,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ej,n−2 the edges incident to vj in Kn according to the circular counterclockwise order around vj starting from ej,0 = (vj, vj−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Each of the zig-zag drawings that form Γ contains a subset of these edges and Γ is a valid packing if there is no edge that belongs to two different zig-zag drawings in the set whose union is Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We denote by Sn the (circular) sequence of slopes si = i · π n, for i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, without loss of generality, we can assume that the convex hull of Γ has a side with slope s0 and, as a consequence, every edge of Γ has a slope in the set Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let vj be a vertex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' if the slope of ej,0 is sij, then the slope of ej,p is sij+p (with indices taken modulo n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' in other words, the edges incident to each vertex have slopes that form a sub-sequence of n − 1 consecutive elements of Sn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we denote such a sequence as ψ(ij), where ij indicates that the first element of ψ(ij) is sij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We say that vj uses the sequence ψ(ij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If we consider two different vertices vj and vj+p and vj uses the sequence ψ(ij), then vj+p uses the sequence ψ(ij − 2p) (with indices taken modulo n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' in other words, the sequence used by a vertex shifts clockwise by two elements moving to the next vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n- vertex ∆2-regular caterpillar with ∆i ≤ n − 2 (for i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ1 be a zig-zag drawing of C1 with starting point vj1 and let Γ2 be a zig-zag drawing of C2 with starting point vj2 with 0 < j2 − j1 < n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If Γ1 ∪ Γ2 has no multiple edges, then any edge of Γ1 ∪ Γ2 is crossed at most 2(∆1 + ∆2) + 4(j2 − j1) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 7 s0 s1 sn−1 sn−2 vj ej,0 ej,n−2 ej+1,0 vj+1 ψ(j) ψ(j + 1) Sn ej+1,n−2 ≡ Figure 3: Illustration for the definition of slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We first observe that the edges of a zig-zag drawing of a ∆-regular cater- pillar are all drawn as segments whose slope belongs to a set of ∆ slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, for every spine vertex v, the edges incident to v are drawn using all these ∆ slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider the starting vertex vj1 of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the edges incident to vj1 are drawn with the first ∆1 slopes of ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Analogously, the edges incident to the starting vertex vj2 of Γ2 are drawn with the first ∆2 slopes of ψ(ij2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The sequence ψ(ij2) is shifted clockwise by 2(j2−j1) units with respect to ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, since j2−j1 < n 2 , the first slope of ψ(ij2) is distinct from the first slope of ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let e = (vi, vg) be an edge of Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove that the number of crossings along e is at most the one given in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let e1 = (vi, va) be the edge of Γ2 incident to vi that forms the smallest angle with e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' analogously, let e2 = (vg, vb) be the edge of Γ2 incident to vg that forms the smallest angle with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, in principle there are four possible clockwise orders of vi, va, vg, and vb (see cases (a)–(d) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 4 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' However the case (b) cannot happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Namely, in case (b) the slopes used to draw the edges of Γ2 would be shifted counterclockwise with respect to those used to represent the edges of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' but, as observed above, the slopes used by Γ2 are shifted clockwise with respect to those used by Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let α1 be the angle between e and e1 and let α2 be the angle between e and e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let V1 be the set of vertices seen by the angle α1 including va and excluding vg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' analogously let V2 be the set of vertices seen by the angle α2 including vb and excluding vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In each of the three cases (a), (c), and (d), at least one of α1 and α2 is such that e sweeps the angle moving clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let αl with l ∈ {1, 2} be the angle that satisfies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, for case (a) αl can be both α1 or α2, in case (c) αl is α2 and in case (d) αl is α1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Every edge that crosses e has an end-vertex in V1 and one end-vertex in V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To count the number of such edges (and therefore the number of crossings along e), we evaluate |Vl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The value of |Vl| is at most the number of slopes of Sn that are encountered in counterclockwise order between the slope s ∈ Sn of el and the slope s′ ∈ Sn of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, in case (a) |Vl| is exactly this number, while in case (c) and (d) |Vl| is less than this number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The slope s′ is at most the last 8 vi vg va vb α1 α2 V1 V2 (a) vi vg va vb α1 α2 (b) vi vg va vb α1 α2 V2 (c) vi vg va vb α1 α2 V1 (d) Figure 4: Illustration for the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The edges of Γ2 are dashed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 9 slope used by Γ1, which is sp with p = j1 + ∆1, while the slope s is at least the first slope used by Γ2, which is sq with q = j1 − 2(j2 − j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, the number of slopes between s′ (included) and s (excluded) is at most p−q = ∆1 +2(j2 −j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence |Vl| ≤ ∆1 + 2(j2 − j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We call a block a subset of consecutive vertices of Vl starting with a spine vertex and containing all the leaves that follow that spine vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of edges of Γ2 incident to the vertices of a block is 2(∆2 − 1) (since ∆2 edges are incident to the spine vertex and ∆2 − 2 is the number of leaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of blocks in Vl is � |Vl| (∆2−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It follows that the number of crossings χe along e is at most � |Vl| (∆2−1) � 2(∆2 − 1) which is less than 2(|Vl| + ∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since |Vl| ≤ ∆1 + 2(j2 − j1), we have χe ≤ 2(∆1 + ∆2) + 4(j2 − j1), which concludes the proof in the case when e belongs to Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The case when the edge e belongs to Γ2 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, when e belongs to Γ2, the cases (b), (c), and (d) apply, while case (a) does not happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='2 Characterization We are now ready to characterize the ∆-regular caterpillars that admit an h- placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C be a ∆-regular caterpillar with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' An h-placement of C exists if and only if: (i) ∆ ≤ n − h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' and (ii) n ≥ 2h + (∆ − 1) · (σ mod 2), where σ is the number of spine vertices of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Further, if an h-placement exists, there exists one that is k-planar for k ∈ O(∆h + h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We first prove the sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch be the h cater- pillars and assume that n ≥ 2h+(∆−1)(σ mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We compute an h-placement of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch starting from a zig-zag drawing Γ1 of C1 and obtaining the drawing Γi of Ci by rotating Γ1 by i − 1 steps, for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, when the number of spine vertices σ of each Ci is even, h ≤ n 2 and therefore each Γi is rotated by less than n 2 steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' when σ is odd h ≤ n−(∆−1) 2 and each Γi is rotated by less than n−(∆−1) 2 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In both cases, each pair of drawings Γi and Γj satisfies the conditions of Lemma 1 and therefore there are no multiple edges, that is, the union of all Γi is a valid h-placement of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove the necessary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If σ is even, then conditions (i) and (ii) are necessary by Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, consider the case when σ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Condition (i) is necessary by Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Assume, by contradiction, that (ii) is not necessary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', there exists an h-placement of h caterpillars C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch such that n < 2h + (∆ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch admit an h-placement, by Property 1 n must be at least 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, it would be 2h ≤ n < 2h + (∆ − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' in other words, n = 2h + α with 0 ≤ α ≤ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let G be the host graph of the h-placement and let v be the vertex of G to which the largest number of spine vertices of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch is mapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let β be the number of spine vertices that are mapped to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' There are other h − β 10 leaf vertices that are mapped to v (because one vertex per caterpillar has to be mapped on each vertex of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The degree of v in G is at most n − 1 and each of the spine vertices mapped to v has degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, the β spine vertices mapped to v have degree β∆ in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Vertex v can have at most other n−1−β∆ edges and therefore it must be n − 1 − β∆ ≥ h − β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', β ≤ n−1−h ∆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, there are σh spine vertices in total and, since G has n vertices, there are at least ⌈ σh n ⌉ spine vertices mapped to v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', β ≥ ⌈ σh n ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Putting together the two conditions on β we obtain: �σh n � ≤ β ≤ n − 1 − h ∆ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since n = 2h + α, we have h = n−α 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing h in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='2, we obtain: �σ 2 − σα 2n � ≤ β ≤ n + α − 2 2(∆ − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since n = σ(∆ − 1) + 2, we have: �σ 2 − σα 2(σ(∆ − 1) + 2) � ≤ β ≤ σ(∆ − 1) + α 2(∆ − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (1) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (1) implies that: �σ 2 − α 2(∆ − 1) + 4 σ � ≤ σ 2 + α 2(∆ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (2) We now prove that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (2) cannot be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since σ is odd, it is σ = 2i+1 for some i ∈ N, and thus: � i + 1 2 − ζ � ≤ k + 1 2 + ζ′, (3) with ζ = α 2(∆−1)+ 4 σ and ζ′ = α 2(∆−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We have ζ < ζ′ and we prove that ζ′ < 1 2: ζ′ = α 2(∆ − 1) ≤ ∆ − 2 2(∆ − 1) < ∆ − 1 2(∆ − 1) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (3) is i + 1 because 0 < 1 2 − ζ < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the second term is less than i + 1 because 0 < 1 2 + ζ′ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It follows that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (3) does not hold and therefore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (2) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove the bound on the number of crossings along an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We consider an edge of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the number of crossings along an edge of the drawing of another caterpillar is bounded by the same number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let e be an edge of the drawing Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Lemma 2, the number of crossings χe along e due to the edges of another drawing Γl (with 2 ≤ l ≤ h) is at most 2(∆1+∆l)+4(jl−j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Summing 11 u1 u2 u3 u4 u5 (a) v17 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 (b) Figure 5: (a) An inner zig-zag drawing and (d) an outer zig-zag drawing of a 4-regular caterpillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' over all drawings distinct from Γ1, we obtain χe ≤ �h l=2(2(∆1+∆l)+4(jl−j1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Considering that ∆l = ∆ for every l and that jl − j1 = l − 1, we have χe ≤ h � l=2 (4∆ + 4(l − 1)) ≤ (4∆ − 2)h + 2h2 − 4∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (4) We conclude by observing that the number of crossings given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (4) can be reduced, although not asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A zig-zag drawing can be embedded inside the circle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 5(a)) or outside the circle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, the number given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (4) can be halved by embedding half of the caterpillars inside the circle and the other half outside the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 4 h-packing of ∆-regular Caterpillars in k-planar Graphs In this section we study h-packings of h ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h-regular caterpillars such that the degree ∆i and the degree ∆j of the spine vertices can differ from one caterpillar to another, for 1 ≤ i, j ≤ h, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n- vertex ∆2-regular caterpillar such that ∆1 > ∆2 and ∆1 ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ1 be a zig-zag drawing of C1 with starting point vj1 and ending point vr1, and let Γ2 be a zig-zag drawing of C2 with starting point vj2 and ending point vr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If ∆2 2 ≤ j2 − j1 < n−(∆1−1) 2 , then Γ1 ∪ Γ2 has no multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' As described in the proof of Lemma 2, the edges of a zig-zag drawing of a ∆-regular caterpillar are all drawn as segments whose slope belongs to a set of ∆ slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, for every spine vertex v, the edges incident to v are drawn using all these ∆ slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Based on this observation, we show that the ∆1 slopes used to represent the edges of Γ1 are distinct from the ∆2 slopes used to represent the edges of Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We use the same notation used in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider the staring vertex vj1 of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the edges incident to vj1 are drawn with the first ∆1 slopes of ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Analogously, the edges incident to the starting vertex vj2 of Γ2 are drawn with the first ∆2 slopes of ψ(ij2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since j2 −j1 ≥ ∆2 2 , the sequence ψ(ij2) is shifted clockwise by ∆2 units with respect to ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, since j2 − j1 ≤ n−(∆1−1) 2 , the sequence of the first ∆2 slopes of ψ(ij2) does not overlap with the first ∆1 slopes of ψ(ij1), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch be h caterpillars such that Ci is ∆i-regular, for 1 ≤ i ≤ h, and ∆h ≤ ∆h−1 ≤ · · · ≤ ∆1 ≤ n − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If �h i=1 ∆i ≤ n − 1 and �h i=2 � ∆i 2 � < n−(∆1−1) 2 , then there exists a k-planar packing with k ∈ O(∆1h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We compute a zig-zag drawing of C1 with starting point vj1, with j1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' for each Ci, with 2 ≤ i ≤ h, we compute a zig-zag drawing Γi with starting vertex vji where ji = ji−1 + � ∆i 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, each vertex v of Γ1 ∪Γ2 ∪· · ·∪Γh has degree at most n−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' namely �h i=1 degCi(v) ≤ �h i=1 ∆i ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Moreover, given two caterpillars Ci and Ci′ with 1 ≤ i < i′ ≤ h, we have that: (i) ji′ − ji ≥ ji′ − ji′−1 = � ∆i′ 2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' and (ii) ji′ − ji ≤ jh − j1 = �h i=2⌈ ∆i 2 ⌉, which gives ji′ − ji < n−(∆1−1) 2 < n−(∆i−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Putting together (i) and (ii), we obtain ∆i′ 2 < ji′ − ji < n−(∆i−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, Lemma 3 holds for every pair of caterpillars and the union of all the zig-zag drawings Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Γh is a valid packing of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove the bound on the number of crossings along an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We consider an edge of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the number of crossings along an edge of another drawing is bounded by the same number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let e be an edge of the drawing Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Lemma 2, the number of crossings χe along e due to the edges of another drawing Γl (with 2 ≤ l ≤ h) is at most 2(∆1 + ∆l) + 4(jl − j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Summing over all drawings distinct form Γ1, we obtain χe ≤ �h l=2(2(∆1 + ∆l) + 4(jl − j1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Considering that jl ≥ jl−1 + � ∆i 2 � , we obtain that jl − j1 = �l i=2 � ∆i 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since ∆l ≤ ∆1 for every 2 ≤ l ≤ h, we have jl − j1 ≤ (l − 1)( ∆1 2 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Therefore, we obtain χe ≤ �h l=2(4∆1 + 4(l − 1)( ∆1 2 + 1)) ≤ (∆1 + 2)h2 + 4∆1(h − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now consider a special case of packing a set of h ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h- regular caterpillars where, for each ∆i (1 ≤ i ≤ h), we have that ∆i − 1 is a multiple of ∆i+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In this case, we show that the sufficient conditions of Theorem 3 can be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For example, consider the packing of a 17-regular caterpillar and two 9-regular caterpillars, each having 34 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' These three caterpillars do not satisfy the sufficient condition of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' However, a k- planar packing of these caterpillars is possible, as proven in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We start 13 vj vi vr vg upper part lower part c = 0 c = 1 c = 2 d = 0 d = 1 d = 2 Figure 6: Illustration for Property 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' σ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For each spine vertex, c and d are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Considering adjacent spine vertices, the sum of c and d is 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with the following property, which immediately follows from the construction of a zig-zag drawing (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 6 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ be a zig-zag drawing of a ∆-regular caterpillar with starting vertex vj and ending vertex vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vi is a spine vertex in the upper part of Γ, then i = j + c(∆ − 1) for some c ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' if vg is a spine vertex in the lower part of Γ, then g = r + d(∆ − 1) for some d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Moreover, if vi and vg are adjacent then either c + d = � σ 2 � − 1 or c + d = � σ 2 � , where σ is the number of spine vertices of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Property 2 is extensively used in the proof of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n- vertex ∆2-regular caterpillar such that ∆1−1 = q(∆2−1), for some q ∈ N+ and ∆i ≤ n − 2 (for i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ1 be a zig-zag drawing of C1 with starting point vj1 and ending point vr1, and let Γ2 be a zig-zag drawing of C2 with starting point vj2 and ending point vr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If 0 < j2 − j1 < n−(∆1−1) 2 , then Γ1 ∪ Γ2 has no multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let (vi1, vg1) be an edge of Γ1 and (vi2, vg2) be an edge of Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Assume that vi1 belongs to the upper part of Γ1 and vi2 belongs to the upper part of Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Note that this implies that vg1 belongs to the lower part of Γ1 and vg2 belongs to the lower part of Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that (vi1, vg1) and (vi2, vg2) do not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We first show that it does not happen that vi1 coincides with vi2 and vg1 coincides with vg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We then show that it does not happen that vi1 coincides with vg2 and vg1 coincides with vi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In the rest of the proof we will express the four indices i1, i2, g1 and g2 in terms of the values j1, j2, r1 and r2, according to Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Without loss of generality, we can assume that r2 ≤ n and j1 ≥ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', that the vertices vr2, vn, v1, and vj1 appear in this clockwise order, with vr2 and vn possibly coincident and with v1 and vj1 possibly coincident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' With these assumptions, we have j1 < j2 < r1 < r2 and vi1 can coincide with vg2 14 only if i1 = g2 − n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', only if the value of g2 is greater than n and coincides with i1 modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, while assuming that vi1 coincides with vi2 implies that i1 = i2, assuming that vi1 coincides with vg2 implies that i1 = g2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 1: It does not happen that vi1 coincides with vi2 and vg1 coincides with vg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' At least one vertex per edge is a spine vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We distinguish four sub-cases depending on which vertex is a spine vertex for each edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since all the cases are very similar, we give here only the first case and the others can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='a: vi1 and vi2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Property 2 we have, for some c1, c2 ∈ N: i1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (5) and i2 = j2 + c2(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (6) If vi1 coincides with vi2, we have i1 = i2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (6) we obtain: j2 − j1 = (qc1 − c2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (7) Concerning vg1 and vg2, we have: gm 1 ≤ g1 ≤ gM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' gm 2 ≤ g2 ≤ gM 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm l = rl + dl(∆l − 1), gM l = rl + (dl + 1)(∆l − 1) for some dl ∈ N such that cl + dl = � σl 2 � − 1, where σl is the number of spine vertices of Cl, for l = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that gM 1 < gm 2 , which implies g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have gM 1 < gm 2 it must be: r1 + (d1 + 1)(∆1 − 1) < r2 + d2(∆2 − 1) r1 + q(d1 + 1)(∆2 − 1) < r2 + d2(∆2 − 1) r2 − r1 > (qd1 + q − d2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (8) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (8) can be rewritten as: j2 − j1 > � (qd1 + q − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 � (∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (9) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (9) we obtain: qc1 − c2 > (qd1 + q − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − c2 > qσ1 2 + q(σ1 mod 2) 2 − qc1 − q + q − σ2 2 − 1(σ2 mod 2) 2 + c2+ + 1 + σ2 mod 2 2 − q(σ1 mod 2) 2 15 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − c2 > 1 2 (10) In summary, to have gM 1 < gm 2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (10) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (7) and from the hypothesis that j2−j1 > 0 we obtain (qc1−c2)(∆2−1) > 0 which, since (∆2 − 1) > 0, implies qc1 − c2 > 0 and, since qc1 − c2 is integer, can be rewritten as qc1 − c2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (10) holds and therefore that gM 1 < gm 2 and g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 2: It does not happen that vi1 coincides with vg2 and vg1 coincides with vi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Also in this case we distinguish four sub-cases depending on which vertex is a spine vertex for each edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' As in Case 1, we give here only the first sub-case, while the others can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='a: vi1 and vi2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2 + d2(∆2 − 1) + α2, for some α2 such that 0 ≤ α2 < ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vg2 coincides with vi1, as explained above, it must be i1 = g2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Combining the expression of g2 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (5) we obtain: r2 − j1 = (qc1 − d2)(∆2 − 1) − α2 + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (11) Concerning vg1, we have: gm 1 ≤ g1 ≤ gM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm 1 = r1 + d1(∆1 − 1), gM 1 = r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such that c1 + d1 = � σ1 2 � − 1, where σ1 is the number of spine vertices of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that i2 < gm 1 , which implies i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have i2 < gm 1 it must be: j2 + c2(∆2 − 1) < r1 + d1(∆1 − 1) j2 + c2(∆2 − 1) < r1 + qd1(∆2 − 1) j2 − r1 < (qd1 − c2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (12) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (11) can be rewritten as: j2 − j1 = (qc1 − d2)(∆2 − 1) − α2 + n 2 + (∆2 − 1)(σ2 mod 2) 2 , (13) while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (12) can be rewritten as: j2 − j1 < � (qd1 − c2) − q(σ1 mod 2) 2 � (∆2 − 1) + n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (14) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (13) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (14) we obtain: qc1 − d2 < (qd1 − c2) − (σ2 mod 2) 2 − q(σ1 mod 2) 2 + α2 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 16 Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − σ2 2 − σ2 mod 2 2 + c2 + 1 < qσ1 2 + q(σ1 mod 2) 2 − qc1 − q − c2− − σ2 mod 2 2 − q(σ1 mod 2) 2 + α2 ∆2 − 1 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − qσ1 2 + c2 < −q + 1 2 + α2 2(∆2 − 1) (15) In summary, to have iM 2 < gm 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (15) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (13) and from the hypothesis that j2 − j1 < n−(∆1−1) 2 = n−q(∆2−1) 2 we obtain: qc1 − d2 + 1 2(σ2 mod 2) < −q 2 + α2 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Replacing again d2 with σ2+1(σ2 mod 2) 2 − c2 − 1, we obtain: qc1 − qσ1 2 + c2 < −q + 2 2 + α2 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (16) We have that − q 2 − 1 + α2 ∆2−1 < − q 2 − 1 2 + α2 2(∆2−1), since α2 2(∆2−1) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In other words, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (16) implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (15) holds and therefore that i2 < gm 1 and i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch be h caterpillars such that Ci is ∆i-regular, ∆i − 1 is a multiple of ∆i+1 − 1, with 1 ≤ i < h, and ∆i ≤ n − h (for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If n ≥ 2h + (∆1 − 1), then there exists a k-planar packing with k ∈ O(∆1h + h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For each Ci, with 1 ≤ i ≤ h, we compute a zig-zag drawing Γi with starting vertex vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, given two caterpillars Cj1 and Cj2 with 1 ≤ j1 < j2 ≤ h, we have that ∆j1 − 1 is a multiple of ∆j2 − 1, and the zig-zag drawings Γj1 and Γj2 have starting vertices vj1 and vj2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, 0 < j2 − j1 < h and by hypothesis h ≤ n−(∆1−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, Lemma 4 holds for every pair of caterpillars and the union of all zig-zag drawings Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Γh is a valid packing of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The proof of the bound on the number of crossings along an edge is the same as the one of Theorem 2, considering that ∆l ≤ ∆1 and that jl − j1 = l − 1 for every 2 ≤ l ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 17 5 Lower bounds In this section we first give a general lower bound on the value of k for k-planar h-packings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we then increase this lower bound for some small values of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Every k-planar h-packing of h graphs with n vertices and m edges is such that k ≥ h2m2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='6n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of edges of a k-planar graph with n vertices is at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='81 √ k ·n, for k ≥ 2 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since the h graphs have h·m edges in total, a k-planar packing of these graphs can exist only if h ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='81 √ k n m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', if k ≥ h2m2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='6n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since for a tree m = n − 1, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Every k-planar h-packing of h trees is such that k ≥ h2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now refine the lower bound above for small values of h in an h-placement of caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Specifically we show that for values of h equal to 3, 4, and 5 the corresponding lower bounds are 2, 3, and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Note that for all these cases the lower bound implied by Corollary 1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For h = 3, 4 there exists a caterpillar C with at least h+7 vertices for which every k-planar h-placement of C is such that k ≥ h − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For h = 5 there exists a caterpillar C with at least 24 vertices for which every k-planar 5-placement of C is such that k ≥ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case h = 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let n be an integer such that n ≥ h + 7, and let Cn,h be the n-vertex caterpillar shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that the vertex of Cn,h denoted as v in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 7 has degree n − h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we call it the center of Cn,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider any h-placement of Cn,h into a graph G and denote as vi the vertex of G which the center of Ci is mapped to (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The vertices v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vh must be distinct because, if two centers were mapped to the same vertex of G then this vertex would have degree larger than n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Namely, if two centers are mapped to the same vertex, this vertex has degree 2n − 2h which is larger than n − 1 if n > 2h − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', if h + 7 > 2h − 1, which is true for h < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since each vi (1 ≤ i ≤ h) has degree n − h in Ci and degree 1 in each of the h − 1 other caterpillars, its degree in G is n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, G contains Kh,n−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, for h = 3, G contains K3,7 (n ≥ 10 in this case), which is not 1-planar [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' for h = 4, G contains K4,7 (n ≥ 11 in this case), which is not 2-planar [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The case h = 5 is analogous with K5,19, which is not 4-planar [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 6 Concluding Remarks and Open Problems This paper studied the placement and the packing of caterpillars into k-planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It proved necessary and sufficient conditions for the h-placement of ∆- regular caterpillars in a k-planar graph and sufficient conditions for the packing of a set of ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h-regular caterpillars with k ∈ O(∆1h2) (∆1 is the 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' v h + 2 h n − h − 2 Cn,h Figure 7: A caterpillar as described in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' maximum vertex degree in the set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The work in this paper contributes to the rich literature concerning the placement and the packing problem in planar and non-planar host graphs and it specifically relates with a recent re-visitation of these questions in the beyond-planar context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Many open problems naturally arise from the research in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We conclude the paper by listing some of those that, in our opinion, are among the most interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Extend the characterization of Theorem 2 to the placement of caterpillars that are not ∆-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorems 4 and 3 give sufficient conditions for the k-planar packing of some families of caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It would be interesting to give a complete characterization of the packability of these families into k-planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 6 improves the lower bound of Theorem 5 for caterpillars that are not ∆-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It would be interesting to find a similar result with ∆-regular caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Finally, we point out that one could investigate what graphs can be packed/placed into a k-planar graph for a given value of k, instead of studying how k varies with the number h and the vertex degree of the caterpillars that are packed/placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' While the interested reader can refer to [3] for results with k = 1, the following theorem gives a preliminary result for k = 2 (see the appendix for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (4) in the proof of Theorem 2 would give upper bounds in the range [86, 137] for the caterpillars considered by the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A ∆-regular caterpillar with 4 ≤ ∆ ≤ 7 admits a 2-planar 3- placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' References [1] Eyal Ackerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On topological graphs with at most four crossings per edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Computational Geometry, 85:101574, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [2] Oswin Aichholzer, Thomas Hackl, Matias Korman, Marc van Kreveld, Maarten L¨offler, Alexander Pilz, Bettina Speckmann, and Emo 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', Keszthely, 1976), volume 1, pages 463–469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' North-Holland New York, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [17] Sean P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Haler and Hong Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Packing four copies of a tree into a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Australas.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hedetniemi, Stephen Hedetniemi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Slater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A note on packing two trees into Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Ars Combinatoria, 11, 01 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [19] Seok-Hee Hong and Takeshi Tokuyama, editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Beyond Planar Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1007/978-981-15-6533-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [20] Michael Kaufmann and Dorothea Wagner, editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Drawing Graphs, Meth- ods and Models, volume 2025 of Lecture Notes in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Springer, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1007/3-540-44969-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [21] Stephen 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European Workshop on Computational Geometry, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [25] Norbert Sauer and Joel Spencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Edge disjoint placement of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theory, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' B, 25(3):295–302, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Teo and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Yap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Packing two graphs of order n having total size at most 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Graphs Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', 6(2):197–205, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [27] Hong Wang and Norbert Sauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Packing three copies of a tree into a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' European Journal of Combinatorics, 14(2):137 – 142, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [28] Mariusz Wozniak and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Pawel Wojda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Triple placement of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Graphs and Combinatorics, 9(1):85–91, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [29] Andrzej Zak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A note on k-placeable graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Discret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', 311(22):2634– 2636, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 21 A Missing cases for the proof of Lemma 4 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='b: vg1 and vg2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Property 2 we have, for some d1, d2 ∈ N: g1 = r1 + d1(∆1 − 1) = r1 + qd1(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (17) and g2 = r2 + d2(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (18) If vg1 coincides with vg2, we have g1 = g2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (17) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (18) we obtain: r2 − r1 = (qd1 − d2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (19) Concerning vi1 and vi2, we have: im 1 ≤ i1 ≤ iM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' im 2 ≤ i2 ≤ iM 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with im l = il + cl(∆l − 1), iM l = il + (cl + 1)(∆l − 1) for some cl ∈ N such that cl + dl = � σl 2 � − 1, where σl is the number of spine vertices of Cl, for l = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that iM 1 < im 2 , which implies i1 ̸= i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have iM 1 < im 2 it must be: j1 + (c1 + 1)(∆1 − 1) < j2 + c2(∆2 − 1) j1 + q(c1 + 1)(∆2 − 1) < j2 + c2(∆2 − 1) j2 − j1 > (qc1 + q − c2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (20) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (19) can be rewritten as: j2 − j1 = � (qd1 − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 � (∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (21) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (21) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (20) we obtain: c2 − qc1 − q > −1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (22) In summary, to have iM 1 < im 2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (22) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (21) and from the hypothesis that j2 − j1 > 0 we obtain c2 − qc1 − q > −1 which, since c2 − qc1 − q is integer, can be rewritten as c2 − qc1 − q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (22) holds and therefore that iM 1 < im 2 and i1 ̸= i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='c: vi1 and vg2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Property 2 we have, for some c1 ∈ N: i1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (23) We also have, for some c2 ∈ N and 0 ≤ α2 < ∆2 − 1: i2 = j2 + c2(∆2 − 1) + α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (24) 22 If vi1 coincides with vi2, we have i1 = i2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (23) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (24) we obtain: j2 − j1 = (qc1 − c2)(∆2 − 1) − α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (25) Concerning vg1, we have: gm 1 ≤ g1 ≤ gM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm 1 = r1 + d1(∆1 − 1), gM 1 = r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such that c1 + d1 = � σ1 2 � − 1, where σ1 is the number of spine vertices of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2 + d2(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that gM 1 < g2, which implies g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have gM 1 < g2 it must be: r1 + (d1 + 1)(∆1 − 1) < r2 + d2(∆2 − 1) r1 + q(d1 + 1)(∆2 − 1) < r2 + d2(∆2 − 1) r2 − r1 > (qd1 + q − d2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (26) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (26) can be rewritten as: j2 − j1 > � (qd1 + q − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 � (∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (27) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (25) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (27) we obtain: qc1 − c2 − α2 ∆2 − 1 > (qd1 + q − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − c2 − α2 ∆2 − 1 > qσ1 2 + q(σ1 mod 2) 2 − qc1 − q + q − σ2 2 − 1(σ2 mod 2) 2 + c2+ + 1 + σ2 mod 2 2 − q(σ1 mod 2) 2 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − c2 > 1 2 + α2 2(∆2 − 1) (28) In summary, to have gM 1 < g2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (28) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (25) and from the hypothesis that j2 −j1 > 0 we obtain (qc1 −c2)(∆2 −1)− α2 > 0 which, since (∆2 − 1) > 0, implies qc1 − c2 > α2 ∆2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since 0 ≤ α2 ∆2−1 < 1 and qc1 − c2 is integer, we have qc1 − c2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (28) holds and therefore that gM 1 < g2 and g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='d: vg1 and vi2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Property 2 we have, for some c2 ∈ N: i2 = j2 + c2(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (29) 23 We also have, for some c1 ∈ N and 0 ≤ α1 < ∆2 − 1: i1 = j1 + c1(∆1 − 1) + α1 = j1 + qc1(∆2 − 1) + α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (30) If vi1 coincides with vi2, we have i1 = i2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (30) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (29) we obtain: j2 − j1 = (qc1 − c2)(∆2 − 1) + α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (31) Concerning vg2, we have: gm 2 ≤ g2 ≤ gM 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm 2 = r2 + d2(∆2 − 1), gM 2 = r2 + (d2 + 1)(∆2 − 1) for some d2 ∈ N such that c2 + d2 = � σ2 2 � − 1, where σ2 is the number of spine vertices of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg1 is a vertex in the lower part of Γ1, it must be g1 = r1 + d1(∆1 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that g1 < gm 2 , which implies g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have g1 < gm 2 it must be: r1 + d1(∆1 − 1) < r2 + d2(∆2 − 1) r1 + qd1(∆2 − 1) < r2 + d2(∆2 − 1) r2 − r1 > (qd1 − d2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (32) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (32) can be rewritten as: j2 − j1 > � (qd1 − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 � (∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (33) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (31) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (33) we obtain: qc1 − c2 + α1 ∆2 − 1 > (qd1 − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − c2 + α1 ∆2 − 1 > qσ1 2 + q(σ1 mod 2) 2 − qc1 − q − σ2 2 − 1(σ2 mod 2) 2 + c2+ + 1 + σ2 mod 2 2 − q(σ1 mod 2) 2 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − c2 > 1 − q 2 − α1 2(∆2 − 1) (34) In summary, to have g1 < gm 2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (34) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (31) and from the hypothesis that j2 −j1 > 0 we obtain (qc1 −c2)(∆2 −1)+ α1 > 0 which, since (∆2−1) > 0, implies qc1−c2 > − α1 ∆2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since 0 ≤ α1 ∆2−1 < 1 and qc1 − c2 is integer, we have qc1 − c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since q is a positive integer, this implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (34) holds and therefore that g1 < gm 2 and g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 24 Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='b: vg1 and vg2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2+d2(∆2−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vg2 coincides with vi1, as explained above, it must be i1 = g2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Combining the expression of g2 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (30) we obtain: r2 − j1 = (qc1 − d2)(∆2 − 1) + α1 + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (35) Concerning vi2, we have: im 2 ≤ i2 ≤ iM 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with iM 2 = j2 + (c2 + 1)(∆2 − 1) for some c2 ∈ N such that c2 + d2 = � σ2 2 � − 1, where σ2 is the number of spine vertices of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that iM 2 < g1, which implies i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have iM 2 < g1 it must be: j2 + (c2 + 1)(∆2 − 1) < r1 + d1(∆1 − 1) j2 + (c2 + 1)(∆2 − 1) < r1 + qd1(∆2 − 1) j2 − r1 < (qd1 − c2 − 1)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (36) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (35) can be rewritten as: j2 − j1 = (qc1 − d2)(∆2 − 1)/α1 + n 2 + (∆2 − 1)(σ2 mod 2) 2 , (37) while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (36) can be rewritten as: j2 − j1 < � (qd1 − c2 − 1) − q(σ1 mod 2) 2 � (∆2 − 1) + n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (38) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (37) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (38) we obtain: qc1 − d2 < (qd1 − c2 − 1) − (σ2 mod 2) 2 − q(σ1 mod 2) 2 − α1 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − σ2 2 − σ2 mod 2 2 + c2 + 1 + σ2 mod 2 2 + α1 ∆2 − 1 < qσ1 2 + q(σ1 mod 2) 2 − − qc1 − q − c2 − q(σ1 mod 2) 2 − 1 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − qσ1 2 + c2 + 1 < −q 2 − α1 2(∆2 − 1) (39) In summary, to have iM 2 < gm 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (39) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (37) and from the hypothesis that j2 − j1 < n−(∆1−1) 2 = n−q(∆2−1) 2 we obtain: qc1 − qσ1 2 + c2 + 1 < −q 2 − α1 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (40) 25 We have that − q 2 − α1 ∆2−1 < − q 2 − α1 2(∆2−1), since α1 2(∆2−1) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In other words, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (40) implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (39) holds and therefore that iM 2 < g1 and i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='c: vi1 and vg2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2+d2(∆2−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vg2 coincides with vi1, as explained above, it must be i1 = g2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Combining the expression of g2 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (23) we obtain: r2 − j1 = (qc1 − d2)(∆2 − 1) + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (41) Concerning vg1, we have: gm 1 ≤ g1 ≤ gM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm 1 = r1 + d1(∆1 − 1), gM 1 = r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such that c1 + d1 = � σ1 2 � − 1, where σ1 is the number of spine vertices of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that iM 2 < gm 1 , which implies i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have iM 2 < gm 1 it must be: j2 + (c2 + 1)(∆2 − 1) < r1 + d1(∆1 − 1) j2 + (c2 + 1)(∆2 − 1) < r1 + qd1(∆2 − 1) j2 − r1 < (qd1 − c2 − 1)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (42) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (41) can be rewritten as: j2 − j1 = (qc1 − d2)(∆2 − 1) + n 2 + (∆2 − 1)(σ2 mod 2) 2 , (43) while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (42) can be rewritten as: j2 − j1 < � (qd1 − c2 − 1) − q(σ1 mod 2) 2 � (∆2 − 1) + n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (44) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (43) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (44) we obtain: qc1 − d2 < (qd1 − c2 − 1) − (σ2 mod 2) 2 − q(σ1 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − σ2 2 − σ2 mod 2 2 + c2 + 1 < qσ1 2 + q(σ1 mod 2) 2 − qc1 − q − c2 − 1− − σ2 mod 2 2 − q(σ1 mod 2) 2 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − qσ1 2 + c2 + 1 < −q 2 (45) 26 In summary, to have iM 2 < gm 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (45) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (43) and from the hypothesis that j2 − j1 < n−(∆1−1) 2 = n−q(∆2−1) 2 we obtain: qc1 − d2 + 1 2(σ2 mod 2) < −q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Replacing again d2 with σ2+1(σ2 mod 2) 2 − c2 − 1, we obtain: qc1 − qσ1 2 + c2 + 1 < −q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (46) Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (45) is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (46), we can conclude that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (45) holds and therefore that iM 2 < gm 1 and i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='d: vg1 and vi2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg1 is a vertex in the lower part of Γ1, it must be g1 = r1 + d1(∆1 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vg1 coincides with vi2, combining the expression of g1 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (6) we obtain: j2 − r1 = (qd1 − c2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (47) Concerning vi1 vg2, we have: im 1 ≤ i1 ≤ iM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' and gm 2 ≤ g2 ≤ gM 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with im 1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1), gM 2 = r2 + (d2 + 1)(∆2 − 1) − n for some d2 ∈ N such that c2 + d2 = � σ2 2 � − 1, where σ2 is the number of spine vertices of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that gM 2 < im 1 , which implies g2 ̸= i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have gM 2 < im 1 it must be: r2 + (d2 + 1)(∆2 − 1) − n < j1 + c1(∆1 − 1) r2 + (d2 + 1)(∆2 − 1) − n < j1 + qc1(∆2 − 1) r2 − j1 < (qc1 − d2 − 1)(∆2 − 1) + n(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (48) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (47) can be rewritten as: j2 − j1 = (qd1 − c2)(∆2 − 1) − q(∆2 − 1)(σ1 mod 2) 2 + n 2 , (49) while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (48) can be rewritten as: j2 − j1 < (qc1 − d2 − 1)(∆2 − 1) + (∆2 − 1)(σ2 mod 2) 2 + n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (50) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (49) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (50) we obtain: qd1 − c2 − q(σ1 mod 2) 2 < (qc1 − d2 − 1) + (σ2 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 27 Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qσ1 2 + q(σ1 mod 2) 2 − qc1 − q − c2 − q(σ1 mod 2) 2 < qc1 − σ2 2 − σ2 mod 2 2 + + c2 + 1 − 1 + σ2 mod 2 2 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: 2 � qc1 − qσ1 2 + c2 � > −q (51) In summary, to have gM 2 < im 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (51) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (49) and from the hypothesis that j2 − j1 < n−(∆1−1) 2 = n−q(∆2−1) 2 we obtain: qd1 − c2 − q 2(σ1 mod 2) < −q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Replacing again d1 with σ1+1(σ1 mod 2) 2 − c1 − 1, we obtain: 2 � qc1 − qσ1 2 + c2 � > −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (52) Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (51) is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (52), we can conclude that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (51) holds and therefore that gM 2 < im 1 and g2 ̸= i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' B Proof of Theorem 7 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A ∆-regular caterpillar with 4 ≤ ∆ ≤ 7 admits a 2-planar 3- placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1, C2, and C3 be three copies (shown in red, blue and green, respec- tively, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 8) of a ∆-regular caterpillar C with 4 ≤ ∆ ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We denote the vertices of caterpillar Cj for j = 1, 2, 3 as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the spine vertices are denoted as vj 0, vj 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vj c−1 in the order they appear along the spine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the leaves adjacent to vertex vj i (for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , c−1) are denoted as uj i,l with l = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , d, where d = ∆ − 2 if i = 0 or i = c − 1 and d = ∆ − 3 if 0 < i < c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let p0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , pn−1 be n points on a circle in clockwise order (with indices taken modulo n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To construct the packing, we compute a drawing for each caterpillar such that the vertices are mapped to points p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , pn and the union of the three drawings is a 2-planar drawing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We describe the construction for ∆ = 4, 5, 6 (see also Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 8(a) to 8(c));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the construction in the case ∆ = 7 is slightly different and it is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 8(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Caterpillar C1 is drawn outside the circle so that vertex v1 0 is mapped to point p0, each vertex v1 i , for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , c − 1 is mapped to pi(∆−1)+1, each leaf u1 0,l is mapped to the point pl+1, and each leaf u1 i,l is mapped to the point pi(∆−1)+2+l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In other words, each vertex of the spine is followed clockwise by 28 ∆ = 4 (a) ∆ = 5 (b) ∆ = 6 (c) ∆ = 7 (d) Figure 8: 2-planar 3-placements of ∆-regular caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' its leaves and the last of these leaves is followed by the next vertex of the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Caterpillar C2 is drawn inside the circle so that vertex v2 i is mapped to the point immediately following clockwise the point hosting v1 i and each leaf u2 i,l is mapped to the point immediately following clockwise u1 i,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Clearly, the drawings of the first two caterpillars do not cross each other because they are on different sides of the circle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' also, their union has no multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Concerning C3, the vertex v3 i , for i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , c−2 is mapped to the point that hosts u1 i,d and u2 i,d−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', the last leaf of v1 i and the second last leaf of v2 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the vertex v3 c−1 is mapped to the point that hosts u1 i,d−1 and u2 i,d−2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', the second last leaf of v1 c−1 and the third last leaf of v2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' About this mapping, observe that if we draw the edges of the spine of C3 outside the circle, each edge of the spine of C3 intersects two consecutive edges of the spine of C1 and each edge of the spine of C1 intersects at most two consecutive edges of the spine of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To complete the drawing, we need to draw the leaves of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider two consecutive spine vertices v3 i and v3 i+1, with 0 ≤ i ≤ c − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' between these two vertices there are ∆ − 2 points not yet used by C3, we connect the first two of these vertices in clockwise order to vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Depending on the value of ∆, there remain 0, 1, or 2 points between vi and vi+1 not yet used by C3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we connect these points to vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, there remain to map ∆ − 3 leaves adjacent to v3 0 and 3 leaves adjacent to v3 c−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, there are ∆ points not yet used by C3 that are between v3 c−1 and v3 0 clockwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we connect the three vertices following clockwise v3 c−1 to v3 c−1, and the remaining ones to v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' All the edges of C3 that are incident to leaves are drawn inside the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This mapping of C3 does not create multiple edges and gives rise to at most two crossings along the edges of C2 and C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} diff --git a/3tE2T4oBgHgl3EQfjgeA/content/tmp_files/2301.03969v1.pdf.txt b/3tE2T4oBgHgl3EQfjgeA/content/tmp_files/2301.03969v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fdfa8c51cbe127e62dc6dac4315d729a88891750 --- /dev/null +++ b/3tE2T4oBgHgl3EQfjgeA/content/tmp_files/2301.03969v1.pdf.txt @@ -0,0 +1,1533 @@ +Exploring bulk viscous unified scenarios with Gravitational Waves Standard Sirens +Weiqiang Yang,1, ∗ Supriya Pan,2, 3, † Eleonora Di Valentino,4, ‡ +Celia Escamilla-Rivera,5, § and Andronikos Paliathanasis3, 6, 7, ¶ +1Department of Physics, Liaoning Normal University, Dalian, 116029, P. R. China +2Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India +3Institute of Systems Science, Durban University of Technology, +PO Box 1334, Durban 4000, Republic of South Africa +4School of Mathematics and Statistics, University of Sheffield, +Hounsfield Road, Sheffield S3 7RH, United Kingdom +5Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, +Circuito Exterior C.U., A.P. 70-543, M´exico D.F. 04510, M´exico +6Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile, Valdivia 5090000, Chile +7Mathematical Physics and Computational Statistics Research Laboratory, +Department of Environment, Ionian University, Zakinthos 29100, Greece +We consider the unified bulk viscous scenarios and constrain them using the Cosmic Microwave +Background observations from Planck 2018 and the Pantheon sample from Type Ia Supernovae. +Then we generate the luminosity distance measurements from O(103) mock Gravitational Wave +Standard Sirens (GWSS) events for the proposed Einstein Telescope. We then combine these mock +luminosity distance measurements from the GWSS with the current cosmological probes in order to +forecast how the mock GWSS data could be effective in constraining these bulk viscous scenarios. +Our results show that a non-zero time dependent bulk viscosity in the universe sector is strongly +preferred by the current cosmological probes and will possibly be confirmed at many standard +deviations by the future GWSS measurements. We further mention that the addition of GWSS +data can significantly reduce the uncertainties of the key cosmological parameters obtained from +the usual cosmological probes employed in this work. +I. +INTRODUCTION +Understanding the nature of dark matter and dark en- +ergy has been a challenge for cosmologists. The standard +cosmological model, namely, the so-called Λ-Cold Dark +Matter (ΛCDM) model representing a mixture of two +non-interacting fluids − a positive cosmological constant +(Λ > 0) and a cold dark matter component, has un- +doubtedly proved its unprecedented success by explain- +ing a large span of astronomical data. +However, this +simplest cosmological scenario has some limitations. For +example, the cosmological constant problem [1] and the +coincidence problem [2] have already questioned the ex- +isting assumptions in the ΛCDM model, e.g. constant +energy density of the vacuum and the non-interacting na- +ture between Λ and CDM. These limitations motivated +the cosmologists to find alternative cosmological scenar- +ios beyond ΛCDM by relaxing the above assumptions, +and as a consequence, several new cosmological models +were introduced, see [3–12] for a review of various dark +energy and modified gravity models. Additionally, the +appearance of cosmological tensions at many standard +deviations between Planck [13] (assuming ΛCDM in the +background) and other cosmological probes, such as dis- +tance ladders [14–24] or weak lensing [25–29] and galaxy +∗ d11102004@163.com +† supriya.maths@presiuniv.ac.in +‡ e.divalentino@sheffield.ac.uk +§ celia.escamilla@nucleares.unam.mx +¶ anpaliat@phys.uoa.gr +cluster data [30–32] has further weakened the confidence +in the ΛCDM cosmological model [33–37]. Thus, the list +of cosmological models aiming to address the cosmolog- +ical tensions is increasing in time, see the review arti- +cles [38–44] and references therein. Given the fact that +the origin of dark matter and dark energy is not clearly +understood yet, thus, there is no reason to favor any par- +ticular cosmological theory over others. As a result, var- +ious ways have been proposed to interpret the dynamics +of the dark sector in terms of dark matter and dark en- +ergy. +The simplest assumption is the consideration of +independent evolution of these dark fluids. The general- +ization of the above consideration is the assumption of a +non-gravitational interaction between these dark sectors. +On the other hand, a heuristic approach is to consider a +unified dark fluid that can explain the dynamics of dark +energy and dark matter at cosmological scales. The at- +tempt to unify the dark sector of the Universe began long +back ago. The most simplest unified dark sector mod- +els can be constructed in the context of Einstein gravity +with the introduction of a generalized equation of state +p = F(ρ), where p and ρ are respectively the pressure and +energy density of the unified dark sector and F is an an- +alytic function of the energy density, ρ. The well known +unified cosmological models, such as the Chaplygin gas +model [45] and its successive generalizations, namely, the +generalized Chaplygin gas, modified Chaplygin gas, see +Refs. [46–57] and some other unified cosmological scenar- +ios as well [58–60] belong to this classification. While it +is essential to mention that a subset of the unified mod- +els has been diagnosed with exponential blowup in the +arXiv:2301.03969v1 [astro-ph.CO] 10 Jan 2023 + +2 +matter power spectrum which is not consistent with the +observations [61], however, this does not rule out the pos- +sibility of unified models aiming to cover a wide region +of the universe evolution because a new kind of unified +fluid may avoid such unphysical activities. The unified +cosmological models can also be developed by considering +a relation like p = G(H) where G is an analytic function +of H, the Hubble function of the Friedmann-Lemaˆıtre- +Robertson-Walker (FLRW) line element. +Apparently, +theories with p = F(ρ) and p = G(H) seem identical, +however, this is only true in spatially flat FLRW uni- +verse. For a curved universe, the two approaches are not +the same. +In the present work we are interested to study a partic- +ular class of unified models endowed with bulk viscosity. +The cosmological fluids allowing bulk viscosity as an ex- +tra ingredient can explain the accelerating expansion of +the universe, and hence they are also enlisted as possi- +ble alternatives to the standard ΛCDM cosmology in the +literature [62, 63]. +Following an earlier work Ref. [64] +where an evidence of non-zero bulk viscosity was pre- +ferred by the current cosmological probes, in the present +article, we use the simulated Gravitational Waves Stan- +dard Sirens (GWSS) measurements from the Einstein +Telescope [65]1 in order to quantify the improvements +of the cosmological parameters, if any, from the future +GWSS measurements. As the gravitational waves (GW) +have opened a new window for astrophysics and cosmol- +ogy, therefore, it will be interesting to investigate the +contribution from the simulated GWSS data, once com- +bined with the current cosmological probes. This moti- +vated many investigators to use the mock GWSS data +matching the expected sensitivity of the Einstein Tele- +scope to constrain a class of cosmological models, see for +instance, [66–77]. In particular, the combined analysis of +simulated GWSS measurements from Einstein Telescope +and the standard cosmological probes has proven to be +very effective for a class of cosmological models, in the +sense that the error bars in the key cosmological param- +eters of these cosmological models are significantly re- +duced thanks to the mock GWSS dataset [70, 71, 74, 78– +81], however, in some specific f(R) theories of gravity, the +generated mock GWSS from the Einstein Telescope may +not be very much helpful to give stringent constraints on +them during its first phase of running +[82]. Thus, one +may expect that the constraining power of the Einstein +Telescope may depend on the underlying cosmological +model. Aside from the future GWSS measurements from +the Einstein Telescope, one can also use the simulated +GWSS measurements from other GW observatories, such +as, Laser Interferometer Space Antenna (LISA) [83–86] +and DECi-heltz Interferometer Gravitational wave Ob- +servatory (DECIGO) [87, 88], TianQin [89]. In this ar- +ticle, we focus only on the simulated GWSS data from +1 https://www.einsteintelescope.nl/en/ +Einstein Telescope to constrain the bulk viscous unified +scenario. +The paper has been organized as follows: in Sec. II we +discuss the gravitational equations for the bulk viscous +scenario. Sec. III describes the observational data that +we have considered for the analysis in this work. Sec. IV +presents the observational constraints on the bulk vis- +cous models, and mainly we discuss how the inclusion +of gravitational waves data from the Einstein Telescope +improves the constraints. Finally, in Sec. V we present +the conclusions. +II. +REVISITING THE BULK VISCOUS +SCENARIOS: BACKGROUND AND +PERTURBATIONS +As usual, we consider the homogeneous and isotropic +space +time +described +by +the +Friedmann-Lemaˆıtre- +Robertson-Walker (FLRW) line element +ds2 = −dt2 + a2(t) +� +dr2 +1 − kr2 + r2 � +dθ2 + sin2 θdφ2�� +, +(1) +where a(t) is the expansion scale factor and k denotes the +spatial curvature of the universe. For k = 0, −1, +1, we +have three different geometries of the universe, namely, +spatially flat, open and closed, respectively. In this paper +we restrict ourselves to the spatially flat scenario where +we assume that (i) the gravitational sector is described +by the Einstein’s gravity, (ii) the matter sector of the uni- +verse consists of the relativistic radiation, non-relativistic +baryons and a unified bulk viscous fluid which combines +the effects of dark matter and dark energy, (iii) all the +fluids are non-interacting with each other. Within this +framework, we can write down the gravitational field +equations as follows (in the units where 8πG = 1) +H2 = 1 +3ρtot, +(2) +2 ˙H + 3H2 = − ptot, +(3) +where an overhead dot indicates the derivative with re- +spect to the cosmic time t; H ≡ ˙a/a is the Hubble ex- +pansion rate; (ρtot, ptot) = (ρr +ρb +ρu, pr +pb +pu) are +the total energy density and total pressure of the cosmic +components in which (ρr, pr), (ρb, pb), (ρu, pu) are the en- +ergy density and pressure of radiation, baryons and the +unified fluid, respectively. The conservation equation for +each fluid follows the usual law ˙ρi + 3H(1 + wi)ρi = 0, +where the subscript i refers to radiation (i = r), baryons +(i = b) and the unified fluid (i = u) and wi are the stan- +dard barotropic state parameters: wr = pr/ρr = 1/3, +wb = pb/ρb = 0 and wu = pu/ρu = (γ − 1), where γ +is a constant parameter. +In general for different val- +ues of γ, say for instance, γ = 0, we realize a cosmo- +logical constant-like fluid endowed with the bulk viscos- +ity and similarly γ = 1 results in a dust-like fluid en- +dowed with the bulk viscosity. +As the nature of the + +3 +fluid is not clearly understood and as the observational +data play an effective role to understand this nature, +thus, in order to be more transparent in this direction +we consider γ lying in the interval [−3, 3] which includes +both exotic (pu/ρu = (γ − 1) < −1/3) and non-exotic +(pu/ρu = (γ − 1) > −1/3 ) fluids. As already mentioned, +since the unified fluid has a bulk viscosity, therefore, it en- +joys an effective pressure [90]: peff = pu−uν +;νη(ρu), where +uµ +;µ is the expansion scalar of this fluid and η(ρu) > 0 is +the coefficient of the bulk viscosity. Thus, in the FLRW +background, the effective pressure of the bulk viscous +fluid reduces to +peff = pu − 3Hη(ρu). +(4) +Since there is no unique selection for the bulk viscous +coefficient, η(ρu), therefore, we consider a well known +choice for it in which the bulk viscous coefficient has a +power law evolution of the form [90–92]: +η(ρu) = αρm +u , +(5) +where α is a positive constant and m is any real number. +Notice that for the case m = 0 we recover the scenario +with a constant bulk viscous coefficient. Now, with the +consideration of the bulk viscous coefficient in (5), the +effective pressure of the unified fluid can be expressed as +peff = (γ − 1)ρu − +√ +3αρ1/2 +tot ρm +u , +(6) +and consequently, one can define the effective equation +of state of the viscous dark fluid as +weff = peff +ρu += (γ − 1) − +√ +3αρ1/2 +tot ρm−1 +u +. +(7) +The adiabatic sound speed for the viscous fluid is given +by +c2 +a,eff = p′ +eff +ρ′u += weff + +w′ +eff +3H(1 + weff). +(8) +where the prime denotes the derivative with respect to +the conformal time τ and H is the conformal Hubble pa- +rameter, H = aH. Note that depending on the nature of +weff, c2 +a,eff could be negative, and hence ca,eff could be an +imaginary quantity. This may invite instabilities in the +perturbations. Thus, in order to avoid this possible un- +physical situation, we consider the entropy perturbations +(non-adiabatic perturbations) in the unified dark fluid +following the analysis of generalized dark matter [93]. +Now we focus on the evolution of the unified bulk vis- +cous fluid at the level of perturbations. In the entropy +perturbation mode, the true pressure perturbation comes +from the effective pressure given by +δpeff = δpu − δη(∇σuσ) − η(δ∇σuσ) += δpu − 3Hδη − η +a +� +θ + h′ +2 +� +. +(9) +The effective sound speed of viscous dark fluid for the +bulk viscous coefficient (5) can be defined as +c2 +s,eff ≡ +�δpeff +δρu +� +rf += c2 +s − +√ +3αmρ1/2 +tot ρm−1 +u +− αρm−1 +u +aδu +� +θ + h′ +2 +� +, +(10) +where ’|rf’ denotes the rest frame. Following the analysis +in [93], the sound speed in the rest frame is assumed to +be zero, i.e. c2 +s = 0. +The density perturbation and the velocity perturba- +tion can also be written as [93] +δ′ +u = −(1 + weff)(θu + h′ +2 ) + +w′ +eff +1 + weff +δu +− 3H(c2 +s,eff − c2 +a,eff) +� +δu + 3H(1 + weff)θD +k2 +� +, +(11) +θ′ +u = −H(1 − 3c2 +s,eff)θu + +c2 +s,eff +1 + weff +k2δu, +(12) +Thus, following the evolution at the background and per- +turbation level prescribed above, one can now be able to +understand the dynamics of the bulk viscous fluid. In +this work we consider two different bulk viscous scenar- +ios characterized as follows: the bulk viscous model 1 +(labeled as BVF1) where we consider γ = 1, and the +bulk viscous model 2 (labeled as BVF2) where we keep +γ as a free parameter. The common parameters in both +BVF1 and BVF2 are α and m. +III. +STANDARD COSMOLOGICAL PROBES, +SIMULATED GWSS DATA, AND +METHODOLOGY +In this section we describe the cosmological data +sets employed to perform the statistical analyses of the +present bulk viscous scenarios. +• Cosmic Microwave Background (CMB): We +use the CMB data from the Planck 2018 data re- +lease. +Precisely, we use the CMB temperature +and polarization angular power spectra plikTT- +TEEE+lowl+lowE [94, 95]. +• Pantheon sample from Type Ia Supernovae +(SNIa) data: Type Ia Supernovae are the first +astronomical data that probed the accelerating ex- +pansion of the universe and hence indicated the +existence of an exotic fluid with negative pressure +(dark energy). Here we use the Pantheon compila- +tion of the SNIa data comprising 1048 data points +spanned in the redshift interval [0.01, 2.3] [96]. +• Gravitational +Waves +Standard +Sirens +(GWSS): We take the mock Gravitational Waves +Standard +Sirens +(GWSS) +data +generated +by +matching +the +expected +sensitivity +of +Einstein + +4 +Telescope in order to understand the constraining +power of the future GWSS data from the Einstein +Telescope. +The Einstein Telescope is a proposed +ground based third-generation (3G) GW detector. +The telescope will take a triangular shape and +its each arm length will be increased to 10 km, +compared to 3 km arm length VIRGO and 4 km +arm length LIGO [65, 97]. Thus, due to such in- +creased arm length, the Einstein Telescope will be +a potential GW detector by reducing all displace- +ment noises [65, 97]. It is expected that after 10 +years of operation, Einstein Telescope will detect +O(103) GWSS events. Although the detection of +O(103) GWSS events is very optimistic while the +number of detections could be low in reality [65]. +As argued in [65], the Einstein Telescope will likely +to detect 20 − 50 events per year, i.e. 200 − 500 +events in 10 years. However, following the earlier +works [66, 67, 70, 71, 74, 77, 79, 81, 82], in this +article, we restrict ourselves to the detection of +O(103) GWSS events by the Einstein Telescope +to constrain the bulk viscous scenarios. For more +features of the Einstein Telescope we refer the +readers to [65]. +We originally generate the mock GWSS luminosity +distance measurements matching the expected sen- +sitivity of the Einstein Telescope after 10 years of +full operation. Specifically we create 1000 triples +(zi, dL(zi), σi) where zi is the redshift of a GW +source, dL(zi) is the measured luminosity distance +at redshift zi and σi is the uncertainty associated +with the luminosity distance dL(zi). Let us briefly +summarize the procedure of generating the mock +GWSS dataset and we refer to Refs. [70, 71, 81] +for more technical details. +The initial step for +generating the mock GWSS dataset is to identify +the expected GW sources. +We consider the GW +events originating from two distinct binary systems, +namely, (i) a combination of a Black Hole (BH) and +a Neutron Star (NS) merger, identified as BHNS +and (ii) the binary neutron star (BNS) merger. +Following the mass distributions as described in +Ref. [81], the ratio of the number of GW events for +the BHNS merger versus BNS merger is taken to be +0.03 as predicted for the Advanced LIGO-VIRGO +network [98]. We then determine the merger rate +R(z) of sources and from the merger rate of the +sources, we determine the redshift distribution of +the sources, P(z) given by [66, 67, 70, 79, 81, 99] +P(z) ∝ 4πd2 +C(z)R(z) +H(z)(1 + z) , +(13) +where dC(z) ≡ +� z +0 H−1(z′)dz′ is the co-moving dis- +tance and for R(z) we take the following piece- +wise linear function estimated in [100] (also see [66, +67, 70, 79, 81, 101]): R(z) = 1 + 2z for z ≤ 1, +R(z) = 3 +4(5 − z), for 1 < z < 5 and R(z) = 0 for +z > 5. After having P(z), we sample 1000 values +of redshifts from this distribution which represent +the redshifts zi of our 1000 mock GWSS data. +The next step is to choose a fiducial model because +while going from the merger rate to the redshift dis- +tributions, a fiducial cosmological model is needed +since the expression for P(z) includes both the co- +moving distance and expansion rate at redshift z, +i.e. dL(z) and H(z) respectively. This H(z) cor- +responds to the fiducial model. +As in this arti- +cle we are interested to investigate how the inclu- +sion of GWSS data improves the constraints on the +BVF models, therefore, we generate two different +mock GWSS datasets choosing BVF1 and BVF2 +as the fiducial models. +We take the fiducial val- +ues of the cosmological parameters given by the +best fit values of the same cosmological parame- +ters of the BVF1 and BVF2 models obtained from +the CMB+Pantheon data analysis. Now, for the +chosen fiducial model(s), one can now estimate the +luminosity distance at the redshift zi using the re- +lation +dL(zi) = (1 + zi) +� zi +0 +dz′ +H(z′) . +(14) +Thus, after having the luminosity distance dL(zi) of +the GW source, our last job is now to determine the +uncertainty σi associated with this luminosity dis- +tance. The determination of the uncertainty σi di- +rectly connects to the waveform of GW because the +GW amplitude depends on the luminosity distance +(also on the so-called chirp mass [66, 67, 79]) and +hence one can extract the information about dL(zi) +and σi. We refer to Refs. [66, 67, 70, 79, 81] for the +technical details to calculate the uncertainties on +the luminosity distance measurements. The lumi- +nosity distance measurement dL(zi) has two kind of +uncertainties, one is the instrumental uncertainty +σinst +i +and the other one is the weak lensing uncer- +tainty σlens +i +. The instrumental error can be derived +to be σinst +i +(≃ 2dL(zi)/S where S is the combined +signal-to-noise ratio of the Einstein Telescope) us- +ing the Fisher matrix approach and assuming that +the uncertainty on dL(zi) is not correlated with +the uncertainties on the remaining GW parame- +ters (see [66, 67, 70, 79, 81]) and the lensing error +is σlens +i +≃ 0.05zidL(zi) [66]. Thus, the total uncer- +tainty due to the instrumental and the weak lensing +uncertainties on dL(zi) is σi = +� +(σinst +i +)2 + (σlens +i +)2. +Finally, let us note that the combined signal-to- +noise ratio of the GW detector is a very crucial +quantity in this context since for the Einstein Tele- +scope, the combined signal-to-noise ratio should be + +5 +Parameter Priors (BVF1) Priors (BVF2) +Ωbh2 +[0.005, 0.1] +[0.005, 0.1] +τ +[0.01, 0.8] +[0.01, 0.8] +ns +[0.5, 1.5] +[0.5, 1.5] +ln(1010As) +[2.4, 4] +[2.4, 4] +100θMC +[0.5, 10] +[0.5, 10] +β +[0, 1] +[0, 1] +m +[−2, 0.5] +[−2, 0.5] +γ +− +[−3, 3] +TABLE I. We show the flat priors on all the free parameters +associated with the bulk viscous models. +at least 8 for a GW detection [99]. Thus, in sum- +mary, we generate 1000 GW sources up to redshift +z = 2 with S > 8. For more technical details we +refer the readers to Refs. [66, 67, 70, 79, 81, 99]. +To constrain the BVF scenarios we modify the pub- +licly available CosmoMC package [102] which is an excel- +lent cosmological code supporting the Planck 2018 like- +lihood [95] and it has a convergence diagnostic follow- +ing the Gelman-Rubin statistic R − 1 [103]. It is essen- +tial to mention that for both BVF1 and BVF2 scenarios, +we have used the dimensionless quantity β = αH0ρm−1 +tot,0 +where ρtot,0 is the present value of ρtot. We further men- +tion here that β = 0 (equivalently, α = 0) implies no vis- +cosity and hence the overall picture behaves like a unified +cosmic fluid without bulk viscosity. Thus, in summary, +the parameter space of BVF1 and BVF2 are as below: +PBVF1 ≡ {Ωbh2, 100θMC, τ, ns, ln(1010As), β, m} +PBVF2 = {Ωbh2, 100θMC, τ, ns, ln(1010As), β, m, γ} +where the description of the free parameters are as fol- +lows: Ωbh2 is the baryons density, 100θMC is the ratio +of the sound horizon to the angular diameter distance; +τ is the optical depth, ns is the scalar spectral index, +As is the amplitude of the initial power spectrum. The +flat priors on both cosmological scenarios are shown in +Table I. +IV. +OBSERVATIONAL CONSTRAINTS: +RESULTS AND ANALYSIS +In this section we present the constraints on the +bulk viscous scenarios considering CMB+Pantheon and +CMB+Pantheon+GWSS. As we are interested to esti- +mate the improvement of the cosmological parameters in +presence of the GWSS measurements, and as the com- +bined standard cosmological probes offer the most strin- +gent constraints on the cosmological parameters, there- +fore, the inclusion of GWSS with the combined standard +cosmological probes is reasonable. As mentioned earlier, +the key common free parameters of BVF1 and BVF2 are +β and m, since β ̸= 0 indicates the preference for a non- +zero bulk viscosity and m ̸= 0 indicates that the coeffi- +cient of the bulk viscosity is not constant in the redshift +range considered. In the following subsections we discuss +the constraints on these two scenarios in detail. +A. +Constraints on the BVF1 scenario +In +Table +II +we +have +presented +the +constraints +on +the +BVF1 +scenario +for +CMB+Pantheon +and +CMB+Pantheon+GWSS. The latter dataset is aimed to +understand the improvement expected from GWSS on +the constraints from CMB+Pantheon. In Fig. 1 we have +compared these datasets graphically by showing the one +dimensional marginalized distribution of some model pa- +rameters and the two dimensional joint contours. +As +discussed, this scenario has two main key parameters, +namely, β, quantifying the existence of bulk viscosity in +the cosmic sector, and m, which tells us whether the bulk +viscosity will have a dynamical nature (corresponding to +m ̸= 0) or not. +Since for CMB+Pantheon, we find an evidence for a +non-zero bulk viscosity in the cosmic sector at many +standard deviations, i.e. β = 0.430+0.017 +−0.016 at 68% CL, +this is further strengthen for CMB+Pantheon+GWSS, +where β = 0.4262+0.0079 +−0.0078 at 68% CL.2 One can clearly +see that the inclusion of GWSS to CMB+Pantheon im- +proves the error bars on β by a factor of at least 2. +This reflects the constraining power of GWSS. On the +other hand, focusing on the parameter m, which quanti- +fies the time evolution of the bulk viscosity, we see that +m remains non-zero at several standard deviations for +CMB+Pantheon, where the 68% CL constraint on m +is m = −0.557+0.068 +−0.059, and becomes m = −0.519+0.038 +−0.035 +for CMB+Pantheon+GWSS. From the constraints on +m, one can clearly see that the uncertainty in m is re- +duced by a factor of ∼ 1.7 − 1.8 when the GWSS data +are included with the combined dataset CMB+Pantheon. +Concerning the Hubble constant, we find that H0 as- +sumes slightly higher values compared to the ΛCDM +based Planck. Actually, we have H0 = 68.1+1.2 +−1.1 at 68% +CL for CMB+Pantheon, while H0 = 68.30+0.46 +−0.45 at 68% +CL for CMB+Pantheon+GWSS, again improving the +uncertainty in H0 by a factor of 2.5. This shows that the +effects of GWSS are clearly visible through these param- +eters. In Fig. 1, one can compare the constraints on the +model parameters obtained from CMB+Pantheon and +CMB+Pantheon+GWSS. +2 It is worthwhile to note here that the mean value of β is not +significantly changed when the GWSS data are included, because +we built the simulated data using the best fit obtained from +CMB+Pantheon. + +6 +Parameters +CMB+Pantheon +CMB+Pantheon+GWSS +Ωbh2 +0.02232+0.00015+0.00029 +−0.00015−0.00028 +0.02253+0.00014+0.00028 +−0.00014−0.00026 +100θMC +1.02780+0.00058+0.0011 +−0.00055−0.0011 +1.02808+0.00037+0.00073 +−0.00038−0.00073 +τ +0.0537+0.0074+0.016 +−0.0075−0.015 +0.0567+0.0079+0.016 +−0.0078−0.015 +ns +0.9641+0.0043+0.0086 +−0.0043−0.0084 +0.9686+0.0041+0.0080 +−0.0040−0.0080 +ln(1010As) +3.046+0.016+0.031 +−0.015−0.031 +3.048+0.016+0.033 +−0.016−0.033 +β +0.430+0.017+0.033 +−0.016−0.034 +0.4262+0.0079+0.016 +−0.0078−0.015 +m +−0.557+0.068+0.12 +−0.059−0.13 +−0.519+0.038+0.074 +−0.035−0.075 +H0 +68.1+1.2+2.2 +−1.1−2.3 +68.30+0.46+0.91 +−0.45−0.85 +TABLE II. We report the observational constraints on the BVF1 scenario at 68% and 95% CL for CMB+Pantheon and +CMB+Pantheon+GWSS datasets. +64 +66 +68 +70 +72 +H0 +−0.8 +−0.7 +−0.6 +−0.5 +−0.4 +m +0.022 +0.0228 +Ωbh 2 +0.36 +0.39 +0.42 +0.45 +0.48 +β +64 +66 +68 +70 +72 +H0 +0.8 +0.7 +0.6 +0.5 +0.4 +m +0.0220 +0.0228 +Ωbh 2 +BVF1: CMB+Pantheon +BVF1: CMB+Pantheon+GWSS +FIG. 1. +For the BVF1 scenario we show the 1-dimensional posterior distribution of some model parameters and the 2- +dimensional joint contours of the model parameters at 68% and 95% CL for CMB+Pantheon and CMB+Pantheon+GWSS. +Finally, through Fig. 2 we examine how the model af- +fects the CMB TT power spectrum for different values +of the model parameters, β and m with respect to the +standard ΛCDM scenario. In the upper panel of Fig. 2 +we depict the evolution in the CMB TT power spectrum +for different values of β while in the lower panel of Fig. 2 +we depict the evolution in the CMB TT power spectrum +for different values of m. From both the graphs, we no- +tice that as long as β or m increases, the model exhibits +significant differences in the lower multipoles (ℓ ≤ 10). +For ℓ ≥ 10, we observe that with the increasing values +of β and m, the peaks of the CMB TT power spectrum +increase significantly, particularly changing their mutual +ratio. + +7 +Parameters +CMB+Pantheon +CMB+Pantheon+GWSS +Ωbh2 +0.02241+0.00016+0.00032 +−0.00016−0.00032 +0.02238+0.00015+0.00030 +−0.00016−0.00031 +100θMC +1.02907+0.00111+0.00180 +−0.00082−0.00198 +1.02921+0.00041+0.00080 +−0.00040−0.00079 +τ +0.0516+0.0074+0.015 +−0.0072−0.015 +0.0521+0.0071+0.015 +−0.0078−0.014 +ns +0.9575+0.0053+0.012 +−0.0066−0.012 +0.9583+0.0038+0.0075 +−0.0039−0.0077 +ln(1010As) +3.038+0.016+0.032 +−0.017−0.032 +3.040+0.015+0.032 +−0.015−0.031 +β +0.447+0.022+0.042 +−0.022−0.042 +0.425+0.018+0.032 +−0.016−0.034 +m +−0.85+0.30+0.46 +−0.19−0.50 +−0.683+0.099+0.18 +−0.089−0.19 +γ +0.9970+0.0015+0.0042 +−0.0024−0.0036 +0.99757+0.00049+0.0011 +−0.00058−0.0011 +H0 +65.2+1.7+4.4 +−2.6−3.9 +64.91+0.59+1.1 +−0.60−1.2 +TABLE III. We report the observational constraints on the BVF2 scenario at 68% and 95% CL for CMB+Pantheon and +CMB+Pantheon+GWSS. +101 +102 +103 +l +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +l(l + 1)C TT +l /(2π)[µK 2] +ΛCDM +BVF1 : β = 0.1 +BVF1 : β = 0.3 +BVF1 : β = 0.5 +101 +102 +103 +l +0 +2000 +4000 +6000 +8000 +10000 +12000 +l(l + 1)C TT +l /(2π)[µK 2] +ΛCDM +BVF1 : m = − 1.5 +BVF1 : m = − 0.5 +BVF1 : m = 0.5 +FIG. 2. +The CMB CT T +l +power spectrum versus multipole +moment l using the best fits values obtained for the BVF1 +model using the join data sets described, with three arbitrary +β and m values. +B. +Constraints on the BVF2 scenario +In Table III we present the constraints on the +BVF2 +scenario +for +both +CMB+Pantheon +and +CMB+Pantheon+GWSS. +And +in +Fig. +3, +we +com- +pare the constraints from these datasets explicitly +showing the one dimensional marginalized distribution +of some model parameters and the two dimensional joint +contours. As already discussed, this scenario has three +main key parameters, namely, β, which quantifies the +existence of bulk viscosity in the cosmic sector, m, which +tells us whether the bulk viscosity enjoys a dynamical +nature (corresponding to m ̸= 0) or not, and finally, the +parameter γ, which indicates the fluid which endows +the bulk viscosity. +We note that γ = 1 refers to the +dust fluid endowing the bulk viscosity in which we are +interested in, for which we recover the previous scenario +BVF1. +For CMB+Pantheon, we find that β ̸= 0 at several +standard deviations yielding β = 0.447 ± 0.022 at 68% +CL which gives a clear indication of a non-zero bulk +viscosity in the cosmic sector. +When the GWSS are +added to this combination, i.e. CMB+Pantheon+GWSS, +the conclusion about β does not change significantly +(β = 0.425+0.018 +−0.016 at 68% CL), indicating that for this +scenario GWSS do not provide any additional constrain- +ing power on β. Looking at the dynamical nature of the +bulk viscosity, we see that for CMB+Pantheon, m re- +mains nonzero at more than 2 standard deviations lead- +ing to m = −0.85+0.30 +−0.19 at 68% CL. However, this evi- +dence could be further strengthened by the inclusion of +the GWSS data, that we forecast to be m = −0.683+0.099 +−0.089 +at 68% CL for CMB+Pantheon+GWSS, improving the +error bars up to a factor of 3. Finally, focusing on the pa- +rameter γ which directly connects with the nature of the +cosmic fluid endowing the bulk viscosity, we can see that +it is consistent with 1, which corresponds to a dust-like +fluid, within 2 standard deviations for CMB+Pantheon +(γ = 0.9970+0.0015 +−0.0024 at 68% CL). Also for this param- +eter, the addition of the GWSS further improves the +constraining power of a factor larger than 3 to 4, that +we forecast to be γ = 0.99757+0.00049 +−0.00058 at 68% CL for +CMB+Pantheon+GWSS. Therefore, with respect to the +BVF1 case, where the inclusion of the forecasted GWSS +was able to improve both β and m, in this BVF2 scenario, +the improvement of the constraining power is displayed +only on m and γ but does not affect anymore β signifi- + +8 +60 +64 +68 +72 +H0 +−1.6 +−1.2 +−0.8 +−0.4 +m +0.993 +0.999 +1.005 +γ +0.022 +0.0228 +Ωbh 2 +0.36 0.40 0.44 0.48 0.52 +β +60 +64 +68 +72 +H0 +1.6 +1.2 +0.8 +0.4 +m +0.993 +0.999 +1.005 +γ +0.0220 +0.0228 +Ωbh 2 +BVF2: CMB+Pantheon +BVF2: CMB+Pantheon+GWSS +FIG. 3. +For the BVF2 scenario we show the 1-dimensional posterior distributions of some model parameters and the 2- +dimensional joint contours of the model parameters at 68% and 95% CL for CMB+Pantheon and CMB+Pantheon+GWSS. +cantly. +Furthermore, we find that for this scenario, the Hub- +ble constant attains a very low value for CMB+Pantheon +compared to Planck’s estimation within the ΛCDM +paradigm. We also note that H0 is correlated to all three +free parameters of this scenario, namely, β, m and γ. +With β and γ, H0 is positively correlated while with m, +it has a strong anti-correlation. +For CMB+Pantheon, +H0 = 65.2+1.7 +−2.6 km/s/Mpc at 68% CL and after the in- +clusion of GWSS it becomes H0 = 64.91+0.59 +−0.60 km/s/Mpc +at 68% CL, reducing the uncertainty in H0 by a factor +of ∼ 4. +Finally, in Fig. 4, we examine the CMB TT power +spectrum for this bulk viscous scenario BVF2 consider- +ing different values of the free parameter γ with respect +to the standard ΛCDM scenario. As γ lies in the region +[−3, 3] and the nature of the cosmic fluid characterized by +its equation of state pu = (γ−1)ρu depends on the sign of +γ, therefore, we have considered two separate plots, one +where γ is non-negative (i.e. γ ≥ 0) and another plot +where γ allows both positive and negative values includ- +ing γ = 0. From both the panels of Fig. 4, we clearly see +that any deviation from γ = 1 makes significant changes +in the amplitude of the CMB TT power spectrum. In +particular, we see that the peaks of the CMB TT spec- +trum significantly increases and shift towards higher mul- +tipoles for any value different from γ = 1 at small scales, +as well as the Integrated Sachs Wolfe (ISW) plateau at +large scales. As γ = 1 indicates a cosmological constant- +like fluid endowed with the bulk viscosity, therefore, for +γ = 1, we replicate an equivalent behaviour of the ΛCDM +scenario. +V. +CONCLUSIONS +Although the ΛCDM cosmological model is extremely +successful in describing a large span of astronomical ob- +servations, it cannot explain several theoretical and ob- +servational issues. +This motivated the scientific com- +munity to construct a variety of cosmological proposals +and testing them with the available astronomical data. +Among these cosmological models, in this article, we fo- +cus on the unified cosmological models allowing bulk vis- + +9 +101 +102 +103 +l +0 +2000 +4000 +6000 +8000 +10000 +12000 +l(l + 1)C TT +l /(2π)[µK 2] +ΛCDM +BVF2 : γ = − 1 +BVF2 : γ = 0 +BVF2 : γ = 1 +101 +102 +103 +l +0 +2000 +4000 +6000 +8000 +10000 +12000 +l(l + 1)C TT +l /(2π)[µK 2] +ΛCDM +BVF2 : γ = 0 +BVF2 : γ = 0.5 +BVF2 : γ = 1 +FIG. 4. +The CMB CT T +l +power spectrum versus multipole +moment l using the best fits values obtained for each BVF2 +models with three γ values, respectively using the join data +sets described. +cosity in the background. However, since these models +do not recover the ΛCDM scenario as a special case, our +only ability in distinguishing them, once the GWSS data +will be available, will rely only on a Bayesian model com- +parison for a better fit of the cosmological observations, +as done in Ref. [64]. +The unified cosmological scenar- +ios endowed with bulk viscosity are appealing from two +different perspectives: the first one is the concept of a +unified picture of dark matter and dark energy, and the +second is the inclusion of bulk viscosity into that unified +picture. Effectively, the unified bulk viscous scenario is +a generalized cosmic picture combining two distinct cos- +mological directions. According to a recent paper on the +unified bulk viscous scenarios [64], current cosmological +probes prefer a non-zero dynamical bulk viscosity in the +dark sector at many standard deviations. So, in light of +the current cosmological probes, unified bulk viscous cos- +mological scenarios are attractive. In this line of thought, +what about the future of such unified bulk viscous sce- +narios? +In this article we have focused on it with an +answer. +Following Ref. [64], in this work we have explored +these scenarios with the GWSS aiming to understand +how the future distance measurements from GWSS may +improve the constraints on such scenarios. In order to +proceed toward this confrontation, we have generated +O(103) mock GWSS luminosity distance measurements +matching the expected sensitivity of the Einstein Tele- +scope and added these mock data to the current cosmo- +logical probes, namely CMB from Planck 2018 release3 +and SNIa Pantheon sample. We find that the inclusion of +GWSS luminosity distance measurements together with +the current cosmological probes makes the possible fu- +ture evidence for new physics stronger, by reducing the +uncertainty in the parameters in a significant way. This +is a potential behaviour of the GWSS luminosity distance +measurements since this makes the parameter much de- +terministic. Overall for both BVF1 and BVF2 scenar- +ios, we find a very strong preference of a non-zero time +dependent bulk viscous coefficient (alternatively, the vis- +cous nature of the unified dark fluid) at many standard +deviations. +In conclusion, in the present paper we demonstrate +that future GWSS distance measurements from the Ein- +stein Telescope might be powerful to extract more infor- +mation about the physics of the dark sector. Therefore, +based on the present results, we feel that it might just +be a matter of time before we convincingly detect the +viscosity in the dark sector, if any. +VI. +ACKNOWLEDGMENTS +The authors thank the referee for some important com- +ments which helped us to improve the article consider- +ably. WY was supported by the National Natural Science +Foundation of China under Grants No. 12175096 and No. +11705079, and Liaoning Revitalization Talents Program +under Grant no. XLYC1907098. SP acknowledges the +financial support from the Department of Science and +Technology (DST), Govt. +of India, under the Scheme +“Fund for Improvement of S&T Infrastructure (FIST)” +[File No. +SR/FST/MS-I/2019/41]. +EDV is supported +by a Royal Society Dorothy Hodgkin Research Fellow- +ship. CE-R is supported by the Royal Astronomical So- +ciety as FRAS 10147 and by PAPIIT UNAM Project +TA100122. This article/publication is based upon work +from COST Action CA21136 Addressing observational +tensions in cosmology with systematics and fundamen- +tal physics (CosmoVerse) supported by COST (European +Cooperation in Science and Technology). AP was sup- +ported in part by the National Research Foundation of +South Africa (Grant Number 131604). 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Gelman and D. B. Rubin, Statist. Sci. 7, 457 (1992). + diff --git a/3tE2T4oBgHgl3EQfjgeA/content/tmp_files/load_file.txt b/3tE2T4oBgHgl3EQfjgeA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9fff1d3533e9021865c7f1445f9f6c159fe28e1c --- /dev/null +++ b/3tE2T4oBgHgl3EQfjgeA/content/tmp_files/load_file.txt @@ -0,0 +1,1488 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf,len=1487 +page_content='Exploring bulk viscous unified scenarios with Gravitational Waves Standard Sirens Weiqiang Yang,1, ∗ Supriya Pan,2, 3, † Eleonora Di Valentino,4, ‡ Celia Escamilla-Rivera,5, § and Andronikos Paliathanasis3, 6, 7, ¶ 1Department of Physics, Liaoning Normal University, Dalian, 116029, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' China 2Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India 3Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of South Africa 4School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom 5Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 70-543, M´exico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 04510, M´exico 6Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile, Valdivia 5090000, Chile 7Mathematical Physics and Computational Statistics Research Laboratory, Department of Environment, Ionian University, Zakinthos 29100, Greece We consider the unified bulk viscous scenarios and constrain them using the Cosmic Microwave Background observations from Planck 2018 and the Pantheon sample from Type Ia Supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Then we generate the luminosity distance measurements from O(103) mock Gravitational Wave Standard Sirens (GWSS) events for the proposed Einstein Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We then combine these mock luminosity distance measurements from the GWSS with the current cosmological probes in order to forecast how the mock GWSS data could be effective in constraining these bulk viscous scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Our results show that a non-zero time dependent bulk viscosity in the universe sector is strongly preferred by the current cosmological probes and will possibly be confirmed at many standard deviations by the future GWSS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We further mention that the addition of GWSS data can significantly reduce the uncertainties of the key cosmological parameters obtained from the usual cosmological probes employed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' INTRODUCTION Understanding the nature of dark matter and dark en- ergy has been a challenge for cosmologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The standard cosmological model, namely, the so-called Λ-Cold Dark Matter (ΛCDM) model representing a mixture of two non-interacting fluids − a positive cosmological constant (Λ > 0) and a cold dark matter component, has un- doubtedly proved its unprecedented success by explain- ing a large span of astronomical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' However, this simplest cosmological scenario has some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For example, the cosmological constant problem [1] and the coincidence problem [2] have already questioned the ex- isting assumptions in the ΛCDM model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' constant energy density of the vacuum and the non-interacting na- ture between Λ and CDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' These limitations motivated the cosmologists to find alternative cosmological scenar- ios beyond ΛCDM by relaxing the above assumptions, and as a consequence, several new cosmological models were introduced, see [3–12] for a review of various dark energy and modified gravity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Additionally, the appearance of cosmological tensions at many standard deviations between Planck [13] (assuming ΛCDM in the background) and other cosmological probes, such as dis- tance ladders [14–24] or weak lensing [25–29] and galaxy ∗ d11102004@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='com † supriya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='maths@presiuniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='in ‡ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='divalentino@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='uk § celia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='escamilla@nucleares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='mx ¶ anpaliat@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='gr cluster data [30–32] has further weakened the confidence in the ΛCDM cosmological model [33–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Thus, the list of cosmological models aiming to address the cosmolog- ical tensions is increasing in time, see the review arti- cles [38–44] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Given the fact that the origin of dark matter and dark energy is not clearly understood yet, thus, there is no reason to favor any par- ticular cosmological theory over others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As a result, var- ious ways have been proposed to interpret the dynamics of the dark sector in terms of dark matter and dark en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The simplest assumption is the consideration of independent evolution of these dark fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The general- ization of the above consideration is the assumption of a non-gravitational interaction between these dark sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' On the other hand, a heuristic approach is to consider a unified dark fluid that can explain the dynamics of dark energy and dark matter at cosmological scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The at- tempt to unify the dark sector of the Universe began long back ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The most simplest unified dark sector mod- els can be constructed in the context of Einstein gravity with the introduction of a generalized equation of state p = F(ρ), where p and ρ are respectively the pressure and energy density of the unified dark sector and F is an an- alytic function of the energy density, ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The well known unified cosmological models, such as the Chaplygin gas model [45] and its successive generalizations, namely, the generalized Chaplygin gas, modified Chaplygin gas, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [46–57] and some other unified cosmological scenar- ios as well [58–60] belong to this classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' While it is essential to mention that a subset of the unified mod- els has been diagnosed with exponential blowup in the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='03969v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='CO] 10 Jan 2023 2 matter power spectrum which is not consistent with the observations [61], however, this does not rule out the pos- sibility of unified models aiming to cover a wide region of the universe evolution because a new kind of unified fluid may avoid such unphysical activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The unified cosmological models can also be developed by considering a relation like p = G(H) where G is an analytic function of H, the Hubble function of the Friedmann-Lemaˆıtre- Robertson-Walker (FLRW) line element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Apparently, theories with p = F(ρ) and p = G(H) seem identical, however, this is only true in spatially flat FLRW uni- verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For a curved universe, the two approaches are not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In the present work we are interested to study a partic- ular class of unified models endowed with bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The cosmological fluids allowing bulk viscosity as an ex- tra ingredient can explain the accelerating expansion of the universe, and hence they are also enlisted as possi- ble alternatives to the standard ΛCDM cosmology in the literature [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Following an earlier work Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [64] where an evidence of non-zero bulk viscosity was pre- ferred by the current cosmological probes, in the present article, we use the simulated Gravitational Waves Stan- dard Sirens (GWSS) measurements from the Einstein Telescope [65]1 in order to quantify the improvements of the cosmological parameters, if any, from the future GWSS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As the gravitational waves (GW) have opened a new window for astrophysics and cosmol- ogy, therefore, it will be interesting to investigate the contribution from the simulated GWSS data, once com- bined with the current cosmological probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' This moti- vated many investigators to use the mock GWSS data matching the expected sensitivity of the Einstein Tele- scope to constrain a class of cosmological models, see for instance, [66–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' the combined analysis of simulated GWSS measurements from Einstein Telescope and the standard cosmological probes has proven to be very effective for a class of cosmological models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' in the sense that the error bars in the key cosmological param- eters of these cosmological models are significantly re- duced thanks to the mock GWSS dataset [70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 74,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 78– 81],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' in some specific f(R) theories of gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' the generated mock GWSS from the Einstein Telescope may not be very much helpful to give stringent constraints on them during its first phase of running [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Thus, one may expect that the constraining power of the Einstein Telescope may depend on the underlying cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Aside from the future GWSS measurements from the Einstein Telescope, one can also use the simulated GWSS measurements from other GW observatories, such as, Laser Interferometer Space Antenna (LISA) [83–86] and DECi-heltz Interferometer Gravitational wave Ob- servatory (DECIGO) [87, 88], TianQin [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In this ar- ticle, we focus only on the simulated GWSS data from 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='einsteintelescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='nl/en/ Einstein Telescope to constrain the bulk viscous unified scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The paper has been organized as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' II we discuss the gravitational equations for the bulk viscous scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' III describes the observational data that we have considered for the analysis in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' IV presents the observational constraints on the bulk vis- cous models, and mainly we discuss how the inclusion of gravitational waves data from the Einstein Telescope improves the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' V we present the conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' REVISITING THE BULK VISCOUS SCENARIOS: BACKGROUND AND PERTURBATIONS As usual, we consider the homogeneous and isotropic space time described by the Friedmann-Lemaˆıtre- Robertson-Walker (FLRW) line element ds2 = −dt2 + a2(t) � dr2 1 − kr2 + r2 � dθ2 + sin2 θdφ2�� , (1) where a(t) is the expansion scale factor and k denotes the spatial curvature of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For k = 0, −1, +1, we have three different geometries of the universe, namely, spatially flat, open and closed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In this paper we restrict ourselves to the spatially flat scenario where we assume that (i) the gravitational sector is described by the Einstein’s gravity, (ii) the matter sector of the uni- verse consists of the relativistic radiation, non-relativistic baryons and a unified bulk viscous fluid which combines the effects of dark matter and dark energy, (iii) all the fluids are non-interacting with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Within this framework, we can write down the gravitational field equations as follows (in the units where 8πG = 1) H2 = 1 3ρtot, (2) 2 ˙H + 3H2 = − ptot, (3) where an overhead dot indicates the derivative with re- spect to the cosmic time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' H ≡ ˙a/a is the Hubble ex- pansion rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' (ρtot, ptot) = (ρr +ρb +ρu, pr +pb +pu) are the total energy density and total pressure of the cosmic components in which (ρr, pr), (ρb, pb), (ρu, pu) are the en- ergy density and pressure of radiation, baryons and the unified fluid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The conservation equation for each fluid follows the usual law ˙ρi + 3H(1 + wi)ρi = 0, where the subscript i refers to radiation (i = r), baryons (i = b) and the unified fluid (i = u) and wi are the stan- dard barotropic state parameters: wr = pr/ρr = 1/3, wb = pb/ρb = 0 and wu = pu/ρu = (γ − 1), where γ is a constant parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In general for different val- ues of γ, say for instance, γ = 0, we realize a cosmo- logical constant-like fluid endowed with the bulk viscos- ity and similarly γ = 1 results in a dust-like fluid en- dowed with the bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As the nature of the 3 fluid is not clearly understood and as the observational data play an effective role to understand this nature, thus, in order to be more transparent in this direction we consider γ lying in the interval [−3, 3] which includes both exotic (pu/ρu = (γ − 1) < −1/3) and non-exotic (pu/ρu = (γ − 1) > −1/3 ) fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As already mentioned, since the unified fluid has a bulk viscosity, therefore, it en- joys an effective pressure [90]: peff = pu−uν ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='νη(ρu), where uµ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='µ is the expansion scalar of this fluid and η(ρu) > 0 is the coefficient of the bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Thus, in the FLRW background, the effective pressure of the bulk viscous fluid reduces to peff = pu − 3Hη(ρu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' (4) Since there is no unique selection for the bulk viscous coefficient, η(ρu), therefore, we consider a well known choice for it in which the bulk viscous coefficient has a power law evolution of the form [90–92]: η(ρu) = αρm u , (5) where α is a positive constant and m is any real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Notice that for the case m = 0 we recover the scenario with a constant bulk viscous coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Now, with the consideration of the bulk viscous coefficient in (5), the effective pressure of the unified fluid can be expressed as peff = (γ − 1)ρu − √ 3αρ1/2 tot ρm u , (6) and consequently, one can define the effective equation of state of the viscous dark fluid as weff = peff ρu = (γ − 1) − √ 3αρ1/2 tot ρm−1 u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' (7) The adiabatic sound speed for the viscous fluid is given by c2 a,eff = p′ eff ρ′u = weff + w′ eff 3H(1 + weff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' (8) where the prime denotes the derivative with respect to the conformal time τ and H is the conformal Hubble pa- rameter, H = aH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Note that depending on the nature of weff, c2 a,eff could be negative, and hence ca,eff could be an imaginary quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' This may invite instabilities in the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Thus, in order to avoid this possible un- physical situation, we consider the entropy perturbations (non-adiabatic perturbations) in the unified dark fluid following the analysis of generalized dark matter [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Now we focus on the evolution of the unified bulk vis- cous fluid at the level of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In the entropy perturbation mode, the true pressure perturbation comes from the effective pressure given by δpeff = δpu − δη(∇σuσ) − η(δ∇σuσ) = δpu − 3Hδη − η a � θ + h′ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' (9) The effective sound speed of viscous dark fluid for the bulk viscous coefficient (5) can be defined as c2 s,eff ≡ �δpeff δρu � rf = c2 s − √ 3αmρ1/2 tot ρm−1 u − αρm−1 u aδu � θ + h′ 2 � , (10) where ’|rf’ denotes the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Following the analysis in [93], the sound speed in the rest frame is assumed to be zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' c2 s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The density perturbation and the velocity perturba- tion can also be written as [93] δ′ u = −(1 + weff)(θu + h′ 2 ) + w′ eff 1 + weff δu − 3H(c2 s,eff − c2 a,eff) � δu + 3H(1 + weff)θD k2 � , (11) θ′ u = −H(1 − 3c2 s,eff)θu + c2 s,eff 1 + weff k2δu, (12) Thus, following the evolution at the background and per- turbation level prescribed above, one can now be able to understand the dynamics of the bulk viscous fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In this work we consider two different bulk viscous scenar- ios characterized as follows: the bulk viscous model 1 (labeled as BVF1) where we consider γ = 1, and the bulk viscous model 2 (labeled as BVF2) where we keep γ as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The common parameters in both BVF1 and BVF2 are α and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' STANDARD COSMOLOGICAL PROBES, SIMULATED GWSS DATA, AND METHODOLOGY In this section we describe the cosmological data sets employed to perform the statistical analyses of the present bulk viscous scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Cosmic Microwave Background (CMB): We use the CMB data from the Planck 2018 data re- lease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Precisely, we use the CMB temperature and polarization angular power spectra plikTT- TEEE+lowl+lowE [94, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Pantheon sample from Type Ia Supernovae (SNIa) data: Type Ia Supernovae are the first astronomical data that probed the accelerating ex- pansion of the universe and hence indicated the existence of an exotic fluid with negative pressure (dark energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Here we use the Pantheon compila- tion of the SNIa data comprising 1048 data points spanned in the redshift interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='01, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='3] [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Gravitational Waves Standard Sirens (GWSS): We take the mock Gravitational Waves Standard Sirens (GWSS) data generated by matching the expected sensitivity of Einstein 4 Telescope in order to understand the constraining power of the future GWSS data from the Einstein Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The Einstein Telescope is a proposed ground based third-generation (3G) GW detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The telescope will take a triangular shape and its each arm length will be increased to 10 km, compared to 3 km arm length VIRGO and 4 km arm length LIGO [65, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Thus, due to such in- creased arm length, the Einstein Telescope will be a potential GW detector by reducing all displace- ment noises [65, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' It is expected that after 10 years of operation, Einstein Telescope will detect O(103) GWSS events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Although the detection of O(103) GWSS events is very optimistic while the number of detections could be low in reality [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As argued in [65], the Einstein Telescope will likely to detect 20 − 50 events per year, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 200 − 500 events in 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' However, following the earlier works [66, 67, 70, 71, 74, 77, 79, 81, 82], in this article, we restrict ourselves to the detection of O(103) GWSS events by the Einstein Telescope to constrain the bulk viscous scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For more features of the Einstein Telescope we refer the readers to [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We originally generate the mock GWSS luminosity distance measurements matching the expected sen- sitivity of the Einstein Telescope after 10 years of full operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Specifically we create 1000 triples (zi, dL(zi), σi) where zi is the redshift of a GW source, dL(zi) is the measured luminosity distance at redshift zi and σi is the uncertainty associated with the luminosity distance dL(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Let us briefly summarize the procedure of generating the mock GWSS dataset and we refer to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [70, 71, 81] for more technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The initial step for generating the mock GWSS dataset is to identify the expected GW sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We consider the GW events originating from two distinct binary systems, namely, (i) a combination of a Black Hole (BH) and a Neutron Star (NS) merger, identified as BHNS and (ii) the binary neutron star (BNS) merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Following the mass distributions as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [81], the ratio of the number of GW events for the BHNS merger versus BNS merger is taken to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='03 as predicted for the Advanced LIGO-VIRGO network [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We then determine the merger rate R(z) of sources and from the merger rate of the sources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' we determine the redshift distribution of the sources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' P(z) given by [66,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 67,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 79,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 99] P(z) ∝ 4πd2 C(z)R(z) H(z)(1 + z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' (13) where dC(z) ≡ � z 0 H−1(z′)dz′ is the co-moving dis- tance and for R(z) we take the following piece- wise linear function estimated in [100] (also see [66,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 67,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 79,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 101]): R(z) = 1 + 2z for z ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' R(z) = 3 4(5 − z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' for 1 < z < 5 and R(z) = 0 for z > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' After having P(z), we sample 1000 values of redshifts from this distribution which represent the redshifts zi of our 1000 mock GWSS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The next step is to choose a fiducial model because while going from the merger rate to the redshift dis- tributions, a fiducial cosmological model is needed since the expression for P(z) includes both the co- moving distance and expansion rate at redshift z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' dL(z) and H(z) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' This H(z) cor- responds to the fiducial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As in this arti- cle we are interested to investigate how the inclu- sion of GWSS data improves the constraints on the BVF models, therefore, we generate two different mock GWSS datasets choosing BVF1 and BVF2 as the fiducial models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We take the fiducial val- ues of the cosmological parameters given by the best fit values of the same cosmological parame- ters of the BVF1 and BVF2 models obtained from the CMB+Pantheon data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Now, for the chosen fiducial model(s), one can now estimate the luminosity distance at the redshift zi using the re- lation dL(zi) = (1 + zi) � zi 0 dz′ H(z′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' (14) Thus, after having the luminosity distance dL(zi) of the GW source, our last job is now to determine the uncertainty σi associated with this luminosity dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The determination of the uncertainty σi di- rectly connects to the waveform of GW because the GW amplitude depends on the luminosity distance (also on the so-called chirp mass [66, 67, 79]) and hence one can extract the information about dL(zi) and σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We refer to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [66, 67, 70, 79, 81] for the technical details to calculate the uncertainties on the luminosity distance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The lumi- nosity distance measurement dL(zi) has two kind of uncertainties, one is the instrumental uncertainty σinst i and the other one is the weak lensing uncer- tainty σlens i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The instrumental error can be derived to be σinst i (≃ 2dL(zi)/S where S is the combined signal-to-noise ratio of the Einstein Telescope) us- ing the Fisher matrix approach and assuming that the uncertainty on dL(zi) is not correlated with the uncertainties on the remaining GW parame- ters (see [66, 67, 70, 79, 81]) and the lensing error is σlens i ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='05zidL(zi) [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Thus, the total uncer- tainty due to the instrumental and the weak lensing uncertainties on dL(zi) is σi = � (σinst i )2 + (σlens i )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Finally, let us note that the combined signal-to- noise ratio of the GW detector is a very crucial quantity in this context since for the Einstein Tele- scope, the combined signal-to-noise ratio should be 5 Parameter Priors (BVF1) Priors (BVF2) Ωbh2 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='1] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='1] τ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='8] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='8] ns [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5] ln(1010As) [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='4, 4] [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='4, 4] 100θMC [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5, 10] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5, 10] β [0, 1] [0, 1] m [−2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5] [−2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5] γ − [−3, 3] TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We show the flat priors on all the free parameters associated with the bulk viscous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' at least 8 for a GW detection [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Thus, in sum- mary, we generate 1000 GW sources up to redshift z = 2 with S > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For more technical details we refer the readers to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [66, 67, 70, 79, 81, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' To constrain the BVF scenarios we modify the pub- licly available CosmoMC package [102] which is an excel- lent cosmological code supporting the Planck 2018 like- lihood [95] and it has a convergence diagnostic follow- ing the Gelman-Rubin statistic R − 1 [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' It is essen- tial to mention that for both BVF1 and BVF2 scenarios, we have used the dimensionless quantity β = αH0ρm−1 tot,0 where ρtot,0 is the present value of ρtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We further men- tion here that β = 0 (equivalently, α = 0) implies no vis- cosity and hence the overall picture behaves like a unified cosmic fluid without bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Thus, in summary, the parameter space of BVF1 and BVF2 are as below: PBVF1 ≡ {Ωbh2, 100θMC, τ, ns, ln(1010As), β, m} PBVF2 = {Ωbh2, 100θMC, τ, ns, ln(1010As), β, m, γ} where the description of the free parameters are as fol- lows: Ωbh2 is the baryons density, 100θMC is the ratio of the sound horizon to the angular diameter distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' τ is the optical depth, ns is the scalar spectral index, As is the amplitude of the initial power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The flat priors on both cosmological scenarios are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' OBSERVATIONAL CONSTRAINTS: RESULTS AND ANALYSIS In this section we present the constraints on the bulk viscous scenarios considering CMB+Pantheon and CMB+Pantheon+GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As we are interested to esti- mate the improvement of the cosmological parameters in presence of the GWSS measurements, and as the com- bined standard cosmological probes offer the most strin- gent constraints on the cosmological parameters, there- fore, the inclusion of GWSS with the combined standard cosmological probes is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As mentioned earlier, the key common free parameters of BVF1 and BVF2 are β and m, since β ̸= 0 indicates the preference for a non- zero bulk viscosity and m ̸= 0 indicates that the coeffi- cient of the bulk viscosity is not constant in the redshift range considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In the following subsections we discuss the constraints on these two scenarios in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Constraints on the BVF1 scenario In Table II we have presented the constraints on the BVF1 scenario for CMB+Pantheon and CMB+Pantheon+GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The latter dataset is aimed to understand the improvement expected from GWSS on the constraints from CMB+Pantheon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 1 we have compared these datasets graphically by showing the one dimensional marginalized distribution of some model pa- rameters and the two dimensional joint contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As discussed, this scenario has two main key parameters, namely, β, quantifying the existence of bulk viscosity in the cosmic sector, and m, which tells us whether the bulk viscosity will have a dynamical nature (corresponding to m ̸= 0) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Since for CMB+Pantheon, we find an evidence for a non-zero bulk viscosity in the cosmic sector at many standard deviations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='430+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='017 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='016 at 68% CL, this is further strengthen for CMB+Pantheon+GWSS, where β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='4262+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0079 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0078 at 68% CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='2 One can clearly see that the inclusion of GWSS to CMB+Pantheon im- proves the error bars on β by a factor of at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' This reflects the constraining power of GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' On the other hand, focusing on the parameter m, which quanti- fies the time evolution of the bulk viscosity, we see that m remains non-zero at several standard deviations for CMB+Pantheon, where the 68% CL constraint on m is m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='557+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='068 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='059, and becomes m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='519+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='038 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='035 for CMB+Pantheon+GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' From the constraints on m, one can clearly see that the uncertainty in m is re- duced by a factor of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='7 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='8 when the GWSS data are included with the combined dataset CMB+Pantheon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Concerning the Hubble constant, we find that H0 as- sumes slightly higher values compared to the ΛCDM based Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Actually, we have H0 = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='1 at 68% CL for CMB+Pantheon, while H0 = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='45 at 68% CL for CMB+Pantheon+GWSS, again improving the uncertainty in H0 by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' This shows that the effects of GWSS are clearly visible through these param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 1, one can compare the constraints on the model parameters obtained from CMB+Pantheon and CMB+Pantheon+GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 2 It is worthwhile to note here that the mean value of β is not significantly changed when the GWSS data are included, because we built the simulated data using the best fit obtained from CMB+Pantheon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 6 Parameters CMB+Pantheon CMB+Pantheon+GWSS Ωbh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='02232+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00015+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00029 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00015−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='02253+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00014+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00028 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00014−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00026 100θMC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='02780+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00058+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0011 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00055−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='02808+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00037+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00073 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00038−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00073 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0537+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0074+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='016 −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0080 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0040−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0080 ln(1010As) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='046+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='016+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='031 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='015−0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='85 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We report the observational constraints on the BVF1 scenario at 68% and 95% CL for CMB+Pantheon and CMB+Pantheon+GWSS datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 64 66 68 70 72 H0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='4 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0228 Ωbh 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='48 β 64 66 68 70 72 H0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='4 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0228 Ωbh 2 BVF1: CMB+Pantheon BVF1: CMB+Pantheon+GWSS FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For the BVF1 scenario we show the 1-dimensional posterior distribution of some model parameters and the 2- dimensional joint contours of the model parameters at 68% and 95% CL for CMB+Pantheon and CMB+Pantheon+GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Finally, through Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 2 we examine how the model af- fects the CMB TT power spectrum for different values of the model parameters, β and m with respect to the standard ΛCDM scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 2 we depict the evolution in the CMB TT power spectrum for different values of β while in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 2 we depict the evolution in the CMB TT power spectrum for different values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' From both the graphs, we no- tice that as long as β or m increases, the model exhibits significant differences in the lower multipoles (ℓ ≤ 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For ℓ ≥ 10, we observe that with the increasing values of β and m, the peaks of the CMB TT power spectrum increase significantly, particularly changing their mutual ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 7 Parameters CMB+Pantheon CMB+Pantheon+GWSS Ωbh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='02241+0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='017−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='032 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='040+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='015+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='032 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='015−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='031 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='447+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='022+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='042 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='022−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='425+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='018+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='032 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='016−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='034 m −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='19−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='683+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='099+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='089−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='19 γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='9970+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0015+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0042 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0024−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='99757+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00049+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0011 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00058−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0011 H0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='7+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='4 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='6−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='59+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='60−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='2 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We report the observational constraints on the BVF2 scenario at 68% and 95% CL for CMB+Pantheon and CMB+Pantheon+GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 101 102 103 l 0 1000 2000 3000 4000 5000 6000 7000 l(l + 1)C TT l /(2π)[µK 2] ΛCDM BVF1 : β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='1 BVF1 : β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='3 BVF1 : β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5 101 102 103 l 0 2000 4000 6000 8000 10000 12000 l(l + 1)C TT l /(2π)[µK 2] ΛCDM BVF1 : m = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5 BVF1 : m = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5 BVF1 : m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The CMB CT T l power spectrum versus multipole moment l using the best fits values obtained for the BVF1 model using the join data sets described, with three arbitrary β and m values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Constraints on the BVF2 scenario In Table III we present the constraints on the BVF2 scenario for both CMB+Pantheon and CMB+Pantheon+GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' And in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 3, we com- pare the constraints from these datasets explicitly showing the one dimensional marginalized distribution of some model parameters and the two dimensional joint contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As already discussed, this scenario has three main key parameters, namely, β, which quantifies the existence of bulk viscosity in the cosmic sector, m, which tells us whether the bulk viscosity enjoys a dynamical nature (corresponding to m ̸= 0) or not, and finally, the parameter γ, which indicates the fluid which endows the bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We note that γ = 1 refers to the dust fluid endowing the bulk viscosity in which we are interested in, for which we recover the previous scenario BVF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For CMB+Pantheon, we find that β ̸= 0 at several standard deviations yielding β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='447 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='022 at 68% CL which gives a clear indication of a non-zero bulk viscosity in the cosmic sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' When the GWSS are added to this combination, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' CMB+Pantheon+GWSS, the conclusion about β does not change significantly (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='425+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='018 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='016 at 68% CL), indicating that for this scenario GWSS do not provide any additional constrain- ing power on β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Looking at the dynamical nature of the bulk viscosity, we see that for CMB+Pantheon, m re- mains nonzero at more than 2 standard deviations lead- ing to m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='19 at 68% CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' However, this evi- dence could be further strengthened by the inclusion of the GWSS data, that we forecast to be m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='683+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='099 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='089 at 68% CL for CMB+Pantheon+GWSS, improving the error bars up to a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Finally, focusing on the pa- rameter γ which directly connects with the nature of the cosmic fluid endowing the bulk viscosity, we can see that it is consistent with 1, which corresponds to a dust-like fluid, within 2 standard deviations for CMB+Pantheon (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='9970+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0024 at 68% CL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Also for this param- eter, the addition of the GWSS further improves the constraining power of a factor larger than 3 to 4, that we forecast to be γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='99757+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00049 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00058 at 68% CL for CMB+Pantheon+GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Therefore, with respect to the BVF1 case, where the inclusion of the forecasted GWSS was able to improve both β and m, in this BVF2 scenario, the improvement of the constraining power is displayed only on m and γ but does not affect anymore β signifi- 8 60 64 68 72 H0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='4 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='005 γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0228 Ωbh 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='52 β 60 64 68 72 H0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='4 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='005 γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='0228 Ωbh 2 BVF2: CMB+Pantheon BVF2: CMB+Pantheon+GWSS FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For the BVF2 scenario we show the 1-dimensional posterior distributions of some model parameters and the 2- dimensional joint contours of the model parameters at 68% and 95% CL for CMB+Pantheon and CMB+Pantheon+GWSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' cantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Furthermore, we find that for this scenario, the Hub- ble constant attains a very low value for CMB+Pantheon compared to Planck’s estimation within the ΛCDM paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We also note that H0 is correlated to all three free parameters of this scenario, namely, β, m and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' With β and γ, H0 is positively correlated while with m, it has a strong anti-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' For CMB+Pantheon, H0 = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='7 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='6 km/s/Mpc at 68% CL and after the in- clusion of GWSS it becomes H0 = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='60 km/s/Mpc at 68% CL, reducing the uncertainty in H0 by a factor of ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 4, we examine the CMB TT power spectrum for this bulk viscous scenario BVF2 consider- ing different values of the free parameter γ with respect to the standard ΛCDM scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As γ lies in the region [−3, 3] and the nature of the cosmic fluid characterized by its equation of state pu = (γ−1)ρu depends on the sign of γ, therefore, we have considered two separate plots, one where γ is non-negative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' γ ≥ 0) and another plot where γ allows both positive and negative values includ- ing γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' From both the panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 4, we clearly see that any deviation from γ = 1 makes significant changes in the amplitude of the CMB TT power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In particular, we see that the peaks of the CMB TT spec- trum significantly increases and shift towards higher mul- tipoles for any value different from γ = 1 at small scales, as well as the Integrated Sachs Wolfe (ISW) plateau at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' As γ = 1 indicates a cosmological constant- like fluid endowed with the bulk viscosity, therefore, for γ = 1, we replicate an equivalent behaviour of the ΛCDM scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' CONCLUSIONS Although the ΛCDM cosmological model is extremely successful in describing a large span of astronomical ob- servations, it cannot explain several theoretical and ob- servational issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' This motivated the scientific com- munity to construct a variety of cosmological proposals and testing them with the available astronomical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Among these cosmological models, in this article, we fo- cus on the unified cosmological models allowing bulk vis- 9 101 102 103 l 0 2000 4000 6000 8000 10000 12000 l(l + 1)C TT l /(2π)[µK 2] ΛCDM BVF2 : γ = − 1 BVF2 : γ = 0 BVF2 : γ = 1 101 102 103 l 0 2000 4000 6000 8000 10000 12000 l(l + 1)C TT l /(2π)[µK 2] ΛCDM BVF2 : γ = 0 BVF2 : γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='5 BVF2 : γ = 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The CMB CT T l power spectrum versus multipole moment l using the best fits values obtained for each BVF2 models with three γ values, respectively using the join data sets described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' cosity in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' However, since these models do not recover the ΛCDM scenario as a special case, our only ability in distinguishing them, once the GWSS data will be available, will rely only on a Bayesian model com- parison for a better fit of the cosmological observations, as done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' The unified cosmological scenar- ios endowed with bulk viscosity are appealing from two different perspectives: the first one is the concept of a unified picture of dark matter and dark energy, and the second is the inclusion of bulk viscosity into that unified picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Effectively, the unified bulk viscous scenario is a generalized cosmic picture combining two distinct cos- mological directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' According to a recent paper on the unified bulk viscous scenarios [64], current cosmological probes prefer a non-zero dynamical bulk viscosity in the dark sector at many standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' So, in light of the current cosmological probes, unified bulk viscous cos- mological scenarios are attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In this line of thought, what about the future of such unified bulk viscous sce- narios?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In this article we have focused on it with an answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [64], in this work we have explored these scenarios with the GWSS aiming to understand how the future distance measurements from GWSS may improve the constraints on such scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In order to proceed toward this confrontation, we have generated O(103) mock GWSS luminosity distance measurements matching the expected sensitivity of the Einstein Tele- scope and added these mock data to the current cosmo- logical probes, namely CMB from Planck 2018 release3 and SNIa Pantheon sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' We find that the inclusion of GWSS luminosity distance measurements together with the current cosmological probes makes the possible fu- ture evidence for new physics stronger, by reducing the uncertainty in the parameters in a significant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' This is a potential behaviour of the GWSS luminosity distance measurements since this makes the parameter much de- terministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Overall for both BVF1 and BVF2 scenar- ios, we find a very strong preference of a non-zero time dependent bulk viscous coefficient (alternatively, the vis- cous nature of the unified dark fluid) at many standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' In conclusion, in the present paper we demonstrate that future GWSS distance measurements from the Ein- stein Telescope might be powerful to extract more infor- mation about the physics of the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Therefore, based on the present results, we feel that it might just be a matter of time before we convincingly detect the viscosity in the dark sector, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank the referee for some important com- ments which helped us to improve the article consider- ably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' WY was supported by the National Natural Science Foundation of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 12175096 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 11705079, and Liaoning Revitalization Talents Program under Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' XLYC1907098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' SP acknowledges the financial support from the Department of Science and Technology (DST), Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' of India, under the Scheme “Fund for Improvement of S&T Infrastructure (FIST)” [File No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' SR/FST/MS-I/2019/41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' EDV is supported by a Royal Society Dorothy Hodgkin Research Fellow- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' CE-R is supported by the Royal Astronomical So- ciety as FRAS 10147 and by PAPIIT UNAM Project TA100122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' This article/publication is based upon work from COST Action CA21136 Addressing observational tensions in cosmology with systematics and fundamen- tal physics (CosmoVerse) supported by COST (European Cooperation in Science and Technology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' AP was sup- ported in part by the National Research Foundation of South Africa (Grant Number 131604).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Also AP thanks the support of Vicerrector´ıa de Investigaci´on y Desarrollo Tecnol´ogico (Vridt) at Universidad Cat´olica del Norte 3 We mention that in the earlier work [64], CMB data from Planck 2015 were used to 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 342, 155 (2012), arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='3421 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Cai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Capozziello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} 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Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 494, 6072 (2020), arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='06306 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 490, 2071 (2019), arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='10436 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [67] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Cai and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Yang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' D 95, 044024 (2017), arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' D 95, 124008 (2017), arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='09853 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [69] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Cai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Wang, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Yang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' D 97, 103005 (2018), arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00952 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [70] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Pan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Di Valentino, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Wang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Wang, JCAP 05, 050 (2020), arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='11980 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [71] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Vagnozzi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Di Valentino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Nunes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Pan, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Mota, JCAP 07, 037 (2019), arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='08286 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [72] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Bachega, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Costa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Abdalla, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Fornazier, JCAP 05, 021 (2020), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='08909 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [73] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Fu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Peng, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' D 100, 123539 (2019), arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='02327 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [74] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Pan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Mota, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Du, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 497, 879 (2020), arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='02180 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [75] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Mitra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Mifsud, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Mota, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Parkin- son, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 502, 5563 (2021), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='00189 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [76] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Escamilla-Rivera and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Quevedo, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 38, 115009 (2021), arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='08784 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [77] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Pan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' He, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Qi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Cao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Liu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Wang, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' 911, 135 (2021), arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='05212 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' [78] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfjgeA/content/2301.03969v1.pdf'} 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Inspired by video representation +learning, these methods mainly focus on designing mod- +ules to extract informative spatial and temporal fea- +tures. However, they are still limited in extracting local +attributes and global identity information, which are +critical for the person re-identification task. In this paper, +we propose a novel Multi-Stage Spatial-Temporal Aggre- +gation Transformer (MSTAT) with two novel designed +proxy embedding modules to address the above issue. +Specifically, MSTAT consists of three stages to encode +the attribute-associated, the identity-associated, and the +attribute-identity-associated information from the video +clips, respectively, achieving the holistic perception of +the input person. We combine the outputs of all the +stages for the final identification. In practice, to save the +computational cost, the Spatial-Temporal Aggregation +(STA) modules are first adopted in each stage to conduct +the self-attention operations along the spatial and tem- +poral dimensions separately. We further introduce the +Attribute-Aware and Identity-Aware Proxy embedding +modules (AAP and IAP) to extract the informative +and discriminative feature representations at different +stages. All of them are realized by employing newly +designed self-attention operations with specific meanings. +Moreover, temporal patch shuffling is also introduced to +further improve the robustness of the model. Extensive +experimental results demonstrate the effectiveness of +the proposed modules in extracting the informative and +discriminative information from the videos, and illustrate +the MSTAT can achieve state-of-the-art accuracies on +various standard benchmarks. +Index +Terms—Video-based +Person +Re-ID, +Trans- +former, Spatial Temporal Modeling, Deep Representation +Learning +I. INTRODUCTION +P +ERSON Re-identification (re-ID) [6], [26], [28], +which aims at matching pedestrians across dif- +ferent camera views at different times, is a critical +Ziyi Tang, Ruimao Zhang, and Jinrui Chen are with The Chi- +nese University of Hong Kong (Shenzhen), and Ziyi Tang is +also with Sun Yat-sen University (e-mail: tangziyi@cuhk.edu.cn, +ruimao.zhang@ieee.org, and 120090765@link.cuhk.edu.cn ). +Zhanglin Peng is with the Department of Computer Science, +The University of Hong Kong, Hong Kong, China (e-mail: +zhanglin.peng@connect.hku.hk ). +Liang Lin is with the School of Computer Science and Engineer- +ing, Sun Yat-sen University (e-mail: linliang@ieee.org). +This paper was done when Ziyi Tang was working as a Research +Assistant at The Chinese University of Hong Kong (Shenzhen). +The Corresponding Author is Ruimao Zhang +Fig. 1: Comparison between different Transformer- +based frameworks for video re-ID. (a) shows the +framework where the Transformer fuse post-CNN fea- +tures of the entire video. (b) is Trigeminal Trans- +former [51], including three separate streams for tem- +poral, spatial, and spatio-temporal feature extraction. +(c) displays a multi-stage spatio-temporal aggregation +Transformer, which consists of three stages, all with a +spatio-temporal view but different meanings. +task of visual surveillance. In the earlier stage, the +studies have mainly focused on image-based person +re-ID [26], [28], [46], which mine the discrimina- +tive information in the spatial domain. With the de- +velopment of the monitoring sensors, multi-modality +information has been introduced to re-ID task [33], +[71], [72]. Numerous methods have been proposed to +break down barriers between modalities regarding their +image styles [86], structural features [81], [84], [97], +or network parameters [33], [82]. +On the other hand, some studies have exploited +multi-frame data and proposed various schemes [40], +[62], [100] to extract informative temporal represen- +tations to pursue video-based person re-ID. In such a +setting, each time a non-labeled query tracklet clip is +given, its discriminative feature representation needs to +be extracted to retrieve the clips of the corresponding +person in the non-labeled gallery. In practice, how to +simultaneously extract such discriminative information +arXiv:2301.00531v1 [cs.CV] 2 Jan 2023 + +Cross-viewTransfomer +Spatio-temporalTransfomer +Temporal +Spatial +Spatio- +temporal +Transfomer +Transfomer +Transformer +CNN +CNN +(a)Post-fusion Transformer +(b)TrigeminalTransformer +i Concatenation +个 +企 +Space-related module/ +Attribute-Aware +Identity-Aware +Identity-Aware +featurerepresentation +Proxy Embedding +ProxyEmbedding +ProxyEmbedding +Module +Module +Module +Time-related module/ +featurerepresentation +AttributeSpatio +Space-time-relatedmodule/ +Spatio-Temporal +Spatio-Temporal +Temporal +featurerepresentation +Aggregation +Aggregation +Aggregation +Attribute-relatedmodule +Attribute-Associated +Identity-Associated +Attribute-ldentity- +/featurerepresentation +Stage +Stage +AssociatedStage +Identity-related module +(c) MSTATTANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION +2 +from spatial and temporal dimensions is the key to +improving the accuracy of video-based re-ID. +To address such an issue, traditional methods [20] +usually employ hierarchically convolutional architec- +tures to update local patterns progressively. Further- +more, some attempts [14], [15], [48], [73], [94] adopt +attention-based modules to dynamically infer discrim- +inative information from videos. For instance, Wu et +al. [72] embed body part prior knowledge inside the +network architecture via dense and non-local region- +based attention. Although recent years have witnessed +the success of convolution-based methods [12], [13], +[20], [38], [43], [74], [94], [104], they have encoun- +tered a bottleneck of accuracy improvement, as con- +volution layers suffer from their intrinsic limitations +of spatial-temporal dependency modeling and infor- +mation aggregation [96]. +Recently, the Transformer architecture [24], [32], +[54], [89] has attracted much attention in the com- +puter vision area because of its excellent context +modeling ability. The core idea of such a model is +to construct interrelationships between local contents +via global attention operation. In the literature, some +hybrid network architectures [19], [34], [51] have +been proposed to tackle long-range context modeling +in video-based re-ID. A widely used paradigm is +to leverage Transformer as the post-processing unit, +coupled with a convolutional neural network (CNN) +as the basic feature extractor. For example, as sum- +marized in Fig. 1 (a), He et al. [35] and Zhang +et al. [95] adopt a monolithic Transformer to fuse +frame-level CNN feature. As shown in Fig. 1 (b), +Liu et al. [51] take a step further and put forward +multi-stream Transformer architecture in which each +stream emphasizes a particular dimension of the video +features. In a hybrid architecture, however, the 2D +CNN bottom encoder restricts the long-range spatio- +temporal interactions among local contents, which +hinders the discovery of contextual cues. Later, to +address this problem, some pure Transformer-based +approaches are introduced to video-based re-ID. Nev- +ertheless, the existing Transformer-based frameworks +are mainly motivated by those in video understanding +and concentrate on designing the architecture to learn +spatial-temporal representations efficiently. Most of +them are still limited in extracting informative and +human-relevant discriminative information from the +video clips, which are critical for large-scale matching +tasks [39], [92], [98], [104]. +To address the above issues, we propose a novel +Multi-stage +Spatial-Temporal +Aggregation +Trans- +former framework, named MSTAT, which consists +of three stages to respectively encode the attribute- +associated, the identity-associated, and the attribute- +identity-associated +information +from +video +clips. +Firstly, to save the computational cost, the Spatial- +Temporal Aggregation (STA) modules [4], [7] are +firstly adopted in each stage as their building blocks +to conduct the self-attention operations along the +spatial and temporal dimensions separately. Further, +as shown in Fig. 1, we introduce the plug-and-play +Attribute-Aware Proxy and Identity-Aware Proxy +(AAP and IAP) embedding modules into different +stages, for the purpose of reserving informative at- +tribute features and aggregating discriminative identity +features respectively. They are both implemented by +self-attention operations but with different learnable +proxy embedding schemes. For the AAP embedding +module, AAPs play the role of attribute queries to +reserve a diversity of implicit attributes of a person. +Arguably, the combination of these attribute repre- +sentations is informative and provides discriminative +power, complementary to the identity-only prediction. +In contrast, the IAP embedding module maintains a +group of IAPs as key-value pairs. With explicit con- +straints, they learn to successively match and aggregate +the discriminative identity-aware features embedded +in patch tokens. During similarity measurement, the +output feature representations of the three stages are +concatenated to form a holistic view of the input +person. +In practice, a Transformer-specific data augmenta- +tion scheme, Temporal Patch Shuffling, is also intro- +duced, which randomly rearranges the patches tem- +porally. With such a scheme, the enriched training +data effectively improve the ability to learn invariant +appearance features, leading to the robustness of the +model. Extensive experiments on three public bench- +marks demonstrate our proposed framework is superior +to the state-of-the-art on different metrics. Concretely, +we achieve the best performance of 91.8% rank-1 +accuracy on MARS, which is the largest video re-ID +dataset at present. +In summary, our contributions are three-fold. (1) We +introduce a Multi-stage Spatial-Temporal Aggregation +Transformer framework (MSTAT) for video-based per- +son re-ID. Compared to existing Transformer-based +frameworks, MSTAT better learns informative attribute +features and discriminative identity features. (2) For +different stages, we devise two different proxy embed- +ding modules, named Attribute-Aware and Identity- +Aware Proxy embedding modules, to extract infor- +mative attribute features and aggregate discriminative +identity features from the entire video, respectively. +(3) A simple yet effective data augmentation scheme, +referred to as Temporal Patch Shuffling, is proposed +to consolidate the network’s invariance to appearance +shifts and enrich training data. +II. RELATED WORKS +A. Image-Based Person Re-ID +Image-based person re-ID mainly focuses on person +representation learning. Early works focus primarily +on carefully designed handcraft features [6], [26], [28], +[44], [46], [103]. Recently, The flourishing deep learn- +ing has become the mainstream method for learning + +3 +IEEE TRANSACTIONS ON MULTIMEDIA +representation in person ReID [43], [65], [67], [74], +[77], [88]. For the last few years, CNN has been a +widely-used feature extractor [1], [17], [41], [43]–[45], +[65], [76], [94]. OSNet [104] fuses multi-scale features +in an attention-style sub-network to obtain informative +omni-scale features. Some works +[18], [87], [98] +focus on extracting and aligning semantic information +to address misalignment caused by pose/viewpoint +variations, imperfect person detection, etc. To avoid +the misleading by noisy labels, Ye et al. [83] presents +a self-label refining strategy, deeply integrating anno- +tation optimization and network training. So far, some +works [19], [34] also explore Image-based person re- +ID based on Vision Transformer [24]. For example, +TransReID [34] adopts Transformer as the backbone +and extracts discriminative features from randomly +sampled patch groups. +B. Video-Based Person ReID +Compared to image-based person re-ID, video- +based person re-ID usually performs better because it +provides temporal information and mitigates occlusion +by taking advantage of multi-frame information. For +capturing more robust and discriminative representa- +tion from frame sequences, traditional video-based re- +ID methods usually focus on two areas: 1) encoding +of temporal information; 2) aggregation of temporal +information. +To encode additional temporal information, early +methods [40], [62], [100] directly use temporal infor- +mation as additional features. Some works [1], [49], +[55], [73] use recurrent models, e.g., RNNs [56] and +LSTM [37], to process the temporal information. Some +other works [1], [12], [13], [53], [55], [60], [105] +go further by introducing the attention mechanism to +apply dynamic temporal feature fusion. Another class +of works [21] introduces optical flow that captures +temporal motion. What is more, some works [2], +[42], [63], [75], [91], [102] directly implement spatio- +temporal pooling to video sequences and generate a +global representation via CNNs. Recently, 3D CNNs +[29], [45] learn to encode video features in a joint +spatio-temporal manner. M3D [41] endows 2D CNN +with multi-scale temporal feature extraction ability via +multi-scale 3D convolutional kernels. +For the sake of aggregation that aims to generate +discriminative features from full video features, a +class of approaches [55], [93], [105] applies average +pooling on the time dimension to aggregate spatio- +temporal feature maps. Recently, some attention-based +methods [2], [15], [72], [80] attained significant per- +formance improvement by dynamically highlighting +different video frames/regions so as to filter more dis- +criminative features from these critical frames/regions. +For instance, Liu et al. [51] introduce cross-attention +to aggregate multi-view video features by pair-wise +interaction between these views. Apart from the explo- +ration of more effective architectural design, a branch +of works study the effect of pedestrian attributes [10], +[61], [101], such as shoes, bag, and down color, or +the gait [11], [57], i.e. walking style of pedestrians, as +a more comprehensive form of pedestrian description. +Chang et al. [11] closely integrate two coherent tasks: +gait recognition and video-based re-ID by using a +hybrid framework including a set-based gait recogni- +tion branch. Some works [61], [101] embed attribute +predictors into the network supported by annotations +obtained from a network pretrained on an attribute +dataset. Chai et al. [10] separate attributes into ID- +relevant and ID-irrelevant ones and propose a novel +pose-invariant and motion-invariant triplet loss to mine +the hardest samples considering the distance of pose +and motion states. +Although the above methods have made significant +progress in performance, Transformer [66], which is +deemed a more powerful architecture to process se- +quence data, may raise the performance ceiling of +video-based re-ID. To illustrate this, Transformer can +readily adapt to video data with the support of the +global attention mechanism to capture spatio-temporal +dependencies and temporal positional encoding to or- +der spatio-temporal positions. In addition, the class +token is off-the-shelf for Transformer-based models +to aggregate spatio-temporal information. However, +Transformer suffers from multiple drawbacks [24], +[70], [89], [90], and few works have been released so +far on video-based person re-ID based on Transformer. +In this work, we attempt to explore the potential of +intractable Transformer in video-based person re-ID. +C. Vision Transformer +Recently, Transformer has shown its ability as an +alternative to CNN. Inspired by the great success of +Transformer in natural language processing, recent +researchers [24], [54], [54], [70] have extended Trans- +former to CV tasks and obtained promising results. +Bertasius et al. [7] explores different video self- +attention schemes considering their cost-performance +trade-off, resulting in a conclusion that the di- +vided space-time self-attention is optimal. Similarly, +ViViT [4] factorizes self-attention to compute self- +attention spatially and then temporally. Inspired by +these works, we divide video self-attention into spa- +tial attention followed by temporal attention, and we +further propose a attribute-aware variant for video- +based re-ID. Furthermore, little research has been +done on Transformer for Video-based person re-ID. +Trigeminal Transformers (TMT) [51] puts the input +patch token sequence through a spatial, a temporal, and +a spatio-temporal minor Transformer, respectively, and +a cross-view interaction module fuses their outputs. +Differently, MSTAT has three stages, all extracting +spatio-temporal features but with different meanings: +(1) attribute features, (2) identity features, (3) attribute- +identity features. + +TANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION +4 +c +Tokenization +Class +Token +Patch +Tokens +Attribute-aware Prxoy +Embedding Module +… +… +L CE + L Tri +N3 +Stage III +2 +Spatio-Temporal +Aggregation +N2 +Stage II +Class Token Re-init +L CE + L Tri +L CE + L Tri +c +Inference +Element-wise +Addition +Spatial Positional +Encoding +Concatenation +Attribute +Representation +c +M +X +E +× +3 +M +× +× +× +A-Spatio-Temporal +Aggregation +Spatio-Temporal +Aggregation +N +× +Stage I +1 +Identity-aware Prxoy +Embedding Module +Identity-aware Prxoy +Embedding Module +Fig. 2: The overall architecture of our proposed MSTAT which consists of three stages, all based on the +Transformer architecture. Stage I updates the spatio-temporal patch token sequence of the input video +and aggregates them into a group of attribute-associated representations. Subsequently, Stage II aggregates +discriminative identity-associated features and Stage III attribute-identity-associated features, relying upon +their stage-specific class tokens. Here, we omit the input and output of each module except the attribute- +aware proxy embedding module in Stage I. At inference time, all these feature representations are combined +through concatenation to infer the pedestrian’s identity jointly. +III. METHOD +In Sec. III-A, we first overview the proposed +MSTAT framework. Then, Spatio-Temporal Aggrega- +tion (STA), the normal spatial-temporal feature extrac- +tor in MSTAT, is formulated in section Sec. III-B. +Along with it, we introduce the proposed Attribute- +Aware Proxy (AAP) and Identity-Aware Proxy (IAP) +embedding modules in Sec. III-C. Finally, Tem- +poral Patch Shuffling (TPS), a newly introduced +Transformer-specific data augmentation scheme, is +presented in section III-E. +A. Overview +This section briefly summarizes the workflow of +MSTAT. The overall MSTAT framework is shown in +Fig. 2. Given a video tracklet V P RT ˆ3ˆHˆW with +T frames and the resolution of each frame is H ˆ W, +the goal of MSTAT is to learn a mapping from a video +tracklet V to a d-dimension representation space in +which each identity is discriminative from the others. +Specifically, as shown on the left of Fig. 2, MSTAT +first linearly projects non-overlapping image patches +of size 3 ˆ P ˆ P into d-dimensional patch tokens, +where d “ 3P 2 denotes the embedded dimension of +tokens. Thus, a patch token sequence X P RT ˆNˆd is +obtained, where the number of patch tokens in each +frame is denoted by N “ HˆW +P 2 . Meanwhile, spatial +positional encoding E P RNˆd is added to X in a +element-wise manner for reserving spatial structure +in each frame. Notably, we do not insert temporal +positional encoding into X, since the temporal order +is usually not conducive to video-based re-ID, which +is also demonstrated in [92]. Finally, a class token +c P Rd is associated with X to aggregate global identity +representation. +Next, we feed the token sequence X into Stage I of +MSTAT. It takes X and c as input, and employs a stack +of eight Spatio-Temporal Aggregation (STA) blocks +for inter-frame and intra-frame correlation modeling. +The output tokens are then fed into an Attribute-Aware +Proxy (AAP) embedding module to mine rich visual +attributes, a composite group of semantic cues that im- +ply identity information, e.g., garments, handbags +and so on. The Stage II includes a series of STA +blocks (three in our experiments), followed by an +Identity-Aware Proxy (IAP) embedding module which +is able to screen out discriminative identity-associated +information by inspecting the entire sequence in par- +allel. In the Stage III, we first introduce a novel class +token to directly aggregate higher-level features. In +addition, a stack of Attribute-STA (A-STA) blocks is +used to fuse attributes from different frames. At last, +an IAP embedding module is adopted to generate a +discriminative representation for the person. In the +training phase, the attribute representations extracted +from Stage I and the class tokens of Stage II and +Stage III are supervised separately by a group of +losses. During the testing, the attribute representations +and the class tokens from the last two stages are +concatenated for similarity measurement. +B. Spatio-temporal Aggregation +To begin with, we make a quick review of the vanilla +Transformer self-attention mechanism first proposed in +[66]. In practice, visual Transformer embeds an image +into a sequence of patch tokens, and self-attention +operation first linearly projects these tokens to the +corresponding query Q, key K and value V respec- +tively. Then, the scaled product of Q and K generates +an attention map A, indicating estimated relationships + +s:/iblog.csdn.net/qqn34182315 +IEEE TRANSACTIONS ON MULTIMEDIA +between token representations in Q and K. Then, V +performs a re-weighting by multiplying the attention +map A, to obtain the output of Transformer self- +Attention. In this way, patch tokens are reconstructed +by leveraging interaction with each other. Formally, +self-Attention operation SAp.q can be formulated as +follows: +Q, K, V “ ˆSWq, ˆSWk, ˆSWv +A “ SoftmaxpQKTq{ +? +d +SApˆSq “ AV +(1) +where ˆS P R ˆ +Nˆd denotes an 2-dimensional input +token sequence, and Wq, Wk, Wv P Rdˆd1 denote +three learnable parameter matrices of size d ˆ d1. In +the multi-head setting, we let d1 “ d{n, where n +indicates the number of attention heads. The function +Softmaxp¨q denotes the softmax operation for each +row. And the scaling operation in Eqn.(1) eliminates +the influence from the scale of embedded dimension +d1. +In our Spatio-Temporal Aggregation block (STA), +self-attention operation along time axis and along +space axis (i.e. temporal attention and spatial attention) +are separately denoted as SAtp¨q and SAsp¨q. Let +S P R ˆT ˆ ˆ +Nˆd denote an input spatio-temporal token +sequence. Formally, SAtp¨q and SAsp¨q can be written +as: +SAtpSq “ SApConcatpS:,0, ..., S:,n, ..., S:,N´1qq +SAspSq “ SApConcatpS0,:, ..., St,:, ..., ST´1,:qq +(2) +where T indicates the total number of frames in +video clip, N is the total spatial position index, and +Concatp¨q denotes the concatenation operation in the +split dimension, e.g., the spatial position dimension in +Eqn.(2). +Given SAtp.q and SAsp.q, the STA block consecu- +tively integrates these two self-attention modules to ex- +tract spatial-temporal features. As illustrated in Fig. 3, +STA further extracts discriminative information from +patch tokens to the class token through spatial attention +SAsp¨q, which can be realized by concatenating the +copies of class token to the token sequence of each +frame before SAsp.q, and taking the average of class +token copies after SAsp.q to further apply the later +temporal aggregation. In this way, the general form +of STA can be presented as: +S1 “ S ` α ˆ SAtpLNpSqq +STApS, cq “ ConcatpS1, cq +`β ˆ SAspLNpConcatpS1, cqqq +(3) +where LNp¨q denotes Layer Normalization [5]. The +hyper-parameter α and β are learnable scalar residual +weights to balance temporal attention and spatial at- +tention. Compared with the space-time joint attention +in [7] and [4], which jointly processes all patches of a +video, STA is more computation-efficient by reducing +Fig. 3: The detailed comparison between (a) Spatio- +Temporal Aggregation block (STA) and (b) Attri- +bution Spatio-Temporal Aggregation block (A-STA). +Two additional Attribute-Aware Proxy (AAP) embed- +ding modules are placed into the latter, before and +after the temporal attention module. The class token +broadcasting operation duplicates the class token for +each frame to attend spatial attention within a specific +frame. Oppositely, class token averaging calculates the +average of all class token copies. Note that the Pre- +Norm [79] layers before temporal attention and spatial +attention are omitted. +Fig. 4: The detailed module design of the Attribute- +Aware +Proxy +(AAP) +embedding +module. +The +Attribute-Aware +Proxy +Embedding +denotes +a +learnable matrix that is used as the query of the +attention operation. For simplicity, this figure only +shows the single-head version of the AAP embedding +module and the scaling operation before the softmax +operation is omitted. +complexity from OpT 2N 2q to OpT 2 ` N 2q. Actually, +it avoids operating on a long sequence, whose length +always leads to quadratic growth of computational +complexity [31], [68]. +C. Attribute-Aware Proxy Embedding Module +Local patch tokens usually contain rich attributive +information, e.g., glasses, umbrellas, logos, +and so on. Even if a single attribute is not discrimina- +tive enough to recover one’s identity, the combinations + +00000 +00000 +个个个个个 +个个个个个 +ClassToken +Class Token +Averaging +- +Averaging +- +- +- +- +SpatialAttention +Spatial Attention +- +- +ClassToken +- +ClassToken +- +Broadcasting +Broadcasting +- +- +- +- +- +- +AAP embeddingmodule +- +- +- +TemporalAttention +- +TemporalAttention +- +- +- +- +- +AAP embeddingmodule +- +个个个个个 +个不不个个 +00000 +00000 +(a) STA +(b) A-STAAttribute +Representation +Linear +Attribute-Aware +i Proxy Embedding +: Module +Softmax +Q +K +V +Attribute-Aware +Linear +Linear +Proxy Embeddings +Class +0000000 +Patch +Token +TokenTANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION +6 +of a pedestrian’s rich attributes should be discrimina- +tive as each attribute eliminates a certain degree of +uncertainty. Rather than directly aggregating into a +“coarse” class token, we introduce the Attribute-Aware +Proxy (AAP) embedding module to directly extract +attribute features from a single-frame or multi-frame +patch token sequence. Practically, AAP embeddings +are formed by a learnable matrix with anisotropic +initialization for the richness of learned attributes. It +can be considered as the “attribute bank” to serve as +the query of the attention operation to match with +the feature representations of the input patch tokens. +Specifically, AAP embeddings interact with the keys +of the patch token sequence. Finally, the resulting +attention map is used to re-weight the value, generating +the attribute representations of the specific video clip +with the same dimension of AAPs. Formally, an AAP +embedding module can be written as follows, +Q, K, V “ PQ, SWk, SWv, +AAPpSq “ SoftmaxpQKTq +? +d +V +(4) +here we use the multi-head version of AAP embedding +module in practice, which has the same multi-head +setting as SAp¨q in Eqn.(1). Note that the spatio- +temporal input S here can also be ˆS P R ˆ +Nˆd for +spatial-only use. Compared with SAp¨q, the newly +proposed AAP module consider the query Q in Eqn.(1) +as the a set of learnable parameters PQ P RNaˆd1, +where Na ! N is a hyper-parameter that indicates the +number of AAPs. By controlling Na, the AAP module +could have a manually defined capacity, which leads +to flexibility for various real applications. +As shown in Fig. 2, both Stage I and Stage +III employ the proposed AAP embedding modules. +Specifically, in Stage I, the proposed AAP module is +firstly used to generate attribute representations from a +multi-frame sequence of patch tokens S P R ˆT ˆ ˆ +Nˆd for +similarity measurement. Although we do not have any +attribute-level annotations, we hope the AAP module +can automatically learn a rich set of implicit attributes +from the entire training dataset, while these resultant +attribute representations could also present discrimina- +tive power complementary to ID-only representations. +To achieve this goal, the ID-level supervision signal is +first imposed on the combination of learned attribute +representations to constrain its discriminative power. +In addition, we initialize the AAPs with anisotropic +distributions to capture diverse implicit attribute rep- +resentations. In practice, we surprisingly find that +such anisotropy can maintain after the model training, +which means such optimized AAP could respond to a +set of differentiated attributes. Moreover, the number +of AAPs can be relatively large compared with the +class token to cover rich attribute information. In +this sense, both the richness and diversity of learned +implicit attributes can be guaranteed. +In Stage III, we further insert two intra-frame +AAP embedding modules before and after the tem- +poral attention of each STA to conduct attribute-aware +temporal interaction. Such a modified STA block is +named A-STA, which is illustrated in Fig. 3. In A-STA, +semantic-related attributes in different frames experi- +ence inter-frame interaction to model their temporal +relations. In the end, after temporal attention, we set +Na equal to N for the second AAP embedding module +so that it has N tokens as output to keep the input- +output consistency. +D. Identity-Aware Proxy Embedding Module +Extracting discriminative identity representation is +also crucial for video-based re-ID. To this end, the +Identity-Aware Proxy (IAP) embedding module is pro- +posed for effective and efficient discriminative repre- +sentation generation. In previous works, joint space- +time attention has shown promising results [4], [7], as +it accelerates information aggregation by applying self- +attention over spatial and temporal dimensions jointly. +However, the quadratic computational overheads limit +its applicability. The IAP embedding module is pro- +posed to address such an issue, which performs joint +space-time attention with high efficiency while main- +taining the discrimination of the identity feature rep- +resentation. +The IAP module contains a set of identity proto- +types, which are presented as two learnable matrics. +In practice, we exploit them to replace the keys +tpi +KuM +i“1 P PK and values tpi +V uM +i“1 P PV of the +attention operation. Both PK, PV +P RMˆd1, where +M P N` denotes the number of identity prototypes +and determine the capacity of the IAP module (usually +M ! N). As shown in Fig. 5, an attention map +A P RMˆN is first calculated to present the affinity +between prototype-patch pairs. Thus each element in A +reflects how close a patch token is to a specific identity +prototype. Then this attention map is sparsified by suc- +cessively applying an L1 normalization and softmax +normalization along M and N, respectively. At last, +the class token c, i.e. the first row of V, is updated +by the multiplication of V and A. Such an operation +aggregates the most discriminative identity features +from the entire patch token sequence. Formally, given +the spatio-temporal token sequence S, the output of +the IAP module can be calculated as follows: +Q, K, V “ SWq, PK, PV +A “ SoftmaxpL1NormpQKTqq +? +d +IAPpSq “ AV +(5) +where K and V are not conditioned on input S +but are learnable parameters. Here we insert an L1 +normalization layer before the softmax operation in +Eqn. (5), resulting in double normalization [30], [31]. +Such a scheme performs patch token re-coding to +reduce the noise of patch representations, leading to + +7 +IEEE TRANSACTIONS ON MULTIMEDIA +Fig. 5: The detailed module design of the Identity- +Aware Proxy (IAP) embedding module. The IAP em- +bedding denotes the learnable matrix used to calculate +the key or value of the attention operation. Here +we only show the single-head version of the IAP +embedding module and omit the scaling operation. +In such a scheme, The output token sequence can +be considered as reconstruction by a group of IAPs, +which tend to reserve the most discriminative identity +features. +robust identification results. Specifically, the learnable +matrix PK matches the input tokens through the double +normalization operation to generate the affinity map +A. Then these input tokens are thereupon re-coded +through a projection of PV along A. Since the numbers +of learnable vectors in PK and PV are much smaller +than the number of input tokens, the above operation +has been able to represent each token in a more +compact space (i.e. linear combination of the vectors +in PV ), effectively suppressing irrelevant information +for re-ID. Moreover, IAPp¨q has OpNq computational +complexity since the number of identity prototypes M +is fixed and is usually much less than the total number +of patch tokens of a specific video tracklet (e.g., 64 +in our experiments). So, the proposed IAP embedding +module allows all spatio-temporal patch tokens to be +processed in parallel for effective and efficient feature +extraction. +E. Temporal Patch Shuffling +To improve the robustness of the model, we propose +a novel data augmentation scheme termed Temporal +Patch Shuffling (TPS). Suppose we have one patch +sequences Ri “ tri1, ..., rit, ..., riT u from the same +video clip, where the sub-index i denotes specific +spatial locations. As shown in Fig. 6, the proposed +TPS randomly permutes the patch tokens in Ri and +refill the shuffled sequence ˆ +Ri to form the new video +clip for training. As illustrated in Fig. 6, we could +simultaneously select multiple spatial regions in one +video clip for shuffling. While in the inference phase, +the original video clip is directly fed into the model +for identification. TPS brings firm appearance shifts +and motion changes from which the network learns +to extract generalizable and invariant visual clues. In +addition, the scale of available training data can be +f1 +f2 +f3 +f4 +f5 +r21 +r22 +r23 +r24 +r25 +r11 +r12 +r13 +r14 +r15 +r15 +r14 +r11 +r12 +r13 +r24 +r23 +r25 +r22 +r21 +X +x’ +shuffling +f1 +f2 +f3 +f4 +f5 +Fig. 6: Visualization of Temporal Patch Shuffling +(TPS). ft represents tth frame, rit the patch in spatial +position i and tth frame. TPS is a built-in data aug- +mentation scheme that randomizes the order of a patch +sequence sampled from spatial position i. As a result, +for example, the patch in the red box is transferred +from the 5th frame to the 1st frame. +greatly extended based on such a scheme, which helps +to prevent the network from overfitting. +In our experiments, we treat TPS as a plug-and-play +operation and implement it at the stem of the network +to promote the entire network for the best performance. +The following section will conduct ablation studies to +explore where to insert TPS and to what extent TPS +should be for optimal training results. +IV. EXPERIMENT +A. Datasets and evaluation protocols +In this paper, we evaluate our proposed MSTAT +on three widely-used video-based person re-ID bench- +marks: +iLIDS-VID +[69], +DukeMTMC-VideoReID +(DukeV) [59], and MARS [102]. +1) iLIDS-VID [69] is comprised of 600 video track- +lets of 300 persons captured from two cameras. In +these video tracklets, frame numbers range from 23 +and 192. The test set shares 150 identities with the +training set. +2) DukeMTMC-VideoReID [59] is a large-scale +video-based benchmark which contains 4, 832 videos +sharing 1, 404 identities. In the following sections, we +use the abbreviation “DukeV” for the DukeMTMC- +VideoReID dataset. The video sequences in the DukeV +dataset are commonly longer than videos in other +datasets, which contain 168 frames on average. +3) MARS [102] is one of the largest video re- +ID benchmarks which collects 1, 261 identities exist- +ing in around 20, 000 video tracklets captured by 6 +cameras. Frames within a video tracklet are relatively +more misaligned since they are obtained by a DPM +detector [27] and a GMMCP tracker [22] rather than +hand drawing. Furthermore, around 3, 200 distractor +tracklets are mixed into the dataset to simulate real- +world scenarios. +For evaluation on MARS and DukeV datasets, we +use two metrics: the Cumulative Matching Character- +istic (CMC) curves [8] and mean Average Precision +(mAP) following previous works [16], [51], [94], [99]. +However, in the gallery set of iLIDS-VID, there is +merely one correct match for each query. For this +benchmark, only cumulative accuracy is reported. + +00000000 +Identity-Aware +: Proxy Embedding +i Module +Softmax +L1Norm +K +V +Identity-Aware +Identity-Aware +Linear +Proxy Embeddings +Proxy Embeddings +Class +Patch +Token +TokensXXTANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION +8 +Method +Source +Backbone +MARS +Duke-V +iLIDS-VID +Rank-1 +Rank-5 +mAP +Rank-1 +Rank-5 +mAP +Rank-1 +Rank-5 +SCAN [94] +TIP19 +Pure-CNN +87.2 +95.2 +77.2 +- +- +- +88.0 +96.7 +VRSTC [39] +CVPR19 +Pure-CNN +- +89.8 +85.1 +96.9 +- +96.2 +86.6 +- +M3D [39] +AAAI19 +Pure-CNN +- +- +- +96.9 +- +96.2 +74.0 +94.3 +MG-RAFA [99] +CVPR20 +Pure-CNN +88.8 +97.0 +85.9 +- +- +- +88.6 +98.0 +AFA [16] +ECCV20 +Pure-CNN +90.2 +96.6 +82.9 +- +- +- +88.5 +96.8 +AP3D [29] +ECCV20 +Pure-CNN +90.7 +- +85.6 +97.2 +- +96.1 +88.7 +- +TCLNet [16] +ECCV20 +Pure-CNN +89.8 +- +85.1 +96.9 +- +96.2 +86.6 +- +A3D [15] +TIP20 +Pure-CNN +86.3 +95.5 +80.4 +- +- +- +86.7 +98.6 +GRL [52] +CVPR21 +Pure-CNN +90.4 +96.7 +84.8 +95.0 +98.7 +93.8 +90.4 +98.3 +STRF [3] +ICCV21 +Pure-CNN +90.3 +- +86.1 +97.4 +- +96.4 +89.3 +- +Fang et al. [25] +WACV21 +Pure-CNN +87.9 +97.2 +83.2 +- +- +- +88.6 +98.6 +TMT [51] +Arxiv21 +CNN-Transformer +91.2 +97.3 +85.8 +- +- +- +91.3 +98.6 +Liu et al. [47] +CVPR21* +CNN-Transformer +91.3 +- +86.5 +96.7 +- +96.2 +- +- +STT [95] +Arxiv21 +CNN-Transformer +88.7 +- +86.3 +97.6 +- +97.4 +87.5 +95.0 +ASANet [10] +TCSVT22 +Pure-CNN +91.1 +97.0 +86.0 +97.6 +99.9 +97.1 +- +- +MSTAT(ours) +- +Pure-Transformer +91.8 +97.4 +85.3 +97.4 +99.3 +96.4 +93.3 +99.3 +TABLE I: Result comparison with state-of-the-art video-based person re-ID methods on MARS, DukeMTMC- +VideoReID, and iLIDS-VID. * denotes the workshop of the conference. +B. Implementation details +Our proposed MSTAT framework is built based on +Pytorch toolbox [58]. In our experiments, it is running +on a single NVIDIA A100 GPU (40G memory). We +resize each video frame to 224 ˆ 112 for the above +benchmarks. Typical data augmentation schemes are +involved in training, including horizontal flipping, ran- +dom cropping, and random erasing. For all stages, +STA modules are pretrained on an action recognition +dataset, K600 [9], while other aforementioned modules +are randomly initialized. +In the training phase, if not specified, we sample +L “ 8 frames each time for a video tracklet and +set the batch size as 24. In each mini-batch, we +randomly sample two video tracklets from different +cameras for each person. We supervise the network +by cross-entropy loss with label smoothing [64] asso- +ciated with widely used BatchHard triplet loss [36]. +Specifically, we impose supervision signals separately +on the concatenated attribute representation from the +AAP embedding module in Stage I, the output class +tokens from Stage II, and Stage III. The learning +rate is initially set to 1e-3, which would be multiplied +by 0.75 after every 25 epochs. The entire network is +updated by an SGD optimizer in which the weight +decay and Nesterov momentum are set to 5 ˆ 10´5 +and 0.9, respectively. +In the test phase, following [29], [95], we randomly +sample 32 frames as a sequence from each original +tracklet in either query or gallery. For each sequence, +The attribute representation from Stage I, the output +class tokens from stage Stage II and Stage III are +concatenated as the overall representation. Following +the widely-used protocol, we compute the cosine sim- +ilarity between each query-gallery pair using their +overall representations. Then, the CMC curves and the +mAP can be calculated based on the predicted ranking +list and the ground truth identity of each query. Note +that we do not use any re-ranking technique. +C. Compared with the state of the arts +In Table I, we make a comparison on three bench- +mark datasets between our method and video-based +person re-ID methods from 2019 to 2021, including +M3D [39], GRL [52], STRF [3], Fang et al. [25], +TMT [51], Liu et al. [47], ASANet [10]. According to +their backbones, these re-ID methods can be roughly +divided into the following types: Pure-CNN, CNN- +Transformer Hybrid, and Pure-Transformer methods. +In real-world applications, rank-1 accuracy [8] re- +flects what extent a method can find the most confident +positive sample [85], and relatively high rank-1 accu- +racy can save time in confirmation. As the first method +based on Pure-Transformer for video-based re-ID so +far, we achieve state-of-the-art results in rank-1 ac- +curacy on three benchmarks. Our approach especially +attains rank-1 accuracy of 91.8% and rank-5 accuracy +of 97.4% on the largest-scale benchmark, MARS. It +is noteworthy that our MSTAT outperforms the best +pure CNN-based methods using ID annotations only +by a margin of 1.1% and a CNN-Transformer hybrid +method, TMT, by 0.6% in MARS rank-1 accuracy. +Compared to our proposed method, TCLNet [16] +explicitly captures complementary features over dif- +ferent frames, and GRL [52] devises a guiding mech- +anism for reciprocating feature learning. However, the +designed modules in these methods commonly take +as input the deep spatial feature maps extracted by +a CNN backbone (e.g. ResNet50) that may overlook +attribute-associated or identity-associated information +without explicit modeling. Similar to ours, TMT [51] +and M3D [3] process video tracklets in multiple +views to extract and fuse multi-view features. No- +tably, in all stages of MSTAT, intermediate features +are spatio-temporal and can be iteratively updated to +capture spatio-temporal cues with different emphases. +ASANet [10] exploits explicit ID-relevant attributes +(e.g., gender, clothes, and hair) and ID-irrelevant at- +tributes (e.g., pose and motion) on a multi-branch net- + +9 +IEEE TRANSACTIONS ON MULTIMEDIA +Method +Test Protocol +Rank-1 +Rank-5 +mAP +MSTAT +Stage I +89.2 +96.7 +82.4 +Stage II +89.2 +96.5 +83.0 +Stage III +89.8 +96.5 +83.0 +Stage I & II +91.2 +97.3 +85.0 +Stage I & III +90.5 +97.2 +83.9 +Stage II & III +90.6 +96.9 +84.6 +Stage I, II, & III +91.8 +97.4 +85.3 +TABLE II: Ablation study on three stages of MSTAT +on MARS. Test Protocol means the final feature rep- +resentation used for similarity measurement. The net- +work architecture and training hyper-parameter setting +remain the same for each experiment. +work. Despite the performance growth, the demand for +attribute annotations may limit its applications in large- +scale scenarios. In comparison with existing methods, +our method aggregates spatio-temporal information in +a unified manner and explicitly capitalizes on implicit +attribute information to improve recognizability un- +der challenging scenarios. Conclusively, our method +achieves the state-of-the-art performance of 91.8% and +93.3% rank-1 accuracy, respectively, on MARS and +iLIDS-VID. +D. Effectiveness of Multi-Stage Framework Architec- +ture +To evaluate the effectiveness of the three stages +in our proposed MSTAT, we carry out a series of +ablation experiments whose results are displayed in +Table II. After the three stages are jointly trained, we +first separately evaluate each stage using its output +feature representation. Then, we concatenate two or +more stages to evaluate whether each is effective. +For three single stages, each has rank-1 accuracy +ranging from 89.2 to 89.8. However, their combina- +tions result in a significant increase of over 0.8%. +Remarkably, while Stage I and Stage II secure only +89.2 rank-1 accuracy, their integration attains up to +91.2%, surpassing them by a 2% margin. One can +attribute such a result to their emphases: one stage on +attribute-associated features and the other on identity- +associated features. Eventually, when all three stages +are used, MSTAT reaches a 91.8% rank-1 accuracy, +higher than all two-stage combinations. Overall, these +experiments demonstrate that the three stages have dif- +ferent preferences toward features and can complement +each other by simple concatenation. +E. Effectiveness of Key Components +To demonstrate the effectiveness of our proposed +MSTAT, we conduct a range of ablative experiments +on the largest public benchmark MARS. +1) Effectiveness of Attribute-Aware Proxy Embed- +ding Module: As shown in Fig. 7, we evaluate MSTAT +with different AAP numbers (i.e. Na in Sec. III-C) in +the AAP embedding module in the last layer of Stage +Fig. 7: Ablation study on the attribute-aware proxy +(AAP) embedding module for attribute extraction in +MARS. ”Base” is the network without attribute ex- +traction using AAP in training and testing. AAP- +k indicates the network where the AAP embedding +module in Stage I has k AAPs. +Fig. 8: Ablation study on A-STA. ”Base” is the net- +work that consists of STA only. A-STA-k represents +the network in which Stage III is equipped with A- +STA layers each of k AAPs. +I. The figure reveals that 24 proxies are optimal for +attributive information extraction as it attains the best +performance in terms of rank-1 and rank-5 accuracy. +In contrast to the baseline, MSTAT has seen over 2% +growth in rank-1 accuracy and around 1% in rank-5 +accuracy. However, a redundant or insufficient number +of AAPs may cause a minor performance drop since +they may pay attention to noisy or useless attributes. +In summary, the AAP embedding module for clue +extraction gives a boost to the performance in rank- +1 and rank-5 accuracy, with negligible computational +overhead. +Attribute-Aware Proxy (AAP) embedding modules +are also used for A-STA, a variant of STA for attribute- +aware temporal feature fusion in Stage III. As shown +in Fig. 8, we conduct a series of experiments to explore +whether A-STA is effective and how many AAPs for +A-STA are appropriate (also corresponding to Na in +III-C)). The experiment results reveal that the baseline +model fails to reach 90% rank-1 accuracy or 97% rank- +5 accuracy. As the number of AAPs increases, these +two metrics grow to 91.8% and 97.4%. +Therefore, we can attribute the performance soar +to A-STA, allowing for attribute-aware temporal in- +teraction. A-STA offers a different viewpoint from +that of Stage II on videos. Moreover, due to the +redundancy of temporal information in many video re- +ID scenarios discussed in [16], A-STA with too many +AAPs incurs meaningless attributes. This can be why +the performance descends once A-STA has too many +AAPs. + +Rank-1 accuracy +Rank-5 accuracy +9160 +0.93 +0.974 +0.92 +0.918 +0.972 +0.908 +0.971 +0.91 +0.907 +0.906 +0L60 +0.969 +0.90 +0.968 +0.968 +0.892 +0.967 +0.B9 +0.966 +Bese +AAP-BAAP-16 AAP-24 AAP-32 +Bease +AAP-BAAP-16 AAP-24 AAP-32Rank-1 accuracy +Rank-5 accuracy +0.976 +0.93 +0.974 +0.92 +0.918 +0.910 +0.912 +0.972 +0.971 +0.91 +0.906 +0.970 +0.970 +0.90 +0.968 +0.968 +0.892 +0.B9 +0.566 +0.966 +BaBBASTA-16 ASTA-32 ASTA4B ASTA-64 +Baiga +ASTA-16 ASTA-32 ASTA-4B ASTA-64TANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION +10 +Fig. 9: Study on the effect of training video sequence +length on MARS. +In conclusion, our proposed AAP embedding mod- +ule can be used for: (1) the extraction of informative +attributes as plugged into any Transformer layer and +(2) attribute-aware temporal interaction when a tempo- +ral attention module is sandwiched between two. Both +of the two functionalities cause a significant increase +in performance, demonstrating their effectiveness. +2) Effectiveness of Identity-Aware Proxy Embedding +Module: In Table III, MSTAT that discards IAP em- +bedding modules leads to only 88.2% rank-1 accuracy +and 96.4% rank-5 accuracy. However, it boosts rank-1 +performance by 2.8% or 2.2% by taking the place of +STA in Stage II or Stage III. Finally, IAP embedding +modules in the last layers in both Stage II and +Stage III further improve 0.8% rank-1 accuracy and +0.4% rank-5 accuracy. The IAP embedding module’s +ablation results demonstrate its ability to generate +discriminative representations efficiently. Intuitively, +we place the IAP embedding module only in the last +few depths because it may discard non-discriminative +features that should be preserved in shallow layers. +3) Effectiveness of Temporal Patch Shuffling: To +evaluate the effectiveness of Temporal Patch Shuffling +(TPS), we assign different probabilities to implement +TPS for each training video sample. Note that in the +following experiments, the number of spatial positions +to shuffle is set to 5 if we implement TPS on this sam- +ple. As shown in Table IV, 20% probability provides +the best result over others, which leads to a growth of +0.3% in rank-1 accuracy. However, the 60% or 80% +probability results in a 0.1% or 0.2% rank-1 accuracy +drop mainly due to heavy noise. In summary, a proper +level of TPS would be an effective data augmentation +method for the Transformer for video-based person +re-ID. Further, rather than reserving temporal motion +(an ordered sequence of patches), TPS stimulates re- +identification accuracy by learning temporal coherence +from shuffled patch tokens. +F. Effect of video sequence length +To investigate how temporal noise influences the +training of MSTAT, we conduct experiments on videos +with varied lengths. In Fig. 9, experiments provide +length-varying video tracklets for training, while all +experiments are implemented under the identical eval- +uation setting with a fixed video length of 32. All +Method +Position +Rank-1 +Rank-5 +w/o IAP embedding module +- +88.2 +96.4 +w/ IAP embedding module +Stage II +91.0 +97.0 +Stage III +90.4 +97.0 +Stage II&III +91.8 +97.4 +TABLE III: Ablation study on the IAP embedding +module. Stage II and Stage III in this table means that +an IAP embedding module is appended to the last layer +of Stage II and Stage III respectively. This table shows +that the IAP embedding module brings improvements +to every single stage. When it is placed on both two +stages, MSTAT shows the best performance. +Methods +Prob. +Rank-1 +Rank-5 +mAP +MSTAT w/o TPS +0% +91.5 +97.5 +85.2 +MSTAT w/ TPS +20% +91.8 +97.4 +85.3 +40% +91.7 +97.3 +85.2 +60% +91.4 +97.5 +85.1 +80% +91.3 +97.1 +85.1 +TABLE IV: Ablation study on Temporal Patch Shuf- +fling. The table shows that the proper level of shuffling +can bring slight improvement. However, it may de- +grade the learning while the shuffling degree becomes +increasingly overwhelming. +experiments shut down until the loss stops decreasing +for ten epochs. +On the one hand, rank-1 accuracy shows an upward +trend as temporal noise gradually decreases, reach- +ing a peak at 8. On the other hand, temporal noise +shows no apparent correlation with rank-5 accuracy +and mAP. These results show that our model gains +up to 0.6% rank-1 accuracy through learning better +temporal features from data. However, rank-5 accuracy +and mAP benefit little from noise reduction, from +which we can speculate that in most cases in video +re-ID, learning temporal features is less important than +learning appearance features as they only account for +0.6% of rank-1 and 0.2% of rank-5 accuracy. Similar +results can be found in [51]. +G. Comparison among metric learning methods +Metric learning aims to regularize the sample distri- +bution on feature space. Usually, metric learning losses +constrain the compactness of intra-class distribution +and sparsity of the overall distribution. To explore +which strategy cooperates with our framework better, +we compare a range of classic metric learning loss +functions on iLIDS-VID, as shown in Table V. Note +that these losses are scaled to the same magnitude +to ensure fairness. Significantly, OIM loss [78] and +BatchHard triplet loss [36], widely used in re-ID, +outperform Arcface [23] and SphereFace [50] losses +by a large margin since the latter two loss functions +suffer from untimely overfitting in our experiments. + +Rank-l accuracy +Rank-5 accuracy +0.920 +0.980 +0.918 +0.918 +0.916 +0.976 +0.975 +0.975 +0.914 - +0.914 +160 +0.974 +0.913 +0.973 +0.912 +0.912 +0.972 +0.91 +6 +0.970 +4 +8 +4 +6 +1 +training video sequence length +training video sequence length11 +IEEE TRANSACTIONS ON MULTIMEDIA +Metric learning loss +Rank-1 +Rank-5 +w/o Metric learning +66.0 +90.0 +Arcface [23] +73.3 +90.7 +SphereFace [50] +66.7 +89.3 +OIM [78] +89.3 +98.3 +BatchHard* [36] +93.3 +99.3 +TABLE V: Comparison among metric learning loss +functions +on +iLIDS-VID, +where +* +denotes +the +method used in our implementation. For Arcface and +SphereFace, we test three margins and report the best +result: (1) by default, (2) 20% larger than the default, +(3) 20% smaller than the default. +Fig. 10: Visualization of the similarity matrix of +attribute-aware proxies trained on MARS. The maxi- +mal similarity between all pairs is around 0.2, demon- +strating that AAPs learn to capture diverse attributes. +H. Visualization +To better understand how the proposed framework +works, we conduct visualization on the AAP embed- +ding module. In Fig. 10, we show the diversity of +implicit attributes by the similarity matrix of 24 AAPs. +This figure implies that AAPs are anisotropic, covering +different attribute features that appear in the given +training dataset. +Specifically, as shown in Fig. 11, we randomly +select two pedestrians’ tracklets. Attention map visu- +alization is adopted as a sign of each AAP’s concen- +tration. In practice, we process the raw attention maps +first by several average filters and then by thresholding +to deliver smooth visual effects instead of grid-like +maps. In these heap maps, the brighter color denotes +the higher attention value. Despite the absence of +attribute-level supervision, Fig.11 shows that some +AAPs learn to pay attention to a local region with +special meanings as an identity cue. For example, the +AAP in white color in video clips (a) automatically +learns to cover the logo in the T-shirt, while the one +in (b) captures the head of the woman. +Moreover, we display the t-SNE visualization result +on iLIDS-VID in Fig. 12. It only contains the first 1/3 +of the IDs in the test set for a better visual effect. +We also provide the corresponding quantitative evalu- +Methods +IntraÓ +IntraÓ +InterÒ +Rank-1Ò +Baseline [7] +0.4572 +0.4495 +0.4704 +0.873 +MSTAT w/o attr. +0.4517 +0.4469 +0.4644 +0.913 +MSTAT (ours) +0.4410 +0.4389 +0.5012 +0.933 +TABLE VI: Quantitative evaluation on iLIDS-VID. +”Intra” denotes the averaged normalized intra-class +distance, and ”Inter” is the minimum inter-class dis- +tance. Here, * means that the metric is computed on +samples with the correct rank-1 match. + + +(a) +(b) +Fig. 11: Visualization of attribute-aware proxies for +two different pedestrians on MARS. Attention heat +maps of four consecutive frames from the AAP em- +bedding module on Stage I are displayed. +ation results in Table VI measured by the normalized +averaged intra-class distance and the minimum inter- +class distance (0-2) on the entire test set. As a result, +MSTAT drops the average intra-class distance from +0.4572 of the baseline to 0.4410 and enlarges the +minimum inter-class distance from 0.4704 to 0.5012. +Further, to eliminate the influence of accuracy, we +measure the intra-class distance between correctly +matched samples, from which we witness a similar +result. These results explain why MSTAT’s t-SNE +visualization seems sparser. +V. CONCLUSION +This paper proposes a novel framework for video- +based person re-ID, referred to as Spatial-Temporal +Aggregation Transformer (MSTAT). To tackle simulta- +neous extraction for local attributes and global identity +information, MSTAT adopts a multi-stage architecture +to extract (1) attribute-associated, (2) the identity- +associated, and (3) the attribute-identity-associated in- +formation from video clips, with all layers inherited +from the vanilla Transformer. Further, for reserving +informative attribute features and aggregating discrim- +inative identity features, we introduce two proxy em- +bedding modules (Attribute-Aware Proxy embedding +module and Identity-Aware Proxy embedding module) +into different stages. In addition, a patch-based data +augmentation scheme, Temporal Patch Shuffling, is +proposed to force the network to learn invariance +to appearance shifts while enriching training data. +Massive experiments show that MSTAT can extract +attribute-aware features consistent across frames while +reserving discriminative global identity information on + +0 +1.0 +2 +3 +4- +0.8 +5 +6 +7 +8- +- 0.6 +9- +10 +11 +12 +13 +0.4 +14 +15 +16 +17 + 0.2 +18- +19 +20 +21 +22 +- +0.0 +23 +456780TANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION +12 +Fig. 12: T-SNE Visualization of the iLIDS-VID test +set. The numbers on the plots indicate person IDs. +MSTAT shows an increase in intra-class compactness +and the minimum inter-class distance over the entire +test set compared to the baseline. +different stages to attain high performance. Finally, +MSTAT outperforms most existing state-of-the-arts on +three public video-based re-ID benchmarks. +Future work may focus on mining the hard instances +or local informative attribute locations to conduct con- +trastive learning to promote the model’s accuracy fur- +ther. Moreover, leveraging more unlabeled and multi- +modal data to improve the model’s effectiveness is also +a potential research direction. +ACKNOWLEDGMENT +The work is supported in part by the Young Sci- +entists Fund of the National Natural Science Foun- +dation of China under grant No. 62106154, by Na- +tional Key R&D Program of China under Grant No. +2021ZD0111600, by Natural Science Foundation of +Guangdong Province, China (General Program) un- +der grant No.2022A1515011524, by Guangdong Ba- +sic and Applied Basic Research Foundation under +Grant No. 2017A030312006, by CCF-Tencent Open +Fund, by Shenzhen Science and Technology Program +ZDSYS20211021111415025, and by the Guangdong +Provincial Key Laboratory of Big Data Computing, +The Chinese Univeristy of Hong Kong (Shenzhen). +REFERENCES +[1] +[2] +[3] A. Aich, M. Zheng, S. Karanam, T. Chen, A. K. Roy- +Chowdhury, and Z. 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Springer, 2016. +[103] W.-S. Zheng, S. Gong, and T. Xiang. Towards open-world +person re-identification by one-shot group-based verification. +IEEE transactions on pattern analysis and machine intelli- +gence, 38(3):591–606, 2015. +[104] K. Zhou, Y. Yang, A. Cavallaro, and T. Xiang. Omni-scale +feature learning for person re-identification. In Proceedings of +the IEEE/CVF International Conference on Computer Vision, +pages 3702–3712, 2019. +[105] Z. Zhou, Y. Huang, W. Wang, L. Wang, and T. Tan. See +the forest for the trees: Joint spatial and temporal recurrent +neural networks for video-based person re-identification. In +Proceedings of the IEEE Conference on Computer Vision and +Pattern Recognition, pages 4747–4756, 2017. + +15 +IEEE TRANSACTIONS ON MULTIMEDIA +Ziyi Tang is now pursuing his Ph.D. +degree at Sun Yat-Sen University. Before +that, he was a research assistant at The +Chinese University of Hong Kong, Shen- +zhen (CUHK-SZ), China. He received the +B.E. degree from South China Agriculture +University (SCAU), Guangzhou, China in +2019 and M.S. degree from The Univer- +sity of Southampton, Southampton, U.K. +in 2020. He has won top places in data +science competitions hosted by Kaggle and +Huawei respectively. His research interests include Computer Vision, +Vision-Language Joint Modeling, and Casual Inference. +Ruimao Zhang is currently a Research +Assistant Professor in the School of Data +Science, The Chinese University of Hong +Kong, Shenzhen (CUHK-SZ), China. He +is also a Research Scientist at Shenzhen +Research Institute of Big Data. He received +the B.E. and Ph.D. degrees from Sun Yat- +sen University, Guangzhou, China in 2011 +and 2016, respectively. From 2017 to 2019, +he was a Post-doctoral Research Fellow in +the Multimedia Lab, The Chinese Univer- +sity of Hong Kong (CUHK), Hong Kong. After that, he joined at +SenseTime Research as a Senior Researcher until 2021. His research +interests include computer vision, deep learning and related multi- +media applications. He has published about 40 peer-reviewed articles +in top-tier conferences and journals such as TPAMI, IJCV, ICML, +ICLR, CVPR, and ICCV. He has won a number of competitions and +awards such as Gold medal in 2017 Youtube 8M Video Classification +Challenge, the first place in 2020 AIM Challenge on Learned Image +Signal Processing Pipeline. He was rated as Outstanding Reviewer +of NeurIPS in 2021. He is a member of IEEE. +Zhanglin Peng is now pursuing her Ph.D. +degree with the Department of Computer +Science, The University of Hong Kong, +Hong Kong, China. She received her B.E. +and M.S. degrees from Sun Yat-Sen Uni- +versity, Guangzhou, China in 2013 and +2016, respectively. From 2016 to 2020, she +was a researcher at SenseTime Research. +Her research interests are computer vision +and machine learning. +Jinrui Chen is currently pursuing the B.A. +degree in Financial Engineering conferred +jointly by the School of Data Science, +the School of Science and Engineering, +and the School of Management and Eco- +nomics, The Chinese University of Hong +Kong, Shenzhen (CUHK-SZ), China. His +research interests include deep learning +and financial technology. +Liang Lin (M’09, SM’15) is a Full Pro- +fessor of computer science at Sun Yat- +sen University. He served as the Exec- +utive Director and Distinguished Scien- +tist of SenseTime Group from 2016 to +2018, leading the R&D teams for cutting- +edge technology transferring. He has au- +thored or co-authored more than 200 pa- +pers in leading academic journals and con- +ferences, and his papers have been cited by +more than 22,000 times. He is an associate +editor of IEEE Trans. Multimedia and IEEE Trans. Neural Networks +and Learning Systems, and served as Area Chairs for numerous +conferences such as CVPR, ICCV, SIGKDD and AAAI. He is the +recipient of numerous awards and honors including Wu Wen-Jun +Artificial Intelligence Award, the First Prize of China Society of +Image and Graphics, ICCV Best Paper Nomination in 2019, Annual +Best Paper Award by Pattern Recognition (Elsevier) in 2018, Best +Paper Dimond Award in IEEE ICME 2017, Google Faculty Award +in 2012. His supervised PhD students received ACM China Doctoral +Dissertation Award, CCF Best Doctoral Dissertation and CAAI Best +Doctoral Dissertation. He is a Fellow of IET and IAPR. + diff --git a/7NAyT4oBgHgl3EQfpviE/content/tmp_files/load_file.txt b/7NAyT4oBgHgl3EQfpviE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..686bc541857c547fce583fa7eb629669d1d80a25 --- /dev/null +++ b/7NAyT4oBgHgl3EQfpviE/content/tmp_files/load_file.txt @@ -0,0 +1,1857 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf,len=1856 +page_content='SUBMISSION TO IEEE TRANSACTION ON MULTIMEDIA 1 Multi-Stage Spatio-Temporal Aggregation Transformer for Video Person Re-identification Ziyi Tang, Ruimao Zhang, Member, IEEE, Zhanglin Peng, Jinrui Chen, Liang Lin, Senior Member, IEEE Abstract—In recent years, the Transformer architec- ture has shown its superiority in the video-based person re-identification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Inspired by video representation learning, these methods mainly focus on designing mod- ules to extract informative spatial and temporal fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, they are still limited in extracting local attributes and global identity information, which are critical for the person re-identification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In this paper, we propose a novel Multi-Stage Spatial-Temporal Aggre- gation Transformer (MSTAT) with two novel designed proxy embedding modules to address the above issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Specifically, MSTAT consists of three stages to encode the attribute-associated, the identity-associated, and the attribute-identity-associated information from the video clips, respectively, achieving the holistic perception of the input person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' We combine the outputs of all the stages for the final identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In practice, to save the computational cost, the Spatial-Temporal Aggregation (STA) modules are first adopted in each stage to conduct the self-attention operations along the spatial and tem- poral dimensions separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' We further introduce the Attribute-Aware and Identity-Aware Proxy embedding modules (AAP and IAP) to extract the informative and discriminative feature representations at different stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' All of them are realized by employing newly designed self-attention operations with specific meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Moreover, temporal patch shuffling is also introduced to further improve the robustness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Extensive experimental results demonstrate the effectiveness of the proposed modules in extracting the informative and discriminative information from the videos, and illustrate the MSTAT can achieve state-of-the-art accuracies on various standard benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Index Terms—Video-based Person Re-ID, Trans- former, Spatial Temporal Modeling, Deep Representation Learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' INTRODUCTION P ERSON Re-identification (re-ID) [6], [26], [28], which aims at matching pedestrians across dif- ferent camera views at different times, is a critical Ziyi Tang, Ruimao Zhang, and Jinrui Chen are with The Chi- nese University of Hong Kong (Shenzhen), and Ziyi Tang is also with Sun Yat-sen University (e-mail: tangziyi@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='cn, ruimao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='zhang@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='org, and 120090765@link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='cn ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Zhanglin Peng is with the Department of Computer Science, The University of Hong Kong, Hong Kong, China (e-mail: zhanglin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='peng@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='hk ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Liang Lin is with the School of Computer Science and Engineer- ing, Sun Yat-sen University (e-mail: linliang@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' This paper was done when Ziyi Tang was working as a Research Assistant at The Chinese University of Hong Kong (Shenzhen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The Corresponding Author is Ruimao Zhang Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 1: Comparison between different Transformer- based frameworks for video re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (a) shows the framework where the Transformer fuse post-CNN fea- tures of the entire video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (b) is Trigeminal Trans- former [51], including three separate streams for tem- poral, spatial, and spatio-temporal feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (c) displays a multi-stage spatio-temporal aggregation Transformer, which consists of three stages, all with a spatio-temporal view but different meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' task of visual surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In the earlier stage, the studies have mainly focused on image-based person re-ID [26], [28], [46], which mine the discrimina- tive information in the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' With the de- velopment of the monitoring sensors, multi-modality information has been introduced to re-ID task [33], [71], [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Numerous methods have been proposed to break down barriers between modalities regarding their image styles [86], structural features [81], [84], [97], or network parameters [33], [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' On the other hand, some studies have exploited multi-frame data and proposed various schemes [40], [62], [100] to extract informative temporal represen- tations to pursue video-based person re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In such a setting, each time a non-labeled query tracklet clip is given, its discriminative feature representation needs to be extracted to retrieve the clips of the corresponding person in the non-labeled gallery.' 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+page_content='featurerepresentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='AttributeSpatio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Space-time-relatedmodule/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Spatio-Temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Spatio-Temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='featurerepresentation ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Attribute-ldentity- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='/featurerepresentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='AssociatedStage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Identity-related module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='(c) MSTATTANG et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' : MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION 2 from spatial and temporal dimensions is the key to improving the accuracy of video-based re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' To address such an issue, traditional methods [20] usually employ hierarchically convolutional architec- tures to update local patterns progressively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Further- more, some attempts [14], [15], [48], [73], [94] adopt attention-based modules to dynamically infer discrim- inative information from videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For instance, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [72] embed body part prior knowledge inside the network architecture via dense and non-local region- based attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Although recent years have witnessed the success of convolution-based methods [12], [13], [20], [38], [43], [74], [94], [104], they have encoun- tered a bottleneck of accuracy improvement, as con- volution layers suffer from their intrinsic limitations of spatial-temporal dependency modeling and infor- mation aggregation [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Recently, the Transformer architecture [24], [32], [54], [89] has attracted much attention in the com- puter vision area because of its excellent context modeling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The core idea of such a model is to construct interrelationships between local contents via global attention operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In the literature, some hybrid network architectures [19], [34], [51] have been proposed to tackle long-range context modeling in video-based re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' A widely used paradigm is to leverage Transformer as the post-processing unit, coupled with a convolutional neural network (CNN) as the basic feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For example, as sum- marized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 1 (a), He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [35] and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [95] adopt a monolithic Transformer to fuse frame-level CNN feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 1 (b), Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [51] take a step further and put forward multi-stream Transformer architecture in which each stream emphasizes a particular dimension of the video features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In a hybrid architecture, however, the 2D CNN bottom encoder restricts the long-range spatio- temporal interactions among local contents, which hinders the discovery of contextual cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Later, to address this problem, some pure Transformer-based approaches are introduced to video-based re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Nev- ertheless, the existing Transformer-based frameworks are mainly motivated by those in video understanding and concentrate on designing the architecture to learn spatial-temporal representations efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Most of them are still limited in extracting informative and human-relevant discriminative information from the video clips, which are critical for large-scale matching tasks [39], [92], [98], [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' To address the above issues, we propose a novel Multi-stage Spatial-Temporal Aggregation Trans- former framework, named MSTAT, which consists of three stages to respectively encode the attribute- associated, the identity-associated, and the attribute- identity-associated information from video clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Firstly, to save the computational cost, the Spatial- Temporal Aggregation (STA) modules [4], [7] are firstly adopted in each stage as their building blocks to conduct the self-attention operations along the spatial and temporal dimensions separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Further, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 1, we introduce the plug-and-play Attribute-Aware Proxy and Identity-Aware Proxy (AAP and IAP) embedding modules into different stages, for the purpose of reserving informative at- tribute features and aggregating discriminative identity features respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' They are both implemented by self-attention operations but with different learnable proxy embedding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For the AAP embedding module, AAPs play the role of attribute queries to reserve a diversity of implicit attributes of a person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Arguably, the combination of these attribute repre- sentations is informative and provides discriminative power, complementary to the identity-only prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In contrast, the IAP embedding module maintains a group of IAPs as key-value pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' With explicit con- straints, they learn to successively match and aggregate the discriminative identity-aware features embedded in patch tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' During similarity measurement, the output feature representations of the three stages are concatenated to form a holistic view of the input person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In practice, a Transformer-specific data augmenta- tion scheme, Temporal Patch Shuffling, is also intro- duced, which randomly rearranges the patches tem- porally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' With such a scheme, the enriched training data effectively improve the ability to learn invariant appearance features, leading to the robustness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Extensive experiments on three public bench- marks demonstrate our proposed framework is superior to the state-of-the-art on different metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Concretely, we achieve the best performance of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8% rank-1 accuracy on MARS, which is the largest video re-ID dataset at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In summary, our contributions are three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (1) We introduce a Multi-stage Spatial-Temporal Aggregation Transformer framework (MSTAT) for video-based per- son re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Compared to existing Transformer-based frameworks, MSTAT better learns informative attribute features and discriminative identity features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (2) For different stages, we devise two different proxy embed- ding modules, named Attribute-Aware and Identity- Aware Proxy embedding modules, to extract infor- mative attribute features and aggregate discriminative identity features from the entire video, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (3) A simple yet effective data augmentation scheme, referred to as Temporal Patch Shuffling, is proposed to consolidate the network’s invariance to appearance shifts and enrich training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' RELATED WORKS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Image-Based Person Re-ID Image-based person re-ID mainly focuses on person representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Early works focus primarily on carefully designed handcraft features [6], [26], [28], [44], [46], [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Recently, The flourishing deep learn- ing has become the mainstream method for learning 3 IEEE TRANSACTIONS ON MULTIMEDIA representation in person ReID [43], [65], [67], [74], [77], [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For the last few years, CNN has been a widely-used feature extractor [1], [17], [41], [43]–[45], [65], [76], [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' OSNet [104] fuses multi-scale features in an attention-style sub-network to obtain informative omni-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Some works [18], [87], [98] focus on extracting and aligning semantic information to address misalignment caused by pose/viewpoint variations, imperfect person detection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' To avoid the misleading by noisy labels, Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [83] presents a self-label refining strategy, deeply integrating anno- tation optimization and network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' So far, some works [19], [34] also explore Image-based person re- ID based on Vision Transformer [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For example, TransReID [34] adopts Transformer as the backbone and extracts discriminative features from randomly sampled patch groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Video-Based Person ReID Compared to image-based person re-ID, video- based person re-ID usually performs better because it provides temporal information and mitigates occlusion by taking advantage of multi-frame information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For capturing more robust and discriminative representa- tion from frame sequences, traditional video-based re- ID methods usually focus on two areas: 1) encoding of temporal information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 2) aggregation of temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' To encode additional temporal information, early methods [40], [62], [100] directly use temporal infor- mation as additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Some works [1], [49], [55], [73] use recurrent models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', RNNs [56] and LSTM [37], to process the temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Some other works [1], [12], [13], [53], [55], [60], [105] go further by introducing the attention mechanism to apply dynamic temporal feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Another class of works [21] introduces optical flow that captures temporal motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' What is more, some works [2], [42], [63], [75], [91], [102] directly implement spatio- temporal pooling to video sequences and generate a global representation via CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Recently, 3D CNNs [29], [45] learn to encode video features in a joint spatio-temporal manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' M3D [41] endows 2D CNN with multi-scale temporal feature extraction ability via multi-scale 3D convolutional kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For the sake of aggregation that aims to generate discriminative features from full video features, a class of approaches [55], [93], [105] applies average pooling on the time dimension to aggregate spatio- temporal feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Recently, some attention-based methods [2], [15], [72], [80] attained significant per- formance improvement by dynamically highlighting different video frames/regions so as to filter more dis- criminative features from these critical frames/regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For instance, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [51] introduce cross-attention to aggregate multi-view video features by pair-wise interaction between these views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Apart from the explo- ration of more effective architectural design, a branch of works study the effect of pedestrian attributes [10], [61], [101], such as shoes, bag, and down color, or the gait [11], [57], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' walking style of pedestrians, as a more comprehensive form of pedestrian description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [11] closely integrate two coherent tasks: gait recognition and video-based re-ID by using a hybrid framework including a set-based gait recogni- tion branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Some works [61], [101] embed attribute predictors into the network supported by annotations obtained from a network pretrained on an attribute dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Chai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [10] separate attributes into ID- relevant and ID-irrelevant ones and propose a novel pose-invariant and motion-invariant triplet loss to mine the hardest samples considering the distance of pose and motion states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Although the above methods have made significant progress in performance, Transformer [66], which is deemed a more powerful architecture to process se- quence data, may raise the performance ceiling of video-based re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' To illustrate this, Transformer can readily adapt to video data with the support of the global attention mechanism to capture spatio-temporal dependencies and temporal positional encoding to or- der spatio-temporal positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In addition, the class token is off-the-shelf for Transformer-based models to aggregate spatio-temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, Transformer suffers from multiple drawbacks [24], [70], [89], [90], and few works have been released so far on video-based person re-ID based on Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In this work, we attempt to explore the potential of intractable Transformer in video-based person re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Vision Transformer Recently, Transformer has shown its ability as an alternative to CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Inspired by the great success of Transformer in natural language processing, recent researchers [24], [54], [54], [70] have extended Trans- former to CV tasks and obtained promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Bertasius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [7] explores different video self- attention schemes considering their cost-performance trade-off, resulting in a conclusion that the di- vided space-time self-attention is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Similarly, ViViT [4] factorizes self-attention to compute self- attention spatially and then temporally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Inspired by these works, we divide video self-attention into spa- tial attention followed by temporal attention, and we further propose a attribute-aware variant for video- based re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Furthermore, little research has been done on Transformer for Video-based person re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Trigeminal Transformers (TMT) [51] puts the input patch token sequence through a spatial, a temporal, and a spatio-temporal minor Transformer, respectively, and a cross-view interaction module fuses their outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Differently, MSTAT has three stages, all extracting spatio-temporal features but with different meanings: (1) attribute features, (2) identity features, (3) attribute- identity features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' TANG et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' : MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Tokenization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Patch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Tokens ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Attribute-aware Prxoy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Embedding Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='L CE + L Tri ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='N3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Stage III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Spatio-Temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Stage II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Class Token Re-init ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='L CE + L Tri ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='L CE + L Tri ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Inference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Element-wise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Addition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Spatial Positional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Encoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Concatenation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Attribute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='A-Spatio-Temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Spatio-Temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Stage I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Identity-aware Prxoy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Embedding Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Identity-aware Prxoy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Embedding Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 2: The overall architecture of our proposed MSTAT which consists of three stages, all based on the Transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Stage I updates the spatio-temporal patch token sequence of the input video and aggregates them into a group of attribute-associated representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Subsequently, Stage II aggregates discriminative identity-associated features and Stage III attribute-identity-associated features, relying upon their stage-specific class tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Here, we omit the input and output of each module except the attribute- aware proxy embedding module in Stage I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' At inference time, all these feature representations are combined through concatenation to infer the pedestrian’s identity jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' METHOD In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' III-A, we first overview the proposed MSTAT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Then, Spatio-Temporal Aggrega- tion (STA), the normal spatial-temporal feature extrac- tor in MSTAT, is formulated in section Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Along with it, we introduce the proposed Attribute- Aware Proxy (AAP) and Identity-Aware Proxy (IAP) embedding modules in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Finally, Tem- poral Patch Shuffling (TPS), a newly introduced Transformer-specific data augmentation scheme, is presented in section III-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Overview This section briefly summarizes the workflow of MSTAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The overall MSTAT framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Given a video tracklet V P RT ˆ3ˆHˆW with T frames and the resolution of each frame is H ˆ W, the goal of MSTAT is to learn a mapping from a video tracklet V to a d-dimension representation space in which each identity is discriminative from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Specifically, as shown on the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 2, MSTAT first linearly projects non-overlapping image patches of size 3 ˆ P ˆ P into d-dimensional patch tokens, where d “ 3P 2 denotes the embedded dimension of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Thus, a patch token sequence X P RT ˆNˆd is obtained, where the number of patch tokens in each frame is denoted by N “ HˆW P 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Meanwhile, spatial positional encoding E P RNˆd is added to X in a element-wise manner for reserving spatial structure in each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Notably, we do not insert temporal positional encoding into X, since the temporal order is usually not conducive to video-based re-ID, which is also demonstrated in [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Finally, a class token c P Rd is associated with X to aggregate global identity representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Next, we feed the token sequence X into Stage I of MSTAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' It takes X and c as input, and employs a stack of eight Spatio-Temporal Aggregation (STA) blocks for inter-frame and intra-frame correlation modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The output tokens are then fed into an Attribute-Aware Proxy (AAP) embedding module to mine rich visual attributes, a composite group of semantic cues that im- ply identity information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', garments, handbags and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The Stage II includes a series of STA blocks (three in our experiments), followed by an Identity-Aware Proxy (IAP) embedding module which is able to screen out discriminative identity-associated information by inspecting the entire sequence in par- allel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In the Stage III, we first introduce a novel class token to directly aggregate higher-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In addition, a stack of Attribute-STA (A-STA) blocks is used to fuse attributes from different frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' At last, an IAP embedding module is adopted to generate a discriminative representation for the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In the training phase, the attribute representations extracted from Stage I and the class tokens of Stage II and Stage III are supervised separately by a group of losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' During the testing, the attribute representations and the class tokens from the last two stages are concatenated for similarity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Spatio-temporal Aggregation To begin with, we make a quick review of the vanilla Transformer self-attention mechanism first proposed in [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In practice, visual Transformer embeds an image into a sequence of patch tokens, and self-attention operation first linearly projects these tokens to the corresponding query Q, key K and value V respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Then, the scaled product of Q and K generates an attention map A, indicating estimated relationships s:/iblog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='csdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='net/qqn34182315 IEEE TRANSACTIONS ON MULTIMEDIA between token representations in Q and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Then, V performs a re-weighting by multiplying the attention map A, to obtain the output of Transformer self- Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In this way, patch tokens are reconstructed by leveraging interaction with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Formally, self-Attention operation SAp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='q can be formulated as follows: Q, K, V “ ˆSWq, ˆSWk, ˆSWv A “ SoftmaxpQKTq{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' d SApˆSq “ AV (1) where ˆS P R ˆ Nˆd denotes an 2-dimensional input token sequence, and Wq, Wk, Wv P Rdˆd1 denote three learnable parameter matrices of size d ˆ d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In the multi-head setting, we let d1 “ d{n, where n indicates the number of attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The function Softmaxp¨q denotes the softmax operation for each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' And the scaling operation in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (1) eliminates the influence from the scale of embedded dimension d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In our Spatio-Temporal Aggregation block (STA), self-attention operation along time axis and along space axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' temporal attention and spatial attention) are separately denoted as SAtp¨q and SAsp¨q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Let S P R ˆT ˆ ˆ Nˆd denote an input spatio-temporal token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Formally, SAtp¨q and SAsp¨q can be written as: SAtpSq “ SApConcatpS:,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', S:,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', S:,N´1qq SAspSq “ SApConcatpS0,:, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', St,:, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', ST´1,:qq (2) where T indicates the total number of frames in video clip, N is the total spatial position index, and Concatp¨q denotes the concatenation operation in the split dimension, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', the spatial position dimension in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Given SAtp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='q and SAsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='q, the STA block consecu- tively integrates these two self-attention modules to ex- tract spatial-temporal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 3, STA further extracts discriminative information from patch tokens to the class token through spatial attention SAsp¨q, which can be realized by concatenating the copies of class token to the token sequence of each frame before SAsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='q, and taking the average of class token copies after SAsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='q to further apply the later temporal aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In this way, the general form of STA can be presented as: S1 “ S ` α ˆ SAtpLNpSqq STApS, cq “ ConcatpS1, cq `β ˆ SAspLNpConcatpS1, cqqq (3) where LNp¨q denotes Layer Normalization [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The hyper-parameter α and β are learnable scalar residual weights to balance temporal attention and spatial at- tention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Compared with the space-time joint attention in [7] and [4], which jointly processes all patches of a video, STA is more computation-efficient by reducing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 3: The detailed comparison between (a) Spatio- Temporal Aggregation block (STA) and (b) Attri- bution Spatio-Temporal Aggregation block (A-STA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Two additional Attribute-Aware Proxy (AAP) embed- ding modules are placed into the latter, before and after the temporal attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The class token broadcasting operation duplicates the class token for each frame to attend spatial attention within a specific frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Oppositely, class token averaging calculates the average of all class token copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Note that the Pre- Norm [79] layers before temporal attention and spatial attention are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 4: The detailed module design of the Attribute- Aware Proxy (AAP) embedding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The Attribute-Aware Proxy Embedding denotes a learnable matrix that is used as the query of the attention operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For simplicity, this figure only shows the single-head version of the AAP embedding module and the scaling operation before the softmax operation is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' complexity from OpT 2N 2q to OpT 2 ` N 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Actually, it avoids operating on a long sequence, whose length always leads to quadratic growth of computational complexity [31], [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Attribute-Aware Proxy Embedding Module Local patch tokens usually contain rich attributive information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', glasses, umbrellas, logos, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Even if a single attribute is not discrimina- tive enough to recover one’s identity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' the combinations 00000 00000 个个个个个 个个个个个 ClassToken Class Token Averaging Averaging SpatialAttention Spatial Attention ClassToken ClassToken Broadcasting Broadcasting AAP embeddingmodule TemporalAttention TemporalAttention AAP embeddingmodule 个个个个个 个不不个个 00000 00000 (a) STA (b) A-STAAttribute Representation Linear Attribute-Aware i Proxy Embedding : Module Softmax Q K V Attribute-Aware Linear Linear Proxy Embeddings Class 0000000 Patch Token TokenTANG et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' : MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION 6 of a pedestrian’s rich attributes should be discrimina- tive as each attribute eliminates a certain degree of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Rather than directly aggregating into a “coarse” class token, we introduce the Attribute-Aware Proxy (AAP) embedding module to directly extract attribute features from a single-frame or multi-frame patch token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Practically, AAP embeddings are formed by a learnable matrix with anisotropic initialization for the richness of learned attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' It can be considered as the “attribute bank” to serve as the query of the attention operation to match with the feature representations of the input patch tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Specifically, AAP embeddings interact with the keys of the patch token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Finally, the resulting attention map is used to re-weight the value, generating the attribute representations of the specific video clip with the same dimension of AAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Formally, an AAP embedding module can be written as follows, Q, K, V “ PQ, SWk, SWv, AAPpSq “ SoftmaxpQKTq ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' d V (4) here we use the multi-head version of AAP embedding module in practice, which has the same multi-head setting as SAp¨q in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Note that the spatio- temporal input S here can also be ˆS P R ˆ Nˆd for spatial-only use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Compared with SAp¨q, the newly proposed AAP module consider the query Q in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (1) as the a set of learnable parameters PQ P RNaˆd1, where Na !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' N is a hyper-parameter that indicates the number of AAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' By controlling Na, the AAP module could have a manually defined capacity, which leads to flexibility for various real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 2, both Stage I and Stage III employ the proposed AAP embedding modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Specifically, in Stage I, the proposed AAP module is firstly used to generate attribute representations from a multi-frame sequence of patch tokens S P R ˆT ˆ ˆ Nˆd for similarity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Although we do not have any attribute-level annotations, we hope the AAP module can automatically learn a rich set of implicit attributes from the entire training dataset, while these resultant attribute representations could also present discrimina- tive power complementary to ID-only representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' To achieve this goal, the ID-level supervision signal is first imposed on the combination of learned attribute representations to constrain its discriminative power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In addition, we initialize the AAPs with anisotropic distributions to capture diverse implicit attribute rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In practice, we surprisingly find that such anisotropy can maintain after the model training, which means such optimized AAP could respond to a set of differentiated attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Moreover, the number of AAPs can be relatively large compared with the class token to cover rich attribute information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In this sense, both the richness and diversity of learned implicit attributes can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In Stage III, we further insert two intra-frame AAP embedding modules before and after the tem- poral attention of each STA to conduct attribute-aware temporal interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Such a modified STA block is named A-STA, which is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In A-STA, semantic-related attributes in different frames experi- ence inter-frame interaction to model their temporal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In the end, after temporal attention, we set Na equal to N for the second AAP embedding module so that it has N tokens as output to keep the input- output consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Identity-Aware Proxy Embedding Module Extracting discriminative identity representation is also crucial for video-based re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' To this end, the Identity-Aware Proxy (IAP) embedding module is pro- posed for effective and efficient discriminative repre- sentation generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In previous works, joint space- time attention has shown promising results [4], [7], as it accelerates information aggregation by applying self- attention over spatial and temporal dimensions jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, the quadratic computational overheads limit its applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The IAP embedding module is pro- posed to address such an issue, which performs joint space-time attention with high efficiency while main- taining the discrimination of the identity feature rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The IAP module contains a set of identity proto- types, which are presented as two learnable matrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In practice, we exploit them to replace the keys tpi KuM i“1 P PK and values tpi V uM i“1 P PV of the attention operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Both PK, PV P RMˆd1, where M P N` denotes the number of identity prototypes and determine the capacity of the IAP module (usually M !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 5, an attention map A P RMˆN is first calculated to present the affinity between prototype-patch pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Thus each element in A reflects how close a patch token is to a specific identity prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Then this attention map is sparsified by suc- cessively applying an L1 normalization and softmax normalization along M and N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' At last, the class token c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' the first row of V, is updated by the multiplication of V and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Such an operation aggregates the most discriminative identity features from the entire patch token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Formally, given the spatio-temporal token sequence S, the output of the IAP module can be calculated as follows: Q, K, V “ SWq, PK, PV A “ SoftmaxpL1NormpQKTqq ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' d IAPpSq “ AV (5) where K and V are not conditioned on input S but are learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Here we insert an L1 normalization layer before the softmax operation in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (5), resulting in double normalization [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Such a scheme performs patch token re-coding to reduce the noise of patch representations, leading to 7 IEEE TRANSACTIONS ON MULTIMEDIA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 5: The detailed module design of the Identity- Aware Proxy (IAP) embedding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The IAP em- bedding denotes the learnable matrix used to calculate the key or value of the attention operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Here we only show the single-head version of the IAP embedding module and omit the scaling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In such a scheme, The output token sequence can be considered as reconstruction by a group of IAPs, which tend to reserve the most discriminative identity features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' robust identification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Specifically, the learnable matrix PK matches the input tokens through the double normalization operation to generate the affinity map A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Then these input tokens are thereupon re-coded through a projection of PV along A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Since the numbers of learnable vectors in PK and PV are much smaller than the number of input tokens, the above operation has been able to represent each token in a more compact space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' linear combination of the vectors in PV ), effectively suppressing irrelevant information for re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Moreover, IAPp¨q has OpNq computational complexity since the number of identity prototypes M is fixed and is usually much less than the total number of patch tokens of a specific video tracklet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', 64 in our experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' So, the proposed IAP embedding module allows all spatio-temporal patch tokens to be processed in parallel for effective and efficient feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Temporal Patch Shuffling To improve the robustness of the model, we propose a novel data augmentation scheme termed Temporal Patch Shuffling (TPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Suppose we have one patch sequences Ri “ tri1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', rit, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', riT u from the same video clip, where the sub-index i denotes specific spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 6, the proposed TPS randomly permutes the patch tokens in Ri and refill the shuffled sequence ˆ Ri to form the new video clip for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 6, we could simultaneously select multiple spatial regions in one video clip for shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' While in the inference phase, the original video clip is directly fed into the model for identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' TPS brings firm appearance shifts and motion changes from which the network learns to extract generalizable and invariant visual clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In addition, the scale of available training data can be f1 f2 f3 f4 f5 r21 r22 r23 r24 r25 r11 r12 r13 r14 r15 r15 r14 r11 r12 r13 r24 r23 r25 r22 r21 X x’ shuffling f1 f2 f3 f4 f5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 6: Visualization of Temporal Patch Shuffling (TPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' ft represents tth frame, rit the patch in spatial position i and tth frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' TPS is a built-in data aug- mentation scheme that randomizes the order of a patch sequence sampled from spatial position i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As a result, for example, the patch in the red box is transferred from the 5th frame to the 1st frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' greatly extended based on such a scheme, which helps to prevent the network from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In our experiments, we treat TPS as a plug-and-play operation and implement it at the stem of the network to promote the entire network for the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The following section will conduct ablation studies to explore where to insert TPS and to what extent TPS should be for optimal training results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' EXPERIMENT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Datasets and evaluation protocols In this paper, we evaluate our proposed MSTAT on three widely-used video-based person re-ID bench- marks: iLIDS-VID [69], DukeMTMC-VideoReID (DukeV) [59], and MARS [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 1) iLIDS-VID [69] is comprised of 600 video track- lets of 300 persons captured from two cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In these video tracklets, frame numbers range from 23 and 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The test set shares 150 identities with the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 2) DukeMTMC-VideoReID [59] is a large-scale video-based benchmark which contains 4, 832 videos sharing 1, 404 identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In the following sections, we use the abbreviation “DukeV” for the DukeMTMC- VideoReID dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The video sequences in the DukeV dataset are commonly longer than videos in other datasets, which contain 168 frames on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 3) MARS [102] is one of the largest video re- ID benchmarks which collects 1, 261 identities exist- ing in around 20, 000 video tracklets captured by 6 cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Frames within a video tracklet are relatively more misaligned since they are obtained by a DPM detector [27] and a GMMCP tracker [22] rather than hand drawing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Furthermore, around 3, 200 distractor tracklets are mixed into the dataset to simulate real- world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For evaluation on MARS and DukeV datasets, we use two metrics: the Cumulative Matching Character- istic (CMC) curves [8] and mean Average Precision (mAP) following previous works [16], [51], [94], [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, in the gallery set of iLIDS-VID, there is merely one correct match for each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For this benchmark, only cumulative accuracy is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 00000000 Identity-Aware : Proxy Embedding i Module Softmax L1Norm K V Identity-Aware Identity-Aware Linear Proxy Embeddings Proxy Embeddings Class Patch Token TokensXXTANG et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' : MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION 8 Method Source Backbone MARS Duke-V iLIDS-VID Rank-1 Rank-5 mAP Rank-1 Rank-5 mAP Rank-1 Rank-5 SCAN [94] TIP19 Pure-CNN 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 VRSTC [39] CVPR19 Pure-CNN 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 M3D [39] AAAI19 Pure-CNN 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 MG-RAFA [99] CVPR20 Pure-CNN 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 AFA [16] ECCV20 Pure-CNN 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 AP3D [29] ECCV20 Pure-CNN 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 TCLNet [16] ECCV20 Pure-CNN 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 A3D [15] TIP20 Pure-CNN 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 GRL [52] CVPR21 Pure-CNN 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 STRF [3] ICCV21 Pure-CNN 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [25] WACV21 Pure-CNN 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 TMT [51] Arxiv21 CNN-Transformer 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [47] CVPR21* CNN-Transformer 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 STT [95] Arxiv21 CNN-Transformer 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 ASANet [10] TCSVT22 Pure-CNN 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 MSTAT(ours) Pure-Transformer 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 TABLE I: Result comparison with state-of-the-art video-based person re-ID methods on MARS, DukeMTMC- VideoReID, and iLIDS-VID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' * denotes the workshop of the conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Implementation details Our proposed MSTAT framework is built based on Pytorch toolbox [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In our experiments, it is running on a single NVIDIA A100 GPU (40G memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' We resize each video frame to 224 ˆ 112 for the above benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Typical data augmentation schemes are involved in training, including horizontal flipping, ran- dom cropping, and random erasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For all stages, STA modules are pretrained on an action recognition dataset, K600 [9], while other aforementioned modules are randomly initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In the training phase, if not specified, we sample L “ 8 frames each time for a video tracklet and set the batch size as 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In each mini-batch, we randomly sample two video tracklets from different cameras for each person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' We supervise the network by cross-entropy loss with label smoothing [64] asso- ciated with widely used BatchHard triplet loss [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Specifically, we impose supervision signals separately on the concatenated attribute representation from the AAP embedding module in Stage I, the output class tokens from Stage II, and Stage III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The learning rate is initially set to 1e-3, which would be multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='75 after every 25 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The entire network is updated by an SGD optimizer in which the weight decay and Nesterov momentum are set to 5 ˆ 10´5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In the test phase, following [29], [95], we randomly sample 32 frames as a sequence from each original tracklet in either query or gallery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For each sequence, The attribute representation from Stage I, the output class tokens from stage Stage II and Stage III are concatenated as the overall representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Following the widely-used protocol, we compute the cosine sim- ilarity between each query-gallery pair using their overall representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Then, the CMC curves and the mAP can be calculated based on the predicted ranking list and the ground truth identity of each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Note that we do not use any re-ranking technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Compared with the state of the arts In Table I, we make a comparison on three bench- mark datasets between our method and video-based person re-ID methods from 2019 to 2021, including M3D [39], GRL [52], STRF [3], Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [25], TMT [51], Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [47], ASANet [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' According to their backbones, these re-ID methods can be roughly divided into the following types: Pure-CNN, CNN- Transformer Hybrid, and Pure-Transformer methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In real-world applications, rank-1 accuracy [8] re- flects what extent a method can find the most confident positive sample [85], and relatively high rank-1 accu- racy can save time in confirmation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As the first method based on Pure-Transformer for video-based re-ID so far, we achieve state-of-the-art results in rank-1 ac- curacy on three benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Our approach especially attains rank-1 accuracy of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8% and rank-5 accuracy of 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4% on the largest-scale benchmark, MARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' It is noteworthy that our MSTAT outperforms the best pure CNN-based methods using ID annotations only by a margin of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1% and a CNN-Transformer hybrid method, TMT, by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6% in MARS rank-1 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Compared to our proposed method, TCLNet [16] explicitly captures complementary features over dif- ferent frames, and GRL [52] devises a guiding mech- anism for reciprocating feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, the designed modules in these methods commonly take as input the deep spatial feature maps extracted by a CNN backbone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' ResNet50) that may overlook attribute-associated or identity-associated information without explicit modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Similar to ours, TMT [51] and M3D [3] process video tracklets in multiple views to extract and fuse multi-view features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' No- tably, in all stages of MSTAT, intermediate features are spatio-temporal and can be iteratively updated to capture spatio-temporal cues with different emphases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' ASANet [10] exploits explicit ID-relevant attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', gender, clothes, and hair) and ID-irrelevant at- tributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=', pose and motion) on a multi-branch net- 9 IEEE TRANSACTIONS ON MULTIMEDIA Method Test Protocol Rank-1 Rank-5 mAP MSTAT Stage I 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 Stage II 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 Stage III 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 Stage I & II 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 Stage I & III 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9 Stage II & III 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 Stage I, II, & III 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 TABLE II: Ablation study on three stages of MSTAT on MARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Test Protocol means the final feature rep- resentation used for similarity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The net- work architecture and training hyper-parameter setting remain the same for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Despite the performance growth, the demand for attribute annotations may limit its applications in large- scale scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In comparison with existing methods, our method aggregates spatio-temporal information in a unified manner and explicitly capitalizes on implicit attribute information to improve recognizability un- der challenging scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Conclusively, our method achieves the state-of-the-art performance of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8% and 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3% rank-1 accuracy, respectively, on MARS and iLIDS-VID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Effectiveness of Multi-Stage Framework Architec- ture To evaluate the effectiveness of the three stages in our proposed MSTAT, we carry out a series of ablation experiments whose results are displayed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' After the three stages are jointly trained, we first separately evaluate each stage using its output feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Then, we concatenate two or more stages to evaluate whether each is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For three single stages, each has rank-1 accuracy ranging from 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 to 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, their combina- tions result in a significant increase of over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Remarkably, while Stage I and Stage II secure only 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 rank-1 accuracy, their integration attains up to 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2%, surpassing them by a 2% margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' One can attribute such a result to their emphases: one stage on attribute-associated features and the other on identity- associated features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Eventually, when all three stages are used, MSTAT reaches a 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8% rank-1 accuracy, higher than all two-stage combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Overall, these experiments demonstrate that the three stages have dif- ferent preferences toward features and can complement each other by simple concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Effectiveness of Key Components To demonstrate the effectiveness of our proposed MSTAT, we conduct a range of ablative experiments on the largest public benchmark MARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 1) Effectiveness of Attribute-Aware Proxy Embed- ding Module: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 7, we evaluate MSTAT with different AAP numbers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Na in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' III-C) in the AAP embedding module in the last layer of Stage Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 7: Ablation study on the attribute-aware proxy (AAP) embedding module for attribute extraction in MARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' ”Base” is the network without attribute ex- traction using AAP in training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' AAP- k indicates the network where the AAP embedding module in Stage I has k AAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 8: Ablation study on A-STA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' ”Base” is the net- work that consists of STA only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' A-STA-k represents the network in which Stage III is equipped with A- STA layers each of k AAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The figure reveals that 24 proxies are optimal for attributive information extraction as it attains the best performance in terms of rank-1 and rank-5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In contrast to the baseline, MSTAT has seen over 2% growth in rank-1 accuracy and around 1% in rank-5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, a redundant or insufficient number of AAPs may cause a minor performance drop since they may pay attention to noisy or useless attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In summary, the AAP embedding module for clue extraction gives a boost to the performance in rank- 1 and rank-5 accuracy, with negligible computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Attribute-Aware Proxy (AAP) embedding modules are also used for A-STA, a variant of STA for attribute- aware temporal feature fusion in Stage III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 8, we conduct a series of experiments to explore whether A-STA is effective and how many AAPs for A-STA are appropriate (also corresponding to Na in III-C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The experiment results reveal that the baseline model fails to reach 90% rank-1 accuracy or 97% rank- 5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As the number of AAPs increases, these two metrics grow to 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Therefore, we can attribute the performance soar to A-STA, allowing for attribute-aware temporal in- teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' A-STA offers a different viewpoint from that of Stage II on videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Moreover, due to the redundancy of temporal information in many video re- ID scenarios discussed in [16], A-STA with too many AAPs incurs meaningless attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' This can be why the performance descends once A-STA has too many AAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Rank-1 accuracy Rank-5 accuracy 9160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='918 0.' metadata={'source': 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ASTA-16 ASTA-32 ASTA-4B ASTA-64TANG et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' : MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 9: Study on the effect of training video sequence length on MARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In conclusion, our proposed AAP embedding mod- ule can be used for: (1) the extraction of informative attributes as plugged into any Transformer layer and (2) attribute-aware temporal interaction when a tempo- ral attention module is sandwiched between two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Both of the two functionalities cause a significant increase in performance, demonstrating their effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 2) Effectiveness of Identity-Aware Proxy Embedding Module: In Table III, MSTAT that discards IAP em- bedding modules leads to only 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2% rank-1 accuracy and 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4% rank-5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, it boosts rank-1 performance by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8% or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2% by taking the place of STA in Stage II or Stage III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Finally, IAP embedding modules in the last layers in both Stage II and Stage III further improve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8% rank-1 accuracy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4% rank-5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The IAP embedding module’s ablation results demonstrate its ability to generate discriminative representations efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Intuitively, we place the IAP embedding module only in the last few depths because it may discard non-discriminative features that should be preserved in shallow layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 3) Effectiveness of Temporal Patch Shuffling: To evaluate the effectiveness of Temporal Patch Shuffling (TPS), we assign different probabilities to implement TPS for each training video sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Note that in the following experiments, the number of spatial positions to shuffle is set to 5 if we implement TPS on this sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As shown in Table IV, 20% probability provides the best result over others, which leads to a growth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3% in rank-1 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, the 60% or 80% probability results in a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1% or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2% rank-1 accuracy drop mainly due to heavy noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In summary, a proper level of TPS would be an effective data augmentation method for the Transformer for video-based person re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Further, rather than reserving temporal motion (an ordered sequence of patches), TPS stimulates re- identification accuracy by learning temporal coherence from shuffled patch tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Effect of video sequence length To investigate how temporal noise influences the training of MSTAT, we conduct experiments on videos with varied lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 9, experiments provide length-varying video tracklets for training, while all experiments are implemented under the identical eval- uation setting with a fixed video length of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' All Method Position Rank-1 Rank-5 w/o IAP embedding module 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 w/ IAP embedding module Stage II 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 Stage III 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 Stage II&III 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 TABLE III: Ablation study on the IAP embedding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Stage II and Stage III in this table means that an IAP embedding module is appended to the last layer of Stage II and Stage III respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' This table shows that the IAP embedding module brings improvements to every single stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' When it is placed on both two stages, MSTAT shows the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Methods Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Rank-1 Rank-5 mAP MSTAT w/o TPS 0% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 MSTAT w/ TPS 20% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 40% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 60% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 80% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='1 TABLE IV: Ablation study on Temporal Patch Shuf- fling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The table shows that the proper level of shuffling can bring slight improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, it may de- grade the learning while the shuffling degree becomes increasingly overwhelming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' experiments shut down until the loss stops decreasing for ten epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' On the one hand, rank-1 accuracy shows an upward trend as temporal noise gradually decreases, reach- ing a peak at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' On the other hand, temporal noise shows no apparent correlation with rank-5 accuracy and mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' These results show that our model gains up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6% rank-1 accuracy through learning better temporal features from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' However, rank-5 accuracy and mAP benefit little from noise reduction, from which we can speculate that in most cases in video re-ID, learning temporal features is less important than learning appearance features as they only account for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6% of rank-1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2% of rank-5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Similar results can be found in [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Comparison among metric learning methods Metric learning aims to regularize the sample distri- bution on feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Usually, metric learning losses constrain the compactness of intra-class distribution and sparsity of the overall distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' To explore which strategy cooperates with our framework better, we compare a range of classic metric learning loss functions on iLIDS-VID, as shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Note that these losses are scaled to the same magnitude to ensure fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Significantly, OIM loss [78] and BatchHard triplet loss [36], widely used in re-ID, outperform Arcface [23] and SphereFace [50] losses by a large margin since the latter two loss functions suffer from untimely overfitting in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Rank-l accuracy Rank-5 accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='914 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='914 160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='91 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='970 4 8 4 6 1 training video sequence length training video sequence length11 IEEE TRANSACTIONS ON MULTIMEDIA Metric learning loss Rank-1 Rank-5 w/o Metric learning 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 Arcface [23] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 SphereFace [50] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 OIM [78] 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 BatchHard* [36] 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='3 TABLE V: Comparison among metric learning loss functions on iLIDS-VID, where denotes the method used in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For Arcface and SphereFace, we test three margins and report the best result: (1) by default, (2) 20% larger than the default, (3) 20% smaller than the default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 10: Visualization of the similarity matrix of attribute-aware proxies trained on MARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The maxi- mal similarity between all pairs is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2, demon- strating that AAPs learn to capture diverse attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Visualization To better understand how the proposed framework works, we conduct visualization on the AAP embed- ding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 10, we show the diversity of implicit attributes by the similarity matrix of 24 AAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' This figure implies that AAPs are anisotropic, covering different attribute features that appear in the given training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Specifically, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 11, we randomly select two pedestrians’ tracklets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Attention map visu- alization is adopted as a sign of each AAP’s concen- tration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In practice, we process the raw attention maps first by several average filters and then by thresholding to deliver smooth visual effects instead of grid-like maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In these heap maps, the brighter color denotes the higher attention value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Despite the absence of attribute-level supervision, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='11 shows that some AAPs learn to pay attention to a local region with special meanings as an identity cue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' For example, the AAP in white color in video clips (a) automatically learns to cover the logo in the T-shirt, while the one in (b) captures the head of the woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Moreover, we display the t-SNE visualization result on iLIDS-VID in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' It only contains the first 1/3 of the IDs in the test set for a better visual effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' We also provide the corresponding quantitative evalu- Methods IntraÓ IntraÓ InterÒ Rank-1Ò Baseline [7] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='873 MSTAT w/o attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4644 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='913 MSTAT (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4410 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='933 TABLE VI: Quantitative evaluation on iLIDS-VID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' ”Intra” denotes the averaged normalized intra-class distance, and ”Inter” is the minimum inter-class dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Here, * means that the metric is computed on samples with the correct rank-1 match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 11: Visualization of attribute-aware proxies for two different pedestrians on MARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Attention heat maps of four consecutive frames from the AAP em- bedding module on Stage I are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' ation results in Table VI measured by the normalized averaged intra-class distance and the minimum inter- class distance (0-2) on the entire test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' As a result, MSTAT drops the average intra-class distance from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4572 of the baseline to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4410 and enlarges the minimum inter-class distance from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4704 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='5012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Further, to eliminate the influence of accuracy, we measure the intra-class distance between correctly matched samples, from which we witness a similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' These results explain why MSTAT’s t-SNE visualization seems sparser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' CONCLUSION This paper proposes a novel framework for video- based person re-ID, referred to as Spatial-Temporal Aggregation Transformer (MSTAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' To tackle simulta- neous extraction for local attributes and global identity information, MSTAT adopts a multi-stage architecture to extract (1) attribute-associated, (2) the identity- associated, and (3) the attribute-identity-associated in- formation from video clips, with all layers inherited from the vanilla Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Further, for reserving informative attribute features and aggregating discrim- inative identity features, we introduce two proxy em- bedding modules (Attribute-Aware Proxy embedding module and Identity-Aware Proxy embedding module) into different stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In addition, a patch-based data augmentation scheme, Temporal Patch Shuffling, is proposed to force the network to learn invariance to appearance shifts while enriching training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Massive experiments show that MSTAT can extract attribute-aware features consistent across frames while reserving discriminative global identity information on 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 2 3 4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='8 5 6 7 8- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='6 9- 10 11 12 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='4 14 15 16 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2 18- 19 20 21 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='0 23 456780TANG et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' : MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 12: T-SNE Visualization of the iLIDS-VID test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' The numbers on the plots indicate person IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' MSTAT shows an increase in intra-class compactness and the minimum inter-class distance over the entire test set compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' different stages to attain high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Finally, MSTAT outperforms most existing state-of-the-arts on three public video-based re-ID benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Future work may focus on mining the hard instances or local informative attribute locations to conduct con- trastive learning to promote the model’s accuracy fur- ther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Moreover, leveraging more unlabeled and multi- modal data to improve the model’s effectiveness is also a potential research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' ACKNOWLEDGMENT The work is supported in part by the Young Sci- entists Fund of the National Natural Science Foun- dation of China under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 62106154, by Na- tional Key R&D Program of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 2021ZD0111600, by Natural Science Foundation of Guangdong Province, China (General Program) un- der grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='2022A1515011524, by Guangdong Ba- sic and Applied Basic Research Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 2017A030312006, by CCF-Tencent Open Fund, by Shenzhen Science and Technology 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2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' [105] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Huang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Wang, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' See the forest for the trees: Joint spatial and temporal recurrent neural networks for video-based person re-identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4747–4756, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' 15 IEEE TRANSACTIONS ON MULTIMEDIA Ziyi Tang is now pursuing his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' degree at Sun Yat-Sen University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Before that, he was a research assistant at The Chinese University of Hong Kong, Shen- zhen (CUHK-SZ), China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' degree from South China Agriculture University (SCAU), Guangzhou, China in 2019 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' degree from The Univer- sity of Southampton, Southampton, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He has won top places in data science competitions hosted by Kaggle and Huawei respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' His research interests include Computer Vision, Vision-Language Joint Modeling, and Casual Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Ruimao Zhang is currently a Research Assistant Professor in the School of Data Science, The Chinese University of Hong Kong, Shenzhen (CUHK-SZ), China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He is also a Research Scientist at Shenzhen Research Institute of Big Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' degrees from Sun Yat- sen University, Guangzhou, China in 2011 and 2016, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' From 2017 to 2019, he was a Post-doctoral Research Fellow in the Multimedia Lab, The Chinese Univer- sity of Hong Kong (CUHK), Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' After that, he joined at SenseTime Research as a Senior Researcher until 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' His research interests include computer vision, deep learning and related multi- media applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He has published about 40 peer-reviewed articles in top-tier conferences and journals such as TPAMI, IJCV, ICML, ICLR, CVPR, and ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He has won a number of competitions and awards such as Gold medal in 2017 Youtube 8M Video Classification Challenge, the first place in 2020 AIM Challenge on Learned Image Signal Processing Pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He was rated as Outstanding Reviewer of NeurIPS in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He is a member of IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Zhanglin Peng is now pursuing her Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' degree with the Department of Computer Science, The University of Hong Kong, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' She received her B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' degrees from Sun Yat-Sen Uni- versity, Guangzhou, China in 2013 and 2016, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' From 2016 to 2020, she was a researcher at SenseTime Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Her research interests are computer vision and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Jinrui Chen is currently pursuing the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' degree in Financial Engineering conferred jointly by the School of Data Science, the School of Science and Engineering, and the School of Management and Eco- nomics, The Chinese University of Hong Kong, Shenzhen (CUHK-SZ), China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' His research interests include deep learning and financial technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Liang Lin (M’09, SM’15) is a Full Pro- fessor of computer science at Sun Yat- sen University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He served as the Exec- utive Director and Distinguished Scien- tist of SenseTime Group from 2016 to 2018, leading the R&D teams for cutting- edge technology transferring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He has au- thored or co-authored more than 200 pa- pers in leading academic journals and con- ferences, and his papers have been cited by more than 22,000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He is an associate editor of IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Multimedia and IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' Neural Networks and Learning Systems, and served as Area Chairs for numerous conferences such as CVPR, ICCV, SIGKDD and AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He is the recipient of numerous awards and honors including Wu Wen-Jun Artificial Intelligence Award, the First Prize of China Society of Image and Graphics, ICCV Best Paper Nomination in 2019, Annual Best Paper Award by Pattern Recognition (Elsevier) in 2018, Best Paper Dimond Award in IEEE ICME 2017, Google Faculty Award in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' His supervised PhD students received ACM China Doctoral Dissertation Award, CCF Best Doctoral Dissertation and CAAI Best Doctoral Dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} +page_content=' He is a Fellow of IET and IAPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfpviE/content/2301.00531v1.pdf'} diff --git a/7tAyT4oBgHgl3EQfc_c0/content/tmp_files/2301.00292v1.pdf.txt b/7tAyT4oBgHgl3EQfc_c0/content/tmp_files/2301.00292v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f84e6d551df72692e04ff823bb509d49b3a63eca --- /dev/null +++ b/7tAyT4oBgHgl3EQfc_c0/content/tmp_files/2301.00292v1.pdf.txt @@ -0,0 +1,2858 @@ +Inference for Large Panel Data with Many Covariates∗ +Markus Pelger† +Jiacheng Zou‡ +December 31, 2022 +Abstract +This paper proposes a new method for covariate selection in large dimensional panels. We de- +velop the inferential theory for large dimensional panel data with many covariates by combining +post-selection inference with a new multiple testing method specifically designed for panel data. +Our novel data-driven hypotheses are conditional on sparse covariate selections and valid for +any regularized estimator. Based on our panel localization procedure, we control for family-wise +error rates for the covariate discovery and can test unordered and nested families of hypothe- +ses for large cross-sections. As an easy-to-use and practically relevant procedure, we propose +Panel-PoSI, which combines the data-driven adjustment for panel multiple testing with valid +post-selection p-values of a generalized LASSO, that allows to incorporate priors. In an empir- +ical study, we select a small number of asset pricing factors that explain a large cross-section of +investment strategies. Our method dominates the benchmarks out-of-sample due to its better +control of false rejections and detections. +Keywords: panel data, high-dimensional data, LASSO, number of covariates, post-selection +inference, multiple testing, adaptive hypothesis, step-down procedures, factor model +JEL classification: C33, C38, C52, C55, G12 +∗We thank conference and seminar participants at Stanford, the California Econometric conference and the NBER-NSF +SBIES conference for helpful comments. Jiacheng Zou gratefully acknowledges the generous support by the MS&E Departmental +Fellowship, and Charles & Katherine Lin Fellowship. +†Stanford University, Department of Management Science & Engineering, Email: mpelger@stanford.edu. +‡Stanford University, Department of Management Science & Engineering, Email: jiachengzou@stanford.edu +arXiv:2301.00292v1 [econ.EM] 31 Dec 2022 + +1 +Introduction +Our goal is the selection of a parsimonious sparse model from a large set of candidate covariates +that explains a large dimensional panel. This problem is common in many social science applica- +tions, where a large number of potential covariates are available to explain the time-series of a large +cross-section of units or individuals. An example is empirical asset pricing, where the literature has +produced a “factor zoo” of potential risk factors to explain the large cross-section of stock returns. +This problem requires a large panel, as a successful asset pricing model should explain the many +available investment strategies, resulting in a large panel of test assets. At the same time, there is +no consensus about which are the appropriate factors, which leads to a statistical selection problem +from a large set of candidate risk factors. So far, the literature has only provided solutions for one +of the two subproblems, while keeping the dimensionality of the other problem small. Our paper +closes this gap. +The inferential theory on a large panel with many covariates is a challenging problem. As a first +step, we have to select a sparse set of covariates from a large pool of candidates with a regularized +estimator. The challenge is to provide valid p-values from this estimation that account for the +post-selection inference. Furthermore, researchers might want to impose economic priors on which +variables should be more likely to be selected. The second challenge is that the panel cross-section +results in a large number of p-values. Hence, some of them are inadvertently very small, which if +left unaddressed leads to “p-hacking”. The multiple testing adjustment conditional on the selected +subset of covariates from the first step is a novel problem, and requires to redesign what hypotheses +should be tested jointly. A naive counting of all tests is overly conservative, and the test design +and simultaneity counts need to be conditional on the covariate selection. +This paper proposes a new method for covariate selection in large dimensional panels, tackling +all of the above challenges. We develop the inferential theory for large dimensional panel data with +many covariates by combining post-selection inference with a new multiple testing method specifi- +cally designed for panel data. Our novel data-driven hypotheses are conditional on sparse covariate +selections and valid for any regularized estimator. Based on our panel localization procedure, we +control for family-wise error rates for the covariate discovery and can test unordered and nested +families of hypotheses for large cross-sections. As an easy-to-use and practically relevant procedure, +we propose Panel-PoSI, which combines the data-driven adjustment for panel multiple testing with +valid post-selection p-values of a generalized LASSO, that allows to incorporate priors. +Our paper proposes the novel conceptual idea of data-driven hypotheses family for panels. This +allows us to put forward a unifying framework of valid post-selection inference and multiple test- +ing. Leveraging our data-driven hypotheses family, we adjust for multiple testing with a localized +simultaneity count, which increases the power, while maintaining false discovery rate control. An +essential step for a formal statistical test is to formulate the hypothesis. This turns out to be +non-trivial for a large panel with a first stage selection step for the covariates. It is a fundamental +insight of our paper, that the hypothesis of our test has to be conditional on the selected set of +1 + +active covariates of the first stage. Once we have defined the appropriate hypothesis, we can deal +with the multiple testing adjustment, which by construction is also conditional on the selection +step. +Our method is a disciplined approach based on formal statistical theory to construct and in- +terpret a parsimonious model. It goes beyond the selection of a sparse set of covariates as it also +provides the inferential theory. +This is important as it allows to rank the covariates based on +their statistical significance and can also be applied for relatively short time horizons, where cross- +validation for tuning a regularization parameter might not be reliable. We answer the question +which covariates are needed to explain the full panel jointly, and can also accommodate “weak” +covariates or factors that only affect a small subset of the cross-sectional units. +Our data-driven hypothesis perspective exploits the geometric structure implied by the first +stage selection step. +Given valid post-selection p-values of a regularized sparse estimator from +time-series regressions, we collect them across the large cross-section into a “matrix” of p-values. +Only active coefficients, that are selected in the first stage, contribute p-value entries, whereas +covariates that were non-active lead to “holes” in this matrix. We leverage the non-trivial shape +of this matrix to form our adaptive hypotheses. This allows us to make valid multiple testing +adjusted inference statements, for which we design a panel modified Bonferroni-type procedure +that can control for the family-wiser error rate (FWER) in discovery of the covariates. As one +loosens the FWER requirements, the inferential thresholds admits more and more explanatory +variables, which suggests that the amount of covariates we expect to admit and the FWER control +level form an “false-discovery control frontier”. We provide a method that allows us to traverse the +inferential results and determine the least number of covariates that have to be included given a +user-specified FWER level. In other words, we provide a statistical significance test for the number +of factors in a panel. +We propose the novel procedure Panel-PoSI, which combines the data-driven adjustment for +panel multiple testing with valid post-selection p-values of a generalized LASSO. While our multiple +testing procedure is valid for any sparsity constrained model, Panel-PoSI is an easy-to-use and prac- +tically relevant special case. We propose Weighted-LASSO for the first stage selection regression and +provide valid p-values through post-selection inference (PoSI), which yields a truncated-Gaussian +distribution for an adjusted LASSO estimator. This geometric perspective is less common in the +LASSO literature, but has the advantage that it avoids the use of infeasible quantities, in particu- +lar the second moment of the large set of potential covariates. The Weighted-LASSO generalizes +LASSO by allowing to put weights onto prior belief sets. For example, a researcher might have +economic knowledge that she wants to include in her statistical selection method, and impose an in- +finite prior weight to include specific covariates in the sparse selection model. Our Weighted-LASSO +makes several contributions. First, the expression for the truncated conditional distribution with +weights become much more complex than for the special case of the conventional LASSO. Second, +we provide a simple, easy-to-use and asymptotically valid conditional distribution in the case of an +estimated noise variance. +2 + +We demonstrate in simulations and empirically that our inferential theory allows us to select +better models. We compare different estimation approaches to select covariates and show that our +approach better trades off false discovery and correct selections and hence results in a better out- +of-sample performance. Our empirical analysis studies the fundamental problem in asset pricing of +selecting a parsimonious factor model from a large set of candidate factors that can jointly explain +the asset prices of a large cross-section of investment strategies. We consider a standard data set +of 114 candidate asset pricing factors to explain 243 double sorted anomaly portfolios. We show +that Panel PoSI selects 3 factors which form the best model to explain out-of-sample the expected +returns and the variations of the test assets. The selected factors are economically meaningful and +we can rank them based on their relative importance. A prior on the Fama-French factors does not +improve the model. Our findings contributes to the discussion about the number of asset pricing +factors. +The rest of the paper is organized as follows. Section 1.1 relates our work to the literature. +Section 2 introduces the model and the Weighted-LASSO. Section 3 discusses the appropriate +hypotheses to be considered for inference on the entire panel. Section 4 proposes a joint unordered +test for the panel using multiple testing adjustment so that we can maintain FWER control, and +shows how to traverse this procedure to acquire the least factor count associated with each FWER +target. In section 5 we consider the case of nested hypotheses, where the covariates observe a fixed +ordering, which is of independent interest, and we propose a step-down procedure for this setting +that maintains false discovery control. Section 6 provides the results of our simulation and Section +7 discusses our empirical studies on a large asset pricing panel data set. Section 8 concludes. The +proofs and more technical details are available in the Online Appendix. +1.1 +Related Literature +The problem of multiple testing is an active area of research with a long history. The statistical +inference community has studied the problem of controlling the classical FWER since Bonferroni +(1935), and controlling for false-discover rate (FDR) going back to Benjamini and Hochberg (1995) +and Benjamini and Yekutieli (2001). Bonferroni (1935) allows for arbitrary correlation in the test +statistics because its validity comes from a simple union bound argument, and is in fact the optimal +test when statistics are “close to independent” under true sparse non-nulls. FDR control on the +other hand requires a discussion about the estimated covariance in the test statistics. +Recent +developments include a stream of papers led by Barber and Cand´es (2015) and Cand´es, Fan, +Janson, and Lv (2018), which constructs a generative model to produce fake data and control +for FDR. Fithian and Lei (2022) is a more recent work that iteratively adjusts the threshold for +each hypothesis in the family to seek finite sample exact FDR control and dominates Benjamini +and Hochberg (1995) and Benjamini and Yekutieli (2001) in terms of power. Another notion on +temporal false discovery control has been revived more recently by Johari, Koomen, Pekelis, and +Walsh (2021), who consider the industry practice of constantly checking p-values and provide an +early stopping in line with Siegmund (1985) that adjusts for bias from sequentially picking favorable +3 + +evidence, whereas we consider a static panel that is not an on-going experiment. +There are cases where the covariates warrant a natural order such that the hypothesis family +possesses a special testing logic. A hierarchical structure in covariates arises when the inclusion of +the next covariate only make sense if the previous covariates is included. An example is the use of +principal component (PC) factors, where PCs are included sequentially from the dominating one +to the least dominating one. We distinguish this from putting weights and assigning importance +on features because this variant of family of hypotheses warrants a new definition of FWER. We +propose a step-down procedure that can be considered as a panel extension of G’Sell, Wager, +Chouldechova, and Tibshirani (2016), relying on an approximation of the R´enyi representation of +p-values. The step-down control for nested FWER is based on Simes (1986), which along with +Bonferroni (1935) can be seen as comparing sorted p-values against linear growth. Our framework +contributes to estimating the number of principal component factors in a panel. There are have been +many studies that provide consistent estimators for the number of PCs based on the divergence in +eigenvalues of the covariance matrix, which include Onatski (2010), Ahn and Horenstein (2013) and +Pelger (2019). Another direction uses sequential testing procedures that presume correct nested +family of hypotheses, which include Kapetanios (2010) and Choi, Taylor, and Tibshirani (2017). +In contrast, we characterize the least amount of factors (which can also be based on principal +components), which should be expected when a FWER rate is provided. The nested version of +our procedure is close in nature to a panel version of “when-to-stop” problem of a multiple testing +procedure. +The problem of post-LASSO statistical testing for small dimensional cross-sections is studied in +a stream of papers including Meinshausen and B¨uhlmann (2006), Zhang and Zhang (2014), van de +Geer, B¨uhlmann, Ritov, and Dezeure (2014) and Javanmard and Montanari (2018), which consider +inference statements by debiasing the LASSO estimator. An alternative stream of post-selection +or post-machine learning inference literature includes Chernozhukov, Hansen, and Spindler (2015), +Kuchibhotla, Brown, Buja, George, and Zhao (2018) and Zrnic and Jordan (2020), who provide +non-parametric post-selection or post-regularization valid confidence intervals and p-values. These +papers do not make conditional statements and presume that the researcher sets the hypotheses +before seeing the data, which we will refer to as data agnostic hypothesis family. We follow a dif- +ferent train of thought that treats LASSO, among a family of conic maximum likelihood estimator, +as a polyhedral constraint on the support of the response variable. This geometric perspective +that provides inferential theory post-LASSO is pioneered by the work of Lee, Sun, Sun, and Taylor +(2016) and followed up by Fithian, Sun, and Taylor (2017) and Tian and Taylor (2018), assum- +ing Gaussian linear model. Markovic, Xia, and Taylor (2018) extends the results to LASSO with +cross-validation, Tian, Loftus, and Taylor (2018) discusses a square-root LASSO variant that takes +unknown covariance into consideration and Tian and Taylor (2017) considers the asymptotic results +when removing the Gaussian assumption. This body literature is often referred to as PoSI, and +traverses the Karush-Kuhn-Tucker (KKT) condition of a LASSO optimization problem to show +that the LASSO fit can be expressed as a polyhedral constraint on the support of the response +4 + +variable. We extend this work by allowing to put weights onto prior belief sets, and by bringing it +to the panel setting with multiple testing adjustment. +2 +Sparse linear models +We consider a large dimensional panel data set Y ∈ RT×N which we want explain with a large +number of potential covariates X ∈ RT×J. The panel data and explanatory variables are both +observed over T time periods.1 The size of the cross-section N and the dimension of the covariate +candidate set J are both large in our problem. We assume a linear relationship between Y and X: +Yt,n = +J +� +j=1 +Xt,jβ(n) +j ++ ϵt,n +for n = 1, ..., N, +which reads in matrix notation as +Y = Xβ + ϵ +(1) +We refer to the coefficients β as loading matrix, where the nth column β(n) corresponds to the nth +unit and β(n) +j +denotes the loading of the nth unit on the jth covariate. The remainder term ϵ is +unexplained noise. +We assume that a sparse linear model can explain jointly the full panel. Formally, a sparse +linear model with s active covariates is +Y = XSβS + ϵ +(2) +where s = |S| is the cardinality of the set of active covariates S = {j : ∃β(j) +n +̸= 0, n ∈ {1, ..., N}, +that is, the set of covariates with non-zero loadings. XS is the subset of covariates that belong to +S. Our goal is to estimate this low dimensional model, that can explain the full panel, from a large +number of candidate covariates, and provide a valid inferential theory. +Note that our sparse model formulation allows for two important properties. First, different +units can be explained by different covariates with different loadings. This means that β(n) ̸= β(m) +for n ̸= m. +For example, a subset of the cross-sectional units might be modeled by different +covariates than the remaining part of the panel. Second, we can accommodate “weak” covariates. A +covariate is included in S if it is required by at least one cross-sectional unit requires as explanatory +variable. In other words, a sparse model can include covariates in XS that explain only a very +small subset of the panel Y . +The first step is to estimate the sparse models over the time-series for each unit separately due +to the heterogeneity in the loadings. In a second step, we provide the valid inferential theory for +the loadings on the full panel. The time-series estimation requires an appropriate regularization to +1Our setting and multiple testing results can be readily extended to the case of unbalanced panel, although we +focus on the balanced panel case for now to highlight the core multiple testing insight of our method. We will further +discuss on this once we introduce our main procedure in Section 4 +5 + +select a small subset of covariates that contains all the relevant covariates for each unit. We allow +for a prior belief weight ω ∈ ¯RJ ++, so that different X can have different relative penalizations, and +a global λ ∈ R+ scalar penalty parameter. For the nth unit, we denote its β(n) estimate as ˆβ(n) +and the active set M(n) = {j : ˆβ(n) +j +̸= 0} as the set of j’s with non-zero loadings ˆβ(n) +j +. A general +regularized linear estimator solves the following optimization problem +ˆβ(j)(λ, ω) = arg min +β +1 +2T ∥Y (j) − Xβ∥2 +2 + λ · f(β, ω) +(3) +for a penalty function f and appropriate weights. In this paper, we consider the weighted LASSO +estimator with the regularization function +f(β, ω) = +J +� +j=1 +fj(βj, ωj) +where fj(βj, ωj) = +� +� +� +|βj| +ωj +ωj < ∞ +0 +o.w. +(4) +and weights ωj > 0 for all j ∈ {1, ..., J} and ∥ω−1∥1 = J. +We assume that the penalty λ is +selected such that the set ∥ˆβ(j)∥0 = |M(j)| is low dimensional. Importantly, we do not need to +assume that the selected set contains all active covariates. Our goal it is provide a valid inferential +theory conditional on the selected set. Our estimator generalizes the conventional LASSO with +the l1 regularization function of Tibshirani (1996) by allowing for different relative weighting in +the penalty. Importantly, we also allow for an infinite weight, which can be interpreted as a prior +on a set of covariates. +This allows researchers to take advantage of prior information and for +example ensure that a specific set of covariates will always be included. The weighted LASSO +will be particularly relevant in our empirical study, where we can answer the question which risk +factors should be added to a given set of economically motivated risk factors. Our weighted LASSO +formulation can also be interpreted as a Bayesian estimator with the canonical Laplacian prior. +Conventional regression theory will not provide correct inferential statements on the weighted- +LASSO estimates. We face two challenges. First, regularized estimation results in a bias, which +needs to be corrected. Second and more challenging, post-selection inference changes the distribu- +tion of the estimators. When we observe an active ˆβ(n) +j +from (3), it would be incorrect to simply +calculate its p-value from a conventional t-distribution. This invalidity stems from the fact that +conditional on observing a LASSO output, β(n) +j +must be large enough in magnitude for its ˆβ(n) +j +to +be active. In other words, the probability distribution of the estimators is truncated. +The correct inference has to be conditional on the covariates being selected by the LASSO +estimator. Hence, valid p-values have to be the tail probability conditional on being in the selection +set. The key to quantify such styles of inference is to recognize that a sparsity constrained estimator +is typically the result of solving Karush-Kuhn-Tucker (KKT) conditions, which can in turn be +geometrically characterized as polyhedral constraints on the support of response variables. This +is first established in Lee, Sun, Sun, and Taylor (2016), who provide the stylized results that +Post-Selection Inference (PoSI) of debiased non-weighted LASSO estimators can be calculated as +6 + +polyhedral truncation on Y . This line of research is also referred to as Selective Inference in other +literature such as Taylor and Tibshirani (2015). We extend this line of literature to allow for the +Weighted-LASSO. We derive these results with assumptions common in the PoSI LASSO literature, +detailed in Appendix A, and referred to as conventional regularity conditions for ease of exhibition. +THEOREM 1. Truncated Gaussian Distribution of Weighted-LASSO +Under conventional regularity conditions, the debiased estimate ¯βi for the i-th Weighted-LASSO +active covariate is conditionally distributed as +¯βi|Weighted-LASSO ∼ T N {η⊤Y :AY ≤b(ω)} +(5) +where T N A is truncated-Gaussian with truncation A, and the weights ω only appear in b(ω) +Theorem 1 has two elements. First, it debiases the LASSO estimate by a shifting argument. +While we use a geometric argument to remove the bias, the bias adjustment takes the usual form +in the LASSO literature as for example in Belloni and Chernozhukov (2013). The debiased LASSO +estimator simply equals a standard OLS estimation on the subset Mn selected by the Weighted- +Lasso. Second, the distribution of the linear coefficients is not a usual Gaussian distribution, but it +is truncated due to studying post-selection coefficients. This geometric perspective is less common +in the LASSO literature, but provides several advantages. One advantage of the geometric approach +is that it avoids the use of infeasible quantities, in particular the second moment of the large set of +potential covariates. Furthermore, the distribution result is not asymptotic in T, but also valid in +finite samples. We can obtain these results because we make the stronger assumption that the data +is normally distributed. Appendix A provides the detailed information on constructing ¯β and the +definitions of η, A, b(ω) along with lemmas that lead up to this result. It also discusses extensions +and the effect of estimating the variance of the noise. The empirical analysis is based on the explicit +form of Theorem 1 formulated in Theorem A.3. +Our Weighted-LASSO results make several contributions. First, the expression for the trun- +cated conditional distribution with weights become much more complex than for the special case +of the conventional LASSO. Second, we provide a simple, easy-to-use and asymptotically valid +conditional distribution in the case of an estimated noise variance. Last but not least, we show the +formal connection with alternative debiased LASSO estimators by showing that debiasing can be +interpreted as one step in a Newton-Ralphson method of solving a constrained optimization. +Theorem 1 allows us to obtain valid p-values for Weighted-LASSO coefficients. We obtain these +p values from the simulated cumulative distribution function of the truncated Gaussian distribution. +Crucially, all results for multiple testing adjustment in panels that we study in the following sections +neither require us to use a weighted Lasso estimator nor to use the p-values implied by Theorem +1. We only require to have a set of valid p-values for sparsity constrained models. These can be +obtained with any suitable regularized estimator and post-selection inference. The key element is +the selection of a low dimensional subset with p-values conditional on this selection. We propose +the weighted LASSO conditional inference results as an example of the type of sparsity constraint +7 + +models we are interested in, and demonstrate a machinery with which we can obtain valid p-values +for sparsity constrained models. In our empirical studies, we use Weighted-LASSO as our sparsity +constrained model since we want to specify strong prior beliefs on a few covariates and it is common +practice to use LASSO in the context of our empirical studies. Nonetheless, the testing methods +in the next sections accommodate any sparse estimator, and can be detached from inference for +Weighted-LASSO. +3 +Data-Driven Hypotheses +Our goal is to provide formal statistical tests that allow us to establish a joint model across a +large cross-section with potentially weak covariates. This requires us to provide a form of statistical +significance test with multiple testing adjustment that properly accounts for covariates that only ex- +plain a small subset of the cross-sectional units. This is important as in many problems in economic +and finance there is substantial cross-sectional variation in the explanatory power of covariates, and +a model that simply minimizes an average error metric might neglect weaker covariates. +An essential step for a formal statistical test is to formulate the hypothesis. This turns out to be +non-trivial for a large panel with a first stage selection step for the covariates. It is a fundamental +insight of our paper, that the hypothesis of our test has to be conditional on the selected set of +active covariates of the first stage. Once we have defined the appropriate hypothesis, we can deal +with the multiple testing adjustment, which by construction is also conditional on the selection +step. +The hypothesis formulation and test construction only requires valid p-values from a first stage +selection estimator. The results of the next two sections do not depend on a specific model for +obtaining these p-values and the active set. The results are valid for any model including non- +linear ones. The input to the analysis is a N ×J matrix, which specifies which covariates are active +for each unit and the corresponding p-values. The Weighted-LASSO is only one possible model, +but it can be replaced by any regularized model. We have introduced the sparse linear model as +it is the horse race model for many problems in economics and finance, and therefore of practical +relevance. +We illustrate the concept of a data-driven hypothesis with a simple example, which we will +use throughout this section. For simplicity we assume that we have J = 4 covariates and want +to explain N = 6 cross-sectional units. In the first stage, we have estimated a Weighted-LASSO +and have obtained the post-selection valid p-values for each of the N units. We collect the fitted +sparse estimator ¯β(n) for the nth unit in the matrix ¯β. Note, that this matrix has “holes” due to +the sparsity for each ¯β(n). Figure 1(a) illustrates ¯β for this example. +Similarly, we collect the corresponding p-values in the matrix P . For the nth unit, we only +have p-values for those covariates that are active in the nth linear sparse model. Thus, Figure +1(b) also has white boxes showing the same pattern of unavailable p-values due to the conditioning +on the output of the linear sparse model. These holes can appear at different positions for each +8 + +Figure 1: Illustrative example of data-driven selection +(a) Matrix ¯β +(b) Matrix P of p-values +This figure illustrates in a simple example the data-driven selection of a linear sparse model. In a first stage, we have +estimated a regularized sparse linear model for each of the N = 6 units with J = 4 covariates. Each row represents +the selected covariates with their estimated coefficients and p-values. The columns represent the J = 4 different +covariates. The grey shaded boxes represent the active set, while white boxes indicate the inactive covariates. The +numbers are purely for demonstrative purposes. +unit, which makes this problem non-trivial. This non-trivial shape of either subplot (a) or (b) +is completely data-driven and a consequence of linear sparse model selection. We show that the +hypothesis should be formed around these non trivial shapes as well, which is why we name it the +data-driven hypothesis family. +We want to test which covariates are jointly insignificant in the full panel. A data-agnostic +approach would simply test if all covariates are jointly insignificant, independent of the data-driven +selection step in the first stage. A data-agnostic hypothesis is unconditional as it does not depend +on any model output. +However, as we will show, this perspective is problematic for the high- +dimensional panel setting with many covariates as it ignores the dimension reduction from the +selection step. Therefore, an unconditional multiple testing adjustment accounts for “too many” +tests, which severely reduces the power. +We propose to form the hypothesis conditional on the first stage selection step. The data-driven +hypothesis only tests the significance of the covariates that were included in the selection, and hence +can drastically reduce the number of hypothesis. However, given the non-trivial shape of the active +set, the multiple testing adjustment for the data-driven hypothesis is more challenging. +Before formally defining the families of hypothesis, we illustrate them in our running example. +9 + +1 +2 +3 +4 +1 +-5.43 +2.15 +2 +-1.10 +4.78 +-0.08 +m +0.19 +4.59 +4 +4.44 +2.10 +5 +1.44 +4.53 +2.10 +6 +-0.46 +4.701 +2 +3 +4 +1 +0.127 +0.587 +2 +0.005 +0.001 +0.871 +3 +0.526 +0.001 +4 +:0.001 +≤0.001 +5 +:0.0010.001 +0.001 +6 +0.102 +0.010The data-agnostic hypothesis HA for explaining the full panel takes the following form: +HA = {HA0,1, HA0,2, HA0,3, HA0,4} += {β(1) +1 +=β(2) +1 += β(3) +1 += β(4) +1 += β(5) +1 += β(6) +1 += 0, +β(1) +2 +=β(2) +2 += β(3) +2 += β(4) +2 += β(5) +2 += β(6) +2 += 0, +β(1) +3 +=β(2) +3 += β(3) +3 += β(4) +3 += β(5) +3 += β(6) +3 += 0, +β(1) +4 +=β(2) +4 += β(3) +4 += β(4) +4 += β(5) +4 += β(6) +4 += 0} +(6) +The data-driven hypothesis HD only includes the active set and hence equals +HD = {β(2) +1 +=0, +β(1) +2 +=β(3) +2 += β(5) +2 += β(6) +2 += 0, +β(1) +3 +=β(2) +3 += β(3) +3 += β(4) +3 += β(5) +3 += β(6) +3 += 0, +β(2) +4 +=β(4) +4 += β(5) +4 += 0} +(7) +Obviously, HA has a larger cardinality of |HA| = 24 > |HD| = 14. This holds in general, unless the +first stage selects all covariates for each unit, in which case the two hypotheses coincide. +Formally, the data-agnostic family of hypothesis is defined as follows: +DEFINITION 1. Data-agnostic family +The data-agnostic family of hypotheses is +HA = {HA0,i|i ∈ [d]} +where HA0,i = +� +j∈[N] +H(j) +A0,i and H(j) +A0,i : β(j) +i += 0. +(8) +It is evident that HA does not need any model output or exploratory analysis, so it is indeed +data-agnostic. +As soon as we use a sparsity constrained model that has censoring capabilities, we no longer +observe (Y , X) from its data generating process. Consequently, unless our hypotheses depend on +how we built the model, or equivalently on how the data was censored, the data-agnostic hypotheses +forgo power without any benefit in false discovery control. Therefore, we formulate the hypothesis +on the ith covariate H(j) +0,i only if i ∈ M(j), that is, it is in the active set. Conditional on observing +the model output, there is no inference statement to be made about H(j) +0,i if i /∈ M(j), because its +estimator is censored by the model. +We denote as Ki the set of units for which the ith covariate is active. We define the cross- +sectional hypothesis for the ith covariate as: +H0,i = +� +j∈Ki +H(j) +0,i +����M, +∀i : Ki ̸= ∅ +(9) +By combining all covariates {i : Ki ̸= ∅} that show up at least once in one of the active sets of our +sparse linear estimators, we arrive at a data-driven hypothesis associated with our panel. This is +defined as follows: +10 + +DEFINITION 2 (Data-driven family). The data-driven family of hypotheses conditional on M +is +HD = {H0,i|i : Ki ̸= ∅} +(10) +This demonstrates the non-trivial nature of writing down a hypothesis in high-dimensional +panel: we can only collect Ki - the set of units for which the ith covariate is active - after seeing +the sparse selection estimation result. +4 +Multiple Testing Adjustment for Data-Driven Hypothesis +4.1 +Simultaneity Counts through Panel Localization +We show how to adjust for multiple testing of data-driven hypotheses. Given the p-values p(j) +i +for i ∈ M and j ∈ Ki, we form the data-driven hypothesis HD. Our goal is to reject members of +HD while controlling the Type I error, and the common way to measure such error is the family- +wise error rate. This is the same underlying logic that is used to define confidence intervals and +determine significance of covariates in a conventional setup. The crucial difference is that we need +to account for multiple testing given the large number of cross-sectional units. The family-wise +error rate (FWER) is defined as follows: +DEFINITION 3. Family-wise error rate +Let V denote the number of rejections of H(j) +0,i |M(j) when the null hypothesis is true. The family- +wise error rate (FWER) is P(V ≥ 1). +Similar to the conventional definition, we simply count the false rejections V and define FWER +as the probability of making at least one false rejection. +Importantly, Definition 3 accounts for the fact that we might repeatedly test on � +j∈[N] |Mj| +rather than a single hypothesis test of the form H(j) +0,i : β(j) +i += 0|M(j). Our contribution to FWER +control in the panel setting is thus to take into consideration both the multiplicities in units and +covariates when we deal with the “matrix” of p-values P . To achieve this goal, we propose a new +simultaneity account for the ith covariate, calculated as +Ni = +� +j∈Ki +|Mj| +(11) +Figure 2 illustrates the simultaneity counting for our running example with N = 6 units and +J = 4 covariates. The blue boxes represent the active set for a specific covariate. The yellow boxes +indicate the “co-active” covariates, which have to be accounted for in a multiple testing adjustment. +In the case of the first covariate j = 1, only the second unit n = 2 has selected this covariate. This +second unit has also selected covariate j = 3 and j = 4, which are jointly tested with the first +covariates. Hence, they are “co-active”, and the simultaneity count equals N1 = 3. Intuitively, +Nj represents all relevant comparisons for the jth covariate because it counts how many covariates +11 + +Figure 2: Simultaneity counts Ni in the illustrative example +(a) N1 = 3 +(b) N2 = 9 +(c) N3 = 14 +(d) N4 = 8 +This figure shows the simultaneity counts Ni in the illustrative example. The subplots represent the simultaneity +counts for the J = 4 covariates. The blue boxes indicate the active set Kj of the j covariates, while yellow boxes +indicate the “co-active” covariates of the jth covariate. The simultaneity counts are the sum of yellow and blue +boxes. +are active with the jth covariate in the regressions. Hence, Nj quantifies the number of “multiple +tests” for each covariate. +In subplot 2(a), we see that K1 = {2} for the 1st covariate, indicated by the blue box, because +it is only active in the second unit’s regression. The multiple testing adjustment needs to consider +all yellow boxes, and N1 = 3 is thus the total count of 1 blue and 2 yellow boxes. Similarly, for +the second covariate, K2 = {1, 3, 5, 6}, so we shade boxes yellow for the 2nd, 3rd and 5th units +and obtain N2 = 9. We can already see that our design of simultaneity count takes all relevant +pairwise comparisons into considerations, but avoids counting the white boxes - which would cause +overcounting and result in over-conservatism. +Our multiplicity counting is a generalization of the classical Bonferroni adjustment for multiple +testing. A conventional Bonferroni method for the data-agnostic hypothesis HA has a simultaneity +count of |HA| = N · J = 24 for testing each covariate. A direct application of a vanilla Bonfer- +roni method to the panel of all selected units and the data-driven hypothesis HD, would use a +simultaneity count of |HD| = 14 for testing each covariate. Our proposed multiplicity counting is +a refinement that leverages the structure of the problem, and takes the heterogeneity of the active +sets for each covariate into account. Our count has only N1 = 3, N2 = 9 and N4 = 8 for the +covariates j = 1, 2 and 4. Only for covariate j = 3 is the simultaneity count the same as a vanilla +Bonferroni count applied to HD, i.e. N3 = 14. +In addition to the simultaneity count of each covariate, we need an additional “global” metric +for our testing procedure. We define a panel cohesion coefficient ρ as a scalar that measures how +12 + +1 +2 +3 +4 +2 +4 +5 +61 +2 +3 +4 +1 +2 +4 +5 +61 +2 +3 +4 +1 +2 +4 +5 +62 +3 +4 +1 +2 +3 +4 +5 +6Figure 3: Illustration of the cohesion coefficient +(a) ρ = J−1 = 0.25 +(b) ρ = 0.44 +(c) ρ = 1 +This figure illustrate the cohesion coefficient ρ in three separate examples. It shows the smallest, largest and in- +between cases of ρ. The columns represent the J = 4 different covariates.The blue boxes indicate the active sets for +each panel. +sparse or de-centralized the proposed hypotheses family is: +ρ = +� +�� +j +|Kj| +Nj +� +� +−1 +(12) +The panel cohesion coefficient ρ is conditional on the data-driven selection of the overall panel. It is +straightforward to compute once we observe the sparse selection of the panel. This coefficient takes +values between J−1 and 1,2 where larger values of ρ imply that the active set is more dependent in +the cross-section. This can be interpreted as that the panel Y has a stronger dependency due to +the covariates X. Intuitively, in the extreme case when ρ = J−1, the panel can be separated into +J smaller problems, each containing a subset of response units explained by only one covariate. +Thus the panel would be very incohesive, and could be studied with J independent tests. In the +other extreme, if ρ approaches 1, the first-stage models include all active covariates for all units. +We consider this as a very cohesive panel. If ρ is between theses bounds, the panel is cohesive in +a non-trivial way such that some units can be explained by some covariates and there is no clear +separation of the panel into independent subproblems. +Figure 3 illustrates the panel cohesion coefficient in three examples. The subplots show three +active sets that are different from our running example. The left subplot 3(a) shows the extreme +case of ρ = J−1, where the panel is the least cohesive. The right subplot 3(c) illustrates the other +extreme for ρ = 1, where the panel is the most cohesive. The middle subplot 3(b) is the complex +case of a medium cohesion coefficient. +2We prove this bound in the Appendix, without leveraging sparsity of first-stage models but rather as an algebraic +result with intuitive interpretations. +13 + +1 +2 +3 +4 +1 +2 +3 +4 +5 +61 +2 +3 +4 +1 +2 +4 +5 +61 +2 +3 +4 +1 +2 +4 +5 +6Our novel simultaneity count and cohesiveness measure are the basis for modifying a Bonferroni +test for FWER-controlled inference. Theorem 2 formally states the FWER control. The proof is +in the Online Appendix. +THEOREM 2. FWER control +The following rejection rule has FWER≤ γ on HD: +min +n∈Kj +� +p(n)(j) +� +≤ ρ γ +Nj +⇒ Reject H0,j +(13) +where p(n)(j) are valid p-values for each univariate unit n, and ρ is the panel cohesion coefficient. +This completes the joint testing procedure. First, we calculate p-values after running a sparse +linear estimator time-series regression. Second, we use the sparse linear estimator output to write +down a hypothesis and, third, we provide a FWER control inference procedure by combining the +p-values across the cross-section and test the hypothesis. +The difference between a naive Bonferroni and our FWER control is particularly pronounced +for weak covariates that affect only a subset of the cross-sectional units. Given a FWER control +level of γ, the rejection threshold for a naive Bonferroni test is +γ +JN for every covariate. The rejection +threshold for our FWER control is always higher, and differs in particular when Nj is small and ρ +is large. This is the case for weak covariates in a cohesive panel. +As it is common in statistical inference, we focus on Type I error control. Type II error rates +require the specification of alternatives. While we do not provide formal theoretical results for the +power of our inference approach, we show comprehensively in the simulation and empirical part, +that our approach has substantially higher power than conventional approaches. +We point out that the validity of our procedure holds for unbalanced panels as well. +This +is because even when there are different number of observations for the nth and mth units, i.e. +Tn ̸= Tm for n ̸= m, they can still be estimated separately in the first stage of the regularized +regression. The hypothesis testing and selection of a parsimonious model only requires the matrix +P of valid p-values, which can be based on different samples. +4.2 +Least Number of Covariates: Traversing the Threshold +The typical logic of statistical inference is to determine which covariates we should admit from +XM, given a significance level γ. We use K to denote the number of selected covariates. When +γ is specified as a lower quantity, we expect K to decrease as well, that is, the rejection becomes +harsher. +As the number of admitted covariates of our procedure is monotone in γ, we want to ask +the following converse question: How low do we need to set γ such that we reject K covariates? +Concretely, we are interested in finding: +14 + +γ∗(K) = sup +� +� +�γ|K = +J +� +j=1 +1 +� +min +n∈Kj +� +p(n) +j +� +≤ ρ γ +Nj +�� +� +� . +(14) +Let pj = minn∈Kj{p(n) +j +} be the 1st order statistic for j = 1, ..., J. Then (14) is simply the K-th +order statistics of Njpj/ρ: +γ∗(K) = min{Nipi/ρ|∃j1, j2, ..., jK ∈ {1, ..., J} : Nipi ≥ Njkpjk}. +(15) +Since this minimization scan is monotone, we can determine how many covariates at least +should be admitted, given a control level, which is similar to the “SimpleStop” procedure described +in Choi, Taylor, and Tibshirani (2017). The following corollary formalizes this inversion method +that finds the least number of covariates to admit: +COROLLARY 1. Least number of covariates +Given the FWER level γ, there exists a unique number K∗(γ) such that +K∗(γ) = +� +� +� +arg max0≤K≤J γ∗(K) ≤ γ +∃K : γ∗(K) ≤ γ +d +o.w. +(16) +The statement simply states that the simplest linear model should have at least K∗(γ) covariates +for a given γ. Note that it is possible that, for example, γ∗(5) and γ∗(6) are both equal to 0.05, +while γ∗(7) > 0.05. In this case the minimum number of covariates is K∗(0.05) = 6 because it does +not hurt FWER-wise to include 6 covariates in the model. Hence, we are making a slightly different +statement than that there would be exactly K∗(γ) covariates in the true linear model. The number +of covariates is obviously conditional on the set of candidate covariates X, and we can only make +statements for this given set. +In our empirical study we consider candidate asset pricing factors X to explain the investment +strategies Y . More generally, the linear model that we consider is often referred to as a factor model. +Therefore, we will also refer to the selected covariates as factors, and use these two expressions as +synonyms moving forward. This directly links our procedure to the literature on estimating the +number of factors to explain a panel. A common approach in this literature is to use statistics based +on the eigenvalues of either Y or X to make statements about the underlying factor structure. Our +approach is different, as it provides significance levels for the selected factors and FWER control +for the number of factors. +Table 1 illustrates the estimation of the number of factors and their ranking with our running +example introduced in Figure 1. We calculate the simultaneity counts Ni’s as given in (11) and +demonstrated in Figure 2, and pi as the smallest p-values associated with the ith covariate. Then, +the rejection rule in Theorem 2 is based on whether a pre-specified level γ satisfies pi < ργ +Ni , which +is equivalent to Ni · piρ < γ. +Thus, the natural ranking of the covariates is to sort all covariates in descending order of the +15 + +Table 1: Sorted p-values for the running example +Factor (j) +pj +Simultaneity count for HD +Conventional Bonferroni for HA +ρ−1 · Nj +ρ−1 · Nj · pj +J · N +J · N · pj +3 +< 0.001 +22.1 +< 0.001 +24 +0.002 +4 +< 0.001 +11.1 +0.001 +24 +0.003 +1 +0.005 +4.7 +0.024 +24 +0.120 +2 +0.002 +14.3 +0.028 +24 +0.051 +This table constructs “significance” levels for the running example introduce in Figure 1. We compare the simul- +taneity count for the data-driven hypotheses HD and a onventional Bonferroni count for data-agnostic hypotheses +HA. The products Nj · pj, respectively J · N · pj, can be interpreted as the significance levels for the corresponding +approach. Given a FWER control γ all factors with ρ−1 · Nj · pj (respectively J · N · pj) below this threshold are +selected. +Ni · pi/ρ values as shown in Table 1. It is then trivial to determine K∗(γ) for any choice of γ. For +example, for γ = 1%, we would select factors 3 and 4, but not 1 and 2. On the other hand, for +γ > 2%, we would include all four factors. Hence, the ranking of Nipi/ρ directly maps into K∗(γ). +The list of Nipi/ρ encompasses more information than just the number of factors. Naturally, it +provides an importance ranking of the factors. Furthermore, the number Ni reveals if significant +factors are “weak”. In our case, factor 1 has N1 = 3, which indicates that it affects only a small +number of hypothesis. Its p-value p1 is sufficiently small to still imply significance in terms of +FWER control. +For comparison, Table 1 also includes the corresponding analysis for the data-agnostic hypoth- +esis and a conventional Bonferroni correction. The Bonferroni analysis uses the same p-values but +a different multiple testing adjustment. In our case, the p values would be multiplied by J ·N = 24 +as this corresponds to the total number of hypothesis tests. This will obviously make the inference +substantially more conservative. Indeed, even for a FWER control of γ = 4%, we would only select +factors 3 and 4. We would need to raise the FWER control to γ = 12% to include factor 1. Hence, +weak factors, like factor 1, are more likely to be discarded by the data-agnostic hypothesis with +conventional multiple testing adjustment. +We want to emphasize that a data-agnostic hypotheses with conventional Bonferroni correction +does provide correct FWER control, but it is overly conservative. By construction, the data-agnostic +Bonferroni approach will test a larger number of hypothesis, which means that the corresponding +“significance levels” will always be lower or equal to our data-driven simultaneity count. Second, +the data-agnostic Bonferroni approach does not differentiate the “strength” of the factors, while +our approach provides a selection-based heterogeneous adjustment of the p-values. This is essential +for detecting weak factors. +Having introduced all building blocks of our novel method to detect covariates, we put the entire +procedure together as “Panel-PoSI”: +PROCEDURE 1. Panel-PoSI +The Panel-PoSI procedure consists of the following steps: +16 + +1. For each unit n = 1, ..., N unit, we fit a linear sparse model ˆβ(n) +X,Y (c, ω) given (X, Y , λ, ω). We +suggest cross-validation to select the LASSO penalty λ. We construct the sparse estimators +¯β(n) and the corresponding p-values for the active covariates for each unit, and collect them +in the “matrix” of p-values P . +2. We collect the panel-level sparse model selection event M and construct the data-driven hy- +pothesis HD. +3. Given the FWER control level γ and based on the the simultaneity counts Nj, we make +inference decision for the sparse model. We can rank covariates in terms of their significance +and select a parsimonious model that explains the full panel. +As we have now all results in place, we can summarize the advantages of our procedure. First, +we want to clarify that our goals and results are different from just some form of optimal shrink- +age selection. Selecting a shrinkage parameter with some form of cross-validation in a regularized +estimator like LASSO does not provide the same insights and model that we do. +A shrinkage +estimator can either be applied to each unit separately, as we do it in our first step, or to the +full panel in a LASSO panel regression. The separate covariate selection for each cross-sectional +unit does not answer the question which covariates are needed to explain the full panel jointly. A +shrinkage selection on the full panel for some form of panel LASSO can neglect weaker factors, +as those receive a low weight in the cross-validation objective function. Second, tuning parameter +selection with cross-validation requires a sufficiently large amount of data. Our approach is attrac- +tive as we can do the complete analysis on the same data. That means, an initial LASSO is used +to first reduce the number of covariates, but this set is then further trimmed down using inferential +theory. Hence, we can construct a parsimonious model even for data with a relatively short time +horizon, but large cross-sectional dimension. Third, the statements that we can make are much +richer than a simple variable selection. We can formally assess the relative importance of factors +in terms of their significance. The model selection is directly linked to a form of significance level, +which allows us to assess the relevance of including more factors. Last but not least, we can also +make statements about the strength of factors. In summary, Panel-PoSI is a disciplined approach +based on formal statistical theory to construct and interpret a parsimonious model. +5 +Ordered Multiple Testing on Nested Hypothesis Family +So far, our hypothesis family HD has no hierarchy and consequently, we have not imposed a +sequential structures on the admission order of covariates of X. However, there are cases where +the covariates or factors warrant a natural order such that the family possesses a special testing +logic. A hierarchical structure in covariates arises when the inclusion of the next covariate only +make sense if the previous covariates is included. One example would be if the next covariates +refines a property of the previous covariate. Another case is the use of principal component (PC) +factors. +The conventional logic is to include PCs sequentially from the dominating one to the +17 + +least dominating one. This is similar to the motivation for Choi, Taylor, and Tibshirani (2017), +but different from them, we treat the PCs as exogenous without taking the estimation of PCs +explicitly into account. In this section, we will use exogenous PCs as hierarchical covariates, as this +is the main example in our empirical study. However, all the results hold for any set of exogenous +hierarchical covariates. +Without loss of generality, we presume X has the jth column as the jth nested factor. A +k-order nested model N(k) is of the following form +N(k) model : Y = X[k]β[k] +(17) +where [k] = {1, ..., k} is the set that includes indices up to k. For example, a hierarchical three +factor model corresponds to X{1,2,3}. When formulating our hypothesis family, we must represent +the sequential testing structure. This is reflected in our definition of nested families of hypotheses: +DEFINITION 4. Data-driven nested family +The data-driven nested family of hypotheses conditional on M is +HN = {HN,k : k = 0, 1, ..., J}, +HN,k = +� +j∈Kk +H(j) +N,k +����M, +H(j) +N,k : {i′ : β(j) +i′ +̸= 0} ≤ k. +(18) +HN,0 completes the case when no rejection on any factor is made. Whenever HN,k is true, then +HN,k′ is also true for k < k′ ≤ J. Moreover, in the cases where Kk = ∅ but Kk′ ̸= ∅ with k < k′, +the notation ensures that the hypothesis HN,k is included in HN simply because Kk′ is present. +In other words, if a less dominating hypothesis HN,k′ is suggested by data (that is, its active set is +non-empty Kk′ ̸= ∅), HN would automatically include all HN,k for k ≤ k′. +The FWER control property needs to be adapted to the nested nature of this family. Choi, +Taylor, and Tibshirani (2017) argue that the proper measurement is to control for ordered factor +count over-estimation with level γ, as follows: +DEFINITION 5. FWER for nested family +For a test that rejects HN,k for k = 1, 2, ..., ˆk of HN, the FWER control at the level γ satisfies +P(ˆk ≥ s) ≤ γ, where s is the true factor count. +Given the hierarchical belief about the model, we need to add the following additional assump- +tion: +ASSUMPTION 1. Tail p-values +Under H(j) +N,k, there is p(j)(i′) iid +∼ Unif [0, 1] if i′ > k. +Assumption 1 only needs to hold for the tail hierarchical covariates. In the case of PCs, it only +applies to the lower order tail PC factors that should not be included for a given null hypothesis. +For example, if the true model is HN,s, we only need p(j)(i) iid +∼ Unif[0, 1] for i > s, which is a +usual type of assumption in this literature such as in G’Sell, Wager, Chouldechova, and Tibshirani +18 + +Figure 4: Example of hierarchical simultaneity counts Norder +k +for HN +(a) N order +4 += 3 +(b) N order +3 += 5 +(c) N order +2 += 8 +(d) N order +1 += 12 +This figure shows the simultaneity counts N order +i +in an illustrative example. The subplots represent the simultaneity +counts for the J = 4 covariates and N = 6 units. The dark blue columns present the active factors, while the light +blue columns capture factors of higher-order. The sub-plots from left-to-right represent our calculation order from +the highest-order factor to the 1st factor. +(2016). Moreover, because the nested nature guarantees that the higher-order PCs are more likely +to be null, a step-down procedure is expected to increase the power relative to a step-up procedure. +As our focus is to control for false discoveries, we also need to adjust our simultaneity counts +to the sequential testing. Concretely, we consider first taking a union to obtain the active unit set +Korder +k +and then calculate conservative simultaneity counts Norder +k +: +Korder +k += +� +i∈{k,k+1,...,J} +Ki, +Norder +k += +� +j∈Korder +k +|Mj|. +(19) +It is possible for some |Mk| to be 0 (that is, the kth PC could be inactive for all units), but its +Norder +k +would be 0 if and only if higher-order PCs all have |Mk′| = 0 for k′ > k. +Figure 4 illustrates the process of our step-down simultaneity count. From the left, we start +with factor k = 4 and move step-wise down to factor k = 1 on the right. The dark blue columns +present the active factors, while the light blue columns capture factors of higher-order. In the +left-most sub-figure, we only need to account for the 4th PC, implying Norder +4 += 3, whereas in the +mid-left sub-figure, the 3rd PC has Norder +3 += 2 + 3 = 5. Eventually, in the right-most sub-figure, +we have swept through the entire panel and the 1st PC has a simultaneity count of Norder +1 += 12. +Now we can introduce a step-down procedure adapted to the nested structure of HN: +PROCEDURE 2. Step-down rejection of nested ordered family HN +The step-down rejection procedure consists of the following steps: +1. For each k ∈ {1, ..., J} calculate the ordered simultaneity count Norder +k +. +19 + +1 +2 +3 +4 +1 +2 +3 +4 +5 +61 +2 +4 +1 +2 +4 +5 +61 +2 +4 +1 +2 +4 +5 +61 +2 +4 +1 +2 +3 +4 +5 +62. For each k ∈ {1, ..., J} calculate the approximated R´enyi representation Zorder +k +and its trans- +formed reversed order statistics qorder +k +: +Zorder +k += +J +� +i=k +� +j∈Ki +ln(p(j)(k)) +Norder +1 +− Norder +i+1 1{i ̸= J}, +qorder +k += exp(−Zorder +k +) +(20) +3. Reject hypothesis 1, 2, ..., ˆk, where ˆk = max{k : qorder +k +≤ γNorder +k +JN +}. +This procedure will have FWER control at level γ as stated in the following theorem: +THEOREM 3. FWER control for ordered hypothesis +Under Assumption 1, Procedure 2 has FWER control of γ for the ordered hypothesis HN. +The proof is deferred to the Online Appendix. This design extends Procedure 2 from G’Sell, +Wager, Chouldechova, and Tibshirani (2016) and “Rank Estimation” from Choi, Taylor, and Tib- +shirani (2017), both of which focus on a single sequence of p-values rather than the panel setting. +In Step 2, we use Assumption 1 to transform p-values into ln(p(j)(k)), which are i.i.d. standard +exponential random variables. Since the family HN has J members, we need to modify our simul- +taneity count and in a sense condense the panel into a sequence of statistics associated with the +ordered covariates. We built a staircase sequence of conservative simultaneity count Norder +k +in Step +1 to accumulate the number of p-values we use up to the kth ordered covariate, starting from the +end. By the R´enyi representation of R´enyi (1953), the Zorder +k +of Step 2 approximate exponential +order statistics and the qorder +k +approximate uniform order statistics. The nature of these approxima- +tions is to create a more conservative rejection, the technical details of which are examined in the +proof in our Online Appendix. Finally, we run the order statistics through a step-down procedure +proposed by Simes (1986) so that we find the ˆk largest number of ordered covariates rejected by +the data with FWER control. Also note that even if the global null, i.e. HN,0, is true, and every +linear sparse model active set is empty, that is Norder +1 += 0, the procedure in Step 3 is still valid +because we do not reject HN,1. +6 +Simulation +We demonstrate in simulations that our inferential theory allows us to select better models. +We compare different estimation approaches to select covariates and show that our approach bet- +ter trades off false discovery and correct selections and hence results in a better out-of-sample +performance. +Table 2 summarizes the benchmark models. Our framework contributes among three dimen- +sions: the selection step for the sparse model, the construction of the hypothesis and the multiple +testing adjustment. We consider variations for these three dimensions which yields in total six +estimation methods. By varying the different elements of the estimators, we can understand the +benefit of each component. +20 + +Table 2: Summary of estimation methods +Name +Abbreviation +Selection +Hypothesis +Multiple Testing +Rejection rule +Naive OLS +N-OLS +OLS without LASSO +Agnostic HA +No adjustment +pOLS < γ +Bonferroni OLS +B-OLS +OLS without LASSO +Agnostic HA +No adjustment +pOLS < +γ +JN +Naive LASSO +N-LASSO +LASSO without PoSI +Agnostic HA +No adjustment +pLASSO < γ +Bonferroni Naive LASSO +B-LASSO +LASSO without PoSI +Agnostic HA +Bonferroni +pLASSO < +γ +JN +Bonferroni PoSI +B-PoSI +LASSO with PoSI +Agnostic HA +Bonferroni +pPoSI < +γ +JN +Panel PoSI +P-PoSI +LASSO with PoSI +Data-driven HD +Simultaneity count +pPoSI < ργ +Ni +This table compares the different methods to estimate a set of covariates from a large dimensional panel. For each +method, we list the name and abbreviation. The selection refers to the regression approach for each univariate +time-series. The hypothesis is either agnostic or data-driven given the selected subset of covariates. The multiple +testing adjustment includes no adjustment, a conventional Bonferroni adjustment and our novel simultaneity count +for a data-driven hypothesis. The rejection rules combine the valid p-values and multiple testing adjustment. pOLS +is the p-value for a conventional t-statistics of an OLS estimator. pLASSO is the p-value without removing the lasso +bias or adjusting for post-selection inference, that is, it is simply the OLS p-values using the selected subset of +regressors. pPoSI is the debiased post-selection adjusted p-value based on Theorem 1. +Our baseline model is Panel PoSI, which uses post-selection inference LASSO, and a simultane- +ity count for a data driven hypothesis. The first component that we modify is the selection of the +sparse model. A simple OLS regression without shrinkage does not produce a sparse model. This +gives us the methods Naive OLS and Bonferroni OLS. A conventional LASSO results in a sparse +selection, but the p-values are not adjusted for the post-selection inference and the bias adjustment. +The corresponding models are the Naive LASSO and the Bonferroni Naive LASSO. The second +component is the hypothesis, which is agnostic for methods besides Panel PoSI. For the comparison +models, we either consider no multiple testing adjustment or the conventional Bonferroni adjust- +ment. Under the multiple testing adjustment we obtain the Bonferroni OLS, the Bonferroni Naive +LASSO and the Bonferroni PoSI. The outcome of all the estimations are adjusted p-values for the +covariates, which we use to select our model for a given target threshold. For a given value of γ we +include a covariate if its adjusted p-value is below the critical values summarized in the last column +of Table 2. +We simulate a simple and transparent model. Our panel follows the linear model +Yt,n = +J +� +j=1 +Xt,jβ(n) +j ++ ϵt,n +for t = 1, ..., T, n = 1, ..., N and j = 1, .., J. +The covariates and errors are sampled independently as normally distributed random variables: +Xt,j +iid +∼ N(0, 1), +ϵt +iid +∼ N(0, Σ). +The noise is either generated as independent noise with covariance matrix Σ = σ2I or as cross- +sectionally dependent noise with non-zero off-diagonal elements Σij = κ and diagonal elements +Σii = σ2. Note that our theorems for PoSI assume homogeneous noise, while dependent noise +violates our assumptions. Hence, the dependent noise allows us to test how robust our method is +21 + +Figure 5: Design of loadings β +This figure demonstrates the setting of our simulations with 10 factors, where loadings are shaded based on whether +they are active. In this staircase setting, the first factor affects all units, the 2nd factor affects 90%, and so on, and +lastly the 10th factor affects 10% of all units. +to misspecification. We set σ2 = 2 and κ = 1, but the results are robust to all these choices. +We construct the active set based on the staircase structure depicted in Figure 5. Of the J +covariates in X, we have K = 10 active independent factors. Figure 5 demonstrates the setting +for the 10 factors, where loadings are shaded based on whether they are active. The first factor +affects all units, the 2nd factor affects 90%, and so on, and lastly the 10th factor affects 10% of all +units. This setting is relevant, and also challenging from a multiple testing perspective. It results +in a large cohesion coefficient ρ, which makes the correct FWER control even more important. The +loadings are sampled from a uniform distribution, if they are in the active set: +β(n) +j +iid +∼ Unif +� +−1 +2, 1 +2 +� +for j in the active set, +β(n) +j += 0 +for j outside the active set. +We simulate a panel of dimension N = 120, J = 100 and T = 300 with K = 10 active factors. +The first half of the time-series observations is used for the in-sample estimation and selection, while +the second half serves for the out-of-sample analysis. All results are averages of 100 simulations. +We use the covariates selected on the in-sample data for regressions out-of-sample. Our focus is +on the inferential theory, and not on the bias correction for shrinkage. Hence, we first use the +inferential theory on the in-sample data to select our set of covariates. Second, we use the selected +subset of covariates in an OLS regression on the in-sample data to obtain the loadings. Last but +not least, we apply the estimated loadings of the selected subset to the out-of-sample data to obtain +the model fit. Note that this procedure helps a Naive LASSO, which in contrast to PoSI LASSO +does not have a bias correction. The out-of-sample explained variation is measured by R2, which +is the sum of explained variation normalized by the total variation. The rejection FWER is set to +γ = 5% or γ = 1%. The LASSO shrinkage penalty λ is selected by 5-fold cross-validation on the +in-sample data. +22 + +." +.". +... +" +" +" +"Table 3: Simulation Comparison between Selection Methods +Independent noise +Method +# Selections +# False Selections +# Correct Selections +OOS R2 +FWER γ = 5% +Panel PoSI +10.8 +2.8 +7.9 +10.0% +Bonferroni PoSI +4.7 +0.0 +4.7 +8.0% +Bonferroni Naive LASSO +0.0 +0.0 +0.0 +0.0% +Naive LASSO +0.2 +0.0 +0.2 +0.4% +Bonferroni OLS +1.0 +0.0 +1.0 +1.7% +Naive OLS +99.2 +89.2 +10.0 +-144.2% +FWER γ = 1% +Panel PoSI +8.6 +1.1 +7.5 +10.6% +Bonferroni PoSI +2.7 +0.0 +2.7 +5.2% +Bonferroni Naive LASSO +0.0 +0.0 +0.0 +0.0% +Naive LASSO +0.1 +0.0 +0.1 +0.3% +Bonferroni OLS +0.2 +0.0 +0.2 +0.5% +Naive OLS +46.4 +36.5 +9.9 +-19.3% +Cross-sectionally dependent noise +Method +# Selections +# False Selections +# Correct Selections +OOS R2 +FWER γ = 5% +Panel PoSI +10.1 +2.2 +7.9 +8.0% +Bonferroni PoSI +4.4 +0.0 +4.4 +7.2% +Bonferroni Naive LASSO +0.0 +0.0 +0.0 +0.0% +Naive LASSO +0.4 +0.0 +0.4 +0.5% +Bonferroni OLS +0.9 +0.0 +0.9 +1.3% +Naive OLS +83.7 +73.7 +10.0 +-83.8% +FWER γ = 1% +Panel PoSI +7.9 +0.6 +7.3 +10.3% +Bonferroni PoSI +2.4 +0.0 +2.4 +3.9% +Bonferroni Naive LASSO +0.0 +0.0 +0.0 +0.0% +Naive LASSO +0.0 +0.0 +0.0 +0.0% +Bonferroni OLS +0.3 +0.0 +0.3 +0.4% +Naive OLS +31.0 +21.2 +9.8 +-6.8% +This table compares the selection results for different methods in a simulation. For each method we report the num- +ber of selected covariates, the number of falsely selected covariates and the number of correctly selected covariates. +We also report the out-of-sample R2 of the models that estimated with the selected covariates on the out-of-sample +data. All results are averages of 100 simulations. The rejection FWER is set to γ = 5% or γ = 1%. We simulate a +panel of dimension N = 120, J = 100, T = 300. The first half of time-series observations is used for the in-sample +estimation and selection, while the second half serves for the out-of-sample analysis. The panel is generated by 10 +independent factors. The active set of the factors follows the staircase structure of Figure 5. The first factor affects +all units, the second 90%, and lastly the 10th factor affects 10%. The unknown error variance is estimated based as +a homogenous sample variance. The noise is either generated as independent noise with covariance matrix Σ = σ2I +or as cross-sectionally dependent noise with Σij = κ and Σii = σ2 for σ2 = 2 and κ = 1. +Table 3 compares the selection results for the different methods. For each method we report the +number of selected covariates, the number of falsely selected covariates and the number of correctly +23 + +selected covariates. We also report the out-of-sample R2. The upper panel shows the results for +independent noise, while the lower panel collects the results for cross-sectionally dependent noise. +PanelPoSI clearly dominates all models. It provides the best trade-off between correct and +false selection, which results in the best out-of-sample performance. In the case of γ = 5% and +independent noise, Panel PoSI selects 10.8 factors in a model generated by 10 factors. +7.9 of +these factors are correct. A simple Bonferroni correction is overly conservative. The Bonferroni +PoSI selects only 4.7 correct factors. While this overly conservative selection protects against false +discovery, it omits over half of the relevant factors which lowers the out-of-sample performance. +Using post-selection inference is important, as a naive lasso provides wrong p-values which makes +the overly conservative selection even worse. The other extreme is to have neither shrinkage nor +multiple testing adjustment. As expected the naive OLS has an extreme number of false selections +with a correspondingly terrible out-of-sample performance. +As expected, tightening the FWER control to 1% lowers the number of false rejections, but +also the number of correct selections. It reveals again that Panel PoSI provides the best inferential +theory among the benchmark models. Panel PoSI selects 7.5 correct covariates, while it controls +the false rejections at 1.1. The overly conservative Bonferroni methods select even fewer correct +covariates, which further deteriorates the out-of-sample performance. The gap in OOS R2 between +Panel PoSI and Bonferroni PoSI widens to 5.4%. All the other approaches cannot be used for a +meaningful selection. +Panel PosI performs well, even when some of the underlying assumptions are not satisfied. The +lower panel of Table 3 shows the results for dependent noise. As the dependence in the noise is +relatively strong, it can be interpreted as omitting a relevant factor in the set of candidate covariates +X. Even thought the PoSI theory is developed for homogeneous noise, Panel PoSI continues to +perform very well. In contrast, the comparison methods perform even worse, and the Bonferroni +approaches select even less correct covariates. +7 +Empirical Analysis +7.1 +Data and Problem +Our empirical analysis studies a fundamental problem in asset pricing. We select a parsimonious +factor model from a large set of candidate factors that can jointly explain the asset prices of a large +cross-section of investment strategies. Our data is standard and obtained from the data libraries +of Kenneth French and Hou, Xue, and Zhang (2018). +We consider monthly excess returns from January 1967 to December 2021, which results in a +time dimension of T = 660. Our test assets are the N = 243 double-sorted portfolios of Kenneth +French’s data library summarized in Table A.1 in the Appendix. The candidate factors are J = 114 +univariate long-short factors based on the data of Hou, Xue, and Zhang (2018). We include all +univariate portfolio sorts from their data library that are available for our time period, and construct +top minus bottom decile factor portfolios. In addition, we include the five Fama-French factors of +24 + +Fama and French (2015) from Kenneth French’s data library. +Our analysis projects out the excess return of the market factor. +We are interested in the +question which factors explain the component that is orthogonal to market movements. Hence, +we regress out the market factor from the test assets and use the residuals as test assets. We also +do not include a market factor in the set of long-short candidate factors. The original test assets +have a market component as they are long only portfolios. Our results are essentially the same +when we include the market component in the test assets, with the only difference that we would +need to include the market factor as an additional factor in our parsimonious models. The market +factor would always be selected by all models as significant, but this by itself is neither a novel nor +interesting result. +We present in-sample and out-of-sample results. +The in-sample analysis uses the first 330 +observations (January, 1967 to June, 1994), while the out-of-sample results are based on the second +330 observations (July, 1994 to December, 2021). As in the simulation, we first use the inferential +theory on the in-sample data to select our set of covariates. Second, we use the selected subset +of covariates in an OLS regression on the in-sample data to obtain the loadings. Last but not +least, we use the estimated loadings on the selected subset of factors for the out-of-sample model. +The LASSO penalty λ is selected via 5-fold cross-validation on the in-sample data to minimize the +squared errors.3 Hence, LASSO represents a first-stage dimension reduction tool, and we need the +inferential theory to select our final sparse model. +We allow our selection to impose a prior on two of the most widely used asset pricing models. +More specifically, we estimate models without a prior, and two specific priors that impose an infinite +weight on the Fama-French 3 factors (FF3) and the Fama-French 5 factors (FF5). This prior as +part of PoSI LASSO enforces that the FF3 and FF5 factors are included in the active set. Note +that because we work with data orthogonal to the market return, we do not include the market +factor in the prior, but only the size and value factors for FF3 and in addition the investment and +profitability factor for FF5. We denote these weights by ωFF3 and ωFF5. This is an example where +the researcher has economic knowledge that she wants to include in her statistical selection method. +We evaluate the models with standard metrics. The root-mean-squared error (RMSE) is based +on the squared residuals relative to the estimated factor models. Hence, in-sample the models are +estimated to minimize the RMSE. The pricing error is the economic quantity of interest. It is +the time-series mean of the residual component of the factor model, and corresponds to the mean +return that is not explained by the risk premia and exposure to the factors. In summary, we obtain +the residuals as ˆϵ = Yt,n − XS ˆβS for the selected factors, where the loadings are estimated on the +in-sample data. The metrics are the RMSE and mean absolute pricing error (MAPE): +RMSE = +� +� +� +� 1 +N T +N +� +i=1 +T +� +t=1 +ˆϵ2, +MAPE = 1 +N +N +� +i=1 +����� +1 +T +T +� +t=1 +ˆϵ +����� . +3We select λ from the grid exp(a) · log J/ +√ +T with a = −8, ..., 8. This grid choice satisfies the Assumptions in +Chatterjee (2014) and hence Assumption A.4. +25 + +In addition to Panel PoSI without and with the FF3 and FF5 priors, we consider the benchmark +methods of Table 2. We compare Panel PoSI (P-PoSI), Panel PoSI with infinite priors on FF3 and +FF5 (P-PoSI ωFF3 respectively ωFF5), Bonferroni Naive LASSO (B-LASSO), Naive LASSO (N- +LASSO), Bonferroni OLS (B-OLS) and Naive OLS (N-OLS). Our main analysis sets the FWER +control to the usual γ = 5%. +7.2 +Asset Pricing Results +Panel PoSI selects parsimonious factor models with the best out-of-sample performance among +the benchmarks. For the FWER rate of γ = 5% the number of factors differs substantially among +the different methods. Panel PoSI selects 3 factors. Imposing infinite priors on FF3 or FF5 results +in 4 and 5 factors for P-PoSI ωFF3 respectively ωFF5. In contrast, the alternative approaches select +too many factors. Bonferroni Naive LASSO includes 10, Naive Lasso 70, Bonferroni OLS 107 and +Naive OLS 114. These over-parametrized models lead to overfitting of the in-sample data. +Figure 6 shows in-sample and out-of-sample RMSE for each set of double-sorts. The composition +of the double sorts is summarized in Table A.1 in the Appendix. The in-sample performance in +the left subfigure has the expected result that more factors mechanically decrease the RMSE. +The important findings are in the right subfigure with the out-of-sample RMSE. The uniformly +best performing model is Panel PoSI without any priors. In fact, imposing a prior on the Fama- +French factors increases the out-of-sample RMSE. The conventional LASSO and OLS estimates +have substantially higher RMSE, which can be more than twice as large. +The Panel PoSI models also explain the average returns the best. In Figure 7, we compare +the mean absolute pricing errors among the benchmarks for each set of double sorts. Importantly, +the pricing errors are not used as in objective function of the estimation, and hence the fact that +the models with the smallest RMSE explain expected returns is an economic finding supporting +arbitrage pricing theory. Our Panel PoSI has the smallest out-of-sample pricing errors, which can +be up to six times smaller compared to the OLS estimates. Including the Fama-French factors as a +prior does not improve the models, except for the profitability and investment double sort, which +uses the same information as two of the Fama-French factors. +The Panel PoSI models select economically meaningful factors. Table 4 reports the ranking of +factors based on their FWER bound without prior and infinite prior weights on the Fama-French +3 and 5 factors. The rows are ordered based on sorted ascending ρ−1Njpj, which corresponds to +the FWER bound. It allows us to infer the number of factors for different levels of FWER control +values. Setting γ = 5% leads to 3, 4 and respectively 5 factors, while a γ = 1% results in 2, 4 and +5 factors, respectively. +In addition to their significance, we can infer the relative importance of factors. The baseline +PoSI with γ = 5% selects a size, dollar trading volume and value factor. The size and value factors +are among the most widely used asset pricing factors. Their selection is in line with their economic +importance and confirms the Fama-French 3 factor model. The dollar trading volume factor is less +conventional, but is correlated with many assets in our cross-sections. The size factor is the most +26 + +Figure 6: RMSE across cross-sections +(a) In-sample +(b) Out-of-sample +This figure shows the in-sample and out-of-sample root-mean-squared errors (RMSE) for each cross-section of test +assets for different factor models. The test assets are the N = 243 double-sorted portfolios, and we show the RMSE +for each set of double-sorts. The rejection FWER is set to γ = 5% The candidate factors are the 114 univariate +factor portfolios. The time dimension is T = 660. We use the first half for the in-sample estimation and selection, +while the second half serves for the out-of-sample analysis. We compare Panel PoSI (P-PoSI), Panel PoSI with +infinite priors on FF3 and FF5 (P-PoSI ωFF3 respectively ωFF5), Bonferroni LASSO (B-LASSO), Naive LASSO +(N-LASSO), Bonferroni OLS (B-OLS) and Naive OLS (N-OLS). +important as measured by the FWER bound, that is, the product of the number of relevant assets +and its minimum p-value are the smallest. The short term reversal factor is less important and +would require a FWER control of 10% to be included. +Imposing a prior affects the p-values of PoSI and the simultaneity count. For example, the +cohesiveness coefficient increases from ρ = 0.16 for no priors to ρ = 0.18 in the case of the two +priors. Hence, the FWER bounds of all factors can change when we impose a prior. The FF3 prior +increases the significance of the short-term reversal factor, which is widely used in asset pricing. +Interestingly, even for a FF5 prior, the profitability and investment factors remain insignificant. +7.3 +Number of Factors +Our method contributes to the discussion about the number of asset pricing factors. Many +popular asset pricing models suggest between three and six factors. Our approach allows a disci- +27 + +OP +1.32 +1.35 +1.49 +1.82 +2.02 +2.01 +2.00 +INV +ME +1.16 +1.18 +1.30 +1.58 +1.74 +1.73 +1.62 +Prior60 +ME +1.13 +1.16 +1.29 +1.88 +1.99 +1.94 +1.93 +Prior12 +ME +1.08 +1.10 +1.23 +1.49 +1.68 +1.53 +1.53 +Priorl +ME +0.95 +0.96 +1.06 +1.32 +1.44 +1.43 +1.42 +OP +ME +0.95 +0.97 +1.07 +1.32 +1.43 +1.42 +1.42 +INV +ME +0.59 +0.60 +0.68 +0.92 +1.02 +1.01 +1.01 +EP +ME +0.62 +0.64 +0.71 +0.98 +1.09 +1.06 +1.06 +DP +ME +0.57 +0.58 +0.66 +0.90 +1.00 +1.00 +1.00 +CFP +BEME +1.53 +1.57 +1.72 +2.11 +2.29 +2.28 +2.27 +OP +BEME +1.37 +1.40 +1.56 +1.91 +2.09 +2.08 +2.07 +INV +BE +0.99 +1.01 +1.11 +1.39 +1.48 +1.47 +1.46 +ME +N-OLS +B-OLS +N-LASSO B-LASSO +P-POSI +P-POSI +P-POSI +WFF3 +WFF5OP +2.72 +2.62 +2.33 +1.56 +0.91 +0.95 +0.96 +INV +ME +3.05 +3.03 +2.80 +2.28 +2.05 +2.10 +2.22 +Prior60 +ME +3.39 +3.35 +3.16 +2.12 +1.82 +2.04 +2.02 +Prior12 +ME +3.14 +3.08 +2.82 +2.29 +1.78 +2.23 +2.24 +Priorl +ME +2.87 +2.84 +2.65 +2.18 +1.91 +1.93 +1.95 +OP +ME +2.83 +2.79 +2.58 +2.10 +1.94 +1.98 +1.99 +INV +ME +2.39 +2.39 +2.31 +2.15 +1.99 +2.01 +2.01 +EP +ME +2.19 +2.14 +2.05 +1.90 +1.82 +1.88 +1.87 +DP +ME +2.35 +2.34 +2.29 +2.16 +1.97 +1.99 +2.00 +CFP +BEME +3.42 +3.30 +2.98 +2.39 +1.62 +1.67 +1.67 +OP +BEME +2.88 +2.78 +2.45 +1.79 +1.46 +1.50 +1.52 +INV +BE +3.04 +3.00 +2.84 +2.39 +2.28 +2.30 +2.31 +ME +N-OLS +B-OLS +N-LASSO B-LASSO +P-POSI +P-POSI +P-POSI +WFF3 +wFF5Figure 7: MAPE across cross-sections +(a) In-sample +(b) Out-of-sample +This figure shows the mean absolute pricing errors (MAPE) for each cross-section of test assets for different factor +models. The test assets are the N = 243 double-sorted portfolios, and we show the average |α| for each set of double +sorts. The rejection FWER is set to γ = 5% The candidate factors are the 114 univariate factor portfolios. The +time dimension is T = 660. We use the first half for the in-sample estimation and selection, while the second half +serves for the out-of-sample analysis. We compare Panel PoSI (P-PoSI), Panel PoSI with infinite priors on FF3 and +FF5 (P-PoSI ωFF3 respectively ωFF5), Bonferroni LASSO (B-LASSO), Naive LASSO (N-LASSO), Bonferroni OLS +(B-OLS) and Naive OLS (N-OLS). +plined estimate for the number of factors based on inferential theory. The level of sparsity of a +linear model also depends on the rotation of the covariates. Therefore, we also study the principal +components (PCs) of the covariates X as candidate factors. In this case, we use the step-down +procedure, which we refer to as “Ordered PoSI” or O-POSI for short. +Figure 8 shows the number of factors for different FWER rates γ. The factor count is obtained +by traversing K∗(γ) equal to 0.01, 0.02, 0.05 and 0.1. Panel PoSI without priors selects 2 factors +for γ = 0.01 and 3 for γ = 0.05. Once, we impose an infinite weight on the Fama-French 3 factors, +we select 4 factors for all FWER levels, while the prior on the Fama-French 5 factors results in a +5 factor model for all FWER levels. The Ordered PoSI with PCA rotated factors selects 3 factors +for all FWER levels. In summary, our results confirm that depending on the desired significance, +the number of asset pricing factors for a good model seems to be between 2 and 4. Note that our +analysis is orthogonal to the market factor, which would also be added to the final model. Thus, +28 + +OP +0.04 +0.04 +0.04 +0.08 +0.17 +0.17 +0.17 +INV +ME +0.04 +0.04 +0.05 +0.12 +0.10 +0.10 +0.11 +Prior60 +ME +0.04 +0.04 +0.05 +0.28 +0.36 +0.39 +0.38 +Prior12 +ME +0.06 +0.06 +0.08 +0.17 +0.21 +0.19 +0.19 +Priorl +ME +0.03 +0.03 +0.03 +0.07 +0.11 +0.11 +0.11 +OP +ME +0.03 +0.03 +0.04 +0.08 +0.11 +0.10 +0.10 +INV +ME +0.02 +0.02 +0.03 +0.05 +0.10 +0.10 +0.10 +EP +ME +0.02 +0.02 +0.03 +0.06 +0.09 +0.09 +0.09 +DP +ME +0.02 +0.02 +0.04 +0.06 +0.11 +0.11 +0.11 +CFP +BEME +0.04 +0.04 +0.05 +0.12 +0.17 +0.17 +0.17 +OP +BEME +0.06 +0.05 +0.06 +0.10 +0.13 +0.13 +0.13 +INV +BE +0.04 +0.04 +0.04 +0.10 +0.11 +0.10 +0.10 +ME +N-OLS +B-OLS +N-LASSO B-LASSO +P-POSI +P-POSI +P-POSI +WFF3 +WFF5Op +0.19 +0.21 +0.17 +0.14 +0.03 +0.03 +0.02 +INV +ME +0.10 +0.11 +0.10 +0.10 +0.09 +0.10 +0.09 +Prior60 +ME +0.18 +0.19 +0.17 +0.11 +0.08 +0.09 +0.08 +Prior12 +ME +0.14 +0.13 +0.09 +0.10 +0.08 +0.09 +0.09 +Priorl +ME +0.18 +0.17 +0.16 +0.14 +0.08 +0.08 +0.08 +OP +ME +0.14 +0.14 +0.13 +0.10 +0.09 +0.08 +0.08 +INV +ME +0.13 +0.14 +0.13 +0.09 +0.08 +0.08 +0.08 +EP +ME +0.12 +0.13 +0.15 +0.11 +0.07 +0.07 +0.08 +DP +ME +0.13 +0.12 +0.12 +0.10 +0.07 +0.07 +0.07 +CFP +BEME +0.27 +0.27 +0.23 +0.15 +0.05 +0.05 +0.05 +OP +BEME +0.17 +0.14 +0.13 +0.09 +0.05 +0.05 +0.05 +INV +BE +0.14 +0.12 +0.14 +0.10 +0.09 +0.09 +0.09 +ME +N-OLS +B-OLS +N-LASSO B-LASSO +P-POSI +P-POSI +P-POSI +WFF3 +wFF5Table 4: Selected factors with Panel PoSI +Factor +Nj +pj +ρ−1Njpj +Order +No prior +Size (SMB) +1824 +<0.00001 +<0.0001 +1 +Dollar Trading Volume (dtv 12) +2099 +<0.00001 +<0.0001 +2 +Value (HML) +1191 +<0.00001 +0.0280 +3 +Short-Term Reversal (srev) +1050 +0.00001 +0.0974 +4 +Forecast Revisions (rev 1) +242 +0.00018 +0.2782 +5 +Investment (CMA) +998 +0.00112 +>0.9999 +6 +Profitability (RMW) +797 +0.00123 +>0.9999 +7 +FF3 prior (ωFF3) +Size (SMB) +2802 +<0.00001 +<0.0001 +1 +Value (HML) +2802 +<0.00001 +<0.0001 +2 +Dollar Trading Volume (dtv 12) +779 +<0.00001 +0.0017 +3 +Short-Term Reversal (srev) +1106 +<0.00001 +0.0049 +4 +Profitability (RMW) +819 +0.00006 +0.2527 +5 +Investment (CMA) +874 +0.00087 +>0.9999 +6 +FF5 prior (ωFF5) +Size (SMB) +2911 +<0.00001 +<0.0001 +1 +Value (HML) +2911 +<0.00001 +<0.0001 +2 +Forecast Revisions (rev 1) +230 +<0.00001 +0.0005 +3 +Short-Term Reversal (srev) +1140 +<0.00001 +0.0052 +4 +Dollar Trading Volume (dtv 12) +661 +<0.00001 +0.0072 +5 +Profitability (RMW) +2911 +0.00001 +0.1937 +6 +Investment (CMA) +2911 +0.00001 +0.1996 +7 +Gross profits-to-assets (gpa) +1151 +0.00013 +0.8382 +8 +This table reports ranking of factors based on their FWER bound for no prior, and infinite weight priors on the +Fama-French 3 and 5 factors. The test assets are the N = 243 double-sorted portfolios and the candidate factors are +J = 114 univariate long-short factors. The rows are ordered based on sorted ascending ρ−1Njpj, which corresponds +to the FWER bound. +the final model would have between 3 and 5 factors. +Table 5 further confirms our findings about the number of asset pricing factors. We compare +the number of factors for γ = 5% selected either from the univariate high-minus-low factors (HL), +their PCA rotation or the combination of the high-minus-low factors and their PCs. Panel PoSI +selects consistently 3 factors from the long-short factors and their PCs. When combined, PoSI +selects 4 factors, which is plausible as the optimal sparse model can be different for this larger set +of candidate factors. The Bonferroni PoSI is overly conservative and selects only 2 HL factors. The +models based on Naive LASSO or OLS select excessively many factors independent of the rotation. +Overall, the findings support that parsimonious asset pricing models can be described by three to +four factors. Of course, any discussion about the number of asset pricing factors is always subject +to the choice of test assets and candidate factors. +29 + +Figure 8: Number of selected factors for different FWER +(a) Univariate factors with priors (P-POSI) +(b) PCA rotated factors (O-POSI) +This figure shows the number of selected factors to explain the test assets of double-sorted portfolios for different +FWER rates γ. The factor count is obtained by traversing K∗(γ) for γ ranging from 0.01 to 0.1. The left subfigure +uses univariate high-minus-low factors as candidate factors. We consider the case of no prior, and the cases of an +infinite weight on the Fama-French 3 factor model (ωFF3) and an infinite weight on the Fama-French 5 factor model +(ωFF5). The right subfigure uses the PCA rotation as candidate factors with the step-down procedure Ordered PoSI +(O-POSI). +Table 5: Number of selected factors for different methods +HL +PCs +HL + PCs +Panel PoSI +3 +3 +4 +Bonferroni PoSI +2 +3 +2 +Bonferroni Naive LASSO +10 +29 +10 +Naive LASSO +70 +50 +76 +Bonferroni OLS +107 +13 +117 +Naive OLS +114 +50 +164 +This figure shows the number of selected factors to explain the test assets of double-sorted portfolios for different +methods and different sets of candidate factors. The rejection FWER is set to γ = 5%. The factor count is obtained +by traversing K∗(γ) for γ. The number of factors is selected on the in-sample data. For PCs, we use the step-down +method for the nested hypothesis. +8 +Conclusion +This paper proposes a new method for covariate selection in large dimensional panels. +We +develop the conditional inferential theory for large dimensional panel data with many covariates +by combining post-selection inference with a new multiple testing method specifically designed for +panel data. Our novel data-driven hypotheses are conditional on sparse covariate selections and +valid for any regularized estimator. +Based on our panel localization procedure, we control for +family-wise error rates for the covariate discovery and can test unordered and nested families of +hypotheses for large cross-sections. We provide a method that allows us to traverse the inferential +30 + +P-PoSI +P-PoSL WE3 +6 +P-PoSI WFF5 +Factor count +5 +5 +5 +5 +5 +4 +4 +4 +44 +4 +3 +3 +2 +2 +2 +0.01 +0.02 +0.05 +0.17 +6 +PC count +5 +4. +3 +3 +3 +3 +3 : +0.01 +0.02 +0.05 +0.1results and determine the least number of covariates that have to be included given a user-specified +FWER level. +As an easy-to-use and practically relevant procedure, we propose Panel-PoSI, which combines +the data-driven adjustment for panel multiple testing with valid post-selection p-values of a gen- +eralized LASSO, that allows to incorporate weights for priors. In an empirical study, we select a +small number of asset pricing factors that explain a large cross-section of investment strategies. +Our method dominates the benchmarks out-of-sample due to its better control of false rejections +and detections. +A +Post-selection Inference with Weighted-LASSO +A.1 +Weighted-LASSO: Linear Truncation Results +This appendix collects the assumptions and formal statements underlying Theorem 1. +We +present the results for the Weighted-LASSO, which includes the conventional LASSO as a special +case. In order to ensure uniqueness of the LASSO solution, we impose the following condition, +which is standard in the LASSO literature: +Definition A.1. General position +The matrix X ∈ RT×J has columns in general position if the affine span of any J0 + 1 points +(σ1Xi1, ..., σJ0+1XiJ0+1) in RT for arbitrary σ1, ...σd0+1 ∈ {±1} does not contain any element of +{±Xi : i /∈ {i1, ..., iJ0+1}}, where J0 < J4 and Xi denotes ith column of X. +This position notion will help us to avoid ambiguity in the LASSO solution. Note that this +condition is a much weaker requirement than full-rank of X, and states that if one constructs a +J0-dimensional subspace, it must contain at most J0 + 1 entries of {±X1, ..., ±XJ}. Even though +this appears to be a complicated and mechanical condition, by a union argument it turns out that +with probability 1, if the entries of X ∈ RT×J are drawn from a continuous probability distribution +on RT×J then X is in general position.5 Then, we will be able to discuss the LASSO solution for +general design with relative ease, thanks to Lemma 3 of Tibshirani (2013). It shows that if X lie +in general position, it is sufficient to have a unique LASSO solution regardless of the penalty scalar +λ. This condition will later be used in establishing our Lemma A.2. +We can now state the formal assumptions: +Assumption A.1. Unique low dimensional model +(a) Low dimensional truth: +The data satisfies Y = XSβS + ϵ where |S| = O(1); +(b) General position design: +The covariates X have columns in general position as given by Definition A.1; +4The original condition needs to hold for J0 < min{T, J} but in the scope of our study, we consider T > J. +5See Donoho (2006) and §2.2 of Tibshirani (2013) for more discussions on uniqueness and general position. +31 + +We start our analysis with the simpler model of known error variance, and later extend it to +the case of estimated unknown variance. +Assumption A.2. Gaussian residual with known variance +The residuals are distributed as ϵ ∼ N(0, Σ) where Σ is known. +Before formalizing the inferential theory, we need to clarify the quantity for which we want +to make inference statements. As stated before, we only test the hypothesis on a covariate if its +LASSO estimate turns out active. This is exactly the approach how researchers in practice conduct +explorations in high-dimensional datasets. In other words, we focus on ˆβM and quantities associated +with it, where M denotes the active set of selected covariates. +We study the inferential theory of the “debiased estimator”, which is a shifted version of the +LASSO fit as defined below. We show that this debiased estimator is unbiased, consistent and +follows a truncated Gaussian distribution, with profound connections to the debiased LASSO lit- +erature such as Javanmard and Montanari (2018), but has different properties by a subtle different +descent direction. More concretely, given M, clearly ˆY = XM ˆβM is the fitted value since ˆβ−M = 0, +where −M is the complement of the set M. We let ˆϵM := Y − XM ˆβM be the residual from the +LASSO estimator. By considering only the partial LASSO loss of ℓ(Y, XM, λ, β) and given we are +currently at the LASSO estimator ˆβ, the Newton step is X+ +Mˆϵ following (Boyd and Vandenberghe, +2004, § 9.5.2), where we denote X+ +M = (X⊤ +MXM)−1X⊤ +M as the pseudo-inverse of the active subma- +trix of X. The invertibility of X⊤ +MXM either is observed when we are in the fixed design regime +or happens almost surely when we are dealing with continuous quantities, as a consequence of +Assumption A.1(b) as argued in Tibshirani (2013) and Lee, Sun, Sun, and Taylor (2016). Now we +can formally define the main object of our inferential theories: +Definition A.2. Debiased Estimator +The debiased Weighted-LASSO estimator ¯βM given M is given by +¯βM = ˆβM + X+ +MˆϵM +(21) +It is now evident why some of the literature refers to the debiased estimator also as the one-step +estimator: given that ˆβM solves the Karush-Kuhn-Tucker (KKT) condition and reaches the optimal +sub-gradient for the full loss ℓ(Y, X, c, β), our debiased estimator ¯βM is the result of moving one +more Newton-Ralphson method step after ˆβM, but only taking XM rather than X as a whole into +the likelihood loss function. Hence, the update step is actually only a partial update from the +LASSO solution point. Intuitively, ¯βM should still be close to solving the KKT conditions, and +would exactly solve the KKT conditions if XM happen to be the true covariates (i.e. XM = XS). +If we were to take a Newton’s method step with gradient and Hessian calculated with the entirety +of data X, or equivalently taking a full update from the stationary point, we will recover the ˆβd +M +proposed in Javanmard and Montanari (2018). The material difference is that the full-update would +require the J ×J precision matrix Ω = Γ−1, where Γ = X⊤X if X assumed fixed or Γ = E[X⊤X] if +X assumed to be generated from a stationary process. Using ℓ(Y, XM, λ, β) instead of ℓ(Y, X, λ, β), +32 + +our debiased estimator would not need the full Hessian, which is leveraging LASSO’s screening +property and uses (X⊤ +MXM)−1X⊤ +M (i.e. X+ +M) as a much lower-dimensional alternative of ΩX⊤. +Without loss of generality, we assume that the covariate indexed i ≤ |M| is part of M, and we can +always rearrange the columns of X to have the first |M| covariates as active. Let η = (X+ +M)⊤ei ∈ RT +be a vector where ei ∈ R|M| is a vector with 1 at ith coordinate and 0 otherwise. Hence, the η +vector is the linear mapping from Y to the ith coordinate of an OLS estimator. In particular, the +debiased estimator and the response satisfy the following relationship: +Lemma A.1. Debiased Estimator is OLS-post-LASSO +The debiased estimator is a linear mapping of Y . Specifically, given η = (X+ +M)⊤ei: +¯βi = η⊤Y +(22) +Moreover, ¯βM is the OLS estimate of regressing XM on Y : +¯βM = arg min +β +1 +2T ∥Y − XMβ∥2 +2. +(23) +The proof of Lemma A.1 is deferred to the Online Appendix. Although its proof is simple, this +lemma reveals that our debiased estimator is the same as the least-square after LASSO estimator +proposed in Belloni and Chernozhukov (2013). +Our strategy to obtain a rigorous statistical inferential theory with p-values is as follows. First +we perform an algebraic manipulation to transform ˆβM into ¯βM in the linear form of (22). Then, we +follow the strategy in Lee, Sun, Sun, and Taylor (2016) to traverse the KKT subgradient optimal +equations for general X by writing it equivalently into a truncation in the form of {AY ≤ b}, as +we will do in Lemma A.2. Finally we will circle back to ˆβM by the linear mapping between ¯βM +and Y and the distributional results induced by the fact that Y is truncated by {AY ≤ b}. +For our Weighted-LASSO, the KKT sub-gradient equations are +X⊤(X ˆβ − Y ) + λ +� +s +v +� +⊙ ω−1 = 0 +where +� +� +� +si = sign(ˆβi) +if ˆβi ̸= 0, ωi < ∞ +vi ∈ [−1, 1] +if ˆβi = 0, ωi < ∞ +(24) +In other words, when ω is specified, the KKT conditions can be identified using the tuple of +{M, s}, where M is the active covariates set and s is the signs of LASSO fit. This is a consequence +of how LASSO KKT condition can separate the slacks into s for active variables and v for inactive +variables. If we have infinite importance weights (J ̸= ∅), we would simply need si < ∞ for i ∈ J +because λsi/ωi = 0 is guaranteed. We rigorously characterize the KKT sub-gradient conditions as a +combinations of signs and infinity norm bounds conditions by the following lemma, which parallels +Lemma 4.1 of Lee, Sun, Sun, and Taylor (2016): +Lemma A.2. Selection in norm equivalency +33 + +Consider the following random variables +w(M, s, ω) = (X⊤ +MXM)−1(X⊤ +MY − λs ⊙ ω−1 +M ) +u(M, s, ω) = ω−M ⊙ +� +X⊤ +−M(X+ +M)⊤s ⊙ ω−1 +M + 1 +λX⊤ +−M(I − PM)Y +� +(25) +where PM = XMX+ +M ∈ RT×|M| is the projection matrix. The Weighted-LASSO selection can be +written equivalently as +{M, s} = {sign(w(M, s, ω)) = s, ∥u(M, s, ω)∥∞ < 1} +(26) +Using this characterization, we are then able to provide the distributional results for the debiased +estimators. Consider ξ = Ση(η⊤Ση)−1 ∈ RT as a covariance-scaled version of our η, and a mapping +of Y using residual projection matrix: z = (I − ξη⊤)Y . Note that z can be calculated once we +observe (X, Y ), so it can be conditioned on were we to do so. We will soon see that the truncation +set will depend on the variable z, but this does not cause any issues thanks to the following lemma, +the proof of which is deferred to the Online Appendix: +Lemma A.3. Ancillarity in truncation +The projected z and the debiased estimator ¯βi are independently distributed. +As a result of Lemma A.3, when describing the distribution of ¯βi, we can use z in its truncation +conditions as long as we condition on z as well. To simplify notation, we can collect all quantities +we need to condition on into +˜ +M := ((M, s), z, ω, X). Now we can assemble the consequences of +Lemmas A.1, A.2, A.3 to arrive at the truncated Gaussian statements for the debiased estimator +similar to Lee, Sun, Sun, and Taylor (2016), but for weighted-LASSO: +Theorem A.1. Truncated Gaussian +Under Assumptions A.1 and A.2 for i ∈ M, ¯βi is conditionally distributed as: +¯βi| ˜ +M ∼ T N(βi, η⊤Ση; [V −(z), V +(z)]) +(27) +where T N is a truncated Gaussian with mean βi, variance η⊤Ση and truncation set [V −(z), V +(z)]. +βi denotes the ith entry of the true β. The vector of signs is s = sign(ˆβM) ∈ R|M| and the truncation +set depends on +A = +� +�� +λ−1X⊤ +−M(I − PM) +−λ−1X⊤ +−M(I − PM) +−diag(s)X+ +M +� +�� ∈ R(2J−|M|)×T , +b = +� +�� +ω−1 +−M − X⊤ +−M(X+ +M)⊤s ⊙ ω−1 +M +ω−1 +−M + X⊤ +−M(X+ +M)⊤s ⊙ ω−1 +M +−λ · diag(s)(X⊤ +MXM)−1s ⊙ ω−1 +M +� +�� ∈ R2J−|M| +V −(z) = +max +j:(Aξ)j<0 +bj − (Az)j +(Aξ)j +, +V +(z) = +min +j:(Aξ)j>0 +bj − (Az)j +(Aξ)j +. +Notice that Theorem A.1 is decoupled across M, which is to say we are able to deal with +1-dimensional statistics. We arrive at this form because the construction of (V −, V +) over the +34 + +extreme points of the linear inequality system (or vertices of the polyhedral) has decomposed the +dimensionality of the truncation. This decoupling is of significant practical value, in that it would +be otherwise a non-trivial task to calculate a statistic of multivariate (in our case |M|-dimensional) +truncated Gaussian and then marginalize over |M| − 1 dimensions. +A.2 +Weighted-LASSO Quasi-Linear Truncation with Estimated Variance +This section generalizes the distribution results to the practically relevant case when the noise +variance is unknown and has to be estimated. This becomes a challenging problem for post-selection +inference. We replace Assumption A.2 by the following assumption: +Assumption A.3. Gaussian residual with simple unknown variance +The residuals are distributed as ϵi +iid +∼ N(0, σ2) where σ2 is unknown. +The simple structure of unknown variance of Assumption A.2 is common in the post-selection +inference literature as for example in Lee, Sun, Sun, and Taylor (2016) and Tian, Loftus, and Taylor +(2018). A feasible conditional distribution replaces σ2 with an estimate. Under Assumption A.2, +we can estimate the variance using LASSO residuals and then reiterate the previous truncation +arguments. The most common standard variance estimator is +ˆσ2(Y ) = ∥Y − X ˆβ∥2 +2/(T − |M|). +(28) +In classical regression analysis, the normally distributed estimated coefficient divided by an +estimated standard deviation follows a t-statistic. Hence, we would expect that a truncated normal +debiased estimator divided by a sample standard deviation might yield a truncated t-distribution. +However, the arguments are substantially more involved. Simply using ˆσ(Y ) of (28) in the expres- +sion η⊤Ση of Theorem A.1 changes the truncation. Specifically, Y having truncated support means +ˆσ(Y )2 is not χ2-distributed supported on the entire R+, which makes the support of ¯β/ˆσ(Y ) non- +trivial. Therefore, in order to correctly assess the truncation of the studentized quantity, we have +to disentangle how much truncation is implied in ˆσ(Y )−1 and ¯β simultaneously. Geometrically, as +ˆσ(Y ) is a non-linear function of Y and ¯β, the truncation on Y is in fact no longer of the simple +linear form {AY ≤ b} such as in Theorem A.1. +Instead of a polyhedral induced by affine constraints, we have a “quasi-affine constraints” form +of {CY ≤ ˆσ(Y )b} because LASSO KKT conditions preserve the estimated variance throughout +the arguments. Thus, both sides of the inequality CY ≤ ˆσ(Y )b have Y , and in right-hand-side +the ˆσ(Y ) is non-linear in Y . A significantly more complex set of arguments are needed compute +the exact truncation, which is equivalent to solve for a |M|-system of non-linear inequalities rather +than linear inequalities that constrain the support of Y for inference on each ¯βi. Theorem A.2 +shows the appropriate truncation based on those arguments: +Theorem A.2. Truncated t-distribution for estimated variance +Under Assumptions A.1 and A.3, and the null hypothesis that βi = 0, the studentized quantity +35 + +¯βi/∥η∥ˆσ(Y ) follows +¯βi/∥η∥ˆσ(Y ) ∼ TTd;Ω, +(29) +where TT is a truncated t-distribution with d degrees of freedom and truncation set Ω. +The +truncation set Ω = � +i∈M{t : t +√ +Wνi + ξi +√ +d + t2 ≤ −θi +√ +W} is an |M|-intersection of simple +inequality-induced intervals based on the following quantities. +The active signs are denoted as +s = sign(ˆβM) ∈ R|M|. The scaled LASSO equivalent penalty is ˜λ2 = +λ2 +ˆσ2(Y )·(T−|M|)+∥(X+ +M)⊤s⊙ω−1 +M ∥2 +2λ2 . +θi = (˜λsi +� +T − |M| +1 − ˜λ2∥(X+ +M)⊤s ⊙ ω−1 +M ∥2 +2 +) · e⊤ +i +� +(X⊤ +MXM)−1s ⊙ ω−1� +for i ∈ M +C = −diag(s)X+ +M ∈ R|M|×T , +ν = Cη ∈ R|M|, +ξ = C(PM − ηη⊤)Y ∈ R|M|, +d = tr(I − PM), +W = ˆσ2(Y ) · d + (η⊤Y )2 +The quantities θ and C describe the quasi-linear constraints, whereas ν and ξ transform them +into the form of Ω. Note that the Ω set is obtained from solving a low-dimensional set of quadratic +inequalities that do not necessarily yield a single interval after intersection. We provide a proof of +this result in the Online Appendix. +Using Theorem A.2 in practice poses several challenges. First, the computations are much more +involved, especially as each βi requires calculation of Ω which includes |M| actual constraints, each +of which involves solving a simple but still non-linear inequality. It is non-trivial to ensure that the +numerical stability holds at every step of the calculations. Second, since Ω is not necessarily an +interval, it is harder to interpret the truncation and also calculate the cumulative density function +through Monte-Carlo simulations when there is a non-trivial truncation structure. Third, in fact, +the authors in Tian, Loftus, and Taylor (2018) recommend a regularized likelihood minimizing +variance estimator that deviates from the simple ˆσ(Y ), which would in turn involves more numerical +integration and optimization steps. Last but not least, this result was proposed initially for studying +scale-LASSO, which is why there has to be a penalty term transformation of λ to ˜λ. Our goal is to +provide a set of tools that can be useful for a wide range of applications including the LASSO with +l2 squared norm loss rather than un-squared norm loss. These implementation difficulties are also +discussed in more detail in the Online Appendix, which provides the accompanying proofs and the +exact forms of the truncations. +We provide a practical solution based on an asymptotic normal argument. +We impose the +standard assumption that we have a consistent estimator of the residual variance: +Assumption A.4. Consistent estimator ˆσ(Y ) +Given λ, the residual variance estimator is consistent ˆσ(Y ) +p→ σ2 as T → ∞. +This general assumption includes many common scenarios such as the results specified in Corol- +lary 6.1 of van de Geer and B¨uhlmann (2011), or in Theorem 2 of Chatterjee (2014). For example, +for diminishing c +� +log(J)/T → 0 as J, T grow and our Assumptions A.1 and A.3, we obtain con- +sistency of ˆσ(Y ) of (28) by Chatterjee (2014). +36 + +Theorem A.3. Asymptotic truncated normal distribution +Suppose Assumptions A.1, A.3 and A.4 hold. Under the null hypothesis that βi = 0 and for T → ∞ +the studentized quantity ¯βi/∥η∥ˆσ(Y ) follows +¯βi/∥η∥ˆσ(Y ) ∼ TNΩ, +(30) +where TN is a truncated normal distribution with truncation Ω = [V −(z)/∥η∥2ˆσ(Y ); V +(z)/∥η∥2ˆσ(Y )], +where V −(z) and V +(z) are the same as in Theorem A.1. +The asymptotic distribution result has several advantages. First, it is intuitive since it parallels +the classical OLS inference with a t-statistic converging to Gaussianity. Secondly, it is computa- +tionally more tractable than results of Appendix Theorem A.2. 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Zhang (2014): “Confidence intervals for low dimensional parameters in high +dimensional linear models,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), +76(1), 217–242. +Zrnic, T., and M. I. Jordan (2020): “Post-Selection Inference via Algorithmic Stability,” Working paper. +39 + diff --git a/7tAyT4oBgHgl3EQfc_c0/content/tmp_files/load_file.txt b/7tAyT4oBgHgl3EQfc_c0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d4bb0ab2f65f40aedef25f0ab880d94d6b9c0e4 --- /dev/null +++ b/7tAyT4oBgHgl3EQfc_c0/content/tmp_files/load_file.txt @@ -0,0 +1,1667 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf,len=1666 +page_content='Inference for Large Panel Data with Many Covariates∗ Markus Pelger† Jiacheng Zou‡ December 31, 2022 Abstract This paper proposes a new method for covariate selection in large dimensional panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We de- velop the inferential theory for large dimensional panel data with many covariates by combining post-selection inference with a new multiple testing method specifically designed for panel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our novel data-driven hypotheses are conditional on sparse covariate selections and valid for any regularized estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Based on our panel localization procedure, we control for family-wise error rates for the covariate discovery and can test unordered and nested families of hypothe- ses for large cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As an easy-to-use and practically relevant procedure, we propose Panel-PoSI, which combines the data-driven adjustment for panel multiple testing with valid post-selection p-values of a generalized LASSO, that allows to incorporate priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In an empir- ical study, we select a small number of asset pricing factors that explain a large cross-section of investment strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our method dominates the benchmarks out-of-sample due to its better control of false rejections and detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Keywords: panel data, high-dimensional data, LASSO, number of covariates, post-selection inference, multiple testing, adaptive hypothesis, step-down procedures, factor model JEL classification: C33, C38, C52, C55, G12 ∗We thank conference and seminar participants at Stanford, the California Econometric conference and the NBER-NSF SBIES conference for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Jiacheng Zou gratefully acknowledges the generous support by the MS&E Departmental Fellowship, and Charles & Katherine Lin Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' †Stanford University, Department of Management Science & Engineering, Email: mpelger@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' ‡Stanford University, Department of Management Science & Engineering, Email: jiachengzou@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00292v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='EM] 31 Dec 2022 1 Introduction Our goal is the selection of a parsimonious sparse model from a large set of candidate covariates that explains a large dimensional panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This problem is common in many social science applica- tions, where a large number of potential covariates are available to explain the time-series of a large cross-section of units or individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' An example is empirical asset pricing, where the literature has produced a “factor zoo” of potential risk factors to explain the large cross-section of stock returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This problem requires a large panel, as a successful asset pricing model should explain the many available investment strategies, resulting in a large panel of test assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' At the same time, there is no consensus about which are the appropriate factors, which leads to a statistical selection problem from a large set of candidate risk factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' So far, the literature has only provided solutions for one of the two subproblems, while keeping the dimensionality of the other problem small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our paper closes this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The inferential theory on a large panel with many covariates is a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As a first step, we have to select a sparse set of covariates from a large pool of candidates with a regularized estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The challenge is to provide valid p-values from this estimation that account for the post-selection inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Furthermore, researchers might want to impose economic priors on which variables should be more likely to be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The second challenge is that the panel cross-section results in a large number of p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, some of them are inadvertently very small, which if left unaddressed leads to “p-hacking”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The multiple testing adjustment conditional on the selected subset of covariates from the first step is a novel problem, and requires to redesign what hypotheses should be tested jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A naive counting of all tests is overly conservative, and the test design and simultaneity counts need to be conditional on the covariate selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This paper proposes a new method for covariate selection in large dimensional panels, tackling all of the above challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We develop the inferential theory for large dimensional panel data with many covariates by combining post-selection inference with a new multiple testing method specifi- cally designed for panel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our novel data-driven hypotheses are conditional on sparse covariate selections and valid for any regularized estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Based on our panel localization procedure, we control for family-wise error rates for the covariate discovery and can test unordered and nested families of hypotheses for large cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As an easy-to-use and practically relevant procedure, we propose Panel-PoSI, which combines the data-driven adjustment for panel multiple testing with valid post-selection p-values of a generalized LASSO, that allows to incorporate priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our paper proposes the novel conceptual idea of data-driven hypotheses family for panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This allows us to put forward a unifying framework of valid post-selection inference and multiple test- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Leveraging our data-driven hypotheses family, we adjust for multiple testing with a localized simultaneity count, which increases the power, while maintaining false discovery rate control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' An essential step for a formal statistical test is to formulate the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This turns out to be non-trivial for a large panel with a first stage selection step for the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It is a fundamental insight of our paper, that the hypothesis of our test has to be conditional on the selected set of 1 active covariates of the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Once we have defined the appropriate hypothesis, we can deal with the multiple testing adjustment, which by construction is also conditional on the selection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our method is a disciplined approach based on formal statistical theory to construct and in- terpret a parsimonious model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It goes beyond the selection of a sparse set of covariates as it also provides the inferential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is important as it allows to rank the covariates based on their statistical significance and can also be applied for relatively short time horizons, where cross- validation for tuning a regularization parameter might not be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We answer the question which covariates are needed to explain the full panel jointly, and can also accommodate “weak” covariates or factors that only affect a small subset of the cross-sectional units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our data-driven hypothesis perspective exploits the geometric structure implied by the first stage selection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Given valid post-selection p-values of a regularized sparse estimator from time-series regressions, we collect them across the large cross-section into a “matrix” of p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Only active coefficients, that are selected in the first stage, contribute p-value entries, whereas covariates that were non-active lead to “holes” in this matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We leverage the non-trivial shape of this matrix to form our adaptive hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This allows us to make valid multiple testing adjusted inference statements, for which we design a panel modified Bonferroni-type procedure that can control for the family-wiser error rate (FWER) in discovery of the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As one loosens the FWER requirements, the inferential thresholds admits more and more explanatory variables, which suggests that the amount of covariates we expect to admit and the FWER control level form an “false-discovery control frontier”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We provide a method that allows us to traverse the inferential results and determine the least number of covariates that have to be included given a user-specified FWER level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In other words, we provide a statistical significance test for the number of factors in a panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We propose the novel procedure Panel-PoSI, which combines the data-driven adjustment for panel multiple testing with valid post-selection p-values of a generalized LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' While our multiple testing procedure is valid for any sparsity constrained model, Panel-PoSI is an easy-to-use and prac- tically relevant special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We propose Weighted-LASSO for the first stage selection regression and provide valid p-values through post-selection inference (PoSI), which yields a truncated-Gaussian distribution for an adjusted LASSO estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This geometric perspective is less common in the LASSO literature, but has the advantage that it avoids the use of infeasible quantities, in particu- lar the second moment of the large set of potential covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The Weighted-LASSO generalizes LASSO by allowing to put weights onto prior belief sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For example, a researcher might have economic knowledge that she wants to include in her statistical selection method, and impose an in- finite prior weight to include specific covariates in the sparse selection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our Weighted-LASSO makes several contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First, the expression for the truncated conditional distribution with weights become much more complex than for the special case of the conventional LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, we provide a simple, easy-to-use and asymptotically valid conditional distribution in the case of an estimated noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 2 We demonstrate in simulations and empirically that our inferential theory allows us to select better models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We compare different estimation approaches to select covariates and show that our approach better trades off false discovery and correct selections and hence results in a better out- of-sample performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our empirical analysis studies the fundamental problem in asset pricing of selecting a parsimonious factor model from a large set of candidate factors that can jointly explain the asset prices of a large cross-section of investment strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We consider a standard data set of 114 candidate asset pricing factors to explain 243 double sorted anomaly portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We show that Panel PoSI selects 3 factors which form the best model to explain out-of-sample the expected returns and the variations of the test assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The selected factors are economically meaningful and we can rank them based on their relative importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A prior on the Fama-French factors does not improve the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our findings contributes to the discussion about the number of asset pricing factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 relates our work to the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Section 2 introduces the model and the Weighted-LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Section 3 discusses the appropriate hypotheses to be considered for inference on the entire panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Section 4 proposes a joint unordered test for the panel using multiple testing adjustment so that we can maintain FWER control, and shows how to traverse this procedure to acquire the least factor count associated with each FWER target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In section 5 we consider the case of nested hypotheses, where the covariates observe a fixed ordering, which is of independent interest, and we propose a step-down procedure for this setting that maintains false discovery control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Section 6 provides the results of our simulation and Section 7 discusses our empirical studies on a large asset pricing panel data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Section 8 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The proofs and more technical details are available in the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 Related Literature The problem of multiple testing is an active area of research with a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The statistical inference community has studied the problem of controlling the classical FWER since Bonferroni (1935), and controlling for false-discover rate (FDR) going back to Benjamini and Hochberg (1995) and Benjamini and Yekutieli (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Bonferroni (1935) allows for arbitrary correlation in the test statistics because its validity comes from a simple union bound argument, and is in fact the optimal test when statistics are “close to independent” under true sparse non-nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' FDR control on the other hand requires a discussion about the estimated covariance in the test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Recent developments include a stream of papers led by Barber and Cand´es (2015) and Cand´es, Fan, Janson, and Lv (2018), which constructs a generative model to produce fake data and control for FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Fithian and Lei (2022) is a more recent work that iteratively adjusts the threshold for each hypothesis in the family to seek finite sample exact FDR control and dominates Benjamini and Hochberg (1995) and Benjamini and Yekutieli (2001) in terms of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Another notion on temporal false discovery control has been revived more recently by Johari, Koomen, Pekelis, and Walsh (2021), who consider the industry practice of constantly checking p-values and provide an early stopping in line with Siegmund (1985) that adjusts for bias from sequentially picking favorable 3 evidence, whereas we consider a static panel that is not an on-going experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' There are cases where the covariates warrant a natural order such that the hypothesis family possesses a special testing logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A hierarchical structure in covariates arises when the inclusion of the next covariate only make sense if the previous covariates is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' An example is the use of principal component (PC) factors, where PCs are included sequentially from the dominating one to the least dominating one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We distinguish this from putting weights and assigning importance on features because this variant of family of hypotheses warrants a new definition of FWER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We propose a step-down procedure that can be considered as a panel extension of G’Sell, Wager, Chouldechova, and Tibshirani (2016), relying on an approximation of the R´enyi representation of p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The step-down control for nested FWER is based on Simes (1986), which along with Bonferroni (1935) can be seen as comparing sorted p-values against linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our framework contributes to estimating the number of principal component factors in a panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' There are have been many studies that provide consistent estimators for the number of PCs based on the divergence in eigenvalues of the covariance matrix, which include Onatski (2010), Ahn and Horenstein (2013) and Pelger (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Another direction uses sequential testing procedures that presume correct nested family of hypotheses, which include Kapetanios (2010) and Choi, Taylor, and Tibshirani (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In contrast, we characterize the least amount of factors (which can also be based on principal components), which should be expected when a FWER rate is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The nested version of our procedure is close in nature to a panel version of “when-to-stop” problem of a multiple testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The problem of post-LASSO statistical testing for small dimensional cross-sections is studied in a stream of papers including Meinshausen and B¨uhlmann (2006), Zhang and Zhang (2014), van de Geer, B¨uhlmann, Ritov, and Dezeure (2014) and Javanmard and Montanari (2018), which consider inference statements by debiasing the LASSO estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' An alternative stream of post-selection or post-machine learning inference literature includes Chernozhukov, Hansen, and Spindler (2015), Kuchibhotla, Brown, Buja, George, and Zhao (2018) and Zrnic and Jordan (2020), who provide non-parametric post-selection or post-regularization valid confidence intervals and p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' These papers do not make conditional statements and presume that the researcher sets the hypotheses before seeing the data, which we will refer to as data agnostic hypothesis family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We follow a dif- ferent train of thought that treats LASSO, among a family of conic maximum likelihood estimator, as a polyhedral constraint on the support of the response variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This geometric perspective that provides inferential theory post-LASSO is pioneered by the work of Lee, Sun, Sun, and Taylor (2016) and followed up by Fithian, Sun, and Taylor (2017) and Tian and Taylor (2018), assum- ing Gaussian linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Markovic, Xia, and Taylor (2018) extends the results to LASSO with cross-validation, Tian, Loftus, and Taylor (2018) discusses a square-root LASSO variant that takes unknown covariance into consideration and Tian and Taylor (2017) considers the asymptotic results when removing the Gaussian assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This body literature is often referred to as PoSI, and traverses the Karush-Kuhn-Tucker (KKT) condition of a LASSO optimization problem to show that the LASSO fit can be expressed as a polyhedral constraint on the support of the response 4 variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We extend this work by allowing to put weights onto prior belief sets, and by bringing it to the panel setting with multiple testing adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 2 Sparse linear models We consider a large dimensional panel data set Y ∈ RT×N which we want explain with a large number of potential covariates X ∈ RT×J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The panel data and explanatory variables are both observed over T time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 The size of the cross-section N and the dimension of the covariate candidate set J are both large in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We assume a linear relationship between Y and X: Yt,n = J � j=1 Xt,jβ(n) j + ϵt,n for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', N, which reads in matrix notation as Y = Xβ + ϵ (1) We refer to the coefficients β as loading matrix, where the nth column β(n) corresponds to the nth unit and β(n) j denotes the loading of the nth unit on the jth covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The remainder term ϵ is unexplained noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We assume that a sparse linear model can explain jointly the full panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Formally, a sparse linear model with s active covariates is Y = XSβS + ϵ (2) where s = |S| is the cardinality of the set of active covariates S = {j : ∃β(j) n ̸= 0, n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', N}, that is, the set of covariates with non-zero loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' XS is the subset of covariates that belong to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our goal is to estimate this low dimensional model, that can explain the full panel, from a large number of candidate covariates, and provide a valid inferential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note that our sparse model formulation allows for two important properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First, different units can be explained by different covariates with different loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This means that β(n) ̸= β(m) for n ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For example, a subset of the cross-sectional units might be modeled by different covariates than the remaining part of the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, we can accommodate “weak” covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A covariate is included in S if it is required by at least one cross-sectional unit requires as explanatory variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In other words, a sparse model can include covariates in XS that explain only a very small subset of the panel Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The first step is to estimate the sparse models over the time-series for each unit separately due to the heterogeneity in the loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In a second step, we provide the valid inferential theory for the loadings on the full panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The time-series estimation requires an appropriate regularization to 1Our setting and multiple testing results can be readily extended to the case of unbalanced panel, although we focus on the balanced panel case for now to highlight the core multiple testing insight of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We will further discuss on this once we introduce our main procedure in Section 4 5 select a small subset of covariates that contains all the relevant covariates for each unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We allow for a prior belief weight ω ∈ ¯RJ +, so that different X can have different relative penalizations, and a global λ ∈ R+ scalar penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For the nth unit, we denote its β(n) estimate as ˆβ(n) and the active set M(n) = {j : ˆβ(n) j ̸= 0} as the set of j’s with non-zero loadings ˆβ(n) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A general regularized linear estimator solves the following optimization problem ˆβ(j)(λ, ω) = arg min β 1 2T ∥Y (j) − Xβ∥2 2 + λ · f(β, ω) (3) for a penalty function f and appropriate weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In this paper, we consider the weighted LASSO estimator with the regularization function f(β, ω) = J � j=1 fj(βj, ωj) where fj(βj, ωj) = � � � |βj| ωj ωj < ∞ 0 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (4) and weights ωj > 0 for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', J} and ∥ω−1∥1 = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We assume that the penalty λ is selected such that the set ∥ˆβ(j)∥0 = |M(j)| is low dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Importantly, we do not need to assume that the selected set contains all active covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our goal it is provide a valid inferential theory conditional on the selected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our estimator generalizes the conventional LASSO with the l1 regularization function of Tibshirani (1996) by allowing for different relative weighting in the penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Importantly, we also allow for an infinite weight, which can be interpreted as a prior on a set of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This allows researchers to take advantage of prior information and for example ensure that a specific set of covariates will always be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The weighted LASSO will be particularly relevant in our empirical study, where we can answer the question which risk factors should be added to a given set of economically motivated risk factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our weighted LASSO formulation can also be interpreted as a Bayesian estimator with the canonical Laplacian prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Conventional regression theory will not provide correct inferential statements on the weighted- LASSO estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We face two challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First, regularized estimation results in a bias, which needs to be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second and more challenging, post-selection inference changes the distribu- tion of the estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' When we observe an active ˆβ(n) j from (3), it would be incorrect to simply calculate its p-value from a conventional t-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This invalidity stems from the fact that conditional on observing a LASSO output, β(n) j must be large enough in magnitude for its ˆβ(n) j to be active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In other words, the probability distribution of the estimators is truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The correct inference has to be conditional on the covariates being selected by the LASSO estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, valid p-values have to be the tail probability conditional on being in the selection set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The key to quantify such styles of inference is to recognize that a sparsity constrained estimator is typically the result of solving Karush-Kuhn-Tucker (KKT) conditions, which can in turn be geometrically characterized as polyhedral constraints on the support of response variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is first established in Lee, Sun, Sun, and Taylor (2016), who provide the stylized results that Post-Selection Inference (PoSI) of debiased non-weighted LASSO estimators can be calculated as 6 polyhedral truncation on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This line of research is also referred to as Selective Inference in other literature such as Taylor and Tibshirani (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We extend this line of literature to allow for the Weighted-LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We derive these results with assumptions common in the PoSI LASSO literature, detailed in Appendix A, and referred to as conventional regularity conditions for ease of exhibition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' THEOREM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Truncated Gaussian Distribution of Weighted-LASSO Under conventional regularity conditions, the debiased estimate ¯βi for the i-th Weighted-LASSO active covariate is conditionally distributed as ¯βi|Weighted-LASSO ∼ T N {η⊤Y :AY ≤b(ω)} (5) where T N A is truncated-Gaussian with truncation A, and the weights ω only appear in b(ω) Theorem 1 has two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First, it debiases the LASSO estimate by a shifting argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' While we use a geometric argument to remove the bias, the bias adjustment takes the usual form in the LASSO literature as for example in Belloni and Chernozhukov (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The debiased LASSO estimator simply equals a standard OLS estimation on the subset Mn selected by the Weighted- Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, the distribution of the linear coefficients is not a usual Gaussian distribution, but it is truncated due to studying post-selection coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This geometric perspective is less common in the LASSO literature, but provides several advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' One advantage of the geometric approach is that it avoids the use of infeasible quantities, in particular the second moment of the large set of potential covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Furthermore, the distribution result is not asymptotic in T, but also valid in finite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We can obtain these results because we make the stronger assumption that the data is normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Appendix A provides the detailed information on constructing ¯β and the definitions of η, A, b(ω) along with lemmas that lead up to this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It also discusses extensions and the effect of estimating the variance of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The empirical analysis is based on the explicit form of Theorem 1 formulated in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our Weighted-LASSO results make several contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First, the expression for the trun- cated conditional distribution with weights become much more complex than for the special case of the conventional LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, we provide a simple, easy-to-use and asymptotically valid conditional distribution in the case of an estimated noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Last but not least, we show the formal connection with alternative debiased LASSO estimators by showing that debiasing can be interpreted as one step in a Newton-Ralphson method of solving a constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Theorem 1 allows us to obtain valid p-values for Weighted-LASSO coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We obtain these p values from the simulated cumulative distribution function of the truncated Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Crucially, all results for multiple testing adjustment in panels that we study in the following sections neither require us to use a weighted Lasso estimator nor to use the p-values implied by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We only require to have a set of valid p-values for sparsity constrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' These can be obtained with any suitable regularized estimator and post-selection inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The key element is the selection of a low dimensional subset with p-values conditional on this selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We propose the weighted LASSO conditional inference results as an example of the type of sparsity constraint 7 models we are interested in, and demonstrate a machinery with which we can obtain valid p-values for sparsity constrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In our empirical studies, we use Weighted-LASSO as our sparsity constrained model since we want to specify strong prior beliefs on a few covariates and it is common practice to use LASSO in the context of our empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Nonetheless, the testing methods in the next sections accommodate any sparse estimator, and can be detached from inference for Weighted-LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 3 Data-Driven Hypotheses Our goal is to provide formal statistical tests that allow us to establish a joint model across a large cross-section with potentially weak covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This requires us to provide a form of statistical significance test with multiple testing adjustment that properly accounts for covariates that only ex- plain a small subset of the cross-sectional units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is important as in many problems in economic and finance there is substantial cross-sectional variation in the explanatory power of covariates, and a model that simply minimizes an average error metric might neglect weaker covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' An essential step for a formal statistical test is to formulate the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This turns out to be non-trivial for a large panel with a first stage selection step for the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It is a fundamental insight of our paper, that the hypothesis of our test has to be conditional on the selected set of active covariates of the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Once we have defined the appropriate hypothesis, we can deal with the multiple testing adjustment, which by construction is also conditional on the selection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The hypothesis formulation and test construction only requires valid p-values from a first stage selection estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The results of the next two sections do not depend on a specific model for obtaining these p-values and the active set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The results are valid for any model including non- linear ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The input to the analysis is a N ×J matrix, which specifies which covariates are active for each unit and the corresponding p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The Weighted-LASSO is only one possible model, but it can be replaced by any regularized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We have introduced the sparse linear model as it is the horse race model for many problems in economics and finance, and therefore of practical relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We illustrate the concept of a data-driven hypothesis with a simple example, which we will use throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For simplicity we assume that we have J = 4 covariates and want to explain N = 6 cross-sectional units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In the first stage, we have estimated a Weighted-LASSO and have obtained the post-selection valid p-values for each of the N units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We collect the fitted sparse estimator ¯β(n) for the nth unit in the matrix ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note, that this matrix has “holes” due to the sparsity for each ¯β(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Figure 1(a) illustrates ¯β for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Similarly, we collect the corresponding p-values in the matrix P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For the nth unit, we only have p-values for those covariates that are active in the nth linear sparse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Thus, Figure 1(b) also has white boxes showing the same pattern of unavailable p-values due to the conditioning on the output of the linear sparse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' These holes can appear at different positions for each 8 Figure 1: Illustrative example of data-driven selection (a) Matrix ¯β (b) Matrix P of p-values This figure illustrates in a simple example the data-driven selection of a linear sparse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In a first stage, we have estimated a regularized sparse linear model for each of the N = 6 units with J = 4 covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Each row represents the selected covariates with their estimated coefficients and p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The columns represent the J = 4 different covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The grey shaded boxes represent the active set, while white boxes indicate the inactive covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The numbers are purely for demonstrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' unit, which makes this problem non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This non-trivial shape of either subplot (a) or (b) is completely data-driven and a consequence of linear sparse model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We show that the hypothesis should be formed around these non trivial shapes as well, which is why we name it the data-driven hypothesis family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We want to test which covariates are jointly insignificant in the full panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A data-agnostic approach would simply test if all covariates are jointly insignificant, independent of the data-driven selection step in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A data-agnostic hypothesis is unconditional as it does not depend on any model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' However, as we will show, this perspective is problematic for the high- dimensional panel setting with many covariates as it ignores the dimension reduction from the selection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Therefore, an unconditional multiple testing adjustment accounts for “too many” tests, which severely reduces the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We propose to form the hypothesis conditional on the first stage selection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The data-driven hypothesis only tests the significance of the covariates that were included in the selection, and hence can drastically reduce the number of hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' However, given the non-trivial shape of the active set, the multiple testing adjustment for the data-driven hypothesis is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Before formally defining the families of hypothesis, we illustrate them in our running example.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='001 4 :0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='001 ≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='001 5 :0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='001 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='010The data-agnostic hypothesis HA for explaining the full panel takes the following form: HA = {HA0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' HA0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' HA0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' HA0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4} = {β(1) 1 =β(2) 1 = β(3) 1 = β(4) 1 = β(5) 1 = β(6) 1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' β(1) 2 =β(2) 2 = β(3) 2 = β(4) 2 = β(5) 2 = β(6) 2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' β(1) 3 =β(2) 3 = β(3) 3 = β(4) 3 = β(5) 3 = β(6) 3 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' β(1) 4 =β(2) 4 = β(3) 4 = β(4) 4 = β(5) 4 = β(6) 4 = 0} (6) The data-driven hypothesis HD only includes the active set and hence equals HD = {β(2) 1 =0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' β(1) 2 =β(3) 2 = β(5) 2 = β(6) 2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' β(1) 3 =β(2) 3 = β(3) 3 = β(4) 3 = β(5) 3 = β(6) 3 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' β(2) 4 =β(4) 4 = β(5) 4 = 0} (7) Obviously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' HA has a larger cardinality of |HA| = 24 > |HD| = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This holds in general, unless the first stage selects all covariates for each unit, in which case the two hypotheses coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Formally, the data-agnostic family of hypothesis is defined as follows: DEFINITION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Data-agnostic family The data-agnostic family of hypotheses is HA = {HA0,i|i ∈ [d]} where HA0,i = � j∈[N] H(j) A0,i and H(j) A0,i : β(j) i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (8) It is evident that HA does not need any model output or exploratory analysis, so it is indeed data-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As soon as we use a sparsity constrained model that has censoring capabilities, we no longer observe (Y , X) from its data generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Consequently, unless our hypotheses depend on how we built the model, or equivalently on how the data was censored, the data-agnostic hypotheses forgo power without any benefit in false discovery control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Therefore, we formulate the hypothesis on the ith covariate H(j) 0,i only if i ∈ M(j), that is, it is in the active set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Conditional on observing the model output, there is no inference statement to be made about H(j) 0,i if i /∈ M(j), because its estimator is censored by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We denote as Ki the set of units for which the ith covariate is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We define the cross- sectional hypothesis for the ith covariate as: H0,i = � j∈Ki H(j) 0,i ����M, ∀i : Ki ̸= ∅ (9) By combining all covariates {i : Ki ̸= ∅} that show up at least once in one of the active sets of our sparse linear estimators, we arrive at a data-driven hypothesis associated with our panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is defined as follows: 10 DEFINITION 2 (Data-driven family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The data-driven family of hypotheses conditional on M is HD = {H0,i|i : Ki ̸= ∅} (10) This demonstrates the non-trivial nature of writing down a hypothesis in high-dimensional panel: we can only collect Ki - the set of units for which the ith covariate is active - after seeing the sparse selection estimation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 4 Multiple Testing Adjustment for Data-Driven Hypothesis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 Simultaneity Counts through Panel Localization We show how to adjust for multiple testing of data-driven hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Given the p-values p(j) i for i ∈ M and j ∈ Ki, we form the data-driven hypothesis HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our goal is to reject members of HD while controlling the Type I error, and the common way to measure such error is the family- wise error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is the same underlying logic that is used to define confidence intervals and determine significance of covariates in a conventional setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The crucial difference is that we need to account for multiple testing given the large number of cross-sectional units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The family-wise error rate (FWER) is defined as follows: DEFINITION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Family-wise error rate Let V denote the number of rejections of H(j) 0,i |M(j) when the null hypothesis is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The family- wise error rate (FWER) is P(V ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Similar to the conventional definition, we simply count the false rejections V and define FWER as the probability of making at least one false rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Importantly, Definition 3 accounts for the fact that we might repeatedly test on � j∈[N] |Mj| rather than a single hypothesis test of the form H(j) 0,i : β(j) i = 0|M(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our contribution to FWER control in the panel setting is thus to take into consideration both the multiplicities in units and covariates when we deal with the “matrix” of p-values P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' To achieve this goal, we propose a new simultaneity account for the ith covariate, calculated as Ni = � j∈Ki |Mj| (11) Figure 2 illustrates the simultaneity counting for our running example with N = 6 units and J = 4 covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The blue boxes represent the active set for a specific covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The yellow boxes indicate the “co-active” covariates, which have to be accounted for in a multiple testing adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In the case of the first covariate j = 1, only the second unit n = 2 has selected this covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This second unit has also selected covariate j = 3 and j = 4, which are jointly tested with the first covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, they are “co-active”, and the simultaneity count equals N1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Intuitively, Nj represents all relevant comparisons for the jth covariate because it counts how many covariates 11 Figure 2: Simultaneity counts Ni in the illustrative example (a) N1 = 3 (b) N2 = 9 (c) N3 = 14 (d) N4 = 8 This figure shows the simultaneity counts Ni in the illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The subplots represent the simultaneity counts for the J = 4 covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The blue boxes indicate the active set Kj of the j covariates, while yellow boxes indicate the “co-active” covariates of the jth covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The simultaneity counts are the sum of yellow and blue boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' are active with the jth covariate in the regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, Nj quantifies the number of “multiple tests” for each covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In subplot 2(a), we see that K1 = {2} for the 1st covariate, indicated by the blue box, because it is only active in the second unit’s regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The multiple testing adjustment needs to consider all yellow boxes, and N1 = 3 is thus the total count of 1 blue and 2 yellow boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Similarly, for the second covariate, K2 = {1, 3, 5, 6}, so we shade boxes yellow for the 2nd, 3rd and 5th units and obtain N2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We can already see that our design of simultaneity count takes all relevant pairwise comparisons into considerations, but avoids counting the white boxes - which would cause overcounting and result in over-conservatism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our multiplicity counting is a generalization of the classical Bonferroni adjustment for multiple testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A conventional Bonferroni method for the data-agnostic hypothesis HA has a simultaneity count of |HA| = N · J = 24 for testing each covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A direct application of a vanilla Bonfer- roni method to the panel of all selected units and the data-driven hypothesis HD, would use a simultaneity count of |HD| = 14 for testing each covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our proposed multiplicity counting is a refinement that leverages the structure of the problem, and takes the heterogeneity of the active sets for each covariate into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our count has only N1 = 3, N2 = 9 and N4 = 8 for the covariates j = 1, 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Only for covariate j = 3 is the simultaneity count the same as a vanilla Bonferroni count applied to HD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' N3 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In addition to the simultaneity count of each covariate, we need an additional “global” metric for our testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We define a panel cohesion coefficient ρ as a scalar that measures how 12 1 2 3 4 2 4 5 61 2 3 4 1 2 4 5 61 2 3 4 1 2 4 5 62 3 4 1 2 3 4 5 6Figure 3: Illustration of the cohesion coefficient (a) ρ = J−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='25 (b) ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='44 (c) ρ = 1 This figure illustrate the cohesion coefficient ρ in three separate examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It shows the smallest, largest and in- between cases of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The columns represent the J = 4 different covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='The blue boxes indicate the active sets for each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' sparse or de-centralized the proposed hypotheses family is: ρ = � �� j |Kj| Nj � � −1 (12) The panel cohesion coefficient ρ is conditional on the data-driven selection of the overall panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It is straightforward to compute once we observe the sparse selection of the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This coefficient takes values between J−1 and 1,2 where larger values of ρ imply that the active set is more dependent in the cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This can be interpreted as that the panel Y has a stronger dependency due to the covariates X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Intuitively, in the extreme case when ρ = J−1, the panel can be separated into J smaller problems, each containing a subset of response units explained by only one covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Thus the panel would be very incohesive, and could be studied with J independent tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In the other extreme, if ρ approaches 1, the first-stage models include all active covariates for all units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We consider this as a very cohesive panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' If ρ is between theses bounds, the panel is cohesive in a non-trivial way such that some units can be explained by some covariates and there is no clear separation of the panel into independent subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Figure 3 illustrates the panel cohesion coefficient in three examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The subplots show three active sets that are different from our running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The left subplot 3(a) shows the extreme case of ρ = J−1, where the panel is the least cohesive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The right subplot 3(c) illustrates the other extreme for ρ = 1, where the panel is the most cohesive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The middle subplot 3(b) is the complex case of a medium cohesion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 2We prove this bound in the Appendix, without leveraging sparsity of first-stage models but rather as an algebraic result with intuitive interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 13 1 2 3 4 1 2 3 4 5 61 2 3 4 1 2 4 5 61 2 3 4 1 2 4 5 6Our novel simultaneity count and cohesiveness measure are the basis for modifying a Bonferroni test for FWER-controlled inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Theorem 2 formally states the FWER control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The proof is in the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' THEOREM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' FWER control The following rejection rule has FWER≤ γ on HD: min n∈Kj � p(n)(j) � ≤ ρ γ Nj ⇒ Reject H0,j (13) where p(n)(j) are valid p-values for each univariate unit n, and ρ is the panel cohesion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This completes the joint testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First, we calculate p-values after running a sparse linear estimator time-series regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, we use the sparse linear estimator output to write down a hypothesis and, third, we provide a FWER control inference procedure by combining the p-values across the cross-section and test the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The difference between a naive Bonferroni and our FWER control is particularly pronounced for weak covariates that affect only a subset of the cross-sectional units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Given a FWER control level of γ, the rejection threshold for a naive Bonferroni test is γ JN for every covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rejection threshold for our FWER control is always higher, and differs in particular when Nj is small and ρ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is the case for weak covariates in a cohesive panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As it is common in statistical inference, we focus on Type I error control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Type II error rates require the specification of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' While we do not provide formal theoretical results for the power of our inference approach, we show comprehensively in the simulation and empirical part, that our approach has substantially higher power than conventional approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We point out that the validity of our procedure holds for unbalanced panels as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is because even when there are different number of observations for the nth and mth units, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Tn ̸= Tm for n ̸= m, they can still be estimated separately in the first stage of the regularized regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The hypothesis testing and selection of a parsimonious model only requires the matrix P of valid p-values, which can be based on different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 Least Number of Covariates: Traversing the Threshold The typical logic of statistical inference is to determine which covariates we should admit from XM, given a significance level γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We use K to denote the number of selected covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' When γ is specified as a lower quantity, we expect K to decrease as well, that is, the rejection becomes harsher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As the number of admitted covariates of our procedure is monotone in γ, we want to ask the following converse question: How low do we need to set γ such that we reject K covariates?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Concretely, we are interested in finding: 14 γ∗(K) = sup � � �γ|K = J � j=1 1 � min n∈Kj � p(n) j � ≤ ρ γ Nj �� � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (14) Let pj = minn∈Kj{p(n) j } be the 1st order statistic for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Then (14) is simply the K-th order statistics of Njpj/ρ: γ∗(K) = min{Nipi/ρ|∃j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', jK ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', J} : Nipi ≥ Njkpjk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (15) Since this minimization scan is monotone, we can determine how many covariates at least should be admitted, given a control level, which is similar to the “SimpleStop” procedure described in Choi, Taylor, and Tibshirani (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The following corollary formalizes this inversion method that finds the least number of covariates to admit: COROLLARY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Least number of covariates Given the FWER level γ, there exists a unique number K∗(γ) such that K∗(γ) = � � � arg max0≤K≤J γ∗(K) ≤ γ ∃K : γ∗(K) ≤ γ d o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (16) The statement simply states that the simplest linear model should have at least K∗(γ) covariates for a given γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note that it is possible that, for example, γ∗(5) and γ∗(6) are both equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05, while γ∗(7) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In this case the minimum number of covariates is K∗(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05) = 6 because it does not hurt FWER-wise to include 6 covariates in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, we are making a slightly different statement than that there would be exactly K∗(γ) covariates in the true linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The number of covariates is obviously conditional on the set of candidate covariates X, and we can only make statements for this given set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In our empirical study we consider candidate asset pricing factors X to explain the investment strategies Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' More generally, the linear model that we consider is often referred to as a factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Therefore, we will also refer to the selected covariates as factors, and use these two expressions as synonyms moving forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This directly links our procedure to the literature on estimating the number of factors to explain a panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A common approach in this literature is to use statistics based on the eigenvalues of either Y or X to make statements about the underlying factor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our approach is different, as it provides significance levels for the selected factors and FWER control for the number of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Table 1 illustrates the estimation of the number of factors and their ranking with our running example introduced in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We calculate the simultaneity counts Ni’s as given in (11) and demonstrated in Figure 2, and pi as the smallest p-values associated with the ith covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Then, the rejection rule in Theorem 2 is based on whether a pre-specified level γ satisfies pi < ργ Ni , which is equivalent to Ni · piρ < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Thus, the natural ranking of the covariates is to sort all covariates in descending order of the 15 Table 1: Sorted p-values for the running example Factor (j) pj Simultaneity count for HD Conventional Bonferroni for HA ρ−1 · Nj ρ−1 · Nj · pj J · N J · N · pj 3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='001 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='001 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='002 4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='001 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='001 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='003 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='005 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='024 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='120 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='002 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='028 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='051 This table constructs “significance” levels for the running example introduce in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We compare the simul- taneity count for the data-driven hypotheses HD and a onventional Bonferroni count for data-agnostic hypotheses HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The products Nj · pj, respectively J · N · pj, can be interpreted as the significance levels for the corresponding approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Given a FWER control γ all factors with ρ−1 · Nj · pj (respectively J · N · pj) below this threshold are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Ni · pi/ρ values as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It is then trivial to determine K∗(γ) for any choice of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For example, for γ = 1%, we would select factors 3 and 4, but not 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' On the other hand, for γ > 2%, we would include all four factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, the ranking of Nipi/ρ directly maps into K∗(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The list of Nipi/ρ encompasses more information than just the number of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Naturally, it provides an importance ranking of the factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Furthermore, the number Ni reveals if significant factors are “weak”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In our case, factor 1 has N1 = 3, which indicates that it affects only a small number of hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Its p-value p1 is sufficiently small to still imply significance in terms of FWER control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For comparison, Table 1 also includes the corresponding analysis for the data-agnostic hypoth- esis and a conventional Bonferroni correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The Bonferroni analysis uses the same p-values but a different multiple testing adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In our case, the p values would be multiplied by J ·N = 24 as this corresponds to the total number of hypothesis tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This will obviously make the inference substantially more conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Indeed, even for a FWER control of γ = 4%, we would only select factors 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We would need to raise the FWER control to γ = 12% to include factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, weak factors, like factor 1, are more likely to be discarded by the data-agnostic hypothesis with conventional multiple testing adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We want to emphasize that a data-agnostic hypotheses with conventional Bonferroni correction does provide correct FWER control, but it is overly conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' By construction, the data-agnostic Bonferroni approach will test a larger number of hypothesis, which means that the corresponding “significance levels” will always be lower or equal to our data-driven simultaneity count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, the data-agnostic Bonferroni approach does not differentiate the “strength” of the factors, while our approach provides a selection-based heterogeneous adjustment of the p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is essential for detecting weak factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Having introduced all building blocks of our novel method to detect covariates, we put the entire procedure together as “Panel-PoSI”: PROCEDURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Panel-PoSI The Panel-PoSI procedure consists of the following steps: 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For each unit n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', N unit, we fit a linear sparse model ˆβ(n) X,Y (c, ω) given (X, Y , λ, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We suggest cross-validation to select the LASSO penalty λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We construct the sparse estimators ¯β(n) and the corresponding p-values for the active covariates for each unit, and collect them in the “matrix” of p-values P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We collect the panel-level sparse model selection event M and construct the data-driven hy- pothesis HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Given the FWER control level γ and based on the the simultaneity counts Nj, we make inference decision for the sparse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We can rank covariates in terms of their significance and select a parsimonious model that explains the full panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As we have now all results in place, we can summarize the advantages of our procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First, we want to clarify that our goals and results are different from just some form of optimal shrink- age selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Selecting a shrinkage parameter with some form of cross-validation in a regularized estimator like LASSO does not provide the same insights and model that we do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A shrinkage estimator can either be applied to each unit separately, as we do it in our first step, or to the full panel in a LASSO panel regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The separate covariate selection for each cross-sectional unit does not answer the question which covariates are needed to explain the full panel jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A shrinkage selection on the full panel for some form of panel LASSO can neglect weaker factors, as those receive a low weight in the cross-validation objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, tuning parameter selection with cross-validation requires a sufficiently large amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our approach is attrac- tive as we can do the complete analysis on the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' That means, an initial LASSO is used to first reduce the number of covariates, but this set is then further trimmed down using inferential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, we can construct a parsimonious model even for data with a relatively short time horizon, but large cross-sectional dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Third, the statements that we can make are much richer than a simple variable selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We can formally assess the relative importance of factors in terms of their significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The model selection is directly linked to a form of significance level, which allows us to assess the relevance of including more factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Last but not least, we can also make statements about the strength of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In summary, Panel-PoSI is a disciplined approach based on formal statistical theory to construct and interpret a parsimonious model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 5 Ordered Multiple Testing on Nested Hypothesis Family So far, our hypothesis family HD has no hierarchy and consequently, we have not imposed a sequential structures on the admission order of covariates of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' However, there are cases where the covariates or factors warrant a natural order such that the family possesses a special testing logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A hierarchical structure in covariates arises when the inclusion of the next covariate only make sense if the previous covariates is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' One example would be if the next covariates refines a property of the previous covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Another case is the use of principal component (PC) factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The conventional logic is to include PCs sequentially from the dominating one to the 17 least dominating one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is similar to the motivation for Choi, Taylor, and Tibshirani (2017), but different from them, we treat the PCs as exogenous without taking the estimation of PCs explicitly into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In this section, we will use exogenous PCs as hierarchical covariates, as this is the main example in our empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' However, all the results hold for any set of exogenous hierarchical covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Without loss of generality, we presume X has the jth column as the jth nested factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A k-order nested model N(k) is of the following form N(k) model : Y = X[k]β[k] (17) where [k] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', k} is the set that includes indices up to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For example, a hierarchical three factor model corresponds to X{1,2,3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' When formulating our hypothesis family, we must represent the sequential testing structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is reflected in our definition of nested families of hypotheses: DEFINITION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Data-driven nested family The data-driven nested family of hypotheses conditional on M is HN = {HN,k : k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', J}, HN,k = � j∈Kk H(j) N,k ����M, H(j) N,k : {i′ : β(j) i′ ̸= 0} ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (18) HN,0 completes the case when no rejection on any factor is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Whenever HN,k is true, then HN,k′ is also true for k < k′ ≤ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Moreover, in the cases where Kk = ∅ but Kk′ ̸= ∅ with k < k′, the notation ensures that the hypothesis HN,k is included in HN simply because Kk′ is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In other words, if a less dominating hypothesis HN,k′ is suggested by data (that is, its active set is non-empty Kk′ ̸= ∅), HN would automatically include all HN,k for k ≤ k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The FWER control property needs to be adapted to the nested nature of this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Choi, Taylor, and Tibshirani (2017) argue that the proper measurement is to control for ordered factor count over-estimation with level γ, as follows: DEFINITION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' FWER for nested family For a test that rejects HN,k for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', ˆk of HN, the FWER control at the level γ satisfies P(ˆk ≥ s) ≤ γ, where s is the true factor count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Given the hierarchical belief about the model, we need to add the following additional assump- tion: ASSUMPTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Tail p-values Under H(j) N,k, there is p(j)(i′) iid ∼ Unif [0, 1] if i′ > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Assumption 1 only needs to hold for the tail hierarchical covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In the case of PCs, it only applies to the lower order tail PC factors that should not be included for a given null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For example, if the true model is HN,s, we only need p(j)(i) iid ∼ Unif[0, 1] for i > s, which is a usual type of assumption in this literature such as in G’Sell, Wager, Chouldechova, and Tibshirani 18 Figure 4: Example of hierarchical simultaneity counts Norder k for HN (a) N order 4 = 3 (b) N order 3 = 5 (c) N order 2 = 8 (d) N order 1 = 12 This figure shows the simultaneity counts N order i in an illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The subplots represent the simultaneity counts for the J = 4 covariates and N = 6 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The dark blue columns present the active factors, while the light blue columns capture factors of higher-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The sub-plots from left-to-right represent our calculation order from the highest-order factor to the 1st factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Moreover, because the nested nature guarantees that the higher-order PCs are more likely to be null, a step-down procedure is expected to increase the power relative to a step-up procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As our focus is to control for false discoveries, we also need to adjust our simultaneity counts to the sequential testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Concretely, we consider first taking a union to obtain the active unit set Korder k and then calculate conservative simultaneity counts Norder k : Korder k = � i∈{k,k+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=',J} Ki, Norder k = � j∈Korder k |Mj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (19) It is possible for some |Mk| to be 0 (that is, the kth PC could be inactive for all units), but its Norder k would be 0 if and only if higher-order PCs all have |Mk′| = 0 for k′ > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Figure 4 illustrates the process of our step-down simultaneity count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' From the left, we start with factor k = 4 and move step-wise down to factor k = 1 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The dark blue columns present the active factors, while the light blue columns capture factors of higher-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In the left-most sub-figure, we only need to account for the 4th PC, implying Norder 4 = 3, whereas in the mid-left sub-figure, the 3rd PC has Norder 3 = 2 + 3 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Eventually, in the right-most sub-figure, we have swept through the entire panel and the 1st PC has a simultaneity count of Norder 1 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Now we can introduce a step-down procedure adapted to the nested structure of HN: PROCEDURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Step-down rejection of nested ordered family HN The step-down rejection procedure consists of the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', J} calculate the ordered simultaneity count Norder k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 19 1 2 3 4 1 2 3 4 5 61 2 4 1 2 4 5 61 2 4 1 2 4 5 61 2 4 1 2 3 4 5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', J} calculate the approximated R´enyi representation Zorder k and its trans- formed reversed order statistics qorder k : Zorder k = J � i=k � j∈Ki ln(p(j)(k)) Norder 1 − Norder i+1 1{i ̸= J}, qorder k = exp(−Zorder k ) (20) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Reject hypothesis 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', ˆk, where ˆk = max{k : qorder k ≤ γNorder k JN }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This procedure will have FWER control at level γ as stated in the following theorem: THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' FWER control for ordered hypothesis Under Assumption 1, Procedure 2 has FWER control of γ for the ordered hypothesis HN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The proof is deferred to the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This design extends Procedure 2 from G’Sell, Wager, Chouldechova, and Tibshirani (2016) and “Rank Estimation” from Choi, Taylor, and Tib- shirani (2017), both of which focus on a single sequence of p-values rather than the panel setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In Step 2, we use Assumption 1 to transform p-values into ln(p(j)(k)), which are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' standard exponential random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Since the family HN has J members, we need to modify our simul- taneity count and in a sense condense the panel into a sequence of statistics associated with the ordered covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We built a staircase sequence of conservative simultaneity count Norder k in Step 1 to accumulate the number of p-values we use up to the kth ordered covariate, starting from the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' By the R´enyi representation of R´enyi (1953), the Zorder k of Step 2 approximate exponential order statistics and the qorder k approximate uniform order statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The nature of these approxima- tions is to create a more conservative rejection, the technical details of which are examined in the proof in our Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Finally, we run the order statistics through a step-down procedure proposed by Simes (1986) so that we find the ˆk largest number of ordered covariates rejected by the data with FWER control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Also note that even if the global null, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' HN,0, is true, and every linear sparse model active set is empty, that is Norder 1 = 0, the procedure in Step 3 is still valid because we do not reject HN,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 6 Simulation We demonstrate in simulations that our inferential theory allows us to select better models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We compare different estimation approaches to select covariates and show that our approach bet- ter trades off false discovery and correct selections and hence results in a better out-of-sample performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Table 2 summarizes the benchmark models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our framework contributes among three dimen- sions: the selection step for the sparse model, the construction of the hypothesis and the multiple testing adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We consider variations for these three dimensions which yields in total six estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' By varying the different elements of the estimators, we can understand the benefit of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Table 2: Summary of estimation methods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Abbreviation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Hypothesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Multiple Testing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Rejection rule ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Naive OLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='N-OLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='OLS without LASSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Agnostic HA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='No adjustment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='pOLS < γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Bonferroni OLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='B-OLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='OLS without LASSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Agnostic HA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='No adjustment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='pOLS < ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='JN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Naive LASSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='N-LASSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='LASSO without PoSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Agnostic HA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='No adjustment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='pLASSO < γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Bonferroni Naive LASSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='B-LASSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='LASSO without PoSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Agnostic HA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Bonferroni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='pLASSO < ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='JN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Bonferroni PoSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='B-PoSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='LASSO with PoSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Agnostic HA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Bonferroni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='pPoSI < ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='JN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Panel PoSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='P-PoSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='LASSO with PoSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Data-driven HD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Simultaneity count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='pPoSI < ργ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Ni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='This table compares the different methods to estimate a set of covariates from a large dimensional panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For each method, we list the name and abbreviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The selection refers to the regression approach for each univariate time-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The hypothesis is either agnostic or data-driven given the selected subset of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The multiple testing adjustment includes no adjustment, a conventional Bonferroni adjustment and our novel simultaneity count for a data-driven hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rejection rules combine the valid p-values and multiple testing adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' pOLS is the p-value for a conventional t-statistics of an OLS estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' pLASSO is the p-value without removing the lasso bias or adjusting for post-selection inference, that is, it is simply the OLS p-values using the selected subset of regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' pPoSI is the debiased post-selection adjusted p-value based on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our baseline model is Panel PoSI, which uses post-selection inference LASSO, and a simultane- ity count for a data driven hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The first component that we modify is the selection of the sparse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A simple OLS regression without shrinkage does not produce a sparse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This gives us the methods Naive OLS and Bonferroni OLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A conventional LASSO results in a sparse selection, but the p-values are not adjusted for the post-selection inference and the bias adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The corresponding models are the Naive LASSO and the Bonferroni Naive LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The second component is the hypothesis, which is agnostic for methods besides Panel PoSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For the comparison models, we either consider no multiple testing adjustment or the conventional Bonferroni adjust- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Under the multiple testing adjustment we obtain the Bonferroni OLS, the Bonferroni Naive LASSO and the Bonferroni PoSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The outcome of all the estimations are adjusted p-values for the covariates, which we use to select our model for a given target threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For a given value of γ we include a covariate if its adjusted p-value is below the critical values summarized in the last column of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We simulate a simple and transparent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our panel follows the linear model Yt,n = J � j=1 Xt,jβ(n) j + ϵt,n for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', T, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', N and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='., J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The covariates and errors are sampled independently as normally distributed random variables: Xt,j iid ∼ N(0, 1), ϵt iid ∼ N(0, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The noise is either generated as independent noise with covariance matrix Σ = σ2I or as cross- sectionally dependent noise with non-zero off-diagonal elements Σij = κ and diagonal elements Σii = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note that our theorems for PoSI assume homogeneous noise, while dependent noise violates our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, the dependent noise allows us to test how robust our method is 21 Figure 5: Design of loadings β This figure demonstrates the setting of our simulations with 10 factors, where loadings are shaded based on whether they are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In this staircase setting, the first factor affects all units, the 2nd factor affects 90%, and so on, and lastly the 10th factor affects 10% of all units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' to misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We set σ2 = 2 and κ = 1, but the results are robust to all these choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We construct the active set based on the staircase structure depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Of the J covariates in X, we have K = 10 active independent factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Figure 5 demonstrates the setting for the 10 factors, where loadings are shaded based on whether they are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The first factor affects all units, the 2nd factor affects 90%, and so on, and lastly the 10th factor affects 10% of all units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This setting is relevant, and also challenging from a multiple testing perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It results in a large cohesion coefficient ρ, which makes the correct FWER control even more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The loadings are sampled from a uniform distribution, if they are in the active set: β(n) j iid ∼ Unif � −1 2, 1 2 � for j in the active set, β(n) j = 0 for j outside the active set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We simulate a panel of dimension N = 120, J = 100 and T = 300 with K = 10 active factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The first half of the time-series observations is used for the in-sample estimation and selection, while the second half serves for the out-of-sample analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' All results are averages of 100 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We use the covariates selected on the in-sample data for regressions out-of-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our focus is on the inferential theory, and not on the bias correction for shrinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, we first use the inferential theory on the in-sample data to select our set of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, we use the selected subset of covariates in an OLS regression on the in-sample data to obtain the loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Last but not least, we apply the estimated loadings of the selected subset to the out-of-sample data to obtain the model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note that this procedure helps a Naive LASSO, which in contrast to PoSI LASSO does not have a bias correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The out-of-sample explained variation is measured by R2, which is the sum of explained variation normalized by the total variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rejection FWER is set to γ = 5% or γ = 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The LASSO shrinkage penalty λ is selected by 5-fold cross-validation on the in-sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' " " " "Table 3: Simulation Comparison between Selection Methods Independent noise Method # Selections # False Selections # Correct Selections OOS R2 FWER γ = 5% Panel PoSI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0% Bonferroni PoSI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0% Bonferroni Naive LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0% Naive LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4% Bonferroni OLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='7% Naive OLS 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2% FWER γ = 1% Panel PoSI 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='6% Bonferroni PoSI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2% Bonferroni Naive LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0% Naive LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3% Bonferroni OLS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='5% Naive OLS 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3% Cross-sectionally dependent noise Method # Selections # False Selections # Correct Selections OOS R2 FWER γ = 5% Panel PoSI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0% Bonferroni PoSI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2% Bonferroni Naive LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0% Naive LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='5% Bonferroni OLS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3% Naive OLS 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='8% FWER γ = 1% Panel PoSI 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3% Bonferroni PoSI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9% Bonferroni Naive LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0% Naive LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0% Bonferroni OLS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4% Naive OLS 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='8% This table compares the selection results for different methods in a simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For each method we report the num- ber of selected covariates, the number of falsely selected covariates and the number of correctly selected covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We also report the out-of-sample R2 of the models that estimated with the selected covariates on the out-of-sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' All results are averages of 100 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rejection FWER is set to γ = 5% or γ = 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We simulate a panel of dimension N = 120, J = 100, T = 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The first half of time-series observations is used for the in-sample estimation and selection, while the second half serves for the out-of-sample analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The panel is generated by 10 independent factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The active set of the factors follows the staircase structure of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The first factor affects all units, the second 90%, and lastly the 10th factor affects 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The unknown error variance is estimated based as a homogenous sample variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The noise is either generated as independent noise with covariance matrix Σ = σ2I or as cross-sectionally dependent noise with Σij = κ and Σii = σ2 for σ2 = 2 and κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Table 3 compares the selection results for the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For each method we report the number of selected covariates, the number of falsely selected covariates and the number of correctly 23 selected covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We also report the out-of-sample R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The upper panel shows the results for independent noise, while the lower panel collects the results for cross-sectionally dependent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' PanelPoSI clearly dominates all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It provides the best trade-off between correct and false selection, which results in the best out-of-sample performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In the case of γ = 5% and independent noise, Panel PoSI selects 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='8 factors in a model generated by 10 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9 of these factors are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A simple Bonferroni correction is overly conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The Bonferroni PoSI selects only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='7 correct factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' While this overly conservative selection protects against false discovery, it omits over half of the relevant factors which lowers the out-of-sample performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Using post-selection inference is important, as a naive lasso provides wrong p-values which makes the overly conservative selection even worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The other extreme is to have neither shrinkage nor multiple testing adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As expected the naive OLS has an extreme number of false selections with a correspondingly terrible out-of-sample performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As expected, tightening the FWER control to 1% lowers the number of false rejections, but also the number of correct selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It reveals again that Panel PoSI provides the best inferential theory among the benchmark models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Panel PoSI selects 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='5 correct covariates, while it controls the false rejections at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The overly conservative Bonferroni methods select even fewer correct covariates, which further deteriorates the out-of-sample performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The gap in OOS R2 between Panel PoSI and Bonferroni PoSI widens to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' All the other approaches cannot be used for a meaningful selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Panel PosI performs well, even when some of the underlying assumptions are not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The lower panel of Table 3 shows the results for dependent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As the dependence in the noise is relatively strong, it can be interpreted as omitting a relevant factor in the set of candidate covariates X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Even thought the PoSI theory is developed for homogeneous noise, Panel PoSI continues to perform very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In contrast, the comparison methods perform even worse, and the Bonferroni approaches select even less correct covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 7 Empirical Analysis 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 Data and Problem Our empirical analysis studies a fundamental problem in asset pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We select a parsimonious factor model from a large set of candidate factors that can jointly explain the asset prices of a large cross-section of investment strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our data is standard and obtained from the data libraries of Kenneth French and Hou, Xue, and Zhang (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We consider monthly excess returns from January 1967 to December 2021, which results in a time dimension of T = 660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our test assets are the N = 243 double-sorted portfolios of Kenneth French’s data library summarized in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The candidate factors are J = 114 univariate long-short factors based on the data of Hou, Xue, and Zhang (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We include all univariate portfolio sorts from their data library that are available for our time period, and construct top minus bottom decile factor portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In addition, we include the five Fama-French factors of 24 Fama and French (2015) from Kenneth French’s data library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our analysis projects out the excess return of the market factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We are interested in the question which factors explain the component that is orthogonal to market movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, we regress out the market factor from the test assets and use the residuals as test assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We also do not include a market factor in the set of long-short candidate factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The original test assets have a market component as they are long only portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our results are essentially the same when we include the market component in the test assets, with the only difference that we would need to include the market factor as an additional factor in our parsimonious models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The market factor would always be selected by all models as significant, but this by itself is neither a novel nor interesting result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We present in-sample and out-of-sample results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The in-sample analysis uses the first 330 observations (January, 1967 to June, 1994), while the out-of-sample results are based on the second 330 observations (July, 1994 to December, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As in the simulation, we first use the inferential theory on the in-sample data to select our set of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, we use the selected subset of covariates in an OLS regression on the in-sample data to obtain the loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Last but not least, we use the estimated loadings on the selected subset of factors for the out-of-sample model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The LASSO penalty λ is selected via 5-fold cross-validation on the in-sample data to minimize the squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3 Hence, LASSO represents a first-stage dimension reduction tool, and we need the inferential theory to select our final sparse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We allow our selection to impose a prior on two of the most widely used asset pricing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' More specifically, we estimate models without a prior, and two specific priors that impose an infinite weight on the Fama-French 3 factors (FF3) and the Fama-French 5 factors (FF5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This prior as part of PoSI LASSO enforces that the FF3 and FF5 factors are included in the active set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note that because we work with data orthogonal to the market return, we do not include the market factor in the prior, but only the size and value factors for FF3 and in addition the investment and profitability factor for FF5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We denote these weights by ωFF3 and ωFF5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is an example where the researcher has economic knowledge that she wants to include in her statistical selection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We evaluate the models with standard metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The root-mean-squared error (RMSE) is based on the squared residuals relative to the estimated factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, in-sample the models are estimated to minimize the RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The pricing error is the economic quantity of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It is the time-series mean of the residual component of the factor model, and corresponds to the mean return that is not explained by the risk premia and exposure to the factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In summary, we obtain the residuals as ˆϵ = Yt,n − XS ˆβS for the selected factors, where the loadings are estimated on the in-sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The metrics are the RMSE and mean absolute pricing error (MAPE): RMSE = � � � � 1 N T N � i=1 T � t=1 ˆϵ2, MAPE = 1 N N � i=1 ����� 1 T T � t=1 ˆϵ ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 3We select λ from the grid exp(a) · log J/ √ T with a = −8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This grid choice satisfies the Assumptions in Chatterjee (2014) and hence Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 25 In addition to Panel PoSI without and with the FF3 and FF5 priors, we consider the benchmark methods of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We compare Panel PoSI (P-PoSI), Panel PoSI with infinite priors on FF3 and FF5 (P-PoSI ωFF3 respectively ωFF5), Bonferroni Naive LASSO (B-LASSO), Naive LASSO (N- LASSO), Bonferroni OLS (B-OLS) and Naive OLS (N-OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our main analysis sets the FWER control to the usual γ = 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 Asset Pricing Results Panel PoSI selects parsimonious factor models with the best out-of-sample performance among the benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For the FWER rate of γ = 5% the number of factors differs substantially among the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Panel PoSI selects 3 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Imposing infinite priors on FF3 or FF5 results in 4 and 5 factors for P-PoSI ωFF3 respectively ωFF5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In contrast, the alternative approaches select too many factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Bonferroni Naive LASSO includes 10, Naive Lasso 70, Bonferroni OLS 107 and Naive OLS 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' These over-parametrized models lead to overfitting of the in-sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Figure 6 shows in-sample and out-of-sample RMSE for each set of double-sorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The composition of the double sorts is summarized in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The in-sample performance in the left subfigure has the expected result that more factors mechanically decrease the RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The important findings are in the right subfigure with the out-of-sample RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The uniformly best performing model is Panel PoSI without any priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In fact, imposing a prior on the Fama- French factors increases the out-of-sample RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The conventional LASSO and OLS estimates have substantially higher RMSE, which can be more than twice as large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The Panel PoSI models also explain the average returns the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In Figure 7, we compare the mean absolute pricing errors among the benchmarks for each set of double sorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Importantly, the pricing errors are not used as in objective function of the estimation, and hence the fact that the models with the smallest RMSE explain expected returns is an economic finding supporting arbitrage pricing theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our Panel PoSI has the smallest out-of-sample pricing errors, which can be up to six times smaller compared to the OLS estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Including the Fama-French factors as a prior does not improve the models, except for the profitability and investment double sort, which uses the same information as two of the Fama-French factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The Panel PoSI models select economically meaningful factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Table 4 reports the ranking of factors based on their FWER bound without prior and infinite prior weights on the Fama-French 3 and 5 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rows are ordered based on sorted ascending ρ−1Njpj, which corresponds to the FWER bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It allows us to infer the number of factors for different levels of FWER control values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Setting γ = 5% leads to 3, 4 and respectively 5 factors, while a γ = 1% results in 2, 4 and 5 factors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In addition to their significance, we can infer the relative importance of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The baseline PoSI with γ = 5% selects a size, dollar trading volume and value factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The size and value factors are among the most widely used asset pricing factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Their selection is in line with their economic importance and confirms the Fama-French 3 factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The dollar trading volume factor is less conventional, but is correlated with many assets in our cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The size factor is the most 26 Figure 6: RMSE across cross-sections (a) In-sample (b) Out-of-sample This figure shows the in-sample and out-of-sample root-mean-squared errors (RMSE) for each cross-section of test assets for different factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The test assets are the N = 243 double-sorted portfolios, and we show the RMSE for each set of double-sorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rejection FWER is set to γ = 5% The candidate factors are the 114 univariate factor portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The time dimension is T = 660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We use the first half for the in-sample estimation and selection, while the second half serves for the out-of-sample analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We compare Panel PoSI (P-PoSI), Panel PoSI with infinite priors on FF3 and FF5 (P-PoSI ωFF3 respectively ωFF5), Bonferroni LASSO (B-LASSO), Naive LASSO (N-LASSO), Bonferroni OLS (B-OLS) and Naive OLS (N-OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' important as measured by the FWER bound, that is, the product of the number of relevant assets and its minimum p-value are the smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The short term reversal factor is less important and would require a FWER control of 10% to be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Imposing a prior affects the p-values of PoSI and the simultaneity count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For example, the cohesiveness coefficient increases from ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='16 for no priors to ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='18 in the case of the two priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, the FWER bounds of all factors can change when we impose a prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The FF3 prior increases the significance of the short-term reversal factor, which is widely used in asset pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Interestingly, even for a FF5 prior, the profitability and investment factors remain insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3 Number of Factors Our method contributes to the discussion about the number of asset pricing factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Many popular asset pricing models suggest between three and six factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our approach allows a disci- 27 OP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00 INV ME 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='62 Prior60 ME 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='93 Prior12 ME 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='53 Priorl ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='42 OP ME 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='67 OP BEME 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='52 INV BE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='31 ME N-OLS B-OLS N-LASSO B-LASSO P-POSI P-POSI P-POSI WFF3 wFF5Figure 7: MAPE across cross-sections (a) In-sample (b) Out-of-sample This figure shows the mean absolute pricing errors (MAPE) for each cross-section of test assets for different factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The test assets are the N = 243 double-sorted portfolios, and we show the average |α| for each set of double sorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rejection FWER is set to γ = 5% The candidate factors are the 114 univariate factor portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The time dimension is T = 660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We use the first half for the in-sample estimation and selection, while the second half serves for the out-of-sample analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We compare Panel PoSI (P-PoSI), Panel PoSI with infinite priors on FF3 and FF5 (P-PoSI ωFF3 respectively ωFF5), Bonferroni LASSO (B-LASSO), Naive LASSO (N-LASSO), Bonferroni OLS (B-OLS) and Naive OLS (N-OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' plined estimate for the number of factors based on inferential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The level of sparsity of a linear model also depends on the rotation of the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Therefore, we also study the principal components (PCs) of the covariates X as candidate factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In this case, we use the step-down procedure, which we refer to as “Ordered PoSI” or O-POSI for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Figure 8 shows the number of factors for different FWER rates γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The factor count is obtained by traversing K∗(γ) equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Panel PoSI without priors selects 2 factors for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='01 and 3 for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Once, we impose an infinite weight on the Fama-French 3 factors, we select 4 factors for all FWER levels, while the prior on the Fama-French 5 factors results in a 5 factor model for all FWER levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The Ordered PoSI with PCA rotated factors selects 3 factors for all FWER levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In summary, our results confirm that depending on the desired significance, the number of asset pricing factors for a good model seems to be between 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note that our analysis is orthogonal to the market factor, which would also be added to the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Thus, 28 OP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='17 INV ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='11 Prior60 ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='38 Prior12 ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='19 Priorl ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='11 OP ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='10 INV ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='10 0.' metadata={'source': 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+page_content='0001 2 Value (HML) 1191 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0280 3 Short-Term Reversal (srev) 1050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0974 4 Forecast Revisions (rev 1) 242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2782 5 Investment (CMA) 998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00112 >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9999 6 Profitability (RMW) 797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00123 >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9999 7 FF3 prior (ωFF3) Size (SMB) 2802 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0001 1 Value (HML) 2802 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0001 2 Dollar Trading Volume (dtv 12) 779 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0017 3 Short-Term Reversal (srev) 1106 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0049 4 Profitability (RMW) 819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2527 5 Investment (CMA) 874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00087 >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='9999 6 FF5 prior (ωFF5) Size (SMB) 2911 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0001 1 Value (HML) 2911 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0001 2 Forecast Revisions (rev 1) 230 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0005 3 Short-Term Reversal (srev) 1140 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0052 4 Dollar Trading Volume (dtv 12) 661 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='0072 5 Profitability (RMW) 2911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1937 6 Investment (CMA) 2911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1996 7 Gross profits-to-assets (gpa) 1151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='00013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='8382 8 This table reports ranking of factors based on their FWER bound for no prior, and infinite weight priors on the Fama-French 3 and 5 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The test assets are the N = 243 double-sorted portfolios and the candidate factors are J = 114 univariate long-short factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rows are ordered based on sorted ascending ρ−1Njpj, which corresponds to the FWER bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' the final model would have between 3 and 5 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Table 5 further confirms our findings about the number of asset pricing factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We compare the number of factors for γ = 5% selected either from the univariate high-minus-low factors (HL), their PCA rotation or the combination of the high-minus-low factors and their PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Panel PoSI selects consistently 3 factors from the long-short factors and their PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' When combined, PoSI selects 4 factors, which is plausible as the optimal sparse model can be different for this larger set of candidate factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The Bonferroni PoSI is overly conservative and selects only 2 HL factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The models based on Naive LASSO or OLS select excessively many factors independent of the rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Overall, the findings support that parsimonious asset pricing models can be described by three to four factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Of course, any discussion about the number of asset pricing factors is always subject to the choice of test assets and candidate factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 29 Figure 8: Number of selected factors for different FWER (a) Univariate factors with priors (P-POSI) (b) PCA rotated factors (O-POSI) This figure shows the number of selected factors to explain the test assets of double-sorted portfolios for different FWER rates γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The factor count is obtained by traversing K∗(γ) for γ ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The left subfigure uses univariate high-minus-low factors as candidate factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We consider the case of no prior, and the cases of an infinite weight on the Fama-French 3 factor model (ωFF3) and an infinite weight on the Fama-French 5 factor model (ωFF5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The right subfigure uses the PCA rotation as candidate factors with the step-down procedure Ordered PoSI (O-POSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Table 5: Number of selected factors for different methods HL PCs HL + PCs Panel PoSI 3 3 4 Bonferroni PoSI 2 3 2 Bonferroni Naive LASSO 10 29 10 Naive LASSO 70 50 76 Bonferroni OLS 107 13 117 Naive OLS 114 50 164 This figure shows the number of selected factors to explain the test assets of double-sorted portfolios for different methods and different sets of candidate factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The rejection FWER is set to γ = 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The factor count is obtained by traversing K∗(γ) for γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The number of factors is selected on the in-sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For PCs, we use the step-down method for the nested hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 8 Conclusion This paper proposes a new method for covariate selection in large dimensional panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We develop the conditional inferential theory for large dimensional panel data with many covariates by combining post-selection inference with a new multiple testing method specifically designed for panel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our novel data-driven hypotheses are conditional on sparse covariate selections and valid for any regularized estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Based on our panel localization procedure, we control for family-wise error rates for the covariate discovery and can test unordered and nested families of hypotheses for large cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We provide a method that allows us to traverse the inferential 30 P-PoSI P-PoSL WE3 6 P-PoSI WFF5 Factor count 5 5 5 5 5 4 4 4 44 4 3 3 2 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='17 6 PC count 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 3 3 3 3 3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1results and determine the least number of covariates that have to be included given a user-specified FWER level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As an easy-to-use and practically relevant procedure, we propose Panel-PoSI, which combines the data-driven adjustment for panel multiple testing with valid post-selection p-values of a gen- eralized LASSO, that allows to incorporate weights for priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In an empirical study, we select a small number of asset pricing factors that explain a large cross-section of investment strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our method dominates the benchmarks out-of-sample due to its better control of false rejections and detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A Post-selection Inference with Weighted-LASSO A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 Weighted-LASSO: Linear Truncation Results This appendix collects the assumptions and formal statements underlying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We present the results for the Weighted-LASSO, which includes the conventional LASSO as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In order to ensure uniqueness of the LASSO solution, we impose the following condition, which is standard in the LASSO literature: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' General position The matrix X ∈ RT×J has columns in general position if the affine span of any J0 + 1 points (σ1Xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', σJ0+1XiJ0+1) in RT for arbitrary σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='σd0+1 ∈ {±1} does not contain any element of {±Xi : i /∈ {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', iJ0+1}}, where J0 < J4 and Xi denotes ith column of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This position notion will help us to avoid ambiguity in the LASSO solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note that this condition is a much weaker requirement than full-rank of X, and states that if one constructs a J0-dimensional subspace, it must contain at most J0 + 1 entries of {±X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=', ±XJ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Even though this appears to be a complicated and mechanical condition, by a union argument it turns out that with probability 1, if the entries of X ∈ RT×J are drawn from a continuous probability distribution on RT×J then X is in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='5 Then, we will be able to discuss the LASSO solution for general design with relative ease, thanks to Lemma 3 of Tibshirani (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It shows that if X lie in general position, it is sufficient to have a unique LASSO solution regardless of the penalty scalar λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This condition will later be used in establishing our Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We can now state the formal assumptions: Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Unique low dimensional model (a) Low dimensional truth: The data satisfies Y = XSβS + ϵ where |S| = O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (b) General position design: The covariates X have columns in general position as given by Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 4The original condition needs to hold for J0 < min{T, J} but in the scope of our study, we consider T > J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 5See Donoho (2006) and §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 of Tibshirani (2013) for more discussions on uniqueness and general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 31 We start our analysis with the simpler model of known error variance, and later extend it to the case of estimated unknown variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Gaussian residual with known variance The residuals are distributed as ϵ ∼ N(0, Σ) where Σ is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Before formalizing the inferential theory, we need to clarify the quantity for which we want to make inference statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As stated before, we only test the hypothesis on a covariate if its LASSO estimate turns out active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is exactly the approach how researchers in practice conduct explorations in high-dimensional datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In other words, we focus on ˆβM and quantities associated with it, where M denotes the active set of selected covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We study the inferential theory of the “debiased estimator”, which is a shifted version of the LASSO fit as defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We show that this debiased estimator is unbiased, consistent and follows a truncated Gaussian distribution, with profound connections to the debiased LASSO lit- erature such as Javanmard and Montanari (2018), but has different properties by a subtle different descent direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' More concretely, given M, clearly ˆY = XM ˆβM is the fitted value since ˆβ−M = 0, where −M is the complement of the set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We let ˆϵM := Y − XM ˆβM be the residual from the LASSO estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' By considering only the partial LASSO loss of ℓ(Y, XM, λ, β) and given we are currently at the LASSO estimator ˆβ, the Newton step is X+ Mˆϵ following (Boyd and Vandenberghe, 2004, § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2), where we denote X+ M = (X⊤ MXM)−1X⊤ M as the pseudo-inverse of the active subma- trix of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The invertibility of X⊤ MXM either is observed when we are in the fixed design regime or happens almost surely when we are dealing with continuous quantities, as a consequence of Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1(b) as argued in Tibshirani (2013) and Lee, Sun, Sun, and Taylor (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Now we can formally define the main object of our inferential theories: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Debiased Estimator The debiased Weighted-LASSO estimator ¯βM given M is given by ¯βM = ˆβM + X+ MˆϵM (21) It is now evident why some of the literature refers to the debiased estimator also as the one-step estimator: given that ˆβM solves the Karush-Kuhn-Tucker (KKT) condition and reaches the optimal sub-gradient for the full loss ℓ(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' our debiased estimator ¯βM is the result of moving one more Newton-Ralphson method step after ˆβM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' but only taking XM rather than X as a whole into the likelihood loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, the update step is actually only a partial update from the LASSO solution point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Intuitively, ¯βM should still be close to solving the KKT conditions, and would exactly solve the KKT conditions if XM happen to be the true covariates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' XM = XS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' If we were to take a Newton’s method step with gradient and Hessian calculated with the entirety of data X, or equivalently taking a full update from the stationary point, we will recover the ˆβd M proposed in Javanmard and Montanari (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The material difference is that the full-update would require the J ×J precision matrix Ω = Γ−1, where Γ = X⊤X if X assumed fixed or Γ = E[X⊤X] if X assumed to be generated from a stationary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Using ℓ(Y, XM, λ, β) instead of ℓ(Y, X, λ, β), 32 our debiased estimator would not need the full Hessian, which is leveraging LASSO’s screening property and uses (X⊤ MXM)−1X⊤ M (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' X+ M) as a much lower-dimensional alternative of ΩX⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Without loss of generality, we assume that the covariate indexed i ≤ |M| is part of M, and we can always rearrange the columns of X to have the first |M| covariates as active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Let η = (X+ M)⊤ei ∈ RT be a vector where ei ∈ R|M| is a vector with 1 at ith coordinate and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, the η vector is the linear mapping from Y to the ith coordinate of an OLS estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' In particular, the debiased estimator and the response satisfy the following relationship: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Debiased Estimator is OLS-post-LASSO The debiased estimator is a linear mapping of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Specifically, given η = (X+ M)⊤ei: ¯βi = η⊤Y (22) Moreover, ¯βM is the OLS estimate of regressing XM on Y : ¯βM = arg min β 1 2T ∥Y − XMβ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (23) The proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 is deferred to the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Although its proof is simple, this lemma reveals that our debiased estimator is the same as the least-square after LASSO estimator proposed in Belloni and Chernozhukov (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our strategy to obtain a rigorous statistical inferential theory with p-values is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First we perform an algebraic manipulation to transform ˆβM into ¯βM in the linear form of (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Then, we follow the strategy in Lee, Sun, Sun, and Taylor (2016) to traverse the KKT subgradient optimal equations for general X by writing it equivalently into a truncation in the form of {AY ≤ b}, as we will do in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Finally we will circle back to ˆβM by the linear mapping between ¯βM and Y and the distributional results induced by the fact that Y is truncated by {AY ≤ b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For our Weighted-LASSO, the KKT sub-gradient equations are X⊤(X ˆβ − Y ) + λ � s v � ⊙ ω−1 = 0 where � � � si = sign(ˆβi) if ˆβi ̸= 0, ωi < ∞ vi ∈ [−1, 1] if ˆβi = 0, ωi < ∞ (24) In other words, when ω is specified, the KKT conditions can be identified using the tuple of {M, s}, where M is the active covariates set and s is the signs of LASSO fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This is a consequence of how LASSO KKT condition can separate the slacks into s for active variables and v for inactive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' If we have infinite importance weights (J ̸= ∅), we would simply need si < ∞ for i ∈ J because λsi/ωi = 0 is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We rigorously characterize the KKT sub-gradient conditions as a combinations of signs and infinity norm bounds conditions by the following lemma, which parallels Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 of Lee, Sun, Sun, and Taylor (2016): Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Selection in norm equivalency 33 Consider the following random variables w(M, s, ω) = (X⊤ MXM)−1(X⊤ MY − λs ⊙ ω−1 M ) u(M, s, ω) = ω−M ⊙ � X⊤ −M(X+ M)⊤s ⊙ ω−1 M + 1 λX⊤ −M(I − PM)Y � (25) where PM = XMX+ M ∈ RT×|M| is the projection matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The Weighted-LASSO selection can be written equivalently as {M, s} = {sign(w(M, s, ω)) = s, ∥u(M, s, ω)∥∞ < 1} (26) Using this characterization, we are then able to provide the distributional results for the debiased estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Consider ξ = Ση(η⊤Ση)−1 ∈ RT as a covariance-scaled version of our η, and a mapping of Y using residual projection matrix: z = (I − ξη⊤)Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note that z can be calculated once we observe (X, Y ), so it can be conditioned on were we to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We will soon see that the truncation set will depend on the variable z, but this does not cause any issues thanks to the following lemma, the proof of which is deferred to the Online Appendix: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Ancillarity in truncation The projected z and the debiased estimator ¯βi are independently distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' As a result of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3, when describing the distribution of ¯βi, we can use z in its truncation conditions as long as we condition on z as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' To simplify notation, we can collect all quantities we need to condition on into ˜ M := ((M, s), z, ω, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Now we can assemble the consequences of Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3 to arrive at the truncated Gaussian statements for the debiased estimator similar to Lee, Sun, Sun, and Taylor (2016), but for weighted-LASSO: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Truncated Gaussian Under Assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 for i ∈ M, ¯βi is conditionally distributed as: ¯βi| ˜ M ∼ T N(βi, η⊤Ση;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' [V −(z), V +(z)]) (27) where T N is a truncated Gaussian with mean βi, variance η⊤Ση and truncation set [V −(z), V +(z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' βi denotes the ith entry of the true β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The vector of signs is s = sign(ˆβM) ∈ R|M| and the truncation set depends on A = � �� λ−1X⊤ −M(I − PM) −λ−1X⊤ −M(I − PM) −diag(s)X+ M � �� ∈ R(2J−|M|)×T , b = � �� ω−1 −M − X⊤ −M(X+ M)⊤s ⊙ ω−1 M ω−1 −M + X⊤ −M(X+ M)⊤s ⊙ ω−1 M −λ · diag(s)(X⊤ MXM)−1s ⊙ ω−1 M � �� ∈ R2J−|M| V −(z) = max j:(Aξ)j<0 bj − (Az)j (Aξ)j , V +(z) = min j:(Aξ)j>0 bj − (Az)j (Aξ)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Notice that Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 is decoupled across M, which is to say we are able to deal with 1-dimensional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We arrive at this form because the construction of (V −, V +) over the 34 extreme points of the linear inequality system (or vertices of the polyhedral) has decomposed the dimensionality of the truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This decoupling is of significant practical value, in that it would be otherwise a non-trivial task to calculate a statistic of multivariate (in our case |M|-dimensional) truncated Gaussian and then marginalize over |M| − 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 Weighted-LASSO Quasi-Linear Truncation with Estimated Variance This section generalizes the distribution results to the practically relevant case when the noise variance is unknown and has to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This becomes a challenging problem for post-selection inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We replace Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 by the following assumption: Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Gaussian residual with simple unknown variance The residuals are distributed as ϵi iid ∼ N(0, σ2) where σ2 is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The simple structure of unknown variance of Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 is common in the post-selection inference literature as for example in Lee, Sun, Sun, and Taylor (2016) and Tian, Loftus, and Taylor (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A feasible conditional distribution replaces σ2 with an estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Under Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2, we can estimate the variance using LASSO residuals and then reiterate the previous truncation arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The most common standard variance estimator is ˆσ2(Y ) = ∥Y − X ˆβ∥2 2/(T − |M|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' (28) In classical regression analysis, the normally distributed estimated coefficient divided by an estimated standard deviation follows a t-statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Hence, we would expect that a truncated normal debiased estimator divided by a sample standard deviation might yield a truncated t-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' However, the arguments are substantially more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Simply using ˆσ(Y ) of (28) in the expres- sion η⊤Ση of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 changes the truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Specifically, Y having truncated support means ˆσ(Y )2 is not χ2-distributed supported on the entire R+, which makes the support of ¯β/ˆσ(Y ) non- trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Therefore, in order to correctly assess the truncation of the studentized quantity, we have to disentangle how much truncation is implied in ˆσ(Y )−1 and ¯β simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Geometrically, as ˆσ(Y ) is a non-linear function of Y and ¯β, the truncation on Y is in fact no longer of the simple linear form {AY ≤ b} such as in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Instead of a polyhedral induced by affine constraints, we have a “quasi-affine constraints” form of {CY ≤ ˆσ(Y )b} because LASSO KKT conditions preserve the estimated variance throughout the arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Thus, both sides of the inequality CY ≤ ˆσ(Y )b have Y , and in right-hand-side the ˆσ(Y ) is non-linear in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' A significantly more complex set of arguments are needed compute the exact truncation, which is equivalent to solve for a |M|-system of non-linear inequalities rather than linear inequalities that constrain the support of Y for inference on each ¯βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 shows the appropriate truncation based on those arguments: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Truncated t-distribution for estimated variance Under Assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3, and the null hypothesis that βi = 0, the studentized quantity 35 ¯βi/∥η∥ˆσ(Y ) follows ¯βi/∥η∥ˆσ(Y ) ∼ TTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='Ω, (29) where TT is a truncated t-distribution with d degrees of freedom and truncation set Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The truncation set Ω = � i∈M{t : t √ Wνi + ξi √ d + t2 ≤ −θi √ W} is an |M|-intersection of simple inequality-induced intervals based on the following quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The active signs are denoted as s = sign(ˆβM) ∈ R|M|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The scaled LASSO equivalent penalty is ˜λ2 = λ2 ˆσ2(Y )·(T−|M|)+∥(X+ M)⊤s⊙ω−1 M ∥2 2λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' θi = (˜λsi � T − |M| 1 − ˜λ2∥(X+ M)⊤s ⊙ ω−1 M ∥2 2 ) · e⊤ i � (X⊤ MXM)−1s ⊙ ω−1� for i ∈ M C = −diag(s)X+ M ∈ R|M|×T , ν = Cη ∈ R|M|, ξ = C(PM − ηη⊤)Y ∈ R|M|, d = tr(I − PM), W = ˆσ2(Y ) · d + (η⊤Y )2 The quantities θ and C describe the quasi-linear constraints, whereas ν and ξ transform them into the form of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Note that the Ω set is obtained from solving a low-dimensional set of quadratic inequalities that do not necessarily yield a single interval after intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We provide a proof of this result in the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Using Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2 in practice poses several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First, the computations are much more involved, especially as each βi requires calculation of Ω which includes |M| actual constraints, each of which involves solving a simple but still non-linear inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' It is non-trivial to ensure that the numerical stability holds at every step of the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Second, since Ω is not necessarily an interval, it is harder to interpret the truncation and also calculate the cumulative density function through Monte-Carlo simulations when there is a non-trivial truncation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Third, in fact, the authors in Tian, Loftus, and Taylor (2018) recommend a regularized likelihood minimizing variance estimator that deviates from the simple ˆσ(Y ), which would in turn involves more numerical integration and optimization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Last but not least, this result was proposed initially for studying scale-LASSO, which is why there has to be a penalty term transformation of λ to ˜λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Our goal is to provide a set of tools that can be useful for a wide range of applications including the LASSO with l2 squared norm loss rather than un-squared norm loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' These implementation difficulties are also discussed in more detail in the Online Appendix, which provides the accompanying proofs and the exact forms of the truncations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We provide a practical solution based on an asymptotic normal argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' We impose the standard assumption that we have a consistent estimator of the residual variance: Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Consistent estimator ˆσ(Y ) Given λ, the residual variance estimator is consistent ˆσ(Y ) p→ σ2 as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' This general assumption includes many common scenarios such as the results specified in Corol- lary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 of van de Geer and B¨uhlmann (2011), or in Theorem 2 of Chatterjee (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' For example, for diminishing c � log(J)/T → 0 as J, T grow and our Assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3, we obtain con- sistency of ˆσ(Y ) of (28) by Chatterjee (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' 36 Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Asymptotic truncated normal distribution Suppose Assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Under the null hypothesis that βi = 0 and for T → ∞ the studentized quantity ¯βi/∥η∥ˆσ(Y ) follows ¯βi/∥η∥ˆσ(Y ) ∼ TNΩ, (30) where TN is a truncated normal distribution with truncation Ω = [V −(z)/∥η∥2ˆσ(Y );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' V +(z)/∥η∥2ˆσ(Y )], where V −(z) and V +(z) are the same as in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' The asymptotic distribution result has several advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' First, it is intuitive since it parallels the classical OLS inference with a t-statistic converging to Gaussianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' Secondly, it is computa- tionally more tractable than results of Appendix Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' With this result, one could obtain asymptotically valid post-selection p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content=' B Appendix: Empirics Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfc_c0/content/2301.00292v1.pdf'} +page_content='1: Compositions of DS portfolios Sorted by # portfolios Sorted by # portfolios Sorted by # portfolios Sorted by # portfolios BEME, INV 25 ME, CFP 6 ME, INV 25 ME, Prior1 25 BEME, OP 25 ME, DP 6 ME, OP 25 ME, Prior12 25 ME, BE 25 ME, EP 6 OP, INV 25 ME, Prior60 25 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Huse1 +1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA +2Department of Physics, Stanford University, Stanford, California 94305, USA +(Dated: January 13, 2023) +Many-body localized (MBL) systems fail to reach thermal equilibrium under their own dynamics, +even though they are interacting, nonintegrable, and in an extensively excited state. One instability +towards thermalization of MBL systems is the so-called “avalanche”, where a locally thermalizing +rare region is able to spread thermalization through the full system. The spreading of the avalanche +may be modeled and numerically studied in finite one-dimensional MBL systems by weakly coupling +an infinite-temperature bath to one end of the system. We find that the avalanche spreads primarily +via strong many-body resonances between rare near-resonant eigenstates of the closed system. Thus +we find and explore a detailed connection between many-body resonances and avalanches in MBL +systems. +Introduction— Many-body localized (MBL) systems +are a class of isolated many-body quantum systems that +fail to thermalize due to their own unitary dynamics, +even though they are interacting, nonintegrable and ex- +tensively excited [1–7]. This happens for one-dimensional +systems with short-range interactions in the presence +of strong enough quenched randomness, which yields a +thermal-to-MBL phase transition of the dynamics. +In +the MBL phase, there are an extensive number of emer- +gent localized conserved operators [8–11]. +One instability of the MBL phase which is believed to +play a central role in the asymptotic, long-time, infinite- +system MBL phase transition is the avalanche [12–14]. +Rare locally thermalizing regions necessarily exist, how- +ever sparse they may be, due to the randomness. Starting +from such a local thermalizing region, this “thermal bub- +ble” spreads through the adjacent typical MBL regions +until the relaxation rate of the adjacent spins becomes +smaller than the many-body level spacing of the ther- +mal bubble, in which case the spreading of this avalanche +halts. If the strength of the randomness is insufficient, +the relaxation rate remains larger than the level spacing +and the avalanche does not stop: the full system then +slowly thermalizes and is no longer in the MBL phase (al- +though it is likely in a prethermal MBL regime [15, 16]). +The avalanche has been numerically simulated in +small-sized +systems +[15, +17–21] +and +experimentally +probed [22]. Recent work shows that the instability of +MBL to avalanches occurs at much stronger randomness +than had been previously thought [15, 20]. This leaves a +large intermediate prethermal-MBL regime in the phase +diagram between the onset of MBL-like behavior in small +samples (or correspondingly short times) and the asymp- +totic MBL phase transition. +Clear numerical evidence +has been obtained for many-body resonances being an +important part of the physics in the near-thermal part +of this regime [15, 16, 23–29], while no such evidence for +the expected thermalizing rare regions has been found +yet. In the part of this intermediate prethermal MBL +regime that is farther from the thermal regime, it re- +Bath (T= +) +0 +After Long time +(a) +(c) +(b) +n +n +n +Slowest Mode ( +) +near-resonance +dominant decay +B +A +B +A +1 +2 +. . . +. . . +. . . +. . . +. . . +. . . +. . . +L +Figure 1. +Schematic illustrations showing (a) the avalanche +model, (b) the long time dynamics governed by the slowest +mode, and (c) the dominant decay processes involving four +eigenstates with a near-resonance. (a) We connect the bath +in the weak-coupling limit with the one-dimensional MBL sys- +tem. Specifically, we analyze the decay of the slowest mode +(ˆτS), which is localized near the end of the system farthest +from the bath; ˆτS is a “localized integral of motion” in the +MBL phase. (b) Thermalization at the latest times is gov- +erned by ˆτS. (c) Schematic decay of ˆτS. A large fraction of +the probability current in the decay of ˆτS passes through four +eigenstates associated with a rare near-resonance. +mains unclear what is the primary mechanism that leads +to thermalization for samples larger than those that can +be diagonalized. +In this work, we explore how an avalanche spreads +through typical MBL regions for systems that are near +the avalanche instability. We do not simulate the rare +region that initiates the avalanche. Instead, we assume +a large avalanche is spreading and we model that as an +infinite-temperature bath (see Fig. 1) weakly coupled to +one end of our MBL spin chain [15, 20]. We find that +particular many-body near-resonances of the closed sys- +tem play a key role in facilitating the spreading of the +avalanche. +These many-body near-resonances are the +arXiv:2301.04658v1 [cond-mat.stat-mech] 11 Jan 2023 + +2 +dominant process by which the bath at one end of the +chain thermalizes the spins at the other end of the chain +and thus propagates the avalanche. +Model— Our model consists of a chain of L spin- +1/2 degrees of freedom (or qubits). +The dynamics of +the closed system is given by the random-circuit Floquet +MBL model studied in Ref. [15], which has unitary Flo- +quet operator ˆUF . The disorder strength in this model +is given by the parameter α, with the MBL regime be- +ing at large α, while the thermal regime is at small α +(see the Appendix for the full description of this model). +To investigate avalanche spreading, we weakly connect +an infinite-temperature Markovian bath to spin L at the +right end of the system [15, 20]. The quantum state of +this open system is the density matrix ˆρ(t). +In our open-system Floquet model, the bath is repre- +sented by the super-operator Sbath that acts once each +time period: +Sbath[ˆρ] = +ˆρ +1 + 3γ + +γ +1 + 3γ +3 +� +j=1 +ˆEj ˆρ ˆE† +j , +(1) +where ( ˆE1, ˆE2, ˆE3) = ( ˆXL, ˆYL, ˆZL) are the jump opera- +tors acting on the last spin at site L (connected to the +bath). We will take the weak coupling limit γ → 0. The +open-system Floquet super-operator Speriod that takes +our system through one full time-period is +Speriod[ˆρ(t)] = Sbath[ ˆUF ˆρ(t) ˆU † +F ] = ˆρ(t + 1) . +(2) +The time evolution of the system’s state is given by +ˆρ(t) = ˆI/2L + p1e−r1tˆτ1 + +� +k≥2 +pke−rktˆτk, +(3) +where e−rk is the kth largest eigenvalue of Speriod with +eigenoperator ˆτk. Note that the largest (0th) eigenvalue +is 1 and is nondegenerate when γ > 0, with eigenoper- +ator proportional to the identity, which is the long-time +steady state of this system. The mode with the slowest +relaxation for γ > 0 is ˆτS := ˆτ1, which relaxes with rate +rS := Re(r1). The relaxation rate rS is proportional to +γ, and we work to first order in γ [20]. +Relaxation of the slowest mode— In the weak- +coupling limit, one can obtain ˆτS as a superposition of +the diagonal terms |n⟩ ⟨n|, where |n⟩ are the eigenstates +of the closed system such that ˆUF |n⟩ = eiθn |n⟩. When +γ = 0, then |m⟩ ⟨n| are eigenoperators of Speriod with +eigenvalues ei(θm−θn), so all diagonal terms |n⟩ ⟨n| are +degenerate, with rk = 0. Therefore, in the γ ≪ 1 limit, +one can obtain ˆτS in degenerate perturbation theory by +diagonalizing the (super)operator S[ˆρ] := 1 +3 +�3 +j=1 ˆEj ˆρ ˆE† +j +in this degenerate subspace [20], where the matrix ele- +ments are +Smn = ⟨m| S[|n⟩ ⟨n|] |m⟩ = 1 +3 +3 +� +j=1 +| ⟨m| ˆEj |n⟩ |2. +(4) +0.5 +Probability Density (a.u.) +4 +L= +MBL +12 +Thermal +Figure 2. +Probability distribution over samples of the ratio +R = Dµν/(2LΓ) (see text). In the thermal regime (dark red to +yellow), the ratio exponentially decays with increasing L. In +comparison, R barely drifts with L for the MBL case (blue). +We observe that R never exceeds the value 0.5 (gray dashed +line), which is explained with the minimal model in the main +text. In this figure, we used α = 30 and α = 1 for MBL and +thermal regimes, respectively. +Note that in the degenerate subspace, S is a symmet- +ric stochastic matrix with real eigenvalues. +In partic- +ular, ˆτS = � +n cn |n⟩ ⟨n| where ⃗c is the eigenvector of +S with the smallest spectral gap Γ from the steady +state (� +m Snmcm = (1 − Γ)cn). The relaxation rate is +rS = 3γΓ. We normalize ˆτS so Tr{ˆτ 2 +S} = � +n |cn|2 = 2L. +Naively, the slowest mode is the local integral of motion +(LIOM) that is farthest from the bath. More precisely, +as we show in the Appendix, the slowest mode, ˆτS, is +a traceless superposition of projectors on to the eigen- +states of the closed system. Among such operators it is +the one with the smallest weight of Pauli strings with +non-identity at site L (connected to the bath). It is a +LIOM that is indeed localized far from the bath, but it +is different in detail from the ℓ-bits and LIOMs discussed +in Refs. [8–10, 30]. +As illustrated in Fig. 1(b), the latest time dynamics +are determined by ˆτS, with ˆρ(t) ≃ ˆI/2L + pSe−rStˆτS for +any initial conditions that contain ˆτS. This assumes a +nonzero gap between (r1/γ) and (r2/γ), which is indeed +the case for all samples examined. We can view the slow +relaxation of ˆτS in terms of probability currents that flow +between the eigenstates of the isolated system, leading to +the final ˆρ = ˆI/2L equilibrium where all eigenstates have +equal weight. Specifically, we may quantify the contribu- +tion Dmn of the pair of eigenstates m, n to the relaxation +of ˆτS as +Dmn := Smn(cm − cn)2 ≥ 0 . +(5) +One can show (see Appendix) that the relaxation rate of +ˆτS is given by the sum of the contributions from all pairs + +3 +of eigenstates of the closed system: +� +m Ours-R2 > Ours-R4. +This finding is intuitive since the more accurate method +should introduce smaller noise to the BP, e.g., the gradi- +ent filtering with patch size 2 × 2 (Ours-R2) introduces less +noise than with patch size 4 × 4 (Ours-R4). In Figure 5, we +92 +93 +94 +Accuracy [%] +ResNet18-CIFAR10 +101 +102 +103 +#MFLOPs +14.1x OPs +2.1x OPs +2.2% Acc +68.3x +OPs +Baseline +Ours-R2 +Ours-R4 +78 +80 +82 +Accuracy [%] +ResNet34-CIFAR100 +101 +102 +103 +18.8x OPs +1% Acc +7.6x OPs +2.0x OPs +5.1% Acc +63.6x OPs +Baseline +Ours-R2 +Ours-R4 +Figure 4. Computation (#MFLOPs, log scale) and model accuracy +[%] under different hyper-parameter selection. “Baseline” means +vanilla BP; “Ours-R2/4” uses gradient filtering with patch size 2× +2/4 × 4 during BP. +evaluate the relationship between accuracy and noise level +introduced by gradient filtering. With a higher SNR (i.e., a +lower noise level), a better accuracy is achieved. +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +0.225 +SNR [db] +71.6 +71.8 +Top 1 Accuracy [%] +Figure 5. Relationship between accuracy and noise level intro- +duced by the gradient filtering. As shown, accuracy increases as +the SNR increases, i.e., noise level decreases. +c. Given the number of computations, the less accurate +method returns the better accuracy by training more +layers, i.e., Ours-R4 > Ours-R2 > baseline. +This finding suggests that for neural network training with +relatively low computational resources, training more layers +with less accurate gradients is preferable than training fewer +layers with more accurate gradients. +5.5. On-device Performance Evaluation +Figure 6 and Table 4 show our evaluation results on real +devices. More results are included in the Supplementary +Section H. As Figure 6 shows, on CPU, most convolution +layers achieve speedups over 20× with less than 50% mem- +ory consumption for gradient filtering with patch sizes 2×2; +for gradient filtering with patch size 4 × 4, the speedups are +much higher, namely over 60×. On GPU, the speedup is +a little bit lower, but still over 10× and 25×, respectively. +Furthermore, as Table 4 shows, our method saves over 95% +7 + +0 +1 +2 +3 +4 +5 +6 +0× +20× +40× +60× +80× +100× +Speedup (×times) +114× +CPU Speedup +Jetson-R2 +Jetson-R4 +11900KF-R2 +11900KF-R4 +RPi3-R2 +RPi3-R4 +0 +1 +2 +3 +4 +5 +6 +0× +10× +20× +30× +40× +50× +60× +GPU Speedup +Jetson-R2 +Jetson-R4 +RTX3090Ti-R2 +RTX3090Ti-R4 +0 +1 +2 +3 +4 +5 +6 +Test Case - Baseline: MKLDNN +0 +20 +40 +60 +80 +100 +Percentage [%] +Baseline: MKLDNN +50% Memory Cost +Normalized CPU Memory Cost +Jetson-R2 +Jetson-R4 +11900KF-R2 +11900KF-R4 +RPi3-R2 +RPi3-R4 +0 +1 +2 +3 +4 +5 +6 +Test Case - Baseline: CUDNN +0 +20 +40 +60 +80 +100 +Baseline: CUDNN +50% Memory Cost +Normalized GPU Memory Cost +Jetson-R2 +Jetson-R4 +RTX3090Ti-R2 +RTX3090Ti-R4 +Figure 6. Speedup and normalized memory consumption results on multiple CPUs and GPUs under different test cases (i.e. different +input sizes, numbers of channels, etc.) Detailed configuration of these test cases are included in the supplementary material. “R2”, “R4” +mean using gradient filtering with 2 × 2 and 4 × 4 patch sizes, respectively. Our method achieves significant speedup with low memory +consumption compared to all baseline methods. For example, on Jetson CPU with patch size 4 × 4 (“Jetson-R4” in left top figure), our +method achieves 114× speedup with only 33% memory consumption for most test cases. +Device +Patch Size +Normalized Energy Cost [STD] +Edge +CPU +2 × 2 +4.13% [0.61%] +4 × 4 +1.15% [0.18%] +Edge +GPU +2 × 2 +3.80% [0.73%] +4 × 4 +1.22% [1.10%] +Table 4. Normalized energy consumption for BP with gradient +filtering for different patch sizes. Results are normalized w.r.t. the +energy cost of standard BP methods. For instance, for edge CPU +with a 4 × 4 patch, only 1.15% of energy in standard BP is used. +Standard deviations are shown within brackets. +energy for both CPU and GPU scenarios, which largely re- +solves one of the most important constraints on edge de- +vices. All these experiments on real devices show that our +method is practical for the real deployment of both high- +performance and IoT applications. +Model +Ratio +Model +Ratio +(Wide)ResNet18-152 +1.462 +VGG(bn)11-19 +1.497 +DenseNet121-201 +2.278 +EfficientNet b0-b7 +1.240 +Table 5. Evaluation of energy ratio defined in Equation (13) on +models published on Torchvision. The ratio greater than 1 empiri- +cally verifies our assumption. +5.6. Main Assumption Verification +We now empirically verify the assumption that the DC +component dominates the frequency spectrum of the convo- +lution kernel (Section 4.1). To this end, we collect the en- +ergy ratio shown in Equation (13) from trained models pub- +lished in Torchvision [24]. As Table 5 shows, for the con- +volution kernels in all these networks, we get a ratio greater +than one, which means that the energy of DC components +is larger than energy of all AC components. Thus, our as- +sumption in Section 4.1 empirically holds true in practice. +6. Conclusions +In this paper, we have addressed the on-device model +training for resource-constrained edge devices. To this end, +a new gradient filtering method has been proposed to sys- +tematically reduce the computation and memory consump- +tion for the back-propagation algorithm, which is the key +bottleneck for efficient model training. +In Section 3, a new gradient filtering approach has been +proposed to reduce the computation required for propagat- +ing gradients through the convolutional layers. 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In Proceedings of the +AAAI Conference on Artificial Intelligence, volume 35, pages +3483–3491, 2021. 2 +10 + +In this supplementary material, we present: +• A: Detailed derivation for gradient filtering described +in Section 3. +• B: Detailed proof for Proposition 1 in Section 4.1. +• C: Visualized computation analysis for ResNet18. +• D: Detailed experimental setup for Section 5.1. +• E: More experimental results for Semantic Segmenta- +tion in Section 5.3. +• F: More experimental results for hyper-parameter ex- +ploration on CIFAR datasets in Section 5.4. +• G: Experimental results for combining gradient filter- +ing (our method) with existing INT8 gradient quanti- +zation approaches [4,7]. +• H: More experimental results for on-device perfor- +mance evaluation in Section 5.5. +A. Gradient Filtering Derivation +In this section, we present the complete derivations for +Equation (3) and Equation (5) in Section 3, namely the back +propagation with gradient filtering. For convenience, Ta- +ble 6 (reproduced from Table 1 in paper) lists commonly +used symbols. +A.1. Gradient Filtering +We have: +˜gy[n, co, h, w] = 1 +r2 +⌈i/r⌉r +� +h=⌊i/r⌋r +⌈j/r⌉r +� +w=⌊j/r⌋r +gy[n, co, i, j] (17) +Cx +Number of channels of x +Wx, Hx +Width and height of x +θ +Convolution kernel +θ′ +Rotated θ, i.e., θ′ = rot180(θ) +r +Patch size (r × r) +gx, gy, gθ +Gradients w.r.t. x, y, θ +˜gy +Approximated gradient gy +˜x, ˜θ′ +Sum of x and θ′ over +spatial dimensions (height and width) +x[n, ci, h, w] +Element for feature map x +at batch n, channel ci, pixel (h, w) +θ[co, ci, u, v] +Element for convolution kernel θ +at output channel co, input channel ci, +position (u, v) +Table 6. Table of symbols we use. +Thus, for any entry in the approximated gradient ˜gy, the +value equals to the average of all neighboring elements +within the same r × r patch, as shown in the Figure 2 in the +main manuscript. For the approximated gradient ˜gy with +batch size n, channel c, resolution (Hy, Wy), there will be +(n × c × ⌈ Hy +r ⌉ × ⌈ Wy +r ⌉) unique numbers in ˜gy. To simplify +the following derivations, we rewrite the approximated gra- +dient ˜gy as follows: +˜gp +y[n, co, hp, wp, i, j] = ˜gy[n, co, hp∗r+i, wp∗r+j] (18) +where (hp, wp) is the position of the patch and (i, j) is the +offset within the patch. Since every element in the same +patch has the exact same value, we denote this unique value +with ˜gu +y , i.e., +˜gu +y [n, co, hp, wp] = ˜gp +y[n, co, hp, wp, i, j], ∀0 ≤ i, j < r +(19) +A.2. Approximation for Rotated Convolution Ker- +nel θ′ +˜θ′[co, ci] = +� +u,v +θ′[co, ci, u, v] += +� +u,v +rot180(θ)[co, ci, u, v] += +� +u,v +θ[co, ci, u, v] +(20) +A.3. Approximation for Input Feature x +˜x[n, ci, h, w] = +⌈i/r⌉r +� +h=⌊i/r⌋r +⌈j/r⌉r +� +w=⌊j/r⌋r +x[n, ci, i, j] +(21) +Thus for every entry in approximated feature map ˜x, the +value equal to the sum of all neighboring elements within +the same r × r patch. Following the definition of the gra- +dient filter in Section A.1, we use the following symbols to +simplify the derivation: +˜xp[n, ci, hp, wp, i, j] = ˜x[n, ci, hp ∗r +i, wp ∗r +j] (22) +and +˜xu[n, ci, hp, wp] = ˜xp[n, ci, hp, wp, i, j], ∀0 ≤ i, j < r +(23) +A.4. Boundary Elements +As mentioned in Section 3, given the structure created +by the gradient filters, the gradient propagation in a con- +volution layer can be simplified to weights summation and +multiplication with few unique gradient values. This is true +11 + +for all elements far away from the patch boundary because +for these elements, the rotated kernel θ′ only covers the ele- +ments from the same patch, which have the same value, thus +the computation can be saved. However, for the elements +close to the boundary, this is not true, since when convolv- +ing with boundary gradient elements, the kernel may cover +multiple patches with multiple unique values instead of just +one. To eliminate the extra computation introduced by the +boundary elements, we pad each patch sufficiently such that +every element is far away from boundary: +˜gp +y[n, ci, hp, wp, i, j] = ˜gu +y [n, ci, hp, wp], ∀i, j ∈ Z +(24) +For example, with the patch size 4 × 4, the element at the +spatial position (3, 3) is on the boundary, so when we calcu- +late ˜gx[n, ci, 3, 3] by convolving the rotated kernel θ′ with +the approximated gradient ˜gy: +˜gx[n, ci, 3, 3] = +� +i,j +θ′[co, ci, i, j]˜gy[n, co, 3+i, 3+j] (25) +values of ˜gy are from multiple patches and have differ- +ent values (e.g., ˜gy[n, co, 3, 3] is from patch (0, 0) while +˜gy[n, co, 4, 4] is from patch (1, 1); they have different val- +ues). +In our method, we simplify the Equation (25) by +rewriting it in the following way: +˜gx[n, ci, 3, 3] +≈ +1 +� +i,j=−1 +θ′[co, ci, i, j]˜gp +y[n, co, ⌊3 +4⌋, ⌊3 +4⌋, 3 + i, 3 + j] +(26) += +1 +� +i,j=−1 +θ′[co, ci, i, j]˜gu +y [n, co, ⌊3 +4⌋, ⌊3 +4⌋] +(27) += +1 +� +i,j=−1 +θ′[co, ci, i, j]˜gu +y [n, co, 0, 0] +(28) +where Equation (26) is derived from Equation (25) by con- +sidering that patch (0, 0) is sufficiently padded so that for +elements with all offsets (3 + i, 3 + j), they have the same +value, which is the unique value gu +y [n, co, 0, 0]. +For approximated input feature map ˜x, we apply the +same approximation for the boundary elements. +A.5. Gradient w.r.t. Input (Equation (3) in the Pa- +per) +˜gx[n, ci, h, w] +(29) += +� +co,u,v +θ[co, ci, −u, −v]˜gy[n, co, h + u, w + v] +(30) +≈ +� +co,u,v +θ[co, ci, −u, −v]· +˜gp +y[n, co, ⌊h +r ⌋, ⌊w +r ⌋, (h mod r) + u, (w mod r) + v] +(31) += +� +co,u,v +θ[co, ci, −u, −v]˜gu +y [n, co, ⌊h +r ⌋, ⌊w +r ⌋] +(32) += +� +co +˜gu +y [n, co, ⌊h +r ⌋, ⌊w +r ⌋] +� +u,v +θ[co, ci, −u, −v] +(33) += +� +co +˜gu +y [n, co, ⌊h +r ⌋, ⌊w +r ⌋]˜θ′[co, ci] +(34) +By expanding ˜gu +y to ˜gy, we have: +˜gx[n, ci, h, w] = +� +co +˜gy[n, co, h, w] ⊙ ˜θ′[co, ci] +(35) +which is the Equation (3) in Section 3 in the paper. +From Equation (30) to Equation (32), we consider that +the patch in the approximated gradient ˜gy is padded suffi- +ciently so they have the same value for all possible offsets +((h mod r) + u, (w mod r) + v). If there is only one in- +put channel and output channel for the convolutional layer +as the Figure 2 in the paper shows, then Equation (34) +become an element-wise multiplication, which is Equa- +tion (35) (also the Equation (3) in the paper). +A.6. Gradient w.r.t. Convolution Kernel (Equation +(5) in the Paper) +˜gθ[co, ci, u, v] +(36) += +� +n,h,w +x[n, ci, h + u, w + v]˜gy[n, co, h, w] +(37) +≈ +� +n,h,w +˜xp[n, ci, ⌊h +r ⌋, ⌊w +r ⌋, (h mod r) + u, (w mod r) + v]· +˜gu +y [n, co, ⌊h +r ⌋, ⌊w +r ⌋] +(38) += +� +n,h,w +˜xu[n, ci, ⌊h +r ⌋, ⌊w +r ⌋]˜gu +y [n, co, ⌊h +r ⌋, ⌊w +r ⌋] +(39) += +� +n,h,w +˜xu[n, ci, ⌊h +r ⌋, ⌊w +r ⌋]˜gu +y [n, co, ⌊h +r ⌋, ⌊w +r ⌋] +(40) +12 + +By expanding ˜xu and ˜gu +y to ˜x and ˜gy, respectively, we have: +˜gθ[co, ci, u, v] = +� +n,i,j +˜x[n, ci, i, j]˜gy[n, co, i, j] +(41) +which is precisely Equation (5) in Section 3. +From Equation (37) to Equation (39), we consider that +the patch in the approximated input feature map ˜x is padded +sufficiently thus they have the same value for all possible +offsets ((h mod r) + u, (w mod r) + v). For every given +input/output channel pair (co, ci), Equation (40) represents +the Frobenius inner product between ˜xu and ˜gu +y . +B. Detailed Proof for Proposition 1 +In this section, we provide more details to the proof in +Section 4.1. We use Gx, Gy and Θ to denote the gradients +gx, gy and the convolution kernel θ in the frequency domain, +respectively. Gx[u, v] is the spectrum value at frequency +(u, v) and δ is the 2D discrete Dirichlet function. Without +losing generality and to simplify the proof, we consider the +batch size is 1, the number of input/output channels is 1, +namely Cx = Cy = 1, and there is only one patch in ˜gy. +The gradient returned by the gradient filtering can be +written as: +˜gy = 1 +r2 1r×r ⊛ gy +(42) +where ⊛ denotes convolution. +By applying the discrete +Fourier transformation, Equation (42) can be rewritten in +the frequency domain as: +˜Gy[u, v] = 1 +r2 δ[u, v]Gy[u, v] +(43) +˜gy is the approximation for gy(so the ground truth for ˜gy is +gy), and the SNR of ˜gy equals to: +SNR˜gy = ( +� +(u,v)(Gy[u, v], − ˜Gy[u, v])2 +� +(u,v) G2y[u, v] +)−1 += ( +� +(u,v)(Gy[u, v] − 1 +r2 δ[u, v]Gy[u, v])2 +� +(u,v) G2y[u, v] +)−1 +(44) +where the numerator can be written as: +� +(u,v) +(Gy[u, v] − 1 +r2 δ[u, v]Gy[u, v])2 += +� +(u,v)̸=(0,0) +(Gy[u, v] − 1 +r2 δ[u, v]Gy[u, v])2 ++ (Gy[0, 0] − 1 +r2 δ[0, 0]Gy[0, 0])2 +(45) +Because δ[u, v] = +� +1 +(u, v) = (0, 0) +0 +(u, v) ̸= (0, 0), Equation (45) can +be written as: +� +(u,v)̸=(0,0) +G2 +y[u, v] + (r2 − 1)2 +r4 +G2 +y[0, 0] += +� +(u,v)̸=(0,0) +G2 +y[u, v] + G2 +y[0, 0] − G2 +y[0, 0] ++ (r2 − 1)2 +r4 +G2 +y[0, 0] += +� +(u,v) +G2 +y[u, v] − 2r2 − 1 +r4 +G2 +y[0, 0] +(46) +By substituting the numerator in Equation (44) with Equa- +tion (46), we have: +SNR˜gy = ( +� +(u,v) G2 +y[u, v] − 2r2−1 +r4 +G2 +y[0, 0] +� +(u,v) G2y[u, v] +)−1 += (1 − 2r2 − 1 +r4 +G2 +y[0, 0] +� +(u,v) G2y[u, v])−1 += (1 − 2r2 − 1 +r4 +Energy of DC Component in Gy +Total Energy4in Gy +)−1 +(47) +For the convolution layer, the gradient w.r.t. approximated +variable ˜x in the frequency domain is: +˜Gx[u, v] = Θ[−u, −v] ˜Gy[u, v] += 1 +r2 Θ[−u, −v]δ[u, v]Gy[u, v] +(48) +and its ground truth is: +Gx[u, v] = Θ[−u, −v]Gy[u, v] +(49) +Similar to Equation (47), the SNR of g˜x is: +SNR˜gx = (1 − 2r2 − 1 +r4 +Θ2[0, 0]G2 +y[0, 0] +� +(u,v) Θ2[u, v]G2y[u, v])−1 += (1 − 2r2 − 1 +r4 +G2 +x[0, 0] +� +(u,v) G2x[u, v])−1 += (1 − 2r2 − 1 +r4 +Energy of DC Component in Gx +Total Energy5in Gx +)−1 +(50) +Equation (50) can be rewritten as: +r4(1 − SNR−1 +˜gx ) +2r2 − 1 += +(Θ[0, 0]Gy[0, 0])2 +� +(u,v)(Θ[−u, −v]Gy[u, v])2 += +G2 +y[0, 0] +� +(u,v)( Θ[−u,−v] +Θ[0,0] Gy[u, v])2 +(51) +4As reminder, the total energy of a signal is the sum of energy in DC +component and energy in AC components. +13 + +Besides, the proposition’s assumption (the DC component +dominates the frequency spectrum of Θ) can be written as: +Θ2[0, 0] +max(u,v)̸=(0,0)Θ2[u, v] ≥ 1 +(52) +which is: +∀(u, v), Θ2[−u, −v] +Θ2[0, 0] +≤ 1 +(53) +thus, by combining Equation (51) and Equation (53), we +have: +r4(1 − SNR−1 +˜gx ) +2r2 − 1 += +G2 +y[0, 0] +� +(u,v)( Θ[−u,−v] +Θ[0,0] Gy[u, v])2 +≥ +G2 +y[0, 0] +� +(u,v)(Gy[u, v])2 += +r4(1 − SNR−1 +˜gy ) +2r2 − 1 +(54) +which means that: SNR˜gx ≥ SNR˜gy. This completes our +proof for error analysis.■ +In conclusion, as the gradient propagates, the noise in- +troduced by the gradient filter becomes weaker and weaker +compared to the real gradient signal. This property ensures +that the error in gradient has only a limited influence on the +quality of BP. +This proof can be extended to the more general case +where batch size and the number of channels are greater +than 1 by introducing more dimensions (i.e., batch dimen- +sion, channel dimension) into all equations listed above. +C. Computation Analysis for ResNet18 +In this section, we provide two more examples for com- +putation analysis in Section 4.2. Figure 7 shows the com- +putation required by the convolution layers from ResNet18 +with different patch sizes for gradient filtering. With re- +duced unique elements, our approach reduces the num- +ber of computations to 1/r2 of standard BP method; with +structured gradient, our approach further reduces the num- +ber of computations to about 1/(r2HθWθ) of standard BP +method. +D. Detailed Experimental Setup +In this supplementary section, we extend the experimen- +tal setup in Section 5.1. +D.1. ImageNet Classification +D.1.1 +Environment +ImageNet related experiments are conducted on IBM Power +System AC922, which is equipped with a 40-core IBM +Power 9 CPU, 256 GB DRAM and 4 NVIDIA Tesla V100 +16GB GPUs. We use PyTorch 1.9.0 compiled with CUDA +10.1 as the deep learning framework. +1 × 1 +3 × 3 +5 × 5 +7 × 7 +Patch Size r × r +1M +10M +100M +FLOPs +Baseline +Reduced +Unique +Elements ++Structured +Gradient +Actual +Minimum +Achievable Computation +(a) Last convolutional layer in block 4 of ResNet18 with 512 input/output +channels; the resolution of input feature map is 7 × 7. +1 × 1 +4 × 4 +8 × 8 +12 × 12 +Patch Size r × r +100K +1M +10M +100M +FLOPs +Baseline +Reduced +Unique +Elements ++Structured +Gradient +Actual +Minimum +Achievable Computation +(b) Last convolutional layer in block 3 of ResNet18 with 256 input/output +channels; the resolution of input feature map is 14 × 14. +1 × 1 +10 × 10 +20 × 20 +Patch Size r × r +100K +1M +10M +100M +FLOPs +Baseline +Reduced +Unique +Elements ++Structured +Gradient +Actual +Minimum +Achievable Computation +(c) Last convolutional layer in block 2 of ResNet18 with 128 input/output +channels; the resolution of input feature map is 28 × 28. +Figure 7. Computation analysis for three convolution layers in +of ResNet18 model. Since convolutional layers in every block +of ResNet18 is similar, we use the last convolutional layer as the +representative of all convolutional layers in the block. Minimum +achievable computation is presented in Equation (16) in the pa- +per. By reducing the number of unique elements, computations +required by our approach drop to about 1/r2 compared with the +standard BP method. By combining it (“+” in the figure) with +structured gradient map, computations required by our approach +drop further. +D.1.2 +Dataset Split +We split the dataset into two non-i.i.d. partitions following +the FedAvg method [25]. The label distribution is shown +in Figure 8. +Among all 1000 classes for the ImageNet, +pretrain and finetune partitions overlap on only 99 classes, +which suggests that our method can efficiently adapt the +14 + +Model +Accuracy +Model +Accuracy +ResNet-18 +73.5% +MobileNet-V2 +74.3% +ResNet-34 +76.4% +MCUNet +71.4% +Table 7. Model pretraining accuracy on ImageNet. +DNN model to data collected from new environments. For +each partition, we randomly select 80% data as training data +and 20% as validation data. +0 +200 +400 +600 +800 +1000 +Class Index +0 +200 +400 +600 +800 +1000 +Image Count +ImageNet Data Split +Pretrain +Finetune +Figure 8. Label distribution for pretraining and finetuning datasets. +Pretraining and finetuning partitions are split from ImageNet +dataset. +D.1.3 +Pretraining +We pretrain ResNet 18, ResNet 34, MobileNet-V2 and +MCUNet with the same configuration. We use SGD opti- +mizer. The learning rate of the optimizer starts at 0.05 and +decays according to cosine annealing method [23] during +training. Additionally, weight decay is set to 1 × 10−4 and +momentum is set to 0.9. We set batch size to 64. We ran- +domly resize, randomly flip and normalize the image for +data augmentation. We use cross entropy as loss function. +Models are trained for 200 epochs and the model with the +highest validation accuracy is kept for finetuning. Table 7 +shows the pretrain accuracy. +D.1.4 +Finetuning +We adopt the hyper-parameter (e.g., momentum, weight de- +cay, etc.) from pretraining. Several changes are made: mod- +els are finetuned for 90 epochs instead of 200; we apply +L2 gradient clipping with threshold 2.0; linear learning rate +warm-up for 4 epochs is introduced at the beginning of fine- +tuning, i.e., for the first 4 epochs, the learning rate grows +linearly up to 0.05, then the learning rate decays accord- +ing to cosine annealing method in the following epochs. Of +note, to ensure a fair comparison, we use the same hyper- +parameters for all experiments, regardless of model type +and training strategy. +D.2. CIFAR Classification +D.2.1 +Environment +CIFAR related experiments are conducted on a GPU work- +station with a 64-core AMD Ryzen Threadripper PRO +3995WX CPU, 512 GB DRAM and 4 NVIDIA RTX A6000 +GPUs. We use PyTorch 1.12.0 compiled with CUDA 11.6 +as the deep learning framework. +D.2.2 +Dataset Split +We split the dataset into two non-i.i.d. partitions following +FedAvg method. The label distribution is shown in Figure +9. For CIFAR10, pretrain and finetune partitions overlap +on 2 classes out of 10 classes in total. For CIFAR100, pre- +train and finetune partitions overlap on 6 classes out of 100 +classes. +0 +2 +4 +6 +8 +Class Index +0 +1000 +2000 +3000 +4000 +Image Count +CIFAR10 Data Split +Pretrain +Finetune +0 +20 +40 +60 +80 +100 +Class Index +0 +100 +200 +300 +400 +Image Count +CIFAR100 Data Split +Pretrain +Finetune +Figure 9. Label distribution for pretraining and finetuning datasets +on CIFAR10 and CIFAR100. Pretraining and finetuning partitions +are split from CIFAR10/100, respectively. +D.2.3 +Pretraining +We pretrain ResNet18 and ResNet34 with the same config- +uration. We use the ADAM optimizer with a learning rate +of 3 × 10−4 and weight decay 1 × 10−4 with no learning +rate scheduling method. We use cross entropy as loss func- +tion. We set batch size to 128, and normalize the data be- +fore feeding it to the model. Models are trained for 30 and +50 epochs for CIFAR10 and CIFAR100, respectively. Then, +the model with the highest accuracy is kept for finetuning. +Table 8 shows the pretrain accuracy. +D.2.4 +Finetuning +We adopt the training configuration from PSQ [7] with +some changes. We use cross entropy loss with SGD opti- +15 + +ResNet18 +ResNet34 +CIFAR10 +95.1% +97.6% +CIFAR100 +75.5% +83.5% +Table 8. Model pretraining accuracy on CIFAR10/100. +mizer for training. The learning rate of the optimizer starts +at 0.05 and decays according to cosine annealing method +during training. Momentum is set to 0 and weight decay +is set to 1 × 10−4. We apply L2 gradient clipping with +a threshold 2.0. Batch normalization layers are fused with +convolution layers before training, which is a common tech- +nique for inference acceleration. +D.3. Semantic Segmentation +D.3.1 +Environment +ImageNet related experiments are conducted on IBM Power +System AC922, which is equipped with a 40-core IBM +Power 9 CPU, 256 GB DRAM and 4 NVIDIA Tesla V100 +16GB GPUs. We use PyTorch 1.9.0 compiled with CUDA +10.1 as the deep learning framework. We implement our +method based on MMSegmentation 0.27.0. +D.3.2 +Pretraining +We use models pretrained by MMSegmentation. Consider- +ing that the numbers of classes, image statistics, and model +hyper-parameters may be different when applying on dif- +ferent datasets, we calibrate the model before finetuning. +We use SGD optimizer. The learning rate of the optimizer +starts at 0.01 and decays exponentially during training. Ad- +ditionally, weight decay is set to 5 × 10−4 and momentum +is set to 0.9. We set batch size to 8. We randomly crop, flip +and photo-metric distort and normalize the image for data +augmentation. We use cross entropy as loss function. For +DeepLabV3, FCN, PSPNet and UPerNet, we calibrate the +classifier (i.e., the last layer) and statistics in batch normal- +ization layers for 1000 steps on the finetuning dataset. For +DeepLabV3-MobileNetV2 and PSPNet-MobileNetV2, be- +cause the number of channels for convolutional layers in +the decoder are different for models applied on different +datasets, we calibrate the decoder and statistics in batch nor- +malization layers for 5000 steps on the finetuning dataset. +D.3.3 +Finetuning +We finetune all models with the same configuration. We use +the SGD optimizer. The learning rate of the optimizer starts +at 0.01 and decays according to cosine anneling method dur- +ing training. Additionally, weight decay is set to 5 × 10−4 +and momentum is set to 0.9. We set batch size to 8. We +randomly crop, flip and photo-metric distort and normalize +the image for data augmentation. We use cross entropy as +loss function. Models are finetuned for 20000 steps. Exper- +iments are repeated three times with random seed 233, 234 +and 235. +D.4. On-device Performance Evaluation +D.4.1 +NVIDIA Jetson Nano +We use NVIDIA Jetson Nano with quad-core Cortex-A57, +4 GB DRAM, 128-core Maxwell edge GPU for perfor- +mance evaluation on both edge CPU and edge GPU. We +use the aarch64-OS Ubuntu 18.04.6 provided by NVIDIA. +During evaluation, the frequencies for CPU and GPU are +1.5 GHz and 921 MHz, respectively. Our code and library +MKLDNN (a.k.a. OneDNN) are compiled on Jetson Nano +with GCC 7.5.0, while libraries CUDA and CUDNN are +compiled by NVIDIA. For CPU evaluations, our code and +baseline are implemented with MKLDNN v2.6. For GPU +evaluations, our code and baseline are implemented with +CUDA 10.2 and CUDNN 8.2.1. +Before the evaluation for every test case, we warm up +the device by running the test once. Then we repeat the test +10 times and report the average value for latency, energy +consumption, etc. +Energy consumption is obtained by reading the embed- +ded power meter in Jetson Nano every 20 ms. +D.4.2 +Raspberry Pi 3b +We use Raspberry Pi 3b with quad-core Cortex-A53, 1 +GB DRAM for performance evaluation on CPU. We use +the aarch64-OS Raspberry Pi OS. During evaluation, the +frequency for CPU is 1.2 GHz. +Our code and library +MKLDNN are compiled on Raspberry Pi with GCC 10.2. +Our code and baseline are implemented with MKLDNN +v2.6. +Before the evaluation for every test case, we warm up the +device by running the test once. Then we repeat the test 10 +times and report the average value for latency, etc. +D.4.3 +Desktop +We use a desktop PC with Intel 11900KF CPU, 32 GB +DRAM and RTX 3090 Ti GPU for perforamce evaluation +on both desktop CPU and desktop GPU. We use x86 64- +OS Ubuntu 20.04. During evaluation, the frequencies for +CPU and GPU are 4.7 GHz and 2.0 GHz respectively. Our +code is compiled with GCC 9.4.0. MKLDNN is compiled +by Anaconda (tag omp h13be974 0). CUDA and CUDNN +are compiled by NVIDIA. For CPU evaluations, our code +and baseline are implemented with MKLDNN v2.6. For +GPU evaluations, our code and baseline are implemented +with CUDA 11.7 and CUDNN 8.2.1. +16 + +Pretrain: ADE20K Finetune: VOC12Aug +UPerNet +#Layers +GFLOPs +mIoU +mAcc +PSPNet-M +#Layers +GFLOPs +mIoU +mAcc +DLV3-M +#Layers +GFLOPs +mIoU +mAcc +Calibration +0 +0 +37.66 +50.03 +Calibration +0 +0 +30.93 +52.01 +Calibration +0 +0 +35.28 +56.98 +Vanilla BP +All +541.0 +67.23[0.24] +79.79[0.45] +Vanilla BP +All +42.41 +53.51[0.27] +67.01[0.19] +Vanilla BP +All +54.35 +60.78[0.21] +74.10[0.40] +5 +503.9 +72.01[0.09] +81.97[0.30] +5 +12.22 +48.88[0.11] +62.67[0.31] +5 +14.77 +51.51[0.09] +66.08[0.44] +10 +507.6 +72.01[0.19] +81.83[0.44] +10 +22.46 +53.71[0.29] +67.93[0.32] +10 +33.10 +57.63[0.10] +71.93[0.41] +Ours +5 +1.97 +71.76[0.11] +81.57[0.07] +Ours +5 +0.11 +48.59[0.08] +62.28[0.30] +Ours +5 +0.26 +49.40[0.00] +64.13[0.54] +10 +2.22 +71.78[0.23] +81.55[0.38] +10 +0.76 +52.77[0.37] +66.82[0.47] +10 +1.40 +55.14[0.15] +69.48[0.26] +Pretrain: ADE20K Finetune: Cityscapes +UPerNet +#Layers +GFLOPs +mIoU +mAcc +PSPNet-M +#Layers +GFLOPs +mIoU +mAcc +DLV3-M +#Layers +GFLOPs +mIoU +mAcc +Calibration +0 +0 +34.15 +42.45 +Calibration +0 +0 +28.83 +34.85 +Calibration +0 +0 +41.33 +48.65 +Vanilla BP +All +1082.1 +73.02[0.14] +81.01[0.20] +Vanilla BP +All +84.82 +60.21[0.40] +67.72[0.68] +Vanilla BP +All +108.7 +71.12[0.14] +79.81[0.04] +5 +1007.7 +62.46[0.19] +72.62[0.27] +5 +24.43 +42.09[0.43] +48.70[0.49] +5 +29.5 +51.00[0.05] +59.20[0.03] +10 +1015.3 +64.01[0.21] +73.11[0.32] +10 +44.90 +54.03[0.24] +61.48[0.10] +10 +66.2 +61.02[0.14] +69.80[0.06] +Ours +5 +3.94 +60.58[0.25] +70.67[0.32] +Ours +5 +0.22 +41.59[0.38] +48.10[0.41] +Ours +5 +0.50 +48.83[0.07] +56.87[0.08] +10 +4.43 +62.14[0.24] +71.41[0.27] +10 +1.51 +49.10[0.49] +56.93[1.43] +10 +2.74 +50.22[1.01] +59.99[0.31] +Table 9. +Experimental results for semantic segmentation task for UPerNet, DeepLabV3-MobileNetV2 (DLV3-M) and PSPNet- +MobileNetV2 (PSPNet-M). Models are pretrained on ADE20K dataset and finetuned on augmentated Pascal VOC12 dataset and Cityscapes +dataset respectively. “#Layers” is short for “the number of active convolutional layers” that are trained. Strategy “Calibration” shows the +accuracy when only the classifier and normalization statistics are updated to adapt differences (e.g. different number of classes) between +pretraining dataset and finetuning dataset. +No. +#Input Channel +#Output Channel +Input Width +Input Height +0 +128 +128 +120 +160 +1 +256 +256 +60 +80 +2 +512 +512 +30 +40 +3 +512 +512 +14 +14 +4 +256 +256 +14 +14 +5 +128 +128 +28 +28 +6 +64 +64 +56 +56 +Table 10. Layer configuration for test cases in Figure 6 in Section +5.5 in the paper. +Before the evaluation for every test case, we warm up the +device by running the 10 times. Then we repeat the test 200 +times and report the average value for latency, etc. +D.4.4 +Test Case Configurations +Table 10 lists the configurations for test cases shown in Fig- +ure 6 in the paper. In addition to the parameters shown in +the table, for all test cases, we set the batch size to 32, kernel +size to 3 × 3, padding and stride to 1. +E. More Results for Semantic Segmentation +In this section, we extend the experimental results shown +in Section 5.3 (Table 3). Table 9 shows the experimental re- +sults for UPerNet, PSPNet-MobileNetV2 (PSPNet-M) and +DeepLabV3-MobileNetV2 (DLV3-M) on two pairs of pre- +traing and finetuning datasets. These results further show +the effectiveness of our method on a dense prediction task. +F. More Results for CIFAR10/100 with Differ- +ent Hyper-Parameter Selections +In this section, we extend the experimental results shown +in Section 5.4 (Figure 4). Table 11 (page 18) shows the ex- +perimental results for ResNet18 and ResNet34 on CIFAR +datasets. For every model, we test our method with differ- +ent patch sizes for gradient filtering and different numbers +of active convolutional layers (#Layers in Table 11, e.g., if +#Layers equals to 2, the last two convolutional layers are +trained while other layers are frozen). These results further +support the qualitative findings in Section 5.4. +G. Results for Combining Gradient Filtering +with Gradient Quantization +In this section, we provide experimental results for com- +bining our method, i.e. +gradient filtering, with gradient +quantization. Table 12 (page 19) shows experimental re- +sults for ResNet18 and ResNet32 with gradient quantiza- +tion methods PTQ [4] and PSQ [7] and different hyper- +parameters. Both forward propagation and backward prop- +agation are quantized to INT8. These results support the +wide applicability of our method. +H. More Results for On-device Performance +Evaluation +In this section, we extend the experimental results shown +in Section 5.5. Figure 10 shows the energy savings and +overhead of our method. For most test cases with patch +4 × 4, we achieve over 80× energy savings with less than +20% overhead on both CPU and GPU. Moreover, for the +test case 1 on Raspberry Pi CPU, the forward propagation +is even faster when applied our method (which results in +negtive overheads). +These results further show that our +method is practical for the real deployment of both high- +performance and IoT applications. +17 + +CIFAR10 +CIFAR100 +ResNet18 +#Layers +ACC[%] +FLOPs +ResNet34 +#Layers +ACC[%] +FLOPs +ResNet18 +#Layers +ACC[%] +FLOPs +ResNet34 +#Layers +ACC[%] +FLOPs +Vanilla +BP +1 +91.7 +128.25M +Vanilla +BP +1 +94.2 +128.25M +Vanilla +BP +1 +73.8 +128.39M +Vanilla +BP +1 +76.9 +128.39M +2 +93.6 +487.68M +2 +96.6 +487.68M +2 +77.6 +487.82M +2 +82.0 +487.82M +3 +93.7 +847.15M +3 +96.6 +847.13M +3 +77.6 +847.29M +3 +82.1 +847.27M +4 +94.4 +1.14G +4 +96.8 +1.21G +4 +78.0 +1.14G +4 +83.0 +1.21G ++Gradient +Filter +R2 +1 +91.5 +8.18M ++Gradient +Filter +R2 +1 +94.2 +8.18M ++Gradient +Filter +R2 +1 +73.7 +8.31M ++Gradient +Filter +R2 +1 +77.0 +8.31M +2 +92.7 +26.80M +2 +96.6 +26.80M +2 +75.6 +26.94M +2 +81.1 +26.94M +3 +92.8 +45.45M +3 +96.5 +45.44M +3 +75.6 +45.59M +3 +81.1 +45.58M +4 +93.9 +60.01M +4 +96.6 +64.07M +4 +76.4 +60.15M +4 +82.0 +64.21M ++Gradient +Filter +R4 +1 +91.4 +1.88M ++Gradient +Filter +R4 +1 +94.3 +1.88M ++Gradient +Filter +R4 +1 +73.7 +2.02M ++Gradient +Filter +R4 +1 +76.9 +2.02M +2 +92.7 +7.93M +2 +96.4 +7.93M +2 +74.9 +8.07M +2 +80.4 +8.07M +3 +92.8 +13.99M +3 +96.4 +13.98M +3 +74.9 +14.12M +3 +80.4 +14.12M +4 +93.3 +19.12M +4 +96.1 +20.04M +4 +75.2 +19.26M +4 +80.5 +20.17M ++Gradient +Filter +R7 +1 +91.5 +303.10K ++Gradient +Filter +R7 +1 +94.2 +303.10K ++Gradient +Filter +R7 +1 +73.7 +441.34K ++Gradient +Filter +R7 +1 +76.9 +441.34K +2 +91.5 +3.21M +2 +95.8 +3.21M +2 +74.1 +3.35M +2 +80.4 +3.35M +3 +91.7 +6.12M +3 +96.0 +6.12M +3 +74.1 +6.26M +3 +80.3 +6.26M +4 +92.6 +8.90M +4 +96.0 +9.03M +4 +75.4 +9.04M +4 +80.3 +9.17M +Table 11. Experimental results on CIFAR10 and CIFAR100 datasets for ResNet18 and ResNet34 with different hyper-parameter selections. +“ACC” is short for accuracy. “#Layers” is short for “the number of active convolution layers”. For example. #Layers equals to 2 means that +only the last two convolutional layers are trained. “Gradient Filter R2/4/7” use proposed gradient filtering method with patch size 2 × 2, +4 × 4 and 7 × 7, respectively. +0 +1 +2 +3 +4 +5 +6 +0× +20× +40× +60× +80× +100× +120× +Energy Savings [×times] +CPU Energy Savings +Jetson-R2 +Jetson-R4 +0 +1 +2 +3 +4 +5 +6 +0× +20× +40× +60× +80× +100× +GPU Energy Savings +Jetson-R2 +Jetson-R4 +0 +1 +2 +3 +4 +5 +6 +Test Case - Baseline: MKLDNN +0 +25 +50 +75 +100 +Percentage [%] +Forward Cost +20% Overhead +Normalized CPU Overhead +Jetson-R2 +Jetson-R4 +11900KF-R2 +11900KF-R4 +RPi3-R2 +RPi3-R4 +0 +1 +2 +3 +4 +5 +6 +Test Case - Baseline: CUDNN +0 +20 +40 +60 +80 +100 +Forward Cost +20% Overhead +Normalized GPU Overhead +Jetson-R2 +Jetson-R4 +RTX3090Ti-R2 +RTX3090Ti-R4 +Figure 10. Energy savings and overhead resuls on multiple CPUs and GPUs under different test cases (i.e., different input sizes, number of +channels, etc..). For test case 4 and 5 with patch size 4 × 4 (Jetson-R4) on GPU, the latency of our method is too small to be captured by +the power meter with a 20 ms sample rate so the energy savings data is not available. For most test cases with patch size 4 × 4, our method +achieves over 80× energy savings with less than 20% overhead. +18 + +CIFAR10 +CIFAR100 +ResNet18 +ResNet34 +ResNet18 +ResNet34 +Strategy +#Layers +ACC[%] +#OPs +Strategy +#Layers +ACC[%] +#OPs +Strategy +#Layers +ACC[%] +#OPs +Strategy +#Layers +ACC[%] +#OPs +PTQ +1 +91.6 +128.25M +PTQ +1 +93.6 +128.25M +PTQ +1 +74.0 +128.39M +PTQ +1 +76.4 +128.39M +2 +93.2 +487.68M +2 +96.2 +487.68M +2 +77.8 +487.82M +2 +80.3 +487.82M +3 +93.5 +847.15M +3 +96.2 +847.13M +3 +77.9 +847.29M +3 +80.5 +847.27M +4 +94.4 +1.14G +4 +96.5 +1.21G +4 +77.9 +1.14G +4 +82.2 +1.21G +PTQ ++Gradient +Filter +R2 +1 +91.4 +8.18M +PTQ ++Gradient +Filter +R2 +1 +93.5 +8.18M +PTQ ++Gradient +Filter +R2 +1 +73.9 +8.31M +PTQ ++Gradient +Filter +R2 +1 +76.5 +8.31M +2 +92.6 +26.80M +2 +95.9 +26.80M +2 +75.7 +26.94M +2 +80.0 +26.94M +3 +92.7 +45.45M +3 +96.0 +45.44M +3 +75.9 +45.59M +3 +80.1 +45.58M +4 +93.7 +60.01M +4 +96.2 +64.07M +4 +76.3 +60.15M +4 +80.9 +64.21M +PTQ ++Gradient +Filter +R4 +1 +91.3 +1.88M +PTQ ++Gradient +Filter +R4 +1 +93.6 +1.88M +PTQ ++Gradient +Filter +R4 +1 +73.7 +2.02M +PTQ ++Gradient +Filter +R4 +1 +76.5 +2.02M +2 +92.5 +7.93M +2 +95.6 +7.93M +2 +75.1 +8.07M +2 +79.5 +8.07M +3 +92.7 +13.99M +3 +95.6 +13.98M +3 +75.4 +14.12M +3 +79.5 +14.12M +4 +93.4 +19.12M +4 +95.6 +20.04M +4 +76.1 +19.26M +4 +80.5 +20.17M +PTQ ++Gradient +Filter +R7 +1 +91.2 +303.10K +PTQ ++Gradient +Filter +R7 +1 +93.6 +303.10K +PTQ ++Gradient +Filter +R7 +1 +73.7 +441.34K +PTQ ++Gradient +Filter +R7 +1 +76.5 +441.34K +2 +91.5 +3.21M +2 +95.5 +3.21M +2 +74.5 +3.35M +2 +79.4 +3.35M +3 +91.6 +6.12M +3 +95.4 +6.12M +3 +74.5 +6.26M +3 +79.5 +6.26M +4 +92.6 +8.90M +4 +95.5 +9.03M +4 +75.3 +9.04M +4 +79.6 +9.17M +PSQ +1 +91.4 +128.25M +PSQ +1 +93.6 +128.25M +PSQ +1 +73.9 +128.39M +PSQ +1 +76.4 +128.39M +2 +93.3 +487.68M +2 +96.1 +487.68M +2 +77.7 +487.82M +2 +80.3 +487.82M +3 +93.4 +847.15M +3 +96.2 +847.13M +3 +77.9 +847.29M +3 +80.5 +847.27M +4 +94.5 +1.14G +4 +96.4 +1.21G +4 +78.0 +1.14G +4 +82.2 +1.21G +PSQ ++Gradient +Filter +R2 +1 +91.3 +8.18M +PSQ ++Gradient +Filter +R2 +1 +93.5 +8.18M +PSQ ++Gradient +Filter +R2 +1 +73.8 +8.31M +PSQ ++Gradient +Filter +R2 +1 +76.4 +8.31M +2 +92.6 +26.80M +2 +96.0 +26.80M +2 +76.0 +26.94M +2 +80.1 +26.94M +3 +92.8 +45.45M +3 +96.1 +45.44M +3 +75.9 +45.59M +3 +80.0 +45.58M +4 +93.7 +60.01M +4 +96.1 +64.07M +4 +76.3 +60.15M +4 +80.9 +64.21M +PSQ ++Gradient +Filter +R4 +1 +91.4 +1.88M +PSQ ++Gradient +Filter +R4 +1 +93.6 +1.88M +PSQ ++Gradient +Filter +R4 +1 +73.5 +2.02M +PSQ ++Gradient +Filter +R4 +1 +76.5 +2.02M +2 +92.6 +7.93M +2 +95.6 +7.93M +2 +75.3 +8.07M +2 +79.5 +8.07M +3 +92.7 +13.99M +3 +95.6 +13.98M +3 +75.1 +14.12M +3 +79.6 +14.12M +4 +93.2 +19.12M +4 +95.5 +20.04M +4 +76.2 +19.26M +4 +80.2 +20.17M +PSQ ++Gradient +Filter +R7 +1 +91.2 +303.10K +PSQ ++Gradient +Filter +R7 +1 +93.6 +303.10K +PSQ ++Gradient +Filter +R7 +1 +73.5 +441.34K +PSQ ++Gradient +Filter +R7 +1 +76.5 +441.34K +2 +91.4 +3.21M +2 +95.5 +3.21M +2 +74.4 +3.35M +2 +79.5 +3.35M +3 +91.6 +6.12M +3 +95.4 +6.12M +3 +74.5 +6.26M +3 +79.6 +6.26M +4 +92.7 +8.90M +4 +95.5 +9.03M +4 +75.5 +9.04M +4 +79.6 +9.17M +Table 12. Experimental results for ResNet18 and ResNet34 with different gradient quantization methods (i.e., PTQ [4] and PSQ [7]) and +hyper-parameter selections on CIFAR10/100. Feature map, activation, weight and gradient are quantized to INT8. “ACC” is short for +accuracy. “#Layers” is short for “the number of active convolution layers”. For example. #Layers equals to 2 means that the last two +convolutional layers are trained. “Gradient Filter R2/4/7” use proposed gradient filtering method with patch size 2 × 2, 4 × 4 and 7 × 7, +respectively. +19 + diff --git a/FNAyT4oBgHgl3EQfe_gi/content/tmp_files/load_file.txt b/FNAyT4oBgHgl3EQfe_gi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..111f5e32a9a14849703506e82b5d8c2e7736ad29 --- /dev/null +++ b/FNAyT4oBgHgl3EQfe_gi/content/tmp_files/load_file.txt @@ -0,0 +1,2242 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf,len=2241 +page_content='Efficient On-device Training via Gradient Filtering Yuedong Yang Guihong Li Radu Marculescu The University of Texas at Austin {albertyoung, lgh, radum}@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='edu Abstract Despite its importance for federated learning, continu- ous learning and many other applications, on-device train- ing remains an open problem for EdgeAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The problem stems from the large number of operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', floating point multiplications and additions) and memory consump- tion required during training by the back-propagation al- gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Consequently, in this paper, we propose a new gradient filtering approach which enables on-device DNN model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More precisely, our approach creates a special structure with fewer unique elements in the gradi- ent map, thus significantly reducing the computational com- plexity and memory consumption of back propagation dur- ing training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Extensive experiments on image classification and semantic segmentation with multiple DNN models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', MobileNet, DeepLabV3, UPerNet) and devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', Rasp- berry Pi and Jetson Nano) demonstrate the effectiveness and wide applicability of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For example, com- pared to SOTA, we achieve up to 19× speedup and 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1% memory savings on ImageNet classification with only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1% accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Finally, our method is easy to implement and deploy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' over 20× speedup and 90% energy savings have been observed compared to highly optimized baselines in MKLDNN and CUDNN on NVIDIA Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Conse- quently, our approach opens up a new direction of research with a huge potential for on-device training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Introduction Existing approaches for on-device training are neither efficient nor practical enough to satisfy the resource con- straints of edge devices (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This is because these methods do not properly address a fundamental problem in on-device training, namely the computational and memory complexity of the back-propagation (BP) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More precisely, although the architecture modification [6] and layer freezing [19, 21] can help skipping the BP for some layers, for other layers, the complexity remains high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gra- dient quantization [4, 7] can reduce the cost of arithmetic operations but cannot reduce the number of operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', multiplications);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' thus, the speedup in training remains lim- ited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Moreover, gradient quantization is not supported by existing deep-learning frameworks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', CUDNN [9], MKLDNN [1], PyTorch [26] and Tensorflow [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' To en- able on-device training, there are two important questions must be addressed: How can we reduce the computational complexity of back propagation through the convolution layers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' How can we reduce the data required by the gradient computation during back propagation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In this paper, we propose gradient filtering, a new research direction, to address both questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' By addressing the first question, we reduce the computational complexity of train- ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' by addressing the second question, we reduce the mem- ory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In general, the gradient propagation through a convolu- tion layer involves multiplying the gradient of the output variable with a Jacobian matrix constructed with data from either the input feature map or the convolution kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We aim at simplifying this process with the new gradient filter- ing approach proposed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Intuitively, if the gradi- ent map w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the output has the same value for all entries, then the computation-intensive matrix multiplication can be greatly simplified, and the data required to construct the Ja- cobian matrix can be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Thus, our gra- dient filtering can approximate the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the output by creating a new gradient map with a special (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', spatial) structure and fewer unique elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' By doing so, the gra- dient propagation through the convolution layers reduces to cheaper operations, while the data required (hence memory) for the forward propagation also lessens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Through this fil- tering process, we trade off the gradient precision against the computation complexity during BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We note that gradi- ent filtering does not necessarily lead to a worse precision, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', models sometimes perform better with filtered gradi- ents when compared against models trained with vanilla BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In summary, our contributions are as follows: We propose gradient filtering, which reduces the com- putation and memory required for BP by more than 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='00330v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='CV] 1 Jan 2023 two orders of magnitude compared to the exact gradi- ent calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We provide a rigorous error analysis which shows that the errors introduced by the gradient filtering have only a limited influence on model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our experiments with multiple DNN models and com- puter vision tasks show that we can train a neural net- work with significantly less computation and memory costs, with only a marginal accuracy loss compared to baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Side-by-side comparisons against other training acceleration techniques also suggest the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our method is easy to deploy with highly optimized deep learning frameworks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', MKLDNN [1] and CUDNN [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Evaluations on resource-constrained edge (Raspberry Pi and Jetson Nano) and high- performance devices (CPU/GPU) show that our method is highly suitable for real life deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Section 2 reviews rel- evant work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Section 3 presents our method in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Sec- tion 4 discusses error analysis, computation and memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Experimental results are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Finally, Section 6 summarizes our main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Related Work Architecture Modification: Authors of [6] propose to at- tach small branches to the original neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Dur- ing training, the attached branches and biases in the orig- inal model are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Though memory consumption is reduced, updating these branches still needs gradient prop- agation through the entire network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' moreover, a large com- putational overhead for inference is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Layer Freezing: Authors of [19, 21] propose to only train parts of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' [19] makes layer selection based on layer importance metrics, while [21] uses evolutionary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' However, the layers selected by all these methods are typ- ically computationally heavy layers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', the last few lay- ers in ResNet [15]) which consume most of the resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Thus, the speedup achieved by these approaches is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gradient Quantization: [3,5] quantize gradient after back- propagation, which means these methods cannot accelerate the training on a single device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Work in [4, 7, 16, 18, 29, 30, 34] accelerates training by reducing the cost for every arithmetic operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' However, these methods do not re- duce the number of operations, which is typically huge for SOTA CNNs, so their achievable speedup is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Also, all these methods are not supported by the popular deep learning frameworks [1,2,9,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In contrast to the prior work, our method opens up a new research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More precisely, we reduce the number of Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Modification Example: [6] Drawbacks: Large overhead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Limited to specific model Layer/Channel Freezing Example: [19, 21] Drawbacks: High search cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Limited to simple models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gradient Quantization Example: [4, 7, 16] Drawbacks: Not supported by existing DL frameworks Gradient Filtering [Ours] Advantages: Very fast and accurate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Well supported by existing DL frameworks Efficiency Applicability Orthogonal Research Directions for On-device Training Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Matrix of orthogonal directions for on-device training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “Arch” is short for “architecture”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our approach opens up a new direction of research for on-device training for EdgeAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' computations and memory consumption required for train- ing a single layer via gradient filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Thus, our method can be combined with any of the methods mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For example, in Section G in the Supplementary, we illus- trate how our method can work together with the gradient quantization methods to enable a higher speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Proposed Method In this section, we introduce our gradient filtering ap- proach to accelerate BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' To this end, we target the most computation and memory heavy operation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', convolution (Figure 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Table 1 lists some symbols we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Cx Number of channels of x Wx, Hx Width and height of x θ Convolution kernel θ′ Rotated θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', θ′ = rot180(θ) r Patch size (r × r ) gx, gy, gθ Gradients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' x, y, θ ˜gy Approximated gradient gy ˜x, ˜θ′ Sum of x and θ′ over spatial dimensions (height and width) x[n, ci, h, w] Element for feature map x at batch n, channel ci, pixel (h, w) θ[co, ci, u, v] Element for convolution kernel θ at output channel co, input channel ci, position (u, v) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Table of symbols we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 2 Loss Loss Gradient Filter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 ������������𝜃 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 �������������𝜃 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01 ������������������������� Ⓕ ⊙ ������������ ������������������������ ������������𝜃 ⊛ ������������ ������������������������ Memory Ⓕ ������������ ������������������������� Memory ������������� Ⓕ ������������ ������������������������� ������������𝜃 �������������𝜃 ⊙ ������������ ������������ ������������������������� ������������������������ ⊛ ������������ ������������ ������������������������ ⊛ Vanilla Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Ⓐ (a) (b) Ⓐ Spatial Sum = ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ������������ × ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ������������ + ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ������������ × ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ������������ + ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ������������ × ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ������������ + −������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ������������ × (−������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ������������) Patch size ������������ × ������������ = 2 × 2 Forward Propagation Backward Propagation Average Filter Ⓐ ������������ Spatial Sum ⊛Convolution ⒻFrobenius Inner Product ⊙Element-wise Product ������������ ������������ ������������ ������������ Average Value Other Layers Other Layers Height Width Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' (a) Computation procedures for vanilla training method (upper) and our method (lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' (b) Example of gradient propagation with gradient filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Numbers in this example are chosen randomly for illustration purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In this case, the patch size selected for the gradient filter is 2 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Thus, the 4 × 4 gradient map gy is approximated by ˜gy, which has four 2 × 2 patches with one unique value for each patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Also, input feature map x and mirrored convolution kernel θ′ are spatial summed to ˜x and ˜θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Since ˜x has fewer unique values than x, memory consumption is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Finally, with ˜gy, ˜x and ˜θ, we compute the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' kernel and input feature map with much fewer operations than the standard back propagation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Problem Setup The computations for both forward and backward paths are shown in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For the standard (vanilla) ap- proach (upper Figure 2(a)), starting with input x, the for- ward propagation convolves the input feature map x with kernel θ and returns output y, which is further processed by the other layers in the neural network (dotted arrow) un- til the loss value l is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' As shown in Figure 2(a), the BP of the convolution layer starts with the gradient map w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' output y (gy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' input (gx) is calcu- lated by convolving gy with the rotated convolution kernel θ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', gx = gy ⊛ rot180(θ) = gy ⊛ θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' convolution kernel, namely gθ, is calculated with the Frobe- nius inner product [17] between x and gy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', gθ = gy F x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The lower half of Figure 2(a) shows our method, where several changes are made: We introduce the gradient filter “ A ” after gy to generate the approximate gradient for BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Also, instead of using the accurate x and θ′ values for gra- dient computation, we sum over spatial dimensions (height and width dimensions), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', ˜x and ˜θ′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Finally, the convolution layer now multiplies the approximate gra- dient ˜gy with spatial kernel ˜θ′ instead of convolving with it to calculate ˜gx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Figure 2(b) shows an example of gradient propagation with our gradient filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Preliminary Analysis Consider the vanilla BP for convolution in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Equation (1) shows the number of computations (#FLOPs) required to calculate gx given gy: #FLOPs = 2CxCy · WyHy · WθHθ (1) The computation requirements in Equation (1) belong to three categories: number of channels, number of unique el- ements per channel in the gradient map, and kernel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our method focuses on the last two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Unique elements: (WyHy) represents the number of unique elements per channel in the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' output variable y (gy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Given the high-resolution images we use, this term is huge, so if we manage to reduce the number of unique elements in the spatial dimensions (height and width), the computations required are greatly reduced too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Kernel size: (WθHθ) represents the number of unique elements in the convolution kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' If the gradient gy has some special structure, for example gy = 1Hy×Wy · v (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', every element in gy has the same value v), then the convolution can be simplified to (� θ′)v1Hy×Wy (with boundary elements ignored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' With such a special structure, only one multiplication and (WθHθ − 1) additions are re- quired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Moreover, � θ′ is independent of data so the result can be shared across multiple images until θ gets updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gradient Filtering To reduce the number of unique elements and create the special structure in the gradient map, we apply the gradi- ent filter after the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' output (gy) is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' During the backward propagation, the gradient filter A ap- proximates the gradient gy by spatially cutting the gradient map into r×r-pixel patches and then replacing all elements in each patch with their average value (Figure 2(b)): ˜gy[n, co, h, w] = 1 r2 ⌈h/r⌉r � i=⌊h/r⌋r ⌈w/r⌉r � j=⌊w/r⌋r gy[n, co, i, j] (2) 3 For instance in Figure 2(b), we replace the 16 distinct values in the gradient map gy with 4 average values in ˜gy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' So given a gradient map gy with N images per batch, C channels, and H × W pixels per channel, the gradient filter returns a structured approximation of the gradient map containing only N × C × ⌈ H r ⌉ × ⌈ W r ⌉ blocks, with one unique value per patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use this matrix of unique values to represent the approximate gradient map ˜gy, as shown in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Back Propagation with Gradient Filtering We describe now the computation procedure used after applying the gradient filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Detailed derivations are pro- vided in Supplementary Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' input: The gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' input is cal- culated by convolving θ′ with gy (Figure 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' With the approximate gradient ˜gy, this convolution simplifies to: ˜gx[n, ci, h, w] = � co ˜gy[n, co, h, w] ⊙ ˜θ′[co, ci] (3) where ˜θ′[co, ci] = � u,v θ′[co, ci, u, v] is the spatial sum of convolution kernel θ, as shown in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' kernel: The gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the kernel is calculated by taking the Frobenius inner product between x and gy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', gθ[co, ci, u, v] = x F gy, namely: gθ[co, ci, u, v] = � n,i,j x[n, ci, i+u, j +v]gy[n, co, i, j] (4) With the approximate gradient ˜gy, the operation can be sim- plified to: ˜gθ[co, ci, u, v] = � n,i,j ˜x[n, ci, i, j]˜gy[n, co, i, j] (5) with ˜x[n, ci, i, j] = �⌈i/r⌉r h=⌊i/r⌋r �⌈j/r⌉r w=⌊j/r⌋r x[n, ci, h, w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' As shown in Figure 2(b), ˜x[n, ci, i, j] is the spatial sum of x elements in the same patch containing pixel (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Analyses of Proposed Approach In this section, we analyze our method from three per- spectives: gradient filtering approximation error, computa- tion reduction, and memory cost reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Error Analysis of Gradient Filtering We prove that the approximation error introduced by our gradient filtering is bounded during the gradient propaga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Without losing generality, we consider that all vari- ables have only one channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', Cx0 = Cx1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Proposition 1: For any input-output channel pair (co, ci) in the convolution kernel θ, assuming the DC component has the largest energy value compared to all components in the spectrum1, then the signal-to-noise-ratio (SNR) of ˜gx is greater than SNR of ˜gy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Proof: We use Gx, Gy and Θ to denote the gradients gx, gy and the convolution kernel θ in the frequency domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gx[u, v] is the spectrum value at frequency (u, v) and δ is the 2D discrete Dirichlet function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' To simplify the discus- sion, we consider only one patch of size r × r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The gradient returned by the gradient filtering can be written as: ˜gy = 1 r2 1r×r ⊛ gy (6) where ⊛ denotes convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' By applying the discrete Fourier transformation, Equation (6) can be rewritten in the frequency domain as: ˜Gy[u, v] = 1 r2 δ[u, v]Gy[u, v] (7) ˜gy is the approximation of gy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', the ground truth for ˜gy is gy), and the SNR of ˜gy equals to: SNR˜gy = � (u,v) G2 y[u, v] � (u,v)(Gy[u, v] − 1 r2 δ[u, v]Gy[u, v])2 = (1 − 2r2 − 1 r4 G2 y[0, 0] � (u,v) G2y[u, v])−1 (8) For the convolution layer, the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the approxi- mate variable ˜x in the frequency domain is2: ˜Gx[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] = Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v] ˜Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] = 1 r2 Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v]δ[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] (9) and its ground truth is: Gx[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] = Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] (10) Similar to Equation (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the SNR of g˜x is: SNR˜gx = (1 − 2r2 − 1 r4 (Θ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0]Gy[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0])2 � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) (Θ[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])2 )−1 (11) Equation (11) can be rewritten as: r4(1 − SNR−1 ˜gx ) 2r2 − 1 = (Θ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0]Gy[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0])2 � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v)(Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])2 = G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v)( Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='−v] Θ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0] Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])2 (12) Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the main assumption (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', the DC component dominates the frequency spectrum of Θ) can be written as: Θ2[0, 0]/max(u,v)̸=(0,0)Θ2[u, v] ≥ 1 (13) 1As a reminder, the energy of a signal is the sum of energy of the DC component and the energy of its AC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 2Because gy is convolved with the rotated kernel θ′, in the frequency domain, we use Θ[−u, −v] instead of Θ[u, v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 4 1 × 1 10 × 10 20 × 20 30 × 30 Patch Size r × r 1M 10M 100M 1G #FLOPs Baseline Reduced Unique Elements Reduced Unique Elements +Structured Gradient Actual Minimum Achievable Computation Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Computation analysis for a specific convolution layer3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Minimum achievable computation is given in Equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' By reducing the number of unique elements, computations required by our approach drop to about 1/r2 compared with the standard BP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' By combining it with structured gradient map, com- putations required by our approach drop further, getting very close to the theoretical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' that is, ∀(u, v), Θ2[−u,−v] Θ2[0,0] ≤ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' thus, by combining Equa- tion (12) and Equation (13), we have: G2 y[0, 0] � (u,v)( Θ[−u,−v] Θ[0,0] Gy[u, v])2 ≥ G2 y[0, 0] � (u,v)(Gy[u, v])2 ⇔ r4(1 − SNR−1 ˜gx ) 2r2 − 1 ≥ r4(1 − SNR−1 ˜gy ) 2r2 − 1 (14) which means that: SNR˜gx ≥ SNR˜gy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This completes our proof for error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ■ In conclusion, as the gradient propagates through the net- work, the noise introduced by our gradient filter becomes weaker compared to the real gradient signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This property ensures that the error in gradient has only a limited influ- ence on the quality of BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We validate Proposition 1 later in the experimental section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Computation and Overhead Analysis In this section, we analyse the computation required to compute gx, the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Figure 3 compares the computation required to propagate the gradient through this convolution layer under different patch sizes r × r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' A patch size 1 × 1 means the vanilla BP algorithm which we use as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' As discussed in the preliminary analysis section (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2), two terms contribute to the computa- tion savings: fewer unique elements in the gradient map and the structured gradient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Fewer unique elements: In vanilla BP, there are HyWy unique elements in the gradient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' After applying gradi- ent filtering with a patch size r × r, the number of unique 3The layer is from U-Net [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The size of the input is assumed to be 120 × 160 pixels with 192 channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the output has the same resolution, but with only 64 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The kernel size of the convolution layer is 3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Analysis for ResNet is included in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' elements reduces to only ⌈ Hy r ⌉⌈ Wy r ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This reduction con- tributes the most to the savings in computation (orange line in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Structured Gradient Map: By creating the structured gra- dient map, the convolution over the gradient map ˜gy is sim- plified to the element-wise multiplication and channel-wise addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Computation is thus reduced to (HθWθ)−1 of its original value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For instance, the example convolution layer in Figure 3 uses a 3 × 3 convolution kernel so around 89% computations are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The blue line in Figure 3 shows the #FLOPs after combining both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Greater reduc- tion is expected when applying our method with larger con- volution kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For instance, FastDepth [31] uses 5 × 5 convolution kernel so as much as 96% reduction in compu- tation can be achieved, in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Minimum Achievable Computation: With the two reduc- tions mentioned above, the computation required to propa- gate the gradient through the convolution layer is: #FLOPs(r) = ⌈Hy r ⌉⌈Wy r ⌉Cx(2Cy −1)+o(HyWy) (15) where o(HyWy) is a constant term which is independent of r and negligible compared to HyWy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' When the patch is as large as the feature map, our method reaches the minimum achievable computation (blue dashed line in Figure 3): minr #FLOPs(r) = 2CxCy − Cx + o(HyWy) (16) In this case, each channel of the gradient map is represented with a single value, so the computation is controlled by the number of input and output channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Overhead: The overhead of our approach comes from ap- proximating the feature map x, gradient gy, and kernel θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' As the lower part of Figure 2(a) shows, the approximation for x is considered as part of forward propagation, while the other two as back propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Indeed, with the patch size r, the ratio of forward propagation overhead is about 1/(2CoWθHθ), while the ratio of backward propagation overhead is about (r2 − 1)/(2Cx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Given the large number of channels and spatial dimen- sions in typical neural networks, both overhead values take less than 1% computation in the U-Net example above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Memory Analysis As Figure 2(a) shows, the standard back propagation for a convolution layer relies on the input feature map x, which needs to be stored in memory during forward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Since every convolution layer requiring gradient for its ker- nel needs to save a copy of feature map x, the memory consumption for storing x is huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' With our method, we simplify the feature map x to approximated ˜x, which has only ⌈ Hx r ⌉⌈ Wx r ⌉ unique elements for every channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Thus, by saving only these unique values, our method achieves around (1 − 1 r2 ) memory savings, overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 5 MobileNetV2 [28] #Layers Accuracy FLOPs Mem ResNet-18 [15] #Layers Accuracy FLOPs Mem No 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+page_content='00KB Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Experimental results for ImageNet classification with four neural networks (MobileNet-V2, ResNet18/34, MCUNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “#Layers” is short for “the number of active convolutional layers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For example, #Layers equals to 2 means that only the last two convolutional layers are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For memory consumption, we only consider the memory for input feature x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Strategy “No Finetuning” shows the accuracy on new datasets without finetuning the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Since TinyTL [6] changes the architecture, “#Layers” is not applicable (N/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' PSPNet [33] #Layers GFLOPs mIoU mAcc PSPNet-M [33] #Layers GFLOPs mIoU mAcc FCN [22] #Layers GFLOPs mIoU mAcc Calibration 0 0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='86 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='74 Calibration 0 0 14.' 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541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='88 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='13 5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='85 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='16 5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='8 38.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='31 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='09 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='33 Ours 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='26 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='47 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='35 Ours 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='97 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='04 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='44 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='96 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='28 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='53 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='99 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='22 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='00 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='07 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Experimental results for semantic segmentation task on augmented Pascal VOC12 dataset [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Model name with postfix “M” means the model uses MobileNetV2 as backbone, otherwise ResNet18 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “#Layers” is short for “the number of active convolutional layers” that are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' All models are pretrained on Cityscapes dataset [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Strategy “Calibration” shows the accuracy when only the classifier and normalization statistics are updated to adapt different numbers of classes between augmented Pascal VOC12 and Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Experiments In this section, we first present our experimental results on ImageNet classification [12] and semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Then, we study the impact of different hyper-parameter se- lections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Furthermore, we present the evaluation result run- ning our method on real hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Lastly, we empirically validate the assumption in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Experimental Setup Classification: Following [25], we split every dataset into two highly non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' partitions with the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Then, we pretrain our models on the first partition with a vanilla training strategy, and finetune the model on the other par- tition with different configurations for the training strat- egy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', with/without gradient filtering, hyper-parameters, number of convolution layers to be trained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', hyper-parameters) are in the Supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Segmentation: Models are pretrained on Cityscapes [11] by MMSegmentation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Then, we calibrate and finetune these models with different training strategies on the aug- mented Pascal-VOC12 dataset following [8], which is the combination of Pascal-VOC12 [13] and SBD [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More details are included in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' On-device Performance Evaluation: For CPU per- formance evaluation, we implement our method with MKLDNN [1] (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' OneDNN) v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 and compare it with the convolution BP method provided by MKLDNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We test on three CPUs, namely Intel 11900KF, Quad-core Cortex- A72 (Jetson Nano) and Quad-core Cortex-A53 (Raspberry Pi 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For GPU performance evaluation, we implement our method on CUDNN v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 [9] and compare with the BP method provided by CUDNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We test on two GPUs, RTX 3090Ti and the edge GPU on Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Since both 6 MKLDNN and CUDNN only support float32 BP, we test float32 BP only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Additionally, for the experiments on Jet- son Nano, we record the energy consumption for CPU and GPU with the embedded power meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', frequency) are included in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ImageNet Classification Table 2 shows our evaluation results on the ImageNet classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' As shown, our method significantly re- duces the FLOPs and memory required for BP, with very little accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For example, for ResNet34, our method achieves 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='9× speedup with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='7% accuracy loss when training four layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' for MobileNetV2, we get a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2% bet- ter accuracy with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0× speedup and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1× memory savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' These results illustrate the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' On most networks, TinyTL has a lower accuracy while consum- ing more resources compared to the baselines methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Semantic Segmentation Table 3 shows our evaluation results on the augmented Pascal-VOC12 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' On a wide range of networks, our method constantly achieves significant speedup with marginal accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For the large network UPerNet, our method achieves 229× speedup with only 1% mIoU loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For the small network PSPNet, our method speedups train- ing by 140× with only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='27% mIoU loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This shows the effectiveness of our method on a dense prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Hyper-Parameter Selection Figure 4 shows our experimental results for ResNets un- der different hyper-parameter selection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' number of con- volution layers and patch size of gradient filter r × r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Of note, the y-axis (MFLOPs) in Figure 4 is log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More results are included in Supplementary Section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We high- light three qualitative findings in Figure 4: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For a similar accuracy, our method greatly reduces the number of operations (1 to 2 orders of magni- tude), while for a similar number of computations, our method achieves a higher accuracy (2% to 5% better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This finding proves the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Given the number of convolution layers to be trained, the more accurate method returns a better accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Baseline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', standard BP) uses the most accurate gra- dient, Ours-R4 (BP with gradient filter with patch size 4 × 4) uses the least accurate gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' thus, Baseline > Ours-R2 > Ours-R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This finding is intuitive since the more accurate method should introduce smaller noise to the BP, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', the gradi- ent filtering with patch size 2 × 2 (Ours-R2) introduces less noise than with patch size 4 × 4 (Ours-R4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In Figure 5, we 92 93 94 Accuracy [%] ResNet18-CIFAR10 101 102 103 #MFLOPs 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1x OPs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1x OPs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2% Acc 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3x OPs Baseline Ours-R2 Ours-R4 78 80 82 Accuracy [%] ResNet34-CIFAR100 101 102 103 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='8x OPs 1% Acc 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6x OPs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0x OPs 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1% Acc 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6x OPs Baseline Ours-R2 Ours-R4 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Computation (#MFLOPs, log scale) and model accuracy [%] under different hyper-parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “Baseline” means vanilla BP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “Ours-R2/4” uses gradient filtering with patch size 2× 2/4 × 4 during BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' evaluate the relationship between accuracy and noise level introduced by gradient filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' With a higher SNR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', a lower noise level), a better accuracy is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='225 SNR [db] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='8 Top 1 Accuracy [%] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Relationship between accuracy and noise level intro- duced by the gradient filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' As shown, accuracy increases as the SNR increases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', noise level decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Given the number of computations, the less accurate method returns the better accuracy by training more layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', Ours-R4 > Ours-R2 > baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This finding suggests that for neural network training with relatively low computational resources, training more layers with less accurate gradients is preferable than training fewer layers with more accurate gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' On-device Performance Evaluation Figure 6 and Table 4 show our evaluation results on real devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More results are included in the Supplementary Section H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' As Figure 6 shows, on CPU, most convolution layers achieve speedups over 20× with less than 50% mem- ory consumption for gradient filtering with patch sizes 2×2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' for gradient filtering with patch size 4 × 4, the speedups are much higher, namely over 60×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' On GPU, the speedup is a little bit lower, but still over 10× and 25×, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' as Table 4 shows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' our method saves over 95% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='60× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='80× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='100× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Speedup (×times) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='114× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='CPU Speedup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11900KF-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11900KF-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RPi3-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RPi3-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='50× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='60× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='GPU Speedup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RTX3090Ti-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RTX3090Ti-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Test Case - Baseline: MKLDNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Percentage [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Baseline: MKLDNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='50% Memory Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Normalized CPU Memory Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11900KF-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11900KF-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RPi3-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RPi3-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Test Case - Baseline: CUDNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Baseline: CUDNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='50% Memory Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Normalized GPU Memory Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RTX3090Ti-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RTX3090Ti-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Speedup and normalized memory consumption results on multiple CPUs and GPUs under different test cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' different input sizes, numbers of channels, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=') Detailed configuration of these test cases are included in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “R2”, “R4” mean using gradient filtering with 2 × 2 and 4 × 4 patch sizes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our method achieves significant speedup with low memory consumption compared to all baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For example, on Jetson CPU with patch size 4 × 4 (“Jetson-R4” in left top figure), our method achieves 114× speedup with only 33% memory consumption for most test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Device Patch Size Normalized Energy Cost [STD] Edge CPU 2 × 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='13% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='61%] 4 × 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='15% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='18%] Edge GPU 2 × 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='80% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='73%] 4 × 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='22% [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='10%] Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Normalized energy consumption for BP with gradient filtering for different patch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Results are normalized w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the energy cost of standard BP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For instance, for edge CPU with a 4 × 4 patch, only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='15% of energy in standard BP is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Standard deviations are shown within brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' energy for both CPU and GPU scenarios, which largely re- solves one of the most important constraints on edge de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' All these experiments on real devices show that our method is practical for the real deployment of both high- performance and IoT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Model Ratio Model Ratio (Wide)ResNet18-152 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='462 VGG(bn)11-19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='497 DenseNet121-201 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='278 EfficientNet b0-b7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='240 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Evaluation of energy ratio defined in Equation (13) on models published on Torchvision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The ratio greater than 1 empiri- cally verifies our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Main Assumption Verification We now empirically verify the assumption that the DC component dominates the frequency spectrum of the convo- lution kernel (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' To this end, we collect the en- ergy ratio shown in Equation (13) from trained models pub- lished in Torchvision [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' As Table 5 shows, for the con- volution kernels in all these networks, we get a ratio greater than one, which means that the energy of DC components is larger than energy of all AC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Thus, our as- sumption in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 empirically holds true in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Conclusions In this paper, we have addressed the on-device model training for resource-constrained edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' To this end, a new gradient filtering method has been proposed to sys- tematically reduce the computation and memory consump- tion for the back-propagation algorithm, which is the key bottleneck for efficient model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In Section 3, a new gradient filtering approach has been proposed to reduce the computation required for propagat- ing gradients through the convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The gradient filtering creates an approximate gradient feature map with fewer unique elements and a special structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' this reduces the computation by more than two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Furthermore, we proved that the error introduced during back-propagation by our gradient filter is bounded so the influence of gradient approximation is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Extensive experiments in Section 5 have demonstrated the efficiency and wide applicability of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Indeed, models can be finetuned with orders of magnitudes fewer computations, while having only a marginal accuracy loss compared to popular baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 8 References [1] Intel® oneapi deep neural network library (onednn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='intel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='com/content/www/us/en/ developer/tools/oneapi/onednn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 1, 2, 6 [2] Mart´ın Abadi, Ashish 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Confer- ence on Robotics and Automation (ICRA), pages 6101–6108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 5 [32] Tete Xiao, Yingcheng Liu, Bolei Zhou, Yuning Jiang, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Unified perceptual parsing for scene understand- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In Proceedings of the European conference on computer vision (ECCV), pages 418–434, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 6 [33] Hengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang Wang, and Jiaya Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Pyramid scene parsing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2881–2890, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 6 [34] Kang Zhao, Sida Huang, Pan Pan, Yinghan Li, Yingya Zhang, Zhenyu Gu, and Yinghui Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Distribution adaptive int8 quantization for training cnns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 3483–3491, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 2 10 In this supplementary material, we present: A: Detailed derivation for gradient filtering described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' B: Detailed proof for Proposition 1 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' C: Visualized computation analysis for ResNet18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D: Detailed experimental setup for Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' E: More experimental results for Semantic Segmenta- tion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' F: More experimental results for hyper-parameter ex- ploration on CIFAR datasets in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' G: Experimental results for combining gradient filter- ing (our method) with existing INT8 gradient quanti- zation approaches [4,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' H: More experimental results for on-device perfor- mance evaluation in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gradient Filtering Derivation In this section, we present the complete derivations for Equation (3) and Equation (5) in Section 3, namely the back propagation with gradient filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For convenience, Ta- ble 6 (reproduced from Table 1 in paper) lists commonly used symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gradient Filtering We have: ˜gy[n, co, h, w] = 1 r2 ⌈i/r⌉r � h=⌊i/r⌋r ⌈j/r⌉r � w=⌊j/r⌋r gy[n, co, i, j] (17) Cx Number of channels of x Wx, Hx Width and height of x θ Convolution kernel θ′ Rotated θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', θ′ = rot180(θ) r Patch size (r × r) gx, gy, gθ Gradients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' x, y, θ ˜gy Approximated gradient gy ˜x, ˜θ′ Sum of x and θ′ over spatial dimensions (height and width) x[n, ci, h, w] Element for feature map x at batch n, channel ci, pixel (h, w) θ[co, ci, u, v] Element for convolution kernel θ at output channel co, input channel ci, position (u, v) Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Table of symbols we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Thus, for any entry in the approximated gradient ˜gy, the value equals to the average of all neighboring elements within the same r × r patch, as shown in the Figure 2 in the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For the approximated gradient ˜gy with batch size n, channel c, resolution (Hy, Wy), there will be (n × c × ⌈ Hy r ⌉ × ⌈ Wy r ⌉) unique numbers in ˜gy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' To simplify the following derivations, we rewrite the approximated gra- dient ˜gy as follows: ˜gp y[n, co, hp, wp, i, j] = ˜gy[n, co, hp∗r+i, wp∗r+j] (18) where (hp, wp) is the position of the patch and (i, j) is the offset within the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Since every element in the same patch has the exact same value, we denote this unique value with ˜gu y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', ˜gu y [n, co, hp, wp] = ˜gp y[n, co, hp, wp, i, j], ∀0 ≤ i, j < r (19) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Approximation for Rotated Convolution Ker- nel θ′ ˜θ′[co, ci] = � u,v θ′[co, ci, u, v] = � u,v rot180(θ)[co, ci, u, v] = � u,v θ[co, ci, u, v] (20) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Approximation for Input Feature x ˜x[n, ci, h, w] = ⌈i/r⌉r � h=⌊i/r⌋r ⌈j/r⌉r � w=⌊j/r⌋r x[n, ci, i, j] (21) Thus for every entry in approximated feature map ˜x, the value equal to the sum of all neighboring elements within the same r × r patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Following the definition of the gra- dient filter in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1, we use the following symbols to simplify the derivation: ˜xp[n, ci, hp, wp, i, j] = ˜x[n, ci, hp ∗r +i, wp ∗r +j] (22) and ˜xu[n, ci, hp, wp] = ˜xp[n, ci, hp, wp, i, j], ∀0 ≤ i, j < r (23) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Boundary Elements As mentioned in Section 3, given the structure created by the gradient filters, the gradient propagation in a con- volution layer can be simplified to weights summation and multiplication with few unique gradient values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This is true 11 for all elements far away from the patch boundary because for these elements, the rotated kernel θ′ only covers the ele- ments from the same patch, which have the same value, thus the computation can be saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' However, for the elements close to the boundary, this is not true, since when convolv- ing with boundary gradient elements, the kernel may cover multiple patches with multiple unique values instead of just one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' To eliminate the extra computation introduced by the boundary elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' we pad each patch sufficiently such that every element is far away from boundary: ˜gp y[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' hp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' wp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' j] = ˜gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' hp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' wp],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ∀i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' j ∈ Z (24) For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' with the patch size 4 × 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the element at the spatial position (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3) is on the boundary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' so when we calcu- late ˜gx[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3] by convolving the rotated kernel θ′ with the approximated gradient ˜gy: ˜gx[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3] = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='j θ′[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' j]˜gy[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3+j] (25) values of ˜gy are from multiple patches and have differ- ent values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', ˜gy[n, co, 3, 3] is from patch (0, 0) while ˜gy[n, co, 4, 4] is from patch (1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' they have different val- ues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In our method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' we simplify the Equation (25) by rewriting it in the following way: ˜gx[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3] ≈ 1 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='j=−1 θ′[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' j]˜gp y[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊3 4⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊3 4⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3 + i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3 + j] (26) = 1 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='j=−1 θ′[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' j]˜gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊3 4⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊3 4⌋] (27) = 1 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='j=−1 θ′[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' j]˜gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] (28) where Equation (26) is derived from Equation (25) by con- sidering that patch (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0) is sufficiently padded so that for elements with all offsets (3 + i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 3 + j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' they have the same value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' which is the unique value gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For approximated input feature map ˜x, we apply the same approximation for the boundary elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Input (Equation (3) in the Pa- per) ˜gx[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' w] (29) = � co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v θ[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v]˜gy[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' h + u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' w + v] (30) ≈ � co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v θ[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v]· ˜gp y[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' (h mod r) + u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' (w mod r) + v] (31) = � co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v θ[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v]˜gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋] (32) = � co ˜gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋] � u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v θ[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v] (33) = � co ˜gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋]˜θ′[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci] (34) By expanding ˜gu y to ˜gy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' we have: ˜gx[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' w] = � co ˜gy[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' w] ⊙ ˜θ′[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci] (35) which is the Equation (3) in Section 3 in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' From Equation (30) to Equation (32), we consider that the patch in the approximated gradient ˜gy is padded suffi- ciently so they have the same value for all possible offsets ((h mod r) + u, (w mod r) + v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' If there is only one in- put channel and output channel for the convolutional layer as the Figure 2 in the paper shows, then Equation (34) become an element-wise multiplication, which is Equa- tion (35) (also the Equation (3) in the paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Convolution Kernel (Equation (5) in the Paper) ˜gθ[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] (36) = � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='w x[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' h + u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' w + v]˜gy[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' w] (37) ≈ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='w ˜xp[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' (h mod r) + u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' (w mod r) + v]· ˜gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋] (38) = � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='w ˜xu[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋]˜gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋] (39) = � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='w ˜xu[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋]˜gu y [n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊h r ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ⌊w r ⌋] (40) 12 By expanding ˜xu and ˜gu y to ˜x and ˜gy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' we have: ˜gθ[co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] = � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='j ˜x[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' j]˜gy[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' j] (41) which is precisely Equation (5) in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' From Equation (37) to Equation (39), we consider that the patch in the approximated input feature map ˜x is padded sufficiently thus they have the same value for all possible offsets ((h mod r) + u, (w mod r) + v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For every given input/output channel pair (co, ci), Equation (40) represents the Frobenius inner product between ˜xu and ˜gu y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Detailed Proof for Proposition 1 In this section, we provide more details to the proof in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use Gx, Gy and Θ to denote the gradients gx, gy and the convolution kernel θ in the frequency domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Gx[u, v] is the spectrum value at frequency (u, v) and δ is the 2D discrete Dirichlet function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Without losing generality and to simplify the proof, we consider the batch size is 1, the number of input/output channels is 1, namely Cx = Cy = 1, and there is only one patch in ˜gy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The gradient returned by the gradient filtering can be written as: ˜gy = 1 r2 1r×r ⊛ gy (42) where ⊛ denotes convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' By applying the discrete Fourier transformation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Equation (42) can be rewritten in the frequency domain as: ˜Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] = 1 r2 δ[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] (43) ˜gy is the approximation for gy(so the ground truth for ˜gy is gy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' and the SNR of ˜gy equals to: SNR˜gy = ( � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v)(Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' − ˜Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])2 � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) G2y[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] )−1 = ( � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v)(Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] − 1 r2 δ[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])2 � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) G2y[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] )−1 (44) where the numerator can be written as: � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) (Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] − 1 r2 δ[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])2 = � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v)̸=(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0) (Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] − 1 r2 δ[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])2 + (Gy[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] − 1 r2 δ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0]Gy[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0])2 (45) Because δ[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] = � 1 (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v) = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0) 0 (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v) ̸= (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Equation (45) can be written as: � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v)̸=(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0) G2 y[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] + (r2 − 1)2 r4 G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] = � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v)̸=(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0) G2 y[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] + G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] − G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] + (r2 − 1)2 r4 G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] = � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) G2 y[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] − 2r2 − 1 r4 G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] (46) By substituting the numerator in Equation (44) with Equa- tion (46),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' we have: SNR˜gy = ( � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) G2 y[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] − 2r2−1 r4 G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) G2y[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] )−1 = (1 − 2r2 − 1 r4 G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) G2y[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])−1 = (1 − 2r2 − 1 r4 Energy of DC Component in Gy Total Energy4in Gy )−1 (47) For the convolution layer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' approximated variable ˜x in the frequency domain is: ˜Gx[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] = Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v] ˜Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] = 1 r2 Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v]δ[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] (48) and its ground truth is: Gx[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] = Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v] (49) Similar to Equation (47),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the SNR of g˜x is: SNR˜gx = (1 − 2r2 − 1 r4 Θ2[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0]G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) Θ2[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v]G2y[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])−1 = (1 − 2r2 − 1 r4 G2 x[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v) G2x[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])−1 = (1 − 2r2 − 1 r4 Energy of DC Component in Gx Total Energy5in Gx )−1 (50) Equation (50) can be rewritten as: r4(1 − SNR−1 ˜gx ) 2r2 − 1 = (Θ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0]Gy[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0])2 � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v)(Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' −v]Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])2 = G2 y[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0] � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='v)( Θ[−u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='−v] Θ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0] Gy[u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' v])2 (51) 4As reminder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the total energy of a signal is the sum of energy in DC component and energy in AC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 13 Besides, the proposition’s assumption (the DC component dominates the frequency spectrum of Θ) can be written as: Θ2[0, 0] max(u,v)̸=(0,0)Θ2[u, v] ≥ 1 (52) which is: ∀(u, v), Θ2[−u, −v] Θ2[0, 0] ≤ 1 (53) thus, by combining Equation (51) and Equation (53), we have: r4(1 − SNR−1 ˜gx ) 2r2 − 1 = G2 y[0, 0] � (u,v)( Θ[−u,−v] Θ[0,0] Gy[u, v])2 ≥ G2 y[0, 0] � (u,v)(Gy[u, v])2 = r4(1 − SNR−1 ˜gy ) 2r2 − 1 (54) which means that: SNR˜gx ≥ SNR˜gy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This completes our proof for error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='■ In conclusion, as the gradient propagates, the noise in- troduced by the gradient filter becomes weaker and weaker compared to the real gradient signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This property ensures that the error in gradient has only a limited influence on the quality of BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' This proof can be extended to the more general case where batch size and the number of channels are greater than 1 by introducing more dimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', batch dimen- sion, channel dimension) into all equations listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Computation Analysis for ResNet18 In this section, we provide two more examples for com- putation analysis in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Figure 7 shows the com- putation required by the convolution layers from ResNet18 with different patch sizes for gradient filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' With re- duced unique elements, our approach reduces the num- ber of computations to 1/r2 of standard BP method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' with structured gradient, our approach further reduces the num- ber of computations to about 1/(r2HθWθ) of standard BP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Detailed Experimental Setup In this supplementary section, we extend the experimen- tal setup in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ImageNet Classification D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 Environment ImageNet related experiments are conducted on IBM Power System AC922, which is equipped with a 40-core IBM Power 9 CPU, 256 GB DRAM and 4 NVIDIA Tesla V100 16GB GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 compiled with CUDA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 as the deep learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 1 × 1 3 × 3 5 × 5 7 × 7 Patch Size r × r 1M 10M 100M FLOPs Baseline Reduced Unique Elements +Structured Gradient Actual Minimum Achievable Computation (a) Last convolutional layer in block 4 of ResNet18 with 512 input/output channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the resolution of input feature map is 7 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 1 × 1 4 × 4 8 × 8 12 × 12 Patch Size r × r 100K 1M 10M 100M FLOPs Baseline Reduced Unique Elements +Structured Gradient Actual Minimum Achievable Computation (b) Last convolutional layer in block 3 of ResNet18 with 256 input/output channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the resolution of input feature map is 14 × 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 1 × 1 10 × 10 20 × 20 Patch Size r × r 100K 1M 10M 100M FLOPs Baseline Reduced Unique Elements +Structured Gradient Actual Minimum Achievable Computation (c) Last convolutional layer in block 2 of ResNet18 with 128 input/output channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' the resolution of input feature map is 28 × 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Computation analysis for three convolution layers in of ResNet18 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Since convolutional layers in every block of ResNet18 is similar, we use the last convolutional layer as the representative of all convolutional layers in the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Minimum achievable computation is presented in Equation (16) in the pa- per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' By reducing the number of unique elements, computations required by our approach drop to about 1/r2 compared with the standard BP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' By combining it (“+” in the figure) with structured gradient map, computations required by our approach drop further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 Dataset Split We split the dataset into two non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' partitions following the FedAvg method [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The label distribution is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Among all 1000 classes for the ImageNet, pretrain and finetune partitions overlap on only 99 classes, which suggests that our method can efficiently adapt the 14 Model Accuracy Model Accuracy ResNet-18 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5% MobileNet-V2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3% ResNet-34 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4% MCUNet 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4% Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Model pretraining accuracy on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' DNN model to data collected from new environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For each partition, we randomly select 80% data as training data and 20% as validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0 200 400 600 800 1000 Class Index 0 200 400 600 800 1000 Image Count ImageNet Data Split Pretrain Finetune Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Label distribution for pretraining and finetuning datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Pretraining and finetuning partitions are split from ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 Pretraining We pretrain ResNet 18, ResNet 34, MobileNet-V2 and MCUNet with the same configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use SGD opti- mizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The learning rate of the optimizer starts at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='05 and decays according to cosine annealing method [23] during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Additionally, weight decay is set to 1 × 10−4 and momentum is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We set batch size to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We ran- domly resize, randomly flip and normalize the image for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use cross entropy as loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Models are trained for 200 epochs and the model with the highest validation accuracy is kept for finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Table 7 shows the pretrain accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 Finetuning We adopt the hyper-parameter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', momentum, weight de- cay, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=') from pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Several changes are made: mod- els are finetuned for 90 epochs instead of 200;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' we apply L2 gradient clipping with threshold 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' linear learning rate warm-up for 4 epochs is introduced at the beginning of fine- tuning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', for the first 4 epochs, the learning rate grows linearly up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='05, then the learning rate decays accord- ing to cosine annealing method in the following epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Of note, to ensure a fair comparison, we use the same hyper- parameters for all experiments, regardless of model type and training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' CIFAR Classification D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 Environment CIFAR related experiments are conducted on a GPU work- station with a 64-core AMD Ryzen Threadripper PRO 3995WX CPU, 512 GB DRAM and 4 NVIDIA RTX A6000 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 compiled with CUDA 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 as the deep learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 Dataset Split We split the dataset into two non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' partitions following FedAvg method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The label distribution is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For CIFAR10, pretrain and finetune partitions overlap on 2 classes out of 10 classes in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For CIFAR100, pre- train and finetune partitions overlap on 6 classes out of 100 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 0 2 4 6 8 Class Index 0 1000 2000 3000 4000 Image Count CIFAR10 Data Split Pretrain Finetune 0 20 40 60 80 100 Class Index 0 100 200 300 400 Image Count CIFAR100 Data Split Pretrain Finetune Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Label distribution for pretraining and finetuning datasets on CIFAR10 and CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Pretraining and finetuning partitions are split from CIFAR10/100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 Pretraining We pretrain ResNet18 and ResNet34 with the same config- uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use the ADAM optimizer with a learning rate of 3 × 10−4 and weight decay 1 × 10−4 with no learning rate scheduling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use cross entropy as loss func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We set batch size to 128, and normalize the data be- fore feeding it to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Models are trained for 30 and 50 epochs for CIFAR10 and CIFAR100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Then, the model with the highest accuracy is kept for finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Table 8 shows the pretrain accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 Finetuning We adopt the training configuration from PSQ [7] with some changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use cross entropy loss with SGD opti- 15 ResNet18 ResNet34 CIFAR10 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6% CIFAR100 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5% Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Model pretraining accuracy on CIFAR10/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' mizer for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The learning rate of the optimizer starts at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='05 and decays according to cosine annealing method during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Momentum is set to 0 and weight decay is set to 1 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We apply L2 gradient clipping with a threshold 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Batch normalization layers are fused with convolution layers before training, which is a common tech- nique for inference acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Semantic Segmentation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 Environment ImageNet related experiments are conducted on IBM Power System AC922, which is equipped with a 40-core IBM Power 9 CPU, 256 GB DRAM and 4 NVIDIA Tesla V100 16GB GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 compiled with CUDA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 as the deep learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We implement our method based on MMSegmentation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 Pretraining We use models pretrained by MMSegmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Consider- ing that the numbers of classes, image statistics, and model hyper-parameters may be different when applying on dif- ferent datasets, we calibrate the model before finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use SGD optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The learning rate of the optimizer starts at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01 and decays exponentially during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Ad- ditionally, weight decay is set to 5 × 10−4 and momentum is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We set batch size to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We randomly crop, flip and photo-metric distort and normalize the image for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use cross entropy as loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For DeepLabV3, FCN, PSPNet and UPerNet, we calibrate the classifier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', the last layer) and statistics in batch normal- ization layers for 1000 steps on the finetuning dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For DeepLabV3-MobileNetV2 and PSPNet-MobileNetV2, be- cause the number of channels for convolutional layers in the decoder are different for models applied on different datasets, we calibrate the decoder and statistics in batch nor- malization layers for 5000 steps on the finetuning dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 Finetuning We finetune all models with the same configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use the SGD optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' The learning rate of the optimizer starts at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01 and decays according to cosine anneling method dur- ing training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Additionally, weight decay is set to 5 × 10−4 and momentum is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We set batch size to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We randomly crop, flip and photo-metric distort and normalize the image for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use cross entropy as loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Models are finetuned for 20000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Exper- iments are repeated three times with random seed 233, 234 and 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' On-device Performance Evaluation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 NVIDIA Jetson Nano We use NVIDIA Jetson Nano with quad-core Cortex-A57, 4 GB DRAM, 128-core Maxwell edge GPU for perfor- mance evaluation on both edge CPU and edge GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use the aarch64-OS Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 provided by NVIDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' During evaluation, the frequencies for CPU and GPU are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 GHz and 921 MHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our code and library MKLDNN (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' OneDNN) are compiled on Jetson Nano with GCC 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0, while libraries CUDA and CUDNN are compiled by NVIDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For CPU evaluations, our code and baseline are implemented with MKLDNN v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For GPU evaluations, our code and baseline are implemented with CUDA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 and CUDNN 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Before the evaluation for every test case, we warm up the device by running the test once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Then we repeat the test 10 times and report the average value for latency, energy consumption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Energy consumption is obtained by reading the embed- ded power meter in Jetson Nano every 20 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 Raspberry Pi 3b We use Raspberry Pi 3b with quad-core Cortex-A53, 1 GB DRAM for performance evaluation on CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use the aarch64-OS Raspberry Pi OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' During evaluation, the frequency for CPU is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our code and library MKLDNN are compiled on Raspberry Pi with GCC 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our code and baseline are implemented with MKLDNN v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Before the evaluation for every test case, we warm up the device by running the test once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Then we repeat the test 10 times and report the average value for latency, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 Desktop We use a desktop PC with Intel 11900KF CPU, 32 GB DRAM and RTX 3090 Ti GPU for perforamce evaluation on both desktop CPU and desktop GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' We use x86 64- OS Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' During evaluation, the frequencies for CPU and GPU are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='7 GHz and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 GHz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Our code is compiled with GCC 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' MKLDNN is compiled by Anaconda (tag omp h13be974 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' CUDA and CUDNN are compiled by NVIDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For CPU evaluations, our code and baseline are implemented with MKLDNN v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For GPU evaluations, our code and baseline are implemented with CUDA 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='7 and CUDNN 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 16 Pretrain: ADE20K Finetune: VOC12Aug UPerNet #Layers GFLOPs mIoU mAcc PSPNet-M #Layers GFLOPs mIoU mAcc DLV3-M #Layers GFLOPs mIoU mAcc Calibration 0 0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='66 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='03 Calibration 0 0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='93 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01 Calibration 0 0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='28 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='98 Vanilla BP All 541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='23[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='24] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='79[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='45] Vanilla BP All 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='41 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='51[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='27] 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='19] Vanilla BP All 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='35 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='78[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='21] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='10[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40] 5 503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='09] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='97[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='30] 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='22 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} 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71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='93[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='41] Ours 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='97 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='76[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='57[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='07] Ours 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='59[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='08] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='28[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='30] Ours 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='26 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='00] 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='13[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='54] 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='22 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='78[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='23] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='55[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='38] 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='76 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='77[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='37] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='82[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='47] 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='14[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='15] 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='48[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='26] Pretrain: ADE20K Finetune: Cityscapes UPerNet #Layers GFLOPs mIoU mAcc PSPNet-M #Layers GFLOPs mIoU mAcc DLV3-M #Layers GFLOPs mIoU mAcc Calibration 0 0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='15 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='45 Calibration 0 0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='83 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='85 Calibration 0 0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='33 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='65 Vanilla BP All 1082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='02[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='14] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20] Vanilla BP All 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='82 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='21[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40] 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='72[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='68] Vanilla BP All 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='12[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='14] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='81[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='04] 5 1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='46[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='19] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='62[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='27] 5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='43 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='09[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='43] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='70[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='49] 5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='00[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='05] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='03] 10 1015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='21] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='32] 10 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='90 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='03[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='24] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='48[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='10] 10 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='02[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='14] 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='80[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='06] Ours 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='94 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='58[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='25] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='67[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='32] Ours 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='22 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='59[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='38] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='10[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='41] Ours 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='50 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='83[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='07] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='87[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='08] 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='43 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='14[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='24] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='41[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='27] 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='51 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='10[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='49] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='93[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='43] 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='74 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='22[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='01] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='99[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='31] Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Experimental results for semantic segmentation task for UPerNet, DeepLabV3-MobileNetV2 (DLV3-M) and PSPNet- MobileNetV2 (PSPNet-M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Models are pretrained on ADE20K dataset and finetuned on augmentated Pascal VOC12 dataset and Cityscapes dataset respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “#Layers” is short for “the number of active convolutional layers” that are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Strategy “Calibration” shows the accuracy when only the classifier and normalization statistics are updated to adapt differences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' different number of classes) between pretraining dataset and finetuning dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' #Input Channel #Output Channel Input Width Input Height 0 128 128 120 160 1 256 256 60 80 2 512 512 30 40 3 512 512 14 14 4 256 256 14 14 5 128 128 28 28 6 64 64 56 56 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Layer configuration for test cases in Figure 6 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Before the evaluation for every test case, we warm up the device by running the 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Then we repeat the test 200 times and report the average value for latency, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 Test Case Configurations Table 10 lists the configurations for test cases shown in Fig- ure 6 in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' In addition to the parameters shown in the table, for all test cases, we set the batch size to 32, kernel size to 3 × 3, padding and stride to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More Results for Semantic Segmentation In this section, we extend the experimental results shown in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Table 9 shows the experimental re- sults for UPerNet, PSPNet-MobileNetV2 (PSPNet-M) and DeepLabV3-MobileNetV2 (DLV3-M) on two pairs of pre- traing and finetuning datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' These results further show the effectiveness of our method on a dense prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More Results for CIFAR10/100 with Differ- ent Hyper-Parameter Selections In this section, we extend the experimental results shown in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Table 11 (page 18) shows the ex- perimental results for ResNet18 and ResNet34 on CIFAR datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For every model, we test our method with differ- ent patch sizes for gradient filtering and different numbers of active convolutional layers (#Layers in Table 11, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', if #Layers equals to 2, the last two convolutional layers are trained while other layers are frozen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' These results further support the qualitative findings in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Results for Combining Gradient Filtering with Gradient Quantization In this section, we provide experimental results for com- bining our method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' gradient filtering, with gradient quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Table 12 (page 19) shows experimental re- sults for ResNet18 and ResNet32 with gradient quantiza- tion methods PTQ [4] and PSQ [7] and different hyper- parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Both forward propagation and backward prop- agation are quantized to INT8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' These results support the wide applicability of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' More Results for On-device Performance Evaluation In this section, we extend the experimental results shown in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Figure 10 shows the energy savings and overhead of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For most test cases with patch 4 × 4, we achieve over 80× energy savings with less than 20% overhead on both CPU and GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Moreover, for the test case 1 on Raspberry Pi CPU, the forward propagation is even faster when applied our method (which results in negtive overheads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' These results further show that our method is practical for the real deployment of both high- performance and IoT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 17 CIFAR10 CIFAR100 ResNet18 #Layers ACC[%] FLOPs 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='35M 2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='35M 3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='12M 3 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='03M 4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='04M 4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='17M Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Experimental results on CIFAR10 and CIFAR100 datasets for ResNet18 and ResNet34 with different hyper-parameter selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “ACC” is short for accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “#Layers” is short for “the number of active convolution layers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' #Layers equals to 2 means that only the last two convolutional layers are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “Gradient Filter R2/4/7” use proposed gradient filtering method with patch size 2 × 2, 4 × 4 and 7 × 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='60× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='80× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='100× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='120× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Energy Savings [×times] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='CPU Energy Savings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='60× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='80× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='100× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='GPU Energy Savings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Test Case - Baseline: MKLDNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Percentage [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Forward Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20% Overhead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Normalized CPU Overhead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11900KF-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='11900KF-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RPi3-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RPi3-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Test Case - Baseline: CUDNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Forward Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='20% Overhead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Normalized GPU Overhead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Jetson-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RTX3090Ti-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='RTX3090Ti-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Energy savings and overhead resuls on multiple CPUs and GPUs under different test cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', different input sizes, number of channels, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='.).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For test case 4 and 5 with patch size 4 × 4 (Jetson-R4) on GPU, the latency of our method is too small to be captured by the power meter with a 20 ms sample rate so the energy savings data is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For most test cases with patch size 4 × 4, our method achieves over 80× energy savings with less than 20% overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 18 CIFAR10 CIFAR100 ResNet18 ResNet34 ResNet18 ResNet34 Strategy #Layers ACC[%] #OPs Strategy #Layers ACC[%] #OPs Strategy #Layers ACC[%] #OPs Strategy #Layers ACC[%] #OPs PTQ 1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='25M PTQ 1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='6 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='25M PTQ 1 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} 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ResNet34 with different gradient quantization methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=', PTQ [4] and PSQ [7]) and hyper-parameter selections on CIFAR10/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' Feature map, activation, weight and gradient are quantized to INT8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “ACC” is short for accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “#Layers” is short for “the number of active convolution layers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' For example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' #Layers equals to 2 means that the last two convolutional layers are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' “Gradient Filter R2/4/7” use proposed gradient filtering method with patch size 2 × 2, 4 × 4 and 7 × 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfe_gi/content/2301.00330v1.pdf'} diff --git a/H9FKT4oBgHgl3EQfdS4O/vector_store/index.faiss b/H9FKT4oBgHgl3EQfdS4O/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..955d3c1efede55f1e90c865fe2186bc10277ae48 --- /dev/null +++ b/H9FKT4oBgHgl3EQfdS4O/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1d0c24255f086e64642849d14acc3ce881f434c5e22edced3d38b932748017c7 +size 4718637 diff --git a/HdE0T4oBgHgl3EQfhgHx/content/tmp_files/2301.02434v1.pdf.txt b/HdE0T4oBgHgl3EQfhgHx/content/tmp_files/2301.02434v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e9d5507a3918a4877391b3e13dc7a1eb933260a8 --- /dev/null +++ b/HdE0T4oBgHgl3EQfhgHx/content/tmp_files/2301.02434v1.pdf.txt @@ -0,0 +1,3387 @@ +Asymptotic decay function of the stationary tail probabilities along +an arbitrary direction in a two-dimensional discrete-time QBD +process +Toshihisa Ozawa +Faculty of Business Administration, Komazawa University +1-23-1 Komazawa, Setagaya-ku, Tokyo 154-8525, Japan +E-mail: toshi@komazawa-u.ac.jp +Abstract +We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD pro- +cess for short) on Z2 ++ ×S0, where S0 is a finite set, and consider a topic remaining unresolved in +our previous paper. In that paper, the asymptotic decay rate of the stationary tail probabilities +along an arbitrary direction has been obtained. It has also been clarified that if the asymptotic +decay rate ξc, where c is a direction vector in N2, is less than a certain value θmax +c +, the sequence +of the stationary tail probabilities along the direction c geometrically decays without power +terms, asymptotically. In this article, we give the function that the sequence asymptotically +decays according to when ξc = θmax +c +, but it contains an unknown parameter. To determine the +value of the parameter is a next challenge. +Keywards: quasi-birth-and-death process, Markov modulated reflecting random walk, Markov +additive process, asymptotic decay rate, asymptotic decay function, stationary distribution, ma- +trix analytic method +Mathematics Subject Classification: 60J10, 60K25 +1 +Introduction +We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for +short) {Y n} = {(Xn, Jn)} on Z2 ++ × S0, where S0 is a finite set. This model is a Markov modulated +reflecting random walk (MMRRW for short) whose transitions are skip free, and the MMRRW is +a kind of reflecting random walk (RRW for short) with a background process, where the transition +probabilities of the RRW vary depending on the state of the background process. One-dimensional +QBD processes have been introduced by Macel Neuts and studied in the literature as one of the +essential stochastic models in the queueing theory (see, for example, [1, 5, 7, 8]). The 2d-QBD +process is a two-dimensional version of one-dimensional QBD process, and it enable us to analyze, +for example, two-node queueing networks and two-node polling models. +Assume the 2d-QBD process {Y n} is positive recurrent and denote by ν = (ν(x,j); (x, j) ∈ +Z2 ++ × S0) the stationary distribution, where ν(x,j) is the stationary probability that the process +is in the state (x, j). Our interest is asymptotics of the stationary distribution ν, especially, tail +asymptotics in an arbitrary direction. Let an integer vector c = (c1, c2) be nonzero and nonnegative. +Two typical objects of our study are the asymptotic decay rate ξc and asymptotic decay function +hc(k) defined as, for j ∈ S0, +ξc = − lim +k→∞ +1 +k log ν(kc,j), +lim +k→∞ +ν(kc,j) +hc(k) = gj, +1 +arXiv:2301.02434v1 [math.PR] 6 Jan 2023 + +where gj is a positive constant. +Under a certain condition, the asymptotic decay rate of the +probability sequence {νx+kc,j; k ≥ 0} does not depend on x and j if it exists, see Proposition 2.3 +of Ozawa [14]. In the case where c = (1, 0) or c = (0, 1), the asymptotic decay rate ξc has been +obtained in Ozawa [10], see Corollary 4.3 of [14], and the asymptotic decay function hc(k) in Ozawa +and Kobayashi [11], see Theorem 2.1 of [11]. The results in the case where c = (c, 0) or c = (0, c) +for c ≥ 2 are automatically obtained from those in [10, 11]. In the case where c = (c1, c2) ≥ (1, 1), +the asymptotic decay rate ξc has been obtained in Ozawa [14], see Theorem 3.2 of [14]. A condition +ensuring the asymptotic decay function is given by hc(k) = e−ξck, an exponential function without +a power term, has also been given in the theorem. +In this article, we give the expression of the asymptotic decay function hc(k) when c = (c1, c2) ≥ +(1, 1). +To this end, we clarify the analytic properties of the vector generating function of the +stationary probabilities along the direction c, ϕc(z). +The point z = eξc is a singular point of +the vector function ϕc(z), and if ξc is equal to a certain value θmax +c +, z = eθmax +c +is a branch point +of ϕc(z) with order one. From this result, we obtain the expression of hc(k), but it contains an +unknown parameter. +To determine the value of the parameter, it suffices to prove that ϕc(z) +diverges elementwise at z = eθmax +c +. It seems to be a hard work and we leave it as a next challenge. +We also generalize a part of existing results. One crucial point in analyzing the asymptotic decay +function is how to analytically extend the G-matrix function appeared in the vector generating +function of the stationary probabilities. In [11], it has been done under the assumption that all +the eigenvalues of the G-matrix function are distinct, see Assumption 4.1 and Lemma 4.5 of [11]. +This assumption is not easy to verify in general. We, therefore, remove the assumption and give a +general formula of the Jordan decomposition of the G-matrix function, see Section 3.1. +The rest of the article is organized as follows. In Section 2, we describe the 2d-QBD process in +detail and state assumptions and main results. In Section 3, an analytic extension of the G-matrix +function is given in a general setting. The definition of G-matrix in the reverse direction and its +properties are also given in the same section. They are used in the following section. The proof +of the main results is given in Sections 4, where we demonstrate that the vector function ϕc(z) +is elementwise analytic in the open disk with radius eξc + ε for some ε > 0, except for the point +z = eξc, and clarify its singularity at the point z = eξc. The asymptotic decay function is obtained +from those results. The paper concludes with some remarks in Section 5. +2 +Model description and main results +2.1 +Model description +We consider the same model as that described in [14] and use the same notation. +Denote by I2 the set of all the subsets of {1, 2}, i.e., I2 = {∅, {1}, {2}, {1, 2}}, and we use it +as an index set. Divide Z2 ++ into 22 = 4 exclusive subsets defined as +Bα = {x = (x1, x2) ∈ Z2 ++; xi > 0 for i ∈ α, xi = 0 for i ∈ {1, 2} \ α}, α ∈ I2. +Let {Y n} = {(Xn, Jn)} be a 2d-QBD process on S = Z2 ++ × S0, where S0 = {1, 2, ..., s0}. Let P be +the transition probability matrix of {Y n} and represent it in block form as P = +� +Px,x′; x, x′ ∈ Z2 ++ +� +, +where Px,x′ = (p(x,j),(x′,j′); j, j′ ∈ S0) and p(x,j),(x′,j′) = P(Y 1 = (x′, j′) | Y 0 = (x, j)). For α ∈ I2 +and i1, i2 ∈ {−1, 0, 1}, let Aα +i1,i2 be a one-step transition probability block from a state in Bα, where +we assume the blocks corresponding to impossible transitions are zero (see Fig. 1). Since the level +process is skip free, for every x, x′ ∈ Z2 ++, Px,x′ is given by +Px,x′ = +� Aα +x′−x, +if x ∈ Bα for some α ∈ I2 and x′ − x ∈ {−1, 0, 1}2, +O, +otherwise. +(2.1) +We assume the following condition throughout the paper. +2 + +Figure 1: Transition probability blocks +Assumption 2.1. The 2d-QBD process {Y n} is irreducible and aperiodic. +Next, we define several Markov chains derived from the 2d-QBD process. For a nonempty set +α ∈ I2, let {Y α +n} = {(Xα +n, Jα +n )} be a process derived from the 2d-QBD process {Y n} by removing +the boundaries that are orthogonal to the xi-axis for each i ∈ α. The process {Y {1} +n } is a Markov +chain on Z × Z+ × S0 whose transition probability matrix P {1} = (P {1} +x,x′; x, x′ ∈ Z × Z+) is given +as +P {1} +x,x′ = +� +� +� +� +� +A{1} +x′−x, +if x ∈ Z × {0} and x′ − x ∈ {−1, 0, 1} × {0, 1}, +A{1,2} +x′−x, +if x ∈ Z × N and x′ − x ∈ {−1, 0, 1}2, +O, +otherwise, +(2.2) +where N is the set of all positive integers. The process {Y {2} +n } on Z+ × Z × S0 and its transition +probability matrix P {2} = (P {2} +x,x′; x, x′ ∈ Z+ × Z) are analogously defined. The process {Y {1,2} +n +} +is a Markov chain on Z2 × S0, whose transition probability matrix P {1,2} = (P {1,2} +x,x′ ; x, x′ ∈ Z2) is +given as +P {1,2} +x,x′ = +� +A{1,2} +x′−x, +if x′ − x ∈ {−1, 0, 1}2, +O, +otherwise. +(2.3) +Regarding X{1} +1,n as the additive part, we see that the process {Y {1} +n } = {(X{1} +1,n , (X{1} +2,n , J{1} +n +))} is +a Markov additive process (MA-process for short) with the background state (X{1} +2,n , J{1} +n +) (with +respect to MA-processes, see, for example, Ney and Nummelin [9]). +The process {Y {2} +n } = +{(X{2} +2,n , (X{2} +1,n , J{2} +n +))} is also an MA-process, where X{2} +2,n is the additive part and (X{2} +1,n , J{2} +n +) the +background state, and {Y {1,2} +n +} = {(X{1,2} +1,n +, X{1,2} +2,n +), J{1,2} +n +)} an MA-process, where (X{1,2} +1,n +, X{1,2} +2,n +) +the additive part and J{1,2} +n +the background state. We call them the induced MA-processes de- +rived from the original 2d-QBD process. Let { ¯A{1} +i +; i ∈ {−1, 0, 1}} be the Markov additive kernel +(MA-kernel for short) of the induced MA-process {Y {1} +n }, which is the set of transition probability +blocks and defined as, for i ∈ {−1, 0, 1}, +¯A{1} +i += +� +¯A{1} +i,(x2,x′ +2); x2, x′ +2 ∈ Z+ +� +, +¯A{1} +i,(x2,x′ +2) = +� +� +� +� +� +A{1} +i,x′ +2−x2, +if x2 = 0 and x′ +2 − x2 ∈ {0, 1}, +A{1,2} +i,x′ +2−x2, +if x2 ≥ 1 and x′ +2 − x2 ∈ {−1, 0, 1}, +O, +otherwise. +3 + +X2 ^ +B(2] +B(1,2] +[2] +[1,2] +12 +i1,i2 +17 +,12 +B(1) +Bo +0 +x1Let { ¯A{2} +i +; i ∈ {−1, 0, 1}} be the MA-kernel of {Y {2} +n }, defined in the same manner. With respect to +{Y {1,2} +n +}, the MA-kernel is given by {A{1,2} +i1,i2 ; i1, i2 ∈ {−1, 0, 1}}. We assume the following condition +throughout the paper. +Assumption 2.2. The induced MA-processes {Y {1} +n }, {Y {2} +n } and {Y {1,2} +n +} are irreducible and +aperiodic. +According to [14], we assume several other technical conditions for the induced MA-process +{Y {1,2} +n +}, concerning irreducibility and aperiodicity on subspaces. Let {Y + +n } = {(X+ +n , J+ +n )} be a +lossy Markov chain derived from the induced MA-process {Y {1,2} +n +} by restricting the state space +of the additive part to N2. The process {Y + +n } is a Markov chain on N2 × S0 whose transition +probability matrix P + is given as P + = (P {1,2} +x,x′ ; x, x′ ∈ N2), where P + is strictly substochastic. +The process {Y + +n } is also a lossy Markov chain derived from the original 2d-QBD process {Y n} +by restricting the state space of the level to N2. We assume the following condition throughout the +paper. +Assumption 2.3. {Y + +n } is irreducible and aperiodic. +For k ∈ Z, let Z≤k and Z≥k be the set of integers less than or equal to k and that of integers +greater than or equal to k, respectively. We also assume the following condition throughout the +paper. For what this assumption implies, see Remark 3.1 of [14]. +Assumption 2.4. +(i) The lossy Markov chain derived from the induced MA-process {Y {1,2} +n +} by +restricting the state space to Z≤0 × Z≥0 × S0 is irreducible and aperiodic. +(ii) The lossy Markov chain derived from {Y {1,2} +n +} by restricting the state space to Z≥0×Z≤0×S0 +is irreducible and aperiodic. +The stability condition of the 2d-QBD process has already been obtained in [12]. Let a{1}, a{2} +and a{1,2} = (a{1,2} +1 +, a{1,2} +2 +) be the mean drifts of the additive part in the induced MA-processes +{Y {1} +n }, {Y {2} +n } and {Y {1,2} +n +}, respectively. By Corollary 3.1 of [12], the stability condition of the +2d-QBD process {Y n} is given as follows: +Lemma 2.1. +(i) In the case where a{1,2} +1 +< 0 and a{1,2} +2 +< 0, the 2d-QBD process {Y n} is +positive recurrent if a{1} < 0 and a{2} < 0, and it is transient if either a{1} > 0 or a{2} > 0. +(ii) In the case where a{1,2} +1 +≥ 0 and a{1,2} +2 +< 0, {Y n} is positive recurrent if a{1} < 0, and it is +transient if a{1} > 0. +(iii) In the case where a{1,2} +1 +< 0 and a{1,2} +2 +≥ 0, {Y n} is positive recurrent if a{2} < 0, and it is +transient if a{2} > 0. +(iv) If one of a{1,2} +1 +and a{1,2} +2 +is positive and the other is non-negative, then {Y n} is transient. +For the explicit expression of the mean drifts, see Section 3.1 of [12] and its related parts. We +assume the following condition throughout the paper. +Assumption 2.5. The condition in Lemma 2.1 that ensures the 2d-QBD process {Y n} is positive +recurrent holds. +Denote by ν the stationary distribution of {Y n}, where ν = (νx, x ∈ Z2 ++), νx = (ν(x,j), j ∈ S0) +and ν(x,j) is the stationary probability that the 2d-QBD process is in the state (x, j). +4 + +Figure 2: Domains Γ{1,2}, Γ{1} and Γ{2} +2.2 +Main results +Let ¯A{1} +∗ +(z) and ¯A{2} +∗ +(z) be the matrix generating functions of the MA-kernels of {Y {1} +n } and +{Y {2} +n }, respectively, defined as +¯A{1} +∗ +(z) = +� +i∈{−1,0,1} +zi ¯A{1} +i +, +¯A{2} +∗ +(z) = +� +i∈{−1,0,1} +zi ¯A{2} +i +. +The matrix generating function of the MA-kernel of {Y {1,2} +n +} is given by A{1,2} +∗,∗ +(z1, z2), defined as +A{1,2} +∗,∗ +(z1, z2) = +� +i1,i2∈{−1,0,1} +zi1 +1 zi2 +2 A{1,2} +i1,i2 . +Let Γ{1}, Γ{2} and Γ{1,2} be regions in which the convergence parameters of ¯A{1} +∗ +(eθ1), ¯A{2} +∗ +(eθ2) +and A{1,2} +∗,∗ +(eθ1, eθ2) are greater than 1, respectively, i.e., +Γ{1} = {(θ1, θ2) ∈ R2; cp( ¯A{1} +∗ +(eθ1)) > 1}, +Γ{2} = {(θ1, θ2) ∈ R2; cp( ¯A{2} +∗ +(eθ2)) > 1}, +Γ{1,2} = {(θ1, θ2) ∈ R2; cp(A{1,2} +∗,∗ +(eθ1, eθ2)) > 1}, +where, for a nonnegative square matrix A with a finite or countable dimension, cp(A) denote the +convergence parameter of A, i.e., cp(A) = sup{r ∈ R+; �∞ +n=0 rnAn < ∞, entry-wise}. We have +cp(A{1,2} +∗,∗ +(eθ1, eθ2)) = spr(A{1,2} +∗,∗ +(eθ1, eθ2))−1, where for a square complex matrix A, spr(A) is the +spectral radius of A. By Lemma A.1 of Ozawa [13], cp( ¯A{1} +∗ +(eθ))−1 and cp( ¯A{2} +∗ +(eθ))−1 are log- +convex in θ, and the closures of Γ{1} and Γ{2} are convex sets; spr( ¯A{1,2} +∗ +(eθ1, eθ2)) is also log-convex +in (θ1, θ2), and the closure of Γ{1,2} is a convex set. Furthermore, by Proposition B.1 of Ozawa +[13], Γ{1,2} is bounded under Assumption 2.2. We depict an example of the domains Γ{1,2}, Γ{1} +and Γ{2} in Fig. 2. +We define several extreme values and several functions with respect to the domains. For i ∈ +{1, 2}, define θmin +i +and θmax +i +as +θmin +i += inf{θi ∈ R : (θ1, θ2) ∈ Γ{1,2}}, +θmax +i += sup{θi ∈ R : (θ1, θ2) ∈ Γ{1,2}}, +and for a direction vector c = (c1, c2) ∈ N2, θmax +c +as +θmax +c += sup{c1θ1 + c2θ2 : (θ1, θ2) ∈ Γ{1,2}}. +For θ1 ∈ [θmin +1 +, θmax +1 +], there exist two real solutions to equation spr(A{1,2} +∗,∗ +(eθ1, eθ2)) = 1, counting +multiplicity. Denote them by θ2 = η2(θ1) and θ2 = ¯η2(θ1), respectively, where η2(θ1) ≤ ¯η2(θ1). For +5 + +C101 + C202 = 0max +02 个 +01 = 0 +02 = 02 +r(1) +spr +amax +n2 (0) +r(2] +r(1,2] +n2(0) +> +0 +0 +1 +0 +01 +0Figure 3: Classification +θ2 ∈ [θmin +2 +, θmax +2 +], also denote by θ1 = η1(θ2) and θ1 = ¯η1(θ2) the two real solutions to the equation +spr(A{1,2} +∗,∗ +(eθ1, eθ2)) = 1, where η1(θ2) ≤ ¯η1(θ2). For i ∈ {1, 2}, define θ∗ +i as +θ∗ +i = sup{θi ∈ R : (θ1, θ2) ∈ Γ{i}}. +For another characterization of θ∗ +i , see Proposition 3.7 of Ozawa [10], where θ∗ +i is denoted by z0. +In terms of these points and functions, we geometrically classify the model into four types +according to Section 4.1 of [14]. Define two points Q1 and Q2 as Q1 = (θ∗ +1, ¯η2(θ∗ +1)) and Q2 = +(¯η1(θ∗ +2), θ∗ +2), respectively. Using these points, we define the following classification (see Fig. 3). +Type 1: θ∗ +1 ≥ ¯η1(θ∗ +2) and ¯η2(θ∗ +1) ≤ θ∗ +2, +Type 2: θ∗ +1 < ¯η1(θ∗ +2) and ¯η2(θ∗ +1) > θ∗ +2, +Type 3: θ∗ +1 ≥ ¯η1(θ∗ +2) and ¯η2(θ∗ +1) > θ∗ +2, +Type 4: θ∗ +1 < ¯η1(θ∗ +2) and ¯η2(θ∗ +1) ≤ θ∗ +2. +By Proposition 2.3 of [14], for any direction vector c = (c1, c2) ∈ N2, the asymptotic decay +rate in the direction c is space homogeneous. Hence, we denote it by ξc, which satisfies, for any +(x, j) ∈ Z2 ++ × S0, +ξc = − lim +k→∞ +1 +n log ν(x+kc.j). +(2.4) +The asymptotic decay rate ξc has already been obtained in [14], and as described in Section 4.1 of +[14], it is given as follows. +Theorem 2.1. Let c = (c1, c2) be an arbitrary direction vector in N2. +Type 1: +ξc = +� +� +� +c1θ∗ +1 + c2¯η2(θ∗ +1) +if − c1 +c2 < ¯η′ +2(θ∗ +1), +θmax +c +if ¯η′ +2(θ∗ +1) ≤ − c1 +c2 ≤ ¯η′ +1(θ∗ +2)−1, +c1¯η1(θ∗ +2) + c2θ∗ +2 +if − c1 +c2 > ¯η′ +1(θ∗ +2)−1, +where ¯η′ +2(x) = +d +dx ¯η2(x) and ¯η′ +1(x) = +d +dx ¯η1(x). +Type 2: +ξc = +� +� +� +c1θ∗ +1 + c2¯η2(θ∗ +1) +if − c1 +c2 ≤ θ∗ +2−¯η2(θ∗ +1) +¯η1(θ∗ +2)−θ∗ +1 , +c1¯η1(θ∗ +2) + c2θ∗ +2 +if − c1 +c2 > θ∗ +2−¯η2(θ∗ +1) +¯η1(θ∗ +2)−θ∗ +1 . +6 + +A. +0* +02 +r(1,2] +> +01 +0* 1 +1 +Type 1 +Type 2 +Type 3 +Type 4Type 3: ξc = c1¯η1(θ∗ +2) + c2θ∗ +2. +Type 4: ξc = c1θ∗ +1 + c2¯η2(θ∗ +1). +The asymptotic decay function hc(k) in the direction c is defined as the function that satisfies, +for some positive vector gc, +lim +k→∞ +νkc +hc(k) = gc. +(2.5) +It is given as follows. +Theorem 2.2. Let c be an arbitrary direction vector in N2. +hc(k) = +� +k− 1 +2 (2l−1)e−ξck +if ¯η′ +2(θ∗ +1) < − c1 +c2 < ¯η′ +1(θ∗ +2)−1 in Type 1, +e−ξck +otherwise, +as k → ∞. +(2.6) +where l is some positive integer. +Except for the case where ¯η′ +2(θ∗ +1) ≤ − c1 +c2 ≤ ¯η′ +1(θ∗ +2)−1 in Type 1, Theorem 2.2 has already been +proved in [14], see Theorem 3.2 of [14]. +Hence, to this end, it suffices to prove the following +proposition. +Proposition 2.1. Assume Type 1 and set c = (c1, c2) = (1, 1). +Then, the asymptotic decay +function hc(k) is given as +hc(k) = +� +k− 1 +2 (2l−1)e−θmax +c +k +if ¯η′ +2(θ∗ +1) < − c1 +c2 = −1 < ¯η′ +1(θ∗ +2)−1, +e−θmax +c +k +if ¯η′ +2(θ∗ +1) = −1 or ¯η′ +1(θ∗ +2) = −1, +(2.7) +where l is some positive integer. +From this proposition, we can obtain the same result for a general direction vector c ∈ N2, by +using the block state process derived from the original 2d-QBD process; See Section 3.3 of [14]. +We, therefore, prove the proposition in Section 4. +Remark 2.1. From the corresponding results for a 2d-RRW without a background process obtained +in Malyshev [6], it is expected that the value of l in Theorem 2.2 is one, i.e., hc(k) = k− 1 +2 e−ξck. +3 +Preliminaries +Let z and w be complex valuables unless otherwise stated. For a positive number r, denote by ∆r +the open disk of center 0 and radius r on the complex plain, and ∂∆r the circle of the same center +and radius. We denote by ¯∆r the closure of ∆r. For a, b ∈ R+ such that a < b, let ∆a,b be an open +annular domain on C defined as ∆a,b = {z ∈ C : a < |z| < b}. We denote by ¯∆a,b the closure of +∆a,b. For r > 0, ε > 0 and θ ∈ [0, π/2), define +˜∆r(ε, θ) = {z ∈ C : |z| < r + ε, z ̸= r, | arg(z − r)| > θ}. +For r > 0, we denote by “ ˜∆r ∋ z → r” that ˜∆r(ε, θ) ∋ z → r for some ε > 0 and some θ ∈ [0, π/2). +In the rest of the paper, instead of proving that a function f(z) is analytic in ˜∆r(ε, θ) for some ε > 0 +and θ ∈ [0, π/2), we often demonstrate that the function f(z) is analytic in ∆r and on ∂∆r \ {r}. +In order to give general results, this section is described independently from other parts of the +article. +7 + +3.1 +Analytic extension of a G-matrix function +First, we define a G-matrix function according to Ozawa and Kobayashi [11]. For i, j ∈ {−1, 0, 1}, +let Ai,j be a substochastic matrix with a finite dimension s0, and define the following matrix +functions: +A∗,j(z) = +� +i∈{−1,0,1} +ziAi,j, j = −1, 0, 1, +A∗,∗(z, w) = +� +i,j∈{−1,0,1} +ziwjAi,j. +We assume the following condition throughout this subsection. +Assumption 3.1. A∗,∗(1, 1) is stochastic. +Let χ(z, w) be the spectral radius of A∗.∗(z, w), i.e., χ(z, w) = spr(A∗,∗(z, w)), and Γ be a +domain on R2 defined as +Γ = {(θ1, θ2) ∈ R2 : χ(eθ1, eθ2) < 1}. +We assume the following condition throughout this subsection. +Assumption 3.2. The Markov modulated random walk on Z2 × {1, 2, ..., s0} that is governed by +{Ai,j; i, j ∈ {−1, 0, 1}} is irreducible and aperiodic. +Under this assumption, A∗,∗(1, 1) is also irreducible and aperiodic. Furthermore, by Lemma 2.2 +of [11], Γ is bounded. Since χ(eθ1, eθ2) is convex in (θ1, θ2) ∈ R2, the closure of Γ is a convex set. +Define extreme points θmin +1 +and θmax +2 +as follows: +θmin +1 += +inf +(θ1,θ2)∈Γ θ1, +θmax +1 += +sup +(θ1,θ2)∈Γ +θ1. +For θ1 ∈ [θmin +1 +, θmax +1 +], let θ2(θ1) and ¯θ2(θ1) be the two real solutions to equation χ(eθ1, eθ2) = 1, +counting multiplicity, where θ2(θ1) ≤ ¯θ2(θ1). We set zmin +1 += eθmin +1 +and zmax +1 += eθmax +1 +. For n ≥ 1, +define the following set of index sequences: +In = +� +i(n) ∈ {−1, 0, 1}n : +k +� +l=1 +il ≥ 0 for k ∈ {1, 2, ..., n − 1} and +n +� +l=1 +il = −1 +� +, +where i(n) = (i1, i2, ..., in), and define the following matrix function: +Dn(z) = +� +i(n)∈In +A∗,i1(z)A∗,i2(z) · · · A∗,in(z). +Define a matrix function G(z) as +G(z) = +∞ +� +n=1 +Dn(z). +By Lemma 4.1 of [11], this matrix series absolutely converges entry-wise in z ∈ ¯∆zmin +1 +,zmax +1 +. We call +this G(z) the G-matrix function generated from {Ai,j; i, j ∈ {−1, 0, 1}}. For z ∈ ¯∆zmin +1 +,zmax +1 +, G(z) +satisfies the inequality |G(z)| ≤ G(|z|) and the following matrix quadratic equation: +A∗,−1(z) + A∗,0(z)G(z) + A∗,1(z)G(z)2 = G(z). +(3.1) +Furthermore, for z ∈ [zmin +1 +, zmax +1 +], it is the minimum nonnegative solution to equation (3.1). Hence, +G(z) is an extension of a usual G-matrix in the queueing theory; see, for example, [7]. By Proposi- +tion 2.5 of [11], we see that, for z ∈ [zmin +1 +, zmax +1 +], the Perron-Frobenius eigenvalue of G(z) is given +by eθ2(log z), i.e., spr(G(z)) = eθ2(log z). By Lemma 4.1 of [11], G(z) satisfies +I − A∗,∗(z, w) = w−1 (I − A∗,0(z) − wA∗,1(z) + A∗,1(z)G(z)) (wI − G(z)). +(3.2) +By Lemma 4.2 of [11], the following property holds true for G(z). +8 + +Lemma 3.1. G(z) is entry-wise analytic in the open annular domain ∆zmin +1 +,zmax +1 +. +We give the eigenvalues of G(z) according to [11]. Note that our final aim in this subsection is +to give an analytic extension of G(z) through its Jordan canonical form without assuming all the +eigenvalues of G(z) are distinct. On the other hand, in [11], the eigenvalues were assumed to be +distinct. Define a matrix function L(z, w) as +L(z, w) = zw(I − A∗,∗(z, w)). +Each entry of L(z, w) is a polynomial in z and w with at most degree 2 for each variable. We use +a notation Ξ, defined as follows. Let f(z, w) be an irreducible polynomial in z and w and assume +its degree with respect to w is m ≥ 1. Let a(z) be the coefficient of wm in f(z, w). Define a point +set Ξ(f) as +Ξ(f) = {z ∈ C : a(z) = 0 or (f(z, w) = 0 and fw(z, w) = 0 for some w ∈ C)}, +where fw(z, w) = (∂/∂w)f(z, w). Each point in Ξ(f) is an algebraic singularity of the algebraic +function w = α(z) defined by polynomial equation f(z, w) = 0. For each point z ∈ C \ Ξ(f), +f(z, w) = 0 has just m distinct solutions, which correspond to the m branches of the algebraic +function. Let φ(z, w) be a polynomial in z and w defined as +φ(z, w) = det L(z, w) +and mφ its degree with respect to w, where s0 ≤ mφ ≤ 2s0. Let α1(z), α2(z), ..., αmφ(z) be the +mφ branches of the algebraic function w = α(z) defined by the polynomial equation φ(z, w) = 0, +counting multiplicity. We number the brunches so that they satisfy the following: +(1) For every z ∈ ¯∆zmin +1 +,zmax +1 +and for every k ∈ {1, 2, ..., s0}, |αk(z)| ≤ eθ2(log |z|). +(2) For every z ∈ ¯∆zmin +1 +,zmax +1 +and for every k ∈ {s0 + 1, s0 + 2, ..., mφ}, |αk(z)| ≥ e¯θ2(log |z|). +(3) For every z ∈ [zmin +1 +, zmax +1 +], αs0(z) = eθ2(log z) and αs0+1(z) = e¯θ2(log z). +This is possible by Lemma 4.3 of [11]. By Lemmas 4.3 and 4.4 of [11], the G-matrix function of +G(z) satisfies the following property. +Lemma 3.2. For every z ∈ ¯∆zmin +1 +,zmax +1 +, the eigenvalues of G(z) are given by α1(z), α2(z), ..., +αs0(z). +Without loss of generality, we assume that, for some nφ ∈ N and l1, l2, ..., lnφ ∈ N, the polynomial +φ(z, w) is factorized as +φ(z, w) = f1(z, w)l1f2(z, w)l2 · · · fnφ(z, w)lnφ, +(3.3) +where fk(z, w), k = 1, 2, ..., nφ, are irreducible polynomials in z and w and they are relatively +prime. Since the field of coefficients of polynomials is C, this factorization is unique. For every +k ∈ {1, 2, ..., mφ}, αk(z) is a branch of the algebraic function w = α(z) defined by the polynomial +equation fn(z, w) = 0 for some n ∈ {1, 2, ..., nφ}. We denote such n by q(k), i.e., fq(k)(z, αk(z)) = 0. +Since αs0(z) is the Perron-Frobenius eigenvalue of G(z) when z ∈ [zmin +1 +, zmax +1 +], the multiplicity of +αs0(z) is one and we have lq(s0) = 1. Define a point set E1 as +E1 = +nφ +� +n=1 +Ξ(fn). +9 + +Since, for every n, the polynomial fn(z, w) is irreducible and not identically zero, the point set E1 +is finite. Every branch αk(z) is analytic in C \ E1. Define a point set E2 as +E2 = {z ∈ C \ E1 : fn(z, w) = fn′(z, w) = 0 +for some n, n′ ∈ {1, 2, ..., nφ} such that n ̸= n′ and for some w ∈ C}. +Since, for any n, n′ such that n ̸= n′, fn(z, w) and fn′(z, w) are relatively prime, the point set E2 +is finite. Note that every branch αk(z) is analytic in a neighborhood of any z0 ∈ E2. For every +k ∈ {1, 2, ..., mφ} and for every z ∈ C \ (E1 ∪ E2), the multiplicity of αk(z) as a zero of det L(z, w) +is equal to lq(k). This means that, for every z ∈ ¯∆zmin +1 +,zmax +1 +\ (E1 ∪ E2), the multiplicity of the +eigenvalue αk(z) of G(z) is lq(k), which does not depend on z. Define a positive integer m0 as +m0 = +s0 +� +k=1 +1 +lq(k) +. +This m0 is the number of different branches in {αi(z) : i = 1, 2, ..., s0} when z ∈ C \ (E1 ∪ E2). +Denote the different branches by ˇαk(z), k = 1, 2, ..., m0, so that ˇαm0(z) = αs0(z). Instead of using +q(k), we define a function ˇq(k) so that lˇq(k) indicates the multiplicity of ˇαk(z) when z ∈ C\(E1∪E2). +We always have lˇq(m0) = 1. +We give the Jordan normal form of G(z). Define a domain Ω as Ω = ∆zmin +1 +,zmax +1 +\ (E1 ∪ E2). For +k ∈ {1, 2, ..., m0} and for i ∈ {1, 2, ..., lˇq(k)}, define a positive integer tk,i as +tk,i = min +z∈Ω dim Ker (ˇαk(z)I − G(z))i +and a point set Gk,i as +Gk,i = {z ∈ Ω : dim Ker (ˇαk(z)I − G(z))i > tk,i}. +Since ˇαk(z) and G(z) are analytic in Ω, we see from the proof of Theorem S6.1 of [3] that each Gk,i +is an empty set or a set of discrete complex numbers. For k ∈ {1, 2, ..., m0} and i ∈ {1, 2, ..., lˇq(k)}, +define a nonnegative integer sk,i as +sk,i = 2tk,i − tk,i+1 − tk,i−1, +where tk,0 = 0 and tk,lˇq(k)+1 = lˇq(k). For k ∈ {1, 2, ..., m0}, define a positive integer mk,0 and point +set EG +k as +mk,0 = tk,1, +EG +k = +lˇq(k) +� +i=1 +Gk,i. +When z ∈ Ω \ EG +k , this mk,0 is the number of Jordan blocks of G(z) with respect to the eigenvalue +ˇαk(z) and, for i ∈ {1, 2, ..., lˇq(k)}, sk,i is the number of Jordan blocks whose dimension is i. Hence, +the Jordan normal form of G(z) takes a common form in z ∈ Ω \ �m0 +k=1 EG +k . For k ∈ {1, 2, ..., m0} +and for i ∈ {1, 2, ..., mk,0}, denote by mk,i the dimension of the i-th Jordan block of G(z) with +respect to the eigenvalue ˇαk(z), where we number the Jordan blocks so that if i ≤ i′, mk,i ≥ mk,i′. +For each k ∈ {1, 2, ..., m0}, they satisfy �mk,0 +i=1 mk,i = lˇq(k). Denote by Jn(λ) the n-dimensional +Jordan block of eigenvalue λ. For z ∈ Ω \ �m0 +k=1 EG +k , the Jordan normal form of G(z), JG(z), is +given by +JG(z) = diag(Jmk,i(ˇαk(z)), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0), +(3.4) +where mm0,0 = 1 and Jmm0,1(ˇαm0(z)) = αs0(z). Note that the matrix function JG(z) is defined on +C and analytic in C \ E1. An analytic extension of G(z) is given by the following theorem. +10 + +Theorem 3.1. There exist vector functions: +ˇvL +k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i, +such that they are analytic in C \ E1 and satisfy for every z ∈ ∆zmin +1 +,zmax +1 +\ (E1 ∪ E0) that +G(z) = T L(z)JG(z)(T L(z))−1, +(3.5) +where E0 is a set of discrete complex numbers and matrix function T L(z) is defined as +T L(z) = +�ˇvL +k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i +� +. +Since the proof of Theorem 3.1 is elementary and very lengthy, we give it in Appendix A. In +Theorem 3.1, {ˇvL +k,i,j(z)} is the set of the generalized eigenvectors of G(z), but we denote them with +superscript L since they are generated from the matrix function L(z, w); see Appendix A. Define a +point set EL +T as +EL +T = {z ∈ C \ E1 : det T L(z) = 0}, +which is an empty set or a set of discrete complex numbers since det T L(z) is not identically zero. +Define a matrix function ˇG(z) as +ˇG(z) = T L(z)JG(z)(T L(z))−1 = T L(z)JG(z) adj(T L(z)) +det(T L(z)) +. +(3.6) +Then, it is entry-wise analytic in C\(E1∪EL +T ). By Theorem 3.1 and the identity theorem for analytic +functions, this ˇG(z) is an analytic extension of the matrix function G(z). Hence, we denote ˇG(z) +by G(z). By Lemma 3.1, G(z) is entry-wise analytic in ∆zmin +1 +,zmax +1 +. The following corollary asserts +that G(z) is also analytic on the outside boundary of ∆zmin +1 +,zmax +1 +except for the point z = zmax +1 +. +Corollary 3.1. The extended G-matrix function G(z) is entry-wise analytic on ∂∆zmax +1 +\ {zmax +1 +}. +Since this corollary can be proved in a manner similar to that used in the proof of Lemma 4.7 +of [11], we omit it. +Denote by ˇuL +m0,1,1(z) the last row of the matrix function (T L(z))−1, and define a diagonal +matrix function Js0(z) as Js0(z) = diag +� +0 +· · · +0 +αs0(z) +� +, where αs0(z) = ˇαm0(z). Then, since +mm0,0 = 1 and mm0,1 = 1, we obtain the following decomposition of G(z) from (3.6): +G(z) = G†(z) + αs0(z)ˇvL +m0,1,1(z)ˇuL +m0,1,1(z), +(3.7) +where +G†(z) = T L(z)(JG(z) − Js0(z))(T L(z))−1. +By the definition, G(z) satisfies, for n ≥ 1, +G(z)n = G†(z)n + αs0(z)nˇvL +m0,1,1(z)ˇuL +m0,1,1(z), +(3.8) +and G†(z), for z ∈ ¯∆zmin +1 +,zmax +1 +, spr(G†(z)) ≤ spr(G†(|z|) < spr(G(|z|)) = αs0(|z|). Furthermore, +in a neighborhood of z = zmax +1 +, we have spr(G†(z)) < αs0(zmax +1 +). Since the point z = zmax +1 +is a +branch point of ˇαm0(z) (= αs0(z)), there exists a function ˜αs0(ζ) being analytic in a neighborhood +of ζ = 0 and satisfying +ˇαm0(z) = αs0(z) = ˜αs0((zmax +1 +− z) +1 +2 ). +11 + +Let ˜vs0(ζ) be a vector function satisfying +L(zmax +1 +− ζ2, ˜αs0(ζ))˜vs0(ζ) = 0, +where ˜vs0(ζ) is elementwise analytic in a neighborhood of ζ = 0. +Denote by ˜T(ζ) the matrix +function given by replacing the last column of T L(zmax +1 +− ζ2) with ˜vs0(ζ) and by ˜us0(ζ) the last +row of ˜T(ζ)−1. By the definition, ˜T(ζ) as well as ˜us0(ζ) is entry-wise analytic in a neighborhood +of ζ = 0. Define a diagonal matrix function ˜Js0(ζ) as ˜Js0(ζ) = diag +� +0 +· · · +0 +˜αs0(ζ) +� +. For later +use, we give the following lemma. +Lemma 3.3. There exists a matrix function ˜G(ζ) being entry-wise analytic in a neighborhood of +ζ = 0 and satisfying G(z) = ˜G((zmax +1 +−z) +1 +2 ) in a neighborhood of z = zmax +1 +. This ˜G(ζ) is represented +as +˜G(ζ) = ˜G†(ζ) + ˜αs0(ζ)˜vs0(ζ)˜us0(ζ), +(3.9) +where ˜G†(ζ) is a matrix function being entry-wise analytic in a neighborhood of ζ = 0 and satisfying +G†(z) = ˜G†((zmax +1 +− z) +1 +2 ) in a neighborhood of z = zmax +1 +. In a neighborhood of ζ = 0, spr( ˜G†(ζ)) < +˜αs0(0) = αs0(zmax +1 +). +Proof. Give ˜G†(ζ) as +˜G†(ζ) = ˜T(ζ)(JG(zmax +1 +− ζ2) − Js0(zmax +1 +− ζ2)) ˜T(ζ)−1. +Then, by (3.7), we obtain the results of the lemma. +The following limit with respect to αs0(z) (= ˇαm0(z)) is given by Proposition 5.5 of [11] (also +see Lemma 10 of [4]). +Lemma 3.4. +lim +˜∆zmax +1 +∋z→zmax +1 +αs0(zmax +1 +) − αs0(z) +(zmax +1 +− z) +1 +2 += −αs0,1 = +√ +2 +� +−¯ζ1,w2(ζ2(zmax +1 +)) +> 0, +(3.10) +where z = ¯ζ1(w) is the larger one of two real solutions to equation χ(z, w) = 1 and ¯ζ1,w2(w) = +(d2/dw2) ¯ζ1(w). +Let R(z) be the rate matrix function generated from {Ai,j; i, j = −1, 0, 1}; for the definition of +R(z), see Section 4.1 of [11]. Define a matrix function N(z) as +N(z) = (I − A∗,0(z) − A∗,1(z)G(z))−1. +N(z) is well defined for every z ∈ ¯∆zmin +1 +,zmax +1 +. The extended G(z) satisfies the following property. +Lemma 3.5. +lim +˜∆zmax +1 +∋z→zmax +1 +G(zmax +1 +) − G(z) +(zmax +1 +− z) +1 +2 += −G1 += −αs0,1N(zmax +1 +)vR(zmax +1 +)uG +s0(zmax +1 +) ≥ O, ̸= O, +(3.11) +where uG +s0(zmax +1 +) is the left eigenvector of G(zmax +1 +) with respect to the eigenvalue eθ2(log zmax +1 +) = +αs0(zmax +1 +), vR(zmax +1 +) the right eigenvector of R(zmax +1 +) with respect to the eigenvalue e−¯θ2(log zmax +1 +) = +e−θ2(log zmax +1 +) and they satisfy uG +s0(zmax +1 +)N(zmax +1 +)vR(zmax +1 +) = 1. +Since this lemma can be proved in a manner similar to that used in the proof of Proposition +5.6 of [11], we omit it. +12 + +3.2 +G-matrix in the reverse direction and its properties +Let A−1, A0 and A1 be square nonnegative matrices with a finite dimension. Define a matrix +function A∗(z) and matrix Q as +A∗(z) = z−1A−1 + A0 + zA1, +(3.12) +Q = +� +� +� +� +� +A0 +A1 +A−1 +A0 +A1 +A−1 +A0 +A1 +... +... +... +� +� +� +� +� . +(3.13) +We assume: +(a1) Q is irreducible. +(a2) The infimum of the maximum eigenvalue of A∗(eθ) in θ ∈ R is less than or equal to 1, i.e., +infθ∈R spr(A∗(eθ)) ≤ 1. +Then, there are two real solutions to equation cp(A∗(eθ)) = 1, counting multiplicity, see comments +to Condition 2.6 in [13]. We denote the solutions by θ and ¯θ, where θ ≤ ¯θ. The rate matrix +and G-matrix generated from the triplet {A−1, A0, A1} also exist; we denote them by R and G, +respectively. R and G are the minimal nonnegative solutions to the following matrix quadratic +equations: +R = R2A−1 + RA0 + A1, +(3.14) +G = A−1 + A0G + A1G2. +(3.15) +We have +I − A∗(z) = (I − zR)(I − H)(I − z−1G), +(3.16) +spr(R) = e−¯θ, +spr(G) = eθ, +(3.17) +where H = A0 + A1G; see, for example, Lemma 2.2 of [13]. We define a rate matrix and G-matrix +in the reverse direction generated from the triplet {A−1, A0, A1}, denoted by Rr and Gr, as the +minimal nonnegative solutions to the following matrix quadratic equations: +Rr = (Rr)2A1 + RrA0 + A−1, +(3.18) +Gr = A1 + A0Gr + A−1(Gr)2. +(3.19) +In other words, Rr and Gr are, respectively, the rate matrix and G-matrix generated from the +triplet by exchanging A−1 and A1. Since z−1A1 + A0 + zA−1 = A∗(z−1), we obtain by (3.16) and +(3.17) that +I − A∗(z−1) = (I − zRr)(I − Hr)(I − z−1Gr), +(3.20) +spr(Rr) = eθ, +spr(Gr) = e−¯θ, +(3.21) +where Hr = A0 + A−1Gr. We use the following property in the proof of Proposition 4.5. +Lemma 3.6. Let v be the right eigenvector of G with respect to the eigenvalue eθ and vr that of +Gr with respect to the eigenvalue e−¯θ, i.e., Gv = eθv and Grvr = e−¯θvr. If θ = ¯θ, we have v = vr, +up to multiplication by a positive constant. +Proof. By (3.16) and (3.20), we obtain +A∗(eθ)v = v, +A∗(e +¯θ)vr = A∗(eθ)vr = vr. +Since spr(A∗(eθ)) = 1 and A∗(eθ) is irreducible, the right eigenvector of A∗(eθ) with respect to the +eigenvalue of 1 is unique, up to multiplication by a positive constant. This implies v = vr. +13 + +4 +Proof of Proposition 2.1 +4.1 +Methodology and outline of the proof +Define the vector generating function of the stationary probabilities in direction c ∈ N2, ϕc(z), as +ϕc(z) = +∞ +� +k=0 +zkνkc. +Also define zmin +c +and zmax +c +as zmin +c += eθmin +c +and zmax +c += eθmax +c +, respectively. +Hereafter, we set +c = (1, 1). In order to obtain the asymptotic function of the stationary tail probability in the +direction c = (1, 1), we apply the following lemma to the vector generating function ϕc(z). +Lemma 4.1 (Theorem VI.4 of Flajolet and Sedgewick [2]). Let f be a generating function of a +sequence of real numbers {an, n ∈ Z+}, i.e., f(z) = �∞ +n=0 anzn. If f(z) is singular at z = z0 > 0 +and analytic in ˜∆z0(ε, θ) for some ε > 0 and some θ ∈ [0, π/2) and if it satisfies +lim +˜∆z0∋z→z0 +(z0 − z)αf(z) = c0 +(4.1) +for α ∈ R \ {0, −1, −2, ...} and some nonzero constant c0 ∈ R, then +lim +n→∞ +�nα−1 +Γ(α) z−n +0 +�−1 +an = c +(4.2) +for some real number c, where Γ(z) is the gamma function. This means that the asymptotic function +of the sequence {an} is given by nα−1z−n +0 . +For the purpose, we prove the following propositions in Section 4.2. +Proposition 4.1. Assume Type 1. If ¯η′ +1(θ∗ +2) ≤ −c1/c2 = −1 ≤ 1/¯η′ +2(θ∗ +1), the vector function ϕc(z) +is elementwise analytic in ˜∆zmax +c +(ε, θ) for some ε > 0 and some θ ∈ [0, π/2). +Proposition 4.2. Assume Type 1. If ¯η′ +1(θ∗ +2) ≤ −c1/c2 = −1 ≤ 1/¯η′ +2(θ∗ +1), there exist a vector +function ˜ϕc(ζ) being meromorphic in a neighborhood of ζ = 0 and satisfying ϕc(z) = ˜ϕc((zmax +c +− +z) +1 +2 ) in a neighborhood of z = zmax +c +. If ¯η′ +1(θ∗ +2) < −1 < 1/¯η′ +2(θ∗ +1), the point ζ = 0 is a pole of ˜ϕc(ζ) +with at most order one; if ¯η′ +1(θ∗ +2) = −1 or ¯η′ +2(θ∗ +1) = −1, it is a pole of ˜ϕc(ζ) with at most order two. +By Proposition 4.2, if ¯η′ +1(θ∗ +2) < −1 < 1/¯η′ +2(θ∗ +1), the Puiseux series of ϕc(z) is represented as +ϕc(z) = +∞ +� +k=−1 +ϕc +1,k(zmax +c +− z) +k +2 , +(4.3) +where {ϕc +1,k} is a series of coefficient vectors; if ¯η′ +1(θ∗ +2) = −1 or ¯η′ +2(θ∗ +1) = −1, it is represented as +ϕc(z) = +∞ +� +k=−2 +ϕc +2,k(zmax +c +− z) +n +2 , +(4.4) +where {ϕc +2,k} is a series of coefficient vectors. Let l be a positive integer such that ϕc +1,l−2 ̸= 0 and +ϕc +1,k−2 = 0 for all positive integer k less than l. Then, applying Lemma 4.1 to (4.3), we obtain +hc(k) = k− 1 +2 (2l−1)(zmax +c +)−k = k− 1 +2 (2l−1)e−θmax +c +k. +This completes the former half of the proof of Proposition 2.1. If ¯η′ +1(θ∗ +2) = −1 or ¯η′ +2(θ∗ +1) = −1, +ϕc(z) satisfies the following property, which will be proved in Section 4.2. +14 + +Proposition 4.3. Assume Type 1. Then, we have, for some positive vectors uc +1 and uc +2, +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z)ϕc(z) = +� +� +� +uc +1 +if ¯η′ +1(θ∗ +2) = −1 and ¯η′ +2(θ∗ +1) < −1, +uc +2 +if ¯η′ +1(θ∗ +2) < −1 and ¯η′ +2(θ∗ +1) = −1, +uc +1 + uc +2 +if ¯η′ +1(θ∗ +2) = ¯η′ +2(θ∗ +1) = −1. +(4.5) +Hence, ϕc +2,−2 is positive, and by Lemma 4.1, we obtain +hc(k) = (zmax +c +)−k = e−θmax +c +k. +This completes the latter half of the proof of Proposition 2.1. +Remark 4.1. Assume Type 1 and ¯η′ +1(θ∗ +2) < −c1/c2 = −1 < 1/¯η′ +2(θ∗ +1). If the vector function ϕc(z) +diverges at z = zmax +c +, the coefficient vector ϕc +1,−1 in (4.3) must be nonzero and , by Lemma 4.1, we +have +hc(k) = k− 1 +2 (zmax +c +)−k = k− 1 +2 e−θmax +c +k. +4.2 +Proof of Propositions 4.1, 4.2 and 4.3 +Recall that the direction vector c is set as c = (1, 1). Notation of this subsection follows [14]. +Denote by Φ{1,2} = (Φ{1,2} +x,x′ ; x, x′ ∈ Z2) the fundamental matrix (potential matrix) of P {1,2}, i.e., +Φ{1,2} = �∞ +n=0(P {1,2})n, where P {1,2} = (P {1,2} +x,x′ ; x, x′ ∈ Z2) is the transition probability matrix of +the induced MA-process {Y {1,2} +n +}. For x ∈ Z2, define the matrix generating function of the blocks +of Φ{1,2} in direction c, Φc +x,∗(z), as +Φc +x,∗(z) = +∞ +� +k=−∞ +zkΦ{1,2} +x,kc . +According to equation (3.3) of [14], we divide ϕc(z) into three parts as follows: +ϕc(z) = ϕc +0(z) + ϕc +1(z) + ϕc +2(z), +(4.6) +where +ϕc +0(z) = +� +i1,i2∈{−1,0,1} +ν(0,0)(A∅ +i1,i2 − A{1,2} +i1,i2 )Φc +(i1,i2),∗(z), +(4.7) +ϕc +1(z) = +∞ +� +k=1 +� +i1,i2∈{−1,0,1} +ν(k,0)(A{1} +i1,i2 − A{1,2} +i1,i2 )Φc +(k+i1,i2),∗(z), +(4.8) +ϕc +2(z) = +∞ +� +k=1 +� +i1,i2∈{−1,0,1} +ν(0,k)(A{2} +i1,i2 − A{1,2} +i1,i2 )Φc +(i1,k+i2),∗(z). +(4.9) +According to [14], we focus on ϕc +2(z) and consider another skip-free MA-process generated from +{Y {1,2} +n +}. The MA-process is { ˆY n} = {( ˆXn, ˆJn)} = {( ˆX1,n, ˆX2,n), ( ˆRn, ˆJn)}, where ˆX1,n = X{1,2} +1,n +, +ˆX2,n and ˆRn are the quotient and remainder of X{1,2} +2,n +− X{1,2} +1,n +divided by 2, respectively, and +ˆJn = J{1,2} +n +. The state space of { ˆY n} is Z2 × {0, 1} × S0 and the additive part { ˆXn} is skip +free. From the definition, if ˆXn = (x1, x2) and ˆRn = r in the new MA-process, it follows that +X{1,2} +1,n += x1, X{1,2} +2,n += x1 + 2x2 + r in the original MA-process. Hence, ˆY n = (k, 0, 0, j) means +15 + +Y {1,2} +n += (k, k, j). Denote by ˆP = ( ˆPx,x′; x, x′ ∈ Z2) the transition probability matrix of { ˆY n}, +which is given as +ˆPx,x′ = +� +ˆA{1,2} +x′−x, +if x′ − x ∈ {−1, 0, 1}2, +O, +otherwise, +where +ˆA{1,2} +−1,1 = +� +A{1,2} +−1,1 +O +A{1,2} +−1,0 +A{1,2} +−1,1 +� +, +ˆA{1,2} +0,1 += +� +O +O +A{1,2} +0,1 +O +� +, +ˆA{1,2} +1,1 += +�O +O +O +O +� +, +ˆA{1,2} +−1,0 = +� +A{1,2} +−1,−1 +A{1,2} +−1,0 +O +A{1,2} +−1,−1 +� +, +ˆA{1,2} +0,0 += +� +A{1,2} +0,0 +A{1,2} +0,1 +A{1,2} +0,−1 +A{1,2} +0,0 +� +, +ˆA{1,2} +1,0 += +� +A{1,2} +1,1 +O +A{1,2} +1,0 +A{1,2} +1,1 +� +, +ˆA{1,2} +−1,−1 = +�O +O +O +O +� +, +ˆA{1,2} +0,−1 = +� +O +A{1,2} +0,−1 +O +O +� +, +ˆA{1,2} +1,−1 = +� +A{1,2} +1,−1 +A{1,2} +1,0 +O +A{1,2} +1,−1 +� +. +Denote by ˆΦ = (ˆΦx,x′; x, x′ ∈ Z2) the fundamental matrix of ˆP, i.e., ˆΦ = �∞ +n=0( ˆP)n, and for +x = (x1, x2) ∈ Z2, define a matrix generating function ˆΦx,∗(z) as +ˆΦx,∗(z) = +∞ +� +k=−∞ +zk ˆΦx,(k,0) = +� +Φc +(x1,x1+2x2),∗(z) +Φc +(x1,x1+2x2−1),∗(z) +Φc +(x1,x1+2x2+1),∗(z) +Φc +(x1,x1+2x2),∗(z) +� +. +(4.10) +We consider analytic properties of the matrix function Φc +(x1,x1+2x2),∗(z) through ˆΦ(x1,x2),∗(z). Define +blocks ˆA{2} +i1,i2, i1, i2 ∈ {−1, 0, 1}, as ˆA{2} +−1,1 = ˆA{2} +−1,0 = ˆA{2} +−1,−1 = O and +ˆA{2} +0,1 = +� +O +O +A{2} +0,1 +O +� +, +ˆA{2} +0,0 = +� +A{2} +0,0 +A{2} +0,1 +A{2} +0,−1 +A{2} +0,0 +� +, +ˆA{2} +0,−1 = +� +O +A{2} +0,−1 +O +O +� +, +ˆA{2} +1,1 = +�O +O +O +O +� +, +ˆA{2} +1,0 = +� +A{2} +1,1 +O +A{2} +1,0 +A{2} +1,1 +� +, +ˆA{2} +1,−1 = +� +A{2} +1,−1 +A{2} +1,0 +O +A{2} +1,−1 +� +. +For i1, i2 ∈ {−1, 0, 1}, define the following matrix generating functions: +ˆA{1,2} +∗,i2 (z) = +� +i∈{−1,0,1} +zi ˆA{1,2} +i,i2 , +ˆA{1,2} +i1,∗ (z) = +� +i∈{−1,0,1} +zi ˆA{1,2} +i1,i , +ˆA{2} +∗,i2(z) = +� +i∈{0,1} +zi ˆA{2} +i,i2, +ˆA{2} +i1,∗(z) = +� +i∈{−1,0,1} +zi ˆA{2} +i1,i. +Define a vector generating function ˆϕ2(z) as +ˆϕ2(z) = +�ˆϕ2,1(z) +ˆϕ2,2(z) +� += +∞ +� +k=1 +� +i1,i2∈{−1,0,1} +ˆν(0,k)( ˆA{2} +i1,i2 − ˆA{1,2} +i1,i2 )ˆΦ(i1,k+i2),∗(z), +(4.11) +where, for x = (x1, x2) ∈ Z2 ++, ˆνx = +� +ν(x1,x1+2x2) +ν(x1,x1+2x2+1) +� +and hence, for k ≥ 0, +ˆν(0,k) = +� +ν(0,2k) +ν(0,2k+1) +� +. +By equation (3.9) of [14], ϕc +2(z) is represented as +ϕc +2(z) = ˆϕ2,1(z) + +� +i1,i2∈{−1,0,1} +ν(0,1)(A{2} +i1,i2 − A{1,2} +i1,i2 )Φc +(i1,i2+1),∗(z). +(4.12) +16 + +Hence, we consider analytic properties of the vector function ϕc +2(z) through ˆϕc +2(z) and ˆΦx,∗(z). +Let ˆG0,∗(z) be the G-matrix function generated from the triplet { ˆA{1,2} +∗,−1 (z), ˆA{1,2} +∗,0 +(z), ˆA{1,2} +∗,1 +(z)}. +By equations (3.11) and (3.13) of [14], we have, for x2 ≥ 0, +ˆΦ(x1,x2),∗(z) = zx1 ˆG0,∗(z)x2 ˆΦ(0,0),∗(z), +(4.13) +and this leads us to +ˆϕ2(z) = +∞ +� +k=1 +� +i2∈{−1,0,1} +ˆν(0,k)( ˆA{2} +∗,i2(z) − ˆA{1,2} +∗,i2 (z)) ˆG0,∗(z)k+i2 ˆΦ(0,0),∗(z). +(4.14) +Hence, analytic properties of the vector function ˆϕ2(z) as well as the matrix function ˆΦx,∗(z) can +be clarified through ˆG0,∗(z) and ˆΦ(0,0),∗(z). +By (4.14), ˆϕ2(z) is represented as +ˆϕ2(z) = ˆa(z, ˆG0,∗(z))ˆΦ(0,0),∗(z), +(4.15) +where +ˆa(z, w) = +∞ +� +k=1 +ˆν(0,k) ˆD(z, ˆG0,∗(z))wk−1, +ˆD(z, w) = ˆA{2} +∗,−1(z) + ˆA{2} +∗,0 (z)w + ˆA{2} +∗,1 (z)w2 − Iw. +First, we consider ˆΦ(0,0),∗(z). Let ˆGr +0,∗(z) be the G-matrix function in the reverse direction generated +from the triplet { ˆA{1,2} +∗,−1 (z), ˆA{1,2} +∗,0 +(z), ˆA{1,2} +∗,1 +(z)}, which means that ˆGr +0,∗(z) is the G-matrix function +generated from the triplet by exchanging ˆA{1,2} +∗,−1 (z) and ˆA{1,2} +∗,1 +(z); see Section 3.2. Define a matrix +function ˆU(z) as +ˆU(z) = ˆA{1,2} +∗,−1 (z) ˆGr +0,∗(z) + ˆA{1,2} +∗,0 +(z) + ˆA{1,2} +∗,1 +(z) ˆG0,∗(z). +(4.16) +Then, ˆΦ(0,0),∗(z) in (4.15) is given as +ˆΦ(0,0),∗(z) = +∞ +� +n=0 +ˆU(z)n = (I − ˆU(z))−1 = adj(I − ˆU(z)) +det(I − ˆU(z)) +. +(4.17) +Recall that zmin +c += eθmin +c +and zmax +c += eθmax +c +. +For θ ∈ [θmin +c +, θmax +c +], let (ηR +c,1(θ), ηR +c,2(θ)) and +(ηL +c,1(θ), ηL +c,2(θ)) be the two real roots of the simultaneous equations: +spr(A{1,2} +∗,∗ +(eθ1, eθ2)) = 1, +θ1 + θ2 = θ, +(4.18) +counting multiplicity, where ηL +c,1(θ) ≤ ηR +c,1(θ) and ηL +c,2(θ)) ≥ ηR +c,2(θ). +Note that ηL +c,1(θmax +c +) = +ηR +c,1(θmax +c +) and ηL +c,2(θmax +c +) = ηR +c,2(θmax +c +). By equations (3.18) and (3.32) of [14], we have +spr( ˆG0,∗(eθ)) = e2ηR +c,2(θ). +(4.19) +Since the eigenvalues of ˆGr +0,∗(z) are coincide with those of the rate matrix function generated from +the same triplet { ˆA{1,2} +∗,−1 (z), ˆA{1,2} +∗,0 +(z), ˆA{1,2} +∗,1 +(z)}, we have +spr( ˆGr +0,∗(eθ)) = e−2ηL +c,2(θ). +(4.20) +By Lemmas 3.1 and 3.3 and Corollary 3.1, ˆG0,∗(z) and ˆGr +0,∗(z) satisfy the following properties. +17 + +Proposition 4.4. +(1) The extended G-matrix functions ˆG0,∗(z) and ˆGr +0,∗(z) are entry-wise an- +alytic in ∆zmin +c +,zmax +c +∪ ∂∆zmax +c +\ {zmax +c +}. The point z = zmax +c +is a common branch point of +ˆG0,∗(z) and ˆGr +0,∗(z) with order one. +(2) There exist matrix functions ˜G0,∗(ζ) and ˜Gr +0,∗(ζ) being analytic in a neighborhood of ζ = 0 +and satisfying ˆG0,∗(z) = ˜G0,∗((zmax +c +− z) +1 +2 ) and ˆGr +0,∗(z) = ˜Gr +0,∗((zmax +c +− z) +1 +2 ), respectively, in +a neighborhood of z = zmax +c +. +In order to investigate singularity of ˆΦ(0,0),∗(z) at z = zmax +c +, we give the following proposition. +Proposition 4.5. The maximum eigenvalue of ˆU(zmax +c +) is 1, and it is simple. +Proof. By equation (3.30) of [14], we have spr( ˆA{1,2} +∗,∗ +(zmax +c +, e2ηR +c,2(θmax +c +))) = 1. Let v be the right +eigenvector of ˆA{1,2} +∗,∗ +(zmax +c +, e2ηR +c,2(θmax +c +)) with respect to eigenvalue 1. +Since spr( ˆG0,∗(zmax +c +)) = +e2ηR +c,2(θmax +c +) and spr( ˆGr +0,∗(zmax +c +)) = e−2ηL +c,2(θmax +c +) = e−2ηR +c,2(θmax +c +), we have, by Lemma 3.6, +ˆG0,∗(zmax +c +)v = e2ηR +c,2(θmax +c +)v, +ˆGr +0,∗(zmax +c +)v = e−2ηR +c,2(θmax +c +)v, +Hence, +ˆU(zmax +c +)v = ˆA{1,2} +∗,∗ +(zmax +c +, e2ηR +c,2(θmax +c +))v = 1. +This means that the value of 1 is an eigenvalue of ˆU(zmax +c +), and we obtain spr( ˆU(zmax +c +)) ≥ 1. +Suppose spr( ˆU(zmax +c +)) > 1. Then, since spr( ˆU(eθ)) is convex in θ ∈ R, there exist a positive +θ0 < θmax +c +such that spr( ˆU(eθ0)) = 1. For this θ0, ˆΦ(0,0),∗(z) diverges at z = eθ0 < zmax +c +. This +contradicts Proposition 3.1 of [14], which asserts that ˆΦ(0,0),∗(z) absolutely convergent in z ∈ +∆zmin +c +,zmax +c +. Hence, spr( ˆU(zmax +c +)) ≤ 1, and this implies the maximum eigenvalue of ˆU(zmax +c +) is 1. +Since ˆU(zmax +c +) is irreducible, it is simple. +Let ˆλU(z) be the eigenvalue of ˆU(z) satisfying ˆλU(z) = spr( ˆU(z)) for z ∈ [zmin +c +, zmax +c +]. Let +ˆuU(z) and ˆvU(z) be the left and right eigenvectors of ˆU(z) with respect to the eigenvalue ˆλU(z), +respectively, satisfying ˆuU(z)ˆvU(z) = 1. Define a matrix function ˜U(ζ) as +˜U(ζ) = ˆA{1,2} +∗,−1 (zmax +c +− ζ2) ˜Gr +0,∗(ζ) + ˆA{1,2} +∗,0 +(zmax +c +− ζ2) + ˆA{1,2} +∗,1 +(zmax +c +− ζ2) ˜G0,∗(ζ). +By Proposition 4.4, ˜U(ζ) is entry-wise analytic in a neighborhood of ζ = 0 and satisfies ˆU(z) = +˜U((zmax +c +− z) +1 +2 ) in a neighborhood of z = zmax +c +. Define a matrix function ˜Φ(0,0),∗(ζ) as +˜Φ(0,0),∗(ζ) = (I − ˜U(ζ))−1 = adj(I − ˜U(ζ)) +det(I − ˜U(ζ)) +. +(4.21) +ˆΦ(0,0),∗(z) and ˜Φ(0,0),∗(ζ) satisfy the following properties. +Proposition 4.6. +(1) The matrix function ˆΦ(0,0),∗(z) is entry-wise analytic in ∆zmin +c +,zmax +c +∪∂∆zmax +c +\ +{zmax +c +}. +(2) ˜Φ(0,0),∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0, and the point ζ = 0 is a pole +of ˜Φ(0,0),∗(ζ) with order one. ˆΦ(0,0),∗(z) is represented as ˆΦ(0,0),∗(z) = ˜Φ(0,0),∗((zmax +c +− z) +1 +2 ) in +a neighborhood of z = zmax +c +. +18 + +(3) ˆΦ(0,0),∗(z) satisfies +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z) +1 +2 ˆΦ(0,0),∗(z) = ˆgΦˆvU(zmax +c +)ˆuU(zmax +c +) > O, +(4.22) +where both ˆvU(zmax +c +) and ˆuU(zmax +c +) are positive, +ˆgΦ = − +� +ˆuU(zmax +c +)( ˆA{1,2} +∗,−1 (zmax +c +) ˆGr +0,∗,1 + ˆA{1,2} +∗,1 +(zmax +c +) ˆG0,∗,1)ˆvU(zmax +c +) +�−1 +> 0, +(4.23) +and ˆGr +0,∗,1 and ˆG0,∗,1 are the limits of ˆGr +0,∗(z) and ˆG0,∗(z), respectively, given by Lemma 3.5. +Proof. By (4.16) and Proposition 4.4, ˆU(z) is entry-wise analytic in ∆zmin +c +,zmax +c +∪ ∂∆zmax +c +\ {zmax +c +}. +Hence, by (4.17), ˆΦ(0,0),∗(z) is entry-wise meromorphic in the same domain. Recall that, under +Assumption 2.2, the induced MA-process {Y {1,2} +n +} is irreducible and aperiodic. Hence, in a manner +similar to that used in the proof of Proposition 5.2 of [11], we obtain by Proposition 4.5 that, for +every z ∈ ∆zmin +c +,zmax +c +∪ ∂∆zmax +c +\ {zmax +c +}, +spr( ˆU(z)) < spr( ˆU(|z|)) < spr( ˆU(zmax +c +)) = 1, +and this leads us to det(I − ˆU(z)) ̸= 0. This completes the proof of statement (1). +By (4.21), ˜Φ(0,0),∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0. Since ˜U(0) = +ˆU(zmax +c +), we see by Proposition 4.5 that det(I − ˜U(0)) = 0 and the multiplicity of zero of det(I − +˜U(ζ)) at ζ = 0 is one. Hence, by the identity theorem for analytic functions, det(I − ˜U(ζ)) is +nonzero in a neighborhood of ζ = 0 except for the point ζ = 0 and the point ζ = 0 is a pole of +˜Φ(0,0),∗(ζ) with order one. This completes the proof of statement (2) since the representation of +ˆΦ(0,0),∗(z) is obvious. +Define a function f(λ, z) as +f(λ, z) = det(λI − ˆU(z)). +By Corollary 2 of Seneta [15] and Proposition 4.5 (also see Proposition 5.11 of [11]), +adj(I − ˆU(zmax +c +)) = fλ(1, zmax +c +)ˆvU(zmax +c +)ˆuU(zmax +c +), +(4.24) +where fλ(λ, z) = +∂ +∂λf(λ, z) and both ˆvU(zmax +c +) and ˆuU(zmax +c +) are positive since ˆU(zmax +c +) is irre- +ducible. Furthermore, in a manner similar to that used in the proof of Proposition 5.9 of [11], we +obtain +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z)− 1 +2 f(1, z) = −c0fλ(1, zmax +c +), +(4.25) +where c0 = ˆuU(zmax +c +)( ˆA{1,2} +∗,−1 (zmax +c +) ˆGr +0,∗,1 + ˆA{1,2} +∗,1 +(zmax +c +) ˆG0,∗,1)ˆvU(zmax +c +) < 0 since, by Lemma 3.5, +both ˆG0,∗,1 and ˆGr +0,∗,1 are nonzero and nonpositive. By (4.21), this completes the proof of statement +(3). +Let αs0(z) be the eigenvalue of ˆG0,∗(z) that satisfies, for z ∈ [zmin +c +, zmax +c +], αs0(z) = spr( ˆG0,∗(z)) = +e2ηR +c,2(log z). Let ˆuG(z) and ˆvG(z) be the left and right eigenvectors of ˆG0,∗(z) with respect to the +eigenvalue αs0(z), satisfying ˆuG(z)ˆvG(z) = 1. By Lemma 3.3, ˜G0,∗(ζ) in Proposition 4.4 satisfies +the following property. +19 + +Proposition 4.7. There exists a matrix function ˜G† +0,∗(ζ) entry-wise analytic in a neighborhood of +ζ = 0 such that ˜G0,∗(ζ) is represented as +˜G0,∗(ζ) = ˜G† +0,∗(ζ) + ˜αs0(ζ)˜vG(ζ)˜uG(ζ), +(4.26) +where function ˜αs0(ζ), row vector function ˜uG(ζ) and column vector ˜vG(ζ) are elementwise analytic +in a neighborhood of ζ = 0 and satisfying αs0(z) = ˜αs0((zmax +c +− z) +1 +2 ), ˆuG(z) = ˜uG((zmax +c +− z) +1 +2 ) +and ˆvG(z) = ˜vG((zmax +c +− z) +1 +2 ), respectively, in a neighborhood of z = zmax +c +. In a neighborhood of +ζ = 0, ˜G† +0,∗(ζ) satisfies spr( ˜G† +0,∗(ζ)) < αs0(zmax +c +) = e2ηR +c,2(θmax +c +). Furthermore, ˜G0,∗(ζ) satisfies, for +n ≥ 1, +˜G0,∗(ζ)n = ˜G† +0,∗(ζ)n + ˜αs0(ζ)n˜vG(ζ)˜uG(ζ). +(4.27) +Let ˆν(0,∗)(z) be the generating function of {ˆν(0,k)} defined as ˆν(0,∗)(z) = �∞ +k=1 zkˆν(0,k). Define +a matrix function ˆU2(z) as +ˆU2(z) = ˆA{2} +0,∗ (z) + ˆA{2} +1,∗ (z) ˆG∗,0(z), +and let ˆuU +2 (z) and ˆvU +2 (z) be the left and right eigenvectors of ˆU2(z) with respect to the maximum +eigenvalue of ˆU2(z), satisfying ˆuU +2 (z)ˆvU +2 (z) = 1. By Lemma 5.3 of [11] (also see Proposition 3.5 of +[14]), ˆν(0,∗)(z) satisfies the following properties. +Proposition 4.8. Assume Type 1. +(1) The vector function ˆν(0,∗)(z) is elementwise analytic in ¯∆e2θ∗ +2 \ {e2θ∗ +2}. +(2) If θ∗ +2 < θmax +2 +, ˆν(0,∗)(z) is elementwise meromorphic in a neighborhood of z = e2θ∗ +2 and the +point z = e2θ∗ +2 is a pole of ˆν(0,∗)(z) with order one. It satisfies, for some positive constant ˆg2, +lim +˜∆ +e2θ∗ +2 ∋z→e2θ∗ +2 +(e2θ∗ +2 − z)ˆϕ2(z) = ˆg2ˆuU +2 (e2θ∗ +2), +(4.28) +where ˆuU +2 (e2θ∗ +2) is positive. +Define a vector function ˜a(ζ, w) as +˜a(ζ, w) = +∞ +� +k=1 +ˆν(0,k) ˆD(zmax +c +− ζ2, ˜G0,∗(ζ))wk−1. +(4.29) +Then, the vector functions ˆa(z, ˆG0,∗(z)) in (4.15) and ˜a(ζ, ˜G0,∗(ζ)) satisfy the following properties. +Proposition 4.9. Assume Type 1. +(1) If ¯η′ +1(θ∗ +2) ≤ −c1/c2 = −1, the vector function ˆa(z, ˆG0,∗(z)) is elementwise analytic in ∆zmin +c +,zmax +c +∪ +∂∆zmax +c +\ {zmax +c +}. +(2) If ¯η′ +1(θ∗ +2) < −1, ˜a(ζ, ˜G0,∗(ζ)) is elementwise analytic in a neighborhood of ζ = 0; if ¯η′ +1(θ∗ +2) = +−1, it is elementwise meromorphic in a neighborhood of ζ = 0 and the point ζ = 0 is a +pole of it with order one. The vector function ˆa(z, ˆG0,∗(z)) is represented as ˆa(z, ˜G0,∗(z)) = +˜a((zmax +c +− z) +1 +2 , ˜G0,∗((zmax +c +− z) +1 +2 )) in a neighborhood of z = zmax +c +. +20 + +(3) If ¯η′ +1(θ∗ +2) = −1, ˆa(z, ˆG0,∗(z)) satisfies, for a positive constant ˆga +2, +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z) +1 +2 ˆa(z, ˆG0,∗(z)) = ˆga +2 ˆuG(zmax +c +) ≥ 0⊤, ̸= 0⊤. +(4.30) +Proof. By Proposition 4.2 of [11], if ¯η′ +1(θ∗ +2) ≤ −1, we have for z ∈ ∆zmin +c +,zmax +c +∪ ∂∆zmax +c +\ {zmax +c +} +that |αs0(z)| < αs0(zmax +c +) = e2ηR +c,2(θmax +c +) ≤ e2θ∗ +2, and this implies spr( ˆG0,∗(z)) < e2θ∗ +2. Hence, by +Lemma 3.2 of [11] and Proposition 4.8, vector series ˆa(z, ˆG0,∗(z)) elementwise converges absolutely +in ∆zmin +c +,zmax +c +∪ ∂∆zmax +c +\ {zmax +c +}. This completes the proof of statement (1). +By Proposition 4.7, we have +˜a(ζ, ˜G0,∗(ζ)) = ˜a(ζ, ˜G† +0,∗(ζ)) ++ (˜αs0(ζ)−1ˆν(0,∗)(˜αs0(ζ)) − ˆν(0,1)) ˆD(zmax +c +− ζ2, ˜αs0(ζ))˜vG(ζ)˜uG(ζ). +(4.31) +If ¯η′ +1(θ∗ +2) ≤ −1, spr( ˜G† +0,∗(ζ)) < e2ηR +c,2(θmax +c +) ≤ e2θ∗ +2 in a neighborhood of ζ = 0. +Hence, vec- +tor series ˜a(ζ, ˜G† +0,∗(ζ)) is elementwise convergent absolutely and analytic in a neighborhood of +ζ = 0. If ¯η′ +1(θ∗ +2) < −1, ˜αs0(0) = αs0(zmax +c +) = e2ηR +c,2(θmax +c +) < e2θ∗ +2, and this implies |˜αs0(ζ)| < e2θ∗ +2 +in a neighborhood of ζ = 0. +Hence, by Proposition 4.8, the vector function ˜a(ζ, ˜G0,∗(ζ)) as +well as ˆν(0,∗)(˜αs0(ζ)) is elementwise analytic in a neighborhood of ζ = 0. +If ¯η′ +1(θ∗ +2) = −1, +˜αs0(0) = e2ηR +c,2(θmax +c +) = e2θ∗ +2. Hence, by Proposition 4.8, the vector function ˜a(ζ, ˜G0,∗(ζ)) as well as +ˆν(0,∗)(˜αs0(ζ)) is meromorphic in a neighborhood of ζ = 0 and the point ζ = 0 is a pole of it with +order one. This completes the proof of statement (2). +If ¯η′ +1(θ∗ +2) = −1, αs0(zmax +c +) = e2ηR +c,2(θmax +c +) = e2θ∗ +2. Hence, by Lemma 3.4 and Proposition 4.8, we +have +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z) +1 +2 ˆν(0,∗)(αs0(z)) += +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z) +1 +2 +αs0(zmax +c +) − αs0(z)(αs0(zmax +c +) − αs0(z))ˆν(0,∗)(αs0(z)) += (−αs0,1)−1ˆg2ˆuU +2 (e2θ∗ +2), +where αs0,1 is the limit of αs0(z) given by (3.10) and it is negative. This leads us to +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z) +1 +2 ˆa(z, ˆG0,∗(z)) += (−αs0,1)−1ˆg2e−2θ∗ +2 ˆuU +2 (e2θ∗ +2)D(zmax +c +, e2θ∗ +2)ˆvG(zmax +c +)ˆuG(zmax +c +). +(4.32) +From this, we see that ˆga +2 ˆuG(zmax +c +) in (4.30) is given by the right-hand side of (4.32). Since ˆuU +2 (e2θ∗ +2) +is positive, ˆga +2 is also positive. This completes the proof of statement (3). +Finally, we give the proof of Propositions 4.1, 4.2 and 4.3. +Proof of Proposition 4.1. Assume Type 1 and ¯η′ +1(θ∗ +2) ≤ −c1/c2 = −1 ≤ 1/¯η′ +2(θ∗ +1). Since ϕc(z) is a +probability vector generating function, it is automatically analytic elementwise in ∆zmax +c +. Hence, +we prove it is elementwise analytic on ∂∆zmax +c +\ {zmax +c +}. For the purpose, we use equations (4.6), +(4.7), (4.10), (4.12), (4.13) and (4.15). +By Propositions 4.4, 4.6 and 4.9, ˆG0,∗(z), ˆa(z, ˆG0,∗(z)) and ˆΦ(0,0),∗(z) are elementwise analytic +on ∂∆zmax +c +\ {zmax +c +}. Hence, by (4.13) and (4.15), ˆΦx,∗(z) and ˆϕ2(z) are also analytic elementwise +on ∂∆zmax +c +\ {zmax +c +}. By (4.10), the analytic property of ˆΦx,∗(z) implies that Φc +x,∗(z) is entry-wise +21 + +analytic on ∂∆zmax +c +\{zmax +c +}. Hence, by (4.12), ϕc +2(z) is elementwise analytic on ∂∆zmax +c +\{zmax +c +}. In +the same way, we can see that if ¯η′ +2(θ∗ +1) ≤ −1, ϕc +1(z) is elementwise analytic on ∂∆zmax +c +\{zmax +c +}. By +(4.7), the analytic property of Φc +x,∗(z) implies that ϕc +0(z) is elementwise analytic on ∂∆zmax +c +\{zmax +c +}. +As a result, we see by (4.6) that ϕc(z) is elementwise analytic on ∂∆zmax +c +\ {zmax +c +}. This completes +the proof. +Proof of Proposition 4.2. Assuming Type 1 and ¯η′ +1(θ∗ +2) ≤ −c1/c2 = −1 ≤ 1/¯η′ +2(θ∗ +1), we also use +equations (4.6), (4.7), (4.10),(4.12), (4.13) and (4.15). +First, we consider about Φc +x,∗(z) and ϕc +0(z), where x = (x1, x2) ∈ Z2. Define ˜Φ(x1,x2),∗(ζ) as +˜Φ(x1,x2),∗(ζ) = (zmax +c +− ζ2)x1 ˜G0,∗(ζ)x2 ˜Φ(0,0),∗(ζ). +Then, by Propositions 4.4 and 4.6, the matrix function ˜Φx,∗(ζ) is entry-wise meromorphic in a +neighborhood of ζ = 0 and satisfies ˆΦx,∗(z) = ˜Φ(x1,x2),∗((zmax +c +− z) +1 +2 ) in a neighborhood of z = +zmax +c +. +The point ζ = 0 is a pole of ˜Φx,∗(ζ) with order one. +Hence, by (4.10), there exists a +matrix function ˜Φc +x,∗(ζ) being entry-wise meromorphic in a neighborhood of ζ = 0 and satisfying +Φc +x,∗(z) = ˜Φc +x,∗((zmax +c +− z) +1 +2 ) in a neighborhood of z = zmax +c +. The point ζ = 0 is a pole of ˜Φc +x,∗(ζ) +with order one. Define ˜ϕc +0(z) as +˜ϕc +0(ζ) = +� +i1,i2∈{−1,0,1} +ν(0,0)(A∅ +i1,i2 − A{1,2} +i1,i2 )˜Φc +(i1,i2),∗(ζ), +which satisfies the same analytic property as ˜Φc +x,∗(ζ). It also satisfies ϕc +0(z) = ˜ϕc +0((zmax +c +− z) +1 +2 ) in +a neighborhood of z = zmax +c +. +Next, we consider about ϕc +2(z). Define ˜ϕ2(ζ) as +˜ϕ2(ζ) = ˜a(ζ, ˜G0,∗(ζ))˜Φ(0,0),∗(ζ) +By Propositions 4.6 and 4.9 and (4.15), ˜ϕ2(ζ) is entry-wise meromorphic in a neighborhood of +ζ = 0 and satisfying ˆϕ2(z) = ˜ϕ2((zmax +c +− z) +1 +2 ) in a neighborhood of z = zmax +c +. If ¯η′ +1(θ∗ +2) < −1, the +point ζ = 0 is a pole of ˜ϕ2(ζ) with at most order one; if ¯η′ +1(θ∗ +2) = −1, it is a pole of ˜ϕ2(ζ) with at +most order two. Represent ˜ϕ2(ζ) in block form as ˜ϕ2(ζ) = +�˜ϕ2,1(ζ) +˜ϕ2,2(ζ) +� +and define ˜ϕc +2(ζ) as +˜ϕc +2(ζ) = ˜ϕ2,1(ζ) + +� +i1,i2∈{−1,0,1} +ν(0,1)(A{2} +i1,i2 − A{1,2} +i1,i2 )˜Φc +(i1,i2+1),∗(ζ). +Then, the vector function ˜ϕc +2(ζ) is elementwise meromorphic in a neighborhood of ζ = 0, and by +(4.12), it satisfies ϕc +2(z) = ˜ϕc +2((zmax +c +− z) +1 +2 ) in a neighborhood of z = zmax +c +. If ¯η′ +1(θ∗ +2) < −1, the +point ζ = 0 is a pole of ˜ϕc +2(ζ) with at most order one; if ¯η′ +1(θ∗ +2) = −1, it is a pole of ˜ϕc +2(ζ) with at +most order two. +Finally, we consider about ϕc(z). In the same way as that used for ϕc +2(z), we can see that +there exists a vector function ˜ϕc +1(ζ) being elementwise meromorphic in a neighborhood of ζ = 0 +and satisfying ϕc +1(z) = ˜ϕc +1((zmax +c +− z) +1 +2 ) in a neighborhood of z = zmax +c +. If ¯η′ +2(θ∗ +1) < −1, the point +ζ = 0 is a pole of ˜ϕc +1(ζ) with at most order one; if ¯η′ +2(θ∗ +1) = −1, it is a pole of ˜ϕc +1(ζ) with at most +order two. Define ˜ϕc(ζ) as +˜ϕc(ζ) = ˜ϕc +0(ζ) + ˜ϕc +1(ζ) + ˜ϕc +2(ζ). +Then, the vector function ˜ϕc(ζ) is elementwise meromorphic in a neighborhood of ζ = 0, and by +(4.6), it satisfies ϕc(z) = ˜ϕc((zmax +c +− z) +1 +2 ) in a neighborhood of z = zmax +c +. If ˜η′ +1(θ∗ +2) < −c1/c2 = +−1 < 1/˜η′ +2(θ∗ +1), the point ζ = 0 is a pole of ˜ϕc(ζ) with at most order one; if ˜η′ +1(θ∗ +2) = −1 or +˜η′ +2(θ∗ +1) = −1, it is a pole of ˜ϕc(ζ) with at most order two. This completes the proof. +22 + +Proof of Proposition 4.3. Assume Type 1. By Proposition 4.6 and equations (4.10) and (4.13), +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z)Φc +x,∗(z) = O. +(4.33) +Hence, by (4.7), +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z)ϕc +0(z) = 0⊤. +(4.34) +If ¯η′ +1(θ∗ +2) = −1, by Propositions 4.6 and 4.9 and equations (4.12) and (4.34), representing ˆuU(zmax +c +) +in block form as ˆuU(zmax +c +) = +� +ˆuU +1 (zmax +c +) +ˆuU +2 (zmax +c +) +� +, we obtain +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z)ϕc +2(z) = uc +2 = ˆga +2ˆgΦˆuG(zmax +c +)ˆvU(zmax +c +)ˆuU +1 (zmax +c +) > 0⊤, +(4.35) +where ˆuG(zmax +c +) is nonzero and nonnegative and other terms on the right-hand side of the equation +are positive; if ¯η′ +1(θ∗ +2) < −1, we have +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z)ϕc +2(z) = 0⊤. +(4.36) +In a manner similar to that used for ϕc +2(z), we can see that if ¯η′ +2(θ∗ +1) = −1, then for some positive +vector uc +1, +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z)ϕc +1(z) = uc +1, +(4.37) +and if ¯η′ +1(θ∗ +2) < −1, +lim +˜∆zmax +c +∋z→zmax +c +(zmax +c +− z)ϕc +1(z) = 0⊤. +(4.38) +As a result, by (4.6), (4.35), (4.36), (4.37) and (4.38), we obtain (4.5) in Proposition 4.3. +5 +Concluding remarks +We consider another topic, which relates to the singularity of the vector generating function ϕc(z) +at z = zmax +c += eθmax +c +, where c ∈ N2. +Recall that P {1,2} = (P {1,2} +x,x′ ; x, x′ ∈ Z2) is the transition probability matrix of the induced +MA-process {Y {1,2} +n +} and Φ{1,2} = (Φ{1,2} +x,x′ ; x, x′ ∈ Z2) the fundamental matrix (potential matrix) +of P {1,2}. Let hΦ +c (k) be the asymptotic decay function of the matrix sequence {Φ{1,2} +x,kc ; k ∈ N}, i.e., +for some positive matrix C, +lim +k→∞ Φ{1,2} +x,kc /hΦ +c (k) = C. +(5.1) +By Proposition 4.6, we obtain +hΦ +c (k) = k− 1 +2 e−θmax +c +k. +(5.2) +Furthermore, recall that P + is a partial matrix of P {1,2} given by restricting the state space of +the level to the positive quadrant, i.e., P + = (P {1,2} +x,x′ ; x, x′ ∈ N2). P + is also a partial matrix of +the transition probability matrix of the original 2d-QBD process, P = (Px,x′; x, x′ ∈ Z2 ++), i.e., +P + = (Px,x′; x, x′ ∈ N2). Let ˜Q = ( ˜Qx,x′; x, x′ ∈ N2) be the fundamental matrix of P +, i.e., +23 + +˜Q = �∞ +n=0(P +)n. For j, j′ ∈ S0, denote by ˜q(x,j),(x′,j′) the (j, j′)-entry of ˜Qx,x′. The entries of ˜Q +are called an occupation measure in [13]. By Theorem 5.1 of [13], the asymptotic decay rate of the +matrix sequence { ˜Qx,kc; k ∈ N} is given by eθmax +c +, i.e., +− lim +k→∞ +1 +k log ˜q(x,j),(kc,j′) = θmax +c +, +(5.3) +which coincides with that of the matrix sequence {Φ{1,2} +x,kc ; k ∈ N}. One question, therefore, arises: +Does the asymptotic decay function of the matrix sequence { ˜Qx,kc; k ∈ N} coincide with that of +the matrix sequence {Φ{1,2} +x,kc ; k ∈ N}? If the answer to the question is yes, we can indicate that the +vector generating function ϕc(z) diverges at z = eθmax +c +. +References +[1] Bini, D.A., Latouche, G. and Meini, B., Oxford University Press, Oxford (2005). +[2] Flajolet, P. and Sedgewick, R., Analytic Combinatorics, Cambridge University Press, Cam- +bridge (2009). +[3] Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials, SIAM, Philadelphia (2009). +[4] Kobayashi, M. and Miyazawa, M., Revisit to the tail asymptotics of the double QBD process: +Refinement and complete solutions for the coordinate and diagonal directions, Matrix-Analytic +Methods in Stochastic Models (2013), 145-185. +[5] Latouche, G. and Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic +Modeling, SIAM, Philadelphia (1999). +[6] Malyshev, V.A., Asymptotic behavior of the stationary probabilities for two-dimensional pos- +itive random walks, Siberian Mathematical Journal 14(1) (1973), 109–118. +[7] Neuts, M.F., Matrix-Geometric Solutions in Stochastic Models, Dover Publications, New York +(1994). +[8] Neuts, M.F., Structured stochastic matrices of M/G/1 type and their applications, Marcel +Dekker, New York (1989). +[9] Ney, P. and Nummelin, E., Markov additive processes I. Eigenvalue properties and limit the- +orems, The Annals of Probability 15(2) (1987), 561–592. +[10] Ozawa, T., Asymptotics for the stationary distribution in a discrete-time two-dimensional +quasi-birth-and-death process, Queueing Systems 74 (2013), 109–149. +[11] Ozawa, T. and Kobayashi, M., Exact asymptotic formulae of the stationary distribution of a +discrete-time two-dimensional QBD process, Queueing Systems 90 (2018), 351-403. +[12] Ozawa, T., Stability condition of a two-dimensional QBD process and its application to esti- +mation of efficiency for two-queue models, Performance Evaluation 130 (2019), 101–118. +[13] Ozawa, +T., +Asymptotic properties of the occupation measure in a multidimensional +skip-free +Markov +modulated +random +walk, +Queueing +Systems +97 +(2021), +125–161. +(DOI:10.1007/s11134-020-09673-9) +24 + +[14] Ozawa, +T., +Tail +Asymptotics +in +any +direction +of +the +stationary +distribution +in +a +two-dimensional discrete-time QBD process, +Queueing Systems +102 (2022), +227–267. +(DOI:10.1007/s11134-022-09860-w) +[15] E. Seneta: Non-negative Matrices and Markov Chains, revised printing. Springer-Verlag, New +York (2006). +A +Proof of Theorem 3.1 +First, we give the generalized eigenvectors of G(z) for z ∈ ∆zmain +1 +,zmax +1 +\E1, then analytically extend +them to z ∈ C \ E1. +For each k ∈ {1, 2, ..., m0} and for each z ∈ Ω \ �m0 +k=1 EG +k , since the Jordan normal form of G(z) +is given by (3.4), there exist linearly independent vectors called the generalized eigenvectors of G(z) +with respect to the eigenvalue ˇαk(z), ˇvk,i,j(z), i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i, satisfying +(ˇαk(z)I − G(z))ˇvk,i,j(z) = ˇvk,i,j+1(z), +(A.1) +where ˇvk,i,mk,i+1(z) = 0. +For each i, ˇvk,i,j(z), j = 1, 2, ..., mk,i, are called a Jordan sequence +of the generalized eigenvectors. +Using the Jordan sequences, we define lˇq(k) × 1 block vectors, +vk,i,j(z), i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i, as +vk,i,j(z) = vec +�ˇvk,i,j(z) +ˇvk,i,j+1(z) +· · · +ˇvk,i,mk,i(z) +0 +· · · +0� +, +where, for a matrix A = +� +a1 +a2 +· · · +an +� +, vec(A) is the column vector given by +vec(A) = +� +� +� +� +� +a1 +a2 +... +an +� +� +� +� +� . +We also define a vector space VG +k (z) as +VG +k (z) = span {vk,i,j(z) : i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i}. +Note that the generalized eigenvectors ˇvk,i,j(z) are not unique but VG +k (z) is. Since the generalized +eigenvectors are linearly independent, vk,i,j(z) are also linearly independent and we have +dim VG +k (z) = +mk,0 +� +i=1 +mk,i = lˇq(k). +For k ∈ {1, 2, ..., m0}, define an lˇq(k) × lˇq(k) block matrix function ΛG +k (z) as +ΛG +k (z) = +� +� +� +� +� +� +� +ˇαk(z)I − G(z) +−I +ˇαk(z)I − G(z) +−I +... +... +ˇαk(z)I − G(z) +−I +ˇαk(z)I − G(z) +� +� +� +� +� +� +� +. +We give the following proposition. +Proposition A.1. For each k ∈ {1, 2, ..., m0} and for each z ∈ Ω \ �m0 +k=1 EG +k , +Ker ΛG +k (z) = VG +k (z). +(A.2) +25 + +Proof. Assume v ∈ VG +k (z). +Then, by the definition of VG +k (z), we have ΛG +k (z)v = 0 and v ∈ +Ker ΛG +k (z). For v = vec +�v1 +v2 +· · · +vlˇq(k) +� +, assume ΛG +k (z)v = 0. If there exists an index i such +that vi = 0, then by the assumption, for every j such that i ≤ j ≤ lˇq(k), we have vj = 0, and this +implies v ∈ VG +k (z). +By Theorem S6.1 of [3], since the matrix function ΛG +k (z) is entry-wise analytic in ∆zmin +1 +,zmax +1 +\E1, +there exist lˇq(k) vector functions vG +k,i(z), i = 1, 2, ..., lˇq(k), that are elementwise analytic and linearly +independent in ∆zmin +1 +,zmax +1 +\ E1 and satisfy +ΛG +k (z)vG +k,i(z) = 0, i = 1, 2, ..., lˇq(k). +Hence, for each z ∈ Ω \ �m0 +k=1 EG +k , vG +k,i(z) ∈ VG +k (z). We select the vectors composed of the Jordan +sequences from {vG +k,i(z), i = 1, 2, ..., lˇq(k)}. Represent each vG +k,i(z) in block form as +vG +k,i(z) = vec +� +vG +k,i,1(z) +vG +k,i,2(z) +· · · +vG +k,i,lˇq(k)(z) +� +. +From the proof of Proposition A.1, we see that, for every i ∈ {1, 2, ..., lˇq(k)}, there exists a positive +integer µk,i such that vG +k,i,j(z) ̸= 0 for every j ∈ {1, 2, ..., µk,i} and vG +k,i,j(z) = 0 for every j ∈ +{µk,i + 1, µk,i + 2, ..., lˇq(k)}. Renumber the elements of {vG +k,i(z)} so that if i ≤ i′, then µk,i ≥ µk,i′. +Define a set of vector functions, ˇVk, according to the following procedure. +(S1) Set ˇVk = ∅ and i = 1. +(S2) If vG +k,i,µk,i(z) is linearly independent of {vG +k,i′,µk,i′(z) : vG +k,i′(z) ∈ ˇVk}, append vG +k,i(z) to ˇVk. +(S3) If i = lˇq(k), stop the procedure; otherwise add 1 to i and go to (S2). +Proposition A.2. For k ∈ {1, 2, ..., m0}, the number of elements of ˇVk is mk,0. +Proof. Since, for every i ∈ {1, 2, ..., lˇq(k)}, (ˇαk(z)I−G(z))vG +k,i,µk,i = 0 and dim Ker (ˇαk(z)I−G(z)) = +mk,0, the number of elements of ˇVk is less than or equal to mk,0. If it is strictly less than mk,0, we +have +dim Ker ΛG +k (z) = dim span {vG +k,i(z), i = 1, 2, ..., lˇq(k)} < dim VG +k (z). +This contradicts (A.2), and we see that the number of elements of ˇVk is just mk,0. +Denote by ˇvG +k,1(z), ˇvG +k,2(z), ..., ˇvG +k,mk,0(z) the elements of ˇVk. For i ∈ {1, 2, ...mk,0}, define ˇµk,i in +a manner similar to that used for defining µk,i. We assume ˇvG +k,i(z), i = 1, 2, ..., mk,0, are numbered +so that if i ≤ i′, then ˇµk,i ≥ ˇµk,i′. +Proposition A.3. For k ∈ {1, 2, ..., m0} and for i ∈ {1, 2, ..., mk,0}, ˇµk,i = mk,i +Proof. For each i ∈ {1, 2, ..., mk,0}, {ˇvG +k,i,1(z), ˇvG +k,i,2(z), ..., ˇvG +k,i,µk,i(z)} is a Jordan sequence of the +generalized eigenvectors of G(z) with respect to the eigenvalue ˇαk(z). +Hence, considering the +procedure defining ˇvG +k,i(z), we see that, for every i ∈ {1, 2, ..., mk,0}, ˇµk,i ≤ mk,i. Suppose there +exists some i0 ∈ {1, 2, ..., mk,0} such that ˇµk,i = mk,i for every i ∈ {1, 2, ..., i0 −1} and ˇµk,i0 < mk,i0. +Then, there exists a vector v = vec +�v1 +v2 +· · · +vmk,i0 +0 +· · · +0� +in VG +k (z) such that vi ̸= 0 +for every i ∈ {1, 2, ..., mk,i0} and v is linearly independent of {vG +k,i(z), i = 1, 2, ..., lˇq(k)}. By the +same reason as that used in the proof of Proposition A.2, this contradicts (A.2) and, for every +i ∈ {1, 2, ..., mk,0}, ˇµk,i must be mk,i. +26 + +From this proposition, we see that, for z ∈ Ω \ �m0 +k=1 EG +k , {ˇvG +k,i,j(z) : k = 1, 2, ..., m0, i = +1, 2, ..., mk,0, j = 1, 2, ..., mk,i} is the set of generalized eigenvectors corresponding to the Jordan +normal form (3.4). Define a matrix function T G(z) as +T G(z) = +�ˇvG +k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i +� +, +which is entry-wise analytic in ∆zmin +1 +,zmax +2 +\ E1. Define a point set EG +T as +EG +T = {z ∈ ∆zmin +1 +,zmax +2 +\ E1 : det T G(z) = 0}, +which is an empty set or a set of discrete complex numbers. Then, for z ∈ Ω \ (�m0 +k=1 EG +k ∪ EG +T ), we +obtain the Jordan decomposition of G(z) as +G(z) = T G(z)JG(z)(T G(z))−1. +(A.3) +Since G(z) is entry-wise analytic in ∆zmin +1 +,zmax +1 +, we see by the identity theorem for analytic functions +that the right hand side of (A.3) is also entry-wise analytic in the same domain. +Next, we analytically extend ˇvG +k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i. Define +matrix functions F1(z, w) and F2(z) as +F1(z, w) = z(I − A∗,0(z) − 2wA∗,1(z)), +F2(z) = zA∗,1(z), +where F1(z, w) is entry-wise analytic on C2 and F2(z) on C. By (3.2), we have +L(z, w) = F1(z, w)(wI − G(z)) + F2(z)(wI − G(z))2. +(A.4) +For k ∈ {1, 2, ..., m0}, define a lˇq(k) × lˇq(k) block matrix function ΛL +k,n(z) as +ΛL +k (z) = +� +� +� +� +� +� +� +� +� +L(z, ˇαk(z)) +−F1(z, ˇαk(z)) +−F2(z) +L(z, ˇαk(z)) +−F1(z, ˇαk(z)) +−F2(z) +... +... +... +L(z, ˇαk(z)) +−F1(z, ˇαk(z)) +−F2(z) +L(z, ˇαk(z)) +−F1(z, ˇαk(z)) +L(z, ˇαk(z)) +� +� +� +� +� +� +� +� +� +, +which is entry-wise analytic in C \ E1. +Proposition A.4. For every k ∈ {1, 2, ..., m0} and for every z ∈ ¯∆zmin +1 +,zmax +1 +, +Ker ΛL +k (z) = Ker ΛG +k (z). +(A.5) +Before proving this proposition, we give another one. +Proposition A.5. For every k ∈ {1, 2, ..., s0} and z ∈ ∆zmin +1 +,zmax +1 +, +F1(z, αk(z)) + F2(z)(αk(z)I − G(z)) = z (I − A∗,0(z) − αk(z)A∗,1(z) + A∗,1(z)G(z)) +is regular (invertible). +Proof. Let R(z) be the rate matrix function generated from {Ai,j; i, j = −1, 0, 1}; for the definition +of R(z), see Subsection 4.1 of [11]. By Lemma 4.3 of [11], nonzero eigenvalues of R(z) are given by +αk(z)−1, k = s0 + 1, s0 + 2, ..., mφ. Since, for every k ∈ {1, 2, ..., s0}, k′ ∈ {s0 + 1, s0 + 2, ..., mφ} +and z ∈ ∆zmin +1 +,zmax +1 +, |αk(z)| ≤ αs0(|z|) < |αk′(z)|, I − αk(z)R(z) is regular. +Define a matrix +function H(z) as H(z) = A∗,0(z) + A∗,1(z)G(z), then by Corollary 4.1 of [11], I − H(z) is regular +in ∆zmin +1 +,zmax +1 +. By Lemma 4.1 of [11], we have +I − A∗,0(z) − αk(z)A∗,1(z) − A∗,1(z)G(z) = (I − αk(z)R(z))(I − H(z)), +(A.6) +and this implies the assertion of the proposition. +27 + +Proof of Proposition A.4. Assume a vector v = vec +�v1 +v2 +· · · +vlˇq(k) +� +satisfies ΛL +k (z)v = 0. +Then, we have for i ∈ {1, 2, ..., lˇq(k)} that +L(z, ˇαk(z))vi = F1(z, ˇαk(z))vi+1 + F2(z)vi+2, +(A.7) +where vlˇq(k)+1 = vlˇq(k)+2 = 0. We prove by induction that this v satisfies, for every i ∈ {1, 2, ..., lˇq(k)}, +(ˇαk(z)I − G(z))vi = vi+1. Let i0 be the maximum integer less than or equal to lˇq(k) that satisfies, +for every i ∈ {i0 + 1, i0 + 2, ..., lˇq(k)}, vi = 0. Then, we have L(z, ˇαk(z))vi0 = 0. By (A.4), we have +L(z, ˇαk(z)) = (F1(z, ˇαk(z)) + F2(z)(ˇαk(z)I − G(z)))(ˇαk(z)I − G(z)). +(A.8) +Hence, by Proposition A.5, we obtain (ˇαk(z)I − G(z))vi0 = 0 = vi0+1. Assume the assumption of +induction holds for a positive integer i less than or equal to i0. Then, +L(z, ˇαk(z))vi−1 = F1(z, ˇαk(z))vi + F2(z)vi+1 += (F1(z, ˇαk(z)) + F2(z)(ˇαk(z)I − G(z)))vi, +(A.9) +and by (A.8), (A.9) and Proposition A.5, we obtain (ˇαk(z)I − G(z))vi−1 = vi. Hence, v satisfies, +for every i ∈ {1, 2, ..., lˇq(k)}, (ˇαk(z)I − G(z))vi = vi+1, and this leads us to ΛG +k (z)v = 0. +Next, assume a vector v = vec +�v1 +v2 +· · · +vlˇq(k) +� +satisfies ΛG +k (z)v = 0. Then, we have for +i ∈ {1, 2, ..., lˇq(k)} that (ˇαk(z)I − G(z))vi = vi+1, where vlˇq(k)+1 = 0. By (A.8), this v satisfies, for +every i ∈ {1, 2, ..., lˇq(k)}, +L(z, ˇαk(z))vi = F1(z, ˇαk(z))vi+1 + F2(z)(ˇαk(z)I − G(z))vi+1 += F1(z, ˇαk(z))vi+1 + F2(z)vi+2, +(A.10) +and this implies ΛL +k (z)v = 0. +Let k be an arbitrary integer in {1, 2, ..., m0}. By Propositions A.1 and A.4, we have +dim Ker ΛL +k (z) = lˇq(k), +except for some discrete points in C. Hence, by Theorem S6.1 of [3], since the matrix function +ΛL +k (z) is entry-wise analytic in C\E1, there exist lˇq(k) vector functions vL +k,i(z), i = 1, 2, ..., lˇq(k), that +are elementwise analytic and linearly independent in C \ E1 and satisfy +ΛL +k (z)vL +k,i(z) = 0, i = 1, 2, ..., lˇq(k). +By Proposition A.4, for each i, vL +k,i(z) also satisfies ΛG +k (z)vL +k,i(z) = 0 for every z ∈ ∆zmin +1 +,zmax +1 +\ E1. +Hence, by the identity theorem, we see that vL +k,i(z) is an analytic extension of vG +k,i(z). By the same +procedure as that used for selecting {ˇvG +k,i(z), i = 1, 2, ..., mk,0} from {vG +k,i(z), i = 1, 2, ..., lˇq(k)}, we +select mk,0 vectors from {vL +k,i(z), i = 1, 2, ..., lˇq(k)} and denote them by {ˇvL +k,i(z), i = 1, 2, ..., mk,0}. +For each i, ˇvL +k,i(z) is represented in block form as +ˇvL +k,i(z) = vec +� +ˇvL +k,i,1(z) +ˇvL +k,i,2(z) +· · · +ˇvL +k,i,mk,i(z) +0 +· · · +0 +� +. +Define a matrix function T L(z) as +T L(z) = +�ˇvL +k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i +� +, +which is entry-wise analytic in C \ E1. Since each ˇvL +k,i,j(z) is an analytic extension of ˇvG +k,i,j(z), we +have for z ∈ Ω \ (�m0 +k=1 EG +k ∪ EG +T ) that +G(z) = T L(z)JG(z)(T L(z))−1, +which is (3.5). Set E0 as E0 = E2 ∪ (�m0 +k=1 EG +k ) ∪ EG +T , then E0 is a set of discrete complex numbers +and we have Ω\(�m0 +k=1 EG +k ∪EG +T ) = ∆zmin +1 +,zmax +1 +\(E1 ∪E0). This completes the proof of Theorem 3.1. +28 + diff --git a/HdE0T4oBgHgl3EQfhgHx/content/tmp_files/load_file.txt b/HdE0T4oBgHgl3EQfhgHx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a5504b0121c3f71de2fb38a3dc547f63796b7f1 --- /dev/null +++ b/HdE0T4oBgHgl3EQfhgHx/content/tmp_files/load_file.txt @@ -0,0 +1,1306 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf,len=1305 +page_content='Asymptotic decay function of the stationary tail probabilities along an arbitrary direction in a two-dimensional discrete-time QBD process Toshihisa Ozawa Faculty of Business Administration, Komazawa University 1-23-1 Komazawa, Setagaya-ku, Tokyo 154-8525, Japan E-mail: toshi@komazawa-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='jp Abstract We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD pro- cess for short) on Z2 + ×S0, where S0 is a finite set, and consider a topic remaining unresolved in our previous paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In that paper, the asymptotic decay rate of the stationary tail probabilities along an arbitrary direction has been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' It has also been clarified that if the asymptotic decay rate ξc, where c is a direction vector in N2, is less than a certain value θmax c , the sequence of the stationary tail probabilities along the direction c geometrically decays without power terms, asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In this article, we give the function that the sequence asymptotically decays according to when ξc = θmax c , but it contains an unknown parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' To determine the value of the parameter is a next challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Keywards: quasi-birth-and-death process, Markov modulated reflecting random walk, Markov additive process, asymptotic decay rate, asymptotic decay function, stationary distribution, ma- trix analytic method Mathematics Subject Classification: 60J10, 60K25 1 Introduction We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) {Y n} = {(Xn, Jn)} on Z2 + × S0, where S0 is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This model is a Markov modulated reflecting random walk (MMRRW for short) whose transitions are skip free, and the MMRRW is a kind of reflecting random walk (RRW for short) with a background process, where the transition probabilities of the RRW vary depending on the state of the background process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' One-dimensional QBD processes have been introduced by Macel Neuts and studied in the literature as one of the essential stochastic models in the queueing theory (see, for example, [1, 5, 7, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The 2d-QBD process is a two-dimensional version of one-dimensional QBD process, and it enable us to analyze, for example, two-node queueing networks and two-node polling models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume the 2d-QBD process {Y n} is positive recurrent and denote by ν = (ν(x,j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (x, j) ∈ Z2 + × S0) the stationary distribution, where ν(x,j) is the stationary probability that the process is in the state (x, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Our interest is asymptotics of the stationary distribution ν, especially, tail asymptotics in an arbitrary direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let an integer vector c = (c1, c2) be nonzero and nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Two typical objects of our study are the asymptotic decay rate ξc and asymptotic decay function hc(k) defined as, for j ∈ S0, ξc = − lim k→∞ 1 k log ν(kc,j), lim k→∞ ν(kc,j) hc(k) = gj, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='02434v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='PR] 6 Jan 2023 where gj is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Under a certain condition, the asymptotic decay rate of the probability sequence {νx+kc,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ≥ 0} does not depend on x and j if it exists, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of Ozawa [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the case where c = (1, 0) or c = (0, 1), the asymptotic decay rate ξc has been obtained in Ozawa [10], see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [14], and the asymptotic decay function hc(k) in Ozawa and Kobayashi [11], see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The results in the case where c = (c, 0) or c = (0, c) for c ≥ 2 are automatically obtained from those in [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the case where c = (c1, c2) ≥ (1, 1), the asymptotic decay rate ξc has been obtained in Ozawa [14], see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' A condition ensuring the asymptotic decay function is given by hc(k) = e−ξck, an exponential function without a power term, has also been given in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In this article, we give the expression of the asymptotic decay function hc(k) when c = (c1, c2) ≥ (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' To this end, we clarify the analytic properties of the vector generating function of the stationary probabilities along the direction c, ϕc(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The point z = eξc is a singular point of the vector function ϕc(z), and if ξc is equal to a certain value θmax c , z = eθmax c is a branch point of ϕc(z) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From this result, we obtain the expression of hc(k), but it contains an unknown parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' To determine the value of the parameter, it suffices to prove that ϕc(z) diverges elementwise at z = eθmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' It seems to be a hard work and we leave it as a next challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We also generalize a part of existing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' One crucial point in analyzing the asymptotic decay function is how to analytically extend the G-matrix function appeared in the vector generating function of the stationary probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In [11], it has been done under the assumption that all the eigenvalues of the G-matrix function are distinct, see Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This assumption is not easy to verify in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We, therefore, remove the assumption and give a general formula of the Jordan decomposition of the G-matrix function, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In Section 2, we describe the 2d-QBD process in detail and state assumptions and main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In Section 3, an analytic extension of the G-matrix function is given in a general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The definition of G-matrix in the reverse direction and its properties are also given in the same section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' They are used in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The proof of the main results is given in Sections 4, where we demonstrate that the vector function ϕc(z) is elementwise analytic in the open disk with radius eξc + ε for some ε > 0, except for the point z = eξc, and clarify its singularity at the point z = eξc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The asymptotic decay function is obtained from those results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The paper concludes with some remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 2 Model description and main results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 Model description We consider the same model as that described in [14] and use the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by I2 the set of all the subsets of {1, 2}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', I2 = {∅, {1}, {2}, {1, 2}}, and we use it as an index set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Divide Z2 + into 22 = 4 exclusive subsets defined as Bα = {x = (x1, x2) ∈ Z2 +;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' xi > 0 for i ∈ α, xi = 0 for i ∈ {1, 2} \\ α}, α ∈ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let {Y n} = {(Xn, Jn)} be a 2d-QBD process on S = Z2 + × S0, where S0 = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let P be the transition probability matrix of {Y n} and represent it in block form as P = � Px,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2 + � , where Px,x′ = (p(x,j),(x′,j′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' j, j′ ∈ S0) and p(x,j),(x′,j′) = P(Y 1 = (x′, j′) | Y 0 = (x, j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For α ∈ I2 and i1, i2 ∈ {−1, 0, 1}, let Aα i1,i2 be a one-step transition probability block from a state in Bα, where we assume the blocks corresponding to impossible transitions are zero (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the level process is skip free, for every x, x′ ∈ Z2 +, Px,x′ is given by Px,x′ = � Aα x′−x, if x ∈ Bα for some α ∈ I2 and x′ − x ∈ {−1, 0, 1}2, O, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) We assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 2 Figure 1: Transition probability blocks Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The 2d-QBD process {Y n} is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Next, we define several Markov chains derived from the 2d-QBD process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For a nonempty set α ∈ I2, let {Y α n} = {(Xα n, Jα n )} be a process derived from the 2d-QBD process {Y n} by removing the boundaries that are orthogonal to the xi-axis for each i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y {1} n } is a Markov chain on Z × Z+ × S0 whose transition probability matrix P {1} = (P {1} x,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z × Z+) is given as P {1} x,x′ = � � � � � A{1} x′−x, if x ∈ Z × {0} and x′ − x ∈ {−1, 0, 1} × {0, 1}, A{1,2} x′−x, if x ∈ Z × N and x′ − x ∈ {−1, 0, 1}2, O, otherwise, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) where N is the set of all positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y {2} n } on Z+ × Z × S0 and its transition probability matrix P {2} = (P {2} x,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z+ × Z) are analogously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y {1,2} n } is a Markov chain on Z2 × S0, whose transition probability matrix P {1,2} = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) is given as P {1,2} x,x′ = � A{1,2} x′−x, if x′ − x ∈ {−1, 0, 1}2, O, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) Regarding X{1} 1,n as the additive part, we see that the process {Y {1} n } = {(X{1} 1,n , (X{1} 2,n , J{1} n ))} is a Markov additive process (MA-process for short) with the background state (X{1} 2,n , J{1} n ) (with respect to MA-processes, see, for example, Ney and Nummelin [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y {2} n } = {(X{2} 2,n , (X{2} 1,n , J{2} n ))} is also an MA-process, where X{2} 2,n is the additive part and (X{2} 1,n , J{2} n ) the background state, and {Y {1,2} n } = {(X{1,2} 1,n , X{1,2} 2,n ), J{1,2} n )} an MA-process, where (X{1,2} 1,n , X{1,2} 2,n ) the additive part and J{1,2} n the background state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We call them the induced MA-processes de- rived from the original 2d-QBD process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let { ¯A{1} i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i ∈ {−1, 0, 1}} be the Markov additive kernel (MA-kernel for short) of the induced MA-process {Y {1} n }, which is the set of transition probability blocks and defined as, for i ∈ {−1, 0, 1}, ¯A{1} i = � ¯A{1} i,(x2,x′ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x2, x′ 2 ∈ Z+ � , ¯A{1} i,(x2,x′ 2) = � � � � � A{1} i,x′ 2−x2, if x2 = 0 and x′ 2 − x2 ∈ {0, 1}, A{1,2} i,x′ 2−x2, if x2 ≥ 1 and x′ 2 − x2 ∈ {−1, 0, 1}, O, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 3 X2 ^ B(2] B(1,2] [2] [1,2] 12 i1,i2 17 ,12 B(1) Bo 0 x1Let { ¯A{2} i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i ∈ {−1, 0, 1}} be the MA-kernel of {Y {2} n }, defined in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' With respect to {Y {1,2} n }, the MA-kernel is given by {A{1,2} i1,i2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i1, i2 ∈ {−1, 0, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The induced MA-processes {Y {1} n }, {Y {2} n } and {Y {1,2} n } are irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' According to [14], we assume several other technical conditions for the induced MA-process {Y {1,2} n }, concerning irreducibility and aperiodicity on subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let {Y + n } = {(X+ n , J+ n )} be a lossy Markov chain derived from the induced MA-process {Y {1,2} n } by restricting the state space of the additive part to N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y + n } is a Markov chain on N2 × S0 whose transition probability matrix P + is given as P + = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ N2), where P + is strictly substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y + n } is also a lossy Markov chain derived from the original 2d-QBD process {Y n} by restricting the state space of the level to N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' {Y + n } is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ Z, let Z≤k and Z≥k be the set of integers less than or equal to k and that of integers greater than or equal to k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We also assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For what this assumption implies, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (i) The lossy Markov chain derived from the induced MA-process {Y {1,2} n } by restricting the state space to Z≤0 × Z≥0 × S0 is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (ii) The lossy Markov chain derived from {Y {1,2} n } by restricting the state space to Z≥0×Z≤0×S0 is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The stability condition of the 2d-QBD process has already been obtained in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let a{1}, a{2} and a{1,2} = (a{1,2} 1 , a{1,2} 2 ) be the mean drifts of the additive part in the induced MA-processes {Y {1} n }, {Y {2} n } and {Y {1,2} n }, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [12], the stability condition of the 2d-QBD process {Y n} is given as follows: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (i) In the case where a{1,2} 1 < 0 and a{1,2} 2 < 0, the 2d-QBD process {Y n} is positive recurrent if a{1} < 0 and a{2} < 0, and it is transient if either a{1} > 0 or a{2} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (ii) In the case where a{1,2} 1 ≥ 0 and a{1,2} 2 < 0, {Y n} is positive recurrent if a{1} < 0, and it is transient if a{1} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (iii) In the case where a{1,2} 1 < 0 and a{1,2} 2 ≥ 0, {Y n} is positive recurrent if a{2} < 0, and it is transient if a{2} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (iv) If one of a{1,2} 1 and a{1,2} 2 is positive and the other is non-negative, then {Y n} is transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For the explicit expression of the mean drifts, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [12] and its related parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The condition in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 that ensures the 2d-QBD process {Y n} is positive recurrent holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ν the stationary distribution of {Y n}, where ν = (νx, x ∈ Z2 +), νx = (ν(x,j), j ∈ S0) and ν(x,j) is the stationary probability that the 2d-QBD process is in the state (x, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 4 Figure 2: Domains Γ{1,2}, Γ{1} and Γ{2} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 Main results Let ¯A{1} ∗ (z) and ¯A{2} ∗ (z) be the matrix generating functions of the MA-kernels of {Y {1} n } and {Y {2} n }, respectively, defined as ¯A{1} ∗ (z) = � i∈{−1,0,1} zi ¯A{1} i , ¯A{2} ∗ (z) = � i∈{−1,0,1} zi ¯A{2} i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The matrix generating function of the MA-kernel of {Y {1,2} n } is given by A{1,2} ∗,∗ (z1, z2), defined as A{1,2} ∗,∗ (z1, z2) = � i1,i2∈{−1,0,1} zi1 1 zi2 2 A{1,2} i1,i2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let Γ{1}, Γ{2} and Γ{1,2} be regions in which the convergence parameters of ¯A{1} ∗ (eθ1), ¯A{2} ∗ (eθ2) and A{1,2} ∗,∗ (eθ1, eθ2) are greater than 1, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Γ{1} = {(θ1, θ2) ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' cp( ¯A{1} ∗ (eθ1)) > 1}, Γ{2} = {(θ1, θ2) ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' cp( ¯A{2} ∗ (eθ2)) > 1}, Γ{1,2} = {(θ1, θ2) ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' cp(A{1,2} ∗,∗ (eθ1, eθ2)) > 1}, where, for a nonnegative square matrix A with a finite or countable dimension, cp(A) denote the convergence parameter of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', cp(A) = sup{r ∈ R+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' �∞ n=0 rnAn < ∞, entry-wise}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We have cp(A{1,2} ∗,∗ (eθ1, eθ2)) = spr(A{1,2} ∗,∗ (eθ1, eθ2))−1, where for a square complex matrix A, spr(A) is the spectral radius of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of Ozawa [13], cp( ¯A{1} ∗ (eθ))−1 and cp( ¯A{2} ∗ (eθ))−1 are log- convex in θ, and the closures of Γ{1} and Γ{2} are convex sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' spr( ¯A{1,2} ∗ (eθ1, eθ2)) is also log-convex in (θ1, θ2), and the closure of Γ{1,2} is a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, by Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of Ozawa [13], Γ{1,2} is bounded under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We depict an example of the domains Γ{1,2}, Γ{1} and Γ{2} in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We define several extreme values and several functions with respect to the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i ∈ {1, 2}, define θmin i and θmax i as θmin i = inf{θi ∈ R : (θ1, θ2) ∈ Γ{1,2}}, θmax i = sup{θi ∈ R : (θ1, θ2) ∈ Γ{1,2}}, and for a direction vector c = (c1, c2) ∈ N2, θmax c as θmax c = sup{c1θ1 + c2θ2 : (θ1, θ2) ∈ Γ{1,2}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For θ1 ∈ [θmin 1 , θmax 1 ], there exist two real solutions to equation spr(A{1,2} ∗,∗ (eθ1, eθ2)) = 1, counting multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote them by θ2 = η2(θ1) and θ2 = ¯η2(θ1), respectively, where η2(θ1) ≤ ¯η2(θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For 5 C101 + C202 = 0max 02 个 01 = 0 02 = 02 r(1) spr amax n2 (0) r(2] r(1,2] n2(0) > 0 0 1 0 01 0Figure 3: Classification θ2 ∈ [θmin 2 , θmax 2 ], also denote by θ1 = η1(θ2) and θ1 = ¯η1(θ2) the two real solutions to the equation spr(A{1,2} ∗,∗ (eθ1, eθ2)) = 1, where η1(θ2) ≤ ¯η1(θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i ∈ {1, 2}, define θ∗ i as θ∗ i = sup{θi ∈ R : (θ1, θ2) ∈ Γ{i}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For another characterization of θ∗ i , see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7 of Ozawa [10], where θ∗ i is denoted by z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In terms of these points and functions, we geometrically classify the model into four types according to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define two points Q1 and Q2 as Q1 = (θ∗ 1, ¯η2(θ∗ 1)) and Q2 = (¯η1(θ∗ 2), θ∗ 2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Using these points, we define the following classification (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Type 1: θ∗ 1 ≥ ¯η1(θ∗ 2) and ¯η2(θ∗ 1) ≤ θ∗ 2, Type 2: θ∗ 1 < ¯η1(θ∗ 2) and ¯η2(θ∗ 1) > θ∗ 2, Type 3: θ∗ 1 ≥ ¯η1(θ∗ 2) and ¯η2(θ∗ 1) > θ∗ 2, Type 4: θ∗ 1 < ¯η1(θ∗ 2) and ¯η2(θ∗ 1) ≤ θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [14], for any direction vector c = (c1, c2) ∈ N2, the asymptotic decay rate in the direction c is space homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, we denote it by ξc, which satisfies, for any (x, j) ∈ Z2 + × S0, ξc = − lim k→∞ 1 n log ν(x+kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4) The asymptotic decay rate ξc has already been obtained in [14], and as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [14], it is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let c = (c1, c2) be an arbitrary direction vector in N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Type 1: ξc = � � � c1θ∗ 1 + c2¯η2(θ∗ 1) if − c1 c2 < ¯η′ 2(θ∗ 1), θmax c if ¯η′ 2(θ∗ 1) ≤ − c1 c2 ≤ ¯η′ 1(θ∗ 2)−1, c1¯η1(θ∗ 2) + c2θ∗ 2 if − c1 c2 > ¯η′ 1(θ∗ 2)−1, where ¯η′ 2(x) = d dx ¯η2(x) and ¯η′ 1(x) = d dx ¯η1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Type 2: ξc = � � � c1θ∗ 1 + c2¯η2(θ∗ 1) if − c1 c2 ≤ θ∗ 2−¯η2(θ∗ 1) ¯η1(θ∗ 2)−θ∗ 1 , c1¯η1(θ∗ 2) + c2θ∗ 2 if − c1 c2 > θ∗ 2−¯η2(θ∗ 1) ¯η1(θ∗ 2)−θ∗ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 0* 02 r(1,2] > 01 0* 1 1 Type 1 Type 2 Type 3 Type 4Type 3: ξc = c1¯η1(θ∗ 2) + c2θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Type 4: ξc = c1θ∗ 1 + c2¯η2(θ∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The asymptotic decay function hc(k) in the direction c is defined as the function that satisfies, for some positive vector gc, lim k→∞ νkc hc(k) = gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) It is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let c be an arbitrary direction vector in N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' hc(k) = � k− 1 2 (2l−1)e−ξck if ¯η′ 2(θ∗ 1) < − c1 c2 < ¯η′ 1(θ∗ 2)−1 in Type 1, e−ξck otherwise, as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) where l is some positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Except for the case where ¯η′ 2(θ∗ 1) ≤ − c1 c2 ≤ ¯η′ 1(θ∗ 2)−1 in Type 1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 has already been proved in [14], see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, to this end, it suffices to prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1 and set c = (c1, c2) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, the asymptotic decay function hc(k) is given as hc(k) = � k− 1 2 (2l−1)e−θmax c k if ¯η′ 2(θ∗ 1) < − c1 c2 = −1 < ¯η′ 1(θ∗ 2)−1, e−θmax c k if ¯η′ 2(θ∗ 1) = −1 or ¯η′ 1(θ∗ 2) = −1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7) where l is some positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From this proposition, we can obtain the same result for a general direction vector c ∈ N2, by using the block state process derived from the original 2d-QBD process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We, therefore, prove the proposition in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From the corresponding results for a 2d-RRW without a background process obtained in Malyshev [6], it is expected that the value of l in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 is one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', hc(k) = k− 1 2 e−ξck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 3 Preliminaries Let z and w be complex valuables unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For a positive number r, denote by ∆r the open disk of center 0 and radius r on the complex plain, and ∂∆r the circle of the same center and radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We denote by ¯∆r the closure of ∆r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For a, b ∈ R+ such that a < b, let ∆a,b be an open annular domain on C defined as ∆a,b = {z ∈ C : a < |z| < b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We denote by ¯∆a,b the closure of ∆a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For r > 0, ε > 0 and θ ∈ [0, π/2), define ˜∆r(ε, θ) = {z ∈ C : |z| < r + ε, z ̸= r, | arg(z − r)| > θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For r > 0, we denote by “ ˜∆r ∋ z → r” that ˜∆r(ε, θ) ∋ z → r for some ε > 0 and some θ ∈ [0, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the rest of the paper, instead of proving that a function f(z) is analytic in ˜∆r(ε, θ) for some ε > 0 and θ ∈ [0, π/2), we often demonstrate that the function f(z) is analytic in ∆r and on ∂∆r \\ {r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In order to give general results, this section is described independently from other parts of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 Analytic extension of a G-matrix function First, we define a G-matrix function according to Ozawa and Kobayashi [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i, j ∈ {−1, 0, 1}, let Ai,j be a substochastic matrix with a finite dimension s0, and define the following matrix functions: A∗,j(z) = � i∈{−1,0,1} ziAi,j, j = −1, 0, 1, A∗,∗(z, w) = � i,j∈{−1,0,1} ziwjAi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' A∗,∗(1, 1) is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let χ(z, w) be the spectral radius of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='∗(z, w), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', χ(z, w) = spr(A∗,∗(z, w)), and Γ be a domain on R2 defined as Γ = {(θ1, θ2) ∈ R2 : χ(eθ1, eθ2) < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The Markov modulated random walk on Z2 × {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0} that is governed by {Ai,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i, j ∈ {−1, 0, 1}} is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Under this assumption, A∗,∗(1, 1) is also irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11], Γ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since χ(eθ1, eθ2) is convex in (θ1, θ2) ∈ R2, the closure of Γ is a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define extreme points θmin 1 and θmax 2 as follows: θmin 1 = inf (θ1,θ2)∈Γ θ1, θmax 1 = sup (θ1,θ2)∈Γ θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For θ1 ∈ [θmin 1 , θmax 1 ], let θ2(θ1) and ¯θ2(θ1) be the two real solutions to equation χ(eθ1, eθ2) = 1, counting multiplicity, where θ2(θ1) ≤ ¯θ2(θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We set zmin 1 = eθmin 1 and zmax 1 = eθmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For n ≥ 1, define the following set of index sequences: In = � i(n) ∈ {−1, 0, 1}n : k � l=1 il ≥ 0 for k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', n − 1} and n � l=1 il = −1 � , where i(n) = (i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', in), and define the following matrix function: Dn(z) = � i(n)∈In A∗,i1(z)A∗,i2(z) · · · A∗,in(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function G(z) as G(z) = ∞ � n=1 Dn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11], this matrix series absolutely converges entry-wise in z ∈ ¯∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We call this G(z) the G-matrix function generated from {Ai,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i, j ∈ {−1, 0, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For z ∈ ¯∆zmin 1 ,zmax 1 , G(z) satisfies the inequality |G(z)| ≤ G(|z|) and the following matrix quadratic equation: A∗,−1(z) + A∗,0(z)G(z) + A∗,1(z)G(z)2 = G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) Furthermore, for z ∈ [zmin 1 , zmax 1 ], it is the minimum nonnegative solution to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, G(z) is an extension of a usual G-matrix in the queueing theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' see, for example, [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 of [11], we see that, for z ∈ [zmin 1 , zmax 1 ], the Perron-Frobenius eigenvalue of G(z) is given by eθ2(log z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', spr(G(z)) = eθ2(log z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11], G(z) satisfies I − A∗,∗(z, w) = w−1 (I − A∗,0(z) − wA∗,1(z) + A∗,1(z)G(z)) (wI − G(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11], the following property holds true for G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 8 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' G(z) is entry-wise analytic in the open annular domain ∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We give the eigenvalues of G(z) according to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that our final aim in this subsection is to give an analytic extension of G(z) through its Jordan canonical form without assuming all the eigenvalues of G(z) are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' On the other hand, in [11], the eigenvalues were assumed to be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function L(z, w) as L(z, w) = zw(I − A∗,∗(z, w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Each entry of L(z, w) is a polynomial in z and w with at most degree 2 for each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We use a notation Ξ, defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let f(z, w) be an irreducible polynomial in z and w and assume its degree with respect to w is m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let a(z) be the coefficient of wm in f(z, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set Ξ(f) as Ξ(f) = {z ∈ C : a(z) = 0 or (f(z, w) = 0 and fw(z, w) = 0 for some w ∈ C)}, where fw(z, w) = (∂/∂w)f(z, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Each point in Ξ(f) is an algebraic singularity of the algebraic function w = α(z) defined by polynomial equation f(z, w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each point z ∈ C \\ Ξ(f), f(z, w) = 0 has just m distinct solutions, which correspond to the m branches of the algebraic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let φ(z, w) be a polynomial in z and w defined as φ(z, w) = det L(z, w) and mφ its degree with respect to w, where s0 ≤ mφ ≤ 2s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let α1(z), α2(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', αmφ(z) be the mφ branches of the algebraic function w = α(z) defined by the polynomial equation φ(z, w) = 0, counting multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We number the brunches so that they satisfy the following: (1) For every z ∈ ¯∆zmin 1 ,zmax 1 and for every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0}, |αk(z)| ≤ eθ2(log |z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) For every z ∈ ¯∆zmin 1 ,zmax 1 and for every k ∈ {s0 + 1, s0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ}, |αk(z)| ≥ e¯θ2(log |z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3) For every z ∈ [zmin 1 , zmax 1 ], αs0(z) = eθ2(log z) and αs0+1(z) = e¯θ2(log z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This is possible by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 of [11], the G-matrix function of G(z) satisfies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every z ∈ ¯∆zmin 1 ,zmax 1 , the eigenvalues of G(z) are given by α1(z), α2(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', αs0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Without loss of generality, we assume that, for some nφ ∈ N and l1, l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lnφ ∈ N, the polynomial φ(z, w) is factorized as φ(z, w) = f1(z, w)l1f2(z, w)l2 · · · fnφ(z, w)lnφ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) where fk(z, w), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', nφ, are irreducible polynomials in z and w and they are relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the field of coefficients of polynomials is C, this factorization is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ}, αk(z) is a branch of the algebraic function w = α(z) defined by the polynomial equation fn(z, w) = 0 for some n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', nφ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We denote such n by q(k), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', fq(k)(z, αk(z)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since αs0(z) is the Perron-Frobenius eigenvalue of G(z) when z ∈ [zmin 1 , zmax 1 ], the multiplicity of αs0(z) is one and we have lq(s0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set E1 as E1 = nφ � n=1 Ξ(fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 9 Since, for every n, the polynomial fn(z, w) is irreducible and not identically zero, the point set E1 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Every branch αk(z) is analytic in C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set E2 as E2 = {z ∈ C \\ E1 : fn(z, w) = fn′(z, w) = 0 for some n, n′ ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', nφ} such that n ̸= n′ and for some w ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since, for any n, n′ such that n ̸= n′, fn(z, w) and fn′(z, w) are relatively prime, the point set E2 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that every branch αk(z) is analytic in a neighborhood of any z0 ∈ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ} and for every z ∈ C \\ (E1 ∪ E2), the multiplicity of αk(z) as a zero of det L(z, w) is equal to lq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This means that, for every z ∈ ¯∆zmin 1 ,zmax 1 \\ (E1 ∪ E2), the multiplicity of the eigenvalue αk(z) of G(z) is lq(k), which does not depend on z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a positive integer m0 as m0 = s0 � k=1 1 lq(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This m0 is the number of different branches in {αi(z) : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0} when z ∈ C \\ (E1 ∪ E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote the different branches by ˇαk(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, so that ˇαm0(z) = αs0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Instead of using q(k), we define a function ˇq(k) so that lˇq(k) indicates the multiplicity of ˇαk(z) when z ∈ C\\(E1∪E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We always have lˇq(m0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We give the Jordan normal form of G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a domain Ω as Ω = ∆zmin 1 ,zmax 1 \\ (E1 ∪ E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, define a positive integer tk,i as tk,i = min z∈Ω dim Ker (ˇαk(z)I − G(z))i and a point set Gk,i as Gk,i = {z ∈ Ω : dim Ker (ˇαk(z)I − G(z))i > tk,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ˇαk(z) and G(z) are analytic in Ω, we see from the proof of Theorem S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [3] that each Gk,i is an empty set or a set of discrete complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, define a nonnegative integer sk,i as sk,i = 2tk,i − tk,i+1 − tk,i−1, where tk,0 = 0 and tk,lˇq(k)+1 = lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, define a positive integer mk,0 and point set EG k as mk,0 = tk,1, EG k = lˇq(k) � i=1 Gk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' When z ∈ Ω \\ EG k , this mk,0 is the number of Jordan blocks of G(z) with respect to the eigenvalue ˇαk(z) and, for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, sk,i is the number of Jordan blocks whose dimension is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, the Jordan normal form of G(z) takes a common form in z ∈ Ω \\ �m0 k=1 EG k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, denote by mk,i the dimension of the i-th Jordan block of G(z) with respect to the eigenvalue ˇαk(z), where we number the Jordan blocks so that if i ≤ i′, mk,i ≥ mk,i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, they satisfy �mk,0 i=1 mk,i = lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by Jn(λ) the n-dimensional Jordan block of eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For z ∈ Ω \\ �m0 k=1 EG k , the Jordan normal form of G(z), JG(z), is given by JG(z) = diag(Jmk,i(ˇαk(z)), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4) where mm0,0 = 1 and Jmm0,1(ˇαm0(z)) = αs0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that the matrix function JG(z) is defined on C and analytic in C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' An analytic extension of G(z) is given by the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 10 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' There exist vector functions: ˇvL k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i, such that they are analytic in C \\ E1 and satisfy for every z ∈ ∆zmin 1 ,zmax 1 \\ (E1 ∪ E0) that G(z) = T L(z)JG(z)(T L(z))−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) where E0 is a set of discrete complex numbers and matrix function T L(z) is defined as T L(z) = �ˇvL k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 is elementary and very lengthy, we give it in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, {ˇvL k,i,j(z)} is the set of the generalized eigenvectors of G(z), but we denote them with superscript L since they are generated from the matrix function L(z, w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set EL T as EL T = {z ∈ C \\ E1 : det T L(z) = 0}, which is an empty set or a set of discrete complex numbers since det T L(z) is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˇG(z) as ˇG(z) = T L(z)JG(z)(T L(z))−1 = T L(z)JG(z) adj(T L(z)) det(T L(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) Then, it is entry-wise analytic in C\\(E1∪EL T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 and the identity theorem for analytic functions, this ˇG(z) is an analytic extension of the matrix function G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, we denote ˇG(z) by G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, G(z) is entry-wise analytic in ∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The following corollary asserts that G(z) is also analytic on the outside boundary of ∆zmin 1 ,zmax 1 except for the point z = zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The extended G-matrix function G(z) is entry-wise analytic on ∂∆zmax 1 \\ {zmax 1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since this corollary can be proved in a manner similar to that used in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7 of [11], we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˇuL m0,1,1(z) the last row of the matrix function (T L(z))−1, and define a diagonal matrix function Js0(z) as Js0(z) = diag � 0 · · 0 αs0(z) � , where αs0(z) = ˇαm0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, since mm0,0 = 1 and mm0,1 = 1, we obtain the following decomposition of G(z) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6): G(z) = G†(z) + αs0(z)ˇvL m0,1,1(z)ˇuL m0,1,1(z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7) where G†(z) = T L(z)(JG(z) − Js0(z))(T L(z))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By the definition, G(z) satisfies, for n ≥ 1, G(z)n = G†(z)n + αs0(z)nˇvL m0,1,1(z)ˇuL m0,1,1(z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8) and G†(z), for z ∈ ¯∆zmin 1 ,zmax 1 , spr(G†(z)) ≤ spr(G†(|z|) < spr(G(|z|)) = αs0(|z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, in a neighborhood of z = zmax 1 , we have spr(G†(z)) < αs0(zmax 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the point z = zmax 1 is a branch point of ˇαm0(z) (= αs0(z)), there exists a function ˜αs0(ζ) being analytic in a neighborhood of ζ = 0 and satisfying ˇαm0(z) = αs0(z) = ˜αs0((zmax 1 − z) 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 11 Let ˜vs0(ζ) be a vector function satisfying L(zmax 1 − ζ2, ˜αs0(ζ))˜vs0(ζ) = 0, where ˜vs0(ζ) is elementwise analytic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˜T(ζ) the matrix function given by replacing the last column of T L(zmax 1 − ζ2) with ˜vs0(ζ) and by ˜us0(ζ) the last row of ˜T(ζ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By the definition, ˜T(ζ) as well as ˜us0(ζ) is entry-wise analytic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a diagonal matrix function ˜Js0(ζ) as ˜Js0(ζ) = diag � 0 · · 0 ˜αs0(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For later use, we give the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' There exists a matrix function ˜G(ζ) being entry-wise analytic in a neighborhood of ζ = 0 and satisfying G(z) = ˜G((zmax 1 −z) 1 2 ) in a neighborhood of z = zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This ˜G(ζ) is represented as ˜G(ζ) = ˜G†(ζ) + ˜αs0(ζ)˜vs0(ζ)˜us0(ζ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) where ˜G†(ζ) is a matrix function being entry-wise analytic in a neighborhood of ζ = 0 and satisfying G†(z) = ˜G†((zmax 1 − z) 1 2 ) in a neighborhood of z = zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In a neighborhood of ζ = 0, spr( ˜G†(ζ)) < ˜αs0(0) = αs0(zmax 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Give ˜G†(ζ) as ˜G†(ζ) = ˜T(ζ)(JG(zmax 1 − ζ2) − Js0(zmax 1 − ζ2)) ˜T(ζ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), we obtain the results of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The following limit with respect to αs0(z) (= ˇαm0(z)) is given by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 of [11] (also see Lemma 10 of [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' lim ˜∆zmax 1 ∋z→zmax 1 αs0(zmax 1 ) − αs0(z) (zmax 1 − z) 1 2 = −αs0,1 = √ 2 � −¯ζ1,w2(ζ2(zmax 1 )) > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) where z = ¯ζ1(w) is the larger one of two real solutions to equation χ(z, w) = 1 and ¯ζ1,w2(w) = (d2/dw2) ¯ζ1(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let R(z) be the rate matrix function generated from {Ai,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i, j = −1, 0, 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' for the definition of R(z), see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function N(z) as N(z) = (I − A∗,0(z) − A∗,1(z)G(z))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' N(z) is well defined for every z ∈ ¯∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The extended G(z) satisfies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' lim ˜∆zmax 1 ∋z→zmax 1 G(zmax 1 ) − G(z) (zmax 1 − z) 1 2 = −G1 = −αs0,1N(zmax 1 )vR(zmax 1 )uG s0(zmax 1 ) ≥ O, ̸= O, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='11) where uG s0(zmax 1 ) is the left eigenvector of G(zmax 1 ) with respect to the eigenvalue eθ2(log zmax 1 ) = αs0(zmax 1 ), vR(zmax 1 ) the right eigenvector of R(zmax 1 ) with respect to the eigenvalue e−¯θ2(log zmax 1 ) = e−θ2(log zmax 1 ) and they satisfy uG s0(zmax 1 )N(zmax 1 )vR(zmax 1 ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since this lemma can be proved in a manner similar to that used in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 of [11], we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 G-matrix in the reverse direction and its properties Let A−1, A0 and A1 be square nonnegative matrices with a finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function A∗(z) and matrix Q as A∗(z) = z−1A−1 + A0 + zA1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12) Q = � � � � � A0 A1 A−1 A0 A1 A−1 A0 A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) We assume: (a1) Q is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (a2) The infimum of the maximum eigenvalue of A∗(eθ) in θ ∈ R is less than or equal to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', infθ∈R spr(A∗(eθ)) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, there are two real solutions to equation cp(A∗(eθ)) = 1, counting multiplicity, see comments to Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We denote the solutions by θ and ¯θ, where θ ≤ ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The rate matrix and G-matrix generated from the triplet {A−1, A0, A1} also exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' we denote them by R and G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' R and G are the minimal nonnegative solutions to the following matrix quadratic equations: R = R2A−1 + RA0 + A1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='14) G = A−1 + A0G + A1G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15) We have I − A∗(z) = (I − zR)(I − H)(I − z−1G), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) spr(R) = e−¯θ, spr(G) = eθ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='17) where H = A0 + A1G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' see, for example, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We define a rate matrix and G-matrix in the reverse direction generated from the triplet {A−1, A0, A1}, denoted by Rr and Gr, as the minimal nonnegative solutions to the following matrix quadratic equations: Rr = (Rr)2A1 + RrA0 + A−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='18) Gr = A1 + A0Gr + A−1(Gr)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='19) In other words, Rr and Gr are, respectively, the rate matrix and G-matrix generated from the triplet by exchanging A−1 and A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since z−1A1 + A0 + zA−1 = A∗(z−1), we obtain by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='17) that I − A∗(z−1) = (I − zRr)(I − Hr)(I − z−1Gr), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='20) spr(Rr) = eθ, spr(Gr) = e−¯θ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='21) where Hr = A0 + A−1Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We use the following property in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let v be the right eigenvector of G with respect to the eigenvalue eθ and vr that of Gr with respect to the eigenvalue e−¯θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Gv = eθv and Grvr = e−¯θvr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If θ = ¯θ, we have v = vr, up to multiplication by a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='20), we obtain A∗(eθ)v = v, A∗(e ¯θ)vr = A∗(eθ)vr = vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since spr(A∗(eθ)) = 1 and A∗(eθ) is irreducible, the right eigenvector of A∗(eθ) with respect to the eigenvalue of 1 is unique, up to multiplication by a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This implies v = vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 13 4 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 Methodology and outline of the proof Define the vector generating function of the stationary probabilities in direction c ∈ N2, ϕc(z), as ϕc(z) = ∞ � k=0 zkνkc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Also define zmin c and zmax c as zmin c = eθmin c and zmax c = eθmax c , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hereafter, we set c = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In order to obtain the asymptotic function of the stationary tail probability in the direction c = (1, 1), we apply the following lemma to the vector generating function ϕc(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 (Theorem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 of Flajolet and Sedgewick [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let f be a generating function of a sequence of real numbers {an, n ∈ Z+}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', f(z) = �∞ n=0 anzn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If f(z) is singular at z = z0 > 0 and analytic in ˜∆z0(ε, θ) for some ε > 0 and some θ ∈ [0, π/2) and if it satisfies lim ˜∆z0∋z→z0 (z0 − z)αf(z) = c0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) for α ∈ R \\ {0, −1, −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='} and some nonzero constant c0 ∈ R, then lim n→∞ �nα−1 Γ(α) z−n 0 �−1 an = c (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) for some real number c, where Γ(z) is the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This means that the asymptotic function of the sequence {an} is given by nα−1z−n 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For the purpose, we prove the following propositions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1 ≤ 1/¯η′ 2(θ∗ 1), the vector function ϕc(z) is elementwise analytic in ˜∆zmax c (ε, θ) for some ε > 0 and some θ ∈ [0, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1 ≤ 1/¯η′ 2(θ∗ 1), there exist a vector function ˜ϕc(ζ) being meromorphic in a neighborhood of ζ = 0 and satisfying ϕc(z) = ˜ϕc((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) < −1 < 1/¯η′ 2(θ∗ 1), the point ζ = 0 is a pole of ˜ϕc(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1 or ¯η′ 2(θ∗ 1) = −1, it is a pole of ˜ϕc(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2, if ¯η′ 1(θ∗ 2) < −1 < 1/¯η′ 2(θ∗ 1), the Puiseux series of ϕc(z) is represented as ϕc(z) = ∞ � k=−1 ϕc 1,k(zmax c − z) k 2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) where {ϕc 1,k} is a series of coefficient vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1 or ¯η′ 2(θ∗ 1) = −1, it is represented as ϕc(z) = ∞ � k=−2 ϕc 2,k(zmax c − z) n 2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4) where {ϕc 2,k} is a series of coefficient vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let l be a positive integer such that ϕc 1,l−2 ̸= 0 and ϕc 1,k−2 = 0 for all positive integer k less than l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3), we obtain hc(k) = k− 1 2 (2l−1)(zmax c )−k = k− 1 2 (2l−1)e−θmax c k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the former half of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) = −1 or ¯η′ 2(θ∗ 1) = −1, ϕc(z) satisfies the following property, which will be proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 14 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, we have, for some positive vectors uc 1 and uc 2, lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc(z) = � � � uc 1 if ¯η′ 1(θ∗ 2) = −1 and ¯η′ 2(θ∗ 1) < −1, uc 2 if ¯η′ 1(θ∗ 2) < −1 and ¯η′ 2(θ∗ 1) = −1, uc 1 + uc 2 if ¯η′ 1(θ∗ 2) = ¯η′ 2(θ∗ 1) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) Hence, ϕc 2,−2 is positive, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, we obtain hc(k) = (zmax c )−k = e−θmax c k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the latter half of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1 and ¯η′ 1(θ∗ 2) < −c1/c2 = −1 < 1/¯η′ 2(θ∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If the vector function ϕc(z) diverges at z = zmax c , the coefficient vector ϕc 1,−1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) must be nonzero and , by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, we have hc(k) = k− 1 2 (zmax c )−k = k− 1 2 e−θmax c k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 Proof of Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 Recall that the direction vector c is set as c = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Notation of this subsection follows [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by Φ{1,2} = (Φ{1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) the fundamental matrix (potential matrix) of P {1,2}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Φ{1,2} = �∞ n=0(P {1,2})n, where P {1,2} = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) is the transition probability matrix of the induced MA-process {Y {1,2} n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For x ∈ Z2, define the matrix generating function of the blocks of Φ{1,2} in direction c, Φc x,∗(z), as Φc x,∗(z) = ∞ � k=−∞ zkΦ{1,2} x,kc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' According to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) of [14], we divide ϕc(z) into three parts as follows: ϕc(z) = ϕc 0(z) + ϕc 1(z) + ϕc 2(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) where ϕc 0(z) = � i1,i2∈{−1,0,1} ν(0,0)(A∅ i1,i2 − A{1,2} i1,i2 )Φc (i1,i2),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7) ϕc 1(z) = ∞ � k=1 � i1,i2∈{−1,0,1} ν(k,0)(A{1} i1,i2 − A{1,2} i1,i2 )Φc (k+i1,i2),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8) ϕc 2(z) = ∞ � k=1 � i1,i2∈{−1,0,1} ν(0,k)(A{2} i1,i2 − A{1,2} i1,i2 )Φc (i1,k+i2),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) According to [14], we focus on ϕc 2(z) and consider another skip-free MA-process generated from {Y {1,2} n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The MA-process is { ˆY n} = {( ˆXn, ˆJn)} = {( ˆX1,n, ˆX2,n), ( ˆRn, ˆJn)}, where ˆX1,n = X{1,2} 1,n , ˆX2,n and ˆRn are the quotient and remainder of X{1,2} 2,n − X{1,2} 1,n divided by 2, respectively, and ˆJn = J{1,2} n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The state space of { ˆY n} is Z2 × {0, 1} × S0 and the additive part { ˆXn} is skip free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From the definition, if ˆXn = (x1, x2) and ˆRn = r in the new MA-process, it follows that X{1,2} 1,n = x1, X{1,2} 2,n = x1 + 2x2 + r in the original MA-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, ˆY n = (k, 0, 0, j) means 15 Y {1,2} n = (k, k, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˆP = ( ˆPx,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x′ ∈ Z2) the transition probability matrix of { ˆY n},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' which is given as ˆPx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='x′ = � ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} x′−x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if x′ − x ∈ {−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 1}2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' where ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 = � O O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 O � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 = �O O O O � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 = �O O O O � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 = � O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 O O � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˆΦ = (ˆΦx,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) the fundamental matrix of ˆP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', ˆΦ = �∞ n=0( ˆP)n, and for x = (x1, x2) ∈ Z2, define a matrix generating function ˆΦx,∗(z) as ˆΦx,∗(z) = ∞ � k=−∞ zk ˆΦx,(k,0) = � Φc (x1,x1+2x2),∗(z) Φc (x1,x1+2x2−1),∗(z) Φc (x1,x1+2x2+1),∗(z) Φc (x1,x1+2x2),∗(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) We consider analytic properties of the matrix function Φc (x1,x1+2x2),∗(z) through ˆΦ(x1,x2),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define blocks ˆA{2} i1,i2, i1, i2 ∈ {−1, 0, 1}, as ˆA{2} −1,1 = ˆA{2} −1,0 = ˆA{2} −1,−1 = O and ˆA{2} 0,1 = � O O A{2} 0,1 O � , ˆA{2} 0,0 = � A{2} 0,0 A{2} 0,1 A{2} 0,−1 A{2} 0,0 � , ˆA{2} 0,−1 = � O A{2} 0,−1 O O � , ˆA{2} 1,1 = �O O O O � , ˆA{2} 1,0 = � A{2} 1,1 O A{2} 1,0 A{2} 1,1 � , ˆA{2} 1,−1 = � A{2} 1,−1 A{2} 1,0 O A{2} 1,−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i1, i2 ∈ {−1, 0, 1}, define the following matrix generating functions: ˆA{1,2} ∗,i2 (z) = � i∈{−1,0,1} zi ˆA{1,2} i,i2 , ˆA{1,2} i1,∗ (z) = � i∈{−1,0,1} zi ˆA{1,2} i1,i , ˆA{2} ∗,i2(z) = � i∈{0,1} zi ˆA{2} i,i2, ˆA{2} i1,∗(z) = � i∈{−1,0,1} zi ˆA{2} i1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a vector generating function ˆϕ2(z) as ˆϕ2(z) = �ˆϕ2,1(z) ˆϕ2,2(z) � = ∞ � k=1 � i1,i2∈{−1,0,1} ˆν(0,k)( ˆA{2} i1,i2 − ˆA{1,2} i1,i2 )ˆΦ(i1,k+i2),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='11) where, for x = (x1, x2) ∈ Z2 +, ˆνx = � ν(x1,x1+2x2) ν(x1,x1+2x2+1) � and hence, for k ≥ 0, ˆν(0,k) = � ν(0,2k) ν(0,2k+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) of [14], ϕc 2(z) is represented as ϕc 2(z) = ˆϕ2,1(z) + � i1,i2∈{−1,0,1} ν(0,1)(A{2} i1,i2 − A{1,2} i1,i2 )Φc (i1,i2+1),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12) 16 Hence, we consider analytic properties of the vector function ϕc 2(z) through ˆϕc 2(z) and ˆΦx,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆG0,∗(z) be the G-matrix function generated from the triplet { ˆA{1,2} ∗,−1 (z), ˆA{1,2} ∗,0 (z), ˆA{1,2} ∗,1 (z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) of [14], we have, for x2 ≥ 0, ˆΦ(x1,x2),∗(z) = zx1 ˆG0,∗(z)x2 ˆΦ(0,0),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) and this leads us to ˆϕ2(z) = ∞ � k=1 � i2∈{−1,0,1} ˆν(0,k)( ˆA{2} ∗,i2(z) − ˆA{1,2} ∗,i2 (z)) ˆG0,∗(z)k+i2 ˆΦ(0,0),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='14) Hence, analytic properties of the vector function ˆϕ2(z) as well as the matrix function ˆΦx,∗(z) can be clarified through ˆG0,∗(z) and ˆΦ(0,0),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='14), ˆϕ2(z) is represented as ˆϕ2(z) = ˆa(z, ˆG0,∗(z))ˆΦ(0,0),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15) where ˆa(z, w) = ∞ � k=1 ˆν(0,k) ˆD(z, ˆG0,∗(z))wk−1, ˆD(z, w) = ˆA{2} ∗,−1(z) + ˆA{2} ∗,0 (z)w + ˆA{2} ∗,1 (z)w2 − Iw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' First, we consider ˆΦ(0,0),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆGr 0,∗(z) be the G-matrix function in the reverse direction generated from the triplet { ˆA{1,2} ∗,−1 (z), ˆA{1,2} ∗,0 (z), ˆA{1,2} ∗,1 (z)}, which means that ˆGr 0,∗(z) is the G-matrix function generated from the triplet by exchanging ˆA{1,2} ∗,−1 (z) and ˆA{1,2} ∗,1 (z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˆU(z) as ˆU(z) = ˆA{1,2} ∗,−1 (z) ˆGr 0,∗(z) + ˆA{1,2} ∗,0 (z) + ˆA{1,2} ∗,1 (z) ˆG0,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) Then, ˆΦ(0,0),∗(z) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15) is given as ˆΦ(0,0),∗(z) = ∞ � n=0 ˆU(z)n = (I − ˆU(z))−1 = adj(I − ˆU(z)) det(I − ˆU(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='17) Recall that zmin c = eθmin c and zmax c = eθmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For θ ∈ [θmin c , θmax c ], let (ηR c,1(θ), ηR c,2(θ)) and (ηL c,1(θ), ηL c,2(θ)) be the two real roots of the simultaneous equations: spr(A{1,2} ∗,∗ (eθ1, eθ2)) = 1, θ1 + θ2 = θ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='18) counting multiplicity, where ηL c,1(θ) ≤ ηR c,1(θ) and ηL c,2(θ)) ≥ ηR c,2(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that ηL c,1(θmax c ) = ηR c,1(θmax c ) and ηL c,2(θmax c ) = ηR c,2(θmax c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='32) of [14], we have spr( ˆG0,∗(eθ)) = e2ηR c,2(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='19) Since the eigenvalues of ˆGr 0,∗(z) are coincide with those of the rate matrix function generated from the same triplet { ˆA{1,2} ∗,−1 (z), ˆA{1,2} ∗,0 (z), ˆA{1,2} ∗,1 (z)}, we have spr( ˆGr 0,∗(eθ)) = e−2ηL c,2(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='20) By Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, ˆG0,∗(z) and ˆGr 0,∗(z) satisfy the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 17 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (1) The extended G-matrix functions ˆG0,∗(z) and ˆGr 0,∗(z) are entry-wise an- alytic in ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The point z = zmax c is a common branch point of ˆG0,∗(z) and ˆGr 0,∗(z) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) There exist matrix functions ˜G0,∗(ζ) and ˜Gr 0,∗(ζ) being analytic in a neighborhood of ζ = 0 and satisfying ˆG0,∗(z) = ˜G0,∗((zmax c − z) 1 2 ) and ˆGr 0,∗(z) = ˜Gr 0,∗((zmax c − z) 1 2 ), respectively, in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In order to investigate singularity of ˆΦ(0,0),∗(z) at z = zmax c , we give the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The maximum eigenvalue of ˆU(zmax c ) is 1, and it is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='30) of [14], we have spr( ˆA{1,2} ∗,∗ (zmax c , e2ηR c,2(θmax c ))) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let v be the right eigenvector of ˆA{1,2} ∗,∗ (zmax c , e2ηR c,2(θmax c )) with respect to eigenvalue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since spr( ˆG0,∗(zmax c )) = e2ηR c,2(θmax c ) and spr( ˆGr 0,∗(zmax c )) = e−2ηL c,2(θmax c ) = e−2ηR c,2(θmax c ), we have, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6, ˆG0,∗(zmax c )v = e2ηR c,2(θmax c )v, ˆGr 0,∗(zmax c )v = e−2ηR c,2(θmax c )v, Hence, ˆU(zmax c )v = ˆA{1,2} ∗,∗ (zmax c , e2ηR c,2(θmax c ))v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This means that the value of 1 is an eigenvalue of ˆU(zmax c ), and we obtain spr( ˆU(zmax c )) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Suppose spr( ˆU(zmax c )) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, since spr( ˆU(eθ)) is convex in θ ∈ R, there exist a positive θ0 < θmax c such that spr( ˆU(eθ0)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For this θ0, ˆΦ(0,0),∗(z) diverges at z = eθ0 < zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [14], which asserts that ˆΦ(0,0),∗(z) absolutely convergent in z ∈ ∆zmin c ,zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, spr( ˆU(zmax c )) ≤ 1, and this implies the maximum eigenvalue of ˆU(zmax c ) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ˆU(zmax c ) is irreducible, it is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆλU(z) be the eigenvalue of ˆU(z) satisfying ˆλU(z) = spr( ˆU(z)) for z ∈ [zmin c , zmax c ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆuU(z) and ˆvU(z) be the left and right eigenvectors of ˆU(z) with respect to the eigenvalue ˆλU(z), respectively, satisfying ˆuU(z)ˆvU(z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˜U(ζ) as ˜U(ζ) = ˆA{1,2} ∗,−1 (zmax c − ζ2) ˜Gr 0,∗(ζ) + ˆA{1,2} ∗,0 (zmax c − ζ2) + ˆA{1,2} ∗,1 (zmax c − ζ2) ˜G0,∗(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, ˜U(ζ) is entry-wise analytic in a neighborhood of ζ = 0 and satisfies ˆU(z) = ˜U((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˜Φ(0,0),∗(ζ) as ˜Φ(0,0),∗(ζ) = (I − ˜U(ζ))−1 = adj(I − ˜U(ζ)) det(I − ˜U(ζ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='21) ˆΦ(0,0),∗(z) and ˜Φ(0,0),∗(ζ) satisfy the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (1) The matrix function ˆΦ(0,0),∗(z) is entry-wise analytic in ∆zmin c ,zmax c ∪∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) ˜Φ(0,0),∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0, and the point ζ = 0 is a pole of ˜Φ(0,0),∗(ζ) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆΦ(0,0),∗(z) is represented as ˆΦ(0,0),∗(z) = ˜Φ(0,0),∗((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 18 (3) ˆΦ(0,0),∗(z) satisfies lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 ˆΦ(0,0),∗(z) = ˆgΦˆvU(zmax c )ˆuU(zmax c ) > O, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='22) where both ˆvU(zmax c ) and ˆuU(zmax c ) are positive, ˆgΦ = − � ˆuU(zmax c )( ˆA{1,2} ∗,−1 (zmax c ) ˆGr 0,∗,1 + ˆA{1,2} ∗,1 (zmax c ) ˆG0,∗,1)ˆvU(zmax c ) �−1 > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='23) and ˆGr 0,∗,1 and ˆG0,∗,1 are the limits of ˆGr 0,∗(z) and ˆG0,∗(z), respectively, given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, ˆU(z) is entry-wise analytic in ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='17), ˆΦ(0,0),∗(z) is entry-wise meromorphic in the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Recall that, under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2, the induced MA-process {Y {1,2} n } is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, in a manner similar to that used in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11], we obtain by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 that, for every z ∈ ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }, spr( ˆU(z)) < spr( ˆU(|z|)) < spr( ˆU(zmax c )) = 1, and this leads us to det(I − ˆU(z)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='21), ˜Φ(0,0),∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ˜U(0) = ˆU(zmax c ), we see by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 that det(I − ˜U(0)) = 0 and the multiplicity of zero of det(I − ˜U(ζ)) at ζ = 0 is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by the identity theorem for analytic functions, det(I − ˜U(ζ)) is nonzero in a neighborhood of ζ = 0 except for the point ζ = 0 and the point ζ = 0 is a pole of ˜Φ(0,0),∗(ζ) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (2) since the representation of ˆΦ(0,0),∗(z) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a function f(λ, z) as f(λ, z) = det(λI − ˆU(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Corollary 2 of Seneta [15] and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 (also see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='11 of [11]), adj(I − ˆU(zmax c )) = fλ(1, zmax c )ˆvU(zmax c )ˆuU(zmax c ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='24) where fλ(λ, z) = ∂ ∂λf(λ, z) and both ˆvU(zmax c ) and ˆuU(zmax c ) are positive since ˆU(zmax c ) is irre- ducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, in a manner similar to that used in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9 of [11], we obtain lim ˜∆zmax c ∋z→zmax c (zmax c − z)− 1 2 f(1, z) = −c0fλ(1, zmax c ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='25) where c0 = ˆuU(zmax c )( ˆA{1,2} ∗,−1 (zmax c ) ˆGr 0,∗,1 + ˆA{1,2} ∗,1 (zmax c ) ˆG0,∗,1)ˆvU(zmax c ) < 0 since, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5, both ˆG0,∗,1 and ˆGr 0,∗,1 are nonzero and nonpositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='21), this completes the proof of statement (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let αs0(z) be the eigenvalue of ˆG0,∗(z) that satisfies, for z ∈ [zmin c , zmax c ], αs0(z) = spr( ˆG0,∗(z)) = e2ηR c,2(log z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆuG(z) and ˆvG(z) be the left and right eigenvectors of ˆG0,∗(z) with respect to the eigenvalue αs0(z), satisfying ˆuG(z)ˆvG(z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3, ˜G0,∗(ζ) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 satisfies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 19 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' There exists a matrix function ˜G† 0,∗(ζ) entry-wise analytic in a neighborhood of ζ = 0 such that ˜G0,∗(ζ) is represented as ˜G0,∗(ζ) = ˜G† 0,∗(ζ) + ˜αs0(ζ)˜vG(ζ)˜uG(ζ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='26) where function ˜αs0(ζ), row vector function ˜uG(ζ) and column vector ˜vG(ζ) are elementwise analytic in a neighborhood of ζ = 0 and satisfying αs0(z) = ˜αs0((zmax c − z) 1 2 ), ˆuG(z) = ˜uG((zmax c − z) 1 2 ) and ˆvG(z) = ˜vG((zmax c − z) 1 2 ), respectively, in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In a neighborhood of ζ = 0, ˜G† 0,∗(ζ) satisfies spr( ˜G† 0,∗(ζ)) < αs0(zmax c ) = e2ηR c,2(θmax c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, ˜G0,∗(ζ) satisfies, for n ≥ 1, ˜G0,∗(ζ)n = ˜G† 0,∗(ζ)n + ˜αs0(ζ)n˜vG(ζ)˜uG(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='27) Let ˆν(0,∗)(z) be the generating function of {ˆν(0,k)} defined as ˆν(0,∗)(z) = �∞ k=1 zkˆν(0,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˆU2(z) as ˆU2(z) = ˆA{2} 0,∗ (z) + ˆA{2} 1,∗ (z) ˆG∗,0(z), and let ˆuU 2 (z) and ˆvU 2 (z) be the left and right eigenvectors of ˆU2(z) with respect to the maximum eigenvalue of ˆU2(z), satisfying ˆuU 2 (z)ˆvU 2 (z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [11] (also see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 of [14]), ˆν(0,∗)(z) satisfies the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (1) The vector function ˆν(0,∗)(z) is elementwise analytic in ¯∆e2θ∗ 2 \\ {e2θ∗ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) If θ∗ 2 < θmax 2 , ˆν(0,∗)(z) is elementwise meromorphic in a neighborhood of z = e2θ∗ 2 and the point z = e2θ∗ 2 is a pole of ˆν(0,∗)(z) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' It satisfies, for some positive constant ˆg2, lim ˜∆ e2θ∗ 2 ∋z→e2θ∗ 2 (e2θ∗ 2 − z)ˆϕ2(z) = ˆg2ˆuU 2 (e2θ∗ 2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='28) where ˆuU 2 (e2θ∗ 2) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a vector function ˜a(ζ, w) as ˜a(ζ, w) = ∞ � k=1 ˆν(0,k) ˆD(zmax c − ζ2, ˜G0,∗(ζ))wk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='29) Then, the vector functions ˆa(z, ˆG0,∗(z)) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15) and ˜a(ζ, ˜G0,∗(ζ)) satisfy the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (1) If ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1, the vector function ˆa(z, ˆG0,∗(z)) is elementwise analytic in ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) If ¯η′ 1(θ∗ 2) < −1, ˜a(ζ, ˜G0,∗(ζ)) is elementwise analytic in a neighborhood of ζ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1, it is elementwise meromorphic in a neighborhood of ζ = 0 and the point ζ = 0 is a pole of it with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The vector function ˆa(z, ˆG0,∗(z)) is represented as ˆa(z, ˜G0,∗(z)) = ˜a((zmax c − z) 1 2 , ˜G0,∗((zmax c − z) 1 2 )) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 20 (3) If ¯η′ 1(θ∗ 2) = −1, ˆa(z, ˆG0,∗(z)) satisfies, for a positive constant ˆga 2, lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 ˆa(z, ˆG0,∗(z)) = ˆga 2 ˆuG(zmax c ) ≥ 0⊤, ̸= 0⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='30) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11], if ¯η′ 1(θ∗ 2) ≤ −1, we have for z ∈ ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c } that |αs0(z)| < αs0(zmax c ) = e2ηR c,2(θmax c ) ≤ e2θ∗ 2, and this implies spr( ˆG0,∗(z)) < e2θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11] and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8, vector series ˆa(z, ˆG0,∗(z)) elementwise converges absolutely in ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7, we have ˜a(ζ, ˜G0,∗(ζ)) = ˜a(ζ, ˜G† 0,∗(ζ)) + (˜αs0(ζ)−1ˆν(0,∗)(˜αs0(ζ)) − ˆν(0,1)) ˆD(zmax c − ζ2, ˜αs0(ζ))˜vG(ζ)˜uG(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='31) If ¯η′ 1(θ∗ 2) ≤ −1, spr( ˜G† 0,∗(ζ)) < e2ηR c,2(θmax c ) ≤ e2θ∗ 2 in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, vec- tor series ˜a(ζ, ˜G† 0,∗(ζ)) is elementwise convergent absolutely and analytic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) < −1, ˜αs0(0) = αs0(zmax c ) = e2ηR c,2(θmax c ) < e2θ∗ 2, and this implies |˜αs0(ζ)| < e2θ∗ 2 in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8, the vector function ˜a(ζ, ˜G0,∗(ζ)) as well as ˆν(0,∗)(˜αs0(ζ)) is elementwise analytic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) = −1, ˜αs0(0) = e2ηR c,2(θmax c ) = e2θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8, the vector function ˜a(ζ, ˜G0,∗(ζ)) as well as ˆν(0,∗)(˜αs0(ζ)) is meromorphic in a neighborhood of ζ = 0 and the point ζ = 0 is a pole of it with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) = −1, αs0(zmax c ) = e2ηR c,2(θmax c ) = e2θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8, we have lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 ˆν(0,∗)(αs0(z)) = lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 αs0(zmax c ) − αs0(z)(αs0(zmax c ) − αs0(z))ˆν(0,∗)(αs0(z)) = (−αs0,1)−1ˆg2ˆuU 2 (e2θ∗ 2), where αs0,1 is the limit of αs0(z) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) and it is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This leads us to lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 ˆa(z, ˆG0,∗(z)) = (−αs0,1)−1ˆg2e−2θ∗ 2 ˆuU 2 (e2θ∗ 2)D(zmax c , e2θ∗ 2)ˆvG(zmax c )ˆuG(zmax c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='32) From this, we see that ˆga 2 ˆuG(zmax c ) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='30) is given by the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ˆuU 2 (e2θ∗ 2) is positive, ˆga 2 is also positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Finally, we give the proof of Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1 and ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1 ≤ 1/¯η′ 2(θ∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ϕc(z) is a probability vector generating function, it is automatically analytic elementwise in ∆zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, we prove it is elementwise analytic on ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For the purpose, we use equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9, ˆG0,∗(z), ˆa(z, ˆG0,∗(z)) and ˆΦ(0,0),∗(z) are elementwise analytic on ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15), ˆΦx,∗(z) and ˆϕ2(z) are also analytic elementwise on ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10), the analytic property of ˆΦx,∗(z) implies that Φc x,∗(z) is entry-wise 21 analytic on ∂∆zmax c \\{zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12), ϕc 2(z) is elementwise analytic on ∂∆zmax c \\{zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the same way, we can see that if ¯η′ 2(θ∗ 1) ≤ −1, ϕc 1(z) is elementwise analytic on ∂∆zmax c \\{zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), the analytic property of Φc x,∗(z) implies that ϕc 0(z) is elementwise analytic on ∂∆zmax c \\{zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' As a result, we see by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) that ϕc(z) is elementwise analytic on ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assuming Type 1 and ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1 ≤ 1/¯η′ 2(θ∗ 1), we also use equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10),(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' First, we consider about Φc x,∗(z) and ϕc 0(z), where x = (x1, x2) ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define ˜Φ(x1,x2),∗(ζ) as ˜Φ(x1,x2),∗(ζ) = (zmax c − ζ2)x1 ˜G0,∗(ζ)x2 ˜Φ(0,0),∗(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, by Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6, the matrix function ˜Φx,∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0 and satisfies ˆΦx,∗(z) = ˜Φ(x1,x2),∗((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The point ζ = 0 is a pole of ˜Φx,∗(ζ) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10), there exists a matrix function ˜Φc x,∗(ζ) being entry-wise meromorphic in a neighborhood of ζ = 0 and satisfying Φc x,∗(z) = ˜Φc x,∗((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The point ζ = 0 is a pole of ˜Φc x,∗(ζ) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define ˜ϕc 0(z) as ˜ϕc 0(ζ) = � i1,i2∈{−1,0,1} ν(0,0)(A∅ i1,i2 − A{1,2} i1,i2 )˜Φc (i1,i2),∗(ζ), which satisfies the same analytic property as ˜Φc x,∗(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' It also satisfies ϕc 0(z) = ˜ϕc 0((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Next, we consider about ϕc 2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define ˜ϕ2(ζ) as ˜ϕ2(ζ) = ˜a(ζ, ˜G0,∗(ζ))˜Φ(0,0),∗(ζ) By Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15), ˜ϕ2(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0 and satisfying ˆϕ2(z) = ˜ϕ2((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) < −1, the point ζ = 0 is a pole of ˜ϕ2(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1, it is a pole of ˜ϕ2(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Represent ˜ϕ2(ζ) in block form as ˜ϕ2(ζ) = �˜ϕ2,1(ζ) ˜ϕ2,2(ζ) � and define ˜ϕc 2(ζ) as ˜ϕc 2(ζ) = ˜ϕ2,1(ζ) + � i1,i2∈{−1,0,1} ν(0,1)(A{2} i1,i2 − A{1,2} i1,i2 )˜Φc (i1,i2+1),∗(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, the vector function ˜ϕc 2(ζ) is elementwise meromorphic in a neighborhood of ζ = 0, and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12), it satisfies ϕc 2(z) = ˜ϕc 2((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) < −1, the point ζ = 0 is a pole of ˜ϕc 2(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1, it is a pole of ˜ϕc 2(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Finally, we consider about ϕc(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the same way as that used for ϕc 2(z), we can see that there exists a vector function ˜ϕc 1(ζ) being elementwise meromorphic in a neighborhood of ζ = 0 and satisfying ϕc 1(z) = ˜ϕc 1((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 2(θ∗ 1) < −1, the point ζ = 0 is a pole of ˜ϕc 1(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 2(θ∗ 1) = −1, it is a pole of ˜ϕc 1(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define ˜ϕc(ζ) as ˜ϕc(ζ) = ˜ϕc 0(ζ) + ˜ϕc 1(ζ) + ˜ϕc 2(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, the vector function ˜ϕc(ζ) is elementwise meromorphic in a neighborhood of ζ = 0, and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6), it satisfies ϕc(z) = ˜ϕc((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ˜η′ 1(θ∗ 2) < −c1/c2 = −1 < 1/˜η′ 2(θ∗ 1), the point ζ = 0 is a pole of ˜ϕc(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ˜η′ 1(θ∗ 2) = −1 or ˜η′ 2(θ∗ 1) = −1, it is a pole of ˜ϕc(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 22 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 and equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13), lim ˜∆zmax c ∋z→zmax c (zmax c − z)Φc x,∗(z) = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='33) Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 0(z) = 0⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='34) If ¯η′ 1(θ∗ 2) = −1, by Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9 and equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='34), representing ˆuU(zmax c ) in block form as ˆuU(zmax c ) = � ˆuU 1 (zmax c ) ˆuU 2 (zmax c ) � , we obtain lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 2(z) = uc 2 = ˆga 2ˆgΦˆuG(zmax c )ˆvU(zmax c )ˆuU 1 (zmax c ) > 0⊤, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='35) where ˆuG(zmax c ) is nonzero and nonnegative and other terms on the right-hand side of the equation are positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) < −1, we have lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 2(z) = 0⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='36) In a manner similar to that used for ϕc 2(z), we can see that if ¯η′ 2(θ∗ 1) = −1, then for some positive vector uc 1, lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 1(z) = uc 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='37) and if ¯η′ 1(θ∗ 2) < −1, lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 1(z) = 0⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='38) As a result, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='35), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='36), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='37) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='38), we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 5 Concluding remarks We consider another topic, which relates to the singularity of the vector generating function ϕc(z) at z = zmax c = eθmax c , where c ∈ N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Recall that P {1,2} = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) is the transition probability matrix of the induced MA-process {Y {1,2} n } and Φ{1,2} = (Φ{1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) the fundamental matrix (potential matrix) of P {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let hΦ c (k) be the asymptotic decay function of the matrix sequence {Φ{1,2} x,kc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', for some positive matrix C, lim k→∞ Φ{1,2} x,kc /hΦ c (k) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6, we obtain hΦ c (k) = k− 1 2 e−θmax c k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) Furthermore, recall that P + is a partial matrix of P {1,2} given by restricting the state space of the level to the positive quadrant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', P + = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' P + is also a partial matrix of the transition probability matrix of the original 2d-QBD process, P = (Px,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2 +), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', P + = (Px,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˜Q = ( ˜Qx,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ N2) be the fundamental matrix of P +, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', 23 ˜Q = �∞ n=0(P +)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For j, j′ ∈ S0, denote by ˜q(x,j),(x′,j′) the (j, j′)-entry of ˜Qx,x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The entries of ˜Q are called an occupation measure in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [13], the asymptotic decay rate of the matrix sequence { ˜Qx,kc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N} is given by eθmax c , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', − lim k→∞ 1 k log ˜q(x,j),(kc,j′) = θmax c , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) which coincides with that of the matrix sequence {Φ{1,2} x,kc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' One question, therefore, arises: Does the asymptotic decay function of the matrix sequence { ˜Qx,kc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N} coincide with that of the matrix sequence {Φ{1,2} x,kc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N}?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If the answer to the question is yes, we can indicate that the vector generating function ϕc(z) diverges at z = eθmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' References [1] Bini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Latouche, G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Seneta: Non-negative Matrices and Markov Chains, revised printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Springer-Verlag, New York (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' A Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 First, we give the generalized eigenvectors of G(z) for z ∈ ∆zmain 1 ,zmax 1 \\E1, then analytically extend them to z ∈ C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for each z ∈ Ω \\ �m0 k=1 EG k , since the Jordan normal form of G(z) is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4), there exist linearly independent vectors called the generalized eigenvectors of G(z) with respect to the eigenvalue ˇαk(z), ˇvk,i,j(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i, satisfying (ˇαk(z)I − G(z))ˇvk,i,j(z) = ˇvk,i,j+1(z), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) where ˇvk,i,mk,i+1(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each i, ˇvk,i,j(z), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i, are called a Jordan sequence of the generalized eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Using the Jordan sequences, we define lˇq(k) × 1 block vectors, vk,i,j(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i, as vk,i,j(z) = vec �ˇvk,i,j(z) ˇvk,i,j+1(z) · · ˇvk,i,mk,i(z) 0 · · 0� , where, for a matrix A = � a1 a2 · · an � , vec(A) is the column vector given by vec(A) = � � � � � a1 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' an � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We also define a vector space VG k (z) as VG k (z) = span {vk,i,j(z) : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that the generalized eigenvectors ˇvk,i,j(z) are not unique but VG k (z) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the generalized eigenvectors are linearly independent, vk,i,j(z) are also linearly independent and we have dim VG k (z) = mk,0 � i=1 mk,i = lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, define an lˇq(k) × lˇq(k) block matrix function ΛG k (z) as ΛG k (z) = � � � � � � � ˇαk(z)I − G(z) −I ˇαk(z)I − G(z) −I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˇαk(z)I − G(z) −I ˇαk(z)I − G(z) � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We give the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for each z ∈ Ω \\ �m0 k=1 EG k , Ker ΛG k (z) = VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume v ∈ VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, by the definition of VG k (z), we have ΛG k (z)v = 0 and v ∈ Ker ΛG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For v = vec �v1 v2 · · vlˇq(k) � , assume ΛG k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If there exists an index i such that vi = 0, then by the assumption, for every j such that i ≤ j ≤ lˇq(k), we have vj = 0, and this implies v ∈ VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Theorem S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [3], since the matrix function ΛG k (z) is entry-wise analytic in ∆zmin 1 ,zmax 1 \\E1, there exist lˇq(k) vector functions vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k), that are elementwise analytic and linearly independent in ∆zmin 1 ,zmax 1 \\ E1 and satisfy ΛG k (z)vG k,i(z) = 0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, for each z ∈ Ω \\ �m0 k=1 EG k , vG k,i(z) ∈ VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We select the vectors composed of the Jordan sequences from {vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Represent each vG k,i(z) in block form as vG k,i(z) = vec � vG k,i,1(z) vG k,i,2(z) · · vG k,i,lˇq(k)(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From the proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, we see that, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, there exists a positive integer µk,i such that vG k,i,j(z) ̸= 0 for every j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', µk,i} and vG k,i,j(z) = 0 for every j ∈ {µk,i + 1, µk,i + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Renumber the elements of {vG k,i(z)} so that if i ≤ i′, then µk,i ≥ µk,i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a set of vector functions, ˇVk, according to the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (S1) Set ˇVk = ∅ and i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (S2) If vG k,i,µk,i(z) is linearly independent of {vG k,i′,µk,i′(z) : vG k,i′(z) ∈ ˇVk}, append vG k,i(z) to ˇVk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (S3) If i = lˇq(k), stop the procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' otherwise add 1 to i and go to (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, the number of elements of ˇVk is mk,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, (ˇαk(z)I−G(z))vG k,i,µk,i = 0 and dim Ker (ˇαk(z)I−G(z)) = mk,0, the number of elements of ˇVk is less than or equal to mk,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If it is strictly less than mk,0, we have dim Ker ΛG k (z) = dim span {vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)} < dim VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This contradicts (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2), and we see that the number of elements of ˇVk is just mk,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˇvG k,1(z), ˇvG k,2(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', ˇvG k,mk,0(z) the elements of ˇVk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='mk,0}, define ˇµk,i in a manner similar to that used for defining µk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume ˇvG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, are numbered so that if i ≤ i′, then ˇµk,i ≥ ˇµk,i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, ˇµk,i = mk,i Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, {ˇvG k,i,1(z), ˇvG k,i,2(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', ˇvG k,i,µk,i(z)} is a Jordan sequence of the generalized eigenvectors of G(z) with respect to the eigenvalue ˇαk(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, considering the procedure defining ˇvG k,i(z), we see that, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, ˇµk,i ≤ mk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Suppose there exists some i0 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0} such that ˇµk,i = mk,i for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', i0 −1} and ˇµk,i0 < mk,i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, there exists a vector v = vec �v1 v2 · · vmk,i0 0 · · 0� in VG k (z) such that vi ̸= 0 for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i0} and v is linearly independent of {vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By the same reason as that used in the proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2, this contradicts (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) and, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, ˇµk,i must be mk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 26 From this proposition, we see that, for z ∈ Ω \\ �m0 k=1 EG k , {ˇvG k,i,j(z) : k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i} is the set of generalized eigenvectors corresponding to the Jordan normal form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function T G(z) as T G(z) = �ˇvG k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i � , which is entry-wise analytic in ∆zmin 1 ,zmax 2 \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set EG T as EG T = {z ∈ ∆zmin 1 ,zmax 2 \\ E1 : det T G(z) = 0}, which is an empty set or a set of discrete complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, for z ∈ Ω \\ (�m0 k=1 EG k ∪ EG T ), we obtain the Jordan decomposition of G(z) as G(z) = T G(z)JG(z)(T G(z))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) Since G(z) is entry-wise analytic in ∆zmin 1 ,zmax 1 , we see by the identity theorem for analytic functions that the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) is also entry-wise analytic in the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Next, we analytically extend ˇvG k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define matrix functions F1(z, w) and F2(z) as F1(z, w) = z(I − A∗,0(z) − 2wA∗,1(z)), F2(z) = zA∗,1(z), where F1(z, w) is entry-wise analytic on C2 and F2(z) on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2), we have L(z, w) = F1(z, w)(wI − G(z)) + F2(z)(wI − G(z))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4) For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, define a lˇq(k) × lˇq(k) block matrix function ΛL k,n(z) as ΛL k (z) = � � � � � � � � � L(z, ˇαk(z)) −F1(z, ˇαk(z)) −F2(z) L(z, ˇαk(z)) −F1(z, ˇαk(z)) −F2(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' L(z, ˇαk(z)) −F1(z, ˇαk(z)) −F2(z) L(z, ˇαk(z)) −F1(z, ˇαk(z)) L(z, ˇαk(z)) � � � � � � � � � , which is entry-wise analytic in C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for every z ∈ ¯∆zmin 1 ,zmax 1 , Ker ΛL k (z) = Ker ΛG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) Before proving this proposition, we give another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0} and z ∈ ∆zmin 1 ,zmax 1 , F1(z, αk(z)) + F2(z)(αk(z)I − G(z)) = z (I − A∗,0(z) − αk(z)A∗,1(z) + A∗,1(z)G(z)) is regular (invertible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let R(z) be the rate matrix function generated from {Ai,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i, j = −1, 0, 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' for the definition of R(z), see Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [11], nonzero eigenvalues of R(z) are given by αk(z)−1, k = s0 + 1, s0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since, for every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0}, k′ ∈ {s0 + 1, s0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ} and z ∈ ∆zmin 1 ,zmax 1 , |αk(z)| ≤ αs0(|z|) < |αk′(z)|, I − αk(z)R(z) is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function H(z) as H(z) = A∗,0(z) + A∗,1(z)G(z), then by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11], I − H(z) is regular in ∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11], we have I − A∗,0(z) − αk(z)A∗,1(z) − A∗,1(z)G(z) = (I − αk(z)R(z))(I − H(z)), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) and this implies the assertion of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 27 Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume a vector v = vec �v1 v2 · · vlˇq(k) � satisfies ΛL k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, we have for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)} that L(z, ˇαk(z))vi = F1(z, ˇαk(z))vi+1 + F2(z)vi+2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7) where vlˇq(k)+1 = vlˇq(k)+2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We prove by induction that this v satisfies, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, (ˇαk(z)I − G(z))vi = vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let i0 be the maximum integer less than or equal to lˇq(k) that satisfies, for every i ∈ {i0 + 1, i0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, vi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, we have L(z, ˇαk(z))vi0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4), we have L(z, ˇαk(z)) = (F1(z, ˇαk(z)) + F2(z)(ˇαk(z)I − G(z)))(ˇαk(z)I − G(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8) Hence, by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5, we obtain (ˇαk(z)I − G(z))vi0 = 0 = vi0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume the assumption of induction holds for a positive integer i less than or equal to i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, L(z, ˇαk(z))vi−1 = F1(z, ˇαk(z))vi + F2(z)vi+1 = (F1(z, ˇαk(z)) + F2(z)(ˇαk(z)I − G(z)))vi, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) and by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5, we obtain (ˇαk(z)I − G(z))vi−1 = vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, v satisfies, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, (ˇαk(z)I − G(z))vi = vi+1, and this leads us to ΛG k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Next, assume a vector v = vec �v1 v2 · · vlˇq(k) � satisfies ΛG k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, we have for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)} that (ˇαk(z)I − G(z))vi = vi+1, where vlˇq(k)+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8), this v satisfies, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, L(z, ˇαk(z))vi = F1(z, ˇαk(z))vi+1 + F2(z)(ˇαk(z)I − G(z))vi+1 = F1(z, ˇαk(z))vi+1 + F2(z)vi+2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) and this implies ΛL k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let k be an arbitrary integer in {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Propositions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, we have dim Ker ΛL k (z) = lˇq(k), except for some discrete points in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Theorem S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [3], since the matrix function ΛL k (z) is entry-wise analytic in C\\E1, there exist lˇq(k) vector functions vL k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k), that are elementwise analytic and linearly independent in C \\ E1 and satisfy ΛL k (z)vL k,i(z) = 0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, for each i, vL k,i(z) also satisfies ΛG k (z)vL k,i(z) = 0 for every z ∈ ∆zmin 1 ,zmax 1 \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by the identity theorem, we see that vL k,i(z) is an analytic extension of vG k,i(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By the same procedure as that used for selecting {ˇvG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0} from {vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, we select mk,0 vectors from {vL k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)} and denote them by {ˇvL k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each i, ˇvL k,i(z) is represented in block form as ˇvL k,i(z) = vec � ˇvL k,i,1(z) ˇvL k,i,2(z) · · ˇvL k,i,mk,i(z) 0 · · 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function T L(z) as T L(z) = �ˇvL k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i � , which is entry-wise analytic in C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since each ˇvL k,i,j(z) is an analytic extension of ˇvG k,i,j(z), we have for z ∈ Ω \\ (�m0 k=1 EG k ∪ EG T ) that G(z) = T L(z)JG(z)(T L(z))−1, which is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Set E0 as E0 = E2 ∪ (�m0 k=1 EG k ) ∪ EG T , then E0 is a set of discrete complex numbers and we have Ω\\(�m0 k=1 EG k ∪EG T ) = ∆zmin 1 ,zmax 1 \\(E1 ∪E0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} diff --git a/INE1T4oBgHgl3EQf_wYq/content/tmp_files/2301.03583v1.pdf.txt b/INE1T4oBgHgl3EQf_wYq/content/tmp_files/2301.03583v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7fca9141498f4919637491f3e65cda9c05fb60db --- /dev/null +++ b/INE1T4oBgHgl3EQf_wYq/content/tmp_files/2301.03583v1.pdf.txt @@ -0,0 +1,1020 @@ +Reversal of quantised Hall drifts at non-interacting and interacting topological +boundaries +Zijie Zhu,∗ Marius G¨achter,∗ Anne-Sophie Walter, Konrad Viebahn,† and Tilman Esslinger +Institute for Quantum Electronics & Quantum Center, ETH Zurich, 8093 Zurich, Switzerland +The transport properties of gapless edge modes at boundaries between topologically distinct +domains are of fundamental and technological importance. Therefore, it is crucial to gain a better +understanding of topological edge states and their response to interparticle interactions. +Here, +we experimentally study long-distance quantised Hall drifts in a harmonically confined topological +pump of non-interacting and interacting ultracold fermionic atoms. We find that quantised drifts +halt and reverse their direction when the atoms reach a critical slope of the confining potential, +revealing the presence of a topological boundary. The drift reversal corresponds to a band transfer +between a band with Chern number C = +1 and a band with C = −1 via a gapless edge mode, +in agreement with the bulk-edge correspondence for non-interacting particles. We establish that a +non-zero repulsive Hubbard interaction leads to the emergence of an additional edge in the system, +relying on a purely interaction-induced mechanism, in which pairs of fermions are split. +The existence of individual edge modes at topo- +logical boundaries plays a crucial role in quantum Hall +physics. More specifically, a non-trivial topology in the +bulk of a material ensures that its edge modes are gapless +and chiral. Gaplessness is related to the bulk-edge corre- +spondence, stating that the number of topological edge +modes is equal to the difference in Chern number across +an interface [1]. +Consequently, a gapless mode should +allow an adiabatic transfer from one band to another, +resulting in a reflection of transverse bulk currents in +the opposite direction if the two bands feature opposite +Chern numbers. However, the coherence time in most +electronic materials is not sufficient to observe this effect, +and edges are generally probed spectroscopically [1–4]. +Moreover, studies of edge physics in engineered quantum +systems, such as ultracold atoms and photonics, have so +far been focussed on chirality [5–9] and localisation [10– +13]. A boundary reflection has not been detected [14, 15], +or it was disregarded [16, 17], and to our knowledge it +has never been studied for variable interaction strength. +Here, we observe the reversal of quantised bulk drifts due +to harmonic trapping in a topological Thouless pump, +the temporal analogue of the quantum Hall effect [18– +20]. +The reflection is a fundamental manifestation of +confined topological matter and directly shows the gap- +less nature of topological edge modes. Going beyond the +non-interacting regime, we discover the emergence of a +second edge for repulsive Hubbard U. +The experiments are performed +with +ultracold +fermionic potassium-40 atoms, which are loaded into the +potential of a generalised optical lattice formed by a com- +bination of standing and running waves of wavelength +λ = 1064 nm [22]. This creates an array of decoupled, +one-dimensional tubes. +Along the tube direction, the +periodically modulated Rice-Mele-Hubbard Hamiltonian +with harmonic confinement is realised, +ˆH(τ) = − +� +j,σ +� +t + (−1)jδ(τ) +� � +ˆc† +jσˆcj+1σ + h.c. +� +(1) ++ ∆(τ) +� +j,σ +(−1)jˆc† +jσˆcjσ + U +� +j +ˆc† +j↑ˆcj↑ˆc† +j↓ˆcj↓ ++ +� +j,σ +Vjˆc† +jσˆcjσ , +where ˆcjσ is the fermionic annihilation operator for spin +σ ∈ {↑, ↓} on site j, and t denotes the average tun- +nelling. +An adiabatic modulation of bond dimerisa- +tion δ(τ) = δ0 cos(2πτ/T) and sublattice offset ∆(τ) = +∆0 sin(2πτ/T) traces a closed trajectory in the δ–∆ plane +around the origin, referred to as critical point. There- +fore, an insulator or homogeneously filled band at U = 0 +describes a topological pump [18, 20] with T being the +pump period. +Experimentally, the topological pump +manifests itself as a quantised drift of the atom position +by one unit cell per pump cycle [23–27]. +The harmonic confinement is characterised by the +trap frequency ν, entering Eq. 1 as Vj = 1 +2m(2πνaj)2 ≡ +V0j2 (a = λ/2, lattice spacing; m, atomic mass). Due to +the confinement, the atoms are initially localised at the +centre of the trap. Topological pumping then leads to a +quantised drift of atoms against the confining potential +(Fig. 1A). Our measurements show that the quantised +drift changes its direction at a certain distance from the +trap centre. We will demonstrate that this happens when +the gradient of the confinement overcomes the band gap +and a boundary between topological and trivial regions +emerges. For repulsive interactions, we observe another +reflection, closer to the trap centre, while a part of the +atoms keeps drifting in the original direction (Fig. 1A). +In the following, we develop a description of the re- +flection in terms of gapless edge modes and the bulk- +edge correspondence within the framework of the Harper- +Hofstadter-Hatsugai (HHH) model with one real (x) and +one synthetic (n) dimension. The model features bulk +arXiv:2301.03583v1 [cond-mat.quant-gas] 9 Jan 2023 + +2 +Floquet +Energy +0 +-π +π +Quasimomentum +Gradient +Interactions +A +B +C +FIG. 1. Reflection of quantised Hall drifts off a topo- +logical interface. (A) Topological Thouless pumping in the +presence of confining potentials. In the non-interacting case +(top), a harmonic trap gives rise to topological trivial (C = 0) +and non-trivial (C ̸= 0) regions, separated by a topological +interface. The atoms exhibit a quantised drift until they are +reflected at the interface. With repulsive on-site interactions +(bottom), the reflection happens closer to the centre, accom- +panied by atoms still drifting in the original direction. (B) +Using Floquet theory, the 1D Rice-Mele pump can be mapped +to a 2D Harper-Hofstadter-Hatsugai model with a linear gra- +dient along the synthetic dimension n which represents the +photon number. The magnetic flux per plaquette is Φ = 1/2 +in units of the magnetic flux quantum [21]. The gradient along +n leads to a transverse Hall drift along x (red arrows) due to +the nontrivial topology of the bands. +(C) Schematic spec- +trum of the mapped 2D Hofstadter model in a semi-infinite +geometry. The lowest two bands have C = ±1, respectively. +The linear gradient induces Bloch oscillations in the synthetic +reciprocal space (dashed arrows). A gapless edge mode (solid +arrow) appears at the topological interface. The reflection of +the Hall drift can be understood as atoms being transported +from the lower band (C = 1) to the higher band (C = −1) +via the topological edge mode. +Chern bands with C = +1 and C = −1. +An ex- +act mapping between the non-interacting 1D Rice-Mele +Hamiltonian (Eq. 1) and the two-dimensional (2D) HHH +model can be obtained using Floquet theory, illustrated +in Fig. 1B (for derivation see, e.g., refs. [19] and [21]). A +linear gradient along the synthetic dimension n appears +in the mapping since the state with n photons acquires +an energy of −nℏω, where ω = 2π/T is the pump fre- +quency. The gradient along n or, equivalently, an exter- +nal force causes Bloch oscillations along the synthetic re- +ciprocal dimension kn which, in turn, lead to a Hall drift +or ‘anomalous velocity’ along the transverse real direc- +tion x [14, 15]. The bulk Hall drift along x corresponds +exactly to the quantised displacement measured in the +topological pump. The trap induces a boundary between +topological (Ccentre = 1) and trivial (Cright = 0) regions +and a single gapless edge mode emerges, according to the +bulk-edge correspondence: Ccentre −Cright = 1. The edge +modes connects two bands of opposite Chern invariant, +as shown in Fig. 1C. Thus, a Bloch oscillation transfers +the atoms from the ground to the first excited band via +that edge mode. Since the first excited band has Chern +number −1 the atoms are now moving ‘backwards’, re- +sulting in a reversal of the quantised Hall drift. +Fig. 2 shows experimental in-situ images of the +atomic cloud as a function of time τ at U = 0. +The +data shows a quantised drift of 1.00(1) × 2a/T up to +about 60 T, which confirms the long coherence time of +Bloch oscillations which induce the transverse drift. At +τ ≃ 75 T the atoms change their drift direction, which +is a key observation of this work. The expected topo- +logical boundary (red dashed line) represents the posi- +tion at which the local tilt from the external harmonic +potential ∆ext (j) ≡ +1 +2 |Vj − Vj−1| = V0 +��j − 1 +2 +�� equals +the maximum sublattice offset ∆0, thus, xedge/ (2a) ≃ +1 +2∆0/V0 = 92(7). +Beyond this position the total sub- +lattice offset ceases to change sign, rendering the region +outside xedge topologically trivial. The boundary caused +by the harmonic confinement is not infinitely sharp, but +smoothened over several lattice sites. +This leads to a +small T–dependence of the reflection position (Fig. S1), +compared to its absolute value, and the calculation above +should be understood as the outermost point of the re- +flective region. The reflected atoms exhibit a quantised +drift of −0.99(3) × 2a/T in the opposite direction, in +agreement with a transfer to the first excited band with +C = −1. +The linear relation between the position of +topological boundary xedge and the maximum sublattice +offset ∆0 is further confirmed by measuring the reflection +in different lattices (Fig. S1). The reflection is observed +under all parameter settings tested in this work, high- +lighting that the existence of the topological boundary is +robust. +In addition to the reflection, we also observe a cloud +of atoms temporarily remaining at the boundary before +gradually dissolving. This process can be understood via +the presence of topologically trivial edge states, which +hybridise with the gapless edge modes. To simplify the +picture, let us consider a sharp domain wall between +C = 1 and C = 0 (Fig. 2C). According to the bulk- +edge correspondence, the topologically nontrivial region +contributes exactly one gapless mode whereas the trivial +region can contribute gapped edge modes. Due to tunnel +coupling at the interface, hybridisation takes place [34] +and gaps on the order of the pump frequency 2π/T +emerge. Bloch oscillations along kn can now lead to non- +adiabatic ‘Landau-Zener’ transfers between topological +and trivial edge modes, causing an incomplete transfer +to the higher band, and atoms remaining at the bound- +ary. +Subsequent Bloch oscillations will transfer atoms +back into the topological domain, leading to the dissolu- + +3 +quasimomentum +energy +trivial +nontrivial +0 +36 +108 +72 +144 +time τ (T) +1st +2nd +2nd +2nd +2nd +Brillouin zone +Dimerisation +Site ofset +A +C +B +FIG. 2. Measuring the reversal of a quantised Hall drift. (A) The time-trace of atomic in-situ images shows a quantised +drift along x before the atoms are reflected at the topological boundary. Each density image is averaged over three individual +measurements with the parameters V0 = 0.0148(9)t, ∆0 = 2.7(1)t, and T = 3 ms = 12.8(2)ℏ/t. The red dashed line indicates +the topological boundary xedge/ (2a) ≈ +1 +2∆0/V0. The white dashed lines are linear fits to the atom drift, yielding slopes of +1.00(1) × 2a/T before, and −0.99(3) × 2a/T after the reflection. Cloud positions, averaged over the transverse direction, are +fitted using Gaussians. The experimental Chern marker (lower panel, points) is determined by the velocity of the right-moving +cloud at different positions. The theoretical Chern marker (line) is calculated in a staggered potential Vj = V0 (−1)j j which +has the same local tilt ∆ext(j) = V0 +��j − 1 +2 +�� as the harmonic trap [21]. In a local density approximation picture, the local tilt +∆ext shifts the δ–∆ pump trajectory upwards. Depending on whether or not the trajectory encloses the critical point, the pump +is rendered topological or trivial. (B) Measured band populations as a function of time τ. Each density image is averaged over +six individual measurements with the parameters V0 = 0.0191(6)t, ∆0 = 3.2(2)t and T = 3 ms = 10.7(2)ℏ/t. The total atom +number remains constant, within error bars, throughout the experiment. Due to the underlying honeycomb lattice geometry +in the x–z plane, the first Brillouin zone has a diamond shape [21]. The band population is inverted when the bulk current +is reflected off the topological interface, manifesting the gapless nature of the topological edge mode. (C) Hybridisation of +the edge modes at the topological interface due to tunnelling. Bloch oscillations along kn can lead to Landau-Zener transfers +between topologically trivial and nontrivial edge modes. The population of trivial edge modes explains the atoms being left at +the boundary. Hybridisation never changes the total number of gapless edge modes at the boundary. +tion of the cloud at the boundary. While the harmonic +confinement leads to a more complex level structure [21], +the underlying process remains qualitatively the same. +We support the in-situ images with measurements +of band population before, during, and after the reflec- +tion, as shown in Fig. 2B [21]. Before the reflection, we +find a filled ground band, which is consistent with the +observation of a quantised Hall drift. At the reflection +(τ ≃ 72 T), we observe an inversion of the population to +the first excited band. After the reflection, the inverted +population remains almost unchanged while the atoms +are travelling back, highlighting the absence of incoher- +ent relaxation to the ground band, even after more than +a hundred Bloch oscillations. +We further explore the effect of attractive and repul- +sive interactions on the topological boundary. For attrac- +tive Hubbard U = −3.48(7)t = 1.27(7)∆0 (Fig. 3A), the +quantised Hall drift is reversed at the same position as +in the non-interacting situation. This can be explained +in terms of the Rice-Mele model in which fermions in the +strongly attractive regime approach the limit of hard-core +bosons +[22], and the condition for the emergence of a +topological boundary ∆ext (j) = ∆0 remains unchanged. +For repulsive Hubbard U = 3.48(7)t (Fig. 3B), we +observe a second reflection in addition to the original one. +Compared to the non-interacting case, this reflection ap- +pears much closer to the trap centre. +The zoomed-in +image (Fig. 3C) shows that a proportion of the atoms +start to move backwards after about 12 cycles. In con- + +4 +Dimerisation +Site ofset +OD +time т (T) +B +D +E +C +A +Time τ ( ) +T +τ0 +τ0 + 0.5 +τ0 + 1 +FIG. 3. +Reflection of quantised Hall drifts from an interacting topological edge. +(A) An attractive Hubbard +interaction of U = −3.48(7)t leads to the same reflection behaviour as observed for U = 0 (measurement parameters otherwise +identical to Fig. 2A). (B) Repulsive Hubbard interactions of U = 3.48(7)t lead to the emergence of a second reflection, closer +to the trap centre, which we attribute to an interacting topological boundary. A zoom-in (C) shows that the early reflection +happens after about twelve cycles. The white dashed lines are guides to the eye, calculated as linear fits to the cloud position, +extracted as the sum of a skewed and a regular Gaussian. (D) Microscopic description of the interaction-induced reflection for +repulsive Hubbard U. When the maximum energy offset between two neighbouring sites 2 (∆0 − ∆ext) becomes smaller than +the Hubbard interaction, formation of double occupancies is prohibited and one atom is left in the higher-energy site of a unit +cell, which then drifts in the opposite direction. (E) The critical point in the δ–∆ plane is split into two in the presence of +repulsive Hubbard U [35]. When the pump trajectory encloses both critical points, a quantised drift is expected, as in the +non-interacting system. The local tilt given by the external potential ∆ext shifts the trajectory along the ∆–axis, eventually +enclosing just one critical point. Since a single split critical point features half the topological charge of the orignal one, the +material’s topology changes and a boundary emerges. +trast to the drift reversal in the non-interacting system, a +large fraction of the atom cloud still undergoes quantised +drifting in the original direction. +In the following, we +develop a microscopic picture of the interaction-induced +partial reflection in the limiting case of two isolated spins +(↑, ↓), which approximates our initial state in a unit cell +(Fig. 3D). As long as the maximum energy offset between +two neighbouring sites 2 (∆0 − ∆ext) is larger than the +Hubbard U, the formation of a double occupancy is en- +ergetically allowed and the quantised drift persists [22], +even in the presence of an external potential. However, +when ∆ext becomes larger, the energy offset between two +neighbouring sites remains always smaller than U, dou- +ble occupancy formation becomes prohibited. In the lat- +ter case, one atom of a singlet pair is transferred to the +energetically excited site of a unit cell, which will subse- +quently drift in the opposite direction. The other atom, +in the lower-energy site, will move onwards because on- +site interactions become irrelevant if there is only one +atom per unit cell. +Since the underlying Hamiltonian +(Eq. 1) is SU(2)–symmetric, spin–↑ and spin–↓ have equal +probability of being reflected and they should remain cor- +related after the splitting process. +The full many-body description of the interaction- +induced reversal requires the development of suitable +topological invariants for smooth confinements and +strong interactions, which goes beyond the scope of this +work. Nevertheless, we obtain an intuition of the bound- +ary’s topological origins using again the idea of shifted +pump trajectories in the δ–∆ plane with a staggered po- +tential (c.f. Fig. 2A). Numerical simulations have shown +that a repulsive Hubbard U can split the critical point at +the origin into two separate ones [35], each retaining half +the original topological charge. +The distance between +the new critical points is approximately U up to a cor- +rection on the order of the tunnelling t [36]. When the +trajectory encloses both critical points, quantised drift of +two spins (↑, ↓) is expected, as in the non-interacting sys- +tem. As the position-dependent local tilt ∆ext(j) shifts +the trajectory upwards, it will enclose only one of the +critical points beyond certain position along x (Fig. 3E). +This indicates a transition of topological properties and +the emergence of an interacting topological edge in real +space. +The estimation of the interacting boundary at +∆ext(j) ≃ ∆0 − U/2 lies close to the centre and agrees +with the microscopic picture discussed above. Similar to +the non-interacting case, this boundary should be con- +sidered as the outermost position of the reflective region. +In conclusion, we have experimentally observed a re- +versal of quantised Hall drifts at a topological boundary +in a harmonic potential. The reflection is a direct mani- +festation of the gapless nature of topological edge modes +between Chern bands of opposite sign. We explore the +effect of Hubbard interactions, both attractive and re- + +5 +pulsive, and find an asymmetric behavior with respect +to U = 0. While on the attractive side the topological +boundary is unaffected, repulsive interactions lead to the +emergence of a second interface, featuring a splitting of +quantised drifts. As a result, our experiments could en- +able the realisation of circular current patterns for con- +structing novel many-body phases [37]. +More broadly, +our work allows the exploration of the bulk-edge corre- +spondence in the presence of interactions [38], as well as +the investigation of edge reconstruction [39] in the quan- +tum Hall effect and in interacting topological insulators. +ACKNOWLEDGMENTS +We would like to thank Jason Ho, Gian-Michele Graf, +Thomas Ihn, Fabian Grusdt, Fabian Heidrich-Meisner, +Armando Aligia, and Eric Bertok for valuable discus- +sions. We also thank Julian L´eonard and Nur ¨Unal for +comments on an earlier version of the manuscript. We +would like to thank Alexander Frank for his contribu- +tions to the electronic part of the experimental setup. +We acknowledge funding by the Swiss National Science +Foundation (Grant Nos. 182650, 212168, NCCR-QSIT, +and TMAG-2 209376) and European Research Council +advanced grant TransQ (Grant No. 742579). +∗ These authors contributed equally. +† viebahnk@phys.ethz.ch +[1] M. Z. Hasan and C. L. Kane, Colloquium: Topological +insulators, Rev. Mod. Phys. 82, 3045 (2010). +[2] M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Tay- +lor, Imaging topological edge states in silicon photonics, +Nature Photon 7, 1001 (2013). +[3] K. +Yatsugi, +T. +Yoshida, +T. +Mizoguchi, +Y. +Kuno, +H. Iizuka, Y. Tadokoro, and Y. Hatsugai, Observation of +bulk-edge correspondence in topological pumping based +on a tunable electric circuit, Commun Phys 5, 180 (2022). +[4] Z.-C. Xiang, K. Huang, Y.-R. Zhang, T. Liu, Y.-H. +Shi, C.-L. Deng, T. Liu, H. Li, G.-H. Liang, Z.-Y. Mei, +H. Yu, G. Xue, Y. 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Hofstetter, Interact- +ing Hofstadter Interface, Phys. Rev. Lett. 122, 010406 +(2019). +[39] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman, +Electrostatics of edge channels, Phys. Rev. B 46, 4026 +(1992). + +7 +SUPPLEMENTAL MATERIALS +Dependence of the drift reversal on experimental +parameters +The expected drift reversal happens when the maxi- +mum local site offset over one pump-cycle ∆0 is equal +to the local tilt given by the harmonic trap. This po- +sition given by xedge ≃ ∆0a/V0 with a = λ/2 and +V0 = 1 +2m(2πνa)2. By measuring the reflection point in +lattices with different ∆0, we verify the relevant scaling +xedge ∝ ∆0, as shown in Fig. S1. The blue line in Fig. +S1A marks the theoretically expected xedge with the un- +certainty propagated from the uncertainty of the trap +frequency ν. The disagreement between theory and ex- +periment for larger values of ∆0 can be explained by the +finite waist of the lattice beams. In order to explore the +edge in our system, the atoms are pumped by almost 100 +unit cells (∼ 100 µm). Due to the Gaussian envelope of +the transverse beams, which are essential to realise the +pump, the lattice is effectively shallower far away from +the centre. +Thus, ∆0 decreases towards the edge and +atoms are reflected sooner. +We also find a small dependence of the reflection point +on the pump period, compared to its absolute value, +which spans roughly 10 unit cells when changing T from +2 ms to 10 ms (Fig. S1B). This can be understood by con- +sidering the energy spectrum of the Harper-Hofstadter- +Hatsugai (HHH) model in a harmonic potential, which +will be discussed below. +Experimental sequence +We start by preparing a degenerate cloud of fermionic +40K in a crossed dipole trap. We have a spin mixture +of mF = {−9/2, −7/2} except for the measurements in +Fig. 2B and Fig. S1, where a spin-polarised cloud in the +magnetic state F = 9/2, mF = −9/2 is used. The spin- +polarised cloud is directly loaded into the pumping lat- +tice, while the spin mixture is first loaded into an inter- +mediate chequerboard lattice with strongly attractive in- +teractions. The two-step loading precludes the presence +of atoms in the higher band and gives a larger fraction +of atoms in doubly occupied unit cells [22]. +After pumping the system for varying times, we either +take a in-situ absorption image to measure the density +or detect the band population with band-mapping. The +latter is implemented with an exponential ramp to switch +off the lattice beam in 500 µs, followed by a time-of-flight +expansion of 25 ms before absorption imaging. +A +B +FIG. S1. +Experimental dependence of the reflection +position. The reflection point is expected to depend linearly +on the maximal site-offset per pump cycle ∆0 which is experi- +mentally verified in (A). Deviations for large values of ∆0 can +be explained by the finite waist of the laser beams forming +the lattice. (B) shows the period dependence of the reflec- +tion position. Changing the pump period T over an order of +magnitude only changes the reflection point by about 10 unit +cells, which is a result of the smooth boundary of a harmonic +potential. +Realisation of a Thouless pump in the Rice-Mele +model +The lattice setup is comprised of non-interfering stand- +ing waves in x, y, and z directions, together with in- +terfering laser beams in the x–z plane. All the lattice +beams come from a single laser source at wavelength +λ = 1064 nm. These potential combine to form a honey- +comb lattice in the x–z plane, which can be considered +as isolated tubes of one-dimensional superlattices along +x in the limit of deep transverse lattices. +In each 1D +tube the potential can be modeled by a one-dimensional +superlattice with two sites per unit cell. With this setup, +we realise the Rice-Mele model [22]. In the tight-binding +limit, the Rice-Mele model can be described with three +numbers: the offset energy ∆ between the two sites of +a unit cell, the averaged nearest-neighbour tunneling t +and the bond dimerisation δ which gives half the differ- +ence between the inter- and intra-dimer tunnellings. By +having a dynamical control of the relative phase ϕ be- +tween the laser beams generating the interfering and the +non-interfering lattice, we manage to shift the two with +respect to each other. This shift modulates ∆ and δ pe- +riodically, which can be depicted as a closed trajectory +in the ∆-δ coordinate (Fig. S2). In the adiabatic limit, +this realises a Thouless pump with its hallmark quan- + +8 +tised transport. In this case, the atomic displacement is +given by the number of revolutions around the origin of +the ∆-δ plane. +φ = 0 +φ = ∏/2 +φ = ∏ +φ = 3∏/2 +x +E +x +E +x +E +x +E +FIG. S2. Realisation of a Thouless pump in the Rice- +Mele model. In our system the 1D lattice potential can be +modelled by a superlattice with two sites per unit cell, which +is depicted in the bottom part of this figure as a function of +relative phase ϕ. The local site offset ∆ as well as the bond +dimerisation δ is depicted in the ∆-δ plots corresponding to +the respective potentials. The resulting pump displacement +corresponds to the number of revolutions around the origin +in the ∆-δ plane. +Mapping a 1D Thouless pump to a 2D Hofstadter +Model with quantum Hall response +A 1D Thouless pump with a period of T, as realised +in our experiment, can be mapped to a 2D topological +tight-binding (HHH) model with an applied electric field +E = 2πℏ +qT where q can be thought of as a fictitious charge +of netural atoms. Due to the topological bandstructure, +this electric field leads to a transverse current Itrans = q +T +of one atom per period, when considering a fully occupied +band. The 2D model therefore has a quantised transverse +conductance σtrans = Itrans/E = +q2 +2πℏ analogous to the +Hall conductance in the Quantum Hall Effect (QHE). +The time-periodicity of the Hamiltonian in Eq. 1 with +ˆH(τ) = ˆH(τ + T) allows us to use Floquet’s theorem. +Solutions of the time-dependent Schr¨odinger equation +iℏ∂τ |Ψ(τ)⟩ = H(τ) |Ψ(τ)⟩ +(S1) +can thus be written as +|Ψ(τ)⟩ = e−iϵτ/ℏ |u(τ)⟩ +(S2) +with |u(τ + T)⟩ = |u(τ)⟩ and ϵ ∈ R. Due to the time- +periodicity of u(τ) we expand it as a Fourier series, +|u(τ)⟩ = +� +n +e−iωnτ |un⟩ , +(S3) +where ω = 2π/T is the pump frequency. +The change +from the time-domain into the Fourier-domain is the key +ingredient to map the 1D Thouless pump to a 2D tight- +binding model. +The index n is also called the photon +number of the mode |un⟩. +Using a multi-index α = (j, σ) we write the T-periodic +1D Hamiltonian for U = 0 in the Fourier-basis: +ˆH(τ) = +� +α,β +hαβ(τ) |α⟩ ⟨β| +(S4) += +� +α,β,m +e−imωτhm +αβ |α⟩ ⟨β| +with hm +αβ = +1 +T +� T +0 eimωτhαβ(τ)dτ and |α⟩ corresponding +to an atom localised on site j with spin σ. +Likewise, +we use Fourier decomposition to express the solutions to +Eq. S1 as +|Ψ(τ)⟩ = e−iϵτ/ℏ � +n,α +e−inωτun,α |α⟩ . +(S5) +where un,α = ⟨α|un⟩. As a result, we obtain an eigen- +value equation for un,α: +ϵun,α = −nℏωun,α + +� +β,m +hm +αβun−m,β ∀n, α +(S6) +which +can +be +understood +as +a +time +independent +Schr¨odinger equation of a 2D tight-binding model with +a tilted potential energy along one axis. +By explicitly +evaluating the hm +αβ, we get +H2D = Hreal + Hsynth + Hdiag + HV + Htilt, +(S7) +with +Hreal = −t +� +j,n,σ +(ˆc† +j,n,σˆcj+1,n,σ + h.c.), +(S8) +Hdiag = −δ0 +2 +� +j,n,σ +e−iπj(ˆc† +j,n,σˆcj+1,n+1,σ ++ ˆc† +j,n,σˆcj+1,n−1,σ + h.c.), +Hsynth = −∆0 +2 +� +j,n,σ +e−iπj(iˆc† +j,n,σˆcj,n+1,σ + h.c.), +HV = +� +j,n,σ +V (j)ˆc† +j,n,σˆcj,n,σ, +Htilt = − +� +j,n,σ +ℏωnˆc† +j,n,σˆcj,n,σ . +Hreal and Hsynth describe tunneling along the real (x) +and synthetic (n) dimension, respectively. +The diagonal tunnelling terms in Hdiag are crucial be- +cause they open a bandgap between the ground band and +the first excited band, characterised by the topological +Chern number C which is further related to the quantised +Hall conductance via σtrans = +q2 +2πℏC. The terms in HV + +9 +describe the external potential along the real-space direc- +tion. Htilt corresponds to a linear tilt in potential energy +along the synthetic dimension which can be thought of as +originating from an electric field E = 2πℏ +qT pointing along +n. +Edge modes and their reflection properties +To illustrate the topological edge modes in the presence +of an external potential, we evaluate the spectrum of H2D +in the adiabatic limit (ω → 0) for different potentials +V (j) = 1 +2m(2πνa)2jκ +(S9) +with m being the mass of 40K, trap frequency ν = 134 Hz, +lattice spacing a = 532 nm, and lattice site j. The pa- +rameter κ, an even integer, characterises steepness of the +trap; the limit κ → ∞ corresponds to the textbook case +of infinitely sharp walls [28]. Fig. S3A-C shows the nu- +merically calculated energy spectra, omitting states on +localised to the left edge for clarity. +Fig. S3A shows the spectrum for a box-like potential +with κ = 24. In this case there is a family of topologi- +cal edge states, marked in red, which connect the lower +and the upper band (black), separated from topologically +trivial states above 5 kHz (also in black). All red states +are localised along the right edge in x-direction. +The +lower and upper band have Chern number 1 and −1, re- +spectively. Considering the dynamics in this model, an +applied electric field along n as defined in Htilt leads to +Bloch oscillations with a period T along kn. At the same +time, the center-of-mass of the atoms moves by one unit +cell per Bloch oscillation period along x, which corre- +sponds to the quantised bulk Hall drift [14, 15]. This drift +can be evaluated in the numerics by following the eigen- +states in Fig. S3A in real space. Since the red edge states +are gapped from the next higher-lying trivial (black) +states, the atoms ‘Bloch-oscillate’ from the ground to +the excited band via the red-marked edge modes over +several periods. Once they are in the excited band they +are transported backwards along x. +Fig. S3B shows the situation for κ = 10. It behaves +similarly to Fig. S3A, except that there are more lo- +calised states marked in red, compared to κ = 2. Like- +wise, these states are transported along x as they undergo +Bloch oscillations. As before, this family of edge states +is gapped from trivial states and connects right-moving +to left-moving states, which leads to the reflection phe- +nomenon. +The experimentally relevant case is a harmonic trap +with κ = 2 (see also refs. [23, 29–31]). Fig. S3C shows +that for κ = 2 the number of localised sates outside of +the bands is even larger than for κ = 10. +As before, +we adiabatically follow these localised states along kn +A +B +C +D +FIG. S3. Energy spectra for different confining poten- +tials. States localised to the left edge are omitted for clarity. +Energy spectra of H2D in the limit ω → 0 for κ = 24 (A), +κ = 10 (B), and κ = 2 (C). Topological edge modes which +connect the two bands with Chern number 1 and −1 in (A) +and (B) are marked in red. The upper inset in (C) marks +the topological boundary where the reflection is observed as +described in the main text. The inset in the center of (C) +shows the tiny avoided crossings which can lead to a slight +period dependence of the observed reflection point (Fig. S1). +(D) Energy spectrum for a linearly increasing staggered po- +tential. The gapless, topological edge mode is marked in red. + +10 +and evaluate their centre-of-mass along x. +By numer- +ically observing these drifts we confirm that the states +describe quantised drifting in a large region, which man- +ifests their nontrivial topological nature. Thus, the κ = 2 +case is ideal to observe the reflection after long-distance +quantised Hall drifts. However, the gaps between topo- +logical (right-moving and left-moving) and trivial (sta- +tionary) states become smaller, compared to the κ = 24 +and κ = 10 cases, as shown in the insets of Fig. S3C +(κ = 2). As a result, the reflection point for κ = 2 is +spread out over several unit cells but the reflection itself +remains intact. Faster pumping leads to non-adiabatic +crossings of the energy gaps between right-moving and +left-moving states, causing the reflection to happen later +in time and further up in energy. We confirm this depen- +dence experimentally in Fig. S1B, which shows a later +reversal for smaller pump periods. +Fig. S3D shows the spectrum for the linearly increasing +staggered potential, described in the following sections. +This potential allows a straightforward identification of +the gapless edge mode (red line). The states correspond- +ing to this gapless edge mode are localised around the +topological boundary. +x +E +FIG. S4. Linearly increasing staggered potential. To +elucidate the topology in our system, a linearly increasing +staggered potential is considered (blue): V (j) = jV0(−1)j +with V0 = 1/2 × m(2πνa)2, as before. It is chosen such that +the local tilt always equals the tilt from the harmonic poten- +tial (orange), but alternates in sign. The staggered potential +allows a simple pictorial representation of the emergence of +the topological boundary. In a local density approximation +the trap linearly shifts the pump trajectory upwards in the +∆-δ plane, as depicted in the upper part of the figure. As +soon as the trajectory ceases to enclose the critical point, a +topological–trivial boundary develops. +Staggered potential +Another possibility to identify the topological bound- +ary in our system makes use of a staggered potential. +First, we consider a potential with uniform staggering, +given by Vc(j) = V (−1)j, where j indexes the lattice-site +and 2V corresponds to the energy difference between ad- +jacent sites. Adding such a potential to the Rice-Mele +Hamiltonian (Eq. 1) changes its trajectory in the ∆-δ +plane. The onsite energy in such a system is given by +(∆(τ) + V )(−1)j, which ranges from −∆ + V to ∆ + V . +The tunnellings are unchanged. Therefore, the trajectory +remains circular and it is simply shifted upwards by an +amount V . +A topological boundary emerges for a linearly increas- +ing staggered potential, given by V (j) = jV0(−1)j, with +V0 = +1 +2m(2πνa)2 as before. +V (j) is chosen such that +it has the same local tilt as the harmonic trap in the +experiment. Within the local density approximation we +assign a ∆-δ trajectory locally to each unit cell. +The +trajectories are thus linearly shifted upwards as func- +tion of j (Fig. S4), describing a change of topology in +real space. We expect the local density approximation +to be valid since the atomic eigenstates in the exper- +iment are strongly localised. +Similar models with lin- +early increasing staggered potential have been studied in +refs. [29, 32, 38]. +Local Chern marker +The mathematical formulation of the Chern number as +a topological invariant requires translational invariance, +which does not apply to realistic experiments. Instead, +we use a local quantity, known as Chern marker [8, 33]. +The local Chern marker depends on the real-space posi- +tion and it is defined by: +c(rγ) = −4π +Ac +Im +� +s=A,B +⟨rγs| ˆP ˆx ˆQˆy ˆP |rγs⟩ , +(S10) +where rγ is the position of the unit cell γ with sub-lattice- +sites at positions rγA and rγB, |rγs⟩ = c† +γs |0⟩ is the state +localised on the corresponding lattice site , Ac is the area +of a real-space unit cell, ˆQ = 1− ˆP and ˆP is the projector +onto the ground band. Defining ˆP is not unambiguously +possible in our system (Eq. S7) because of the energy +shift from the harmonic confinement. +Instead, we use +a linearly increasing staggered potential, as described in +the previous paragraph. This model leaves the bands in- +tact and a ground band can be unambiguously defined. +Experimentally, a local probe of the band topology is the +velocity of the Hall drift, plotted in Fig. 2A. Theory and +experiment agree approximately with one another. The +local velocity is extracted from the atomic positions by +fitting linear functions to groups of three adjacent dat- +apoints in ten pump cycles. The resulting velocities are +plotted against position and smoothed through a running +average of width three (ten cycles). + diff --git a/INE1T4oBgHgl3EQf_wYq/content/tmp_files/load_file.txt b/INE1T4oBgHgl3EQf_wYq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c84c236c8e97a7831faa98d0860914e9a21c6b2 --- /dev/null +++ b/INE1T4oBgHgl3EQf_wYq/content/tmp_files/load_file.txt @@ -0,0 +1,671 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf,len=670 +page_content='Reversal of quantised Hall drifts at non-interacting and interacting topological boundaries Zijie Zhu,∗ Marius G¨achter,∗ Anne-Sophie Walter, Konrad Viebahn,† and Tilman Esslinger Institute for Quantum Electronics & Quantum Center, ETH Zurich, 8093 Zurich, Switzerland The transport properties of gapless edge modes at boundaries between topologically distinct domains are of fundamental and technological importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Therefore, it is crucial to gain a better understanding of topological edge states and their response to interparticle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Here, we experimentally study long-distance quantised Hall drifts in a harmonically confined topological pump of non-interacting and interacting ultracold fermionic atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We find that quantised drifts halt and reverse their direction when the atoms reach a critical slope of the confining potential, revealing the presence of a topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The drift reversal corresponds to a band transfer between a band with Chern number C = +1 and a band with C = −1 via a gapless edge mode, in agreement with the bulk-edge correspondence for non-interacting particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We establish that a non-zero repulsive Hubbard interaction leads to the emergence of an additional edge in the system, relying on a purely interaction-induced mechanism, in which pairs of fermions are split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The existence of individual edge modes at topo- logical boundaries plays a crucial role in quantum Hall physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' More specifically, a non-trivial topology in the bulk of a material ensures that its edge modes are gapless and chiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Gaplessness is related to the bulk-edge corre- spondence, stating that the number of topological edge modes is equal to the difference in Chern number across an interface [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Consequently, a gapless mode should allow an adiabatic transfer from one band to another, resulting in a reflection of transverse bulk currents in the opposite direction if the two bands feature opposite Chern numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' However, the coherence time in most electronic materials is not sufficient to observe this effect, and edges are generally probed spectroscopically [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Moreover, studies of edge physics in engineered quantum systems, such as ultracold atoms and photonics, have so far been focussed on chirality [5–9] and localisation [10– 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A boundary reflection has not been detected [14, 15], or it was disregarded [16, 17], and to our knowledge it has never been studied for variable interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Here, we observe the reversal of quantised bulk drifts due to harmonic trapping in a topological Thouless pump, the temporal analogue of the quantum Hall effect [18– 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection is a fundamental manifestation of confined topological matter and directly shows the gap- less nature of topological edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Going beyond the non-interacting regime, we discover the emergence of a second edge for repulsive Hubbard U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The experiments are performed with ultracold fermionic potassium-40 atoms, which are loaded into the potential of a generalised optical lattice formed by a com- bination of standing and running waves of wavelength λ = 1064 nm [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This creates an array of decoupled, one-dimensional tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Along the tube direction, the periodically modulated Rice-Mele-Hubbard Hamiltonian with harmonic confinement is realised, ˆH(τ) = − � j,σ � t + (−1)jδ(τ) � � ˆc† jσˆcj+1σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' � (1) + ∆(τ) � j,σ (−1)jˆc† jσˆcjσ + U � j ˆc† j↑ˆcj↑ˆc† j↓ˆcj↓ + � j,σ Vjˆc† jσˆcjσ , where ˆcjσ is the fermionic annihilation operator for spin σ ∈ {↑, ↓} on site j, and t denotes the average tun- nelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' An adiabatic modulation of bond dimerisa- tion δ(τ) = δ0 cos(2πτ/T) and sublattice offset ∆(τ) = ∆0 sin(2πτ/T) traces a closed trajectory in the δ–∆ plane around the origin, referred to as critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' There- fore, an insulator or homogeneously filled band at U = 0 describes a topological pump [18, 20] with T being the pump period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Experimentally, the topological pump manifests itself as a quantised drift of the atom position by one unit cell per pump cycle [23–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The harmonic confinement is characterised by the trap frequency ν, entering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1 as Vj = 1 2m(2πνaj)2 ≡ V0j2 (a = λ/2, lattice spacing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' m, atomic mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the confinement, the atoms are initially localised at the centre of the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Topological pumping then leads to a quantised drift of atoms against the confining potential (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Our measurements show that the quantised drift changes its direction at a certain distance from the trap centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We will demonstrate that this happens when the gradient of the confinement overcomes the band gap and a boundary between topological and trivial regions emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' For repulsive interactions, we observe another reflection, closer to the trap centre, while a part of the atoms keeps drifting in the original direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the following, we develop a description of the re- flection in terms of gapless edge modes and the bulk- edge correspondence within the framework of the Harper- Hofstadter-Hatsugai (HHH) model with one real (x) and one synthetic (n) dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The model features bulk arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='03583v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='quant-gas] 9 Jan 2023 2 Floquet Energy 0 π π Quasimomentum Gradient Interactions A B C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Reflection of quantised Hall drifts off a topo- logical interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (A) Topological Thouless pumping in the presence of confining potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the non-interacting case (top), a harmonic trap gives rise to topological trivial (C = 0) and non-trivial (C ̸= 0) regions, separated by a topological interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The atoms exhibit a quantised drift until they are reflected at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' With repulsive on-site interactions (bottom), the reflection happens closer to the centre, accom- panied by atoms still drifting in the original direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (B) Using Floquet theory, the 1D Rice-Mele pump can be mapped to a 2D Harper-Hofstadter-Hatsugai model with a linear gra- dient along the synthetic dimension n which represents the photon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The magnetic flux per plaquette is Φ = 1/2 in units of the magnetic flux quantum [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The gradient along n leads to a transverse Hall drift along x (red arrows) due to the nontrivial topology of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (C) Schematic spec- trum of the mapped 2D Hofstadter model in a semi-infinite geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The lowest two bands have C = ±1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The linear gradient induces Bloch oscillations in the synthetic reciprocal space (dashed arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A gapless edge mode (solid arrow) appears at the topological interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection of the Hall drift can be understood as atoms being transported from the lower band (C = 1) to the higher band (C = −1) via the topological edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Chern bands with C = +1 and C = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' An ex- act mapping between the non-interacting 1D Rice-Mele Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1) and the two-dimensional (2D) HHH model can be obtained using Floquet theory, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1B (for derivation see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=', refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' [19] and [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A linear gradient along the synthetic dimension n appears in the mapping since the state with n photons acquires an energy of −nℏω, where ω = 2π/T is the pump fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The gradient along n or, equivalently, an exter- nal force causes Bloch oscillations along the synthetic re- ciprocal dimension kn which, in turn, lead to a Hall drift or ‘anomalous velocity’ along the transverse real direc- tion x [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The bulk Hall drift along x corresponds exactly to the quantised displacement measured in the topological pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The trap induces a boundary between topological (Ccentre = 1) and trivial (Cright = 0) regions and a single gapless edge mode emerges, according to the bulk-edge correspondence: Ccentre −Cright = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The edge modes connects two bands of opposite Chern invariant, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Thus, a Bloch oscillation transfers the atoms from the ground to the first excited band via that edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Since the first excited band has Chern number −1 the atoms are now moving ‘backwards’, re- sulting in a reversal of the quantised Hall drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2 shows experimental in-situ images of the atomic cloud as a function of time τ at U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The data shows a quantised drift of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='00(1) × 2a/T up to about 60 T, which confirms the long coherence time of Bloch oscillations which induce the transverse drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' At τ ≃ 75 T the atoms change their drift direction, which is a key observation of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The expected topo- logical boundary (red dashed line) represents the posi- tion at which the local tilt from the external harmonic potential ∆ext (j) ≡ 1 2 |Vj − Vj−1| = V0 ��j − 1 2 �� equals the maximum sublattice offset ∆0, thus, xedge/ (2a) ≃ 1 2∆0/V0 = 92(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Beyond this position the total sub- lattice offset ceases to change sign, rendering the region outside xedge topologically trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The boundary caused by the harmonic confinement is not infinitely sharp, but smoothened over several lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This leads to a small T–dependence of the reflection position (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1), compared to its absolute value, and the calculation above should be understood as the outermost point of the re- flective region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflected atoms exhibit a quantised drift of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='99(3) × 2a/T in the opposite direction, in agreement with a transfer to the first excited band with C = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The linear relation between the position of topological boundary xedge and the maximum sublattice offset ∆0 is further confirmed by measuring the reflection in different lattices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection is observed under all parameter settings tested in this work, high- lighting that the existence of the topological boundary is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In addition to the reflection, we also observe a cloud of atoms temporarily remaining at the boundary before gradually dissolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This process can be understood via the presence of topologically trivial edge states, which hybridise with the gapless edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' To simplify the picture, let us consider a sharp domain wall between C = 1 and C = 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' According to the bulk- edge correspondence, the topologically nontrivial region contributes exactly one gapless mode whereas the trivial region can contribute gapped edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to tunnel coupling at the interface, hybridisation takes place [34] and gaps on the order of the pump frequency 2π/T emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Bloch oscillations along kn can now lead to non- adiabatic ‘Landau-Zener’ transfers between topological and trivial edge modes, causing an incomplete transfer to the higher band, and atoms remaining at the bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Subsequent Bloch oscillations will transfer atoms back into the topological domain, leading to the dissolu- 3 quasimomentum energy trivial nontrivial 0 36 108 72 144 time τ (T) 1st 2nd 2nd 2nd 2nd Brillouin zone Dimerisation Site ofset A C B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Measuring the reversal of a quantised Hall drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (A) The time-trace of atomic in-situ images shows a quantised drift along x before the atoms are reflected at the topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Each density image is averaged over three individual measurements with the parameters V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='0148(9)t, ∆0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='7(1)t, and T = 3 ms = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='8(2)ℏ/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The red dashed line indicates the topological boundary xedge/ (2a) ≈ 1 2∆0/V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The white dashed lines are linear fits to the atom drift, yielding slopes of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='00(1) × 2a/T before, and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='99(3) × 2a/T after the reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Cloud positions, averaged over the transverse direction, are fitted using Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The experimental Chern marker (lower panel, points) is determined by the velocity of the right-moving cloud at different positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The theoretical Chern marker (line) is calculated in a staggered potential Vj = V0 (−1)j j which has the same local tilt ∆ext(j) = V0 ��j − 1 2 �� as the harmonic trap [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In a local density approximation picture, the local tilt ∆ext shifts the δ–∆ pump trajectory upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Depending on whether or not the trajectory encloses the critical point, the pump is rendered topological or trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (B) Measured band populations as a function of time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Each density image is averaged over six individual measurements with the parameters V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='0191(6)t, ∆0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='2(2)t and T = 3 ms = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='7(2)ℏ/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The total atom number remains constant, within error bars, throughout the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the underlying honeycomb lattice geometry in the x–z plane, the first Brillouin zone has a diamond shape [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The band population is inverted when the bulk current is reflected off the topological interface, manifesting the gapless nature of the topological edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (C) Hybridisation of the edge modes at the topological interface due to tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Bloch oscillations along kn can lead to Landau-Zener transfers between topologically trivial and nontrivial edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The population of trivial edge modes explains the atoms being left at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Hybridisation never changes the total number of gapless edge modes at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' tion of the cloud at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' While the harmonic confinement leads to a more complex level structure [21], the underlying process remains qualitatively the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We support the in-situ images with measurements of band population before, during, and after the reflec- tion, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2B [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Before the reflection, we find a filled ground band, which is consistent with the observation of a quantised Hall drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' At the reflection (τ ≃ 72 T), we observe an inversion of the population to the first excited band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' After the reflection, the inverted population remains almost unchanged while the atoms are travelling back, highlighting the absence of incoher- ent relaxation to the ground band, even after more than a hundred Bloch oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We further explore the effect of attractive and repul- sive interactions on the topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' For attrac- tive Hubbard U = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='48(7)t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='27(7)∆0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3A), the quantised Hall drift is reversed at the same position as in the non-interacting situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This can be explained in terms of the Rice-Mele model in which fermions in the strongly attractive regime approach the limit of hard-core bosons [22], and the condition for the emergence of a topological boundary ∆ext (j) = ∆0 remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' For repulsive Hubbard U = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='48(7)t (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3B), we observe a second reflection in addition to the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Compared to the non-interacting case, this reflection ap- pears much closer to the trap centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The zoomed-in image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3C) shows that a proportion of the atoms start to move backwards after about 12 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In con- 4 Dimerisation Site ofset OD time т (T) B D E C A Time τ ( ) T τ0 τ0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='5 τ0 + 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Reflection of quantised Hall drifts from an interacting topological edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (A) An attractive Hubbard interaction of U = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='48(7)t leads to the same reflection behaviour as observed for U = 0 (measurement parameters otherwise identical to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (B) Repulsive Hubbard interactions of U = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='48(7)t lead to the emergence of a second reflection, closer to the trap centre, which we attribute to an interacting topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A zoom-in (C) shows that the early reflection happens after about twelve cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The white dashed lines are guides to the eye, calculated as linear fits to the cloud position, extracted as the sum of a skewed and a regular Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (D) Microscopic description of the interaction-induced reflection for repulsive Hubbard U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' When the maximum energy offset between two neighbouring sites 2 (∆0 − ∆ext) becomes smaller than the Hubbard interaction, formation of double occupancies is prohibited and one atom is left in the higher-energy site of a unit cell, which then drifts in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (E) The critical point in the δ–∆ plane is split into two in the presence of repulsive Hubbard U [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' When the pump trajectory encloses both critical points, a quantised drift is expected, as in the non-interacting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The local tilt given by the external potential ∆ext shifts the trajectory along the ∆–axis, eventually enclosing just one critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Since a single split critical point features half the topological charge of the orignal one, the material’s topology changes and a boundary emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' trast to the drift reversal in the non-interacting system, a large fraction of the atom cloud still undergoes quantised drifting in the original direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the following, we develop a microscopic picture of the interaction-induced partial reflection in the limiting case of two isolated spins (↑, ↓), which approximates our initial state in a unit cell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As long as the maximum energy offset between two neighbouring sites 2 (∆0 − ∆ext) is larger than the Hubbard U, the formation of a double occupancy is en- ergetically allowed and the quantised drift persists [22], even in the presence of an external potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' However, when ∆ext becomes larger, the energy offset between two neighbouring sites remains always smaller than U, dou- ble occupancy formation becomes prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the lat- ter case, one atom of a singlet pair is transferred to the energetically excited site of a unit cell, which will subse- quently drift in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The other atom, in the lower-energy site, will move onwards because on- site interactions become irrelevant if there is only one atom per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Since the underlying Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1) is SU(2)–symmetric, spin–↑ and spin–↓ have equal probability of being reflected and they should remain cor- related after the splitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The full many-body description of the interaction- induced reversal requires the development of suitable topological invariants for smooth confinements and strong interactions, which goes beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Nevertheless, we obtain an intuition of the bound- ary’s topological origins using again the idea of shifted pump trajectories in the δ–∆ plane with a staggered po- tential (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Numerical simulations have shown that a repulsive Hubbard U can split the critical point at the origin into two separate ones [35], each retaining half the original topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The distance between the new critical points is approximately U up to a cor- rection on the order of the tunnelling t [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' When the trajectory encloses both critical points, quantised drift of two spins (↑, ↓) is expected, as in the non-interacting sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As the position-dependent local tilt ∆ext(j) shifts the trajectory upwards, it will enclose only one of the critical points beyond certain position along x (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This indicates a transition of topological properties and the emergence of an interacting topological edge in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The estimation of the interacting boundary at ∆ext(j) ≃ ∆0 − U/2 lies close to the centre and agrees with the microscopic picture discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Similar to the non-interacting case, this boundary should be con- sidered as the outermost position of the reflective region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In conclusion, we have experimentally observed a re- versal of quantised Hall drifts at a topological boundary in a harmonic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection is a direct mani- festation of the gapless nature of topological edge modes between Chern bands of opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We explore the effect of Hubbard interactions, both attractive and re- 5 pulsive, and find an asymmetric behavior with respect to U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' While on the attractive side the topological boundary is unaffected, repulsive interactions lead to the emergence of a second interface, featuring a splitting of quantised drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As a result, our experiments could en- able the realisation of circular current patterns for con- structing novel many-body phases [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' More broadly, our work allows the exploration of the bulk-edge corre- spondence in the presence of interactions [38], as well as the investigation of edge reconstruction [39] in the quan- tum Hall effect and in interacting topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank Jason Ho, Gian-Michele Graf, Thomas Ihn, Fabian Grusdt, Fabian Heidrich-Meisner, Armando Aligia, and Eric Bertok for valuable discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We also thank Julian L´eonard and Nur ¨Unal for comments on an earlier version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We would like to thank Alexander Frank for his contribu- tions to the electronic part of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We acknowledge funding by the Swiss National Science Foundation (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 182650, 212168, NCCR-QSIT, and TMAG-2 209376) and European Research Council advanced grant TransQ (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 742579).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' ∗ These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' † viebahnk@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='ch [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' B 46, 4026 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 7 SUPPLEMENTAL MATERIALS Dependence of the drift reversal on experimental parameters The expected drift reversal happens when the maxi- mum local site offset over one pump-cycle ∆0 is equal to the local tilt given by the harmonic trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This po- sition given by xedge ≃ ∆0a/V0 with a = λ/2 and V0 = 1 2m(2πνa)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' By measuring the reflection point in lattices with different ∆0, we verify the relevant scaling xedge ∝ ∆0, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1A marks the theoretically expected xedge with the un- certainty propagated from the uncertainty of the trap frequency ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The disagreement between theory and ex- periment for larger values of ∆0 can be explained by the finite waist of the lattice beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In order to explore the edge in our system, the atoms are pumped by almost 100 unit cells (∼ 100 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the Gaussian envelope of the transverse beams, which are essential to realise the pump, the lattice is effectively shallower far away from the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Thus, ∆0 decreases towards the edge and atoms are reflected sooner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We also find a small dependence of the reflection point on the pump period, compared to its absolute value, which spans roughly 10 unit cells when changing T from 2 ms to 10 ms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This can be understood by con- sidering the energy spectrum of the Harper-Hofstadter- Hatsugai (HHH) model in a harmonic potential, which will be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Experimental sequence We start by preparing a degenerate cloud of fermionic 40K in a crossed dipole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We have a spin mixture of mF = {−9/2, −7/2} except for the measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2B and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1, where a spin-polarised cloud in the magnetic state F = 9/2, mF = −9/2 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The spin- polarised cloud is directly loaded into the pumping lat- tice, while the spin mixture is first loaded into an inter- mediate chequerboard lattice with strongly attractive in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The two-step loading precludes the presence of atoms in the higher band and gives a larger fraction of atoms in doubly occupied unit cells [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' After pumping the system for varying times, we either take a in-situ absorption image to measure the density or detect the band population with band-mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The latter is implemented with an exponential ramp to switch off the lattice beam in 500 µs, followed by a time-of-flight expansion of 25 ms before absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Experimental dependence of the reflection position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection point is expected to depend linearly on the maximal site-offset per pump cycle ∆0 which is experi- mentally verified in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Deviations for large values of ∆0 can be explained by the finite waist of the laser beams forming the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (B) shows the period dependence of the reflec- tion position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Changing the pump period T over an order of magnitude only changes the reflection point by about 10 unit cells, which is a result of the smooth boundary of a harmonic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Realisation of a Thouless pump in the Rice-Mele model The lattice setup is comprised of non-interfering stand- ing waves in x, y, and z directions, together with in- terfering laser beams in the x–z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' All the lattice beams come from a single laser source at wavelength λ = 1064 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' These potential combine to form a honey- comb lattice in the x–z plane, which can be considered as isolated tubes of one-dimensional superlattices along x in the limit of deep transverse lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In each 1D tube the potential can be modeled by a one-dimensional superlattice with two sites per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' With this setup, we realise the Rice-Mele model [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the tight-binding limit, the Rice-Mele model can be described with three numbers: the offset energy ∆ between the two sites of a unit cell, the averaged nearest-neighbour tunneling t and the bond dimerisation δ which gives half the differ- ence between the inter- and intra-dimer tunnellings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' By having a dynamical control of the relative phase ϕ be- tween the laser beams generating the interfering and the non-interfering lattice, we manage to shift the two with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This shift modulates ∆ and δ pe- riodically, which can be depicted as a closed trajectory in the ∆-δ coordinate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the adiabatic limit, this realises a Thouless pump with its hallmark quan- 8 tised transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In this case, the atomic displacement is given by the number of revolutions around the origin of the ∆-δ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' φ = 0 φ = ∏/2 φ = ∏ φ = 3∏/2 x E x E x E x E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Realisation of a Thouless pump in the Rice- Mele model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In our system the 1D lattice potential can be modelled by a superlattice with two sites per unit cell, which is depicted in the bottom part of this figure as a function of relative phase ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The local site offset ∆ as well as the bond dimerisation δ is depicted in the ∆-δ plots corresponding to the respective potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The resulting pump displacement corresponds to the number of revolutions around the origin in the ∆-δ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Mapping a 1D Thouless pump to a 2D Hofstadter Model with quantum Hall response A 1D Thouless pump with a period of T, as realised in our experiment, can be mapped to a 2D topological tight-binding (HHH) model with an applied electric field E = 2πℏ qT where q can be thought of as a fictitious charge of netural atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the topological bandstructure, this electric field leads to a transverse current Itrans = q T of one atom per period, when considering a fully occupied band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The 2D model therefore has a quantised transverse conductance σtrans = Itrans/E = q2 2πℏ analogous to the Hall conductance in the Quantum Hall Effect (QHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The time-periodicity of the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1 with ˆH(τ) = ˆH(τ + T) allows us to use Floquet’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Solutions of the time-dependent Schr¨odinger equation iℏ∂τ |Ψ(τ)⟩ = H(τ) |Ψ(τ)⟩ (S1) can thus be written as |Ψ(τ)⟩ = e−iϵτ/ℏ |u(τ)⟩ (S2) with |u(τ + T)⟩ = |u(τ)⟩ and ϵ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the time- periodicity of u(τ) we expand it as a Fourier series, |u(τ)⟩ = � n e−iωnτ |un⟩ , (S3) where ω = 2π/T is the pump frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The change from the time-domain into the Fourier-domain is the key ingredient to map the 1D Thouless pump to a 2D tight- binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The index n is also called the photon number of the mode |un⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Using a multi-index α = (j, σ) we write the T-periodic 1D Hamiltonian for U = 0 in the Fourier-basis: ˆH(τ) = � α,β hαβ(τ) |α⟩ ⟨β| (S4) = � α,β,m e−imωτhm αβ |α⟩ ⟨β| with hm αβ = 1 T � T 0 eimωτhαβ(τ)dτ and |α⟩ corresponding to an atom localised on site j with spin σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Likewise, we use Fourier decomposition to express the solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1 as |Ψ(τ)⟩ = e−iϵτ/ℏ � n,α e−inωτun,α |α⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (S5) where un,α = ⟨α|un⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As a result, we obtain an eigen- value equation for un,α: ϵun,α = −nℏωun,α + � β,m hm αβun−m,β ∀n, α (S6) which can be understood as a time independent Schr¨odinger equation of a 2D tight-binding model with a tilted potential energy along one axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' By explicitly evaluating the hm αβ, we get H2D = Hreal + Hsynth + Hdiag + HV + Htilt, (S7) with Hreal = −t � j,n,σ (ˆc† j,n,σˆcj+1,n,σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='), (S8) Hdiag = −δ0 2 � j,n,σ e−iπj(ˆc† j,n,σˆcj+1,n+1,σ + ˆc† j,n,σˆcj+1,n−1,σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='), Hsynth = −∆0 2 � j,n,σ e−iπj(iˆc† j,n,σˆcj,n+1,σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='), HV = � j,n,σ V (j)ˆc† j,n,σˆcj,n,σ, Htilt = − � j,n,σ ℏωnˆc† j,n,σˆcj,n,σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Hreal and Hsynth describe tunneling along the real (x) and synthetic (n) dimension, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The diagonal tunnelling terms in Hdiag are crucial be- cause they open a bandgap between the ground band and the first excited band, characterised by the topological Chern number C which is further related to the quantised Hall conductance via σtrans = q2 2πℏC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The terms in HV 9 describe the external potential along the real-space direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Htilt corresponds to a linear tilt in potential energy along the synthetic dimension which can be thought of as originating from an electric field E = 2πℏ qT pointing along n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Edge modes and their reflection properties To illustrate the topological edge modes in the presence of an external potential, we evaluate the spectrum of H2D in the adiabatic limit (ω → 0) for different potentials V (j) = 1 2m(2πνa)2jκ (S9) with m being the mass of 40K, trap frequency ν = 134 Hz, lattice spacing a = 532 nm, and lattice site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The pa- rameter κ, an even integer, characterises steepness of the trap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' the limit κ → ∞ corresponds to the textbook case of infinitely sharp walls [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3A-C shows the nu- merically calculated energy spectra, omitting states on localised to the left edge for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3A shows the spectrum for a box-like potential with κ = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In this case there is a family of topologi- cal edge states, marked in red, which connect the lower and the upper band (black), separated from topologically trivial states above 5 kHz (also in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' All red states are localised along the right edge in x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The lower and upper band have Chern number 1 and −1, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Considering the dynamics in this model, an applied electric field along n as defined in Htilt leads to Bloch oscillations with a period T along kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' At the same time, the center-of-mass of the atoms moves by one unit cell per Bloch oscillation period along x, which corre- sponds to the quantised bulk Hall drift [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This drift can be evaluated in the numerics by following the eigen- states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3A in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Since the red edge states are gapped from the next higher-lying trivial (black) states, the atoms ‘Bloch-oscillate’ from the ground to the excited band via the red-marked edge modes over several periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Once they are in the excited band they are transported backwards along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3B shows the situation for κ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' It behaves similarly to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3A, except that there are more lo- calised states marked in red, compared to κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Like- wise, these states are transported along x as they undergo Bloch oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As before, this family of edge states is gapped from trivial states and connects right-moving to left-moving states, which leads to the reflection phe- nomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The experimentally relevant case is a harmonic trap with κ = 2 (see also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' [23, 29–31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3C shows that for κ = 2 the number of localised sates outside of the bands is even larger than for κ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As before, we adiabatically follow these localised states along kn A B C D FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Energy spectra for different confining poten- tials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' States localised to the left edge are omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Energy spectra of H2D in the limit ω → 0 for κ = 24 (A), κ = 10 (B), and κ = 2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Topological edge modes which connect the two bands with Chern number 1 and −1 in (A) and (B) are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The upper inset in (C) marks the topological boundary where the reflection is observed as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The inset in the center of (C) shows the tiny avoided crossings which can lead to a slight period dependence of the observed reflection point (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (D) Energy spectrum for a linearly increasing staggered po- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The gapless, topological edge mode is marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 10 and evaluate their centre-of-mass along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' By numer- ically observing these drifts we confirm that the states describe quantised drifting in a large region, which man- ifests their nontrivial topological nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Thus, the κ = 2 case is ideal to observe the reflection after long-distance quantised Hall drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' However, the gaps between topo- logical (right-moving and left-moving) and trivial (sta- tionary) states become smaller, compared to the κ = 24 and κ = 10 cases, as shown in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3C (κ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As a result, the reflection point for κ = 2 is spread out over several unit cells but the reflection itself remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Faster pumping leads to non-adiabatic crossings of the energy gaps between right-moving and left-moving states, causing the reflection to happen later in time and further up in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We confirm this depen- dence experimentally in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1B, which shows a later reversal for smaller pump periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3D shows the spectrum for the linearly increasing staggered potential, described in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This potential allows a straightforward identification of the gapless edge mode (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The states correspond- ing to this gapless edge mode are localised around the topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' x E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Linearly increasing staggered potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' To elucidate the topology in our system, a linearly increasing staggered potential is considered (blue): V (j) = jV0(−1)j with V0 = 1/2 × m(2πνa)2, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' It is chosen such that the local tilt always equals the tilt from the harmonic poten- tial (orange), but alternates in sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The staggered potential allows a simple pictorial representation of the emergence of the topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In a local density approximation the trap linearly shifts the pump trajectory upwards in the ∆-δ plane, as depicted in the upper part of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As soon as the trajectory ceases to enclose the critical point, a topological–trivial boundary develops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Staggered potential Another possibility to identify the topological bound- ary in our system makes use of a staggered potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' First, we consider a potential with uniform staggering, given by Vc(j) = V (−1)j, where j indexes the lattice-site and 2V corresponds to the energy difference between ad- jacent sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Adding such a potential to the Rice-Mele Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1) changes its trajectory in the ∆-δ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The onsite energy in such a system is given by (∆(τ) + V )(−1)j, which ranges from −∆ + V to ∆ + V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The tunnellings are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Therefore, the trajectory remains circular and it is simply shifted upwards by an amount V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A topological boundary emerges for a linearly increas- ing staggered potential, given by V (j) = jV0(−1)j, with V0 = 1 2m(2πνa)2 as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' V (j) is chosen such that it has the same local tilt as the harmonic trap in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Within the local density approximation we assign a ∆-δ trajectory locally to each unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The trajectories are thus linearly shifted upwards as func- tion of j (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S4), describing a change of topology in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We expect the local density approximation to be valid since the atomic eigenstates in the exper- iment are strongly localised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Similar models with lin- early increasing staggered potential have been studied in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' [29, 32, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Local Chern marker The mathematical formulation of the Chern number as a topological invariant requires translational invariance, which does not apply to realistic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Instead, we use a local quantity, known as Chern marker [8, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The local Chern marker depends on the real-space posi- tion and it is defined by: c(rγ) = −4π Ac Im � s=A,B ⟨rγs| ˆP ˆx ˆQˆy ˆP |rγs⟩ , (S10) where rγ is the position of the unit cell γ with sub-lattice- sites at positions rγA and rγB, |rγs⟩ = c† γs |0⟩ is the state localised on the corresponding lattice site , Ac is the area of a real-space unit cell, ˆQ = 1− ˆP and ˆP is the projector onto the ground band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Defining ˆP is not unambiguously possible in our system (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S7) because of the energy shift from the harmonic confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Instead, we use a linearly increasing staggered potential, as described in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This model leaves the bands in- tact and a ground band can be unambiguously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Experimentally, a local probe of the band topology is the velocity of the Hall drift, plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Theory and experiment agree approximately with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The local velocity is extracted from the atomic positions by fitting linear functions to groups of three adjacent dat- apoints in ten pump cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The resulting velocities are plotted against position and smoothed through a running average of width three (ten cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} diff --git a/IdE2T4oBgHgl3EQfUQce/vector_store/index.faiss b/IdE2T4oBgHgl3EQfUQce/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..831dbec7d539afe94624b7edb9555cfb692fb456 --- /dev/null +++ b/IdE2T4oBgHgl3EQfUQce/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:04a1ed9a029d36b42ddbcd02733929db4d341e874e8781f550239f227fe90056 +size 8585261 diff --git a/JNE0T4oBgHgl3EQfiAH8/content/tmp_files/2301.02441v1.pdf.txt b/JNE0T4oBgHgl3EQfiAH8/content/tmp_files/2301.02441v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b72545ac612e5655121a6b8d806dafbc96475b6 --- /dev/null +++ b/JNE0T4oBgHgl3EQfiAH8/content/tmp_files/2301.02441v1.pdf.txt @@ -0,0 +1,4576 @@ +1 + +Theory of Edge Effects and Conductunce for Applications in Graphene- +based Nanoantennas +Tomer Berghau , Touvia Milo , Oded Gottlie and Gregory Ya. Slepya + + 1 +1School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel +2 Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, 3200003, Israel +3 School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel +*Correspondence: berghaus@gmail.com (T.B); gregory_slepyan@yahoo.com (G.S.) + +Abstract: In this paper, we develop a theory of edge effects in graphene for its applications +to nanoantennas in the THz, infrared, and visible frequency ranges. Its characteristic feature +is self-consistence reached due the formulation in terms of dynamical conductance instead of +ordinary used surface conductivity. The physical model of edge effects is based on using the +concept of Dirac fermions. The surface conductance is considered as a general susceptibility +and is calculated via the Kubo approach. In contrast with earlier models, the surface +conductance becomes non-homogeneous and non-local. The spatial behavior of the surface +conductance depends on the length of the sheet and the electrochemical potential. Results of +numerical simulations are presented for lengths in the range of 2.1 - 800nm and +electrochemical potentials ranging between 0.1 – 1.0 eV. It is shown that if the length +exceeded 800 nm, our model agrees with the classical Drude conductivity model with a +relatively high degree of accuracy. For rather short lengths, the conductance usually exhibits +spatial oscillations, which absent in conductivity and strongly affect the properties of +graphene based antennas. The period and amplitude of such spatial oscillations, strongly +depend on the electrochemical potential. The new theory opens the way for realizing +electrically controlled nanoantennas by changing the electrochemical potential may of the +gate voltage. The obtained results may be applicable for the design of carbon based +nanodevices in modern quantum technologies. + +Keywords: graphene; edge effects; optical conductance; nanoantennas + + + +1. Introduction + +Innovative electromagnetic nanoantennas, which generally function at wide range of +frequencies (from THz until video frequencies), play a vital role in the emerging field of +photonics and plasmonics [1-6]. These antennas can be used as promising tools for +transforming near-field light into far-field and vice versa. Their excellent capabilities are +usually utilized in a wide scope of applications. Among them, are traditional applications for +basic elements in electronics, high-speed communications, informatics, and quantum +computing [7] (in particular quantum nanomechanical qubit on carbon nanotube [8]). As far +as commercial applications of carbon-based nanostructures [9], one can list some novel +applications such as: i) high-resolved spectroscopy [10]; ii) high-speed communication [11]; +iii) light emission and detection; iv) identification of biomolecules and medical diagnostic +applications [12,13]. The design of the devices mentioned above, requires taking into account +edge effects and their correct physical description. It should cover a wide frequency range +from microwave until optical. These recent innovations have stimulated huge interest in the +theory of nanoantennas. Microwave standards become invalid starting from the THz region, +as the size of the antennas is miniaturized to micrometer scale [6]. Therefore, the common + +2 + +model of perfect electric conductor, which is widely used in microwaves, is not suitable in +the range from THz to optical frequencies. By manipulating the different types of finite-size +(edge) effects [14-16], the conventional antenna configurations can be also used in the range +from microwave to optical frequencies [1-6]. However, antenna devices which operate in the +terahertz range (0.1–10 THz), have some fundamental limitations on their applicability in +practical devices [6]. Recent studies have pointed out that graphene is one of the best +candidates for overcoming the limitations mentioned above [6]. +A nanoantenna in its theoretical analysis, is generally considered as a system +terminated by a well-defined edge. The edge geometry is one of the widely used idealizations +in modern physics. For example, in all branches of classical mechanics and physics (e.g., +elasticity, hydrodynamics, acoustics, electrodynamics), the edge has a configurational +character. It corresponds to a point or contour at the surface of the body on which the normal +vector is usually undefined (half-plane, wedge, cone, etc.). Taking the edge position into +consideration, allows us to simplify the boundary conditions and employ certain analytical +techniques such as separation of variables, using for example special orthogonal coordinate +systems [17] or the Wiener-Hopf method [18]. Such analytical solutions are especially +attractive for the qualitative analysis of novel types of problems involving interacting fields +of different physical origin (for example, plasmonics and optoelectrofluidics [19]). However, +such idealization may lead to the loss of uniqueness of the problem statement. The reason for +it is that the placement of an arbitrary point singularity at the edge, generally does not violate +the governing wave equation as well as the boundary and radiation conditions. The analytic +solutions thus obtained, are correct from the mathematical point of view. However, they may +turn to be physically incorrect because due to lack of uniqueness, they may correspond to +another source of the field. In order to obtain the corresponding of unique solution, one must +enforce the condition of finite (bounded) energy over an arbitrary area (finite extend) of +space (Meixner condition) [20]. Such an approach usually leads to the creation of field +singularities at the vicinity of the edge. Note, that the edge-like configuration does not change +the constitutive properties of the medium (for example, such as a perfect conductor or a +perfect insulator in classical electrodynamics). + +The extensive recent progress in nanotechnologies, lead to the discovery of novel +artificial types of condensed matter, such as graphene [21], carbon nanotubes (CNTs) [22], +Weyl and Dirac semimetals [23] and topological insulators [24,25]. Consequently, a great +number of fundamental physical problems were reconsidered and in particular, but +nevertheless most of them failed to consider the important edge concept. The edge for the +different types of nanostructures corresponds to the spatial area in the vicinity of the sample +boundary which size is rather large compared with the inter-atomic distance. However, the +mechanism of electronic transport at the edge is dramatically different from the +corresponding properties of the bulk region. One of the reasons for this disparity is the +existence of new types of quantum states strongly confined to the vicinity of the edge, which +are able to produce novel physical properties such as topological order (so called, edge states +[16,21,25]). Such transport mechanisms manifest themselves in the special optical and +optomechanical properties of different types of metamaterials (e.g., carbon based +nanostructures). + +The electrical properties of macroscopic structures may be described both in terms of +conductivity and conductance [16], which are equivalent and coupled via the simple constant +coefficient defined by the size values of the sample. As it was noted in [16], the conductivity +for nanomaterials (including graphene) is a well-defined value only when the sample is +enough large. It is clear from intuitive point of view, that the electric current becomes +homogeneous and insensitive to boundaries of the sample. When the sample’s size is +reduced, the current becomes non-homogeneous and non-local with respect to the field + +3 + +variations in the material. Then, the concept of conductivity loses its meaning. The behavior +of the electrons in nanoantennas becomes sensitive to the feeding lines, detectors, modulators +and the edges due to the quantum-mechanical interference. As we will show, exactly optical +conductance will be suitable parameter for formulation of the effective boundary conditions +for electromagnetic field in nanoantennas. In contrast with conductivity, it is non- +homogeneous and not coupled via simple relation with optical conductivity (which is +homogeneous value). It is described by the Kubo approach instead of Boltzmann’s transport +equation. +The edge areas play the main role in forming the emission in classical radio-frequency +antennas [26] and optical nanoantennas [27]. One of the promising types of nanoantennas that +operate in the THz and optical frequency range, are based on carbon-based nanostructures +(graphene, CNTs) [28-31]. The physical mechanism of their radiation, is based on the +existence of a strongly retarded surface plasmon predicted in [28, 32], experimentally +observed in [33] and used for interpretation of the intriguing measurements of the THz +conductivity peak [34]. These works use different models for the charge transport [28, 32, 34, +35], but all of them are not self-consistent. In another words, the boundary-value problem for +the Maxwell equations is formulated for the real geometry of the object, while the value of +conductivity is introduced as a phenomenological parameter which is determined by using a +model corresponding to an infinitely large structure. Of course, such an approach appears to +be attractive since it allows to reach, simplification due to the separation of the EM-field and +charge transport equations. Such an approach keeps the self-consistent requirement for planar +structures with the Fresnel transmission-reflection condition of EM-fields (graphene films, +semitransparent mirrors, etc.) [37-40]. However, it may not be adequate for the detailed +description of EM-field scattering and antenna emission in the general case. One can clearly +expect that the error of such a non-consistent approach may be ignored as being very small, +providing the size of the system is rather large. However, it means, that the effect of the +Fresnel transmission-reflection dominates the field forming, whereas the antenna efficiency +decreases. Nevertheless, it is impossible to say a priori what is the validity bound of such a +simplification. +This problem becomes especially relevant for graphene-based structures, because of +the corresponding large geometrical size disparity for applications in different graphene +devices. The advanced graphene synthesis methods make it possible to grow graphene +samples from 2.1nm to few centimeters [6,41-46]. The detail description of optical graphene +conductivity requires a self-consistent analysis. Such a self-consistent approach determines +the field acting on electrons produced over their motion, by taking into consideration the +finite-size configuration and using an appropriate microscopic model for the edge. The +formulation of such a self-consistent analysis is one of the main contributions of this paper. +One of the important results of this paper, is analyzing the effect of the edges (shape of finite +extend) and determine the corresponding error when self-consistency is not prevailed. It is +important to note; that edge effects do not only add up to the quantitative differences in the +value of conductivity. Edge effects are also able to dramatically affect the special physical +mechanism of charge transport. It makes required the formulation of the theory in terms of +optical conductance, which is important for applications in antenna design. + +The paper is organized as follows: models of the edge of a graphene ribbon based on +the Dirac-fermion concept, are first discussed in Section 2. Thereafter, based on using the +Kubo approach and the concept of general susceptibility [47], we analytically obtain an +expression for the surface conductance of a terminated (finite) graphene sheet. Results of +numerical simulations together with a discussion are presented in Section 3. Conclusion and +outlook are finally given in Section 4. + +4 + +2. Optical conductance of a terminated graphene sheet. + + + + + + + + + + + + + + + + + + + +Figure 1. Problem statement: (a) Geometry of a graphene sheet with zigzag edge configuration; (b) +Fermi-Dirac cones for the model of pseudospin dispersion. + +2.1 Kubo approach for optical conductance of graphene + +In the following we will use the Kubo approach for conductance calculation as a universal +technique which couples the generalized forces of arbitrary physical origin with the responses +of correspondent origin via general susceptibilities [47]. Towards this goal it is convenient to +define the forces as tangential components of electric field and responses as components of +current densities. Thus, the general susceptibilities are defined by a 2 +2 + +matrix where + +  + + +  +, +, +1,2 +; , +a +ab +b +b +j +K +E +d + + + + + + + +   +x +x x +x +x (1) +and + + + + + + + + + + +2 +0 +; , +ˆ +ˆ +ˆ +ˆ +, +0, +0, +, +ab +i t +a +b +b +a +K +e +e +x +t +x +x +x +t +dt + + + + +  + + + + +x x +x +x +x +x + (2) + + +Here +  +  +ˆ +a +a +j +j + +x +x +, + + + + +ˆ +ˆ +, +, +a +a +j +t +i ex +t + + +x +x , +  +ˆ +aj +x + represents the observable current +density, + + +ˆ +, +ax +t x is an operator of charge displacement per unit area. The general +susceptibility tensor + + +; , +ab +K + + +x x depends on the geometry of the sample as well as the +electronic properties of the medium. Therefore, exactly it has a physical meaning of non-local +optical conductance (in the units +2 / +e +) similar to the dc conductance of graphene in [16] . + +The next step is the transformation of Equation (2) to a form that is convenient for +applications in graphene electrodynamics. Thus one can write +(b) +(a) + +Energy +Valley K' +Valley KA-atoms +B-atomsremoved +B-atoms +1 +A-atoms removed5 + + + + + +0 +ˆ +ˆ +; , +i t +ab +ab +ab +ie +K +e +A +B +dt + + + +   + + +x x + (3) + +where + + + + + + + + +ˆ +ˆ +ˆ +ˆ +ˆ +ˆ +, +0, +, +0, +, +ab +ab +a +b +b +a +A +j +t +x +B +x +j +t + + + + +x +x +x +x +and + + + + +0 +0 +ˆ +ˆ +ˆ +ˆ +ab +ab +ab +ss +s +ss +A +Tr A +A + + + +  + (4) +where +0ˆ is the density matrix of free motion. + The diagonal matrix elements of the operator ˆ ab +A are defined by + + + + + + + + + + + +ˆ +ˆ +ˆ +, +0, +ab +a +b +s s +ss +s +ss +A +j +t +x + + + + +  +x +x + (5) + +and following Equation (4) one gets + + + + + + + + + +0 +ˆ +ˆ +ˆ +, +0, +ab +a +b +ss +s s +s +s +ss +A +j +t +x + + + + + + + + +x +x + (6) +Here + + +, +y +s +p k + +denotes the combinative discrete-continuous index (p is the discrete number +of the state and +yk is its continuous wave-number over the y-axis). + +The temporal behavior in the linear approximation can be expressed as + + + + + + + + + + / +ˆ +ˆ +, +0, +s +s +i +t +a +a +ss +ss +j +t +j +e + +  + + + + + +x +x + and as a result we obtain + + + + + + + + + + +0 +ˆ +ˆ +ˆ +0, +0, +s +s +i +t +ab +a +b +ss +s s +s +s +ss +A +j +x +e + + + + + + + + + + + + + +x +x + (7) +A similar relation may be also obtained for ˆ ab +B , namely + + + + + + + + + + +0 +ˆ +ˆ +ˆ +0, +0, +s +s +i +t +ab +b +a +ss +s s +s +s +ss +B +x +j +e + + + + + + + + + + + + +x +x + (8) +Combining Equations (7) and (8) with Equation (3) lead to + + + + + + + + + + + + + + + + + + + + + + +0 +0 +; , +4 +ˆ +ˆ +ˆ +ˆ +0, +0, +0, +0, +s +s +s +s +ab +s +s +i +i +t +t +i t +a +b +b +a +ss +s s +ss +ss +s s +i +K +e +j +x +e +x +j +e +dt + + + + + + + + + + + + + + + + + + + +  + + + + + + + + + + + + +x x +x +x +x +x + + +(9) +For shortness, we will omit the initial value at time t=0 in the matrix elements of +( + + + + +  + + +' +' +ˆ +ˆ +0, +a +a +ss +ss +j +j + +x +x + +, etc.). Using elementary integration yields + +6 + + + +  + + +  + + + + + + +  + + +  + + + + + + +0 +ˆ +ˆ +ˆ +ˆ +; , +a +b +b +a +s s +ss +ss +s s +ab +ss +s +s +s +s +s +s +j +x +x +j +ie +K + + + + + + + + + + + + + + + + + + + + + +   + + + + + + + + + + + + +x +x +x +x +x x + +(10) + +Introducing the following change +, +s +s s +s +  + + in the second term of Equation (10) and +using the relation +  + + + + +  + + +' +' +' +ˆ +ˆ +/ +a +s +s +a +ss +ss +j +ie +x + + + + +x +x + , the latter can be written as + + +  + + +  + + + + + +  +  + +0 +0 +ˆ +ˆ +1 +; , +a +b +ss +s s +ab +ss +s s +s +s +s +s +s +s +j +j +K +i + + + + + + + + + + +  + + + + +   + + + + + +x +x +x x + +(11) +where + + + + +/ +0 +( ) +1/ +1 +B +k T +ss +f +e +  + + + + + + + denotes the Fermi-distribution and  is the +electrochemical potential. The electrochemical potential is defined here by the concentration +of electrons/holes according to the following relation; + + +2 +1 +0 +/ +F +n +v + + + + +[37-40]. This +potential vanishes for a perfectly clean graphene at zero temperature [16] and may be +effectively controlled via the gate voltage [37-40]. + +Because of the Hermittivity nature of the current operator, we have +  + + +  + + +ˆ +ˆ +b +b +s s +ss +j +j + + + +x +x + where the upper asterisk denotes complex conjugate. Therefore, one +can also write Equation (11) in the . compact form + + +  +  + + +; , ' +0 +ab +ab +ab +i +K + + + +  + +  +x x + (12) +where +  +  + + +  + + + + + + + + + + + + +' +ˆ +ˆ +. +a +b +ss +s s +ab +s +s +s +s +s +s +j +j +f +f + + + + + + + + + + + + + + + + + +x +x + +(13) + + +2.2. Zigzag edges + +In this Subsection we will apply the general result to the edge of a zigzag type. The electronic +properties of the ribbon are described by the model of Dirac fermions corresponding to a +tight-binding model on a two-dimensional honeycomb lattice [16,25,48]. We will use the +zigzag-type of boundary conditions for Dirac fermions on a terminated (finite) lattice derived +in [16,25,48], since it was demonstrated that this type of boundary condition can be generally +applied to a terminated honeycomb lattice in the case of electron-hole symmetry [25]). +The A and B atoms are coupled in every spinor modes. From physical point of view, +the interpretation of electronic transport may be considered as a motion from A atoms to B +ones and vice versa, while A-A and B-B motions are forbidden. The electron transport for a +zigzag edge in valleys K and K’ is independent and will be considered separately. The wave +function is expressed as a linear combination of the eigen 4D spinors + + +7 + +The pseudo-spinor modes for valley K have the following form; + + + + + + + +  +  +, +( +) +, +, +, +, +0 +0 +0 +0 +y +s +A s +s +i k y +t +B s +s +Ks +t +u +x +t +v +x +t +e + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +x +x +Ψ +x + (14a) + + + + + + +  +  +( +) +, +, +0 +0 +0 +0 +, +, +, +y +s +i k y +t +K s +A s +s +B s +s +t +e +t +u +x +t +v +x + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Ψ +x +x +x + +(14b) + + + +The functions   +  +, +u x +v x in the valley K (and in K’ respectively) take the following form for +bulk and edge states + +  +  +  +  +  +  +  +  +1 +sin +2 +2 +1 +sinh +2 +2 +1 +sin +2 +2 +1 +sinh +2 +2 +Bulk +b +s +s +p +px +Edge +e +s +s +p +Bulk +b +s +s +p +px +Edge +e +s +s +p +L +u +x +u +x +i +B +x +lL +L +u +x +u +x +B +x +lL +L +v +x +v +x +B +x +lL +L +v +x +v +x +i +B +x +lL + + + + + + + + + + + + + + + + + + + + +  + + + + +  + +  + + + + + + +  + +  + + + +  + +  + + + + + + + + + + + + + + + +  + + + + + + + +  + + + +(14c) + + + +where, + + + +1 +sin 2 +1 +2 +px +b +s +px +L +B +L + + + + + + + + + + + + + (14d) + + + + +1 +1 +2 +sinh 2 +1 +2 +L +e +s +L +B +e +L + + + + + + + + + + + + + (14e) + + + + +where l represents the normalization length in the y-direction (which goes to infinity in the +final result), +sA is a normalization constant, +, +y +xp +k k +are the wavenumbers which are defined +by the dispersive relation + + +2 +xp +ik L +y +xp +y +xp +k +ik +k +ik +e + + + + +and + + +, +y +s +p k + +is the combinative +discrete-continuous index (p is the discrete number with respect to the x-axis and +yk is the + +8 + +continuous wave-number over the y-axis). This dispersion relation has an infinite number of +real roots with a single point of concentration at infinity (confined modes) and one imaginary +root that corresponds to the edge state [16]. Transformation to the edge state from confined +modes in Equations (14a)-(14e) and characteristic equation may be done via exchange +px + + + +.The two signs in (14) correspond to electrons and holes respectively. As one can see +Eq. (14) satisfies the Dirac equation and is subject to the following boundary conditions; + + + + +/2 +/2 +0 +s +s +u +L +v +L + + + +[48].The components u(x) and v(x) are separately orthogonal, while +non-orthogonal mutually due to their coupling over electron motion between the atoms of A +and B sub-lattices. Since the expression for the ac-conductivity with a zigzag edge is +isotropic, we have + + + + +; , +; , +ab +ab +K +K + + + + + + +x x +x x ,where the general susceptibility is + + + +  +  + + +1 +; , ' +0 +K +i + + + + +  + +  +x x +, + + +with +  +  + + +  + + + + + + + + + + + + +' +ˆ +ˆ +ss +s s +s +s +s +s +s +s +f +f + + + + + + + + + + + + + + + + + +j x +j x + (15) +Here represents the matrix element of the current operator, σ denotes the xy-vector of the Pauli +matrices, +2 +2 +s +F +xp +y +v +k +k +   + +is the energy of the s-th state and , +F +e v correspond to the +electron charge and Fermi velocity respectively. + +The general susceptibility may be presented here as the sum of two components of +different +origin + + + + + + +; , ' +; , ' +; , ' +Inter +Intra +K +K +K + + + + + +x x +x x +x x +, +where + + +  +1 +; , ' +Inter +K +i + + + + +  + +x x + corresponds to the interband motion which may be ignored +omitted for rather low (THz) frequencies. The second term + + +  +1 +; , ' +0 +Intra +K +i + + + + +x x +corresponds to the intraband motion and leads to the common conductivity law +  +    +Intra +i +x + +  +j x +E x , where the surface conductance is found by substituting the +pseudospinor modes (14) into (15), which renders +  + + + + +  +  + + +2 +2 +2 +2 +( ) +2 +0 +s +n +y +Intra +F +s +s +y +n +k +ie v +f +x +u +x +v +x +dk +i +  + + + + + + + + + + + +  + + + + + + + +(16) + +(for details of deriving Equation (16) see Appendix A). The explicit expression for the +conductivity given in (16) is spatially inhomogeneous (x-depended) and is a manifestation of +edge effect and incorporating a self-consistent description. +2.3. Approximation for zero temperature. +Let us next consider the important case of the conductance at zero temperature, which opens +the way for further simplification of Equation (16). In this case the Fermi distribution may be +replaced by a step function and its derivative in (16) can be transformed into a Dirac delta +function + + +/ +f + +  + + +   + +, which allows an integration of (16). Performing the integration +in + + +9 + + + + + + + + + + + +Figure 2. Illustration with respect to the zero-temperature approximation. Black lines correspond to +confined modes, red line corresponds to the edge mode, and blue line corresponds to the first confined +mode. The black point corresponds to the critical wavenumber in which the mutual transformation of +edge mode to the first confined mode takes place. Dotted horizontal line corresponds to the given +electrochemical potential. The values +yn +k +denote the roots of Equation (17). + +possible in terms of the discrete number of roots of the following transcendental equation + + +yk + + + +( qualitatively depicted in Figure 2), which may be also written as + + +2 +2 +2 +2 +tg +yn +yn +yn +k +k +L +k + + + + + + (17) +The value +/ +F +v + + + +means normalizing the electrochemical potential by the electron energy. +The corresponding edge mode may be obtained by the formal exchange +yn +k +ik + +(such root +exists only under special conditions). For the integration over +yk we use the following +property of the Dirac delta-function + + + +  + + + + + +yn +y +y +y +n +yn +F k +k +F k +dk +k +  + + + + + + + + + + +(18) + +where + + +yk + +denotes the y-component of the group velocity of pseudospin (prime means the +derivative). The final explicit result for the conductivity can be written as +  + + + + +2 +1 +2 +2 +2 +2 +2 +2 +1 +2 +2 +sin +sin +2 +2 +0 +N +n +yn +yn +n +yn +Intra +B +L +L +k +x +k +x +L +k +e +x +i +i + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +(19) + +kynL +kynL + + + +20 +18 +16 +14 +12 +10 +8 +6 +4 +2 +0 +-25 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +25 +k,L10 + +The values of +yn +k depend on the electrochemical potential and satisfy the characteristic +Equation (17). The normalized coefficients +n +B in (19) are defined by + + + + +1 +2 +2 +2 +2 +2 +2 +sin 2 +1 +2 +2 +yn +n +n +n +yn +yn +k L +B +L +k +k L + + + + + + + + + + + + + + + + + + + + + + +(20) + Equation (19) may be finally transformed to +  + + +  + + +2 +1 +2 +2 +2 +2 +2 +2 +1 +2 +2 +1 +sin +sin +2 +2 +1 +2 +0 +N +yn +yn +n +yn +Intra +L +L +L +k +x +k +x +k +L +e +x +i +i + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +(21) +(for details see Appendix B). +2.4. The limit of an infinite sheet. +In this Subsection we show that our model reduces to the familiar Drude relation for the +conductivity in infinitely wide sheet. Taking the limit L   , and using the following +transformation + +  + + +  +1 +1 +2 +... +2 +... +x +n +L +dk + + + + + + + + +. Equation (16) yields +  + + + + + + + + + + + + +2 +2 +2 +0 +( ) +1 +1 +1 +cos +2 +cos +2 +2 +2 +s +n +y +Intra +F +x +x +x +y +k +ie v +x +i +f +k +x +L +k +x +L +dk dk +  + + + + + + + + + + +  +  + + + + + + + + + + + + + + +  + +(22) +Introducing the polar variables +sin +xk + + +, +cos +yk + + +, we obtain +x +y +dk dk +d d +  + + . Using the +zero temperature approximation, we exchange the Fermi-distribution derivative by the Dirac +delta-function and integrate over  , so that Equation (22) becomes +  + + + + + + + + + + +2 +2 +2 +2 +0 +2 +0 +1 +1 +1 +cos +2 +cos +cos +2 +cos +2 +2 +Intra +ie +x +i +x +L +x +L +d + + + + + + + + + + + + + + +  + + + + + + + + + (23) +Furthermore, using the addtion theorem for Bessel functions [49] +  +cos +i +m +im +m +m +e +i +J +e + + + + + + + +  + (24) +we integrate (23) and obtain + +11 + +  + + + + + + + + + + +2 +0 +0 +2 +1 +1 +1 +2 +2 +0 +2 +2 +Intra +ie +x +J +x +L +J +x +L +i + + + + + + + + + + + + + + + + + + (25) +If the observation point x (of under the assumption of large L), is placed rather far from the +edges, the arguments of Bessel functions become large, so that the asymptotic relations [49] . +In this case, the last two terms in (25) become indefinitely small. The first term, which is +exactly equal to the Drude conductivity + + +2 +2 +/ +0 +Drude +G +ie +i +  + + + +, becomes dominate over the +whole area of the ribbon excluding the narrow vicinities of the edges. Thus, it is +demonstrated that in this limit our model asymptotically reduces to the classical expression +for the Drude conductivity. +2.5. Optical conductance for the edge with infinite-mass boundary condition + +This problem is of special interest in connection with a suspended sheet. The interaction +between the edges of the sheet with the electrodes, results in the creation of electrostatic +potential. Following [13], the electron-hole symmetry, which is generally restricted for +boundary conditions of a zigzag or armchair types is broken. It may be considered as a +manifestation of a staggered potential at zigzag boundaries, which may change the nature of +the boundary condition. For, infinitely large value of potential it leads to an “infinite-mass” +boundary condition, which may be written as + + + + +/2 +/2 +s +s +u +L +v +L + +  + +. The corresponding +eigen pseudospins are then given by Equation (14) with + +  + + +2 +2 +1 +1 +2 2 +s +s +xn +xn +i k +x +i k +x +n +su +x +e +e +L + + + + + + + + + + + + + + + + + + + + +  + + + + + + + (26) +  + + +2 +2 +1 +1 +2 2 +s +s +xn +xn +i k +x +i k +x +n +sv +x +e +e +L + + + + + + + + + + + + + + + + + + + + +  + + + + + + + (27) +where +/ +xn +k +n +L + + +and +2 +2 +s +y +xn +i +y +xn +k +ik +e +k +k + + + + +. +In order to calculate the conductance, we use the Kubo approach with the pseudo spinors +defined in (26) and (27). The difference from the zigzag edge formulation, is due to the lack +of orthogonality inf (26) and (27), which leads to a non-local conductance (spatial +dispersion). The conductance operator does not add up to the convolution form because of the +non-homogeneity of the finite-length structure. In the case of a rather wide sheet the non- +local component becomes relatively small and may be usually ignored. In this particular case +the conductance relates to Equation (16) with the pseudospins given given in (26) and (27). + +3. Numerical results and discussion +The optical properties of graphene are generally defined by the geometrical size of the sample +and the value of the electrochemical potential. These parameters may vary over a wide range +and thus are of practical interest to nanoantenna design. The recent advances in graphene +technology make it possible to produce graphene samples from few nanometers to few +centimeters [6,7,21]. The electrochemical potential may also vary over the interval 0 +1.0 + + + +eV, by doping the sample or by applying gate voltage [6,7,21]. In this Section we will + +12 + +present some numerical results of conductivity simulations for a wide range of physical +parameters, based on the theory developed above. + +One of the main results of the present study according to Equations (16) and (19), is +that the non-homogeneity of conductance is mainly due to the edge effects. The conductance +distribution is controlled by the parameter +/ +F +L +L +v + + + +, which defines the number of modes +supported by the conductance value. Figures 3-5 present the normalized conductance +distribution for a rather large length +800 +L  +nm and for different values of the +electrochemical potential (increasing  corresponds to increasing number of modes). The +qualitative behavior of the conductance is the same in all these Figures. The conductance +oscillates with respect to the spatial variable and decreases near the edges. The amplitude and +period of oscillations decrease with increasing of the electrochemical potential  . The +average value of the conductance (to a high degree of accuracy), corresponds to the classical +Drude model of conductivity, as discussed in Section 2.4. One can anticipate that such small +oscillations are unable to manifest themselves in the scattering of electromagnetic waves, due +to the smallness of their period compared with the wavelength. Thus, we may conclude that +the Drude model can be applied for this range of parameters. +The considered scenario changes dramatically with shortening of the sheet, as shown +in Figures 6-8 (each Figure corresponds to a different value of the length for the same value +of electrochemical potential). One can clearly see the enhancement of oscillations, which +makes the Drude model invalid for such values of parameters. In fact, it becomes impossible +to introduce the conductivity concept in its ordinary meaning. As it was mentioned above, the +value defined by Equation (16) has the meaning of optical conductance, which strongly +depend on the geometrical size of the sheet. It may be coupled with Drude conductivity by +the Relation + + +  +  +1 +Drude +Intra +x +L +x +G + + + + +(28) +where + +  +  + + +2 +1 +2 +2 +2 +2 +2 +2 +1 +1 +sin +sin +2 +2 +1 +N +yn +yn +n +yn +L +L +x +k +x +k +x +k +L + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +(29) + + +is x-dependent coefficient, in which the configuration of the sample manifests itself. The +situation is rather similar to the electron transport in graphene in dc field (the concepts of +conductance and conductivity discussed and compared in [16,50]). +The physical mechanism for the conductivity oscillations is the interference between +the pseudospin modes due to the reflection from the sheet boundaries. The important features +are demonstrated in the single-mode conductance (Figures 6 (a),(b)). For the rather small +electrochemical potential the active mode is an edge type. The sign of the conductance is +negative, which corresponds to its inductive origin. The increase in the electrochemical +potential leads to the transformation from inductive to capacitive one (sign exchange) due to +the transformation from the edge mode to the bulk one. +As one can see, the conductivity of a graphene sheet changes its qualitative behavior +for rather small values of length. However, graphene antennas generally exhibit a resonate +behavior at much lower frequencies as well as their metallic counterparts, which is +experimentally implemented in the THz range [43,45,46,51]. Thus one can effectively exploit +the electrically tunable conductance of graphene exactly for such small sizes, where the +conventional models of conductivity become invalid due to the importance of edge effects. In + +13 + +summary, it is important to note that including edge effects in the physical modeling, opens a +new way for electrical controlling of resonant graphene antennas via the overturned +electrochemical potential by means of the gate voltage. In some cases where the +electrochemical potential varies adiabatically slow in time, it will also produce a modulation +of the THz emission. + + + + + + + + + + + + +Figure 3. The spatial distribution of the conductance in the units +/ +Drude +G +Ll . L=800nm,  =0.1eV. + +The concept of the optical conductance developed in this paper allows formulating the +effective boundary conditions for electromagnetic field at the surface of graphene sheet in the +form + + + +  + +0 +0 +, +, +, +Drude +z +z +G +x + + + + + + + + + +n H +H +n +n E (30) + +Their using leads to modification of integral equations of antenna theory and methods of their +solution. + + + + + + + + + + + + + + + + +?10~3 +2.5 +www +2 +0.5 +o.Drudemodel +0 +0.5 +-0.4 +0.3 +-0.2 +-0.1 +0.1 +0.2 +0.3 +0.4 +0.5 +X/L14 + + + + + + + + + + +Figure 4. The spatial distribution of the conductance in the units +/ +Drude +G +Ll . L=800nm,  =0.5eV. + + + + + + + + +Figure 5. The spatial distribution of the conductance in the units +/ +Drude +G +Ll . L=800nm,  =1.0eV. + +The considered scenario changes dramatically with shortening of the sheet, as shown +in Figures 6-8 (each Figure corresponds to a different value of the length for the same value +of electrochemical potential). One can clearly see the enhancement of oscillations, which +makes the Drude model invalid for such values of parameters. In fact, it becomes impossible +to introduce the conductivity concept in its ordinary meaning. The value defined by Equation +(16), has the meaning of an “effective” conductivity, which strongly depend on the +geometrical size of the sheet. The situation is rather similar to attempt to describe the optical +properties of semiconductor quantum dots via the dielectric function in the limit of weak +conferment [50] (the “effective” dielectric function strongly depends on the sample +configuration). The physical mechanism for the conductivity oscillations, is the interference +between the pseudospin modes due to the reflection from the sheet boundaries. The important +features are demonstrated in the single- mode conductivity (Figures 6 (a), (b)). For the rather +small electrochemical potential the active mode is an edge type. The sign of the conductivity +is negative, which corresponds to its inductive origin. The increase in the electrochemical +potential leads to the transformation from inductive to capacitive one (sign exchange), due to +the transformation from the edge mode to the bulk one. + +12 ×10-3 +10 +6 +Drudemodel +5 +3 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +x/ L2.02 +0.018 +0.016 +0.012 +0.01 +0.008 +0.006 +0.004 +.0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0.1 +0.2 +0.3 +0.4 +0.5 +x/L15 + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Figure 6. The spatial distribution of the conductance in the units +/ +Drude +G +Ll for different values of +length and  =0.1 eV; (a) L=2nm (single-mode regime; edge mode); (b) L=10nm (single-mode +regime; bulk mode); (c) L=50nm (three-mode regime; all modes are of bulk type). +(a) +(b) +(c) + +2.5×10-3 +XX +Drude model +2 +1.5 +0.5 +-0.5 +-0.4 +-0.3 +-0.2 +0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +X/L0.005 +0 +-0.005 +-0.01 +-0.015 +-0.02 +XX +o.- Drude model +-0.025 +-0.03 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +x/L2.7×10~3 +2.6 +2.5 +2.4 +2.3 +2.2 +xX +2.1 +e--Drude model +2 +1.9 +1.8 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +X/L16 + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Figure 7. The spatial distribution of the conductance in the units +/ +Drude +G +Ll for different values of +length.  =0.3 eV; (a) L=2nm; (b) L=10nm; (c) L=50nm. + +(a) +(b) +(c) + +0.015 +0.014 +Drude model +0.013 +0.012 +0.011 +0.01 +0.009 +0.008 +0.007 +0.006 +0.005 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +x/ L9×10-3 +8 +e--Drudemodel +7 +6 +5 +4 +3 +1 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +X/ L8 +6 +5 +3 +2 +model +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +X/L17 + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Figure 8. The spatial distribution of the conductance in the units +/ +Drude +G +Ll for different values of +length.  =1.0 eV; (a) L=2nm; (b) L=10nm; (c) L=50nm. + +(a) +(b) +(c) + +0.12 +Drude model +0.1 +0.08 + 0.06 +0.04 +0.02. +0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +X/ L0.03 +0.025 +0.02 +0.01 +0.005 +Drude model +0. +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +X/L0.03 +0.025 +0.02 + 0.015 +0.01 +0.005 +Drude +%.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +X/ L18 + +4. Conclusion and outlook +The main results of the paper can be summarized as follows: + +1) We have developed a new theory of interaction of electromagnetic field with graphene +sheet for nanoantenna applications in the THz, infrared and optical frequency ranges. The +main characteristic feature of our theory is accounting for edge effects in a self-consistent +manner. It is based on the concept of optical conductance considered as a general +susceptibility and calculated by Kubo approach. The model is based on the concept of +Dirac pseudo-spins founded via solving the boundary-value problem for the Dirac +equation with the appropriate boundary conditions satisfying the physical model, +including edge effects of the sheet; +2) The main manifestation of the importance of edge effects is demonstrated by the +inhomogeneity of the optical conductance. The amplitude and period of its oscillations +depend on the length of the sheet and on the electrochemical potential. It is defined by the +number of pseudo-spin modes supporting the conductance; +3) The developed theory is applied for the simulation of the sheet conductance in a wide +range of sample parameters ( length 2.1nm – 800nm and electrochemical potential 0.1 – +1.0 eV). It is shown, that for a length exceeding 800nm our model and the widely used +Drude model of conductivity agree to a high degree of accuracy. However, for small +geometric sizes (i.e., smaller than 50nm), the physical picture of conductivity with respect +to the Drude model changes dramatically due to the influence of edge effects. This +circumstance should be accounted for in the design of graphene-based resonant THz +antennas and other types of photonic and plasmonic nanodevices; +4) It is shown, that the qualitative distribution of the conductivity along the sheet strongly +depends on the electrochemical potential. Thus, it is possible to control the conductivity +and performance of graphene nanoantennas, by means of varying the gate voltage; +Our theory allows reformulation of the effective boundary conditions for the electromagnetic +field at the surface of the graphene sheet with accounting of the edge effects. It requires the +modification of integral equations of antenna theory and the methods of their solution. This +should be one of the subjects of future research activity as well as their application to +nanoantennas and other nanodevices. +Author Contributions: Developments of the physical models, derivation of the basis equations, +interpretation of the physical results and righting the paper have been done by T.B., T.M., O.G. and +G.S. jointly. The numerical simulations were produced by T.B. +Funding: This research was funded by NATO grant number NATO SPS-G5860 and by +H2020,project TERASSE 823878. +Institutional Review Board Statement: Not applicable. +Informed Consent Statement: Not applicable. +Data Availability Statement: Not applicable. +Conflicts of Interest: The authors declare no conflict of interest + + + + + +19 + +Appendix A. Derivation of Equation (16) + +In this Appendix we discuss the boundary-value problem for pseudo-spin defined by +Equation (14). As it was mentioned above, the pseudo-spin satisfies the Dirac equation with +the following boundary conditions; + + + + +/2 +/2 +0 +s +s +u +L +v +L + + + + [25]. It may be also +transformed into the Helmholtz equation with two special sets of boundary conditions. We +have the Dirichlet condition at the left-hand side and the impedance condition + + +/2 +0 +y +x +x L +k +u + +  + +at the right-hand side for the components u(x)(determined by the Dirac +Equation). The situation is precisely inverted for the second type, namely a Dirichlet +condition at the right-hand side and an impedance condition  + +/2 +0 +y +x +x +L +k +v + +  + +at the left- +hand side. These problems are Hermitian, whereby the eigenmodes form a complete basis. +The components u(x) and v(x) are both separately orthogonal, but are mutually non- +orthogonal, due to their coupling over the electron motion between the atoms of A and B +sublattices. The property of orthogonality is shown at Appendix C. Using completeness, +orthogonality and normalization conditions +  +  +/2 +/2 +2 +2 +/2 +/2 +1 +L +L +p +p +L +L +u +x +dx +v +x +dx + + + + + + + , we obtain +  +  + + +2 +p +p +p +u +x u +x +x +x + + + + + + + + + + + +(A.1) +  +  + + +2 +p +p +p +v +x v +x +x +x + + + + + + + + + + + +(A.2) + Starting from the xx-component and using the basis relation (11) for a=x, b=x, the matrix +element of the current density operator, one gets + + + + + + +  +  +  +  + + + + +1 +ˆ +,0 +y +y +i k +k +y +x +F +n +n +n +n +ss +j +ev +u +x v +x +v +x u +x +e +l + + + + +  + +x + +(A.3) +Next we examine the limit of + +l  by making the exchange + + +1 +1 +2 +s +n +l + + + + + + + + + + +and +the same for +, +s n + + . Summing over s,s’ and taking into account both electrons and holes +  + + +in + +nv +x + +as well and the charge carriers in two valleys K and K’, leads to. + + + + + + + + + +  + +2 +2 +2 +( , +; ) +4 +0 +( ) +; +y +y +n +y +i k +k +y y +F +xx +y +y +xx +nn +y +n +n +k +ie v +K +dk dk +e +i +f +k +  + + + + + + + + + + + +  + + + + + +  + + + + + + + + +  + + +x x +x,x + + +(A.4) +where + + +20 + + +  + +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +, +; +xx +pp +y +p +p +p +p +p +p +p +p +p +p +p +p +p +p +p +p +p +p +p +p +p +x x k +v +x v +x +u +x u +x +u +x u +x +v +x v +x +u +x v +x +v +x u +x +v +x u +x +u +x v +x + + + + + + + + + + + + + + + + + + +  + +  + +  + + + + + + + + +(A.5) + +Note, that the summation in (A.5), means including the contribution from both electrons and +holes and the valleys K, K’. The sum over electrons and holes may be transformed to the sum +over the electron states by means of their doubling. The sum of the two other terms is zero +due to the opposite sign of +  +nv +x (subject to the same +  +nu +x ). The sums over the electron +states are decomposed into the components over the two valleys, implying that +  +  +  +  +... +... +... +2 +... +K +K +K +n +n +n +n + + + + + + + + + (A.6) + Finally +invoking +these +transformations +and +using +the +well-known +identity + + +  +1 +2 +ihy +e dh +y + + + + + + + + , we obtain + + + +  + + + +  +  + + +2 +2 +2 +2 +0( ) +; +' +' +2 +0 +n +y +F +xx +y +n +n +n +k +f +ie v +K +dk +u +x +v +x +i +  + + + +  + + + + + +  + + + + + + + +x,x +x +x + +(A.7) +The above equation relates to the only existing spin-state (a real physical spin rather than a +pseudospin). Therefore, the total conductivity must be doubled, which corresponds to the +value of the conductivity given by Equation (16). Other components of the conductivity +tensor may be obtained in a similar way. For example, we have + + + + +; , +; , +yy +xx +K +K + + + + + +x x +x x and + + + + +; , +; , +0 +xy +yx +K +K + + + + + + +x x +x x +. + +Appendix B: Group velocity of pseudospins in graphene sheet + + Here we calculate the normalized group velocity of the pseudospins defined as + + +/ +g +y +y +v +k +k + + + + +  + +,based on the characteristic equation + + + + +1 +tg += +x +x +y +x +k L +f k +k +k + + +. The y- +component of the wavevector is considered as an independent variable . By taking derivative +with respect to +yk , one gets + +21 + + +2 +1 +x +y +x +y +y +f +f +k +k +k +k +k + + + + + +  + + + + (A.8) + +And by using the relation + + +2 +2 +x +y +y +k +k +k + + + +, we obtain + + + + +2 +2 +2 +2 +y +y +g +y +x +y +y +y +y +y +k +k +v +k +k +k +k +k +k +k + + + + + + + + + + + + + + + + (A.9) +For the derivative +/ +x +f +k + + +we have + + + + +2 +2 +tg +cos +x +x +x +x +x +k L +k L +k L +f +k +k + + + + + (A.10) +The trigonometric functions may be expressed through the algebraic ones using the +characteristic equation which gives + + +2 +2 +y +y +x +y +x +k +L +k +f +k +k k + + + + + + (A.11) +Expressing the group velocity from (A.9) and using (A.10), (A.11), we obtain + + + + + + + + + +2 +1 +y +y +g +y +y +y +k +k L +v +k +k +L +k + + + + + + (A.12) + +In order to determine the renormalization coefficient +n +B , we can make a similar +transformation: express + + +sin 2 +xk L thought + + +tg +xk L and apply the characteristic Equation (17), +which renders by virtue of Equation (20), + + + + +1 +2 +1 +y +yn +yn +n +y +yn +k +k +k +B +k +k L + + + + + + + + + + + + + + + + + (A.13) + with the conductivity given explicitly in Equation (21). +Appendix C: Orthogonality of pseudo-spin modes +The Equations for pseudo-spin mode with number s has the form +, +s +y +s +s +s +s +y +s +s +s +v +k v +u +x +u +k u +v +x + + + + +  + + + + + + + + + + + +, (A.14) +The similar Relation may be formulated for another mode with the number s: + +22 + +, +s +y +s +s +s +s +y +s +s +s +v +k v +u +x +u +k u +v +x + + + + + + + + + + + + +  + + + + + + + + + + + + (A.15) +Let us multiply first Equation (A.14) to +su and second Equation multiply to +sv  . Summarize +them and integrate over the interval +/2 +/2 +L +x +L + + + + . Using the boundary conditions, we +obtain +  +  +  +  +/ 2 +/ 2 +/ 2 +/ 2 +0 +L +L +s +s +s +s +s +s +L +L +u +x u +x dx +v +x v +x dx + + + + + + + + + + + + (A.16) +The similar Relation may be obtained from (A.16) by rearranging the indexes s and s. We +have +  +  +  +  +/ 2 +/ 2 +/ 2 +/ 2 +0 +L +L +s +s +s +s +s +s +L +L +u +x u +x dx +v +x v +x dx + + + + + + + + + + + + (A.17) +The eigenvalues with the different indexes are non-degenerate. The pair of Equations +(A.16),(A.17) may be considered as a system of linear algebraic equations with respect to the +integrals. The determinant of this system +s +s +s +s + + + + + + + + +is non-zero. 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Optomechanical antennas for on- +chip beam-steering, Optics Express, 2018, 26, 17 | 20 + + + diff --git a/JNE0T4oBgHgl3EQfiAH8/content/tmp_files/load_file.txt b/JNE0T4oBgHgl3EQfiAH8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..02fb5f2b5478ad11e340005aa8f8a2389caef910 --- /dev/null +++ b/JNE0T4oBgHgl3EQfiAH8/content/tmp_files/load_file.txt @@ -0,0 +1,1448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf,len=1447 +page_content='1 Theory of Edge Effects and Conductunce for Applications in Graphene- based Nanoantennas Tomer Berghau , Touvia Milo , Oded Gottlie and Gregory Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Slepya 1 1School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel 2 Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, 3200003, Israel 3 School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel Correspondence: berghaus@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='com (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' gregory_slepyan@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='com (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=') Abstract: In this paper, we develop a theory of edge effects in graphene for its applications to nanoantennas in the THz, infrared, and visible frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Its characteristic feature is self-consistence reached due the formulation in terms of dynamical conductance instead of ordinary used surface conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The physical model of edge effects is based on using the concept of Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The surface conductance is considered as a general susceptibility and is calculated via the Kubo approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In contrast with earlier models, the surface conductance becomes non-homogeneous and non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The spatial behavior of the surface conductance depends on the length of the sheet and the electrochemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Results of numerical simulations are presented for lengths in the range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 - 800nm and electrochemical potentials ranging between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It is shown that if the length exceeded 800 nm, our model agrees with the classical Drude conductivity model with a relatively high degree of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' For rather short lengths, the conductance usually exhibits spatial oscillations, which absent in conductivity and strongly affect the properties of graphene based antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The period and amplitude of such spatial oscillations, strongly depend on the electrochemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The new theory opens the way for realizing electrically controlled nanoantennas by changing the electrochemical potential may of the gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The obtained results may be applicable for the design of carbon based nanodevices in modern quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Keywords: graphene;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' edge effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' optical conductance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' nanoantennas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Introduction Innovative electromagnetic nanoantennas, which generally function at wide range of frequencies (from THz until video frequencies), play a vital role in the emerging field of photonics and plasmonics [1-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' These antennas can be used as promising tools for transforming near-field light into far-field and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Their excellent capabilities are usually utilized in a wide scope of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Among them, are traditional applications for basic elements in electronics, high-speed communications, informatics, and quantum computing [7] (in particular quantum nanomechanical qubit on carbon nanotube [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' As far as commercial applications of carbon-based nanostructures [9], one can list some novel applications such as: i) high-resolved spectroscopy [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ii) high-speed communication [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' iii) light emission and detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' iv) identification of biomolecules and medical diagnostic applications [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The design of the devices mentioned above, requires taking into account edge effects and their correct physical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It should cover a wide frequency range from microwave until optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' These recent innovations have stimulated huge interest in the theory of nanoantennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Microwave standards become invalid starting from the THz region, as the size of the antennas is miniaturized to micrometer scale [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Therefore, the common 2 model of perfect electric conductor, which is widely used in microwaves, is not suitable in the range from THz to optical frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' By manipulating the different types of finite-size (edge) effects [14-16], the conventional antenna configurations can be also used in the range from microwave to optical frequencies [1-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' However, antenna devices which operate in the terahertz range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1–10 THz), have some fundamental limitations on their applicability in practical devices [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Recent studies have pointed out that graphene is one of the best candidates for overcoming the limitations mentioned above [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' A nanoantenna in its theoretical analysis, is generally considered as a system terminated by a well-defined edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The edge geometry is one of the widely used idealizations in modern physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' For example, in all branches of classical mechanics and physics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=', elasticity, hydrodynamics, acoustics, electrodynamics), the edge has a configurational character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It corresponds to a point or contour at the surface of the body on which the normal vector is usually undefined (half-plane, wedge, cone, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Taking the edge position into consideration, allows us to simplify the boundary conditions and employ certain analytical techniques such as separation of variables, using for example special orthogonal coordinate systems [17] or the Wiener-Hopf method [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Such analytical solutions are especially attractive for the qualitative analysis of novel types of problems involving interacting fields of different physical origin (for example, plasmonics and optoelectrofluidics [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' However, such idealization may lead to the loss of uniqueness of the problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The reason for it is that the placement of an arbitrary point singularity at the edge, generally does not violate the governing wave equation as well as the boundary and radiation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The analytic solutions thus obtained, are correct from the mathematical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' However, they may turn to be physically incorrect because due to lack of uniqueness, they may correspond to another source of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In order to obtain the corresponding of unique solution, one must enforce the condition of finite (bounded) energy over an arbitrary area (finite extend) of space (Meixner condition) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Such an approach usually leads to the creation of field singularities at the vicinity of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Note, that the edge-like configuration does not change the constitutive properties of the medium (for example, such as a perfect conductor or a perfect insulator in classical electrodynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The extensive recent progress in nanotechnologies, lead to the discovery of novel artificial types of condensed matter, such as graphene [21], carbon nanotubes (CNTs) [22], Weyl and Dirac semimetals [23] and topological insulators [24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Consequently, a great number of fundamental physical problems were reconsidered and in particular, but nevertheless most of them failed to consider the important edge concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The edge for the different types of nanostructures corresponds to the spatial area in the vicinity of the sample boundary which size is rather large compared with the inter-atomic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' However, the mechanism of electronic transport at the edge is dramatically different from the corresponding properties of the bulk region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' One of the reasons for this disparity is the existence of new types of quantum states strongly confined to the vicinity of the edge, which are able to produce novel physical properties such as topological order (so called, edge states [16,21,25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Such transport mechanisms manifest themselves in the special optical and optomechanical properties of different types of metamaterials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=', carbon based nanostructures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The electrical properties of macroscopic structures may be described both in terms of conductivity and conductance [16], which are equivalent and coupled via the simple constant coefficient defined by the size values of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' As it was noted in [16], the conductivity for nanomaterials (including graphene) is a well-defined value only when the sample is enough large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It is clear from intuitive point of view, that the electric current becomes homogeneous and insensitive to boundaries of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' When the sample’s size is reduced, the current becomes non-homogeneous and non-local with respect to the field 3 variations in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Then, the concept of conductivity loses its meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The behavior of the electrons in nanoantennas becomes sensitive to the feeding lines, detectors, modulators and the edges due to the quantum-mechanical interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' As we will show, exactly optical conductance will be suitable parameter for formulation of the effective boundary conditions for electromagnetic field in nanoantennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In contrast with conductivity, it is non- homogeneous and not coupled via simple relation with optical conductivity (which is homogeneous value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It is described by the Kubo approach instead of Boltzmann’s transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The edge areas play the main role in forming the emission in classical radio-frequency antennas [26] and optical nanoantennas [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' One of the promising types of nanoantennas that operate in the THz and optical frequency range, are based on carbon-based nanostructures (graphene, CNTs) [28-31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The physical mechanism of their radiation, is based on the existence of a strongly retarded surface plasmon predicted in [28, 32], experimentally observed in [33] and used for interpretation of the intriguing measurements of the THz conductivity peak [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' These works use different models for the charge transport [28, 32, 34, 35], but all of them are not self-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In another words, the boundary-value problem for the Maxwell equations is formulated for the real geometry of the object, while the value of conductivity is introduced as a phenomenological parameter which is determined by using a model corresponding to an infinitely large structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Of course, such an approach appears to be attractive since it allows to reach, simplification due to the separation of the EM-field and charge transport equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Such an approach keeps the self-consistent requirement for planar structures with the Fresnel transmission-reflection condition of EM-fields (graphene films, semitransparent mirrors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=') [37-40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' However, it may not be adequate for the detailed description of EM-field scattering and antenna emission in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' One can clearly expect that the error of such a non-consistent approach may be ignored as being very small, providing the size of the system is rather large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' However, it means, that the effect of the Fresnel transmission-reflection dominates the field forming, whereas the antenna efficiency decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Nevertheless, it is impossible to say a priori what is the validity bound of such a simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' This problem becomes especially relevant for graphene-based structures, because of the corresponding large geometrical size disparity for applications in different graphene devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The advanced graphene synthesis methods make it possible to grow graphene samples from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1nm to few centimeters [6,41-46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The detail description of optical graphene conductivity requires a self-consistent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Such a self-consistent approach determines the field acting on electrons produced over their motion, by taking into consideration the finite-size configuration and using an appropriate microscopic model for the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The formulation of such a self-consistent analysis is one of the main contributions of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' One of the important results of this paper, is analyzing the effect of the edges (shape of finite extend) and determine the corresponding error when self-consistency is not prevailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It is important to note;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' that edge effects do not only add up to the quantitative differences in the value of conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Edge effects are also able to dramatically affect the special physical mechanism of charge transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It makes required the formulation of the theory in terms of optical conductance, which is important for applications in antenna design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The paper is organized as follows: models of the edge of a graphene ribbon based on the Dirac-fermion concept, are first discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Thereafter, based on using the Kubo approach and the concept of general susceptibility [47], we analytically obtain an expression for the surface conductance of a terminated (finite) graphene sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Results of numerical simulations together with a discussion are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Conclusion and outlook are finally given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Optical conductance of a terminated graphene sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Problem statement: (a) Geometry of a graphene sheet with zigzag edge configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' (b) Fermi-Dirac cones for the model of pseudospin dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 Kubo approach for optical conductance of graphene In the following we will use the Kubo approach for conductance calculation as a universal technique which couples the generalized forces of arbitrary physical origin with the responses of correspondent origin via general susceptibilities [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Towards this goal it is convenient to define the forces as tangential components of electric field and responses as components of current densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Thus, the general susceptibilities are defined by a 2 2 \uf0b4 matrix where \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 , , 1,2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , a ab b b j K E d \uf077 \uf077 \uf077 \uf03d \uf0a2 \uf0a2 \uf0a2 \uf03d \uf02d\uf0e5 \uf0f2 x x x x x (1) and \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , ˆ ˆ ˆ ˆ , 0, 0, , ab i t a b b a K e e x t x x x t dt \uf077 \uf077 \uf077 \uf0a5 \uf0a2 \uf03d \uf0a2 \uf0a2 \uf02d \uf0f2 x x x x x x (2) Here \uf028 \uf029 \uf028 \uf029 ˆ a a j j \uf03d x x , \uf028 \uf029 \uf028 \uf029 ˆ ˆ , , a a j t i ex t \uf077 \uf03d x x , \uf028 \uf029 ˆ aj x represents the observable current density, \uf028 \uf029 ˆ , ax t x is an operator of charge displacement per unit area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The general susceptibility tensor \uf028 \uf029 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , ab K \uf077 \uf0a2 x x depends on the geometry of the sample as well as the electronic properties of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Therefore, exactly it has a physical meaning of non-local optical conductance (in the units 2 / e ) similar to the dc conductance of graphene in [16] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The next step is the transformation of Equation (2) to a form that is convenient for applications in graphene electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" Thus one can write (b) (a) Energy Valley K' Valley KA-atoms B-atomsremoved B-atoms 1 A-atoms removed5 \uf028 \uf029 \uf028 \uf029 0 ˆ ˆ ;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , i t ab ab ab ie K e A B dt \uf077 \uf077 \uf0a5 \uf0a2 \uf03d \uf02d \uf02d \uf0f2 x x (3) where \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ˆ ˆ ˆ ˆ ˆ ˆ , 0, , 0, , ab ab a b b a A j t x B x j t \uf0a2 \uf0a2 \uf03d \uf03d x x x x and \uf028 \uf029 \uf028 \uf029 0 0 ˆ ˆ ˆ ˆ ab ab ab ss s ss A Tr A A \uf072 \uf072 \uf03d \uf03d \uf0e5 (4) where 0ˆ\uf072 is the density matrix of free motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The diagonal matrix elements of the operator ˆ ab A are defined by \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ˆ ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ab a b s s ss s ss A j t x \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf03d \uf0e5 x x (5) and following Equation (4) one gets \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 0 ˆ ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ab a b ss s s s s ss A j t x \uf072 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf03d \uf0d7 \uf0e5\uf0e5 x x (6) Here \uf07b \uf07d ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' y s p k \uf03d denotes the combinative discrete-continuous index (p is the discrete number of the state and yk is its continuous wave-number over the y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The temporal behavior in the linear approximation can be expressed as \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 / ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' s s i t a a ss ss j t j e \uf065 \uf065 \uf0a2 \uf02d \uf02d \uf0a2 \uf0a2 \uf03d x x and as a result we obtain \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 0 ˆ ˆ ˆ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' s s i t ab a b ss s s s s ss A j x e \uf065 \uf065 \uf072 \uf0a2 \uf02d \uf02d \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf03d \uf0d7 \uf0e5\uf0e5 x x (7) A similar relation may be also obtained for ˆ ab B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' namely \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 0 ˆ ˆ ˆ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' s s i t ab b a ss s s s s ss B x j e \uf065 \uf065 \uf072 \uf0a2 \uf02d \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf03d \uf0d7 \uf0e5\uf0e5 x x (8) Combining Equations (7) and (8) with Equation (3) lead to \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 0 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" , 4 ˆ ˆ ˆ ˆ 0, 0, 0, 0, s s s s ab s s i i t t i t a b b a ss s s ss ss s s i K e j x e x j e dt \uf065 \uf065 \uf065 \uf065 \uf077 \uf077 \uf077 \uf072 \uf0a2 \uf0a2 \uf0a2 \uf0a5 \uf02d \uf02d \uf02d \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf03d \uf0d7 \uf0ec \uf0fc \uf0a2 \uf0a2 \uf02d \uf0ed \uf0fd \uf0ee \uf0fe \uf0e5\uf0e5 \uf0f2 x x x x x x (9) For shortness, we will omit the initial value at time t=0 in the matrix elements of ( \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ' ' ˆ ˆ 0, a a ss ss j j \uf0ae x x , etc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Using elementary integration yields 6 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 0 ˆ ˆ ˆ ˆ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' a b b a s s ss ss s s ab ss s s s s s s j x x j ie K \uf077 \uf072 \uf077 \uf065 \uf065 \uf077 \uf065 \uf065 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0ec \uf0fc \uf0a2 \uf0a2 \uf0ef \uf0ef \uf0a2 \uf03d \uf02d \uf02d \uf0ed \uf0fd \uf02d \uf02d \uf02b \uf02d \uf0ef \uf0ef \uf0ee \uf0fe \uf0e5\uf0e5 x x x x x x (10) Introducing the following change ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" s s s s \uf0a2 \uf0a2 \uf0ae \uf0ae in the second term of Equation (10) and using the relation \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ' ' ' ˆ ˆ / a s s a ss ss j ie x \uf065 \uf065 \uf03d \uf02d x x ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' the latter can be written as \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 0 0 ˆ ˆ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , a b ss s s ab ss s s s s s s s s j j K i \uf077 \uf072 \uf072 \uf077 \uf065 \uf065 \uf065 \uf065 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf03d \uf02d \uf02d \uf02d \uf02d \uf02d \uf0e5\uf0e5 x x x x (11) where \uf028 \uf029 \uf028 \uf029 / 0 ( ) 1/ 1 B k T ss f e \uf065 \uf06d \uf072 \uf065 \uf02d \uf03d \uf03d \uf02b denotes the Fermi-distribution and \uf06d is the electrochemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The electrochemical potential is defined here by the concentration of electrons/holes according to the following relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' \uf028 \uf029 2 1 0 / F n v \uf070 \uf06d \uf02d \uf03d [37-40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' This potential vanishes for a perfectly clean graphene at zero temperature [16] and may be effectively controlled via the gate voltage [37-40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Because of the Hermittivity nature of the current operator, we have \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ˆ ˆ b b s s ss j j\uf02a \uf0a2 \uf0a2 \uf03d x x where the upper asterisk denotes complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Therefore, one can also write Equation (11) in the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' compact form \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" , ' 0 ab ab ab i K \uf077 \uf077 \uf077 \uf03d \uf02d \uf050 \uf02d \uf050 x x (12) where \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ' ˆ ˆ ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' a b ss s s ab s s s s s s j j f f \uf077 \uf065 \uf065 \uf077 \uf065 \uf065 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf050 \uf03d \uf02d \uf02d \uf02d \uf0e5\uf0e5 x x (13) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Zigzag edges In this Subsection we will apply the general result to the edge of a zigzag type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The electronic properties of the ribbon are described by the model of Dirac fermions corresponding to a tight-binding model on a two-dimensional honeycomb lattice [16,25,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' We will use the zigzag-type of boundary conditions for Dirac fermions on a terminated (finite) lattice derived in [16,25,48], since it was demonstrated that this type of boundary condition can be generally applied to a terminated honeycomb lattice in the case of electron-hole symmetry [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The A and B atoms are coupled in every spinor modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' From physical point of view, the interpretation of electronic transport may be considered as a motion from A atoms to B ones and vice versa, while A-A and B-B motions are forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The electron transport for a zigzag edge in valleys K and K’ is independent and will be considered separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The wave function is expressed as a linear combination of the eigen 4D spinors 7 The pseudo-spinor modes for valley K have the following form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 0 0 0 0 y s A s s i k y t B s s Ks t u x t v x t e \uf077 \uf02d \uf059 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf059 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf03d \uf03d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 x x Ψ x (14a) \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 0 0 0 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' y s i k y t K s A s s B s s t e t u x t v x \uf077 \uf02d \uf0a2 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf03d \uf03d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf02d\uf059 \uf02d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf02d\uf059 \uf02d \uf0e8 \uf0f8 \uf0e8 \uf0f8 Ψ x x x (14b) The 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0e6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf03d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf03d \uf0b1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf02d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0e7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0e8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0f8 \uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0fb\uf0fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='(14c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' \uf028 \uf029 1 sin 2 1 2 px b s px L B L \uf06b \uf06b \uf02d \uf0e6 \uf0f6 \uf03d \uf02d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 (14d) \uf028 \uf029 1 1 2 sinh 2 1 2 L e s L B e L \uf068 \uf068 \uf068 \uf02d \uf0e6 \uf0f6 \uf03d \uf02d \uf0e7 \uf0f7 \uf0e8 \uf0f8 (14e) where l represents the normalization length in the y-direction (which goes to infinity in the final result),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' sA is a normalization constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' y xp k k are the wavenumbers which are defined by the dispersive relation \uf028 \uf029 2 xp ik L y xp y xp k ik k ik e \uf02d \uf02d \uf03d \uf02b and \uf07b \uf07d ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' y s p k \uf03d is the combinative discrete-continuous index (p is the discrete number with respect to the x-axis and yk is the 8 continuous wave-number over the y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' This dispersion relation has an infinite number of real roots with a single point of concentration at infinity (confined modes) and one imaginary root that corresponds to the edge state [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Transformation to the edge state from confined modes in Equations (14a)-(14e) and characteristic equation may be done via exchange px \uf06b \uf068 \uf0ae .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='The two signs in (14) correspond to electrons and holes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' As one can see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' (14) satisfies the Dirac equation and is subject to the following boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 /2 /2 0 s s u L v L \uf02d \uf03d \uf03d [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='The components u(x) and v(x) are separately orthogonal, while non-orthogonal mutually due to their coupling over electron motion between the atoms of A and B sub-lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Since the expression for the ac-conductivity with a zigzag edge is isotropic, we have \uf028 \uf029 \uf028 \uf029 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , ab ab K K \uf077 \uf064 \uf077 \uf0a2 \uf0a2 \uf03d x x x x ,where the general susceptibility is \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" , ' 0 K i \uf077 \uf077 \uf077 \uf02d \uf03d \uf02d \uf050 \uf02d \uf050 x x , with \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ' ˆ ˆ ss s s s s s s s s f f \uf077 \uf065 \uf065 \uf077 \uf065 \uf065 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf050 \uf03d \uf02d \uf02d \uf02d \uf0e5\uf0e5 j x j x (15) Here represents the matrix element of the current operator, σ denotes the xy-vector of the Pauli matrices, 2 2 s F xp y v k k \uf065 \uf03d \uf0b1 \uf02b is the energy of the s-th state and , F e v correspond to the electron charge and Fermi velocity respectively." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The general susceptibility may be presented here as the sum of two components of different origin \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" , ' ;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" , ' ;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" , ' Inter Intra K K K \uf077 \uf077 \uf077 \uf03d \uf02b x x x x x x , where \uf028 \uf029 \uf028 \uf029 1 ;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" , ' Inter K i \uf077 \uf077 \uf077 \uf02d \uf03d \uf02d \uf050 x x corresponds to the interband motion which may be ignored omitted for rather low (THz) frequencies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The second term \uf028 \uf029 \uf028 \uf029 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" ' 0 Intra K i \uf077 \uf077\uf02d \uf03d \uf050 x x corresponds to the intraband motion and leads to the common conductivity law \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 Intra i x \uf073 \uf03d \uf02d j x E x ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' where the surface conductance is found by substituting the pseudospinor modes (14) into (15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' which renders \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 2 2 ( ) 2 0 s n y Intra F s s y n k ie v f x u x v x dk i \uf065 \uf065 \uf065 \uf065 \uf073 \uf070 \uf077 \uf065 \uf0a5 \uf03d \uf03d \uf02d\uf0a5 \uf0b6 \uf0bb \uf02d \uf0d7 \uf02b \uf02b \uf0b6 \uf0e5 \uf0f2 (16) (for details of deriving Equation (16) see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The explicit expression for the conductivity given in (16) is spatially inhomogeneous (x-depended) and is a manifestation of edge effect and incorporating a self-consistent description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Approximation for zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Let us next consider the important case of the conductance at zero temperature, which opens the way for further simplification of Equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In this case the Fermi distribution may be replaced by a step function and its derivative in (16) can be transformed into a Dirac delta function \uf028 \uf029 / f \uf065 \uf064 \uf065 \uf06d \uf0b6 \uf0b6 \uf03d \uf02d \uf02d , which allows an integration of (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Performing the integration in 9 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Illustration with respect to the zero-temperature approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Black lines correspond to confined modes, red line corresponds to the edge mode, and blue line corresponds to the first confined mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The black point corresponds to the critical wavenumber in which the mutual transformation of edge mode to the first confined mode takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Dotted horizontal line corresponds to the given electrochemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The values yn k denote the roots of Equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' possible in terms of the discrete number of roots of the following transcendental equation \uf028 \uf029 yk \uf065 \uf06d \uf03d ( qualitatively depicted in Figure 2), which may be also written as \uf028 \uf029 2 2 2 2 tg yn yn yn k k L k \uf06d \uf06d \uf02d \uf02d \uf03d (17) The value / F v \uf06d \uf06d \uf03d means normalizing the electrochemical potential by the electron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The corresponding edge mode may be obtained by the formal exchange yn k ik \uf0ae (such root exists only under special conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' For the integration over yk we use the following property of the Dirac delta-function \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 yn y y y n yn F k k F k dk k \uf064 \uf065 \uf06d \uf065 \uf0a5 \uf02d\uf0a5 \uf02d \uf03d \uf0a2 \uf0e5 \uf0f2 (18) where \uf028 \uf029 yk \uf065\uf0a2 denotes the y-component of the group velocity of pseudospin (prime means the derivative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The final explicit result for the conductivity can be written as \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 1 2 2 2 2 2 2 1 2 2 sin sin 2 2 0 N n yn yn n yn Intra B L L k x k x L k e x i i \uf06d \uf06d \uf065 \uf06d \uf073 \uf070 \uf077 \uf02b \uf03d \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf02d \uf02d \uf02b \uf02d \uf02b \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0a2 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0eb \uf0fb \uf03d \uf02b \uf0e5 (19) kynL kynL 20 18 16 14 12 10 8 6 4 2 0 25 20 15 10 5 0 5 10 15 20 25 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='L10 The values of yn k depend on the electrochemical potential and satisfy the characteristic Equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The normalized coefficients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='in (19) are defined by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf028 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf029 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf028 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf029 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='sin 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='yn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='yn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='yn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='k L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='k L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf06d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf065 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf065 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf06d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf02d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0e6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf02d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0e7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf03d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf02d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf03d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0e7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf02d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf02d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0e7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0e8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='(20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='Equation (19) may be finally transformed to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf028 \uf029 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf028 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf029 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf028 \uf029 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf028 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf029 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='yn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='yn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='yn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='Intra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf06d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf06d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf06d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf06d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf073 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf070 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf077 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf02b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf03d ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf03d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf02b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='\uf0e5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='(21) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='(for details see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The limit of an infinite sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In this Subsection we show that our model reduces to the familiar Drude relation for the conductivity in infinitely wide sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Taking the limit L \uf0ae \uf0a5 , and using the following transformation\uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' x n L dk \uf070 \uf0a5 \uf02d \uf02d \uf02d\uf0a5 \uf0ae \uf0e5 \uf0f2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Equation (16) yields \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 2 0 ( ) 1 1 1 cos 2 cos 2 2 2 s n y Intra F x x x y k ie v x i f k x L k x L dk dk \uf065 \uf065 \uf065 \uf073 \uf070 \uf077 \uf065 \uf065 \uf0a5 \uf0a5 \uf03d \uf03d \uf02d\uf0a5 \uf02d\uf0a5 \uf0bb \uf02d \uf0d7 \uf02b \uf0b6 \uf0e9 \uf0f9 \uf02d \uf02d \uf02d \uf02b \uf0ea \uf0fa \uf0b6 \uf0eb \uf0fb \uf0f2 \uf0f2 (22) Introducing the polar variables sin xk \uf065 \uf03d \uf046, cos yk \uf065 \uf03d \uf046, we obtain x y dk dk d d \uf065 \uf065 \uf03d \uf046 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Using the zero temperature approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' we exchange the Fermi-distribution derivative by the Dirac delta-function and integrate over \uf065 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' so that Equation (22) becomes \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 2 2 0 2 0 1 1 1 cos 2 cos cos 2 cos 2 2 Intra ie x i x L x L d \uf070 \uf06d \uf073 \uf070 \uf077 \uf06d \uf06d \uf0bb \uf0d7 \uf02b \uf0e9 \uf0f9 \uf02d \uf02d \uf046 \uf02d \uf02b \uf046 \uf046 \uf0ea \uf0fa \uf0eb \uf0fb \uf0f2 (23) Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' using the addtion theorem for Bessel functions [49] \uf028 \uf029 cos i m im m m e i J e \uf072 \uf072 \uf0a5 \uf046 \uf0b1 \uf046 \uf03d\uf02d\uf0a5 \uf03d \uf0e5 (24) we integrate (23) and obtain 11 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 0 0 2 1 1 1 2 2 0 2 2 Intra ie x J x L J x L i \uf06d \uf073 \uf06d \uf06d \uf070 \uf077 \uf0e9 \uf0f9 \uf0bb \uf02d \uf02d \uf02d \uf02b \uf0ea \uf0fa \uf02b \uf0eb \uf0fb (25) If the observation point x (of under the assumption of large L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' is placed rather far from the edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' the arguments of Bessel functions become large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' so that the asymptotic relations [49] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In this case, the last two terms in (25) become indefinitely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The first term, which is exactly equal to the Drude conductivity \uf028 \uf029 2 2 / 0 Drude G ie i \uf06d \uf070 \uf077 \uf03d \uf02b , becomes dominate over the whole area of the ribbon excluding the narrow vicinities of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Thus, it is demonstrated that in this limit our model asymptotically reduces to the classical expression for the Drude conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Optical conductance for the edge with infinite-mass boundary condition This problem is of special interest in connection with a suspended sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The interaction between the edges of the sheet with the electrodes, results in the creation of electrostatic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Following [13], the electron-hole symmetry, which is generally restricted for boundary conditions of a zigzag or armchair types is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It may be considered as a manifestation of a staggered potential at zigzag boundaries, which may change the nature of the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' For, infinitely large value of potential it leads to an “infinite-mass” boundary condition, which may be written as \uf028 \uf029 \uf028 \uf029 /2 /2 s s u L v L \uf0b1 \uf03d \uf02d \uf0b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The corresponding eigen pseudospins are then given by Equation (14) with \uf028 \uf029 \uf028 \uf029 2 2 1 1 2 2 s s xn xn i k x i k x n su x e e L \uf071 \uf071 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf02d \uf02d \uf02d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e6 \uf0f6 \uf03d \uf02d \uf02d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 (26) \uf028 \uf029 \uf028 \uf029 2 2 1 1 2 2 s s xn xn i k x i k x n sv x e e L \uf071 \uf071 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf02d \uf02b \uf02b \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e6 \uf0f6 \uf03d \uf02d \uf02d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 (27) where / xn k n L \uf070 \uf03d and 2 2 s y xn i y xn k ik e k k \uf071 \uf02b \uf03d \uf02b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In order to calculate the conductance, we use the Kubo approach with the pseudo spinors defined in (26) and (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The difference from the zigzag edge formulation, is due to the lack of orthogonality inf (26) and (27), which leads to a non-local conductance (spatial dispersion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The conductance operator does not add up to the convolution form because of the non-homogeneity of the finite-length structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In the case of a rather wide sheet the non- local component becomes relatively small and may be usually ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In this particular case the conductance relates to Equation (16) with the pseudospins given given in (26) and (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Numerical results and discussion The optical properties of graphene are generally defined by the geometrical size of the sample and the value of the electrochemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' These parameters may vary over a wide range and thus are of practical interest to nanoantenna design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The recent advances in graphene technology make it possible to produce graphene samples from few nanometers to few centimeters [6,7,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The electrochemical potential may also vary over the interval 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='0 \uf06d \uf03c \uf03c eV, by doping the sample or by applying gate voltage [6,7,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In this Section we will 12 present some numerical results of conductivity simulations for a wide range of physical parameters, based on the theory developed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' One of the main results of the present study according to Equations (16) and (19), is that the non-homogeneity of conductance is mainly due to the edge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The conductance distribution is controlled by the parameter / F L L v \uf06d \uf06d \uf03d , which defines the number of modes supported by the conductance value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Figures 3-5 present the normalized conductance distribution for a rather large length 800 L \uf03d nm and for different values of the electrochemical potential (increasing \uf06d corresponds to increasing number of modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The qualitative behavior of the conductance is the same in all these Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The conductance oscillates with respect to the spatial variable and decreases near the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The amplitude and period of oscillations decrease with increasing of the electrochemical potential \uf06d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The average value of the conductance (to a high degree of accuracy), corresponds to the classical Drude model of conductivity, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' One can anticipate that such small oscillations are unable to manifest themselves in the scattering of electromagnetic waves, due to the smallness of their period compared with the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Thus, we may conclude that the Drude model can be applied for this range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The considered scenario changes dramatically with shortening of the sheet, as shown in Figures 6-8 (each Figure corresponds to a different value of the length for the same value of electrochemical potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' One can clearly see the enhancement of oscillations, which makes the Drude model invalid for such values of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In fact, it becomes impossible to introduce the conductivity concept in its ordinary meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' As it was mentioned above, the value defined by Equation (16) has the meaning of optical conductance, which strongly depend on the geometrical size of the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It may be coupled with Drude conductivity by the Relation \uf028 \uf029 \uf028 \uf029 1 Drude Intra x L x G \uf073 \uf051 \uf03d (28) where \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 1 2 2 2 2 2 2 1 1 sin sin 2 2 1 N yn yn n yn L L x k x k x k L \uf06d \uf06d \uf06d \uf02b \uf03d \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf051 \uf03d \uf02d \uf02d \uf02b \uf02d \uf02b \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf02d \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0eb \uf0fb \uf0e5 (29) is x-dependent coefficient, in which the configuration of the sample manifests itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The situation is rather similar to the electron transport in graphene in dc field (the concepts of conductance and conductivity discussed and compared in [16,50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The physical mechanism for the conductivity oscillations is the interference between the pseudospin modes due to the reflection from the sheet boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The important features are demonstrated in the single-mode conductance (Figures 6 (a),(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' For the rather small electrochemical potential the active mode is an edge type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The sign of the conductance is negative, which corresponds to its inductive origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The increase in the electrochemical potential leads to the transformation from inductive to capacitive one (sign exchange) due to the transformation from the edge mode to the bulk one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' As one can see, the conductivity of a graphene sheet changes its qualitative behavior for rather small values of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' However, graphene antennas generally exhibit a resonate behavior at much lower frequencies as well as their metallic counterparts, which is experimentally implemented in the THz range [43,45,46,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Thus one can effectively exploit the electrically tunable conductance of graphene exactly for such small sizes, where the conventional models of conductivity become invalid due to the importance of edge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In 13 summary, it is important to note that including edge effects in the physical modeling, opens a new way for electrical controlling of resonant graphene antennas via the overturned electrochemical potential by means of the gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In some cases where the electrochemical potential varies adiabatically slow in time, it will also produce a modulation of the THz emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The spatial distribution of the conductance in the units / Drude G Ll .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' L=800nm, \uf06d =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The concept of the optical conductance developed in this paper allows formulating the effective boundary conditions for electromagnetic field at the surface of graphene sheet in the form \uf05b \uf05d \uf028 \uf029\uf05b \uf05d 0 0 , , , Drude z z G x \uf03d\uf02b \uf03d\uf02d \uf02d \uf03d \uf051 \uf0e9 \uf0f9 \uf0eb \uf0fb n H H n n E (30) Their using leads to modification of integral equations of antenna theory and methods of their solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='10~3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 www 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='Drudemodel 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 X/L14 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The spatial distribution of the conductance in the units / Drude G Ll .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' L=800nm, \uf06d =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The spatial distribution of the conductance in the units / Drude G Ll .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' L=800nm, \uf06d =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='0eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The considered scenario changes dramatically with shortening of the sheet, as shown in Figures 6-8 (each Figure corresponds to a different value of the length for the same value of electrochemical potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' One can clearly see the enhancement of oscillations, which makes the Drude model invalid for such values of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' In fact, it becomes impossible to introduce the conductivity concept in its ordinary meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The value defined by Equation (16), has the meaning of an “effective” conductivity, which strongly depend on the geometrical size of the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The situation is rather similar to attempt to describe the optical properties of semiconductor quantum dots via the dielectric function in the limit of weak conferment [50] (the “effective” dielectric function strongly depends on the sample configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The physical mechanism for the conductivity oscillations, is the interference between the pseudospin modes due to the reflection from the sheet boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The important features are demonstrated in the single- mode conductivity (Figures 6 (a), (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' For the rather small electrochemical potential the active mode is an edge type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The sign of the conductivity is negative, which corresponds to its inductive origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The increase in the electrochemical potential leads to the transformation from inductive to capacitive one (sign exchange), due to the transformation from the edge mode to the bulk one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 12 ×10-3 10 6 Drudemodel 5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 x/ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='004 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 x/L15 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The spatial distribution of the conductance in the units / Drude G Ll for different values of length and \uf06d =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 eV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' (a) L=2nm (single-mode regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' edge mode);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' (b) L=10nm (single-mode regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' bulk mode);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' (c) L=50nm (three-mode regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' all modes are of bulk type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' (a) (b) (c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5×10-3 XX Drude model 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 X/L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='005 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 X/ L8 6 5 3 2 model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 X/ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='005 Drude model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 X/L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='005 Drude %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5 X/ L18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Conclusion and outlook The main results of the paper can be summarized as follows: 1) We have developed a new theory of interaction of electromagnetic field with graphene sheet for nanoantenna applications in the THz, infrared and optical frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The main characteristic feature of our theory is accounting for edge effects in a self-consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It is based on the concept of optical conductance considered as a general susceptibility and calculated by Kubo approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The model is based on the concept of Dirac pseudo-spins founded via solving the boundary-value problem for the Dirac equation with the appropriate boundary conditions satisfying the physical model, including edge effects of the sheet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 2) The main manifestation of the importance of edge effects is demonstrated by the inhomogeneity of the optical conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The amplitude and period of its oscillations depend on the length of the sheet and on the electrochemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It is defined by the number of pseudo-spin modes supporting the conductance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 3) The developed theory is applied for the simulation of the sheet conductance in a wide range of sample parameters ( length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1nm – 800nm and electrochemical potential 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='0 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It is shown, that for a length exceeding 800nm our model and the widely used Drude model of conductivity agree to a high degree of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' However, for small geometric sizes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=', smaller than 50nm), the physical picture of conductivity with respect to the Drude model changes dramatically due to the influence of edge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' This circumstance should be accounted for in the design of graphene-based resonant THz antennas and other types of photonic and plasmonic nanodevices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 4) It is shown, that the qualitative distribution of the conductivity along the sheet strongly depends on the electrochemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Thus, it is possible to control the conductivity and performance of graphene nanoantennas, by means of varying the gate voltage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Our theory allows reformulation of the effective boundary conditions for the electromagnetic field at the surface of the graphene sheet with accounting of the edge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It requires the modification of integral equations of antenna theory and the methods of their solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' This should be one of the subjects of future research activity as well as their application to nanoantennas and other nanodevices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Author Contributions: Developments of the physical models, derivation of the basis equations, interpretation of the physical results and righting the paper have been done by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=', O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The numerical simulations were produced by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Funding: This research was funded by NATO grant number NATO SPS-G5860 and by H2020,project TERASSE 823878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Institutional Review Board Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Informed Consent Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Data Availability Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Conflicts of Interest: The authors declare no conflict of interest 19 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Derivation of Equation (16) In this Appendix we discuss the boundary-value problem for pseudo-spin defined by Equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' As it was mentioned above, the pseudo-spin satisfies the Dirac equation with the following boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 /2 /2 0 s s u L v L \uf02d \uf03d \uf03d [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It may be also transformed into the Helmholtz equation with two special sets of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' We have the Dirichlet condition at the left-hand side and the impedance condition \uf028 \uf029 /2 0 y x x L k u \uf03d \uf02d \uf0b6 \uf03d at the right-hand side for the components u(x)(determined by the Dirac Equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The situation is precisely inverted for the second type, namely a Dirichlet condition at the right-hand side and an impedance condition \uf028 \uf029 /2 0 y x x L k v \uf03d\uf02d \uf02b \uf0b6 \uf03d at the left- hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' These problems are Hermitian, whereby the eigenmodes form a complete basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The components u(x) and v(x) are both separately orthogonal, but are mutually non- orthogonal, due to their coupling over the electron motion between the atoms of A and B sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The property of orthogonality is shown at Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Using completeness, orthogonality and normalization conditions \uf028 \uf029 \uf028 \uf029 /2 /2 2 2 /2 /2 1 L L p p L L u x dx v x dx \uf02d \uf02d \uf02b \uf03d \uf0f2 \uf0f2 , we obtain \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 p p p u x u x x x \uf064 \uf02a \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf03d \uf02d \uf0e5 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='1) \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 p p p v x v x x x \uf064 \uf02a \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf03d \uf02d \uf0e5 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='2) Starting from the xx-component and using the basis relation (11) for a=x, b=x, the matrix element of the current density operator, one gets \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 ˆ ,0 y y i k k y x F n n n n ss j ev u x v x v x u x e l \uf0a2 \uf02d \uf0a2 \uf0a2 \uf0a2 \uf03d \uf02b x (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='3) Next we examine the limit of l \uf0ae \uf0a5by making the exchange \uf028 \uf029 1 1 2 s n l \uf070 \uf0a5 \uf02d \uf02d \uf02d\uf0a5 \uf0ae \uf0e5 \uf0e5 \uf0f2 and the same for , s n \uf0a2 \uf0a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Summing over s,s’ and taking into account both electrons and holes \uf028 \uf029 \uf028 \uf029 in nv x \uf0b1 as well and the charge carriers in two valleys K and K’, leads to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029\uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 2 ( , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' ) 4 0 ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' y y n y i k k y y F xx y y xx nn y n n k ie v K dk dk e i f k \uf065 \uf065 \uf077 \uf070 \uf077 \uf065 \uf065 \uf0a5 \uf0a5 \uf0a2 \uf0a2 \uf02d \uf02d \uf02d\uf0a5 \uf02d\uf0a5 \uf0a2 \uf0a2 \uf03d \uf0a2 \uf0a2 \uf0bb \uf02d \uf0d7 \uf0d7 \uf02b \uf0b6 \uf0a2 \uf0d7 \uf04c \uf0b6 \uf0f2 \uf0f2 \uf0e5 \uf0e5 x x x,x (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='4) where 20 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' xx pp y p p p p p p p p p p p p p p p p p p p p p x x k v x v x u x u x u x u x v x v x u x v x v x u x v x u x u x v x \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf04c \uf03d \uf0a2 \uf0a2 \uf02b \uf0a2 \uf0a2 \uf02b \uf0a2 \uf0a2 \uf02b \uf0a2 \uf0a2 \uf0e5 \uf0e5 \uf0e5 \uf0e5 \uf0e5 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5) Note, that the summation in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='5), means including the contribution from both electrons and holes and the valleys K, K’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The sum over electrons and holes may be transformed to the sum over the electron states by means of their doubling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The sum of the two other terms is zero due to the opposite sign of \uf028 \uf029 nv x (subject to the same \uf028 \uf029 nu x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The sums over the electron states are decomposed into the components over the two valleys, implying that \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' K K K n n n n \uf0a2 \uf03d \uf02b \uf03d \uf0e5 \uf0e5 \uf0e5 \uf0e5 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='6) Finally invoking these transformations and using the well-known identity \uf028 \uf029 \uf028 \uf029 1 2 ihy e dh y \uf070 \uf064 \uf0a5 \uf02d \uf02d\uf0a5 \uf03d \uf0f2 , we obtain \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 2 2 0( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=" ' ' 2 0 n y F xx y n n n k f ie v K dk u x v x i \uf065 \uf065 \uf065 \uf077 \uf064 \uf070 \uf077 \uf065 \uf0a5 \uf03d \uf02d\uf0a5 \uf0b6 \uf0bb \uf02d \uf02d \uf0d7 \uf02b \uf02b \uf0b6 \uf0e5 \uf0f2 x,x x x (A." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='7) The above equation relates to the only existing spin-state (a real physical spin rather than a pseudospin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Therefore, the total conductivity must be doubled, which corresponds to the value of the conductivity given by Equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Other components of the conductivity tensor may be obtained in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' For example, we have \uf028 \uf029 \uf028 \uf029 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , yy xx K K \uf077 \uf077 \uf0a2 \uf0a2 \uf03d x x x x and \uf028 \uf029 \uf028 \uf029 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' , 0 xy yx K K \uf077 \uf077 \uf0a2 \uf0a2 \uf03d \uf03d x x x x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Appendix B: Group velocity of pseudospins in graphene sheet Here we calculate the normalized group velocity of the pseudospins defined as \uf028 \uf029 / g y y v k k \uf065 \uf065 \uf0a2 \uf03d \uf03d \uf0b6 \uf0b6 ,based on the characteristic equation \uf028 \uf029 \uf028 \uf029 1 tg = x x y x k L f k k k \uf02d \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The y- component of the wavevector is considered as an independent variable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' By taking derivative with respect to yk , one gets 21 2 1 x y x y y f f k k k k k \uf0b6 \uf0b6 \uf0b6 \uf03d \uf0d7 \uf03d \uf02d \uf0b6 \uf0b6 \uf0b6 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='8) And by using the relation \uf028 \uf029 2 2 x y y k k k \uf065 \uf03d \uf02d , we obtain \uf028 \uf029 \uf028 \uf029 2 2 2 2 y y g y x y y y y y k k v k k k k k k k \uf065 \uf065 \uf065 \uf065 \uf065 \uf0b6 \uf02d \uf0b6 \uf02d \uf0b6 \uf03d \uf03d \uf0b6 \uf02d \uf02d (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='9) For the derivative / x f k \uf0b6 \uf0b6 we have \uf028 \uf029 \uf028 \uf029 2 2 tg cos x x x x x k L k L k L f k k \uf02d \uf0b6 \uf03d \uf0b6 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='10) The trigonometric functions may be expressed through the algebraic ones using the characteristic equation which gives \uf028 \uf029 2 2 y y x y x k L k f k k k \uf065 \uf02d \uf0b6 \uf03d \uf0b6 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='11) Expressing the group velocity from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='9) and using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='10), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='11), we obtain \uf028 \uf029 \uf028 \uf029\uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 1 y y g y y y k k L v k k L k \uf065 \uf065 \uf02d \uf03d \uf02d (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='12) In order to determine the renormalization coefficient n B , we can make a similar transformation: express \uf028 \uf029 sin 2 xk L thought \uf028 \uf029 tg xk L and apply the characteristic Equation (17), which renders by virtue of Equation (20), \uf028 \uf029 \uf028 \uf029 1 2 1 y yn yn n y yn k k k B k k L \uf065 \uf065 \uf02d \uf03d \uf0e6 \uf0f6 \uf0b6 \uf0e7 \uf0f7 \uf03d \uf0e7 \uf0f7 \uf0b6 \uf02d \uf0e8 \uf0f8 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='13) with the conductivity given explicitly in Equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Appendix C: Orthogonality of pseudo-spin modes The Equations for pseudo-spin mode with number s has the form , s y s s s s y s s s v k v u x u k u v x \uf065 \uf065 \uf0b6 \uf0fc \uf03d \uf02d \uf02d \uf0ef\uf0ef \uf0b6 \uf0fd \uf0b6 \uf0ef \uf03d \uf02b \uf0ef \uf0b6 \uf0fe , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='14) The similar Relation may be formulated for another mode with the number s\uf0a2: 22 , s y s s s s y s s s v k v u x u k u v x \uf065 \uf065 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf0b6 \uf0fc \uf03d \uf02d \uf02d \uf0ef\uf0ef \uf0b6 \uf0fd \uf0b6 \uf0ef \uf03d \uf02b \uf0ef \uf0b6 \uf0fe (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='15) Let us multiply first Equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='14) to su \uf0a2and second Equation multiply to sv \uf0a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Summarize them and integrate over the interval /2 /2 L x L \uf02d \uf03c \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Using the boundary conditions, we obtain \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 / 2 / 2 / 2 / 2 0 L L s s s s s s L L u x u x dx v x v x dx \uf065 \uf065 \uf0a2 \uf0a2 \uf0a2 \uf02d \uf02d \uf02d \uf03d \uf0f2 \uf0f2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='16) The similar Relation may be obtained from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='16) by rearranging the indexes s and s\uf0a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' We have \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 / 2 / 2 / 2 / 2 0 L L s s s s s s L L u x u x dx v x v x dx \uf065 \uf065 \uf0a2 \uf0a2 \uf0a2 \uf02d \uf02d \uf02d \uf03d \uf0f2 \uf0f2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='17) The eigenvalues with the different indexes are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The pair of Equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='16),(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='17) may be considered as a system of linear algebraic equations with respect to the integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' The determinant of this system s s s s \uf065 \uf065 \uf065 \uf065 \uf0a2 \uf0a2 \uf02d \uf02d is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' It means, that for s s\uf0a2 \uf0b9 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 / 2 / 2 / 2 / 2 0 L L s s s s L L u x u x dx v x v x dx \uf0a2 \uf0a2 \uf02d \uf02d \uf03d \uf03d \uf0f2 \uf0f2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content='18) which gives the orthogonality relation in the required form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE0T4oBgHgl3EQfiAH8/content/2301.02441v1.pdf'} +page_content=' Novotny, L.' metadata={'source': 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Technology, SE-412 96 Gothenburg, Sweden +2 University of Gothenburg, SE-405 30 Gothenburg, Sweden +Abstract. [Context and motivation] The development and opera- +tion of critical software that contains machine learning (ML) models +requires diligence and established processes. Especially the training data +used during the development of ML models have major influences on +the later behaviour of the system. Runtime monitors are used to pro- +vide guarantees for that behaviour. [Question / problem] We see ma- +jor uncertainty in how to specify training data and runtime monitoring +for critical ML models and by this specifying the final functionality of +the system. In this interview-based study we investigate the underlying +challenges for these difficulties. [Principal ideas/results] Based on ten +interviews with practitioners who develop ML models for critical appli- +cations in the automotive and telecommunication sector, we identified 17 +underlying challenges in 6 challenge groups that relate to the challenge of +specifying training data and runtime monitoring. [Contribution] The +article provides a list of the identified underlying challenges related to +the difficulties practitioners experience when specifying training data +and runtime monitoring for ML models. Furthermore, interconnection +between the challenges were found and based on these connections rec- +ommendation proposed to overcome the root causes for the challenges. +Keywords: artificial intelligence · context · data requirements · machine +learning · requirements engineering · runtime monitoring +1 +Introduction +With constant regularity, unexpected and undesirable behaviour of machine +learning (ML) models are reported in academia [9,24,26,51,52], the press, and +by NGOs3. These problems become especially apparent, and reported upon, +when ML models violate ethical principles. Racial, religious, or gender biases +are introduced through a lack of insight into the (sometimes immensely large +⋆ This project has received funding from the European Union’s Horizon 2020 research +and innovation program under grant agreement No 957197. +3 non-governmental organisations, e.g., https://algorithmwatch.org/en/stories/ +arXiv:2301.13476v1 [cs.SE] 31 Jan 2023 + +2 +H.-M. Heyn et al. +set of) training data and missing runtime checks for example in large language +models such as GPT-3 [1], or facial recognition software based on deep learn- +ing [36]. Unfortunately, improving the performance of deep learning models often +requires an exponential growth in training data [3]. Data requirements can help +in preventing unnecessarily large and biased datasets [48]. Due to changes in the +environment, ML models can become “stale”, i.e., the context changes so signif- +icantly that the performance of the model decreases below acceptable levels [5]. +Runtime monitors collect performance data and indicate the need for re-training +of the model with updated training data. However, these monitors need to be +specified at design time. Data requirements can support the specification of run- +time monitor [7]. The lack of specifications becomes specifically apparent with +ML models that are part of critical software 4 because it is not possible to estab- +lish traceability from system requirements (e.g., functional safety requirements) +to requirements set on the training data and the runtime monitoring [35]. +Challenges of specifying  +training data (RQ 1) +5 Unclear design +domain +3 Missing guidelines for +data selection +6 Unsuitable safety +standards +Challenges of specifying  +runtime monitoring (RQ 2) +1 Lack of explainability +about ML decisions +2 Missing conditions for +runtime checks +4 Overhead for +monitoring solution +Fig. 1: Overview of identified challenge categories +Scope and research questions +The purpose of this study is to highlight current challenges experienced by prac- +titioners in specifying training data and runtime monitoring for ML in safety +critical software. +The paper contributes a practitioner’s point of view on the challenges re- +ported in academic literature. The aim is to identify starting-points for a future +engineering research on the use of runtime monitors for critical ML systems. The +following research questions guided this study: +RQ1: What are challenges encountered by practitioners when specifying train- +ing data for ML models in safety critical software? +RQ2: What are challenges encountered by practitioners when specifying run- +time monitors especially in relation to fulfilling safety requirements? +4 We define critical software as software that is safety, privacy, ethically, and/or mission +critical, i.e., a failure in the software can cause significant injury or the loss of life, +invasion of personal privacy, violation of human rights, and/or significant economic +or environmental consequences [31]. + +Challenges when specifying data and runtime monitors +3 +Figure 1 shows the main themes we found in answering the research ques- +tions. Concerning RQ1, the interviewees reported on several problems: the data +selection process is nontransparent and guidelines especially towards defining +suitable measures for data variety are missing. There are no clear context def- +initions that help in defining data needs, and current safety standards provide +little guidance. Concerning RQ2, we found that the problem of defining suitable +metrics and the lack of guidance from safety standards also inhibits the ability to +specify runtime monitors. Furthermore, practitioners reported on challenges re- +garding explainability of ML decisions, and the processing and memory overhead +caused by runtime monitors in safety critical embedded systems. +The remaining sections of this paper are structured as follows: Section 2 +outlines and argues for the research methods of this study; Section 3 presents the +results amd answers to the research questions; Section 4 discusses the findings, +provides recommendations to practitioners and for further research, identifies +related literature, elaborates on threats to validity, and provides a conclusion. +2 +Research Method +We applied a qualitative interview-based survey with open-ended semi-structured +interviews for data collection. Following the suggestions of Creswell and Creswell +[13] the qualitative study was conducted in four steps: Preparation of interviews, +data collection through interviews, data analysis, and result validation. +Preparations of interviews Based on the a-priori formulated research ques- +tions, two of the researchers of this study created an interview guide5 which was +validated and improved by the remaining two researchers. The interview guide +contains four sections of questions: the first section includes questions about +the interviewees’ current role, background and previous experiences. The second +section focuses on questions that try to understand challenges when specifying +and selecting training data for ML models and how training data affect the per- +formance of these models. The third section investigates challenges when ML +models are incorporated in critical systems and how they affect the ability to +specify training data. The fourth section concentrates on the run time monitor- +ing aspect of the ML model and contains questions on challenges when specifying +runtime monitors. +Sampling strategy: We chose the participants for this study purposefully using +a maximum variation strategy [14]. We were able to recruit interviewees from +five different companies, ranging from a local start-up to a multinational world +leading communication company. An overview is given in Table 1. +A selection criteria for the company was that they must work with safety-critical +systems and ML. Within the companies we tried to find interview candidates +with different roles and work experiences to obtain a view beyond the developers’ +perspective. Besides function developers and ML model developers, we were +5 The interview guide is available at https://doi.org/10.7910/DVN/WJ8TKY. + +4 +H.-M. Heyn et al. +Table 1: Companies participating in the study +Company +Area of operations +Employees Countries +1 +Telecommunication networks +> 10.000 +World +2 +Automotive OEM +> 10.000 +World +3 +Automatic Driving +> 1.000 +Europe +4 +Industrial camera systems +> 1000 +USA +5 +Deep Learning optimisation for IoT > 100 +Sweden +Table 2: Participants of the study +Inter- +viewee +Role +Experience +A +Researcher (Academic) +Functional Safety for ADAS +B +Function developer +Sensor and perception systems +C +Principal engineer +ML model integration +D +ML model developer +Distributed and edge systems +E +Function owner +ADAS perception functions +F +Function developers +and test engineer +Automatic driving systems +G +Data Scientist +Distributed systems +H +Requirement Engineer +Perception systems +I +Researcher (Academic) +Neural Network development +J +Functional Safety Manager Sensor systems +ADAS: Advanced Driver Assistance Systems +interested in interviewing requirement engineers and product / function owners +because they represent key roles in deriving system or function specifications. +We provided the companies with a list of roles that we identified beforehand as +interesting for interviewing6. Additionally, we interviewed two researchers from +academia who participate in a joint industry EU Horizon 2020 project called +VEDLIoT7. Both researchers worked also with ML models in industry before. +Therefore, they could provide insights into both the academic and the industry +perspective. A list of the ten interviewees for this study is provided in Table 2. +Data collection through interviews All interviews were conducted remotely +using either the conference software Zoom or Microsoft Teams and took between +60 - 90 minutes. The a-priori defined interview guide was only available to the +interviewers and was not distributed to the participants beforehand. Each par- +ticipant was interviewed by two interviewers who alternated in asking questions +and observing. At the start of each interview, the interviewers provided some +background information about the study’s purpose. Then, the interview guide +was followed. However, as we encouraged discussions with the interviewees, we +allowed deviations from the interview guide by asking additional questions, or +changing the order of the questions when it was appropriate [30]. All interviews +were recorded and semi-automatically transcribed. The interviewers manually +checked and anonymised the results. +6 The list included functional safety experts, requirement engineers, product owners +or function owners, function or model developers, and data engineers. +7 Very efficient deep learning in the Internet of Things + +Challenges when specifying data and runtime monitors +5 +Data analysis The data analysis followed suggestions by Saldana [41] and +consisted of two cycles of coding and validation of the themes through a workshop +and member checking. +First coding cycle: Attribute coding was used to extract information about the +participants’ role and previous experiences. Afterwards, the two interviewers +independently applied structural coding to collect phrases in the interviews that +represent topics relevant to answering the research questions. The researchers +compared the individually assigned codes and applied descriptive coding with the +aim of identifying phrases that describe common themes across the interviews. +Theme validation: In a focus group, the identified themes were presented and dis- +cussed. Thirteen researchers from both industry and academia in the VEDLIoT +project participated. Three of the participants also were interviewed for this +study. The aim of the focus group was to reduce bias in the selection of themes +and to identify any additional themes that the researchers might have missed. +Second coding cycle: After the themes were identified and validated, the second +coding cycle was used to map the statements of the interviewees to the themes, +and consequently identify the answers to the research questions. The second +cycle was conducted by the two researchers who did not conduct the first cycle +coding in order to reduce confirmation bias. The mapping was then confirmed +and agreed upon by all involved researchers. +Result validation Member checking, as described in [14, Ch. 9] was used to +validate the identified themes that answer RQ 1 and RQ 2. Additionally, we +presented the results in a 60 minute focus group to an industry partner and +allowed for feedback and comments on the conclusions we drew from the data. +3 +Results +During the first coding cycle, structural coding resulted in 117 statements for +RQ1 and 77 statements for RQ2. Through descriptive coding preliminary themes +were found. The statements and preliminary themes were discussed during a +workshop. Based on the feedback from the workshop, 117 statements for RQ1 +were categorised into eight final challenge themes and three challenge categories +relating to the challenge of specifying training data. Similar, the 77 original +statements for RQ2 were categorised into 13 final challenge themes in five chal- +lenge categories relating to the challenge of specifying runtime monitoring. A +total of six challenge categories emerged for both RQs, out of which two cate- +gories contain challenges relating to both training data and runtime monitoring +specification, and three challenge themes base on statements from both RQs. +The categories and final challenge themes are listed in Table 3. +3.1 +Answer to RQ1: Challenges practitioners experience when +specifying training data +The interviewees were asked to share their experiences in selecting training data, +the influence of the selection of training data on the system’s performance and + +6 +H.-M. Heyn et al. +Table 3: Challenge groups (bold) and themes found in the interview data. Data.: +Challenges related to specifying training data (RQ1). Monitor.: Challenges re- +lated to specifying runtime monitoring (RQ2). +Relates to +Related +ID +Challenge Theme +Data. Monitor. Literature +1 +Lack of explainability about ML decisions +✓ +1.1 No access to inner states of ML models +✓ +[18] +1.2 No failure models for ML models +✓ +[51] +1.3 IP protection +✓ +2 +Missing conditions for runtime checks +✓ +2.1 Unclear metrics and/or boundary conditions +✓ +[11,21,43] +2.2 Unclear measure of confidence +✓ +[17,34] +3 +Missing guidelines for data selection +✓ +✓ +3.1 Disconnection from requirements +✓ +[16,42] +3.2 Grown data selection habits +✓ +[20,33] +3.3 Unclear completeness criteria +✓ +[49] +3.4 Unclear measure of variety +✓ +✓ +[45,50] +4 +Overhead for monitoring solution +✓ +4.1 Limited resources in embedded systems +✓ +[38] +4.2 Meeting timing requirements +✓ +4.3 Reduction of true positive rate +✓ +5 +Unclear design domain +✓ +5.1 Design domain depends on available data +✓ +[6] +5.2 Uncertainty in context +✓ +[22] +6 +Unsuitable safety standards +✓ +✓ +6.1 Focus on processes instead of technical solution +✓ +✓ +[10] +6.2 No guidelines for probabilistic effects in software +✓ +[28,43] +6.3 Safety case only through monitoring solution +✓ +[31,46] +safety, and any experiences and thoughts on defining specifications for training +data for ML. Based on the interview data, we identified three challenge groups +related to specifying training data: missing guidelines for data selection, unclear +design domain, and unsuitable safety standards +Missing guidelines for data selection Four interviewees reported on a lack +of guidelines and processes related to the selection of training data. A reason +can be that data selection bases on “grown habits” that are not properly doc- +umented. Unlike conventional software development, the training of ML is an +iterative process of discovering the necessary training data based on experience +and experimentation. Requirements set on the data are described as discon- +nected and unclear for the data selection process. For example, one interviewee +stated that if a requirements is set that images shall contain a road, it remains +unclear what specific properties this road should have. Six interviewees described +missing requirements on the data variety and missing completeness criteria as a +reason for the disconnection of requirements from data selection. +“How much of it (the data) should be in darkness? How much in rainy conditions, +and how much should be in snowy situations?” - Interview F +“For example, we said that we shall collect data under varying weather conditions. +What does that mean?” - Interview B +Another interviewee stated that it is not clear how to measure variety, which +could be a reason why it is difficult to define requirements on data variety. + +Challenges when specifying data and runtime monitors +7 +“What [is] include[d] in variety of data? Is there a good measure of variety?” - +Interview A +Unclear design domain Three interviewees describe uncertainty in the design +domain as a reason for why it is difficult to specify training data. If the design +domain is unclear, it will be challenging to specify the necessary training data. +“We need to understand for what context the training data can be used.” - Interview J +“ODD [(Operational Design Domain)]? Yes, of course it translates into data require- +ments.” - Interview F +Unsuitable safety standards Because we were specifically investigating ML +in safety critical applications, we asked the participants if they find guidance in +safety standards towards specifying training data. Five interviewees stated that +current safety standards used in their companies do not provide suitable guid- +ance for the development of ML models. While for example ISO 26262 provides +guidance on how to handle probabilistic effects in hardware, no such guidance is +provided for software related probabilistic faults. +“The ISO 26262 gives guidance on the hardware design; [...] how many faults per +hour [are acceptable] and how you achieve that. For the software side, it doesn’t +give any failure rates or anything like that. It takes a completely process oriented +approach [...].” - Interview C +One interviewee mentioned that safety standards should emphasise more the +data selection to prevent faults in the ML model due to insufficient training. +“To understand that you have the right data and that the data is representative, +ISO 26262 is not covering that right now which is a challenge.” - Interview H +3.2 +Answer to RQ2: Challenges practitioners experience when +specifying runtime monitors +We asked the interviewees on the role of runtime monitoring for the systems they +develop, their experience with specifying runtime monitoring, and the relation of +runtime monitoring to fulfilling safety requirements on the system. We identified +five challenge groups related to runtime monitoring: lack of explainability about +ML decisions, missing conditions for runtime checks, missing guidelines for data +selection, overhead for monitoring solution, and unsuitable safety standards. +Lack of explainability about ML A reason why it is difficult to specify +runtime monitors for ML models is the inability to produce failure models for +ML. In normal software development, causal cascades describe how a fault in a +software components propagates trough the systems and eventually leads to a +failure. This requires the ability to break down the ML model into smaller com- +ponents and analyse their potential failure behaviour. Four interviewees however +reported that they can only see the ML model as a “black-box” with no access +to the inner states of the ML model. As a consequence, there is no insight into +the failure behaviour of the ML model. + +8 +H.-M. Heyn et al. +“[Our insight is] limited because it’s a black box. We can only see what goes in +and then what comes out to the other side. And so if there is some error in the +behavior, then we don’t know if it’s because [of a] classification error, planning +error, execution error?” - Interview F +The reason for opacity of ML models is not necessarily due to technology limita- +tions, but also due to constraints from protection of intellectual property (IP). +“Why is it a black box? That’s not our choice. That’s because we have a supplier +and they don’t want to tell us [all details].” - Interview F +Missing conditions for runtime checks A problem of specifying runtime +monitors is the need for finding suitable monitoring conditions. This requires +the definition of metrics, goals and boundary conditions. Five interviewees report +that they face challenges when defining these metrics for ML models. +“What is like a confidence score, accuracy score, something like that? Which score +do you need to ensure [that you] classified [correctly]?” - Interview F +Especially the relation between correct behaviour of the ML model and measure +of confidence is unclear, and therefore impede runtime monitoring specification. +“We say confidence, that’s really important. But what can actually go wrong here?” +- Interview J +Missing guidelines for data selection The inability to specify the meaning +of data variety also relates to missing conditions for runtime checks. For example, +runtime monitors can be used to collect additional training data, but without a +measure of data variety it is difficult to find the required data points. +Overhead for monitoring solution An often overlooked problem seems to +be the induced (processing) overhead from a monitoring solution. Especially in +the automotive sector, many software components run on embedded computer +devices with limited resources. +“You don’t have that much compute power in the car, so the monitoring needs to +be very light in its memory and compute footprint on the device, maybe even a +separate device that sits next to the device.” - Interview F +And due to the limited resources in embedded systems, monitoring solutions can +compromise timing requirements of the system. Additionally, one interviewee +reported concerns regarding the reduction of the ML model’s performance. +“[. . . ] the true positive rate is actually decreasing when you have to pass it through +this second opinion goal. It’s good from a coverage and safety point of view, but it +reduces the overall system performance.” - Interview F + +Challenges when specifying data and runtime monitors +9 +Unsuitable safety standards Safety standards are mostly not suitable for be- +ing applied to ML model development. Therefore, safety is often ensured through +non-ML monitoring solutions. Interviewees reported that it is not a good solution +to rely only on the monitors for safety criticality: +“[. . . ] so the safety is now moved from the model to the monitor instead, and it +shouldn’t be. It should be the combination of the two that makes up safety.” - +Interview B +One reason is that freedom of inference between a non-safety critical component +(the ML model), and a safety critical component (the monitor) must be ensured +which can complicate the system design. +“And then especially if you have mixed critical systems [it] means you have ASIL +[(Automotive Safety Integrity Level)] and QM [(Quality Management)] components +in your design [and] you want to achieve freedom from interference in your system. +You have to think about safe communication and memory protection.” - Interview J +4 +Discussion and Conclusion +The results reveal connections between the challenges. Not all theme groups re- +late exclusively to one of the two challenges. For example, themes in the groups +unsuitable safety standards and missing guidelines for data selection relate to +both challenges of specifying training data and runtime monitoring. Further- +more, we identified cause-effect relations between different themes and across +different group of themes. For example IP protection is a cause for the inability +of accessing the inner states and for creating failure models for ML model. We +based this assessment on a semantic analyses of the words used in the statements +relating to these themes. For example, Interviewee F stated that: +“That neural network is something [of a] black box in itself. You don’t know why it +do[es] things. Well, you cannot say anything about its inner behavior” - Interview F +Later in the interview, the same participants states: +“Why is it a black box? That’s not our choice. That’s because we have a supplier +and they don’t want to tell us [all details].” - Interview F +Figure 2 illustrates the identified cause-effect relations, relations between the +themes, and how the different themes relate to the challenges. +Recommendations to practitioners and for further research The iden- +tified root causes of the challenges described by the participants allowed us to +formulate recommendations listed in Table 4. Because these recommendations +try to solve root causes described by the participants of the interview study, +we think they are a useful first step towards solving the challenges related to +specifying training data and runtime monitoring. + +10 +H.-M. Heyn et al. +5 Unclear design +domain +3 Missing guidelines for +data selection +2 Missing conditions for +runtime checks +2.2 Unclear mea- +sure of confidence +6 Unsuitable safety +standards +Challenges of +specifying +runtime +monitoring +Challenges of +specifying +training data +3.1 Disconnection +from requirements +3.3 Unclear comple- +teness criteria +5.1 Data require- +ments depend on +design domain +5.2 Uncertainty in +context +1 Lack of explainability +about ML decisions +1.1 No access to +inner states +6.1 Focus on +processes instead of +technical solutions +6.2 No guidelines for +probabilistic effects +in software +6.3 Safety case only +through monitoring +solution +2.1 Unclear +metrics / bound- +ary conditions +4 Overhead for +monitoring solution +4.2 Meeting timing +requirements +4.1 Limited +resources in +embedded +systems +3.2 Grown data +selection habits +1.2 No failure +models +1.3 IP protection +4.3 Reduction of +true positive rate +3.4 Unclear +measure of variety +Fig. 2: Connection between the identified challenge themes. Enclosed themes +have been identified as causes for the surrounding themes. Furthermore, dotted +lines indicate relations between different themes. +4.1 +Related Literature +The problem of finding the “right” data: For acquiring data, data scientists have +to rely on data mining with little to no quality checking and potential biases [4]. +Biased datasets are a common cause for erroneous or unexpected behaviour of +ML models in critical environments, such as in medical diagnostic [8], in the +juridical system [19,37], or in safety-critical applications [15,46]. +There are attempts to create “unbiased” datasets. One approach is to curate +manually the dataset, such as in the FairFace dataset [29], the CASIA-SURF +CeFaA dataset [32], or Fairbatch [40]. An alternative road is to use data augmen- +tation techniques to “rebalance” the dataset [27,45]. However, it was discovered +that it is not sufficient for avoiding bias to use an assumed balanced datasets +during training [20,49,50] because it is often unclear which features in the data +need to be balanced. Approaches for curating or manipulating the dataset re- +quire information on the target domain, i.e., one needs to set requirements on +the dataset depending on the desired operational context [6,16,22]. But deriving +a data specification for ML is not common practise [25,33,42]. + +Challenges when specifying data and runtime monitors +11 +Table 4: Recommendations for practitioners and suggestions for further research +ID +Recommendation +I +Avoid restrictive IP protection. IP protection is a cause for the inability of accessing the +inner states of the ML models (black-box model). This causes a nontransparent measure of +confidence, and an inability to formulate failure models. To our knowledge, no studies have yet +been performed on the consequences of IP protection of ML models on the ability to monitor +and reason (e.g., in a safety case) for the correctness of ML model decisions. +II Relate measures of confidence to actual performance metrics. For runtime monitoring, +the measure of confidence is often used to evaluate the reliability of the ML model’s results. +But without understanding and relating that measure to clearly defined performance metrics of +the ML model first, the measure of confidence provides little insight for runtime monitoring. In +general, defining suitable metrics and boundary conditions should become an integral part of +RE for machine learning as it affects both the ability to define data requirements and runtime +monitoring requirements. +III Overcome grown data selection habits. Grown data selection habits have been mentioned +as a reason for a lack of clear completeness criteria and a disconnection from requirements. +Based on our results, we argue that more systematic data selection processes need to be es- +tablished in companies. This would allow for a better connection of the data selection process +to requirement engineering and it creates a traceability between system requirements, com- +pleteness criteria and data requirements. Additionally, it might also reduce the amount of data +needed for training, and therefore cost of development. +IV Balance hardware limitation in embedded systems. Runtime monitoring causes a pro- +cessing and memory overhead that can compromise timing requirements and reduce the ML +model’s performance. Today, safety criticality of systems with ML is mostly ensured through +monitoring solutions. By decomposing the safety requirements instead onto both the monitor- +ing and the ML model, the monitors might become more resource efficient, faster, and less +constraining in regards to the decisions of the ML model. However, safety requirements on the +ML models might trigger requirements on the training data. +The problem of finding the “right” runtime monitor: Through clever test strate- +gies, some uncertainty can be eliminated in regards to the behaviour of the +model [11]. However, ML components are often part of systems of systems and +their behaviour is hard to predict and analyse at design time [47]. DevOps prin- +ciples from software engineering give promising ideas on how to tackle remaining +uncertainty at runtime [34]. As part of the operation of the model, runtime mod- +els that “augment information available at design-time with information moni- +tored at runtime” help in detecting deviations from the expected behaviour [17]. +These runtime models for ML can take the form of model assertions, i.e., check- +ing of explicitly defined attributes of the model at runtime [28]. However, the +authors state that “bias in training sets are out of scope for model assertion”. An- +other model based approach can be the creation of neuron activation patterns for +runtime monitoring [12]. Other approaches treat the ML model as “black-box”, +and only check for anomalies and drifts in the input data [39] the output [43], +or both [18]. However, similar to the aforementioned challenges when specifying +data for ML, runtime monitoring needs an understanding on how to “define, re- +fine, and measure quality of ML solutions” [23], i.e., in relation to non-functional +requirements one needs to understand which quality aspects are relevant, and +how to measure them [21]. Most commonly applied safety standards emphasise +processes and traceability to mitigate systematic mistakes during the develop- +ment of critical systems. Therefore, if the training data and runtime monitoring +cannot be specified, a traceability between safety goals and the deployed system +cannot be established [10]. + +12 +H.-M. Heyn et al. +For many researchers and practitioners, runtime verification and monitoring +is a promising road to assuring safety and robustness for ML in critical soft- +ware [2,11]. However, runtime monitoring also creates a processing and memory +overhead that needs to be considered especially in resource-limited environments +such as embedded devices [38]. +The related work has been mapped to the challenges identified in the inter- +view study in Table 3. +4.2 +Threats to validity +A lack of rigour (i.e., degree of control) in the study design can cause confound- +ing which can manifest bias in the results [44]. The following mechanisms in this +study tried to reduce confounding: The interview guide was peer-reviewed by an +independent researcher, and a test session of the interview was conducted. To +reduce personal bias, at least two authors were present during all interviews, and +the authors took turn in leading the interviews. To confirm the initial findings +from the interview study and reduce the risk of researchers’ bias, a workshop was +organised which was also visited by participants who were not part of the inter- +view study. Another potential bias can arise from the sampling of participants. +Although we applied purposeful sampling, we still had to rely on the contact +persons of the companies to provide us with suitable interview candidates. We +could not directly see a list of employees and choose the candidates ourselves. +Regarding generalisability of the findings, the limited number of companies in- +volved in the study can pose a threat to external validity. However, two of the +companies are world-leading companies in their fields, which, in our opinion, +gives them a deep understanding and experience of the discussed problems. Fur- +thermore, we included companies from a variety of different fields to establish +better generalisability. Furthermore, our data includes only results valid for the +development of safety-critical ML models. We assume that the findings are ap- +plicable also to other forms of criticality, such as privacy-critical, but we cannot +conclude on that generalisability based on the available data. +4.3 +Conclusion +This paper reported on a interview-based study that identified challenges related +to specifying training data needs and runtime monitoring for safety critical ML +models. Through interviews conducted at five companies we identified 17 chal- +lenges in six groups. Furthermore, we performed a semantic analysis to identify +the underlying root-causes. We saw that several underlying challenges affect +both the ability to specify training data and runtime monitoring. For example, +we concluded that restrictive IP protection can cause an inability to access and +understand the inner states of a ML model. Without insight into the ML model’s +state, the measure of confidence cannot be related to actual performance metrics. +Without clear performance metrics, it is difficult to define the necessary degree of +variety in the training data. Furthermore, grown data selection impedes proper +requirement engineering for training data. Finally, safety requirements should be + +Challenges when specifying data and runtime monitors +13 +distributed on both the ML model which can cause requirements on the training +data, and on runtime monitors which can reduce the overhead by the moni- +toring solution. These recommendations will serve as starting point for further +engineering research. +References +1. Abid, A., Farooqi, M., Zou, J.: Persistent anti-muslim bias in large language mod- +els. In: Proceedings of the 2021 AAAI/ACM Conference on AI, Ethics, and Society. +pp. 298–306 (2021) +2. 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Zhang, X., Xie, X., Ma, L., Du, X., Hu, Q., Liu, Y., Zhao, J., Sun, M.: Towards +characterizing adversarial defects of deep learning software from the lens of uncer- +tainty. 2020 IEEE/ACM 42nd International Conference on Software Engineering +pp. 739–751 (2020) + diff --git a/LNFRT4oBgHgl3EQfEjcG/content/tmp_files/load_file.txt b/LNFRT4oBgHgl3EQfEjcG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..36b4c8007d067372949394df861a37649d27a40d --- /dev/null +++ b/LNFRT4oBgHgl3EQfEjcG/content/tmp_files/load_file.txt @@ -0,0 +1,792 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf,len=791 +page_content='An investigation of challenges encountered when specifying training data and runtime monitors for safety critical ML applications⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Hans-Martin Heyn1,2[0000−0002−2427−6875], Eric Knauss1,2[0000−0002−6631−872X], Iswarya Malleswaran1, and Shruthi Dinakaran1 1 Chalmers University of Technology, SE-412 96 Gothenburg, Sweden 2 University of Gothenburg, SE-405 30 Gothenburg, Sweden Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' [Context and motivation] The development and opera- tion of critical software that contains machine learning (ML) models requires diligence and established processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Especially the training data used during the development of ML models have major influences on the later behaviour of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Runtime monitors are used to pro- vide guarantees for that behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' [Question / problem] We see ma- jor uncertainty in how to specify training data and runtime monitoring for critical ML models and by this specifying the final functionality of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' In this interview-based study we investigate the underlying challenges for these difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' [Principal ideas/results] Based on ten interviews with practitioners who develop ML models for critical appli- cations in the automotive and telecommunication sector, we identified 17 underlying challenges in 6 challenge groups that relate to the challenge of specifying training data and runtime monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' [Contribution] The article provides a list of the identified underlying challenges related to the difficulties practitioners experience when specifying training data and runtime monitoring for ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Furthermore, interconnection between the challenges were found and based on these connections rec- ommendation proposed to overcome the root causes for the challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Keywords: artificial intelligence · context · data requirements · machine learning · requirements engineering · runtime monitoring 1 Introduction With constant regularity, unexpected and undesirable behaviour of machine learning (ML) models are reported in academia [9,24,26,51,52], the press, and by NGOs3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' These problems become especially apparent, and reported upon, when ML models violate ethical principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Racial, religious, or gender biases are introduced through a lack of insight into the (sometimes immensely large ⋆ This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 957197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 3 non-governmental organisations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=', https://algorithmwatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='org/en/stories/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='13476v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='SE] 31 Jan 2023 2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Heyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' set of) training data and missing runtime checks for example in large language models such as GPT-3 [1], or facial recognition software based on deep learn- ing [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Unfortunately, improving the performance of deep learning models often requires an exponential growth in training data [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Data requirements can help in preventing unnecessarily large and biased datasets [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Due to changes in the environment, ML models can become “stale”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=', the context changes so signif- icantly that the performance of the model decreases below acceptable levels [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Runtime monitors collect performance data and indicate the need for re-training of the model with updated training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' However, these monitors need to be specified at design time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Data requirements can support the specification of run- time monitor [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The lack of specifications becomes specifically apparent with ML models that are part of critical software 4 because it is not possible to estab- lish traceability from system requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=', functional safety requirements) to requirements set on the training data and the runtime monitoring [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Challenges of specifying training data (RQ 1) 5 Unclear design domain 3 Missing guidelines for data selection 6 Unsuitable safety standards Challenges of specifying runtime monitoring (RQ 2) 1 Lack of explainability about ML decisions 2 Missing conditions for runtime checks 4 Overhead for monitoring solution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 1: Overview of identified challenge categories Scope and research questions The purpose of this study is to highlight current challenges experienced by prac- titioners in specifying training data and runtime monitoring for ML in safety critical software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The paper contributes a practitioner’s point of view on the challenges re- ported in academic literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The aim is to identify starting-points for a future engineering research on the use of runtime monitors for critical ML systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The following research questions guided this study: RQ1: What are challenges encountered by practitioners when specifying train- ing data for ML models in safety critical software?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' RQ2: What are challenges encountered by practitioners when specifying run- time monitors especially in relation to fulfilling safety requirements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 4 We define critical software as software that is safety, privacy, ethically, and/or mission critical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=', a failure in the software can cause significant injury or the loss of life, invasion of personal privacy, violation of human rights, and/or significant economic or environmental consequences [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Challenges when specifying data and runtime monitors 3 Figure 1 shows the main themes we found in answering the research ques- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Concerning RQ1, the interviewees reported on several problems: the data selection process is nontransparent and guidelines especially towards defining suitable measures for data variety are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' There are no clear context def- initions that help in defining data needs, and current safety standards provide little guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Concerning RQ2, we found that the problem of defining suitable metrics and the lack of guidance from safety standards also inhibits the ability to specify runtime monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Furthermore, practitioners reported on challenges re- garding explainability of ML decisions, and the processing and memory overhead caused by runtime monitors in safety critical embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The remaining sections of this paper are structured as follows: Section 2 outlines and argues for the research methods of this study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Section 3 presents the results amd answers to the research questions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Section 4 discusses the findings, provides recommendations to practitioners and for further research, identifies related literature, elaborates on threats to validity, and provides a conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 2 Research Method We applied a qualitative interview-based survey with open-ended semi-structured interviews for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Following the suggestions of Creswell and Creswell [13] the qualitative study was conducted in four steps: Preparation of interviews, data collection through interviews, data analysis, and result validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Preparations of interviews Based on the a-priori formulated research ques- tions, two of the researchers of this study created an interview guide5 which was validated and improved by the remaining two researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The interview guide contains four sections of questions: the first section includes questions about the interviewees’ current role, background and previous experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The second section focuses on questions that try to understand challenges when specifying and selecting training data for ML models and how training data affect the per- formance of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The third section investigates challenges when ML models are incorporated in critical systems and how they affect the ability to specify training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The fourth section concentrates on the run time monitor- ing aspect of the ML model and contains questions on challenges when specifying runtime monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Sampling strategy: We chose the participants for this study purposefully using a maximum variation strategy [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' We were able to recruit interviewees from five different companies, ranging from a local start-up to a multinational world leading communication company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' An overview is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' A selection criteria for the company was that they must work with safety-critical systems and ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Within the companies we tried to find interview candidates with different roles and work experiences to obtain a view beyond the developers’ perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Besides function developers and ML model developers, we were 5 The interview guide is available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='7910/DVN/WJ8TKY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 4 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Heyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Table 1: Companies participating in the study Company Area of operations Employees Countries 1 Telecommunication networks > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='000 World 2 Automotive OEM > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='000 World 3 Automatic Driving > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Europe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Industrial camera systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='> 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='USA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Deep Learning optimisation for IoT > 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Sweden ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Table 2: Participants of the study ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Inter- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='viewee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Role ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Experience ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Researcher (Academic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Functional Safety for ADAS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Function developer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Sensor and perception systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Principal engineer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='ML model integration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='ML model developer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Distributed and edge systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Function owner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='ADAS perception functions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Function developers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='and test engineer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Automatic driving systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Data Scientist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Distributed systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Requirement Engineer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Perception systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Researcher (Academic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Neural Network development ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='Functional Safety Manager Sensor systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='ADAS: Advanced Driver Assistance Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='interested in interviewing requirement engineers and product / function owners ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='because they represent key roles in deriving system or function specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' We provided the companies with a list of roles that we identified beforehand as interesting for interviewing6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Additionally, we interviewed two researchers from academia who participate in a joint industry EU Horizon 2020 project called VEDLIoT7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Both researchers worked also with ML models in industry before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Therefore, they could provide insights into both the academic and the industry perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' A list of the ten interviewees for this study is provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Data collection through interviews All interviews were conducted remotely using either the conference software Zoom or Microsoft Teams and took between 60 - 90 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The a-priori defined interview guide was only available to the interviewers and was not distributed to the participants beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Each par- ticipant was interviewed by two interviewers who alternated in asking questions and observing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' At the start of each interview, the interviewers provided some background information about the study’s purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Then, the interview guide was followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' However, as we encouraged discussions with the interviewees, we allowed deviations from the interview guide by asking additional questions, or changing the order of the questions when it was appropriate [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' All interviews were recorded and semi-automatically transcribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The interviewers manually checked and anonymised the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 6 The list included functional safety experts, requirement engineers, product owners or function owners, function or model developers, and data engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 7 Very efficient deep learning in the Internet of Things Challenges when specifying data and runtime monitors 5 Data analysis The data analysis followed suggestions by Saldana [41] and consisted of two cycles of coding and validation of the themes through a workshop and member checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' First coding cycle: Attribute coding was used to extract information about the participants’ role and previous experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Afterwards, the two interviewers independently applied structural coding to collect phrases in the interviews that represent topics relevant to answering the research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The researchers compared the individually assigned codes and applied descriptive coding with the aim of identifying phrases that describe common themes across the interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Theme validation: In a focus group, the identified themes were presented and dis- cussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Thirteen researchers from both industry and academia in the VEDLIoT project participated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Three of the participants also were interviewed for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The aim of the focus group was to reduce bias in the selection of themes and to identify any additional themes that the researchers might have missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Second coding cycle: After the themes were identified and validated, the second coding cycle was used to map the statements of the interviewees to the themes, and consequently identify the answers to the research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The second cycle was conducted by the two researchers who did not conduct the first cycle coding in order to reduce confirmation bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The mapping was then confirmed and agreed upon by all involved researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Result validation Member checking, as described in [14, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 9] was used to validate the identified themes that answer RQ 1 and RQ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Additionally, we presented the results in a 60 minute focus group to an industry partner and allowed for feedback and comments on the conclusions we drew from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 3 Results During the first coding cycle, structural coding resulted in 117 statements for RQ1 and 77 statements for RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Through descriptive coding preliminary themes were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The statements and preliminary themes were discussed during a workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Based on the feedback from the workshop, 117 statements for RQ1 were categorised into eight final challenge themes and three challenge categories relating to the challenge of specifying training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Similar, the 77 original statements for RQ2 were categorised into 13 final challenge themes in five chal- lenge categories relating to the challenge of specifying runtime monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' A total of six challenge categories emerged for both RQs, out of which two cate- gories contain challenges relating to both training data and runtime monitoring specification, and three challenge themes base on statements from both RQs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The categories and final challenge themes are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Answer to RQ1: Challenges practitioners experience when specifying training data The interviewees were asked to share their experiences in selecting training data, the influence of the selection of training data on the system’s performance and 6 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Heyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Table 3: Challenge groups (bold) and themes found in the interview data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' : Challenges related to specifying training data (RQ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' : Challenges re- lated to specifying runtime monitoring (RQ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Relates to Related ID Challenge Theme Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Literature 1 Lack of explainability about ML decisions ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 No access to inner states of ML models ✓ [18] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 No failure models for ML models ✓ [51] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='3 IP protection ✓ 2 Missing conditions for runtime checks ✓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Unclear metrics and/or boundary conditions ✓ [11,21,43] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Unclear measure of confidence ✓ [17,34] 3 Missing guidelines for data selection ✓ ✓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Disconnection from requirements ✓ [16,42] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Grown data selection habits ✓ [20,33] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='3 Unclear completeness criteria ✓ [49] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='4 Unclear measure of variety ✓ ✓ [45,50] 4 Overhead for monitoring solution ✓ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Limited resources in embedded systems ✓ [38] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Meeting timing requirements ✓ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='3 Reduction of true positive rate ✓ 5 Unclear design domain ✓ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Design domain depends on available data ✓ [6] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Uncertainty in context ✓ [22] 6 Unsuitable safety standards ✓ ✓ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Focus on processes instead of technical solution ✓ ✓ [10] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 No guidelines for probabilistic effects in software ✓ [28,43] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='3 Safety case only through monitoring solution ✓ [31,46] safety, and any experiences and thoughts on defining specifications for training data for ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Based on the interview data, we identified three challenge groups related to specifying training data: missing guidelines for data selection, unclear design domain, and unsuitable safety standards Missing guidelines for data selection Four interviewees reported on a lack of guidelines and processes related to the selection of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' A reason can be that data selection bases on “grown habits” that are not properly doc- umented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Unlike conventional software development, the training of ML is an iterative process of discovering the necessary training data based on experience and experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Requirements set on the data are described as discon- nected and unclear for the data selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' For example, one interviewee stated that if a requirements is set that images shall contain a road, it remains unclear what specific properties this road should have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Six interviewees described missing requirements on the data variety and missing completeness criteria as a reason for the disconnection of requirements from data selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “How much of it (the data) should be in darkness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' How much in rainy conditions, and how much should be in snowy situations?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' - Interview F “For example, we said that we shall collect data under varying weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' What does that mean?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' - Interview B Another interviewee stated that it is not clear how to measure variety, which could be a reason why it is difficult to define requirements on data variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Challenges when specifying data and runtime monitors 7 “What [is] include[d] in variety of data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Is there a good measure of variety?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' - Interview A Unclear design domain Three interviewees describe uncertainty in the design domain as a reason for why it is difficult to specify training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' If the design domain is unclear, it will be challenging to specify the necessary training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “We need to understand for what context the training data can be used.” - Interview J “ODD [(Operational Design Domain)]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Yes, of course it translates into data require- ments.” - Interview F Unsuitable safety standards Because we were specifically investigating ML in safety critical applications, we asked the participants if they find guidance in safety standards towards specifying training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Five interviewees stated that current safety standards used in their companies do not provide suitable guid- ance for the development of ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' While for example ISO 26262 provides guidance on how to handle probabilistic effects in hardware, no such guidance is provided for software related probabilistic faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “The ISO 26262 gives guidance on the hardware design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='] how many faults per hour [are acceptable] and how you achieve that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' For the software side, it doesn’t give any failure rates or anything like that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' It takes a completely process oriented approach [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='].” - Interview C One interviewee mentioned that safety standards should emphasise more the data selection to prevent faults in the ML model due to insufficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “To understand that you have the right data and that the data is representative, ISO 26262 is not covering that right now which is a challenge.” - Interview H 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Answer to RQ2: Challenges practitioners experience when specifying runtime monitors We asked the interviewees on the role of runtime monitoring for the systems they develop, their experience with specifying runtime monitoring, and the relation of runtime monitoring to fulfilling safety requirements on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' We identified five challenge groups related to runtime monitoring: lack of explainability about ML decisions, missing conditions for runtime checks, missing guidelines for data selection, overhead for monitoring solution, and unsuitable safety standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Lack of explainability about ML A reason why it is difficult to specify runtime monitors for ML models is the inability to produce failure models for ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' In normal software development, causal cascades describe how a fault in a software components propagates trough the systems and eventually leads to a failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' This requires the ability to break down the ML model into smaller com- ponents and analyse their potential failure behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Four interviewees however reported that they can only see the ML model as a “black-box” with no access to the inner states of the ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' As a consequence, there is no insight into the failure behaviour of the ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 8 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Heyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “[Our insight is] limited because it’s a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' We can only see what goes in and then what comes out to the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' And so if there is some error in the behavior, then we don’t know if it’s because [of a] classification error, planning error, execution error?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' - Interview F The reason for opacity of ML models is not necessarily due to technology limita- tions, but also due to constraints from protection of intellectual property (IP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “Why is it a black box?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' That’s not our choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' That’s because we have a supplier and they don’t want to tell us [all details].” - Interview F Missing conditions for runtime checks A problem of specifying runtime monitors is the need for finding suitable monitoring conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' This requires the definition of metrics, goals and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Five interviewees report that they face challenges when defining these metrics for ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “What is like a confidence score, accuracy score, something like that?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Which score do you need to ensure [that you] classified [correctly]?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' - Interview F Especially the relation between correct behaviour of the ML model and measure of confidence is unclear, and therefore impede runtime monitoring specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “We say confidence, that’s really important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' But what can actually go wrong here?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Interview J Missing guidelines for data selection The inability to specify the meaning of data variety also relates to missing conditions for runtime checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' For example, runtime monitors can be used to collect additional training data, but without a measure of data variety it is difficult to find the required data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Overhead for monitoring solution An often overlooked problem seems to be the induced (processing) overhead from a monitoring solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Especially in the automotive sector, many software components run on embedded computer devices with limited resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “You don’t have that much compute power in the car, so the monitoring needs to be very light in its memory and compute footprint on the device, maybe even a separate device that sits next to the device.” - Interview F And due to the limited resources in embedded systems, monitoring solutions can compromise timing requirements of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Additionally, one interviewee reported concerns regarding the reduction of the ML model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' ] the true positive rate is actually decreasing when you have to pass it through this second opinion goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' It’s good from a coverage and safety point of view, but it reduces the overall system performance.” - Interview F Challenges when specifying data and runtime monitors 9 Unsuitable safety standards Safety standards are mostly not suitable for be- ing applied to ML model development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Therefore, safety is often ensured through non-ML monitoring solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Interviewees reported that it is not a good solution to rely only on the monitors for safety criticality: “[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' ] so the safety is now moved from the model to the monitor instead, and it shouldn’t be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' It should be the combination of the two that makes up safety.” - Interview B One reason is that freedom of inference between a non-safety critical component (the ML model), and a safety critical component (the monitor) must be ensured which can complicate the system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' “And then especially if you have mixed critical systems [it] means you have ASIL [(Automotive Safety Integrity Level)] and QM [(Quality Management)] components in your design [and] you want to achieve freedom from interference in your system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' You have to think about safe communication and memory protection.” - Interview J 4 Discussion and Conclusion The results reveal connections between the challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Not all theme groups re- late exclusively to one of the two challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' For example, themes in the groups unsuitable safety standards and missing guidelines for data selection relate to both challenges of specifying training data and runtime monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Further- more, we identified cause-effect relations between different themes and across different group of themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' For example IP protection is a cause for the inability of accessing the inner states and for creating failure models for ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' We based this assessment on a semantic analyses of the words used in the statements relating to these themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' For example, Interviewee F stated that: “That neural network is something [of a] black box in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' You don’t know why it do[es] things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Well, you cannot say anything about its inner behavior” - Interview F Later in the interview, the same participants states: “Why is it a black box?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' That’s not our choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' That’s because we have a supplier and they don’t want to tell us [all details].” - Interview F Figure 2 illustrates the identified cause-effect relations, relations between the themes, and how the different themes relate to the challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Recommendations to practitioners and for further research The iden- tified root causes of the challenges described by the participants allowed us to formulate recommendations listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Because these recommendations try to solve root causes described by the participants of the interview study, we think they are a useful first step towards solving the challenges related to specifying training data and runtime monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 10 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Heyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 5 Unclear design domain 3 Missing guidelines for data selection 2 Missing conditions for runtime checks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Unclear mea- sure of confidence 6 Unsuitable safety standards Challenges of specifying runtime monitoring Challenges of specifying training data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Disconnection from requirements 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='3 Unclear comple- teness criteria 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Data require- ments depend on design domain 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Uncertainty in context 1 Lack of explainability about ML decisions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 No access to inner states 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Focus on processes instead of technical solutions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 No guidelines for probabilistic effects in software 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='3 Safety case only through monitoring solution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Unclear metrics / bound- ary conditions 4 Overhead for monitoring solution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Meeting timing requirements 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Limited resources in embedded systems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Grown data selection habits 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 No failure models 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='3 IP protection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='3 Reduction of true positive rate 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='4 Unclear measure of variety Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 2: Connection between the identified challenge themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Enclosed themes have been identified as causes for the surrounding themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Furthermore, dotted lines indicate relations between different themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='1 Related Literature The problem of finding the “right” data: For acquiring data, data scientists have to rely on data mining with little to no quality checking and potential biases [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Biased datasets are a common cause for erroneous or unexpected behaviour of ML models in critical environments, such as in medical diagnostic [8], in the juridical system [19,37], or in safety-critical applications [15,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' There are attempts to create “unbiased” datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' One approach is to curate manually the dataset, such as in the FairFace dataset [29], the CASIA-SURF CeFaA dataset [32], or Fairbatch [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' An alternative road is to use data augmen- tation techniques to “rebalance” the dataset [27,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' However, it was discovered that it is not sufficient for avoiding bias to use an assumed balanced datasets during training [20,49,50] because it is often unclear which features in the data need to be balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Approaches for curating or manipulating the dataset re- quire information on the target domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=', one needs to set requirements on the dataset depending on the desired operational context [6,16,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' But deriving a data specification for ML is not common practise [25,33,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Challenges when specifying data and runtime monitors 11 Table 4: Recommendations for practitioners and suggestions for further research ID Recommendation I Avoid restrictive IP protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' IP protection is a cause for the inability of accessing the inner states of the ML models (black-box model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' This causes a nontransparent measure of confidence, and an inability to formulate failure models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' To our knowledge, no studies have yet been performed on the consequences of IP protection of ML models on the ability to monitor and reason (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=', in a safety case) for the correctness of ML model decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' II Relate measures of confidence to actual performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' For runtime monitoring, the measure of confidence is often used to evaluate the reliability of the ML model’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' But without understanding and relating that measure to clearly defined performance metrics of the ML model first, the measure of confidence provides little insight for runtime monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' In general, defining suitable metrics and boundary conditions should become an integral part of RE for machine learning as it affects both the ability to define data requirements and runtime monitoring requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' III Overcome grown data selection habits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Grown data selection habits have been mentioned as a reason for a lack of clear completeness criteria and a disconnection from requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Based on our results, we argue that more systematic data selection processes need to be es- tablished in companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' This would allow for a better connection of the data selection process to requirement engineering and it creates a traceability between system requirements, com- pleteness criteria and data requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Additionally, it might also reduce the amount of data needed for training, and therefore cost of development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' IV Balance hardware limitation in embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Runtime monitoring causes a pro- cessing and memory overhead that can compromise timing requirements and reduce the ML model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Today, safety criticality of systems with ML is mostly ensured through monitoring solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' By decomposing the safety requirements instead onto both the monitor- ing and the ML model, the monitors might become more resource efficient, faster, and less constraining in regards to the decisions of the ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' However, safety requirements on the ML models might trigger requirements on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The problem of finding the “right” runtime monitor: Through clever test strate- gies, some uncertainty can be eliminated in regards to the behaviour of the model [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' However, ML components are often part of systems of systems and their behaviour is hard to predict and analyse at design time [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' DevOps prin- ciples from software engineering give promising ideas on how to tackle remaining uncertainty at runtime [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' As part of the operation of the model, runtime mod- els that “augment information available at design-time with information moni- tored at runtime” help in detecting deviations from the expected behaviour [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' These runtime models for ML can take the form of model assertions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=', check- ing of explicitly defined attributes of the model at runtime [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' However, the authors state that “bias in training sets are out of scope for model assertion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' An- other model based approach can be the creation of neuron activation patterns for runtime monitoring [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Other approaches treat the ML model as “black-box”, and only check for anomalies and drifts in the input data [39] the output [43], or both [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' However, similar to the aforementioned challenges when specifying data for ML, runtime monitoring needs an understanding on how to “define, re- fine, and measure quality of ML solutions” [23], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=', in relation to non-functional requirements one needs to understand which quality aspects are relevant, and how to measure them [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Most commonly applied safety standards emphasise processes and traceability to mitigate systematic mistakes during the develop- ment of critical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Therefore, if the training data and runtime monitoring cannot be specified, a traceability between safety goals and the deployed system cannot be established [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 12 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Heyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' For many researchers and practitioners, runtime verification and monitoring is a promising road to assuring safety and robustness for ML in critical soft- ware [2,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' However, runtime monitoring also creates a processing and memory overhead that needs to be considered especially in resource-limited environments such as embedded devices [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The related work has been mapped to the challenges identified in the inter- view study in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='2 Threats to validity A lack of rigour (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=', degree of control) in the study design can cause confound- ing which can manifest bias in the results [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' The following mechanisms in this study tried to reduce confounding: The interview guide was peer-reviewed by an independent researcher, and a test session of the interview was conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' To reduce personal bias, at least two authors were present during all interviews, and the authors took turn in leading the interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' To confirm the initial findings from the interview study and reduce the risk of researchers’ bias, a workshop was organised which was also visited by participants who were not part of the inter- view study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Another potential bias can arise from the sampling of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Although we applied purposeful sampling, we still had to rely on the contact persons of the companies to provide us with suitable interview candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' We could not directly see a list of employees and choose the candidates ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Regarding generalisability of the findings, the limited number of companies in- volved in the study can pose a threat to external validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' However, two of the companies are world-leading companies in their fields, which, in our opinion, gives them a deep understanding and experience of the discussed problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Fur- thermore, we included companies from a variety of different fields to establish better generalisability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Furthermore, our data includes only results valid for the development of safety-critical ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' We assume that the findings are ap- plicable also to other forms of criticality, such as privacy-critical, but we cannot conclude on that generalisability based on the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content='3 Conclusion This paper reported on a interview-based study that identified challenges related to specifying training data needs and runtime monitoring for safety critical ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Through interviews conducted at five companies we identified 17 chal- lenges in six groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Furthermore, we performed a semantic analysis to identify the underlying root-causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' We saw that several underlying challenges affect both the ability to specify training data and runtime monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' For example, we concluded that restrictive IP protection can cause an inability to access and understand the inner states of a ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Without insight into the ML model’s state, the measure of confidence cannot be related to actual performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Without clear performance metrics, it is difficult to define the necessary degree of variety in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Furthermore, grown data selection impedes proper requirement engineering for training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Finally, safety requirements should be Challenges when specifying data and runtime monitors 13 distributed on both the ML model which can cause requirements on the training data, and on runtime monitors which can reduce the overhead by the moni- toring solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' These recommendations will serve as starting point for further engineering research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' Abid, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=': Towards characterizing adversarial defects of deep learning software from the lens of uncer- tainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 2020 IEEE/ACM 42nd International Conference on Software Engineering pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} +page_content=' 739–751 (2020)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFRT4oBgHgl3EQfEjcG/content/2301.13476v1.pdf'} diff --git a/LdE4T4oBgHgl3EQfJww5/content/tmp_files/2301.04923v1.pdf.txt b/LdE4T4oBgHgl3EQfJww5/content/tmp_files/2301.04923v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b83d4ec48f2e89f968ae1351a495a260212844d1 --- /dev/null +++ b/LdE4T4oBgHgl3EQfJww5/content/tmp_files/2301.04923v1.pdf.txt @@ -0,0 +1,2046 @@ +Semi-Lagrangian Finite-Element Exterior +Calculus for Incompressible Flows +Wouter Tonnon* +Ralf Hiptmair† +January 13, 2023 +1 +Incompressible Navier-Stokes Equations +We consider the incompressible Navier-Stokes equations—a standard hydrodynamic +model for the motion of an incompressible, potentially-viscous fluid—in a container +with rigid walls, where we impose so-called “free boundary conditions” in the par- +lance of [31, p. 346] and [43, p. 502], see the latter article for further references. We +search the fluid velocity field u(t, x) and the pressure p(t, x) as functions of time t and +space x on a bounded, Lipschitz domain Ω ⊂ Rd such that they solve the evolution +boundary-value problem +∂tu + u · ∇u − ε∆u + ∇p = f, +on ]0, T[×Ω, +(1a) +∇ · u = 0, +on ]0, T[×Ω, +(1b) +u · n = 0, +on ]0, T[×∂Ω, +(1c) +εn × ∇ × u = 0, +on ]0, T[×∂Ω, +(1d) +u = u0, +on {0} × Ω, +(1e) +where ε ≥ 0 denotes a (non-dimensional) viscosity, f a given source term, T > 0 the +final time, ∂Ω the boundary of Ω, and n(x) the outward normal vector at x ∈ ∂Ω. The +initial condition u0 is to satisfy ∇ · u0 = 0 in Ω and u0 · n = 0, εn × ∇ × u0 = 0 on +∂Ω. Based on the variational description of the Navier-Stokes equations as described +in [5], u can be interpreted as a differential 1-form [33] and we can recast system (1) +in the following way. Let Λk(Ω) for k ∈ N denote the space of differential k-forms on +*SAM, ETH Zürich, CH-8092 Zürich, wouter.tonnon@sam.math.ethz.ch +†SAM, ETH Zürich, CH-8092 Zürich, ralf.hiptmair@sam.math.ethz.ch +1 +arXiv:2301.04923v1 [math.NA] 12 Jan 2023 + +Ω. Then we search ω ∈ Λ1(Ω) and p ∈ Λ0(Ω) such that +Duω + εδdω + dp = f, +on ]0, T[×Ω, +(2a) +δω = 0, +on ]0, T[×Ω, +(2b) +tr ⋆ω = 0, +on ]0, T[×∂Ω, +(2c) +ε tr ⋆dω = 0, +on ]0, T[×∂Ω, +(2d) +ω = ω0, +on{0} × Ω, +(2e) +where Duω denotes the material derivative of ω with respect to u, d : Λk−1(Ω) �→ +Λk(Ω) the exterior derivative, δ : Λk(Ω) �→ Λk−1(Ω) the exterior coderivative, and the +trace tr is the pullback under the embedding ∂Ω ⊂ ¯Ω. Here, u is related to ω through +ω := uZ, i.e. u is the vector proxy of ω w.r.t. the Euclidean metric. Similarly, we have +that Λ1(Ω) ∋ ω0 := u0Z and Λ1(Ω) ∋ f := fZ. Note that (2) can be derived through +classical vector calculus for vector proxies as shown in Appendix A for d = 3. +As shown in [18, 19], sufficiently-smooth solutions ω :]0, T[�→ Λ1(Ω) of the in- +compressible Navier-Stokes equations as given in system (2) satisfy an energy relation, +that is, +dE +dt (t) := d +dt +1 +2 +� +Ω +ω(t) ∧ ⋆ω(t) = −ε +� +Ω +dω(t) ∧ ⋆dω(t). +(3) +This relation implies energy conservation for ε = 0. In the case of ε = 0, we also have +helicity conservation, that is, +ε = 0 =⇒ dH +dt (t) := d +dt +� +Ω +dω(t) ∧ ω(t) = 0. +(4) +Note that the Onsager conjecture tells us that in the case ε = 0 the solutions need to +be at least Hölder regular with exponent 1 +3 for energy conservation to hold [28]. +Remark 1 We acknowledge that the boundary condition (1d) is non-standard. This +boundary condition was chosen because it is the natural boundary condition associ- +ated to system (2). To enforce the standard no-slip boundary conditions, (1d) could +be replaced by εu × n = u on ]0, T[×∂Ω, that is, we impose an essential instead +of natural boundary condition to system (2). Unfortunately, in this case, the scheme +presented in this work leads to an ill-posed system. In the case ε = 0, the only imposed +boundary condition (1c) is standard. +Remark 2 Boundary conditions (1c),(1d) can be interpreted as slip boundary con- +ditions. However, on smooth domains Ω, they are only equivalent to Navier’s slip +boundary conditions if the Weingarten map related to ∂Ω vanishes [31, section 2]. +2 +Outline and Related Work +We propose a semi-Lagrangian approach to the discretization of the reformulated +Navier-Stokes boundary value problem (2). This method revolves around the dis- +cretization of the material derivative Duω in the framework of a finite-element Galerkin +2 + +discretization on a fixed spatial mesh. The main idea is to approximate Duω by back- +ward difference quotients involving transported snapshots of the 1-form ω, which can +be computed via the pullback induced by the flow of the velocity vector field u. +Semi-Lagrangian methods for transient transport equations like (2) are well-es- +tablished for the linear case when u is a given Lipschitz-continuous velocity field. +In particular, for ω a 0-form, that is, a plain scalar-valued function, plenty of semi- +Lagrangian approaches have been proposed and investigated, see, for instance, [8, 7, +20, 21, 23, 35, 39, 40, 6, 12, 45]. We refer to [24, Chapter 5] for a comprehensive +pre-2013 literature review on the analysis of general semi-Lagrangian schemes. Most +of these methods focus on mapping point values under the flow, with the exception +of a particularly interesting class of semi-Lagrangian methods known as Lagrange- +Galerkin methods. +Lagrange-Galerkin methods do not transport point values, but +rather triangles (in 2D) or tetrahedra (in 3D). Refer to [10] for a review of those meth- +ods. +Meanwhile semi-Lagrangian methods for transport problems for differential forms +of any order have been developed [26, 25, 24]. The next section will review these +semi-Lagrangian methods for linear transport problems with emphasis on 1-forms. +We will also introduce a new scheme which is second-order in space and time based +on so-called “small edges”, see section 3.1.2 for details. +Semi-Lagrangian schemes for the incompressible Navier-Stokes equations are also +well-documented in literature, with emphasis on the Lagrange-Galerkin method [10, +14, 15, 30, 1, 11, 9]. A survey of the application of Lagrange-Galerkin methods to the +incompressible Navier-Stokes equations is given in [10]. It is important to note that +these methods require the evaluation of integrals of transported quantities and, in case +these integrals cannot be computed exactly, instabilities can occur [12, 32]. A possi- +ble remedy is to add an additional stabilization term that includes artificial diffusion +[10]. Other semi-Lagrangian methods for incompressible Navier-Stokes equations di- +rectly transport point values with the nodes of a mesh instead of evaluating integrals +of transported quantities, see [34, 29, 47, 46, 13] and [16], where the last work makes +use of exponential integrators [17]. Most authors employ spectral elements for the +discretization in space [34, 29], but any type of finite-element space with degrees-of- +freedom relying on point evaluations can be used. The methods proposed in [47, 46] +are also based on finite-element spaces with degrees-of-freedom on nodes, but em- +ploy backward-difference approximations for the material derivative. The work [13] +proposes an explicit semi-Lagrangian method still built around the transport of point +values in the nodes of the mesh. The diffusion term is also taken into account in a +semi-Lagrangian fashion and the incompressibility constraint is enforced by means of +a Chorin projection. Also [13] proposes an explicit semi-Lagrangian scheme using the +same principles, but based on the vorticity-streamfunction form of the incompressible +Navier-Stokes equations. +All the mentioned semi-Lagrangian schemes rely on the transport of point val- +ues of continuous vector fields, which is the perspective embraced in formulation (1). +However, we believe that, in particular in the case of free boundary conditions (1c) +3 + +and (1d), the semi-Lagrangian method based on (2) offers benefits similar to the bene- +fits bestowed by the use of discrete differential forms (finite-element exterior calculus, +FEEC [3, 4]) for the discretization of electromagnetic fields. Section 4 will convey that +the boundary conditions (2c), (2d), and the incompressiblity constraint can very natu- +rally be incorporated into a variational formulation of (2) posed in spaces of 1-forms. +This has been the main motivation for pursuing the new idea of a semi-Lagrangian +method for (2) that employs discrete 1-forms. Another motivation has been the ex- +pected excellent robustness of the semi-Lagrangian discretization in the limit ε �→ 0. +Numerical tests reported in section 5 will confirm this. +Two more aspects of our method are worth noting: Firstly, a discrete 1-form ωh will +not immediately spawn a continuous velocity field uh = ωZ +h, However, continuity is +essential for defining a meaningful flow. We need an additional averaging step, which +we present in section 4.1. Secondly, since semi-Lagrangian methods fail to respect the +decay/conservation laws (3) and (4) exactly, we present a way how to enforce them in +section 4.3. +3 +Semi-Lagrangian Advection of differential forms +3.1 +Discrete differential forms +We start from a simplicial triangulation Kh(Ω) of Ω, which may rely on a piecewise +linear approximation of ∂Ω so that it covers a slightly perturbed domain. +3.1.1 +Lowest-order case: Whitney forms +For Λ0(Ω)—the space of 0-forms on Ω, which is just a space of real-valued func- +tions—the usual (Lagrange) finite-element space of continuous, piecewise-linear, poly- +nomial functions provides the space Λ0 +h,1(Ω) of lowest-order discrete 0-forms. +Let d ∈ {2, 3}, K a d-simplex with edges {e1, .., e3(d−1)}. To construct lowest- +order discrete 1-forms on K, we associate to every edge ei a local shape function +wei. Let the edge ei be directed from vertex v1 +i to v2 +i , then the local shape function +wei ∈ Λ1(K) associated with edge ei is +wei := λv1 +i dλv2 +i − λv2 +i dλv1 +i , +(5) +where λv represents the barycentric coordinate associated with vertex v. We define +the lowest-order, local space of discrete 1-forms +Λ1 +h,1(K) := span{we; e an edge of K}. +(6) +Using these local spaces, we can define the global space of lowest-order, discrete 1- +forms +Λ1 +h,1(Ω) := {ω ∈ Λ1(Ω); ∀K ∈ Kh(Ω) : ω|K ∈ Λ1 +h,1(K)}, +(7) +4 + +(0,0) +(1,0) +(0,1) +1 +2 +3 +4 +5 +6 +7 +8 +9 +(a) 9 small edges of a second-order element in 2D. All the edges between the different +connection points are small edges. In 3D, we simply have all these small edges on +the faces of the simplex. +edge no. +l.s.f. +edge no. +l.s.f. +1 +[ x +y ] �→ +� +x(x+y−1) +−(x−1)(x+y−1) +� +6 +[ x +y ] �→ +� +(y−1)(x+y−1) +x(1−x−y) +� +2 +[ x +y ] �→ +� +−y2 +y(x−1) +� +7 +[ x +y ] �→ +� +−xy +x(x−1) +� +3 +[ x +y ] �→ +� −y2 +xy +� +8 +[ x +y ] �→ +� y(1−y) +xy +� +4 +[ x +y ] �→ +� −xy +x2 +� +9 +[ x +y ] �→ +� +y(x+y−1) +x(1−x−y) +� +5 +[ x +y ] �→ +� +x(1−y) +x2 +� +(b) Local shape functions (l.s.f.) for the unit triangle associated with second-order, +discrete differential forms in 2D as proposed in [38]. Each shape function corre- +sponds to the small edge in (a) with the same numbering. +Figure 1: Illustration of small edges (a) and corresponding local shape functions (b) +for the unit triangle. +5 + +where Λ1(Ω) again denotes the space of differential 1-forms on Ω. We demand that +for every ω ∈ Λ1(Ω) integration along any smooth oriented path yields a unique value. +Thus, the requirement ω ∈ Λ1(Ω) imposes tangential continuity on the vector proxy +of ω. +3.1.2 +Second-order discrete forms +Similar to the lowest-order case, the space Λ0 +h,2(Ω) of second-order discrete 0-forms +is spawned by the usual (Lagrange) finite-element space of continuous, piecewise- +quadratic, polynomial functions. +Let d ∈ {2, 3}, K a d-simplex with edges {e1, .., e3(d−1)} and vertices {v1, .., vd+1}. +To construct second-order discrete 1-forms, we associate local shape functions to +"small edges". We can construct 3(d + 1)(d − 1) small edges [38, Definition 3.2] +by defining ∀i ∈ {1, .., d + 1} and ∀j ∈ {1, .., 3(d − 1)} +{vi, ej} := {vi + 1 +2(x − vi); x ∈ ej}, +where {vi, ej} denotes the small edge. In Figure 1a we illustrate the 9 small edges of a +2-simplex. For example, we see that small edge 9 can be written as {(0, 0), [(1, 0), (0, 1)]}. +To make the difference between small edges and edges of the mesh explicit, we will +sometimes refer to the latter as "big edges". +The local shape function [38, Definition 3.3] associated with {vi, ej} is given by +w{vi,ej} := λviwej, +where wej denotes the Whitney 1-form associated with the big edge ej as defined in +(5). In Figure 1b we give explicit expressions for the shape functions associated with +the small edges in Figure 1a. Note that the local shape functions of the form w{v,e} +associated with small edges in the interior (d = 2) or on the same face (d = 3) of +the form {v, e} such that v /∈ ∂e (example: small edge 7, 8, and 9 in Figure 1a) are +linearly dependent. We define the second-order, local space of discrete 1-forms [38, +Definition 3.3] +Λ1 +h,2(K) := span{w{v,e}; v a vertex of K, e a (big) edge of K}. +(8) +Using these local spaces, we can define the global space of second-order, discrete 1- +forms +Λ1 +h,2(Ω) := {ω ∈ Λ1(Ω); ∀K ∈ Kh(Ω) : ω|K ∈ Λ1 +h,2(K)}, +(9) +where again we have tangential continuity by a similar argument as in section 3.1.1. +3.1.3 +Projection operators +We denote by Eh,p(Ω) the global set of big edges (p = 1) or small edges (p = 2) +associated with Kh(Ω). We will define the projection operator Ih,p : Λ1(Ω) �→ Λ1 +h,p(Ω) +6 + +as the unique operator that maps ω ∈ Λ1(Ω) to ωh ∈ Λ1 +h,p(Ω) such that the mismatch +� +e∈Eh,p(Ω) +�� +e +ω − +� +e +ωh +�2 +(10) +is minimized. Note that for p = 1, this mismatch can be made to vanish. In this case, +Ih,1 agrees with the usual edge-based nodal projection operator [27, Eq. (3.11)]. +In practice, we can compute the projection locally as follows. Let K ∈ Kh(Ω) be +a d-simplex, d ∈ {2, 3}, and let {s1, .., sNp,d} and {ws1, .., wsNp,d} denote the corre- +sponding big (p = 1) or small (p = 2) edges and corresponding shape functions as +introduced above. Specifically, we have N1,2 = 3, N1,3 = 6, N2,2 = 9, and N2,3 = 24. +We can define the matrix +(M)i,j = +� +si +wsj, +1 ≤ i, j ≤ Np,d. +(11) +We will say that there is an interaction from edge sj to si if (M)i,j ̸= 0. Note that +for p = 1, M is the identity matrix. For p = 2 the local shape functions are linearly +dependent and, thus, the above matrix is not invertible. However, we can decompose +M into invertible and singular sub-matrices. For illustrative purposes we display for +p = 2 and d = 2 the decomposition of M in Figure 2b. The three top-left sub-matrices +in Figure 2b are invertible 2 × 2 matrices that describe the interaction between the +two small edges that lie on the same big edge, that is, the blue, red, and green sub- +matrix in Figure 2b correspond to the blue, red, and green small edges in Figure 2a, +respectively. The orange sub-matrix in Figure 2b is a 3 × 3 matrix with rank 2 that +describes the interaction between the three small edges that lie in the interior of the +simplex in Figure 2a, that is, the orange small edges. The gray sub-matrix encodes the +one-directional interaction from the the small edges that lie on a big edge to the small +edges in the interior. Note that the decomposition of M as given in Figure 2b is not +limited to d = 2. The idea can be extended to d = 3 by considering each face of a +3-simplex as a 2-simplex. This is sufficient, since for d = 3 we have no small edges in +the interior and there is no interaction between small edges that do not lie on the same +face. We give the general structure of M in Figure 2c. Note that the small, purple +sub-matrices represent invertible 2 × 2 matrices and the bigger, orange sub-matrices +represent 3 × 3 matrices with rank 2. +In order to find ωh +�� +K ∈ Λ1 +h,p(K) such that ωh +�� +K = Ih,pω +�� +K, let ⃗ηK be a vector of +coefficients η1 +K, .., η +Np,d +K +such that +ωh|K = +Np,d +� +i=1 +ηi +Kwsi. +(12) +We can then compute ⃗ηK as a least-squares solution of +M⃗ηK = +�� +si +ω +� +1≤i≤Np,d +. +(13) +7 + +(a) 2-simplex K +(b) matrix M (d = 2) +(c) matrix M (d = 3) +Figure 2: For p = 2 and d = 2 the matrix M corresponding to the 2-simplex K in (a) +has the form given in (b). Each row and column in M is associated to a small edge +in (a). Each sub-matrix in (b) describes the interactions between edges with the same +color in (a). The gray sub-matrix is an exception as it describes the one-directional +interaction between the small edges that lie on a big edge and the small edges that lie +in the interior. For d = 3, we M has the structure as shown in Figure 2c, where the +purple sub-matrixs are 2 × 2 invertible matrices and the orange sub-matrixs are 3 × 3 +matrices of rank 2. +Without loss of generality we assume that M has the form as given in Figure 2c. Then, +we solve (13) as follows: +1. The local shape functions related to small edges that lie on a big edge of the +simplex are linearly independent. We solve for their coefficients first, that is, we +solve the system corresponding to the invertible blue sub-matrices in Figure 2c +first. +2. Using the results from step 1, we can move the gray sub-matrix in Figure 2c +to the right-hand side. Then, we solve the matrix-system corresponding to the +orange sub-matrices in Figure 2c in a least-squares sense. +If we perform the above steps for all K ∈ Kh(Ω), we find ωh = Ih,pω ∈ Λ1 +h,p(Ω). +Note that only the shape functions associated to small edges on a face contribute to +the tangential fields on that face. Therefore, the above procedure yields tangential +continuity. +Remark 3 For p = 1, (13) reduces to +ηi +K = +� +si +ω, +∀i ∈ {1, .., 3(d − 1)} +(14) +with si a big edge of the 3-simplex K for all i ∈ {1, .., 3(d − 1)}. This yields the +standard nodal interpolation operator of [27, Eq. (3.11)]. +8 + +3.2 +Semi-Lagrangian material derivative +The method described in this section is largely based on [24, 26]. Throughout this +section, unless stated otherwise, we fix the stationary, Lipschitz-continuous velocity +field u ∈ W 1,∞(Ω) with u · n = 0 on ∂Ω. This means that we consider a linear trans- +port problem and our main concern will be the discretization of the material derivative +Duω for a 1-form ω. We can define the flow ]0, T[×Ω ∋ (τ, x) �→ Xτ(x) ∈ Rd as the +solution of the initial value problems +∂ +∂τ Xt,t+τ(x) = u(Xt,t+τ(x)), +Xt(x) = x. +(15) +Given that flow we can define the material derivative for a time-dependent differential +1-form ω +Duω(t) := ∂ +∂τ X∗ +t,t+τω(t + τ) +���� +τ=0 +. +(16) +We employ a first- or second-order, backward-difference method to approximate the +derivative. Writing X∗ +t,t−τ for the pullback of forms under the flow, we obtain for +sufficiently-smooth t �→ ω(t) and a timestep 0 < τ → 0 +Duω(t) = 1 +τ +� +ω(t) − X∗ +t,t−τω(t − τ) +� ++ O(τ 2) +(17) +or +Duω(t) = 1 +2τ +� +3ω(t) − 4X∗ +t,t−τω(t − τ) + X∗ +t,t−2τω(t − 2τ) +� ++ O(τ 3), +(18) +respectively. Note that both backward-difference methods are A-stable [41]. In the +remainder of this section we restrict ourselves to (17), but exactly the same considera- +tions apply to (18). +Given a temporal mesh .. < tn < tn+1 < .., we approximate ω(tn, ·) ∈ Λ1(Ω) +by a discrete differential form ωn +h ∈ Λ1 +h,p(Ω) with p ∈ {1, 2}. Using the backward- +difference quotient (17), we can define the discrete material derivative for fixed timestep +τ > 0 +(Dβω)(tn) ≈ 1 +τ +� +ωn +h − Ih,pX∗ +t,t−τωn−1 +h +� +∈ Λ1 +h,p(Ω), +(19) +where we need to use the projection operator Ih,p : Λ1(Ω) �→ Λh,p(Ω) since X∗ +t,t−τωn−1 +h +/∈ +Λ1 +h,p(Ω) in general. Recall from section 3.1 that the degrees of freedom for discrete +1-forms are associated to small (p = 2) or big (p = 1) edges. As discussed in sec- +tion 3.1.3, evaluating the interpolation operator entails integrating X∗ +t,t−τωn−1 +h +over +small (p = 2) or big (p = 1) edges. We can approximate these integrals as follows +� +e +X∗ +t,t−τωn−1 +h += +� +Xt,t−τ(e) +ωn−1 +h +≈ +� +¯ +Xt,t−τ(e) +ωn−1 +h +, +(20) +where e is a small or big edge and +¯Xt,t−τ(e) = +� +(1 − ξ)Xt,t−τ(v1) + ξXt,t−τ(v2); 0 ≤ ξ ≤ 1 +� +(21) +9 + +Figure 3: Edge e (in red) is transported using the flow β (in blue). The exact trans- +ported edge Xτ(e) and the approximate transported edge ¯Xτ(e) are given in orange +and green. +with v1, v2 the vertices of e. Instead of transporting the edge e using the exact flow +Xt,t−τ, we follow [12, 24, 26] and only transport the vertices of the small edges (p = 2) +or big edges (p = 1) and obtain a piecewise linear (second-order) approximation +¯Xt,t−τ(e) of the transported edge Xt,t−τ(e) as illustrated in Figure 3. We can approxi- +mate the movement of the endpoints of e under the flow as defined by (15) by solving +(15) using the explicit Euler method or Heun’s method for the first- and second-order +case, respectively. We will elaborate on this further in section 4.1. +In Figure 3, we can also see that the approximate transported edge may intersect +several different elements of the mesh. When we evaluate the integral in (20), it can +happen that there are discontinuities of ωn−1 +h +along ¯Xt,t−τ(e). Therefore, we cannot +apply a global quadrature rule to the entire integral. Instead, we split ¯Xt,t−τ(e) into +several segments defined by the intersection of ¯Xt,t−τ(e) with cells of the mesh. In +our implementation, for the sake of stability, we find the intersection points by trans- +forming back to a reference element as visualised in Figure 4. Algorithm 1 gives all +details. Note that we can forgo the treament of any special cases (e.g. intersection +with vertices) without jeopardizing stability. After we split the transported edge into +segments, we can evaluate the integrals over these individual pieces exactly, because +we know that ωn−1 +h +is of polynomial form when restricted to individual elements of the +mesh (see section 3.1). +When simulating the fluid model (2), we will not have access to an exact velocity +field. Instead we only have access to an approximation of the velocity field. This ap- +proximation may not satisfy exact vanishing normal boundary conditions. Therefore, +a part of ¯Xt,t−τ(e) may end up outside the domain. This can also happen due to an +approximation of the flow by explicit timestepping. Since ωn−1 +h +is not defined outside +10 + +Algorithm 1 Splitting 1-simplex over mesh elements (see Figure 4 for illustration). +Here, Kref denotes the reference simplex. +Input: x0 ∈ K0 ∈ Kh(Ω) and x1 vertices of a 1-simplex e. +Output: Number of elements N, elements {K0, .., KN−1} ∈ Kh(Ω)N. +1: K ← K0 +2: Fold ← NULL +3: Kold ← NULL +4: N ← 1 +5: E ← {K} +6: while x1 /∈ K do +7: +Find the isoparametric mapping ϕK : Kref �→ K +8: +Find face F ⊂ ∂K s.t. F ̸= Fold and ϕ−1 +K (e) ∩ ϕ−1 +K (F) ̸= ∅ +9: +K ← K ∈ Kh(Ω) s.t. F ⊂ ∂K and K ̸= Kold (K on the other side of face F) +10: +Fold ← F +11: +N ← N + 1 +12: +E ← E ∪ {K} +13: end while +ϕK : Kref �→ K +x0 +e +K +ϕ−1 +K (e) +Kref +K0 +x1 +Figure 4: The red line indicates the line that spans multiple elements. On the left we +see the reference triangle associated with the yellow element in the mesh on the right. +11 + +Figure 5: A coarse triangulation of Ω = {x ∈ R2; ||x|| < 1} with the velocity field +u = [−y, x]T satisfying u · n = 0. Despite the vanishing normal components of the +velocity, the blue edge gets transported out of the domain to the green edge. +the domain, we set +� +¯ +Xt,t−τ(e)|Rd\Ω +ωn−1 +h +:= +len +� +¯Xt,t−τ(e) +�� +Rd\Ω +� +len +� +¯Xt,t−τ(e) +� +� +e +ωn−1 +h +, +(22) +where ¯Xt,t−τ(e) +�� +Rd\Ω is the part of ¯Xt,t−τ(e) that lies outside the domain Ω and len +� +·) +gives the arclength of the argument. This is motivated by the situation displayed in +Figure 5—a case where an edge gets transported out of the domain due to the use of +approximate flow maps despite vanishing normal components of the velocity. If we +set the value defined in (22) to zero in this case, it would be equivalent to applying +vanishing tangential boundary conditions, which is inconsistent with (1). Instead, (22) +just takes the tangential components from the previous timestep. +We arrive at the following approximation of the material derivative +(Dβω)(tn) ≈ 1 +τ +� +ωn +h − Ih,p ¯X∗ +t,t−τωn−1 +h +� +, +(23) +where the only difference between (19) and (23) is that Xt,t−τ was replaced by ¯Xt,t−τ +and Ih,p is implemented based on (22). Note that in our scheme ¯X∗ +t,t−τ is always +evaluated in conjunction with Ih,p, which means that we need define ¯Xt,t−τ only on +small (p = 2) or big (p = 1) edges. In fact, ¯Xt,t−τ is defined through (21) for all points +that lie on small (p = 2) or big (p = 1) edges. +Given a velocity field u ∈ W 1,∞(Ω) with u · n = 0, it was shown in [26, section +4] that using a first-order backward difference scheme and lowest-order elements for +the spatial discretization, we can approximate a smooth solution ω ∈ Λ1(Ω) of +Duω = 0, +(24) +12 + +with an L2-error of O(τ − 1 +2h), where h is the spatial meshwidth and τ > 0 is the +timestep size. However, numerical experiments [26, section 6] performed with τ = +O(h) show an error of O(h)—a slight improvement over the a-priori estimates. This +motivates us to link the timestep to the mesh width as τ = O(h). +4 +Semi-Lagrangian Advection applied to the Incom- +pressible Navier-Stokes Equations +Given a temporal mesh t0 < t1 < ... < tN−1 < tN, we elaborate a single timestep +tn−1 �→ tn of size τ := tn − tn−1, n ≤ N. We assume that approximations ωk +h ∈ +Λ1 +h,p(Ω) of ω(tk, ·) ∈ Λ1(Ω) are available for k < n with [0, T] �→ ω ∈ Λ1(Ω) a +solution of (2). +4.1 +Approximation of the flow map +In the Navier-Stokes equations, the flow is induced by the unknown, time-dependent +velocity field u(t, x). Therefore, (15) becomes +∂ +∂τ Xt,t+τ(x) = u(t + τ, Xt,t+τ(x)), +Xt(x) = x , +t ∈ (0, T). +(25) +The discretization of the material derivative requires us to approximate the flow map +Xt,t−τ in order to evaluate (21). +4.1.1 +A first-order scheme +We use the explicit Euler method to approximate the (backward) flow according to +Xtn,tn−τ(x) ≈ x − τu(tn − τ, x), +(26) +where tn is a node in the temporal mesh and τ denotes the timestep size. We only have +access to an approximation un−1 +h +:= (ωn−1 +h +)\ of u at time tn−1, which gives +Xtn,tn−τ(x) ≈ x − τun−1 +h +(x). +(27) +Note that a direct application of the explicit Euler method would require an evaluation +of the velocity field at tn. Instead, we perform a constant extrapolation and evaluate +the velocity field at tn−1, that is, we use un−1 +h +in (27). +The approximation un−1 +h +resides in the space of vector proxies for discrete dif- +ferential 1-forms as discussed in section 3.1. This means that only tangential conti- +nuity of un−1 +h +across faces of elements of the mesh is guaranteed, while discontinu- +ities may appear in the normal direction of the faces. Therefore, un−1 +h +is not defined +point-wise—even though (27) requires point-wise evaluation. For that reason, we will +13 + +replace un−1 +h +by a globally-continuous, smoothened velocity field ¯un−1 +h +that approxi- +mates un−1 +h +(see section 4.1.3 for the construction). We then have +Xtn,tn−τ(x) ≈ x − τ ¯un−1 +h +(x) +(28) +which yields a first-order-in-time approximation of Xtn,tn−τ(x), provided that ¯un−1 +h +is +a first-order approximation of u(tn−1, ·). +4.1.2 +A second-order scheme +A second-order approximation can be achieved by using Heun’s method [44] instead +of explicit Euler. We find the following second-order in time approximations +Xtn,tn−τ(x) ≈ x − τ +2 +� +u∗ +h(x) + un−1 +h +(x − τu∗ +h(x)) +� +, +(29) +Xtntn−2τ(x) ≈ x − τ +� +u∗ +h(x) + un−2 +h +(x − 2τu∗ +h(x)) +� +, +(30) +where we approximate the velocity field at tn by the linear extrapolation u∗ +h = 2un−1 +h +− +un−2 +h +. As described in section 4.1.1, we replace the velocity fields by suitable smooth +approximations. We obtain +Xtn,tn−τ(x) ≈ ¯Xt−τ(x) := x − τ +2 +�¯u∗ +h(x) + ¯un−1 +h +(x − τ ¯u∗ +h(x)) +� +, +(31) +Xtn,tn−2τ(x) ≈ ¯Xt−2τ(x) := x − τ +�¯u∗ +h(x) + ¯un−2 +h +(x − 2τ ¯u∗ +h(x)) +� +, +(32) +where ¯u• +h with • = ∗, n − 1, n − 2 denotes the smoothened version of u• +h as it will be +constructed in the next section. +4.1.3 +Smooth reconstruction of the velocity field +Given a discrete velocity field uZ +h ∈ Λ1 +h,p(Ω), we can define a smoothened version ¯uh +of uh that is +• Lipschitz continuous to ensure stable evaluation of (28), +• well-defined on every point of the meshed domain, +• practically computable, and +• second-order accurate. +We introduce ¯uh as follows. Let hmin denote the length of the shortest edge of the +mesh and (ui +h)i=1,..,d the components of uh. Then, +¯ui +h(x) = +1 +hmin +� xi+ 1 +2 hmin +xi− 1 +2 hmin +ui +h([x1, . . . , xi−1, ξ, xi+1, . . . , xd]T)dξ +(33) +14 + +provides a second-order, Lipschitz-continuous approximation of uh. In the above def- +inition, we can also replace hmin by a localized parameter that scales as O(h) with h +the length of the edges "close" to x. Note that the above integral can be evaluated up +to machine precision using the algorithm as described in section 3.2 for (20). The av- +eraging (33) provides a second-order approximation of uh on every point in the mesh. +4.2 +A first- and second-order SL scheme +We are now ready to turn the ideas of section 3 into a concrete numerical scheme +for the incompressible Navier-Stokes equations as given in (2). We cast (2a) and +(2b) into weak form and, subsequently, do Galerkin discretization in space relying +on those spaces of discrete differential forms introduced in section 3.1. For the first- +order scheme, we have the following discrete variational formulation. Given ωn−1 +h +∈ +Λ1 +h,1(Ω), we search pn +h ∈ Λ0 +h,1(Ω), ωn +h ∈ Λ1 +h,1(Ω) such that +�1 +τ +� +ωn +h − Ih,1 ¯X∗ +tn,tn−τωn−1 +h +� +, ηh +� +Ω ++ε (dωn +h, dηh)Ω + (dpn +h, ηh)Ω = (f n, ηh)Ω , +(34a) +(ωn +h, dψh)Ω = 0 +(34b) +for all ηh ∈ Λ1 +h,1(Ω) and ψh ∈ Λ0 +h,1(Ω). Ih,p denotes the projection operator as defined +in section 3.1. For the second-order scheme, we use second-order timestepping and +second-order discrete differential forms. Given ωn−2 +h +, ωn−1 +h +∈ Λ1 +h,2(Ω), we search pn +h ∈ +Λ0 +h,2(Ω), ωn +h ∈ Λ1 +h,2(Ω) such that +� 1 +2τ +� +3ωn +h − 4Ih,2 ¯X∗ +tn,tn−τωn−1 +h ++ Ih,2 ¯X∗ +tn,tn−2τωn−2 +h +� +, ηh +� +Ω ++ε (dωn +h, dηh)Ω + (dpn +h, ηh)Ω = (f n, ηh)Ω , +(35a) +(ωn +h, dψh)Ω = 0 +(35b) +for all ηh ∈ Λ1 +h,2(Ω) and ψh ∈ Λ0 +h,2(Ω). Numerical experiments reported in section 5 +give evidence that these schemes indeed do provide first- and second-order conver- +gence for smooth solutions. Note that the schemes presented in this section only re- +quire solving symmetric, linear systems of equations at every time-step. +4.3 +Conservative SL schemes +In order to enforce energy-tracking—the correct behavior of the total energy E(t) over +time as expressed in (3)—we add a suitable constraint plus a Lagrange multiplier to +15 + +the discrete variational problems proposed in section 4.2. Given ωn−1 +h +∈ Λ1 +h,1(Ω), we +search pn +h ∈ Λ0 +h,1(Ω), ωn +h ∈ Λ1 +h,1(Ω), and µn ∈ R such that +�1 +τ +� +ωn +h − Ih,1 ¯X∗ +tn,tn−τωn−1 +h +� +, ηh +� +Ω ++ (dpn +h, ηh)Ω + ε(dωn +h,dηh)Ω ++µn [(ωn +h, ηh)Ω + 2ετ(dωn +h, dηh)Ω − τ(f n, ηh)Ω] = (f n, ηh)Ω , +(36a) +(ωn +h, dψh)Ω = 0, +(36b) +(ωn +h, ωn +h)Ω + 2ετ(dωn +h, dωn +h)Ω − τ(f n, ωn +h)Ω = +� +ωn−1 +h +, ωn−1 +h +� +Ω (36c) +for all ηh ∈ Λ1 +h,1(Ω) and ψh ∈ Λ0 +h,1(Ω). Note that the last scalar equation enforces +energy conservation for ε = 0 and f = 0. To solve the nonlinear system (36) for +ωn +h, pn +h, µn, we propose the following iterative scheme. Assume that we have a se- +quence (ωn +h,k)k∈N with ωn +h,k → ωn +h(k → ∞). Then we can employ the Newton-like +linearization +(ωn +h, ωn +h)Ω ← +� +ωn +h,k, ωn +h,k +� +Ω += +� +ωn +h,k−1, ωn +h,k−1 +� +Ω + 2 +� +ωn +h,k−1, ωn +h,k − ωn +h,k−1 +� +Ω + O +� +||ωn +h,k − ωn +h,k−1||2 +Ω +� +. +(37) +We use the above expansion to replace the quadratic terms (ωn, ωn)Ω and (dωn, dωn)Ω +and arrive at the following linear variational problem to be solved in every step of +the inner iteration. Given ωn−1 +h +, ωn +h,k−1 ∈ Λ1 +h,1(Ω), we search pn +h,k ∈ Λ0 +h,1(Ω), ωn +h,k ∈ +Λ1 +h,1(Ω), and µn +k ∈ R such that +�1 +τ +� +ωn +h,k − Ih,1 ¯X∗ +tn,tn−τωn−1 +h +� +, ηh +� +Ω ++ +� +dpn +h,k, ηh +� +Ω + ε(dωn +h,k, dηh)Ω ++µn +k +� +(ωn +h,k−1, ηh)Ω + 2ετ(dωn +h,k−1, dηh)Ω − τ(f n, ηh)Ω +� += (f n, ηh)Ω , +(38a) +� +ωn +h,k, dψh +� +Ω = 0, +(38b) +� +ωn +h,k−1, ωn +h,k−1 +� +Ω + 2 +� +ωn +h,k−1, ωn +h,k − ωn +h,k−1 +� ++2ετ[(dωn +h,k−1, dωn +h,k−1)Ω + 2(dωn +h,k−1, dωn +h,k − dωn +h,k−1)] +−τ(f n, ωn +h,k−1)Ω = +� +ωn−1 +h +, ωn−1 +h +� +Ω +(38c) +for all ηh ∈ Λ1 +h,1(Ω) and ψh ∈ Λ0 +h,1(Ω). This is a symmetric, linear system that +is equivalent to the original system in the limit (ωn +h,k, pn +h,k, µn +k) → (ωn +h, pn +h, µn). In +numerical experiments we observe that it takes around 2-3 steps of the inner iteration +to converge to machine precision using an initial guess ωn +h,0 = ωn−1 +h +. We can apply the +same idea for energy-tracking to our second-order scheme as proposed in section 4.2. +16 + +Remark 4 For the case ε = 0 and f = 0, we can also enforce helicity conservation by +adding a suitable Lagrange multiplier to the discrete system. Given ωn−1 +h +∈ Λ1 +h,1(Ω), +we search pn +h ∈ Λ0 +h,1, ωn +h ∈ Λ1 +h,1(Ω), and λn ∈ R such that +�1 +τ +� +ωn +h − Ih,1 ¯X∗ +tn,tn−τωn−1 +h +� +, ηh +� +Ω ++ (dpn +h, ηh)Ω + ε(dωn +h, dηh)Ω ++λn(ωn +h, dηh)Ω + λn(dωn +h, ηh)Ω + µn(ωn +h, ηh)Ω = 0, +(39a) +(ωn +h, dψh)Ω = 0, +(39b) +(ωn +h, ωn +h)Ω = +� +ωn−1 +h +, ωn−1 +h +� +Ω , +(39c) +(ωn +h, dωn +h)Ω = +� +ωn−1 +h +, dωn−1 +h +� +Ω +(39d) +for all ηh ∈ Λ1 +h,1(Ω) and ψh ∈ Λ0 +h,1(Ω). By linearization as for energy-tracking, we +obtain the following system. Given ωn−1, ωn +h,k−1 ∈ Λ1 +h,1(Ω), we search pn +h,k ∈ Λ0 +h,1(Ω), +ωn +h,k ∈ Λ1 +h,1(Ω), and λn +k ∈ R such that +�1 +τ +� +ωn +h,k − Ih,1 ¯X∗ +tn,tn−τωn−1� +, ηh +� +Ω ++ +� +dpn +h,k, ηh +� +Ω + ε(dωn +h,k, dηh)Ω ++λn +k(ωn +h,k−1, dηh)Ω + λn +k(dωn +h,k−1, ηh)Ω + µn +k(ωn +h,k−1, ηh)Ω = 0, +(40a) +� +ωn +h,k, dψh +� +Ω = 0, +(40b) +� +ωn +h,k−1, ωn +h,k−1 +� +Ω + 2 +� +ωn +h,k−1, ωn +h,k − ωn +h,k−1 +� += +� +ωn−1 +h +, ωn−1 +h +� +Ω , (40c) +� +ωn +h,k−1, dωn +h,k +� +Ω + +� +ωn +h,k, dωn +h,k−1 +� +Ω − +� +ωn +h,k−1, dωn +h,k−1 +� +Ω = +� +ωn−1 +h +, dωn−1 +h +� +Ω (40d) +for all ηh ∈ Λ1 +h,1(Ω) and ψh ∈ Λ0 +h,1(Ω). Again, this is a symmetric, linear system +that is equivalent to the original system in the limit (ωn +h,k, pn +h,k, µn +k) → (ωn +h, pn +h, µn). +Also, numerical experiments hint that it takes around 2-3 steps of the inner iteration to +converge to machine precision using an initial guess ωn +h,0 = ωn−1 +h +. +5 +Numerical Results +In this section, we present multiple numerical experiments to validate the new scheme. +In the following, we will always consider schemes that include energy-tracking as in- +troduced in section 4.3 unless stated otherwise. We only include helicity-conservation +as introduced in Remark 4 for domains in R3 when ε = 0 and f = 0. The ex- +periments are based on a C++ code that heavily relies on MFEM [2]. The source +code is published under the GNU General Public License in the online code repository +https://gitlab.com/WouterTonnon/semi-lagrangian-tools. +17 + +10−1 +mesh width h +10−3 +10−2 +10−1 +L2 Error u +first-order, +ε = 0 +first-order, +ε = 0.01π−2 +first-order, +ε = 0.1π−2 +second-order, +ε = 0 +second-order, +ε = 0.01π−2 +second-order, +ε = 0.1π−2 +Figure 6: Convergence results for Experiment 1 using the first- and second-order +schemes on simplicial meshes with mesh width h, timestep τ = 0.065804h. We ob- +serve first- and second-order algebraic convergence for all values of ε. +5.1 +Experiment 1: Decaying Taylor-Green Vortex +We consider the incompressible Navier-Stokes equations with Ω = [− 1 +2, 1 +2]2, T = 1, +varying ε ≥ 0, f = 0, and vanishing boundary conditions. An exact, classical solution +is the following Taylor-Green vortex [42] +u(t, x) = +� cos(πx1) sin(πx2) +− sin(πx1) cos(πx2) +� +e−2π2εt. +(41) +We ran a h-convergence analysis for different values of ε ≥ 0 and summarize the +results in Figure 6. We also track the energy for different values of ε and compare the +energy to the exact solution in Figure 7. +5.2 +Experiment 2: Taylor-Green Vortex +We consider the incompressible Navier-Stokes equations with Ω = [−1, 1]2, T = 1, +varying ε ≥ 0, f and the boundary conditions chosen such that +u(t, x) = +� cos(πx1) sin(πx2) +− sin(πx1) cos(πx2) +� +(42) +is an exact, classical solution. We ran a h-convergence analysis for all parameters +and summarize the results in Figure 8. We observe first- and second-order algebraic +convergence for the corresponding schemes. Note that the error of the scheme is stable +18 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +time t +0.58 +0.60 +0.62 +0.64 +0.66 +0.68 +0.70 +L2 Norm u +exact, +ε = 0 +first-order, +ε = 0 +exact, +ε = 0.01π−2 +first-order, +ε = 0.01π−2 +exact, +ε = 0.1π−2 +first-order, +ε = 0.1π−2 +(a) first-order +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +time t +0.58 +0.60 +0.62 +0.64 +0.66 +0.68 +0.70 +L2 Norm u +exact, +ε = 0 +second-order, +ε = 0 +exact, +ε = 0.01π−2 +second-order, +ε = 0.01π−2 +exact, +ε = 0.1π−2 +second-order, +ε = 0.1π−2 +(b) second-order +Figure 7: Energy of the discrete and exact solution for Experiment 1 using the first- +and second-order, energy-tracking schemes on a simplicial mesh with mesh width h = +0.0949795, timestep τ = 0.00625. +19 + +10−1 +100 +mesh width h +10−2 +10−1 +100 +L2 Error u +first-order, +ε = 0 +first-order, +ε = 0.01 +first-order, +ε = 1 +second-order, +ε = 0 +second-order, +ε = 0.01 +second-order, +ε = 1 +Figure 8: Convergence results for Experiment 2 using the first- and second-order, +non-conservative schemes on simplicial meshes with mesh width h, timestep τ = +0.032902h. As ε → 0 the error remains bounded. +as ε → 0. This is in agreement with the analysis performed on the vectorial advection +equations presented in [24]. This experiment thus suggests that this analysis can be +extended to the scheme presented in this work. +5.3 +Experiment 3: A rotating hump problem +The Taylor-Green vortices provide exact solutions to the incompressible Navier-Stokes +equations, but they are rather "static" solutions. In this experiment, we consider a more +dynamic solution. Let us consider the incompressible Navier-Stokes equations with +Ω = [− 1 +2, 1 +2]2, T = 1, ε = 0, f = 0, and vanishing normal boundary conditions. We +consider the following initial condition +u0(x) = +� +−πex1 cos(πx1) sin(πx2) +πex1 sin(πx1) cos(πx2) − ex1 cos(πx1) cos(πx2) +� +. +(43) +The exact solution to this problem is unknown, so we compare the solution computed +by our scheme to the solution produced by the incompressible Euler solver Gerris [37]. +The algorithm used in this solver is described in [36]. We computed the solution to this +problem using the second-order, energy-tracking scheme presented in this work. Then, +we plotted the magnitude of the computed velocity vector field for different mesh-sizes +and time-steps at different time instances in Figures 10 to 13. Note that different visu- +alisation tools were used to visualize the fields computed using the different solvers, +but we observe that the solution computed by the semi-Lagrangian scheme comes vi- +sually closer to the solution computed by Gerris as we decrease the mesh width and +20 + +10−1 +mesh width h +10−2 +10−1 +100 +L2 Error u +second-order, +cons +Figure 9: Convergence results for Experiment 2 using the second-order, conservative +scheme on simplicial meshes with mesh width h, timestep τ = 0.06580h, and final +time T = 1. The reference solution is a solution computed by Gerris [36] +time step. This is confirmed by Figure 9, where we display the L2 error between the +solution computed using the semi-Lagrangian scheme and the solution computed using +Gerris. In Figure 15, we display the vector field as computed using the second-order, +conservative semi-Lagrangian scheme. +Also, in Figure 16 we display the values of the L2 norm over time of the solu- +tions produced using our first- and second-, energy-tracking and non-energy-tracking +schemes. Note that the energy-tracking schemes preserve the L2 norm as expected. +The first-order, non-conservative scheme seems unstable at first, but in reality the or- +dinate axis spans a very small range and it turns out that the L2 norm converges to a +bounded value for longer run-times. Note that the helicity has no meaning in R2. +5.4 +Experiment 4: Taylor-Green Vortex in 3D +To observe conservation of helicity, we need to consider a problem in 3D. We con- +sider the incompressible Navier-Stokes equations with Ω = [− 1 +2, 1 +2]3, T = 1, ε = 0, +vanishing normal boundary conditions, and f chosen such that +u(t, x) = +� +� +cos(πx1) sin(πx2) sin(πx3) +− 1 +2 sin(πx1) cos(πx2) sin(πx3) +− 1 +2 sin(πx1) sin(πx2) cos(πx3) +� +� +(44) +is a solution. Note that since the solution is static, we can enforce helicity conser- +vation despite f ̸= 0. We run several experiments using the first- and second-order, +21 + +(a) t = 0.25 +(b) t = 0.5 +(c) t = 0.75 +(d) t = 1 +Figure 10: Experiment 3: mesh width h = 0.379918 and time-step τ = 0.025. +(a) t = 0.25 +(b) t = 0.5 +(c) t = 0.75 +(d) t = 1 +Figure 11: Experiment 3: mesh width h = 0.0949795 and time-step τ = 0.00625. +(a) t = 0.25 +(b) t = 0.5 +(c) t = 0.75 +(d) t = 1 +Figure 12: Experiment 3: mesh width h = 0.023744875 and time-step τ = 0.0015625. +(a) t = 0.25 +(b) t = 0.5 +(c) t = 0.75 +(d) t = 1 +Figure 13: Reference solution Experiment 3 computed using [37]. +0 +1 +2 +3 +4 +5 +Figure 14: Colorbar associated with Figures 10 to 13 +22 + +(a) t = 0.25 +(b) t = 0.5 +(c) t = 0.75 +(d) t = 1 +0 +1 +2 +3 +4 +5 +Figure 15: Velocity field for Experiment 3 computed using the second-order, conser- +vative semi-Lagrangian scheme on a simplicial mesh with mesh width h = 0.189959 +and time-step τ = 0.0125. The colors indicate the magnitude of the vector. +23 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +time t +2.350 +2.355 +2.360 +2.365 +2.370 +2.375 +2.380 +L2 Norm u +second-order, +cons +first-order, +cons +second-order, +non-cons +first-order, +non-cons +Figure 16: The L2 norm of the computed solutions for Experiment 3 using differ- +ent variants of the semi-Lagrangian scheme on a simplicial mesh with mesh width +h = 0.04748975 and time-step τ = 0.003125. In the legend, ’cons’ is short for ’con- +servative’ and refers to energy-tracking schemes. We use ε = 0 and f = 0. +conservative semi-Lagrangian schemes. We summarize the results in Figure 17 and we +observe first- and second-order algebraic convergence for the corresponding schemes. +In Figure 18 and Figure 19 we plot the L2 norm and helicity over time of the discrete +solution for both the first- and second-order scheme. We observe that both quantities +are conserved up to machine precision. +5.5 +Experiment 5: A transient solution in 3D +To verify the scheme for transient solutions in 3D, we consider the incompressible +Navier-Stokes equations with Ω = [− 1 +2, 1 +2]3, T = 1, ε = 0, vanishing normal boundary +conditions and f is chosen such that +u(t, x) = +� +� +−x2π cos( t +4 + πx2x3) cos(πx2) +−x3π cos( t +4 + πx1x3) cos(πx3) +−x1π cos( t +4 + πx1x2) cos(πx1) +� +� +(45) +is a solution. We ran a simulation for different mesh-sizes with time-steps determined +by a suitable CFL condition. We summarize the results in Figure 20. We observe +second-order convergence for the second-order scheme. The first-order scheme seems +to achieve an order of convergence that is between first- and second-order, but this may +be pre-asymptotic behaviour. +24 + +10−1 +2 × 10−1 +3 × 10−1 +4 × 10−1 +6 × 10−1 +mesh width h +10−2 +10−1 +L2 Error u +first-order +second-order +Figure 17: Convergence results for Experiment 4 using the first- and second-order, +conservative schemes on simplicial meshes with mesh width h, timestep τ = +1 +√ +2h. +0.2 +0.4 +0.6 +0.8 +1.0 +time t +0.430 +0.435 +0.440 +0.445 +0.450 +0.455 +L2 Norm u +second-order, +cons +first-order, +cons +second-order, +non-cons +first-order, +non-cons +Figure 18: The L2 norm of the computed solutions for Experiment 4 using differ- +ent variants of the semi-Lagrangian scheme on a simplicial mesh with mesh width +h = 0.08838834764 and time-step τ = 0.0625. In the legend, ’cons’ is short for ’con- +servative’ and refers to energy-tracking schemes and helicity-conserving schemes. We +use ε = 0 and f = 0. +25 + +0.2 +0.4 +0.6 +0.8 +1.0 +time t +−0.006 +−0.004 +−0.002 +0.000 +Helicity u +second-order, +cons +first-order, +cons +second-order, +non-cons +first-order, +non-cons +Figure 19: The helicity of the computed solutions for Experiment 4 using differ- +ent variants of the semi-Lagrangian scheme on a simplicial mesh with mesh width +h = 0.08838834764 and time-step τ = 0.0625. In the legend, ’cons’ is short for +’conservative’ and refers to energy-tracking and helicity-conserving schemes. We use +ε = 0 and f = 0. +10−1 +2 × 10−1 +3 × 10−1 +mesh width h +10−1 +L2 Error u +first-order +second-order +Figure 20: Convergence results for Experiment 5 using the first- and second-order +schemes without energy-tracking and helicity-conservation on simplicial meshes with +mesh width h, timestep τ = +1 +√ +2h. +26 + +0 +0.5 +1 +1.5 +2 +2.5 +Figure 21: Velocity field at T = 7.93s of Experiment 6 computed using the second- +order, non-conservative semi-Lagrangian scheme on a simplicial mesh with mesh +width h = 0.189959 and τ = 0.01. +5.6 +Experiment 6: Lid-driven cavity with slippery walls +In this section, we simulate a situation that resembles a lid-driven cavity problem. +Consider the incompressible Navier-Stokes with Ω = [− 1 +2, 1 +2]2, T = 7.93, ε = 0, +vanishing normal boundary conditions and the initial velocity field is set equal to zero. +Then, to simulate a moving lid at the top, we apply the force-field f(t, x) = [v(x), 0]T +with +v(x) = +� +exp +� +1 − +1 +1−100(0.5−x2)2 +� +, +if 1 − 100(0.5 − x2)2 > 0, +0, +else. +(46) +This force field gives a strong force in the x1-direction close to the top lid, but quickly +tapers off to zero as we go further from the top lid. In Figure 21, we display the +computed velocity field. Note that, because we apply slip boundary conditions, we do +not expect to observe vortices. The numerical solution reproduces this expectation. +27 + +5.7 +Experiment 7: A more complicated domain +The numerical experiments given above, show the convergence and conservative prop- +erties of the introduced schemes. However, these experiments are all performed on +very simple, rectangular domains. In this experiment, we consider a more complicated +domain and mesh (generated using [22]) as shown in Figure 22. +We consider the case of the incompressible Navier-Stokes equations on the domain +as given in Figure 22, T = 100, ε = 0, f = 0 and vanishing normal boundary condi- +tions. We need to construct an initial condition that is divergence-free with vanishing +normal boundary conditions. Following an approach close to a Chorin projection, we +start with +w(x, y) = +�sin +� +2 cos( +� +x2 + y2) − atan2(y, x) +� +sin +� +cos( +� +x2 + y2) − 2 atan2(y, x) +� +� +. +(47) +We use this definition to define a scalar function, ϕ, as +∆ϕ = ∇ · w +in Ω, +(48) +∇ϕ · ˆn = w · ˆn +on ∂Ω. +(49) +We can define our initial condition, u0, as +u0 = w − ∇ϕ +(50) +Note that u0 is divergence-free and has vanishing normal boundary conditions. The +above system of equations can be solved using an appropriate finite-element imple- +mentation. +Note that in this experiment, the field outside the domain is unknown. This is +well-defined on a continuous level, since vanishing boundary conditions imply that +no particle will flow in from outside the domain. However, on the discrete level we +cannot guarantee that the same will happen. It could happen that a part of a transported +edge (as discussed in section 3.2) ends up outside the domain. In this case, we will +assume that the average of the vector field along the part of the edge that lies outside +the domain, will have the same value as the average of the corresponding edge in its +original location (before transport) at the previous timestep. +The first ten seconds were simulated and a video of the results can be found at +https://youtu.be/Eica8XHLtxY. For the different schemes, we also tracked +the energy in Figure 23. +6 +Conclusion +We have developed a mesh-based semi-Lagrangian discretization of the time-dependent +incompressible Navier-Stokes equations with free boundary conditions recast as a non- +linear transport problem for a momentum 1-form. A linearly implicit fully discrete +version of the scheme enjoys excellent stability properties in the vanishing-viscosity +28 + +Figure 22: Domain and mesh associated with Experiment 7. +0 +10 +20 +30 +40 +50 +60 +time t +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +L2 Norm u +first-order, +non-cons +second-order, +non-cons +second-order, +cons +first-order, +cons +Figure 23: The L2 norm of the computed solutions for Experiment 7 using different +variants of the semi-Lagrangian scheme on a simplicial mesh as given in Figure 22 +and time-step τ = 0.01. In the legend, ’cons’ is short for ’conservative’ and refers to +energy-tracking schemes. +29 + +limit and is applicable to inviscid incompressible Euler flows. However, in this case +conservation of energy and helicity have to be enforced separately. Making the reason- +able choice of a time-step size proportional to the mesh width, the algorithm involves +only local computations. Yet, these are significantly more expensive compared to those +required for purely Eulerian finite-element and finite-volume methods. At this point +the verdict on the competitiveness of our semi-Lagrangian scheme is still open. +A +Two formulations of the momentum equation +Consider the momentum equation in (1) +∂tu + u · ∇u − ε∆u + ∇p = 0. +Note that we have by standard vector calculus identities +∆u = ∇(∇ · u) − ∇ × ∇ × u, +where we can use ∇ · u = 0 to obtain +∆u = −∇ × ∇ × u. +This allows us to rewrite the momentum equation as +∂tu + u · ∇u + ε∇ × ∇ × u + ∇p = 0. +Using the gradient of the dot-product, we find +∇(u · u) = 2u · ∇u + 2u × (∇ × u). +This identity allows us to rewrite the momentum equation to +∂tu + ∇(u · u) − u × (∇ × u) + ε∇ × ∇ × u + ∇ +� +−1 +2u · u + p +� += 0. +From [27, 24], we obtain the identity +(Luω)\ = ∇(u · u) − u × (∇ × u) +where ω is the differential 1-form such that u = ω\. 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DOI: 10.1006/jcph. +2001.6847. +35 + diff --git a/LdE4T4oBgHgl3EQfJww5/content/tmp_files/load_file.txt b/LdE4T4oBgHgl3EQfJww5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e06bdef9dfd01f3a12cee35b444d7a00fc299f12 --- /dev/null +++ b/LdE4T4oBgHgl3EQfJww5/content/tmp_files/load_file.txt @@ -0,0 +1,1097 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf,len=1096 +page_content='Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows Wouter Tonnon* Ralf Hiptmair† January 13, 2023 1 Incompressible Navier-Stokes Equations We consider the incompressible Navier-Stokes equations—a standard hydrodynamic model for the motion of an incompressible, potentially-viscous fluid—in a container with rigid walls, where we impose so-called “free boundary conditions” in the par- lance of [31, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 346] and [43, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 502], see the latter article for further references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We search the fluid velocity field u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' x) and the pressure p(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' x) as functions of time t and space x on a bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Lipschitz domain Ω ⊂ Rd such that they solve the evolution boundary-value problem ∂tu + u · ∇u − ε∆u + ∇p = f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' on ]0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' T[×Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (1a) ∇ · u = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' on ]0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' T[×Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (1b) u · n = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' on ]0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' T[×∂Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (1c) εn × ∇ × u = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' on ]0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' T[×∂Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (1d) u = u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' on {0} × Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (1e) where ε ≥ 0 denotes a (non-dimensional) viscosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' f a given source term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' T > 0 the final time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ∂Ω the boundary of Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' and n(x) the outward normal vector at x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The initial condition u0 is to satisfy ∇ · u0 = 0 in Ω and u0 · n = 0, εn × ∇ × u0 = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Based on the variational description of the Navier-Stokes equations as described in [5], u can be interpreted as a differential 1-form [33] and we can recast system (1) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Let Λk(Ω) for k ∈ N denote the space of differential k-forms on SAM, ETH Zürich, CH-8092 Zürich, wouter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='tonnon@sam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='ch †SAM, ETH Zürich, CH-8092 Zürich, ralf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='hiptmair@sam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='ch 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='04923v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='NA] 12 Jan 2023 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Then we search ω ∈ Λ1(Ω) and p ∈ Λ0(Ω) such that Duω + εδdω + dp = f, on ]0, T[×Ω, (2a) δω = 0, on ]0, T[×Ω, (2b) tr ⋆ω = 0, on ]0, T[×∂Ω, (2c) ε tr ⋆dω = 0, on ]0, T[×∂Ω, (2d) ω = ω0, on{0} × Ω, (2e) where Duω denotes the material derivative of ω with respect to u, d : Λk−1(Ω) �→ Λk(Ω) the exterior derivative, δ : Λk(Ω) �→ Λk−1(Ω) the exterior coderivative, and the trace tr is the pullback under the embedding ∂Ω ⊂ ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Here, u is related to ω through ω := uZ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' u is the vector proxy of ω w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Similarly, we have that Λ1(Ω) ∋ ω0 := u0Z and Λ1(Ω) ∋ f := fZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that (2) can be derived through classical vector calculus for vector proxies as shown in Appendix A for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' As shown in [18, 19], sufficiently-smooth solutions ω :]0, T[�→ Λ1(Ω) of the in- compressible Navier-Stokes equations as given in system (2) satisfy an energy relation, that is, dE dt (t) := d dt 1 2 � Ω ω(t) ∧ ⋆ω(t) = −ε � Ω dω(t) ∧ ⋆dω(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (3) This relation implies energy conservation for ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In the case of ε = 0, we also have helicity conservation, that is, ε = 0 =⇒ dH dt (t) := d dt � Ω dω(t) ∧ ω(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (4) Note that the Onsager conjecture tells us that in the case ε = 0 the solutions need to be at least Hölder regular with exponent 1 3 for energy conservation to hold [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Remark 1 We acknowledge that the boundary condition (1d) is non-standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This boundary condition was chosen because it is the natural boundary condition associ- ated to system (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' To enforce the standard no-slip boundary conditions, (1d) could be replaced by εu × n = u on ]0, T[×∂Ω, that is, we impose an essential instead of natural boundary condition to system (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Unfortunately, in this case, the scheme presented in this work leads to an ill-posed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In the case ε = 0, the only imposed boundary condition (1c) is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Remark 2 Boundary conditions (1c),(1d) can be interpreted as slip boundary con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' However, on smooth domains Ω, they are only equivalent to Navier’s slip boundary conditions if the Weingarten map related to ∂Ω vanishes [31, section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 2 Outline and Related Work We propose a semi-Lagrangian approach to the discretization of the reformulated Navier-Stokes boundary value problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This method revolves around the dis- cretization of the material derivative Duω in the framework of a finite-element Galerkin 2 discretization on a fixed spatial mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The main idea is to approximate Duω by back- ward difference quotients involving transported snapshots of the 1-form ω, which can be computed via the pullback induced by the flow of the velocity vector field u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Semi-Lagrangian methods for transient transport equations like (2) are well-es- tablished for the linear case when u is a given Lipschitz-continuous velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In particular, for ω a 0-form, that is, a plain scalar-valued function, plenty of semi- Lagrangian approaches have been proposed and investigated, see, for instance, [8, 7, 20, 21, 23, 35, 39, 40, 6, 12, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We refer to [24, Chapter 5] for a comprehensive pre-2013 literature review on the analysis of general semi-Lagrangian schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Most of these methods focus on mapping point values under the flow, with the exception of a particularly interesting class of semi-Lagrangian methods known as Lagrange- Galerkin methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Lagrange-Galerkin methods do not transport point values, but rather triangles (in 2D) or tetrahedra (in 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Refer to [10] for a review of those meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Meanwhile semi-Lagrangian methods for transport problems for differential forms of any order have been developed [26, 25, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The next section will review these semi-Lagrangian methods for linear transport problems with emphasis on 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We will also introduce a new scheme which is second-order in space and time based on so-called “small edges”, see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Semi-Lagrangian schemes for the incompressible Navier-Stokes equations are also well-documented in literature, with emphasis on the Lagrange-Galerkin method [10, 14, 15, 30, 1, 11, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' A survey of the application of Lagrange-Galerkin methods to the incompressible Navier-Stokes equations is given in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' It is important to note that these methods require the evaluation of integrals of transported quantities and, in case these integrals cannot be computed exactly, instabilities can occur [12, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' A possi- ble remedy is to add an additional stabilization term that includes artificial diffusion [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Other semi-Lagrangian methods for incompressible Navier-Stokes equations di- rectly transport point values with the nodes of a mesh instead of evaluating integrals of transported quantities, see [34, 29, 47, 46, 13] and [16], where the last work makes use of exponential integrators [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Most authors employ spectral elements for the discretization in space [34, 29], but any type of finite-element space with degrees-of- freedom relying on point evaluations can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The methods proposed in [47, 46] are also based on finite-element spaces with degrees-of-freedom on nodes, but em- ploy backward-difference approximations for the material derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The work [13] proposes an explicit semi-Lagrangian method still built around the transport of point values in the nodes of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The diffusion term is also taken into account in a semi-Lagrangian fashion and the incompressibility constraint is enforced by means of a Chorin projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Also [13] proposes an explicit semi-Lagrangian scheme using the same principles, but based on the vorticity-streamfunction form of the incompressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' All the mentioned semi-Lagrangian schemes rely on the transport of point val- ues of continuous vector fields, which is the perspective embraced in formulation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' However, we believe that, in particular in the case of free boundary conditions (1c) 3 and (1d), the semi-Lagrangian method based on (2) offers benefits similar to the bene- fits bestowed by the use of discrete differential forms (finite-element exterior calculus, FEEC [3, 4]) for the discretization of electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Section 4 will convey that the boundary conditions (2c), (2d), and the incompressiblity constraint can very natu- rally be incorporated into a variational formulation of (2) posed in spaces of 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This has been the main motivation for pursuing the new idea of a semi-Lagrangian method for (2) that employs discrete 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Another motivation has been the ex- pected excellent robustness of the semi-Lagrangian discretization in the limit ε �→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Numerical tests reported in section 5 will confirm this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Two more aspects of our method are worth noting: Firstly, a discrete 1-form ωh will not immediately spawn a continuous velocity field uh = ωZ h, However, continuity is essential for defining a meaningful flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We need an additional averaging step, which we present in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Secondly, since semi-Lagrangian methods fail to respect the decay/conservation laws (3) and (4) exactly, we present a way how to enforce them in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 3 Semi-Lagrangian Advection of differential forms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1 Discrete differential forms We start from a simplicial triangulation Kh(Ω) of Ω, which may rely on a piecewise linear approximation of ∂Ω so that it covers a slightly perturbed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1 Lowest-order case: Whitney forms For Λ0(Ω)—the space of 0-forms on Ω, which is just a space of real-valued func- tions—the usual (Lagrange) finite-element space of continuous, piecewise-linear, poly- nomial functions provides the space Λ0 h,1(Ω) of lowest-order discrete 0-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Let d ∈ {2, 3}, K a d-simplex with edges {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., e3(d−1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' To construct lowest- order discrete 1-forms on K, we associate to every edge ei a local shape function wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Let the edge ei be directed from vertex v1 i to v2 i , then the local shape function wei ∈ Λ1(K) associated with edge ei is wei := λv1 i dλv2 i − λv2 i dλv1 i , (5) where λv represents the barycentric coordinate associated with vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We define the lowest-order, local space of discrete 1-forms Λ1 h,1(K) := span{we;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' e an edge of K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (6) Using these local spaces, we can define the global space of lowest-order, discrete 1- forms Λ1 h,1(Ω) := {ω ∈ Λ1(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ∀K ∈ Kh(Ω) : ω|K ∈ Λ1 h,1(K)}, (7) 4 (0,0) (1,0) (0,1) 1 2 3 4 5 6 7 8 9 (a) 9 small edges of a second-order element in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' All the edges between the different connection points are small edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In 3D, we simply have all these small edges on the faces of the simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' edge no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' edge no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 1 [ x y ] �→ � x(x+y−1) −(x−1)(x+y−1) � 6 [ x y ] �→ � (y−1)(x+y−1) x(1−x−y) � 2 [ x y ] �→ � −y2 y(x−1) � 7 [ x y ] �→ � −xy x(x−1) � 3 [ x y ] �→ � −y2 xy � 8 [ x y ] �→ � y(1−y) xy � 4 [ x y ] �→ � −xy x2 � 9 [ x y ] �→ � y(x+y−1) x(1−x−y) � 5 [ x y ] �→ � x(1−y) x2 � (b) Local shape functions (l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=') for the unit triangle associated with second-order, discrete differential forms in 2D as proposed in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Each shape function corre- sponds to the small edge in (a) with the same numbering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Figure 1: Illustration of small edges (a) and corresponding local shape functions (b) for the unit triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 5 where Λ1(Ω) again denotes the space of differential 1-forms on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We demand that for every ω ∈ Λ1(Ω) integration along any smooth oriented path yields a unique value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Thus, the requirement ω ∈ Λ1(Ω) imposes tangential continuity on the vector proxy of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 Second-order discrete forms Similar to the lowest-order case, the space Λ0 h,2(Ω) of second-order discrete 0-forms is spawned by the usual (Lagrange) finite-element space of continuous, piecewise- quadratic, polynomial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Let d ∈ {2, 3}, K a d-simplex with edges {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., e3(d−1)} and vertices {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., vd+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' To construct second-order discrete 1-forms, we associate local shape functions to "small edges".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We can construct 3(d + 1)(d − 1) small edges [38, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2] by defining ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., d + 1} and ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., 3(d − 1)} {vi, ej} := {vi + 1 2(x − vi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' x ∈ ej}, where {vi, ej} denotes the small edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In Figure 1a we illustrate the 9 small edges of a 2-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' For example, we see that small edge 9 can be written as {(0, 0), [(1, 0), (0, 1)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' To make the difference between small edges and edges of the mesh explicit, we will sometimes refer to the latter as "big edges".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The local shape function [38, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3] associated with {vi, ej} is given by w{vi,ej} := λviwej, where wej denotes the Whitney 1-form associated with the big edge ej as defined in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In Figure 1b we give explicit expressions for the shape functions associated with the small edges in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that the local shape functions of the form w{v,e} associated with small edges in the interior (d = 2) or on the same face (d = 3) of the form {v, e} such that v /∈ ∂e (example: small edge 7, 8, and 9 in Figure 1a) are linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We define the second-order, local space of discrete 1-forms [38, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3] Λ1 h,2(K) := span{w{v,e};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' v a vertex of K, e a (big) edge of K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (8) Using these local spaces, we can define the global space of second-order, discrete 1- forms Λ1 h,2(Ω) := {ω ∈ Λ1(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ∀K ∈ Kh(Ω) : ω|K ∈ Λ1 h,2(K)}, (9) where again we have tangential continuity by a similar argument as in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3 Projection operators We denote by Eh,p(Ω) the global set of big edges (p = 1) or small edges (p = 2) associated with Kh(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We will define the projection operator Ih,p : Λ1(Ω) �→ Λ1 h,p(Ω) 6 as the unique operator that maps ω ∈ Λ1(Ω) to ωh ∈ Λ1 h,p(Ω) such that the mismatch � e∈Eh,p(Ω) �� e ω − � e ωh �2 (10) is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that for p = 1, this mismatch can be made to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In this case, Ih,1 agrees with the usual edge-based nodal projection operator [27, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='11)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In practice, we can compute the projection locally as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Let K ∈ Kh(Ω) be a d-simplex, d ∈ {2, 3}, and let {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., sNp,d} and {ws1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., wsNp,d} denote the corre- sponding big (p = 1) or small (p = 2) edges and corresponding shape functions as introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Specifically, we have N1,2 = 3, N1,3 = 6, N2,2 = 9, and N2,3 = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We can define the matrix (M)i,j = � si wsj, 1 ≤ i, j ≤ Np,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (11) We will say that there is an interaction from edge sj to si if (M)i,j ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that for p = 1, M is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' For p = 2 the local shape functions are linearly dependent and, thus, the above matrix is not invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' However, we can decompose M into invertible and singular sub-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' For illustrative purposes we display for p = 2 and d = 2 the decomposition of M in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The three top-left sub-matrices in Figure 2b are invertible 2 × 2 matrices that describe the interaction between the two small edges that lie on the same big edge, that is, the blue, red, and green sub- matrix in Figure 2b correspond to the blue, red, and green small edges in Figure 2a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The orange sub-matrix in Figure 2b is a 3 × 3 matrix with rank 2 that describes the interaction between the three small edges that lie in the interior of the simplex in Figure 2a, that is, the orange small edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The gray sub-matrix encodes the one-directional interaction from the the small edges that lie on a big edge to the small edges in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that the decomposition of M as given in Figure 2b is not limited to d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The idea can be extended to d = 3 by considering each face of a 3-simplex as a 2-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This is sufficient, since for d = 3 we have no small edges in the interior and there is no interaction between small edges that do not lie on the same face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We give the general structure of M in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that the small, purple sub-matrices represent invertible 2 × 2 matrices and the bigger, orange sub-matrices represent 3 × 3 matrices with rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In order to find ωh �� K ∈ Λ1 h,p(K) such that ωh �� K = Ih,pω �� K, let ⃗ηK be a vector of coefficients η1 K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., η Np,d K such that ωh|K = Np,d � i=1 ηi Kwsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (12) We can then compute ⃗ηK as a least-squares solution of M⃗ηK = �� si ω � 1≤i≤Np,d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (13) 7 (a) 2-simplex K (b) matrix M (d = 2) (c) matrix M (d = 3) Figure 2: For p = 2 and d = 2 the matrix M corresponding to the 2-simplex K in (a) has the form given in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Each row and column in M is associated to a small edge in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Each sub-matrix in (b) describes the interactions between edges with the same color in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The gray sub-matrix is an exception as it describes the one-directional interaction between the small edges that lie on a big edge and the small edges that lie in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' For d = 3, we M has the structure as shown in Figure 2c, where the purple sub-matrixs are 2 × 2 invertible matrices and the orange sub-matrixs are 3 × 3 matrices of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Without loss of generality we assume that M has the form as given in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Then, we solve (13) as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The local shape functions related to small edges that lie on a big edge of the simplex are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We solve for their coefficients first, that is, we solve the system corresponding to the invertible blue sub-matrices in Figure 2c first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Using the results from step 1, we can move the gray sub-matrix in Figure 2c to the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Then, we solve the matrix-system corresponding to the orange sub-matrices in Figure 2c in a least-squares sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' If we perform the above steps for all K ∈ Kh(Ω), we find ωh = Ih,pω ∈ Λ1 h,p(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that only the shape functions associated to small edges on a face contribute to the tangential fields on that face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Therefore, the above procedure yields tangential continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Remark 3 For p = 1, (13) reduces to ηi K = � si ω, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., 3(d − 1)} (14) with si a big edge of the 3-simplex K for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., 3(d − 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This yields the standard nodal interpolation operator of [27, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='11)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 Semi-Lagrangian material derivative The method described in this section is largely based on [24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Throughout this section, unless stated otherwise, we fix the stationary, Lipschitz-continuous velocity field u ∈ W 1,∞(Ω) with u · n = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This means that we consider a linear trans- port problem and our main concern will be the discretization of the material derivative Duω for a 1-form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We can define the flow ]0, T[×Ω ∋ (τ, x) �→ Xτ(x) ∈ Rd as the solution of the initial value problems ∂ ∂τ Xt,t+τ(x) = u(Xt,t+τ(x)), Xt(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (15) Given that flow we can define the material derivative for a time-dependent differential 1-form ω Duω(t) := ∂ ∂τ X∗ t,t+τω(t + τ) ���� τ=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (16) We employ a first- or second-order, backward-difference method to approximate the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Writing X∗ t,t−τ for the pullback of forms under the flow, we obtain for sufficiently-smooth t �→ ω(t) and a timestep 0 < τ → 0 Duω(t) = 1 τ � ω(t) − X∗ t,t−τω(t − τ) � + O(τ 2) (17) or Duω(t) = 1 2τ � 3ω(t) − 4X∗ t,t−τω(t − τ) + X∗ t,t−2τω(t − 2τ) � + O(τ 3), (18) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that both backward-difference methods are A-stable [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In the remainder of this section we restrict ourselves to (17), but exactly the same considera- tions apply to (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Given a temporal mesh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='. < tn < tn+1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., we approximate ω(tn, ·) ∈ Λ1(Ω) by a discrete differential form ωn h ∈ Λ1 h,p(Ω) with p ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Using the backward- difference quotient (17), we can define the discrete material derivative for fixed timestep τ > 0 (Dβω)(tn) ≈ 1 τ � ωn h − Ih,pX∗ t,t−τωn−1 h � ∈ Λ1 h,p(Ω), (19) where we need to use the projection operator Ih,p : Λ1(Ω) �→ Λh,p(Ω) since X∗ t,t−τωn−1 h /∈ Λ1 h,p(Ω) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Recall from section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1 that the degrees of freedom for discrete 1-forms are associated to small (p = 2) or big (p = 1) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' As discussed in sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3, evaluating the interpolation operator entails integrating X∗ t,t−τωn−1 h over small (p = 2) or big (p = 1) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We can approximate these integrals as follows � e X∗ t,t−τωn−1 h = � Xt,t−τ(e) ωn−1 h ≈ � ¯ Xt,t−τ(e) ωn−1 h , (20) where e is a small or big edge and ¯Xt,t−τ(e) = � (1 − ξ)Xt,t−τ(v1) + ξXt,t−τ(v2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 0 ≤ ξ ≤ 1 � (21) 9 Figure 3: Edge e (in red) is transported using the flow β (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The exact trans- ported edge Xτ(e) and the approximate transported edge ¯Xτ(e) are given in orange and green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' with v1, v2 the vertices of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Instead of transporting the edge e using the exact flow Xt,t−τ, we follow [12, 24, 26] and only transport the vertices of the small edges (p = 2) or big edges (p = 1) and obtain a piecewise linear (second-order) approximation ¯Xt,t−τ(e) of the transported edge Xt,t−τ(e) as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We can approxi- mate the movement of the endpoints of e under the flow as defined by (15) by solving (15) using the explicit Euler method or Heun’s method for the first- and second-order case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We will elaborate on this further in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In Figure 3, we can also see that the approximate transported edge may intersect several different elements of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' When we evaluate the integral in (20), it can happen that there are discontinuities of ωn−1 h along ¯Xt,t−τ(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Therefore, we cannot apply a global quadrature rule to the entire integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Instead, we split ¯Xt,t−τ(e) into several segments defined by the intersection of ¯Xt,t−τ(e) with cells of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In our implementation, for the sake of stability, we find the intersection points by trans- forming back to a reference element as visualised in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Algorithm 1 gives all details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that we can forgo the treament of any special cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' intersection with vertices) without jeopardizing stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' After we split the transported edge into segments, we can evaluate the integrals over these individual pieces exactly, because we know that ωn−1 h is of polynomial form when restricted to individual elements of the mesh (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' When simulating the fluid model (2), we will not have access to an exact velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Instead we only have access to an approximation of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This ap- proximation may not satisfy exact vanishing normal boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Therefore, a part of ¯Xt,t−τ(e) may end up outside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This can also happen due to an approximation of the flow by explicit timestepping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Since ωn−1 h is not defined outside 10 Algorithm 1 Splitting 1-simplex over mesh elements (see Figure 4 for illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Here, Kref denotes the reference simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Input: x0 ∈ K0 ∈ Kh(Ω) and x1 vertices of a 1-simplex e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Output: Number of elements N, elements {K0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='., KN−1} ∈ Kh(Ω)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 1: K ← K0 2: Fold ← NULL 3: Kold ← NULL 4: N ← 1 5: E ← {K} 6: while x1 /∈ K do 7: Find the isoparametric mapping ϕK : Kref �→ K 8: Find face F ⊂ ∂K s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' F ̸= Fold and ϕ−1 K (e) ∩ ϕ−1 K (F) ̸= ∅ 9: K ← K ∈ Kh(Ω) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' F ⊂ ∂K and K ̸= Kold (K on the other side of face F) 10: Fold ← F 11: N ← N + 1 12: E ← E ∪ {K} 13: end while ϕK : Kref �→ K x0 e K ϕ−1 K (e) Kref K0 x1 Figure 4: The red line indicates the line that spans multiple elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' On the left we see the reference triangle associated with the yellow element in the mesh on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 11 Figure 5: A coarse triangulation of Ω = {x ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ||x|| < 1} with the velocity field u = [−y, x]T satisfying u · n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Despite the vanishing normal components of the velocity, the blue edge gets transported out of the domain to the green edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' the domain, we set � ¯ Xt,t−τ(e)|Rd\\Ω ωn−1 h := len � ¯Xt,t−τ(e) �� Rd\\Ω � len � ¯Xt,t−τ(e) � � e ωn−1 h , (22) where ¯Xt,t−τ(e) �� Rd\\Ω is the part of ¯Xt,t−τ(e) that lies outside the domain Ω and len � ) gives the arclength of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This is motivated by the situation displayed in Figure 5—a case where an edge gets transported out of the domain due to the use of approximate flow maps despite vanishing normal components of the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' If we set the value defined in (22) to zero in this case, it would be equivalent to applying vanishing tangential boundary conditions, which is inconsistent with (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Instead, (22) just takes the tangential components from the previous timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We arrive at the following approximation of the material derivative (Dβω)(tn) ≈ 1 τ � ωn h − Ih,p ¯X∗ t,t−τωn−1 h � , (23) where the only difference between (19) and (23) is that Xt,t−τ was replaced by ¯Xt,t−τ and Ih,p is implemented based on (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that in our scheme ¯X∗ t,t−τ is always evaluated in conjunction with Ih,p, which means that we need define ¯Xt,t−τ only on small (p = 2) or big (p = 1) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In fact, ¯Xt,t−τ is defined through (21) for all points that lie on small (p = 2) or big (p = 1) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Given a velocity field u ∈ W 1,∞(Ω) with u · n = 0, it was shown in [26, section 4] that using a first-order backward difference scheme and lowest-order elements for the spatial discretization, we can approximate a smooth solution ω ∈ Λ1(Ω) of Duω = 0, (24) 12 with an L2-error of O(τ − 1 2h), where h is the spatial meshwidth and τ > 0 is the timestep size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' However, numerical experiments [26, section 6] performed with τ = O(h) show an error of O(h)—a slight improvement over the a-priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This motivates us to link the timestep to the mesh width as τ = O(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 4 Semi-Lagrangian Advection applied to the Incom- pressible Navier-Stokes Equations Given a temporal mesh t0 < t1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' < tN−1 < tN, we elaborate a single timestep tn−1 �→ tn of size τ := tn − tn−1, n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We assume that approximations ωk h ∈ Λ1 h,p(Ω) of ω(tk, ·) ∈ Λ1(Ω) are available for k < n with [0, T] �→ ω ∈ Λ1(Ω) a solution of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1 Approximation of the flow map In the Navier-Stokes equations, the flow is induced by the unknown, time-dependent velocity field u(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Therefore, (15) becomes ∂ ∂τ Xt,t+τ(x) = u(t + τ, Xt,t+τ(x)), Xt(x) = x , t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (25) The discretization of the material derivative requires us to approximate the flow map Xt,t−τ in order to evaluate (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1 A first-order scheme We use the explicit Euler method to approximate the (backward) flow according to Xtn,tn−τ(x) ≈ x − τu(tn − τ, x), (26) where tn is a node in the temporal mesh and τ denotes the timestep size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We only have access to an approximation un−1 h := (ωn−1 h )\\ of u at time tn−1, which gives Xtn,tn−τ(x) ≈ x − τun−1 h (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (27) Note that a direct application of the explicit Euler method would require an evaluation of the velocity field at tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Instead, we perform a constant extrapolation and evaluate the velocity field at tn−1, that is, we use un−1 h in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The approximation un−1 h resides in the space of vector proxies for discrete dif- ferential 1-forms as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This means that only tangential conti- nuity of un−1 h across faces of elements of the mesh is guaranteed, while discontinu- ities may appear in the normal direction of the faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Therefore, un−1 h is not defined point-wise—even though (27) requires point-wise evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' For that reason, we will 13 replace un−1 h by a globally-continuous, smoothened velocity field ¯un−1 h that approxi- mates un−1 h (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3 for the construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We then have Xtn,tn−τ(x) ≈ x − τ ¯un−1 h (x) (28) which yields a first-order-in-time approximation of Xtn,tn−τ(x), provided that ¯un−1 h is a first-order approximation of u(tn−1, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 A second-order scheme A second-order approximation can be achieved by using Heun’s method [44] instead of explicit Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We find the following second-order in time approximations Xtn,tn−τ(x) ≈ x − τ 2 � u∗ h(x) + un−1 h (x − τu∗ h(x)) � , (29) Xtntn−2τ(x) ≈ x − τ � u∗ h(x) + un−2 h (x − 2τu∗ h(x)) � , (30) where we approximate the velocity field at tn by the linear extrapolation u∗ h = 2un−1 h − un−2 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' As described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1, we replace the velocity fields by suitable smooth approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We obtain Xtn,tn−τ(x) ≈ ¯Xt−τ(x) := x − τ 2 �¯u∗ h(x) + ¯un−1 h (x − τ ¯u∗ h(x)) � , (31) Xtn,tn−2τ(x) ≈ ¯Xt−2τ(x) := x − τ �¯u∗ h(x) + ¯un−2 h (x − 2τ ¯u∗ h(x)) � , (32) where ¯u• h with • = ∗, n − 1, n − 2 denotes the smoothened version of u• h as it will be constructed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3 Smooth reconstruction of the velocity field Given a discrete velocity field uZ h ∈ Λ1 h,p(Ω), we can define a smoothened version ¯uh of uh that is Lipschitz continuous to ensure stable evaluation of (28), well-defined on every point of the meshed domain, practically computable, and second-order accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We introduce ¯uh as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Let hmin denote the length of the shortest edge of the mesh and (ui h)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='.,d the components of uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Then, ¯ui h(x) = 1 hmin � xi+ 1 2 hmin xi− 1 2 hmin ui h([x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' , xi−1, ξ, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' , xd]T)dξ (33) 14 provides a second-order, Lipschitz-continuous approximation of uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In the above def- inition, we can also replace hmin by a localized parameter that scales as O(h) with h the length of the edges "close" to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that the above integral can be evaluated up to machine precision using the algorithm as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 for (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The av- eraging (33) provides a second-order approximation of uh on every point in the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 A first- and second-order SL scheme We are now ready to turn the ideas of section 3 into a concrete numerical scheme for the incompressible Navier-Stokes equations as given in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We cast (2a) and (2b) into weak form and, subsequently, do Galerkin discretization in space relying on those spaces of discrete differential forms introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' For the first- order scheme, we have the following discrete variational formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Given ωn−1 h ∈ Λ1 h,1(Ω), we search pn h ∈ Λ0 h,1(Ω), ωn h ∈ Λ1 h,1(Ω) such that �1 τ � ωn h − Ih,1 ¯X∗ tn,tn−τωn−1 h � , ηh � Ω +ε (dωn h, dηh)Ω + (dpn h, ηh)Ω = (f n, ηh)Ω , (34a) (ωn h, dψh)Ω = 0 (34b) for all ηh ∈ Λ1 h,1(Ω) and ψh ∈ Λ0 h,1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Ih,p denotes the projection operator as defined in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' For the second-order scheme, we use second-order timestepping and second-order discrete differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Given ωn−2 h , ωn−1 h ∈ Λ1 h,2(Ω), we search pn h ∈ Λ0 h,2(Ω), ωn h ∈ Λ1 h,2(Ω) such that � 1 2τ � 3ωn h − 4Ih,2 ¯X∗ tn,tn−τωn−1 h + Ih,2 ¯X∗ tn,tn−2τωn−2 h � , ηh � Ω +ε (dωn h, dηh)Ω + (dpn h, ηh)Ω = (f n, ηh)Ω , (35a) (ωn h, dψh)Ω = 0 (35b) for all ηh ∈ Λ1 h,2(Ω) and ψh ∈ Λ0 h,2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Numerical experiments reported in section 5 give evidence that these schemes indeed do provide first- and second-order conver- gence for smooth solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that the schemes presented in this section only re- quire solving symmetric, linear systems of equations at every time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3 Conservative SL schemes In order to enforce energy-tracking—the correct behavior of the total energy E(t) over time as expressed in (3)—we add a suitable constraint plus a Lagrange multiplier to 15 the discrete variational problems proposed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Given ωn−1 h ∈ Λ1 h,1(Ω), we search pn h ∈ Λ0 h,1(Ω), ωn h ∈ Λ1 h,1(Ω), and µn ∈ R such that �1 τ � ωn h − Ih,1 ¯X∗ tn,tn−τωn−1 h � , ηh � Ω + (dpn h, ηh)Ω + ε(dωn h,dηh)Ω +µn [(ωn h, ηh)Ω + 2ετ(dωn h, dηh)Ω − τ(f n, ηh)Ω] = (f n, ηh)Ω , (36a) (ωn h, dψh)Ω = 0, (36b) (ωn h, ωn h)Ω + 2ετ(dωn h, dωn h)Ω − τ(f n, ωn h)Ω = � ωn−1 h , ωn−1 h � Ω (36c) for all ηh ∈ Λ1 h,1(Ω) and ψh ∈ Λ0 h,1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that the last scalar equation enforces energy conservation for ε = 0 and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' To solve the nonlinear system (36) for ωn h, pn h, µn, we propose the following iterative scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Assume that we have a se- quence (ωn h,k)k∈N with ωn h,k → ωn h(k → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Then we can employ the Newton-like linearization (ωn h, ωn h)Ω ← � ωn h,k, ωn h,k � Ω = � ωn h,k−1, ωn h,k−1 � Ω + 2 � ωn h,k−1, ωn h,k − ωn h,k−1 � Ω + O � ||ωn h,k − ωn h,k−1||2 Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (37) We use the above expansion to replace the quadratic terms (ωn, ωn)Ω and (dωn, dωn)Ω and arrive at the following linear variational problem to be solved in every step of the inner iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Given ωn−1 h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1 ∈ Λ1 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' we search pn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k ∈ Λ0 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k ∈ Λ1 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' and µn k ∈ R such that �1 τ � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k − Ih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1 ¯X∗ tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='tn−τωn−1 h � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ηh � Ω + � dpn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ηh � Ω + ε(dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dηh)Ω +µn k � (ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ηh)Ω + 2ετ(dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dηh)Ω − τ(f n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ηh)Ω � = (f n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ηh)Ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (38a) � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dψh � Ω = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (38b) � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1 � Ω + 2 � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k − ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1 � +2ετ[(dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1)Ω + 2(dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k − dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1)] −τ(f n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1)Ω = � ωn−1 h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn−1 h � Ω (38c) for all ηh ∈ Λ1 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω) and ψh ∈ Λ0 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This is a symmetric, linear system that is equivalent to the original system in the limit (ωn h,k, pn h,k, µn k) → (ωn h, pn h, µn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In numerical experiments we observe that it takes around 2-3 steps of the inner iteration to converge to machine precision using an initial guess ωn h,0 = ωn−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We can apply the same idea for energy-tracking to our second-order scheme as proposed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 16 Remark 4 For the case ε = 0 and f = 0, we can also enforce helicity conservation by adding a suitable Lagrange multiplier to the discrete system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Given ωn−1 h ∈ Λ1 h,1(Ω), we search pn h ∈ Λ0 h,1, ωn h ∈ Λ1 h,1(Ω), and λn ∈ R such that �1 τ � ωn h − Ih,1 ¯X∗ tn,tn−τωn−1 h � , ηh � Ω + (dpn h, ηh)Ω + ε(dωn h, dηh)Ω +λn(ωn h, dηh)Ω + λn(dωn h, ηh)Ω + µn(ωn h, ηh)Ω = 0, (39a) (ωn h, dψh)Ω = 0, (39b) (ωn h, ωn h)Ω = � ωn−1 h , ωn−1 h � Ω , (39c) (ωn h, dωn h)Ω = � ωn−1 h , dωn−1 h � Ω (39d) for all ηh ∈ Λ1 h,1(Ω) and ψh ∈ Λ0 h,1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' By linearization as for energy-tracking, we obtain the following system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Given ωn−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1 ∈ Λ1 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' we search pn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k ∈ Λ0 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k ∈ Λ1 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' and λn k ∈ R such that �1 τ � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k − Ih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1 ¯X∗ tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='tn−τωn−1� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ηh � Ω + � dpn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ηh � Ω + ε(dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dηh)Ω +λn k(ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dηh)Ω + λn k(dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ηh)Ω + µn k(ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ηh)Ω = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (40a) � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dψh � Ω = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (40b) � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1 � Ω + 2 � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k − ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1 � = � ωn−1 h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ωn−1 h � Ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (40c) � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k � Ω + � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1 � Ω − � ωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dωn h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='k−1 � Ω = � ωn−1 h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' dωn−1 h � Ω (40d) for all ηh ∈ Λ1 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω) and ψh ∈ Λ0 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Again, this is a symmetric, linear system that is equivalent to the original system in the limit (ωn h,k, pn h,k, µn k) → (ωn h, pn h, µn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Also, numerical experiments hint that it takes around 2-3 steps of the inner iteration to converge to machine precision using an initial guess ωn h,0 = ωn−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 5 Numerical Results In this section, we present multiple numerical experiments to validate the new scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In the following, we will always consider schemes that include energy-tracking as in- troduced in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3 unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We only include helicity-conservation as introduced in Remark 4 for domains in R3 when ε = 0 and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The ex- periments are based on a C++ code that heavily relies on MFEM [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The source code is published under the GNU General Public License in the online code repository https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='com/WouterTonnon/semi-lagrangian-tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 17 10−1 mesh width h 10−3 10−2 10−1 L2 Error u first-order, ε = 0 first-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01π−2 first-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1π−2 second-order, ε = 0 second-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01π−2 second-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1π−2 Figure 6: Convergence results for Experiment 1 using the first- and second-order schemes on simplicial meshes with mesh width h, timestep τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='065804h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We ob- serve first- and second-order algebraic convergence for all values of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1 Experiment 1: Decaying Taylor-Green Vortex We consider the incompressible Navier-Stokes equations with Ω = [− 1 2, 1 2]2, T = 1, varying ε ≥ 0, f = 0, and vanishing boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' An exact, classical solution is the following Taylor-Green vortex [42] u(t, x) = � cos(πx1) sin(πx2) − sin(πx1) cos(πx2) � e−2π2εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (41) We ran a h-convergence analysis for different values of ε ≥ 0 and summarize the results in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We also track the energy for different values of ε and compare the energy to the exact solution in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 Experiment 2: Taylor-Green Vortex We consider the incompressible Navier-Stokes equations with Ω = [−1, 1]2, T = 1, varying ε ≥ 0, f and the boundary conditions chosen such that u(t, x) = � cos(πx1) sin(πx2) − sin(πx1) cos(πx2) � (42) is an exact, classical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We ran a h-convergence analysis for all parameters and summarize the results in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We observe first- and second-order algebraic convergence for the corresponding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that the error of the scheme is stable 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 time t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='70 L2 Norm u exact, ε = 0 first-order, ε = 0 exact, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01π−2 first-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01π−2 exact, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1π−2 first-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1π−2 (a) first-order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 time t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='70 L2 Norm u exact, ε = 0 second-order, ε = 0 exact, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01π−2 second-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01π−2 exact, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1π−2 second-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1π−2 (b) second-order Figure 7: Energy of the discrete and exact solution for Experiment 1 using the first- and second-order, energy-tracking schemes on a simplicial mesh with mesh width h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0949795, timestep τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='00625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 19 10−1 100 mesh width h 10−2 10−1 100 L2 Error u first-order, ε = 0 first-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01 first-order, ε = 1 second-order, ε = 0 second-order, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01 second-order, ε = 1 Figure 8: Convergence results for Experiment 2 using the first- and second-order, non-conservative schemes on simplicial meshes with mesh width h, timestep τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='032902h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' As ε → 0 the error remains bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This is in agreement with the analysis performed on the vectorial advection equations presented in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This experiment thus suggests that this analysis can be extended to the scheme presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3 Experiment 3: A rotating hump problem The Taylor-Green vortices provide exact solutions to the incompressible Navier-Stokes equations, but they are rather "static" solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In this experiment, we consider a more dynamic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Let us consider the incompressible Navier-Stokes equations with Ω = [− 1 2, 1 2]2, T = 1, ε = 0, f = 0, and vanishing normal boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We consider the following initial condition u0(x) = � −πex1 cos(πx1) sin(πx2) πex1 sin(πx1) cos(πx2) − ex1 cos(πx1) cos(πx2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (43) The exact solution to this problem is unknown, so we compare the solution computed by our scheme to the solution produced by the incompressible Euler solver Gerris [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The algorithm used in this solver is described in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We computed the solution to this problem using the second-order, energy-tracking scheme presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Then, we plotted the magnitude of the computed velocity vector field for different mesh-sizes and time-steps at different time instances in Figures 10 to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that different visu- alisation tools were used to visualize the fields computed using the different solvers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' but we observe that the solution computed by the semi-Lagrangian scheme comes vi- sually closer to the solution computed by Gerris as we decrease the mesh width and 20 10−1 mesh width h 10−2 10−1 100 L2 Error u second-order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' cons Figure 9: Convergence results for Experiment 2 using the second-order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' conservative scheme on simplicial meshes with mesh width h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' timestep τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='06580h, and final time T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The reference solution is a solution computed by Gerris [36] time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This is confirmed by Figure 9, where we display the L2 error between the solution computed using the semi-Lagrangian scheme and the solution computed using Gerris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In Figure 15, we display the vector field as computed using the second-order, conservative semi-Lagrangian scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Also, in Figure 16 we display the values of the L2 norm over time of the solu- tions produced using our first- and second-, energy-tracking and non-energy-tracking schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that the energy-tracking schemes preserve the L2 norm as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The first-order, non-conservative scheme seems unstable at first, but in reality the or- dinate axis spans a very small range and it turns out that the L2 norm converges to a bounded value for longer run-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that the helicity has no meaning in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='4 Experiment 4: Taylor-Green Vortex in 3D To observe conservation of helicity, we need to consider a problem in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We con- sider the incompressible Navier-Stokes equations with Ω = [− 1 2, 1 2]3, T = 1, ε = 0, vanishing normal boundary conditions, and f chosen such that u(t, x) = � � cos(πx1) sin(πx2) sin(πx3) − 1 2 sin(πx1) cos(πx2) sin(πx3) − 1 2 sin(πx1) sin(πx2) cos(πx3) � � (44) is a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that since the solution is static, we can enforce helicity conser- vation despite f ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We run several experiments using the first- and second-order, 21 (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='25 (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 (c) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='75 (d) t = 1 Figure 10: Experiment 3: mesh width h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='379918 and time-step τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='25 (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 (c) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='75 (d) t = 1 Figure 11: Experiment 3: mesh width h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0949795 and time-step τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='00625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='25 (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 (c) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='75 (d) t = 1 Figure 12: Experiment 3: mesh width h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='023744875 and time-step τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0015625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='25 (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 (c) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='75 (d) t = 1 Figure 13: Reference solution Experiment 3 computed using [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 0 1 2 3 4 5 Figure 14: Colorbar associated with Figures 10 to 13 22 (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='25 (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 (c) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='75 (d) t = 1 0 1 2 3 4 5 Figure 15: Velocity field for Experiment 3 computed using the second-order, conser- vative semi-Lagrangian scheme on a simplicial mesh with mesh width h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='189959 and time-step τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The colors indicate the magnitude of the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 time t 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='350 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='355 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='360 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='365 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='370 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='375 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='380 L2 Norm u second-order, cons first-order, cons second-order, non-cons first-order, non-cons Figure 16: The L2 norm of the computed solutions for Experiment 3 using differ- ent variants of the semi-Lagrangian scheme on a simplicial mesh with mesh width h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='04748975 and time-step τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='003125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In the legend, ’cons’ is short for ’con- servative’ and refers to energy-tracking schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We use ε = 0 and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' conservative semi-Lagrangian schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We summarize the results in Figure 17 and we observe first- and second-order algebraic convergence for the corresponding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In Figure 18 and Figure 19 we plot the L2 norm and helicity over time of the discrete solution for both the first- and second-order scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We observe that both quantities are conserved up to machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 Experiment 5: A transient solution in 3D To verify the scheme for transient solutions in 3D, we consider the incompressible Navier-Stokes equations with Ω = [− 1 2, 1 2]3, T = 1, ε = 0, vanishing normal boundary conditions and f is chosen such that u(t, x) = � � −x2π cos( t 4 + πx2x3) cos(πx2) −x3π cos( t 4 + πx1x3) cos(πx3) −x1π cos( t 4 + πx1x2) cos(πx1) � � (45) is a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We ran a simulation for different mesh-sizes with time-steps determined by a suitable CFL condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We summarize the results in Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We observe second-order convergence for the second-order scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The first-order scheme seems to achieve an order of convergence that is between first- and second-order, but this may be pre-asymptotic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 24 10−1 2 × 10−1 3 × 10−1 4 × 10−1 6 × 10−1 mesh width h 10−2 10−1 L2 Error u first-order second-order Figure 17: Convergence results for Experiment 4 using the first- and second-order, conservative schemes on simplicial meshes with mesh width h, timestep τ = 1 √ 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 time t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='435 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='455 L2 Norm u second-order, cons first-order, cons second-order, non-cons first-order, non-cons Figure 18: The L2 norm of the computed solutions for Experiment 4 using differ- ent variants of the semi-Lagrangian scheme on a simplicial mesh with mesh width h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='08838834764 and time-step τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In the legend, ’cons’ is short for ’con- servative’ and refers to energy-tracking schemes and helicity-conserving schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We use ε = 0 and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 time t −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='004 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='000 Helicity u second-order, cons first-order, cons second-order, non-cons first-order, non-cons Figure 19: The helicity of the computed solutions for Experiment 4 using differ- ent variants of the semi-Lagrangian scheme on a simplicial mesh with mesh width h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='08838834764 and time-step τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In the legend, ’cons’ is short for ’conservative’ and refers to energy-tracking and helicity-conserving schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We use ε = 0 and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 10−1 2 × 10−1 3 × 10−1 mesh width h 10−1 L2 Error u first-order second-order Figure 20: Convergence results for Experiment 5 using the first- and second-order schemes without energy-tracking and helicity-conservation on simplicial meshes with mesh width h, timestep τ = 1 √ 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 26 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 Figure 21: Velocity field at T = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='93s of Experiment 6 computed using the second- order, non-conservative semi-Lagrangian scheme on a simplicial mesh with mesh width h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='189959 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6 Experiment 6: Lid-driven cavity with slippery walls In this section, we simulate a situation that resembles a lid-driven cavity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Consider the incompressible Navier-Stokes with Ω = [− 1 2, 1 2]2, T = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='93, ε = 0, vanishing normal boundary conditions and the initial velocity field is set equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Then, to simulate a moving lid at the top, we apply the force-field f(t, x) = [v(x), 0]T with v(x) = � exp � 1 − 1 1−100(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5−x2)2 � , if 1 − 100(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5 − x2)2 > 0, 0, else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (46) This force field gives a strong force in the x1-direction close to the top lid, but quickly tapers off to zero as we go further from the top lid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In Figure 21, we display the computed velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that, because we apply slip boundary conditions, we do not expect to observe vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The numerical solution reproduces this expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='7 Experiment 7: A more complicated domain The numerical experiments given above, show the convergence and conservative prop- erties of the introduced schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' However, these experiments are all performed on very simple, rectangular domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In this experiment, we consider a more complicated domain and mesh (generated using [22]) as shown in Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We consider the case of the incompressible Navier-Stokes equations on the domain as given in Figure 22, T = 100, ε = 0, f = 0 and vanishing normal boundary condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' We need to construct an initial condition that is divergence-free with vanishing normal boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Following an approach close to a Chorin projection, we start with w(x, y) = �sin � 2 cos( � x2 + y2) − atan2(y, x) � sin � cos( � x2 + y2) − 2 atan2(y, x) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (47) We use this definition to define a scalar function, ϕ, as ∆ϕ = ∇ · w in Ω, (48) ∇ϕ · ˆn = w · ˆn on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' (49) We can define our initial condition, u0, as u0 = w − ∇ϕ (50) Note that u0 is divergence-free and has vanishing normal boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The above system of equations can be solved using an appropriate finite-element imple- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that in this experiment, the field outside the domain is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This is well-defined on a continuous level, since vanishing boundary conditions imply that no particle will flow in from outside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' However, on the discrete level we cannot guarantee that the same will happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' It could happen that a part of a transported edge (as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2) ends up outside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In this case, we will assume that the average of the vector field along the part of the edge that lies outside the domain, will have the same value as the average of the corresponding edge in its original location (before transport) at the previous timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The first ten seconds were simulated and a video of the results can be found at https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='be/Eica8XHLtxY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' For the different schemes, we also tracked the energy in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 6 Conclusion We have developed a mesh-based semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions recast as a non- linear transport problem for a momentum 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' A linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing-viscosity 28 Figure 22: Domain and mesh associated with Experiment 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 0 10 20 30 40 50 60 time t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='0 L2 Norm u first-order, non-cons second-order, non-cons second-order, cons first-order, cons Figure 23: The L2 norm of the computed solutions for Experiment 7 using different variants of the semi-Lagrangian scheme on a simplicial mesh as given in Figure 22 and time-step τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In the legend, ’cons’ is short for ’conservative’ and refers to energy-tracking schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 29 limit and is applicable to inviscid incompressible Euler flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' However, in this case conservation of energy and helicity have to be enforced separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Making the reason- able choice of a time-step size proportional to the mesh width, the algorithm involves only local computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Yet, these are significantly more expensive compared to those required for purely Eulerian finite-element and finite-volume methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' At this point the verdict on the competitiveness of our semi-Lagrangian scheme is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' A Two formulations of the momentum equation Consider the momentum equation in (1) ∂tu + u · ∇u − ε∆u + ∇p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Note that we have by standard vector calculus identities ∆u = ∇(∇ · u) − ∇ × ∇ × u, where we can use ∇ · u = 0 to obtain ∆u = −∇ × ∇ × u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This allows us to rewrite the momentum equation as ∂tu + u · ∇u + ε∇ × ∇ × u + ∇p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Using the gradient of the dot-product, we find ∇(u · u) = 2u · ∇u + 2u × (∇ × u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' This identity allows us to rewrite the momentum equation to ∂tu + ∇(u · u) − u × (∇ × u) + ε∇ × ∇ × u + ∇ � −1 2u · u + p � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' From [27, 24], we obtain the identity (Luω)\\ = ∇(u · u) − u × (∇ × u) where ω is the differential 1-form such that u = ω\\.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Since the material derivative for this 1-form is Duω := ∂tω + Luω, we find that the momentum equation can be written as Duω + εδdω + d˜p = 0, where ˜p = − 1 2u · u + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 30 References [1] Mofdi El-Amrani and Mohammed Seaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' “An L2-projection for the Galerkin- characteristic solution of incompressible flows”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In: SIAM Journal on Scientific Computing 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 3110–3131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ISSN: 10648275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='9781611975543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' [4] Douglas N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Arnold, Richard S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Falk, and Ragnar Winther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' “Finite element exte- rior calculus, homological techniques, and applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In: Acta Numerica 15 (May 2006), pp.' 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fluides parfaits”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In: An- nales de L’Institut Fourier 16 (1966), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 319–361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' URL: http : / / www .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' numdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='org/articles/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='5802/aif.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ISSN: 00361429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1137/ S0036142900367478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Bercovier, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' Pironneau, and V.' metadata={'source': 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Ap- plied and Industrial Mathematics 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='3 (Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 26–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ISSN: 2038-0909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1515/caim-2016-0021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' [11] Rodolfo Bermejo and Laura Saavedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' “Modified Lagrange–Galerkin Meth- ods to Integrate Time Dependent Incompressible Navier–Stokes Equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In: SIAM Journal on Scientific Computing 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 2015), B779–B803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' In: Computer Methods in Applied Mechanics and Engineering 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1-4 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 1994), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 211– 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ISSN: 00457825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' DOI: 10.' metadata={'source': 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Methods in Fluids 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1 (July 1992), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 23–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ISSN: 0271-2091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1002/fld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1650150103.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 2003), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 341–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ISSN: 0167739X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1016/S0167-739X(02)00161- 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' [18] Alexandre J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1016/S0021- 9991(03)00298-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' [37] Stéphane Popinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' The Gerris Flow Solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' URL: https : / / gfs .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 2001), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 658–684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' ISSN: 00219991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='1006/jcph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfJww5/content/2301.04923v1.pdf'} +page_content='6847.' 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[math.DS] 12 Jan 2023 +A NOTE ON THE MARGINAL INSTABILITY RATES OF +TWO-DIMENSIONAL LINEAR COCYCLES +IAN D. MORRIS AND JONAH VARNEY +Abstract. A theorem of Guglielmi and Zennaro implies that if the uniform +norm growth of a locally constant GL2(R)-cocycle on the full shift is not ex- +ponential then it must be either bounded or linear, with no other possibilities +occurring. We give an alternative proof of this result and demonstrate that its +conclusions do not hold for Lipschitz continuous cocycles over the full shift on +two symbols. +Keywords: discrete linear inclusion, ergodic optimisation, joint spectral ra- +dius, linear cocycle, marginal stability, marginal instability. MSC2020 codes: +37H15 (primary), 37D35, 93C30 (secondary) +1. Introduction and statement of results +Define ΣN := {1, . . ., N}Z and equip this set with the infinite product topol- +ogy, with respect to which it is a compact metrisable topological space. Define +T : ΣN → ΣN to be the shift transformation T [(xn)n∈Z] := (xn+1)n∈Z, which is a +homeomorphism. If a continuous function A: ΣN → GLd(R) is specified, one may +be interested in the growth of the sequence (an) defined by +an := sup +x∈ΣN +��A(T n−1x) · · · A(T x)A(x) +�� . +This sequence is easily seen to be submultiplicative in the sense that an+m ≤ anam +for all n, m ≥ 1, which guarantees the existence of the limit +̺(A) := lim +n→∞ sup +x∈ΣN +��A(T n−1x) · · · A(T x)A(x) +�� +1 +n . +By replacing A with ̺(A)−1 · A we may without loss of generality assume that +̺(A) = 1, and we will make this assumption for the remainder of this note. In this +note we will be interested in the behaviour of the sequence (an) in the reduced case +̺(A) = 1. +Let us say that A: ΣN → GLd(R) is locally constant if for x = (xn)n∈Z the +matrix A(x) is determined by the symbol x0 only. In this case, if A denotes the +range of the function A, then one simply has +(1) +sup +x∈ΣN +��A(T n−1x) · · · A(T x)A(x) +�� = +sup +A1,...,An∈A +∥An · · · A1∥ . +The case in which A is locally constant has been studied extensively due to its +relevance to marginally unstable discrete-time linear switching systems in control +theory, and investigations of sequences (an) of the above form may be found in +numerous works such as [6, 15, 16, 19, 20, 25, 26, 27]. The same problem has also +been studied in [2, 3] based on quite different motivations relating to the notion +of k-regular sequences in symbolic dynamics. In the works just cited the simpler +1 + +2 +IAN D. MORRIS AND JONAH VARNEY +formulation (1) corresponding to the locally constant case is the only case studied, +but the more general case in which A is not assumed locally constant has been +touched upon in the ergodic optimisation literature, notably [4] in which criteria +for (an) to be a bounded sequence are investigated. +An early result describing some possible behaviours of such sequences (an) is the +following, which is essentially due to N. Guglielmi and M. Zennaro: +Theorem 1. Let A: ΣN → GL2(R) be locally constant and define A := {A(x): x ∈ +ΣN}. Suppose that +lim +n→∞ sup +x∈ΣN +��A(T n−1x) · · · A(x) +�� +1 +n = 1. +Then one of the following holds: either +(2) +lim +n→∞ +1 +n sup +x∈ΣN +��A(T n−1x) · · · A(x) +�� > 0, +or we instead have +sup +n≥1 +sup +x∈ΣN +��A(T n−1x) · · · A(x) +�� < ∞. +Moreover, the first case occurs if and only if the semigroup generated by A contains +a nontrivial Jordan matrix with unit determinant, if and only if both of the following +two conditions are met: A is simultaneously triangularisable, and the set of matrices +in A with determinant ±1 is nonempty and is not simultaneously diagonalisable. +We remark that the situation described in Theorem 1 is quite delicate: if the +dimension of the linear maps is raised from 2 to 3, or if a shift over a compact infinite +alphabet is allowed in place of the finite alphabet {1, . . ., N}, then the conclusion +no longer holds and the above sequences may grow at a rate strictly intermediate +between linear growth and boundedness (see [12, 19, 20]). In this article we give +an alternative proof of the above result which is due to the second named author +and which was previously presented in the thesis [26]. We remark that the actual +existence of the limit (2) is a new contribution originating in this article: in [12, 26] +it was shown that the limit inferior and limit superior of this sequence are finite +and nonzero, but it was not shown that they are equal to one another. +The second contribution of this article is to show that if the condition of being +locally constant is relaxed then the dichotomy asserted in Theorem 1 ceases to hold. +We prove: +Theorem 2. Let T : Σ2 → Σ2 be the full shift on two symbols and let d be any +metric which generates the infinite product topology on Σ2. Then there exist Lips- +chitz continuous functions f, g : Σ2 → (0, 1] and φ: Σ2 → R such that the function +A: Σ2 → GL2(R) defined by +A(x) := +� +f(x) +φ(x) +0 +g(x) +� +satisfies +lim +n→∞ sup +x∈Σ2 +��A(T n−1x) · · · A(x) +�� +1 +n = 1, +lim +n→∞ +1 +n sup +x∈Σ2 +��A(T n−1x) · · · A(x) +�� = 0 + +MARGINAL INSTABILITY OF LINEAR COCYCLES +3 +and +sup +n≥1 +sup +x∈Σ2 +��A(T n−1x) · · · A(x) +�� = ∞. +We emphasise that the metric d is not assumed to have any properties other +than generating the usual topology on Σ2. When working with shift spaces it is +usual to consider metrics on Σ2 such that +max +y∈Σ2 diam {(xn)n∈Z ∈ Σ2 : xi = yi for all i such that |i| ≤ n} = O(θn) +for some θ ∈ (0, 1), but in Theorem 2 this sequence may be allowed to tend to zero +arbitrarily slowly or quickly. The functions f, g, φ may therefore be freely taken to +be “super-continuous” in the sense of [5, 21]. +The proof of Theorem 1 is direct, and proceeds by considering the semigroup +generated by the set {A(x): x ∈ ΣN}. The proof of Theorem 2 is more technically +subtle and makes use of ergodic optimisation. The two proofs are presented in +sections 2 and 3 below. +2. Proof of Theorem 1 +In view of the identity (1) it is sufficent to prove the following: if A is a finite +set of real 2 × 2 matrices which satisfies +lim +n→∞ +max +A1,...,An∈A ∥An · · · A1∥ +1 +n = 1, +then the limit +lim +n→∞ +1 +n +max +A1,...,An∈A ∥An · · · A1∥ +exists; if A is simultaneously upper triangularisable and the set {A ∈ A: | det A| = +1} is not simultaneously diagonalisable, then the above limit is nonzero; and if the +two conditions just mentioned do not both hold, then the semigroup generated by +A is bounded. +We will begin the proof by establishing the boundedness of the semigroup gen- +erated by A in a certain special case. The following result is closely related to [12, +Lemma 5.1] but its proof is entirely different: +Proposition 2.1. Let A be any finite set of 2 × 2 real upper triangular matrices +all of which have spectral radius at most 1 and have determinant strictly less than +1 in absolute value. Then the semigroup generated by A is bounded. +Proof. Clearly we may freely assume that A is nonempty. Since every element of A +has spectral radius at most 1, its diagonal entries are at most 1 in absolute value. +By the finiteness of A there exist β ∈ [0, 1) and M ≥ 0 with the following property: +for every A ∈ A, one of the every diagonal entries of A is at most 1 in absolute value, +the other diagonal entry is at most β in absolute value, and the upper-right entry +is at most M in absolute value. (In particular we may take β := maxA∈A | det A|1/2 +and M := maxA∈A ∥A∥.) Let |||·|||1 denote the norm on 2 × 2 real matrices given by +the sum of the absolute values of the matrix entries. If A1, . . . , An are arbitrary real +2×2 matrices, A′ +1, . . . , A′ +n are non-negative 2×2 matrices, and for every i = 1, . . . , n +every entry of A′ +i is greater than or equal to the absolute value of the corresponding +entry of Ai, then it is easily seen that +|||An · · · A1|||1 ≤ |||A′ +n · · · A′ +1|||1. + +4 +IAN D. MORRIS AND JONAH VARNEY +(This fact is particularly easily demonstrated in the context of upper-triangular +matrices, which is the only case which we shall need.) Consequently, in order to +demonstrate that +sup +n≥1 +max +A1,...,An∈A |||An · · · A1|||1 < ∞ +it is sufficient to demonstrate that +(3) +sup +n≥1 +max +i1,...,in∈{1,2} |||Bin · · · Bi1|||1 < ∞ +where +B1 := +� +1 +M +0 +β +� +, +B2 := +� +β +M +0 +1 +� +, +since every A ∈ A either has the absolute values of all of its entries less than or +equal to the corresponding entry of B1, or has the same property with respect to +the matrix B2. +We therefore demonstrate (3). Fix an arbitrary n ≥ 1 and consider a prod- +uct Bin · · · Bi1 of the matrices B1, B2 which maximises the value of |||Bin · · · Bi1|||1 +among all products of B1 and B2 of length n. We claim that necessarily Bin · · · Bi1 = +Bm +1 Bn−m +2 +for some integer m ∈ {0, 1, . . ., n}. Suppose for a contradiction that this +is not the case: then Bin · · · Bi1 may be factorised as +Bin · · · Bi1 = Bin · · · Bik+1B2B1Bik−1 · · · Bi1 = X1B2B1X2, +say, where X1, X2 are invertible non-negative upper triangular matrices (one or +both of which might be the identity matrix). Now, the matrix +Z := B1B2 − B2B1 = +� +0 +2(1 − β)M +0 +0 +� +is non-negative and nonzero, so X1ZX2 is also non-negative and nonzero. It follows +that +|||X1B1B2X2|||1 = |||X1B2B1X2 + X1ZX2|||1 > |||X1B2B1X2|||1 +using the non-negativity of all of these matrices and the fact that X1ZX2 is nonzero. +Since X1B1B2X2 also has the form Bjn · · · Bj1 for some j1, . . . , jn ∈ {1, 2} this +contradicts the presumed maximality of |||Bin · · · Bi1|||1. The claim is proved. Since +for every n ≥ 1 +max +i1,...,in∈{1,2} |||Bin · · · Bi1|||1 = max +0≤m≤n +������Bm +1 Bn−m +2 +������ +1 +by the preceding claim, it follows that +sup +n≥1 +max +i1,...,in∈{1,2} |||Bin · · · Bi1|||1 ≤ sup +n,m≥0 +|||Bn +1 Bm +2 |||1 +≤ +� +sup +n≥0 +|||Bn +1 |||1 +� � +sup +m≥0 +|||Bm +2 |||1 +� +. +Since B1 and B2 are diagonalisable with spectral radius 1, the sets {|||Bn +1 |||1 : n ≥ 0} +and {|||Bm +2 |||1 : n ≥ 0} are bounded, and the result follows. +□ +We now prove Theorem 1 in the form described at the beginning of the section. +Suppose first that A is not simultaneously upper triangularisable. This is equivalent +to the statement that there does not exist a basis for R2 whose first element is an +eigenvector for every A ∈ A, and this in turn is equivalent to the statement that +no one-dimensional vector subspace of R2 is preserved by every element of A. It is + +MARGINAL INSTABILITY OF LINEAR COCYCLES +5 +by now well established (see e.g. [15, Theorem 2.2]) that under this last condition +there necessarily exists a norm |||·||| on R2 such that maxA∈A |||Av||| = |||v||| for every +A ∈ A, in which case in particular maxA∈A |||A||| ≤ 1 in the associated operator +norm. This clearly implies that |||A||| ≤ 1 for every A in the semigroup generated +by A and the result follows in this case. The existence of the limit (2) in this case +is trivial. +For the remainder of the proof we may assume that A is simultaneously upper +triangularisable. By an orthogonal change of basis for R2 we may assume without +loss of generality (and without affecting either the existence or the value of the +desired limit (2)) that every A ∈ A is upper triangular. If any B ∈ A had spectral +radius strictly greater than 1 then we would have +1 = lim +n→∞ +max +A1,...,An∈A ∥An · · · A1∥ +1 +n ≥ lim +n→∞ ∥Bn∥ +1 +n > 1 +which is a contradiction, so every element of A has spectral radius at most 1 and +therefore the absolute values of its diagonal entries are at most 1. +The latter +property is clearly inherited by all products of elements of A. +We now wish to show that the limit +(4) +lim +n→∞ +1 +n +max +A1,...,An∈A ∥An · · · A1∥ ≥ 0 +exists. Define a seminorm | · | on the vector space of real 2 × 2 matrices by defining +|A| to be the absolute value of the upper-right entry of A. An easy direct calculation +demonstrates that if A and B are 2 × 2 upper-triangular matrices whose diagonal +entries have absolute value at most 1, then |AB| ≤ |A| + |B|. It follows directly +that the sequence +n �→ +max +A1,...,An∈A |An · · · A1| +is subadditive and therefore the limit +lim +n→∞ +1 +n +max +A1,...,An∈A |An · · · A1| ≥ 0 +exists. On the other hand, for every product +An · · · A1 = +� +a +b +0 +c +� +of elements of A we clearly have +|b| ≤ +���� +� +a +b +0 +c +����� ≤ +���� +� +a +0 +0 +c +����� + +���� +� +0 +b +0 +0 +����� = max{|a|, |c|} + |b| ≤ 1 + |b| +and therefore +max +A1,...,An∈A |An · · · A1| ≤ +max +A1,...,An∈A ∥An · · · A1∥ ≤ 1 + +max +A1,...,An∈A |An · · · A1| +for every n ≥ 1. We conclude that +lim +n→∞ +1 +n +max +A1,...,An∈A ∥An · · · A1∥ = lim +n→∞ +1 +n +max +A1,...,An∈A |An · · · A1| +and in particular the former limit exists as required. +We now demonstrate that either the limit (4) is positive, or the semigroup gen- +erated by A is bounded. Define +A0 := {A ∈ A: | det A| < 1}, + +6 +IAN D. MORRIS AND JONAH VARNEY +A1 := {A ∈ A: | det A| = 1}. +We observe that every matrix in A1 is of one of the following three types: either +it is equal to plus or minus the identity matrix; or it is a nontrivial Jordan matrix +with determinant 1; or it is an upper triangular matrix with determinant −1. If A1 +contains a matrix B of the second type then clearly +max +A1,...,An∈A ∥An · · · A1∥ ≥ ∥Bn∥ ≥ n|B| > 0 +for every n ≥ 1. If it contains two matrices of the third type which are not scalar +multiples of one another then it is easy to check that their product is a nontrivial +Jordan matrix B, and therefore for all n ≥ 1 +max +A1,...,A2n∈A ∥A2n · · · A1∥ ≥ ∥Bn∥ ≥ n|B| > 0. +In either of these two cases it is obvious that A1 is not simultaneously diagonalisable +and that the limit (4) is positive. If neither of these cases holds then for some upper +triangular matrix X ∈ GL2(R) which has determinant −1 and diagonal entries ±1, +we have A1 ⊆ {I, −I, X, −X}. It is clear that in this case A1 is simultaneously +diagonalisable. Let X := {I, −I, X, −X}. By the Cayley-Hamilton Theorem X2 = +I and it follows easily that {I, −I, X, −X} is a group. Consequently every element +of the semigroup generated by A either is an element of X or is contained in the +semigroup generated by the set +ˆA := A0 ∪ {BA: B ∈ X and A ∈ A0} ∪ {AB : B ∈ X and A ∈ A0} . +But ˆA satisfies the hypotheses of Proposition 2.1, so the semigroup which it gen- +erates is bounded. Thus the semigroup generated by A is bounded. We observe +that unboundedness held precisely in those cases in which the semigroup generated +by A contained a nontrivial Jordan matrix with unit determinant. The proof is +complete. +3. Proof of Theorem 2 +3.1. Ergodic-theory preliminaries and a technical result. We will deduce +Theorem 2 from a more general ergodic-theoretic statement. We begin with some +necessary definitions. If X is a compact metrisable topological space then we let +M(X) denote the set of all Borel probability measures on X. By the Riesz Repre- +sentation Theorem we may identify M(X) with the set of all non-negative elements +of the unit sphere of C(X)∗, and we equip M(X) with the topology which it in- +herits as a subspace of C(X)∗ in its weak-* topology via this identification. This +topology makes M(X) compact and metrisable, and in this topology the function +µ �→ +� +φ dµ is continuous for every φ ∈ C(X). +If T : X → X is a continuous transformation of a compact metric space then we +let MT (X) ⊆ M(X) denote the set of all T -invariant Borel probability measures +on X and ET (X) ⊆ MT (X) the set of all such measures with respect to which T is +ergodic. Both sets are nonempty and the former is closed in the weak-* topology +on M(X), hence is also a compact metrisable topological space. If φ: X → R is +continuous we define β(φ) := supµ∈MT (X) +� +φ dµ. We also define Mmax(φ) to be the +set of all µ ∈ MT (X) which attain this supremum, i.e. which satisfy +� +φ dµ = β(φ). +The set Mmax(φ) is nonempty by elementary considerations of compactness and +continuity, and moreover has nonempty intersection with ET (X) (see for example +[14, Proposition 2.4]). + +MARGINAL INSTABILITY OF LINEAR COCYCLES +7 +The following result will be applied to prove Theorem 2: +Theorem 3. Let T : X → X be a homeomorphism of a compact metric space. +Let f, g : Σ → (0, 1] be continuous, let φ: Σ → R be continuous, and suppose that +β(log f) = β(log g) = 0. For every x ∈ X define +A(x) := +� +f(x) +φ(x) +0 +g(x) +� +∈ GL2(R). +If Mmax(log f) ∩ Mmax(log g) ̸= ∅ then +lim +n→∞ +1 +n sup +x∈X +��A(T n−1x) · · · A(x) +�� = +sup +µ∈Mmax(log f)∩Mmax(log g) +���� +� +φ dµ +���� ≥ 0 +and if Mmax(log f) ∩ Mmax(log g) = ∅ then +lim +n→∞ +1 +n sup +x∈X +��A(T n−1x) · · · A(x) +�� = 0. +3.2. Overview of the proof of Theorem 3 and its relationship with earlier +work. The proof of Theorem 3 begins with the observation that for every n ≥ 1 +and x ∈ X, the product A(T n−1x) · · · A(x) has the form +��n−1 +j=0 f(T jx) +�n−1 +k=0 +��n−1 +j=k+1 f(T jx) +� +φ(T kx) +��k−1 +j=0 g(T jx) +� +0 +�n−1 +j=0 g(T jx) +� +. +Since the diagonal entries necessarily belong to (0, 1], if the norm of this product is +to grow to infinity then it must do so at the same rate as the off-diagonal term +Φn(x) := +n−1 +� +k=0 + + +n−1 +� +j=k+1 +f(T jx) + + φ(T kx) + + +k−1 +� +j=0 +g(T jx) + + +and therefore the problem reduces to showing that +lim +n→∞ +1 +n sup +x∈X +|Φn(x)| = +sup +µ∈Mmax(log f)∩Mmax(log g) +���� +� +φ dµ +���� +if Mmax(log f)∩Mmax(log f) is nonempty, and that the same limit is equal to zero +otherwise. +In the second named author’s PhD thesis this problem was approached as follows. +In order to obtain examples sufficient to prove Theorem 2 it is enough to consider +the case in which f is the constant function, in which case we need only study the +somewhat simpler expression +(5) +Ψn(x) := +n−1 +� +k=0 +φ(T kx) + + +k−1 +� +j=0 +g(T jx) + + . +This expression may be seen as a weighted Birkhoff average reminiscent of those +appearing in the Wiener-Wintner-type ergodic theorems found in such works as +[1, 22, 28, 29] and can be studied using a modification of the strategy used in +[22, 28]. Specifically, one may re-express (5) in terms of an extended dynamical + +8 +IAN D. MORRIS AND JONAH VARNEY +system Tg : X ×[0, 1] → X ×[0, 1] defined by Tg(x, y) := (T x, g(x)y) and continuous +function ψ(x, y) := φ(x)y, obtaining by a simple induction +Ψn(x) = +n−1 +� +k=0 +φ(T kx) + + +k−1 +� +j=0 +g(T jx) + + = +n−1 +� +k=0 +ψ(T k +g (x, 1)) +and therefore +|Ψn(x)| = sup +y∈[0,1] +����� +n−1 +� +k=0 +ψ(T k +g (x, y)) +����� . +The identity +lim +n→∞ +1 +n sup +x∈X +|Ψn(x)| = lim +n→∞ +1 +n +sup +(x,y)∈X×[0,1] +����� +n−1 +� +k=0 +ψ(T k +g (x, y)) +����� +(6) += +sup +µ∈MTg (X×[0,1]) +���� +� +ψ dµ +���� +may then by obtained using appropriate results from the ergodic optimisation litera- +ture (see e.g. [14, Proposition 2.2]). By describing carefully the invariant measures +of Tg and relating them to invariant measures of T a formula similar to that in +Theorem 3 may be deduced. This approach can be developed further to allow for +the possibility that g takes values in [−1, 1], although in this case the resulting +description of the limit +lim +n→∞ +1 +n sup +x∈X +��A(T n−1x) · · · A(x) +�� +becomes an inequality and not an equality, due to the difficulties in treating additive +cancellations in (5) arising from changes in the sign of g. When T and X have addi- +tional regularity properties Theorem 3 may also be extended to allow the condition +sup g ≤ 1 to be removed, since by the use of results such as [14, Theorem 4.7] the +condition sup g ≤ 1 can be obtained automatically from the condition β(log g) = 0 +at the cost of a change of co-ordinates in R2 which depends continuously on the +base point x ∈ X: see [26, §5]. +In the treatment of Theorem 3 in this work, we will take a different approach +by exploiting the fact that the functions Φn defined above have the subadditivity +property +|Φn+m| ≤ |Φn ◦ T m| + |Φm| +and applying techniques from subadditive ergodic optimisation (specifically, from +the appendix to [18]) rather than the additive techniques of [14]. +This has the +advantages that it allows f to be nonconstant and results in a shorter proof, but +has the disadvantage that f and g are constrained to take values in (0, 1] and not +in [−1, 1] as is the case in [26]. +Proofs of the following standard result may be found in e.g. [9, 17]. +Theorem 4 (Subadditive ergodic theorem). Let T be an ergodic measure-preserving +transformation of a probability space (X, F, µ) and let (ψn)∞ +n=1 be a sequence of in- +tegrable functions X → R such that ψn+m ≤ ψn ◦ T m + ψm a.e. for every n, m ≥ 1. +Then +lim +n→∞ +1 +nψn(x) = lim +n→∞ +1 +n +� +ψn dµ = inf +n≥1 +1 +n +� +ψn dµ +for µ-a.e. x ∈ X. + +MARGINAL INSTABILITY OF LINEAR COCYCLES +9 +We also require the following subadditive analogue of (6) which may be found +in the appendix to [18]. For some closely-related earlier results see also [23, 24]. +Theorem 5. Let T : X → X be a continuous transformation of a compact metric +space and let (ψn)∞ +n=1 be a sequence of continuous functions X → R such that +ψn+m(x) ≤ ψm(T nx) + ψn(x) for every x ∈ X and n, m ≥ 1. Then +inf +n≥1 +sup +µ∈MT (X) +1 +n +� +ψn dµ = +sup +µ∈MT (X) +inf +n≥1 +1 +n +� +ψn dµ += inf +n≥1 sup +x∈X +1 +nψn(x) = sup +x∈X +inf +n≥1 +1 +nψn(x). +In the first three expressions the infimum over n ≥ 1 is equal to the limit as n → ∞ +of the same expression. In all cases, every supremum over µ ∈ MT (X) is equal +to the corresponding supremum over µ ∈ ET (X) and is attained by an element of +ET (X). +We now proceed with the proof of Theorem 3 and thence that of Theorem 2. +3.3. Proof of Theorem 3. Consider the sequence of continuous functions Φn : X → +R defined by +Φn(x) := +n−1 +� +j=0 +f(T n−1x) · · · f(T j+1x)φ(T jx)g(T j−1x) · · · g(x). +We note that the relation +Φn+m(x) = Φm(T nx)g(T n−1x) · · · g(x) + f(T n+m−1x) · · · f(T nx)Φn(x) +is satisfied for every x ∈ X and n, m ≥ 1, and consequently +|Φn+m(x)| ≤ |Φm(T nx)| + |Φn(x)| +for all such n, m and x. It follows that Theorem 5 is applicable to the sequence of +functions |Φn|, and therefore +lim +n→∞ +1 +n sup +x∈X +|Φn(x)| = +sup +µ∈ET (X) +inf +n≥1 +1 +n +� +|Φn| dµ. +Now, for every x ∈ X and n ≥ 1 we have +A(T n−1x) · · · A(x) = +��n−1 +j=0 f(T jx) +Φn(x) +0 +�n−1 +j=0 g(T jx) +� +and so in particular +����A(T n−1x) · · · A(x) +�� − |Φn(x)| +�� += +����� +����� +��n−1 +j=0 f(T jx) +Φn(x) +0 +�n−1 +j=0 g(T jx) +������ − +���� +� +0 +Φn(x) +0 +0 +����� +����� +≤ +����� +��n−1 +j=0 f(T jx) +0 +0 +�n−1 +j=0 g(T jx) +������ += max + + + +������ +n−1 +� +j=0 +f(T jx) +������ +, +������ +n−1 +� +j=0 +g(T jx) +������ + + + ≤ 1 + +10 +IAN D. MORRIS AND JONAH VARNEY +by the reverse triangle inequality. Consequently +lim +n→∞ +1 +n sup +x∈X +��A(T n−1x) · · · A(x) +�� = lim +n→∞ +1 +n sup +x∈X +|Φn(x)| = +sup +µ∈ET (X) +inf +n≥1 +1 +n +� +|Φn| dµ +and to prove the theorem we will evaluate the latter. We consider in turn the cases +where µ ∈ ET (X) fails to belong to Mmax(log g), fails to belong to Mmax(log f), +or belongs to both sets. +If µ ∈ ET (X) does not belong to Mmax(log g) then for µ-a.e. x ∈ X +lim +k→∞ +1 +k log +� +g(T k−1x) · · · g(x) +� += lim +k→∞ +1 +k +k−1 +� +j=0 +log g(T jx) = +� +log g dµ < β(log g) = 0 +and hence for µ-a.e. x ∈ X +|Φn(x)| = +����� +n−1 +� +k=0 +f(T n−1x) · · · f(T k+1x)φ(T kx)g(T k−1x) · · · g(x) +����� +≤ +n−1 +� +k=0 +��φ(T kx) +�� · g(T k−1x) · · · g(x) +≤ ∥φ∥∞ +∞ +� +k=0 +g(T k−1x) · · · g(x) < ∞ +for every n ≥ 1. It follows that +1 +n|Φn| → 0 µ-a.e. and hence by the subadditive +ergodic theorem +lim +n→∞ +1 +n +� +|Φn| dµ = inf +n≥1 +1 +n +� +|Φn| dµ = 0. +Now suppose instead that µ ∈ ET (X) does not belong to Mmax(log f). Since µ is +T -invariant it is also T −1-invariant, so by the Birkhoff ergodic theorem applied to +T −1 we likewise have +lim +ℓ→∞ +1 +ℓ log f(x)f(T −1x) · · · f(T −(ℓ−1)x) = +� +log f dµ < β(log f) = 0 +and hence for µ-a.e. x ∈ X +���Φn(T −(n−1)x) +��� = +����� +n−1 +� +k=0 +f(x) · · · f(T k+2−nx)φ(T k+1−nx)g(T k−nx) · · · g(T −(n−1)x) +����� += +����� +n−1 +� +ℓ=0 +f(x) · · · f(T −(ℓ−1)x)φ(T −ℓx)g(T −ℓ−1x) · · · g(T −(n−1)x) +����� +≤ +n +� +ℓ=1 +f(x) · · · f(T −(ℓ−1)x) · +��φ(T −ℓx) +�� +≤ ∥φ∥∞ +∞ +� +ℓ=1 +f(x) · · · f(T −(ℓ−1)x) < ∞ +for every n ≥ 1. It follows that for µ-a.e. x ∈ X +lim +n→∞ +1 +n +���Φn +� +T −(n−1)x +���� = 0 + +MARGINAL INSTABILITY OF LINEAR COCYCLES +11 +and hence in particular 1 +n|Φn◦T −(n−1)| → 0 in the sense of convergence in measure; +but since µ is T -invariant, this implies that 1 +n|Φn| → 0 in the sense of convergence +in measure. In combination with the subadditive ergodic theorem this fact yields +lim +n→∞ +1 +n +� +|Φn| dµ = inf +n≥1 +1 +n +� +|Φn| dµ = 0. +Combining the two facts just demonstrated, we have shown that if µ ∈ ET (X) does +not belong to Mmax(log f) ∩ Mmax(log g) then +inf +n≥1 +1 +n +� +|Φn| dµ = lim +n→∞ +1 +n +� +|Φn| dµ = 0. +If Mmax(log f) ∩ Mmax(log g) is empty, this demonstrates that +sup +µ∈ET (X) +inf +n≥1 +1 +n +� +|Φn| dµ = 0 +which completes the proof of the theorem in that case. Otherwise, suppose that +Mmax(log f) ∩ Mmax(log g) is nonempty. +Since by hypothesis 0 = β(log f) ≤ +sup log f ≤ 0 and 0 = β(log g) ≤ sup log g ≤ 0 it is easily seen that the set +Z := {x ∈ X : log f(x) = log g(x) = 0} = {x ∈ X : f(x) = g(x) = 1} +satisfies µ(Z) = 1 for every µ ∈ Mmax(log f) ∩ Mmax(log g) and in particular is +nonempty. Moreover it is easily verified that +Mmax(log f) ∩ Mmax(log g) = {µ ∈ MT (X): µ(Z) = 1} = MT (Z) +and therefore +Mmax(log f) ∩ Mmax(log g) ∩ ET (X) = MT (Z) ∩ ET (X) = ET (Z). +Now, if µ ∈ Mmax(log f) ∩ Mmax(log g) ∩ ET (X) = ET (Z) then for µ-a.e. x ∈ X +we simply have +lim +n→∞ +1 +n|Φn(x)| = lim +n→∞ +1 +n +����� +n−1 +� +k=0 +φ(T kx) +����� = +���� +� +φ dµ +���� +using the Birkhoff ergodic theorem and the fact that f and g are identically equal +to 1 on Z. Thus +sup +µ∈Mmax(log f)∩Mmax(log g)∩ET (X) +���� +� +φ dµ +���� = +sup +µ∈ET (Z) +���� +� +φ dµ +���� . +Since we have already established that +sup +µ∈ET (X)\(Mmax(log f)∩Mmax(log g)) +inf +n≥1 +1 +n +� +|Φn| dµ = 0 + +12 +IAN D. MORRIS AND JONAH VARNEY +it follows that +sup +µ∈ET (X) +inf +n≥1 +1 +n +� +|Φn| dµ = +sup +µ∈Mmax(log f)∩Mmax(log g)∩ET (X) +���� +� +φ dµ +���� += +sup +µ∈ET (Z) +���� +� +φ dµ +���� += +sup +µ∈MT (Z) +���� +� +φ dµ +���� += +sup +µ∈Mmax(log f)∩Mmax(log g) +���� +� +φ dµ +���� +as required. The proof of the theorem is complete. +3.4. Proof of Theorem 2. Fix a metric d on Σ2 which generates the infinite +product topology. +Let [1] ⊂ Σ2 denote the set of all (xn)n∈Z ∈ Σ2 such that +x0 = 1, which by the definition of the infinite product topology on Σ2 is both +closed and open. By an appropriate version of the Jewett-Krieger Theorem (see +for example [8, §29]), or by various direct constructions such as [7, 10, 11], there +exists a compact T -invariant set Z ⊂ Σ2 with the following properties: there exists +a unique ν ∈ MT such that ν(Z) = 1; T is weak-mixing with respect to this unique +measure ν; the support of ν is precisely Z; and Z is not a singleton set. Since Z +is not a singleton we have 0 < ν([1]) < 1 and therefore e2πiν([1]) ̸= 1, a property +which will be significant later. Define f(x) = g(x) = e− dist(x,Z) for all x ∈ Σ2, +where dist(x, Z) := infy∈Z d(x, y). Clearly f and g are Lipschitz continuous and +satisfy β(log f) = β(log g) = 0 and Mmax(log f) = Mmax(log g) = {ν}. Define +also φ(x) := χ[1](x) − ν([1]). By the definition of the infinite product topology, +φ: Σ2 → R is continuous; since Σ2 is a compact metric space with respect to +d, φ is uniformly continuous with respect to d; and since φ takes exactly two +values, this implies that φ is Lipschitz continuous with respect to d as required. +Define A: Σ2 → GL2(R) as in the statement of the theorem. Clearly Theorem 3 is +applicable. Since +sup +µ∈Mmax(log f)∩Mmax(log g) +���� +� +φ dµ +���� = +���� +� +φ dν +���� = |ν([1]) − ν([1])| = 0, +Theorem 3 yields +(7) +lim +n→∞ +1 +n sup +x∈Σ2 +��A(T n−1x) · · · A(x) +�� = 0. +Suppose for a contradiction that +sup +n≥1 +sup +x∈Σ2 +��A(T n−1x) · · · A(x) +�� < ∞, + +MARGINAL INSTABILITY OF LINEAR COCYCLES +13 +in which case since f ≡ g ≡ 1 on Z, +sup +n≥1 +sup +x∈Z +������ +n−1 +� +j=0 +φ(T jx) +������ +≤ sup +n≥1 +sup +x∈Z +���� +� +1 +�n−1 +j=0 φ(T jx) +0 +1 +����� += sup +n≥1 +sup +x∈Z +���� +� +1 +φ(T n−1x) +0 +1 +� +· · · +� +1 +φ(T x) +0 +1 +� � +1 +φ(x) +0 +1 +����� += sup +n≥1 +sup +x∈Z +��A(T n−1x) · · · A(x) +�� +≤ sup +n≥1 +sup +x∈Σ2 +��A(T n−1x) · · · A(x) +�� < ∞. +We borrow an argument of Hal´asz [13] to deduce that T cannot be weak mix- +ing with respect to ν, giving us the required contradiction. In view of the above +bound we may define a bounded Borel measurable function ψ: Z → R by ψ(x) := +lim supn→∞ +�n−1 +j=0 φ(T jx). Clearly ψ(x) = φ(x) + ψ(T x) for every x ∈ Z, so +e2πiψ(x) = e2πiφ(x)e2πiψ(T x) = e2πiχ[1](x)e−2πiν([1])e2πiψ(T x) = e−2πiν([1])e2πiψ(T x) +for every x ∈ Z, where we have used the fact that the function χ[1] takes only integer +values. In particular e2πiψ ◦ T = e2πiν([1])e2πiψ ν-a.e, so e2πiψ is an eigenfunction of +the composition operator h �→ h ◦ T on L2(ν) with eigenvalue e2πiν([1]) ̸= 1, which +contradicts the fact that T is weak-mixing with respect to ν. We have obtained the +desired contradiction and deduce that necessarily +(8) +sup +n≥1 +sup +x∈Σ2 +��A(T n−1x) · · · A(x) +�� = ∞. +The limit +lim +n→∞ +1 +n log sup +x∈Σ2 +��A(T n−1x) · · · A(x) +�� +clearly exists by subadditivity. In view of (7) this limit cannot be strictly greater +than zero, and in view of (8) it cannot be strictly less than zero. It is therefore +zero, which is to say that +lim +n→∞ sup +x∈Σ2 +��A(T n−1x) · · · A(x) +�� +1 +n = 1 +as required by the statement of the theorem. The proof of the theorem is complete. +4. Acknowledgements +Versions of Theorems 1, 2 and 3, and of Proposition 2.1, previously appeared in +the second named author’s PhD thesis [26]. J. Varney was supported by EPRSC +Doctoral Training Partnership grant EP/R513350/1. +I.D. Morris was partially +supported by Leverhulme Trust Research Project Grant RPG-2016-194. +References +[1] Assani, I. Wiener Wintner ergodic theorems. World Scientific Publishing Co., Inc., River +Edge, NJ, 2003. +[2] Bell, J. P., Coons, M., and Hare, K. G. The minimal growth of a k-regular sequence. +Bull. Aust. Math. Soc. 90, 2 (2014), 195–203. +[3] Bell, J. P., Coons, M., and Hare, K. G. Growth degree classification for finitely generated +semigroups of integer matrices. Semigroup Forum 92, 1 (2016), 23–44. + +14 +IAN D. MORRIS AND JONAH VARNEY +[4] Bochi, J., and Garibaldi, E. Extremal norms for fiber-bunched cocycles. J. ´Ec. polytech. +Math. 6 (2019), 947–1004. +[5] Bochi, J., and Zhang, Y. 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Wahrscheinlichkeitstheorie und Verw. Gebiete 33, 3 (1975/76), 215–217. +[12] Guglielmi, N., and Zennaro, M. On the asymptotic properties of a family of matrices. +Linear Algebra Appl. 322, 1-3 (2001), 169–192. +[13] Hal´asz, G. Remarks on the remainder in Birkhoff’s ergodic theorem. Acta Math. Acad. Sci. +Hungar. 28, 3-4 (1976), 389–395. +[14] Jenkinson, O. Ergodic optimization. Discrete Contin. Dyn. Syst. 15, 1 (2006), 197–224. +[15] Jungers, R. The joint spectral radius: theory and applications, vol. 385 of Lecture Notes in +Control and Information Sciences. Springer-Verlag, Berlin, 2009. +[16] Jungers, R. M., Protasov, V., and Blondel, V. D. Efficient algorithms for deciding the +type of growth of products of integer matrices. Linear Algebra Appl. 428, 10 (2008), 2296– +2311. +[17] Katznelson, Y., and Weiss, B. A simple proof of some ergodic theorems. Israel J. Math. +42, 4 (1982), 291–296. +[18] Morris, I. D. Mather sets for sequences of matrices and applications to the study of joint +spectral radii. Proc. Lond. Math. Soc. (3) 107, 1 (2013), 121–150. +[19] Morris, I. D. Marginally unstable discrete-time linear switched systems with highly irregular +trajectory growth. Systems Control Lett. 163 (2022), 105216. +[20] Protasov, V. Y., and Jungers, R. M. Resonance and marginal instability of switching +systems. Nonlinear Anal. Hybrid Syst. 17 (2015), 81–93. +[21] Quas, A., and Siefken, J. Ergodic optimization of super-continuous functions on shift +spaces. Ergodic Theory Dynam. Systems 32, 6 (2012), 2071–2082. +[22] Santos, S. I., and Walkden, C. Topological Wiener-Wintner ergodic theorems via non- +abelian Lie group extensions. Ergodic Theory Dynam. Systems 27, 5 (2007), 1633–1650. +[23] Schreiber, S. J. On growth rates of subadditive functions for semiflows. J. Differential +Equations 148, 2 (1998), 334–350. +[24] Sturman, R., and Stark, J. Semi-uniform ergodic theorems and applications to forced +systems. Nonlinearity 13, 1 (2000), 113–143. +[25] Sun, Z. A note on marginal stability of switched systems. IEEE Trans. Automat. Control +53, 2 (2008), 625–631. +[26] Varney, J. Marginal instability of matrix systems. PhD thesis, University of Surrey, Guild- +ford, U.K., 2022. +[27] Varney, J., and Morris, I. D. On marginal growth rates of matrix products. Preprint: +arXiv:2209.00449, 2022. +[28] Walters, P. Topological Wiener-Wintner ergodic theorems and a random L2 ergodic theo- +rem. Ergodic Theory Dynam. Systems 16, 1 (1996), 179–206. +[29] Wiener, N., and Wintner, A. Harmonic analysis and ergodic theory. Amer. J. Math. 63 +(1941), 415–426. +I. D. Morris: School of Mathematical Sciences, Queen Mary University of London, +Mile End Road, London E1 4NS, United Kingdom +Email address: i.morris@qmul.ac.uk + +MARGINAL INSTABILITY OF LINEAR COCYCLES +15 +J. Varney: Mathematics Department, University of Surrey, Guildford GU2 7XH, +United Kingdom +Email address: jonahvarney@gmail.com + diff --git a/P9E4T4oBgHgl3EQfkg23/content/tmp_files/load_file.txt b/P9E4T4oBgHgl3EQfkg23/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c6662a9273eaee68293058987977e467199620b --- /dev/null +++ b/P9E4T4oBgHgl3EQfkg23/content/tmp_files/load_file.txt @@ -0,0 +1,499 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf,len=498 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='05152v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='DS] 12 Jan 2023 A NOTE ON THE MARGINAL INSTABILITY RATES OF TWO-DIMENSIONAL LINEAR COCYCLES IAN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MORRIS AND JONAH VARNEY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' A theorem of Guglielmi and Zennaro implies that if the uniform norm growth of a locally constant GL2(R)-cocycle on the full shift is not ex- ponential then it must be either bounded or linear, with no other possibilities occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We give an alternative proof of this result and demonstrate that its conclusions do not hold for Lipschitz continuous cocycles over the full shift on two symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Keywords: discrete linear inclusion, ergodic optimisation, joint spectral ra- dius, linear cocycle, marginal stability, marginal instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MSC2020 codes: 37H15 (primary), 37D35, 93C30 (secondary) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Introduction and statement of results Define ΣN := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=', N}Z and equip this set with the infinite product topol- ogy, with respect to which it is a compact metrisable topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Define T : ΣN → ΣN to be the shift transformation T [(xn)n∈Z] := (xn+1)n∈Z, which is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If a continuous function A: ΣN → GLd(R) is specified, one may be interested in the growth of the sequence (an) defined by an := sup x∈ΣN ��A(T n−1x) · · · A(T x)A(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' This sequence is easily seen to be submultiplicative in the sense that an+m ≤ anam for all n, m ≥ 1, which guarantees the existence of the limit ̺(A) := lim n→∞ sup x∈ΣN ��A(T n−1x) · · · A(T x)A(x) �� 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' By replacing A with ̺(A)−1 · A we may without loss of generality assume that ̺(A) = 1, and we will make this assumption for the remainder of this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In this note we will be interested in the behaviour of the sequence (an) in the reduced case ̺(A) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let us say that A: ΣN → GLd(R) is locally constant if for x = (xn)n∈Z the matrix A(x) is determined by the symbol x0 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In this case, if A denotes the range of the function A, then one simply has (1) sup x∈ΣN ��A(T n−1x) · · · A(T x)A(x) �� = sup A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A ∥An · · · A1∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The case in which A is locally constant has been studied extensively due to its relevance to marginally unstable discrete-time linear switching systems in control theory, and investigations of sequences (an) of the above form may be found in numerous works such as [6, 15, 16, 19, 20, 25, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The same problem has also been studied in [2, 3] based on quite different motivations relating to the notion of k-regular sequences in symbolic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In the works just cited the simpler 1 2 IAN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MORRIS AND JONAH VARNEY formulation (1) corresponding to the locally constant case is the only case studied, but the more general case in which A is not assumed locally constant has been touched upon in the ergodic optimisation literature, notably [4] in which criteria for (an) to be a bounded sequence are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' An early result describing some possible behaviours of such sequences (an) is the following, which is essentially due to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Guglielmi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Zennaro: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let A: ΣN → GL2(R) be locally constant and define A := {A(x): x ∈ ΣN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Suppose that lim n→∞ sup x∈ΣN ��A(T n−1x) · · · A(x) �� 1 n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Then one of the following holds: either (2) lim n→∞ 1 n sup x∈ΣN ��A(T n−1x) · · · A(x) �� > 0, or we instead have sup n≥1 sup x∈ΣN ��A(T n−1x) · · · A(x) �� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Moreover, the first case occurs if and only if the semigroup generated by A contains a nontrivial Jordan matrix with unit determinant, if and only if both of the following two conditions are met: A is simultaneously triangularisable, and the set of matrices in A with determinant ±1 is nonempty and is not simultaneously diagonalisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We remark that the situation described in Theorem 1 is quite delicate: if the dimension of the linear maps is raised from 2 to 3, or if a shift over a compact infinite alphabet is allowed in place of the finite alphabet {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=', N}, then the conclusion no longer holds and the above sequences may grow at a rate strictly intermediate between linear growth and boundedness (see [12, 19, 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In this article we give an alternative proof of the above result which is due to the second named author and which was previously presented in the thesis [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We remark that the actual existence of the limit (2) is a new contribution originating in this article: in [12, 26] it was shown that the limit inferior and limit superior of this sequence are finite and nonzero, but it was not shown that they are equal to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The second contribution of this article is to show that if the condition of being locally constant is relaxed then the dichotomy asserted in Theorem 1 ceases to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We prove: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let T : Σ2 → Σ2 be the full shift on two symbols and let d be any metric which generates the infinite product topology on Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Then there exist Lips- chitz continuous functions f, g : Σ2 → (0, 1] and φ: Σ2 → R such that the function A: Σ2 → GL2(R) defined by A(x) := � f(x) φ(x) 0 g(x) � satisfies lim n→∞ sup x∈Σ2 ��A(T n−1x) · · · A(x) �� 1 n = 1, lim n→∞ 1 n sup x∈Σ2 ��A(T n−1x) · · · A(x) �� = 0 MARGINAL INSTABILITY OF LINEAR COCYCLES 3 and sup n≥1 sup x∈Σ2 ��A(T n−1x) · · · A(x) �� = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We emphasise that the metric d is not assumed to have any properties other than generating the usual topology on Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' When working with shift spaces it is usual to consider metrics on Σ2 such that max y∈Σ2 diam {(xn)n∈Z ∈ Σ2 : xi = yi for all i such that |i| ≤ n} = O(θn) for some θ ∈ (0, 1), but in Theorem 2 this sequence may be allowed to tend to zero arbitrarily slowly or quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The functions f, g, φ may therefore be freely taken to be “super-continuous” in the sense of [5, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The proof of Theorem 1 is direct, and proceeds by considering the semigroup generated by the set {A(x): x ∈ ΣN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The proof of Theorem 2 is more technically subtle and makes use of ergodic optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The two proofs are presented in sections 2 and 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Proof of Theorem 1 In view of the identity (1) it is sufficent to prove the following: if A is a finite set of real 2 × 2 matrices which satisfies lim n→∞ max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A ∥An · · · A1∥ 1 n = 1, then the limit lim n→∞ 1 n max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A ∥An · · · A1∥ exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' if A is simultaneously upper triangularisable and the set {A ∈ A: | det A| = 1} is not simultaneously diagonalisable, then the above limit is nonzero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' and if the two conditions just mentioned do not both hold, then the semigroup generated by A is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We will begin the proof by establishing the boundedness of the semigroup gen- erated by A in a certain special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The following result is closely related to [12, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='1] but its proof is entirely different: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let A be any finite set of 2 × 2 real upper triangular matrices all of which have spectral radius at most 1 and have determinant strictly less than 1 in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Then the semigroup generated by A is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Clearly we may freely assume that A is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since every element of A has spectral radius at most 1, its diagonal entries are at most 1 in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' By the finiteness of A there exist β ∈ [0, 1) and M ≥ 0 with the following property: for every A ∈ A, one of the every diagonal entries of A is at most 1 in absolute value, the other diagonal entry is at most β in absolute value, and the upper-right entry is at most M in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' (In particular we may take β := maxA∈A | det A|1/2 and M := maxA∈A ∥A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=') Let |||·|||1 denote the norm on 2 × 2 real matrices given by the sum of the absolute values of the matrix entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' , An are arbitrary real 2×2 matrices, A′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' , A′ n are non-negative 2×2 matrices, and for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' , n every entry of A′ i is greater than or equal to the absolute value of the corresponding entry of Ai, then it is easily seen that |||An · · · A1|||1 ≤ |||A′ n · · · A′ 1|||1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' 4 IAN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MORRIS AND JONAH VARNEY (This fact is particularly easily demonstrated in the context of upper-triangular matrices, which is the only case which we shall need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=') Consequently, in order to demonstrate that sup n≥1 max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A |||An · · · A1|||1 < ∞ it is sufficient to demonstrate that (3) sup n≥1 max i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',in∈{1,2} |||Bin · · · Bi1|||1 < ∞ where B1 := � 1 M 0 β � , B2 := � β M 0 1 � , since every A ∈ A either has the absolute values of all of its entries less than or equal to the corresponding entry of B1, or has the same property with respect to the matrix B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We therefore demonstrate (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Fix an arbitrary n ≥ 1 and consider a prod- uct Bin · · · Bi1 of the matrices B1, B2 which maximises the value of |||Bin · · · Bi1|||1 among all products of B1 and B2 of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We claim that necessarily Bin · · · Bi1 = Bm 1 Bn−m 2 for some integer m ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Suppose for a contradiction that this is not the case: then Bin · · · Bi1 may be factorised as Bin · · · Bi1 = Bin · · · Bik+1B2B1Bik−1 · · · Bi1 = X1B2B1X2, say, where X1, X2 are invertible non-negative upper triangular matrices (one or both of which might be the identity matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Now, the matrix Z := B1B2 − B2B1 = � 0 2(1 − β)M 0 0 � is non-negative and nonzero, so X1ZX2 is also non-negative and nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' It follows that |||X1B1B2X2|||1 = |||X1B2B1X2 + X1ZX2|||1 > |||X1B2B1X2|||1 using the non-negativity of all of these matrices and the fact that X1ZX2 is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since X1B1B2X2 also has the form Bjn · · · Bj1 for some j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' , jn ∈ {1, 2} this contradicts the presumed maximality of |||Bin · · · Bi1|||1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since for every n ≥ 1 max i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',in∈{1,2} |||Bin · · · Bi1|||1 = max 0≤m≤n ������Bm 1 Bn−m 2 ������ 1 by the preceding claim, it follows that sup n≥1 max i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',in∈{1,2} |||Bin · · · Bi1|||1 ≤ sup n,m≥0 |||Bn 1 Bm 2 |||1 ≤ � sup n≥0 |||Bn 1 |||1 � � sup m≥0 |||Bm 2 |||1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since B1 and B2 are diagonalisable with spectral radius 1, the sets {|||Bn 1 |||1 : n ≥ 0} and {|||Bm 2 |||1 : n ≥ 0} are bounded, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' □ We now prove Theorem 1 in the form described at the beginning of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Suppose first that A is not simultaneously upper triangularisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' This is equivalent to the statement that there does not exist a basis for R2 whose first element is an eigenvector for every A ∈ A, and this in turn is equivalent to the statement that no one-dimensional vector subspace of R2 is preserved by every element of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' It is MARGINAL INSTABILITY OF LINEAR COCYCLES 5 by now well established (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='2]) that under this last condition there necessarily exists a norm |||·||| on R2 such that maxA∈A |||Av||| = |||v||| for every A ∈ A, in which case in particular maxA∈A |||A||| ≤ 1 in the associated operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' This clearly implies that |||A||| ≤ 1 for every A in the semigroup generated by A and the result follows in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The existence of the limit (2) in this case is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' For the remainder of the proof we may assume that A is simultaneously upper triangularisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' By an orthogonal change of basis for R2 we may assume without loss of generality (and without affecting either the existence or the value of the desired limit (2)) that every A ∈ A is upper triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If any B ∈ A had spectral radius strictly greater than 1 then we would have 1 = lim n→∞ max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A ∥An · · · A1∥ 1 n ≥ lim n→∞ ∥Bn∥ 1 n > 1 which is a contradiction, so every element of A has spectral radius at most 1 and therefore the absolute values of its diagonal entries are at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The latter property is clearly inherited by all products of elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We now wish to show that the limit (4) lim n→∞ 1 n max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A ∥An · · · A1∥ ≥ 0 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Define a seminorm | · | on the vector space of real 2 × 2 matrices by defining |A| to be the absolute value of the upper-right entry of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' An easy direct calculation demonstrates that if A and B are 2 × 2 upper-triangular matrices whose diagonal entries have absolute value at most 1, then |AB| ≤ |A| + |B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' It follows directly that the sequence n �→ max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A |An · · · A1| is subadditive and therefore the limit lim n→∞ 1 n max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A |An · · · A1| ≥ 0 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' On the other hand, for every product An · · · A1 = � a b 0 c � of elements of A we clearly have |b| ≤ ���� � a b 0 c ����� ≤ ���� � a 0 0 c ����� + ���� � 0 b 0 0 ����� = max{|a|, |c|} + |b| ≤ 1 + |b| and therefore max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A |An · · · A1| ≤ max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A ∥An · · · A1∥ ≤ 1 + max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A |An · · · A1| for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We conclude that lim n→∞ 1 n max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A ∥An · · · A1∥ = lim n→∞ 1 n max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A |An · · · A1| and in particular the former limit exists as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We now demonstrate that either the limit (4) is positive, or the semigroup gen- erated by A is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Define A0 := {A ∈ A: | det A| < 1}, 6 IAN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MORRIS AND JONAH VARNEY A1 := {A ∈ A: | det A| = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We observe that every matrix in A1 is of one of the following three types: either it is equal to plus or minus the identity matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' or it is a nontrivial Jordan matrix with determinant 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' or it is an upper triangular matrix with determinant −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If A1 contains a matrix B of the second type then clearly max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',An∈A ∥An · · · A1∥ ≥ ∥Bn∥ ≥ n|B| > 0 for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If it contains two matrices of the third type which are not scalar multiples of one another then it is easy to check that their product is a nontrivial Jordan matrix B, and therefore for all n ≥ 1 max A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=',A2n∈A ∥A2n · · · A1∥ ≥ ∥Bn∥ ≥ n|B| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In either of these two cases it is obvious that A1 is not simultaneously diagonalisable and that the limit (4) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If neither of these cases holds then for some upper triangular matrix X ∈ GL2(R) which has determinant −1 and diagonal entries ±1, we have A1 ⊆ {I, −I, X, −X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' It is clear that in this case A1 is simultaneously diagonalisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let X := {I, −I, X, −X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' By the Cayley-Hamilton Theorem X2 = I and it follows easily that {I, −I, X, −X} is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Consequently every element of the semigroup generated by A either is an element of X or is contained in the semigroup generated by the set ˆA := A0 ∪ {BA: B ∈ X and A ∈ A0} ∪ {AB : B ∈ X and A ∈ A0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' But ˆA satisfies the hypotheses of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='1, so the semigroup which it gen- erates is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Thus the semigroup generated by A is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We observe that unboundedness held precisely in those cases in which the semigroup generated by A contained a nontrivial Jordan matrix with unit determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Proof of Theorem 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Ergodic-theory preliminaries and a technical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We will deduce Theorem 2 from a more general ergodic-theoretic statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We begin with some necessary definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If X is a compact metrisable topological space then we let M(X) denote the set of all Borel probability measures on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' By the Riesz Repre- sentation Theorem we may identify M(X) with the set of all non-negative elements of the unit sphere of C(X)∗, and we equip M(X) with the topology which it in- herits as a subspace of C(X)∗ in its weak-* topology via this identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' This topology makes M(X) compact and metrisable, and in this topology the function µ �→ � φ dµ is continuous for every φ ∈ C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If T : X → X is a continuous transformation of a compact metric space then we let MT (X) ⊆ M(X) denote the set of all T -invariant Borel probability measures on X and ET (X) ⊆ MT (X) the set of all such measures with respect to which T is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Both sets are nonempty and the former is closed in the weak-* topology on M(X), hence is also a compact metrisable topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If φ: X → R is continuous we define β(φ) := supµ∈MT (X) � φ dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We also define Mmax(φ) to be the set of all µ ∈ MT (X) which attain this supremum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' which satisfy � φ dµ = β(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The set Mmax(φ) is nonempty by elementary considerations of compactness and continuity, and moreover has nonempty intersection with ET (X) (see for example [14, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MARGINAL INSTABILITY OF LINEAR COCYCLES 7 The following result will be applied to prove Theorem 2: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let T : X → X be a homeomorphism of a compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let f, g : Σ → (0, 1] be continuous, let φ: Σ → R be continuous, and suppose that β(log f) = β(log g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' For every x ∈ X define A(x) := � f(x) φ(x) 0 g(x) � ∈ GL2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If Mmax(log f) ∩ Mmax(log g) ̸= ∅ then lim n→∞ 1 n sup x∈X ��A(T n−1x) · · · A(x) �� = sup µ∈Mmax(log f)∩Mmax(log g) ���� � φ dµ ���� ≥ 0 and if Mmax(log f) ∩ Mmax(log g) = ∅ then lim n→∞ 1 n sup x∈X ��A(T n−1x) · · · A(x) �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Overview of the proof of Theorem 3 and its relationship with earlier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The proof of Theorem 3 begins with the observation that for every n ≥ 1 and x ∈ X, the product A(T n−1x) · · · A(x) has the form ��n−1 j=0 f(T jx) �n−1 k=0 ��n−1 j=k+1 f(T jx) � φ(T kx) ��k−1 j=0 g(T jx) � 0 �n−1 j=0 g(T jx) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since the diagonal entries necessarily belong to (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' if the norm of this product is to grow to infinity then it must do so at the same rate as the off-diagonal term Φn(x) := n−1 � k=0 \uf8eb \uf8ed n−1 � j=k+1 f(T jx) \uf8f6 \uf8f8 φ(T kx) \uf8eb \uf8ed k−1 � j=0 g(T jx) \uf8f6 \uf8f8 and therefore the problem reduces to showing that lim n→∞ 1 n sup x∈X |Φn(x)| = sup µ∈Mmax(log f)∩Mmax(log g) ���� � φ dµ ���� if Mmax(log f)∩Mmax(log f) is nonempty,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' and that the same limit is equal to zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In the second named author’s PhD thesis this problem was approached as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In order to obtain examples sufficient to prove Theorem 2 it is enough to consider the case in which f is the constant function, in which case we need only study the somewhat simpler expression (5) Ψn(x) := n−1 � k=0 φ(T kx) \uf8eb \uf8ed k−1 � j=0 g(T jx) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' This expression may be seen as a weighted Birkhoff average reminiscent of those appearing in the Wiener-Wintner-type ergodic theorems found in such works as [1, 22, 28, 29] and can be studied using a modification of the strategy used in [22, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Specifically, one may re-express (5) in terms of an extended dynamical 8 IAN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MORRIS AND JONAH VARNEY system Tg : X ×[0, 1] → X ×[0, 1] defined by Tg(x, y) := (T x, g(x)y) and continuous function ψ(x, y) := φ(x)y, obtaining by a simple induction Ψn(x) = n−1 � k=0 φ(T kx) \uf8eb \uf8ed k−1 � j=0 g(T jx) \uf8f6 \uf8f8 = n−1 � k=0 ψ(T k g (x, 1)) and therefore |Ψn(x)| = sup y∈[0,1] ����� n−1 � k=0 ψ(T k g (x, y)) ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The identity lim n→∞ 1 n sup x∈X |Ψn(x)| = lim n→∞ 1 n sup (x,y)∈X×[0,1] ����� n−1 � k=0 ψ(T k g (x, y)) ����� (6) = sup µ∈MTg (X×[0,1]) ���� � ψ dµ ���� may then by obtained using appropriate results from the ergodic optimisation litera- ture (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' [14, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' By describing carefully the invariant measures of Tg and relating them to invariant measures of T a formula similar to that in Theorem 3 may be deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' This approach can be developed further to allow for the possibility that g takes values in [−1, 1], although in this case the resulting description of the limit lim n→∞ 1 n sup x∈X ��A(T n−1x) · · · A(x) �� becomes an inequality and not an equality, due to the difficulties in treating additive cancellations in (5) arising from changes in the sign of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' When T and X have addi- tional regularity properties Theorem 3 may also be extended to allow the condition sup g ≤ 1 to be removed, since by the use of results such as [14, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='7] the condition sup g ≤ 1 can be obtained automatically from the condition β(log g) = 0 at the cost of a change of co-ordinates in R2 which depends continuously on the base point x ∈ X: see [26, §5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In the treatment of Theorem 3 in this work, we will take a different approach by exploiting the fact that the functions Φn defined above have the subadditivity property |Φn+m| ≤ |Φn ◦ T m| + |Φm| and applying techniques from subadditive ergodic optimisation (specifically, from the appendix to [18]) rather than the additive techniques of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' This has the advantages that it allows f to be nonconstant and results in a shorter proof, but has the disadvantage that f and g are constrained to take values in (0, 1] and not in [−1, 1] as is the case in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Proofs of the following standard result may be found in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' [9, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Theorem 4 (Subadditive ergodic theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let T be an ergodic measure-preserving transformation of a probability space (X, F, µ) and let (ψn)∞ n=1 be a sequence of in- tegrable functions X → R such that ψn+m ≤ ψn ◦ T m + ψm a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' for every n, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Then lim n→∞ 1 nψn(x) = lim n→∞ 1 n � ψn dµ = inf n≥1 1 n � ψn dµ for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MARGINAL INSTABILITY OF LINEAR COCYCLES 9 We also require the following subadditive analogue of (6) which may be found in the appendix to [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' For some closely-related earlier results see also [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let T : X → X be a continuous transformation of a compact metric space and let (ψn)∞ n=1 be a sequence of continuous functions X → R such that ψn+m(x) ≤ ψm(T nx) + ψn(x) for every x ∈ X and n, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Then inf n≥1 sup µ∈MT (X) 1 n � ψn dµ = sup µ∈MT (X) inf n≥1 1 n � ψn dµ = inf n≥1 sup x∈X 1 nψn(x) = sup x∈X inf n≥1 1 nψn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In the first three expressions the infimum over n ≥ 1 is equal to the limit as n → ∞ of the same expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In all cases, every supremum over µ ∈ MT (X) is equal to the corresponding supremum over µ ∈ ET (X) and is attained by an element of ET (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We now proceed with the proof of Theorem 3 and thence that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Consider the sequence of continuous functions Φn : X → R defined by Φn(x) := n−1 � j=0 f(T n−1x) · · · f(T j+1x)φ(T jx)g(T j−1x) · · · g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We note that the relation Φn+m(x) = Φm(T nx)g(T n−1x) · · · g(x) + f(T n+m−1x) · · · f(T nx)Φn(x) is satisfied for every x ∈ X and n, m ≥ 1, and consequently |Φn+m(x)| ≤ |Φm(T nx)| + |Φn(x)| for all such n, m and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' It follows that Theorem 5 is applicable to the sequence of functions |Φn|, and therefore lim n→∞ 1 n sup x∈X |Φn(x)| = sup µ∈ET (X) inf n≥1 1 n � |Φn| dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Now, for every x ∈ X and n ≥ 1 we have A(T n−1x) · · · A(x) = ��n−1 j=0 f(T jx) Φn(x) 0 �n−1 j=0 g(T jx) � and so in particular ����A(T n−1x) · · · A(x) �� − |Φn(x)| �� = ����� ����� ��n−1 j=0 f(T jx) Φn(x) 0 �n−1 j=0 g(T jx) ������ − ���� � 0 Φn(x) 0 0 ����� ����� ≤ ����� ��n−1 j=0 f(T jx) 0 0 �n−1 j=0 g(T jx) ������ = max \uf8f1 \uf8f2 \uf8f3 ������ n−1 � j=0 f(T jx) ������ , ������ n−1 � j=0 g(T jx) ������ \uf8fc \uf8fd \uf8fe ≤ 1 10 IAN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MORRIS AND JONAH VARNEY by the reverse triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Consequently lim n→∞ 1 n sup x∈X ��A(T n−1x) · · · A(x) �� = lim n→∞ 1 n sup x∈X |Φn(x)| = sup µ∈ET (X) inf n≥1 1 n � |Φn| dµ and to prove the theorem we will evaluate the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We consider in turn the cases where µ ∈ ET (X) fails to belong to Mmax(log g), fails to belong to Mmax(log f), or belongs to both sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If µ ∈ ET (X) does not belong to Mmax(log g) then for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' x ∈ X lim k→∞ 1 k log � g(T k−1x) · · · g(x) � = lim k→∞ 1 k k−1 � j=0 log g(T jx) = � log g dµ < β(log g) = 0 and hence for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' x ∈ X |Φn(x)| = ����� n−1 � k=0 f(T n−1x) · · · f(T k+1x)φ(T kx)g(T k−1x) · · · g(x) ����� ≤ n−1 � k=0 ��φ(T kx) �� · g(T k−1x) · · · g(x) ≤ ∥φ∥∞ ∞ � k=0 g(T k−1x) · · · g(x) < ∞ for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' It follows that 1 n|Φn| → 0 µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' and hence by the subadditive ergodic theorem lim n→∞ 1 n � |Φn| dµ = inf n≥1 1 n � |Φn| dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Now suppose instead that µ ∈ ET (X) does not belong to Mmax(log f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since µ is T -invariant it is also T −1-invariant, so by the Birkhoff ergodic theorem applied to T −1 we likewise have lim ℓ→∞ 1 ℓ log f(x)f(T −1x) · · · f(T −(ℓ−1)x) = � log f dµ < β(log f) = 0 and hence for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' x ∈ X ���Φn(T −(n−1)x) ��� = ����� n−1 � k=0 f(x) · · · f(T k+2−nx)φ(T k+1−nx)g(T k−nx) · · · g(T −(n−1)x) ����� = ����� n−1 � ℓ=0 f(x) · · · f(T −(ℓ−1)x)φ(T −ℓx)g(T −ℓ−1x) · · · g(T −(n−1)x) ����� ≤ n � ℓ=1 f(x) · · · f(T −(ℓ−1)x) · ��φ(T −ℓx) �� ≤ ∥φ∥∞ ∞ � ℓ=1 f(x) · · · f(T −(ℓ−1)x) < ∞ for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' It follows that for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' x ∈ X lim n→∞ 1 n ���Φn � T −(n−1)x ���� = 0 MARGINAL INSTABILITY OF LINEAR COCYCLES 11 and hence in particular 1 n|Φn◦T −(n−1)| → 0 in the sense of convergence in measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' but since µ is T -invariant, this implies that 1 n|Φn| → 0 in the sense of convergence in measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In combination with the subadditive ergodic theorem this fact yields lim n→∞ 1 n � |Φn| dµ = inf n≥1 1 n � |Φn| dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Combining the two facts just demonstrated, we have shown that if µ ∈ ET (X) does not belong to Mmax(log f) ∩ Mmax(log g) then inf n≥1 1 n � |Φn| dµ = lim n→∞ 1 n � |Φn| dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' If Mmax(log f) ∩ Mmax(log g) is empty, this demonstrates that sup µ∈ET (X) inf n≥1 1 n � |Φn| dµ = 0 which completes the proof of the theorem in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Otherwise, suppose that Mmax(log f) ∩ Mmax(log g) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since by hypothesis 0 = β(log f) ≤ sup log f ≤ 0 and 0 = β(log g) ≤ sup log g ≤ 0 it is easily seen that the set Z := {x ∈ X : log f(x) = log g(x) = 0} = {x ∈ X : f(x) = g(x) = 1} satisfies µ(Z) = 1 for every µ ∈ Mmax(log f) ∩ Mmax(log g) and in particular is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Moreover it is easily verified that Mmax(log f) ∩ Mmax(log g) = {µ ∈ MT (X): µ(Z) = 1} = MT (Z) and therefore Mmax(log f) ∩ Mmax(log g) ∩ ET (X) = MT (Z) ∩ ET (X) = ET (Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Now, if µ ∈ Mmax(log f) ∩ Mmax(log g) ∩ ET (X) = ET (Z) then for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' x ∈ X we simply have lim n→∞ 1 n|Φn(x)| = lim n→∞ 1 n ����� n−1 � k=0 φ(T kx) ����� = ���� � φ dµ ���� using the Birkhoff ergodic theorem and the fact that f and g are identically equal to 1 on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Thus sup µ∈Mmax(log f)∩Mmax(log g)∩ET (X) ���� � φ dµ ���� = sup µ∈ET (Z) ���� � φ dµ ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since we have already established that sup µ∈ET (X)\\(Mmax(log f)∩Mmax(log g)) inf n≥1 1 n � |Φn| dµ = 0 12 IAN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MORRIS AND JONAH VARNEY it follows that sup µ∈ET (X) inf n≥1 1 n � |Φn| dµ = sup µ∈Mmax(log f)∩Mmax(log g)∩ET (X) ���� � φ dµ ���� = sup µ∈ET (Z) ���� � φ dµ ���� = sup µ∈MT (Z) ���� � φ dµ ���� = sup µ∈Mmax(log f)∩Mmax(log g) ���� � φ dµ ���� as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The proof of the theorem is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Fix a metric d on Σ2 which generates the infinite product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Let [1] ⊂ Σ2 denote the set of all (xn)n∈Z ∈ Σ2 such that x0 = 1, which by the definition of the infinite product topology on Σ2 is both closed and open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' By an appropriate version of the Jewett-Krieger Theorem (see for example [8, §29]), or by various direct constructions such as [7, 10, 11], there exists a compact T -invariant set Z ⊂ Σ2 with the following properties: there exists a unique ν ∈ MT such that ν(Z) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' T is weak-mixing with respect to this unique measure ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' the support of ν is precisely Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' and Z is not a singleton set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since Z is not a singleton we have 0 < ν([1]) < 1 and therefore e2πiν([1]) ̸= 1, a property which will be significant later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Define f(x) = g(x) = e− dist(x,Z) for all x ∈ Σ2, where dist(x, Z) := infy∈Z d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Clearly f and g are Lipschitz continuous and satisfy β(log f) = β(log g) = 0 and Mmax(log f) = Mmax(log g) = {ν}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Define also φ(x) := χ[1](x) − ν([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' By the definition of the infinite product topology, φ: Σ2 → R is continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' since Σ2 is a compact metric space with respect to d, φ is uniformly continuous with respect to d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' and since φ takes exactly two values, this implies that φ is Lipschitz continuous with respect to d as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Define A: Σ2 → GL2(R) as in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Clearly Theorem 3 is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Since sup µ∈Mmax(log f)∩Mmax(log g) ���� � φ dµ ���� = ���� � φ dν ���� = |ν([1]) − ν([1])| = 0, Theorem 3 yields (7) lim n→∞ 1 n sup x∈Σ2 ��A(T n−1x) · · · A(x) �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Suppose for a contradiction that sup n≥1 sup x∈Σ2 ��A(T n−1x) · · · A(x) �� < ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' MARGINAL INSTABILITY OF LINEAR COCYCLES 13 in which case since f ≡ g ≡ 1 on Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' sup n≥1 sup x∈Z ������ n−1 � j=0 φ(T jx) ������ ≤ sup n≥1 sup x∈Z ���� � 1 �n−1 j=0 φ(T jx) 0 1 ����� = sup n≥1 sup x∈Z ���� � 1 φ(T n−1x) 0 1 � · · � 1 φ(T x) 0 1 � � 1 φ(x) 0 1 ����� = sup n≥1 sup x∈Z ��A(T n−1x) · · · A(x) �� ≤ sup n≥1 sup x∈Σ2 ��A(T n−1x) · · · A(x) �� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We borrow an argument of Hal´asz [13] to deduce that T cannot be weak mix- ing with respect to ν, giving us the required contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In view of the above bound we may define a bounded Borel measurable function ψ: Z → R by ψ(x) := lim supn→∞ �n−1 j=0 φ(T jx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Clearly ψ(x) = φ(x) + ψ(T x) for every x ∈ Z, so e2πiψ(x) = e2πiφ(x)e2πiψ(T x) = e2πiχ[1](x)e−2πiν([1])e2πiψ(T x) = e−2πiν([1])e2πiψ(T x) for every x ∈ Z, where we have used the fact that the function χ[1] takes only integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In particular e2πiψ ◦ T = e2πiν([1])e2πiψ ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='e, so e2πiψ is an eigenfunction of the composition operator h �→ h ◦ T on L2(ν) with eigenvalue e2πiν([1]) ̸= 1, which contradicts the fact that T is weak-mixing with respect to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' We have obtained the desired contradiction and deduce that necessarily (8) sup n≥1 sup x∈Σ2 ��A(T n−1x) · · · A(x) �� = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The limit lim n→∞ 1 n log sup x∈Σ2 ��A(T n−1x) · · · A(x) �� clearly exists by subadditivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' In view of (7) this limit cannot be strictly greater than zero, and in view of (8) it cannot be strictly less than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' It is therefore zero, which is to say that lim n→∞ sup x∈Σ2 ��A(T n−1x) · · · A(x) �� 1 n = 1 as required by the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' The proof of the theorem is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Acknowledgements Versions of Theorems 1, 2 and 3, and of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='1, previously appeared in the second named author’s PhD thesis [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Varney was supported by EPRSC Doctoral Training Partnership grant EP/R513350/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Morris: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom Email address: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='morris@qmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='uk MARGINAL INSTABILITY OF LINEAR COCYCLES 15 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content=' Varney: Mathematics Department, University of Surrey, Guildford GU2 7XH, United Kingdom Email address: jonahvarney@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQfkg23/content/2301.05152v1.pdf'} diff --git a/PNFIT4oBgHgl3EQfeiuo/content/tmp_files/2301.11275v1.pdf.txt b/PNFIT4oBgHgl3EQfeiuo/content/tmp_files/2301.11275v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3dd57cde7835431a044935546aef1e40ee61323 --- /dev/null +++ b/PNFIT4oBgHgl3EQfeiuo/content/tmp_files/2301.11275v1.pdf.txt @@ -0,0 +1,1158 @@ +A Review of Scene Representations for Robot Manipulators +Carter Sifferman +sifferman@wisc.edu +Department of Computer Sciences +University of Wisconsin - Madison +1. Introduction +For a robot to act intelligently, it needs to sense the world around it. Increasingly, robots +build an internal representation of the world from sensor readings. This representation can +then be used to inform downstream tasks, such as manipulation, collision avoidance, or +human interaction. In practice, scene representations vary widely depending on the type +of robot, the sensing modality, and the task that the robot is designed to do. This review +provides an overview of the scene representations used for robot manipulators (robot arms). +We focus primarily on representations which are built from real world sensing and are used +to inform some downstream robotics task. +Building an intermediate scene representation is not necessary for a robotics system. +It is completely possible for a robotics system to act directly on sensor data (e.g. predict +appropriate grasps directly from RGB images), and we will look at many such systems within +this review. However, we argue that intermediate scene representations are beneficial for +robot manipulators as they: +• act as spatial memory +• are efficient storage of past memories +• allow long-horizon planning +• can act as regularization and encode spatial priors for learning systems +In this review, we organize scene representations into three categories depending on the +task that the representation supports. These categories make up the sections of our review: +collision avoidance (section 2), manipulation (section 3), and teleoperation (section 4). +Within each section, we provide a review of the existing literature, summarize the challenges +in the area, and consider directions for future research. +In section 5, we look at scene +representations for manipulators as a whole, and consider directions for future research +which cut across our three categories. +1.1 Literature Survey Process +Our literature survey consisted of two phases. In phase 1, we performed a broad search of +existing reviews in order to gain context and understand the broader landscape of robotics. +1 +arXiv:2301.11275v1 [cs.RO] 22 Dec 2022 + +Figure 1: This review focuses on robotics applications which build some intermediate scene +representation between sensor data and a downstream robotics task. Image sources: [1, 2, 3] +Category +Snowball Seeds +# Papers Found +Collision avoidance +[2, 18, 3] +26 +Manipulation +[19, 20, 21] +28 +Teleoperation +[22, 23, 14] +16 +Table 1: The three sections that this review is organized into, and the “snowball seeds” +which began the literature review process. +To find these reviews, we use Google Scholar search with the “Review Articles” filter en- +abled. The result of this search is 13 reviews spanning a broad spectrum: manipulation and +grasping [4, 5, 6, 7, 8, 9, 10], SLAM [11], human-robot interaction [12, 13], teleoperation +[14], motion planning [15, 16] and inverse kinematics [17]. Knowledge gained from these +reviews was used to determine our taxonomy of scene representations, and the scope of this +review. +The goal of phase 2 of our survey process was to find directly relevant papers which +employ a scene representation for robot manipulators. As the papers that we search span +a wide range of topics, keyword searches did not prove effective. Instead, we “snowballed” +through references, beginning at seed papers which were found through keyword search, +reviews, or consulting with colleagues. These seeds are shown in table 1. We snowballed +through references both through traditional reference chasing, and using Google Scholar’s +“Cited By” page to find papers published after the seed paper. We also use the Abstract +Viewer Project1, a system for finding related papers developed as a research project at UW- +Madison. Abstract Viewer does not use citations to find related papers, instead relying +on analysis of textual content. This proved helpful for finding still-related papers when +references ran dry. In total, 69 papers were collected in phase 2. +This review is not meant to act as a comprehensive survey of any one subject. Instead, +we hope to give a broad overview of the field and provide jumping-off points for further +reading. Articles were not selected to be a representative sample of the entire field; papers +which use an identical scene representation to existing work may be excluded while those +with a unique scene representation are generally included. +1. https://pages.graphics.cs.wisc.edu/AbstractsViewer/ +2 + +Sensing data +Intermediate +Downstream task +Representation2. Collision Avoidance +A central challenge in robotics is being able to avoid collisions, which can potentially be very +costly with a powerful, fast, and expensive robot. Some approaches for collision avoidance +respond directly to measurements from robot-mounted sensors, without building any inter- +mediate representation [24, 25, 26]. A downside of this approach is that it has no memory: +the robot can only act based on what it currently observes, and cannot plan based on its +previous observations. As a result, these approaches are overly cautious and perform poorly +in challenging conditions. +Modern approaches for collision avoidance can be broadly grouped into two categories: +motion planning [27], in which the start and goal position of the robot end effector are +known, and the goal is to find a viable, collision-free path between the two, and inverse +kinematics, in which the end effector position is known, and the goal is to find a viable +joint configuration of the robot which matches that end effector position. This distinction +is unimportant for this review, as both of these approaches have the same requirements +of their scene representations: fast collision checks and (sometimes) fast calculation of the +distance to the nearest obstacle or direction to the nearest obstacle. As a result, scene +representations for collision avoidance are similar whether the problem is formulated as +inverse kinematics or motion planning. +Potential Fields– Early approaches to collision avoidance in a motion planning context +represented the environment with a potential field, first proposed in [28]. This potential field +can be evaluated at any point to yield a scalar “potential” value. This value is determined +by both the scene geometry (which have high potential around them) and the desired +location (which has a low potential). In order to move through space, the robot simply +follows the negative gradient of this potential field. The potential field was heavily utilized +in early motion planning work [29, 30, 31]. While effective for the time, potential fields +suffer from a few problems: the potential function is prone to local minima, and can be +very difficult and computationally intensive to construct [15]. Additionally, it is impractical +to construct potential fields in real-time from sensor measurements, both because they are +slow to construct, and because they require scene geometry to be described in a friendly +closed-form, which sensors cannot provide natively. Later work [32] improved on the local +minimum problem by taking into account the relative starting position as well as scene +geometry during potential field construction, but nonetheless potential fields have fallen +out of favor in collision avoidance applications since the early 2000s. +Signed Distance Fields– A signed distance field is a mapping between 3D points in space +x and the scalar distance d to the nearest obstacle: +SDF(x) = d +The SDF has the nice property that taking −∇SDF(x) yields a vector pointing towards the +nearest obstacle. This representation has been used in computer graphics since at least 1998 +[33, 34], and became popular for robot collision avoidance with the introduction of the highly +influential CHOMP motion planner in 2009 [3]. CHOMP uses the signed distance field, +along with pre-computed gradients to perform optimization over the robot configuration. +Subsequently, the popular STOMP [35], TrajOpt [36], and ITOMP [37] motion planners +also used a signed distance field to represent their environment, and used gradients in a +3 + +similar way. In each of these works, the signed distance field is computed at fixed points on a +regular voxel grid as a pre-processing step. To do such a computation, a precise model of the +underlying geometry is needed, typically in the form of a mesh. Similarly to potential fields, +computing a signed distance field is computationally expensive, and generating it from noisy +sensor data is difficult. In practice, collision avoidance approaches which use an SDF are +constrained to simulations, where the SDF can be pre-computed, or static environments in +the real world. Regardless, SDFs are the most popular scene representation for collision +avoidance, largely because they are supported by popular and effective motion planners. +There exist libraries such as VoxBlox [38] and FIESTA [39] which efficiently compute and +store discretely sampled SDFs for this purpose. +Collections of Primitives– A less common method for representing geometry for col- +lision avoidance is with a collection of primitive shapes, such as spheres, cylinders, and +cubes. These primitives are usually stored parametrically, so that collision checking can +be done quickly and the objects represented natively in optimization solvers. Toussaint et. +al. [40] proposed Logic-Geometric Programming, in which the scene is composed entirely +of parametric cylinders, blocks, and planes. This paper has been influential for its elegant +optimization-centered formulation, but the scene representation used within has not been +heavily utilized; it serves more as a demonstration of the approach’s capability. Similarly, +Gaertner et. al. [41] considers collision avoidance with humanoid robots, and uses a col- +lection of primitives to represent dynamic scenes. Similarly to Toussaint, the collection of +primitives is used primarily to demonstrate the capabilities of the system. In contrast, Zim- +merman et. al. [42] proposes a method for using collections of primitives in gradient-based +optimization methods such as TrajOpt [36]. They provide a unified method for dealing +with many types of primitives, and a method for taking the derivative of the distance to +the nearest primitive, making a collection of primitives a drop-in replacement for SDFs. +However, this approach has not seen widespread adoption. +Collections of Convex Hulls– A collection of convex hulls is another less common way +to represent scene geometry for collision avoidance. Similarly to primitive shapes, convex +hulls allow for fast collision checking and natural representation in optimization solvers. +Convex hulls have the additional benefit that any shape can be broken down to a collection +of convex hulls via an algorithm like QuickHull [43]. Schulman et. al. [44] uses a set of +convex hulls to represent a scene, and outperforms the SDF-based motion planners of the +time like CHOMP [3] and STOMP [35]. CollisionIK [18] introduces an optimization-based +method for inverse kinematics, which is able to operate in real-time (e.g. +for mimicry +control) and avoid collisions with dynamic obstacles. CollisionIK mentions that point cloud +objects could be broken down into convex hulls in real time, but does not demonstrate such +a process. To our knowledge, no existing approach constructs convex hulls in real time from +sensor data. +Learned Representations– Within the last year, one approach has emerged which uses +a learned environment representation to enable real-time collision avoidance with dynamic +obstacles and real-world sensing. This paper is somewhat influenced by the growing lit- +erature around learned representations in computer vision [45], graphics [46], and SLAM +[47, 48]. RCIK [2] proposes a collision cost prediction network, a neural network which +takes as input features extracted from an occupancy grid, as well as a 3D point in that grid; +from this input the network predicts the collision cost, which is an approximation of the +4 + +Figure 2: The collision cost prediction network of RCIK [2] is trained on simulated data, +and outputs a collsion cost Fcol which approximates the signed distance function to enable +real-world real-time collision avoidance. +SDF evaluated at that 3D point. The occupancy grid can be generated in real time with +one or more depth cameras. The network is trained on one million simulated examples of +random environments and joint configurations. While this method does not have the col- +lision avoidance guarantees provided in theory by other methods, it is the first method to +perform collision-free inverse kinematics in real time with real sensing. This same approach +was later evaluated under real-time control [49]. +2.1 Future Directions in Collision Avoidance +Real-time generation of signed distance fields. Signed distance fields have proven +highly effective for enabling collision avoidance. However, SDFs are very costly to generate, +and require a very accurate representation of the underlying environment. Because of this, +the vast majority of work on collision avoidance with robot manipulators does so only in +simulation, or in manually recreated static environments. A pressing challenge is finding +ways to bring these methods to the real world by constructing an SDF in real time. Recent +work on neural representations has enabled real-time construction of a neural SDF from +depth imagery called iSDF [50]. Similarly to RCIK [2], the SDF produced by iSDF is only +an approximation, however it is generated over time from multiple sensor observations, +and does not rely on simulated pre-training. +Adapting a similar approach for static or +manipulator-mounted depth cameras could enable real-world operation of the many collision +avoidance approaches which rely on SDFs. +Moving beyond signed distance fields. Signed distance fields alone are great for +collision avoidance, but offer some limitations. For example, not all collisions are equally +costly. Colliding with a pillow might be admissible if it means avoiding collision with a +human. Future representations, and algorithms which act on them, could store semantic +information along with scene geometry, to enable such decisions to be made. Learning such +semantic scene properties may be possible via extensive pre-training, or via interaction. +5 + +Joint configuration +Joint configurations +feature extractor +col +Regression +module +obs +Obstacle +Occupancy grid +feature extractor3. Manipulation +Arguably the most important task for a robot manipulator is to manipulate things. Manipu- +lation can mean grasping with a simple one degree-of-freedom gripper, articulated grasping, +or simple pushing and nudging of objects with any part of the robot. Robot manipulation +is a vast field, with approaches specialized for dealing with many specific challenges. To +keep the scope of this review reasonable, we focus on scene representations which are used +for: +• Basic grasping with a 1DoF gripper +• Generalizable grasping +• Articulated grasping +• Predicting scene flow +Direct Action on Images– While this review focuses on intermediate scene representa- +tions, it is clear that, for the purposes of robot manipulation, an intermediate representation +is not necessary. A seminal work in this area was Saxena et. al. [21] in 2006, which was +the first work to directly predict grasping points from an image. They use a neural network +to, given an RGB image of an object to be grasped, predict pixels in the image at which +the object is most suitable for grasping. To train their neural network, they use supervised +learning on a large fully synthetic dataset. Two years later, the same authors improved +on the approach by using RGB-D imagery as input to the network [51]. Lenz et. al. [52] +improved upon previous work by aiming to predict the single best grasp, rather than listing +many viable ones, and predicting the orientation and extent of the grasp along with the +location. Later, the “DexNet” series of papers offered iterative improvements by improving +training data and tweaking the neural network outputs, as well as considering alternative +grip types such as articulated hands and suction cups [53, 54, 55]. Other work aims to grasp +objects given some semantic label, e.g. “coffee mug” or “plate”, this is sometimes called +semantic grasping. Jang et. al. [56] proposes a two-stream approach to semantic grasping, +in which one stream identifies objects while another finds suitable grasps. Schwarz et. al. +[57] demonstrates a semantic grasping pipeline which uses a suction cup gripper and works +by simply segmenting out objects and finding their center of mass. This approach per- +formed well at the highly competitive Amazon Picking Challenge2. These direct prediction +approaches are highly effective for real world operation, however they are fairly limited in +their potential. These approaches can only perform single-shot grasping, meaning they are +unable to, for example, rotate an object, let go of it, and grab it again. They also have a +limited ability to reason about novel shapes in 3D. Lastly, these approaches typically rely +on some synthetic training data, which must be generated via other 3D-aware methods. +Meshes– A number of works use a 3D triangular mesh to represent objects for manipu- +lation. The problem of finding suitable grasps given a 3D mesh is a long-standing problem +with active research [58, 59, 60, 61, 62]. This paragraph focuses not on those algorithms, +but on real-world systems for manipulation which represent objects as meshes. The first +of such real-world systems was Berenson et. al. [63] in 2007. This work makes grasping +2. https://robohub.org/amazon-picking-challenge/ +6 + +Figure 3: An early approach for real world grasping, Berenson et. al. [63], relied on a +motion capture system and pre-defined meshes to perform grasping in the real world. +possible in the real world by incorporating information about not only the object to be +grasped, but also nearby obstacles, such as the table or other objects, into the grasp selec- +tion algorithm. In order to sense the positions of objects in real-world tests, this approach +relies on motion capture markers being placed on each object, as shown in fig. 3. Goldfeder +et. al. [64] introduced a method for finding good grasps given a mesh, and tested their +method by scanning real objects. This approach was somewhat effective, but the conversion +from scan to mesh does not happen in real time. Collet et. al. [65] circumvents the prob- +lem of real-time mesh construction by modeling each object as a primitive, and fitting that +primitive to a point cloud in real time. Later work by the same authors [66] extends this +idea to arbitrary meshes by using a 6D pose recognition algorithm. Assuming that a mesh +of the object is known, this approach enables prediction of the mesh’s pose in real time. +Papazov et. al. [67] takes a similar approach, but assumes that the object is represented +with a set of points and surface normals, rather than an RGB image. Varley et. al. [68] +removes the requirement of a pre-made mesh by teaching a neural network to complete a +point cloud. From the completed point cloud, a mesh can be built and that mesh passed +off to a mesh-based grasp planner in real time. The steps in Varley et. al.’s pipeline are +shown in fig. 4. In each of these approaches, we see that the limiting factor is not the grasp +planning algorithms themselves, but the generation of meshes in real time. +Point Clouds– Point clouds have been used as the basis for grasp planning in a similar +manner to meshes. In constrast to meshes, point clouds are closer to the native output of +commonly used depth cameras, making them more practical for real world construction. +Approaches exist for grasp planning directly on point clouds [69, 70, 71, 72], although less +numerous than those for meshes. +Florence et. +al. +[73] (covered in more depth in the +following paragraph) introduces a method for finding corresponding grasps between similar +objects, and uses point clouds to perform the grasping in real-world examples. Simeonov +et. al. [74] considers the problem of manipulation planning, and is able to predict the +movement of scene objects directly from their point clouds. This prediction is used to plan +for manipulations which move the scene towards a goal state. +7 + +2Figure 4: Varley et. al. [68] uses a neural network to enable shape completion on partial +point cloud observations. The completed shapes can then be transformed to a mesh and +used in any mesh-based grasp planner. Image from Varley et. al. +Voxel Grids– Zhang et. al. [75] constructs a gripper-sized voxel grid from a point cloud +and a given gripper position, with each voxel encoding whether the space is occupied by +a point in the point cloud. This voxel grid can then be compared (via nearest neighbors) +to previously simulated voxel grids to determine whether it corresponds to a good gripper +position. +Learned Representations– Dense object nets [73] are an approach for generalizable grasp- +ing. For a given input image, a neural network is trained to produce a dense pixel-level +feature map which produces similar output feature vectors for semantically similar points +in multiple images of similar objects. For example, for any two images of a mug, the goal is +that pixels corresponding to the same point on the mug handle will have the same feature +vector in the output representation. This representation is trained to be consistent across +object instance, pose, and deformation, enabling generalizable grasping. They demonstrate +that this approach is effective at generalizable grasping in the real world, using merged point +cloud data from multiple depth cameras. Simeonov et. al. [19] expands upon this approach +by taking a 3D point as input to the neural network, rather than a 2D pixel position. The +object properties are encoded within the weights of a multi-layer perceptron, similarly to +NeRF [45]. They demonstrate that this approach can be used to perform pick-and-place +tasks of novel objects of a given class given fewer than 10 demonstrations. Additionally, the +architecture of their MLP ensures that the descriptor fields are SE(3)-equivariant, making +them robust to arbitrary object poses. Van Der Merwe et. al. [76] uses a learned represen- +tation for articulated grasping. A neural network is trained to, given a point cloud and a +3D query point, approximate the signed distance function at that point. The latent space +of this network is concatenated with information about desired grasp qualities and robot +8 + +15 +Kinect +430 +98499824 +26 +(a) Image of Occluded Side +(b) Point Cloud +(c) Segmented and Meshed +(d) CNN Input +403530252015105 +15 +25 +30 +10 +15 +20 +(e) CNN Output +(f) Fast Mesh +(g) Detailed Mesh +(h) Grasp Planningconfiguration, and fed into another network which predicts the success rate of the proposed +grasp. Xu et. al. [20] builds a visual predictive model for robotics. Their goal is to, given +some representation of the scene along with a robot action, predict the scene after the action +has occurred. Rather than using a manually constructed scene representation, they allow +the scene representation to be learned in an end-to-end manner, as a 128x128x48 feature +grid with an 8-dimensional vector at each point in the grid. Using this representation, they +are able to predict 3D scene flow and plan manipulation tasks. +3.1 Future Directions in Manipulation +Performing 3D grasp planning based on real-world sensor data. There is a dis- +connect between the way scenes are represented for grasp planning (primarily meshes) and +the way that the most effective robotics systems, such as those used in the Amazon Pick- +ing Challenge, are operated (primarily direct prediction). An important direction for future +work is bridging this gap. Direct prediction methods are severely limited in their spatial rea- +soning, while mesh-based methods are severely limited in their real-world operation. Point +cloud manipulation and learned representations may bridge the gap, but do not currently +offer the high grasping accuracy of mesh based approaches. +Representing complex object properties. Any object may have many properties +which are relevant to manipulation: its center of mass, deformation properties, or con- +straints on its motion (e.g. hold a mug full of coffee upright, pull a drawer straight out). +Learning-based approaches, such as Dense PhysNet [77] have shown promise towards being +able to learn these properties autonomously. A needed direction for future research is de- +termining ways to learn these properties, generalize them to novel objects, and store these +properties alongside their geometric representations. +4. Teleoperation +In a teleoperation scenario, a human controls a robot remotely, and relies on an intermediate +interface, such as a monitor or VR headset, to understand the robot’s surroundings. Much +recent work on teleoperation places the human operator in virtual reality (VR); studies have +shown increased task performance in virtual reality, due to the ease of controlling the user’s +view and all six degrees-of-freedom of the robot [78]. However, the scene representations +used in these VR approaches can generally applied to any sort of teleoperation scenario. +4.1 Human Control of Mobile Robots +Mobile robots are much more likely than robot manipulators to venture into unknown +environments. +Because of this, the foundational work in representing scenes to remote +operators has taken place in mobile robotics. +For context, we offer a brief overview of +such work here. Nielsen et. al. [79] was an early attempt at incorporating data beyond +RGB camera feeds into robot teleoperation. They create a dashboard which shows an RGB +camera feed along with LiDaR scans and a rough map of the environment. While all of the +information is presented to the user, it is not combined to create an intuitive display. Kelly +et. al. [80] builds a 3D map of the environment by using LiDaR scans to create a 3D map, +and coloring that map with data from RGB images. A CAD model of the mobile robot is +9 + +(a) Dashboard-based interface +from [82] (2004) +(b) Augmented interface from +[79] (2007) +(c) +3D +interface +from +[80] +(2011) +Figure 5: A history of the typical information displays used in mobile robot teleoperation +placed in the environment. This allows rendering arbitrary viewpoints, such as an overview +of the entire scene, an overhead view, or a third person view, as would be seen in a racing +video game. Stotko et. al. [81] builds a 3D mesh-based scene representation for a mobile +manipulator, and displays the mesh to the user interactively in virtual reality. They find +that users have fewer collisions, and report a greater level of immersion and awareness than +with a 2D interface. Livantino et. al. [23] augments robot-attached camera views with 3D +data to overlay data such as desired path, destination, and label traversable terrain. For +mobile robot teleoperation, the trend has been towards a free floating, user controllable +view and natural display of information. These same goals apply to robot manipulator +teleoperation. +4.2 Human Control of Robot Manipulators +No intermediate representation– A simple baseline for teleoperation is the use of one or +more static cameras, which the user can either see all at once, or switch between manually. +While simple to implement, a static camera approach is limiting: they present the user +with limited geometric information, and don’t leverage computation to enhance human +perception. Static cameras are also prone to being blocked by the robot manipulator itself. +There are a number of approaches which improve on the static camera baseline without +building an intermediate representation. Murata et. al. [83] considers mobile manipulators +(manipulators mounted on mobile robots), and renders a CAD model of the robot on top +of a background made of stitched, observed images. The CAD model is kept updated to +represent the robot’s current state, and the user is able to position the camera to generate +an arbitrary view. A different approach is taken by Rakita et. al. [84], in which a robot +arm is used as a camera operator for another robot arm. The camera operator’s pose is +automatically optimized in real time to present a clear view of the manipulating robot’s +end effector. +Point Clouds– Point clouds are a common choice for representing scenes for real time +operation, because they are the native output of depth cameras, and can be displayed in +real time with little processing. Brizzi et. al. [85] considers augmented reality for VR +teleoperation. +Operators see an RGB image of the scene, along with features, such as +distance to the target, direction to the target, or gripper state. Point clouds from a depth +10 + +QuitProgram +Select Camera +Home Camere +Zoom In +ZoomOut +X +Feedbock +Resistonce Limit H +Velocity Limit +Auto +ON +Robot: +Sensorstatus +Health +Sensor +Status +Heoding +Roll +Anitude +Sonar +OK +ojps +02LF +Laser +OK +Pitch +Camera +05DN +OK +Pressfor +Sensor +Tele +Velocity +Inertials +OK +Power +Status +0.0 +24.5Volts +Inclinometer +OK +Update +Turn +n/ +2025 +Compass +Escape +OK +12 +0.0 +m/s +Thermometer +OK +Bump +OK +Motion +Pursuit +CommunicationsHealth OK +Enobled +FLIR +OK +StopVideo +Map +Map +Robota +2.92kph +655.1 m +UnknownFigure 6: Kohn et. al. [86] uses meshes to represent known objects (table and robot) and +point clouds to represent unknown objects (puzzle box) for teleoperation. Image from Kohn +et. al. +camera are used to calculate these features. Kohn et. al. [86] displays a combination of +point clouds from RGBD cameras and meshes to a user in VR. Meshes are used for known +objects (such as the floor, table, and robot) and point clouds are used for the unknown. +The unknown objects are filtered in real time according to the meshes, as shown in fig. 6. +A similar approach is taken by Su et. al. [87]. Wei et. al. [22] similarly displays point +clouds to users, but unlike previous approaches, they do not use meshes to represent known +objects, aside from the robot gripper. They perform a user study, comparing the point +cloud to multi-view RGB and to a point cloud projected onto an RGB image. In their +experiments, the hybrid point cloud and RGB view performs best. +Meshes– While point clouds are native and fast to display, meshes may be preferred +due to their potentially higher visual fidelity. +Wonsick et. +al. +[88] takes an approach +similar to the mesh-based approaches in section 3; a point cloud of the scene is semantically +segmented to identify its class amongst known objects. A known 3D model is then fit to +each point cloud segment, and that 3D model is displayed to the user in virtual reality. +Models are able to be constructed in much higher fidelity, and the scene can be entirely +rendered, enabling control of lighting, texture, etc. They run a user study and say that users +find their approach more usable and less cognitive load than point-cloud streaming alone. +However their approach relies on having known 3D models of objects in the environment, +and has no fallback if such a model does not exist, as such it is not suitable to unknown +environments. Piyavichayanon et. al. [89] generates a “depth mesh” by combining the view +of multiple depth cameras. This mesh is used to display augmented reality features, such +as distance to a collision state, on a handheld smartphone. Merwe et. al. [90] performs +a user study to investigate how the type of information presented to the operator changes +their performance. They compare a full information 3D model to a “representative” mesh +based model, which only displays crucial information. They find that task completion time +is lower with the full model, but cognitive load is higher. +Occupancy Grids– Omarali et. al. [91] considers multiple modes of visualization for +VR teleoperation. Alongside the depth camera-based baselines, they present a hybrid view, +which displays the current output from the depth camera, alongside a translucent occupancy +map in the previously observed areas. This occupancy map gives the user some context for +the greater environment, while indicating that the context is not as reliable as the currently +11 + +observed region. They find that users prefer the hybrid depth camera + occupancy map +approach. +4.3 Future Directions in Teleoperation +Improving real-time visualization of the robot’s environment. Existing approaches +tend to rely on depth cameras, which can render the world in real time, but are low in detail +and high in noise. Approaches which attempt to process this data into something more +appealing, like a mesh, require heavy computation and require meshes of potential objects +to be known a priori. Future work should consider ways to improve the fidelity and detail +of reconstruction in real time and in unknown environments. To some extent, advances in +imaging, such as high resolution depth sensors may improve this problem. Another potential +solution is dynamically directing sensing towards areas that need high detail, such as around +the end effector. The problem of representing the environment from a novel perspective is +known as novel view synthesis and is well studied in the computer vision literature (see the +recent review [92]). Incorporating techniques from novel view synthesis could allow a more +complete environment representation to be displayed to the operator. +Building representations suited to the operator. Currently, the vast majority of +approaches focus on building representations which represent the environment faithfully. +However, it has long been known that realistic displays are not always ideal, as they can +make it difficult to parse what is relevant [93]. +Some work covered in this review has +considered building a sparse representation of only relevant features [90, 88]. +However, +these approaches require such a representation to be specified by hand prior to operation. +Future work should consider ways of filtering the scene representation to represent only +what is important to the user in an unknown scene. +5. Future Directions +Quantifying Uncertainty. Present systems for robot manipulation build a most likely +map of the environment. In poor sensing conditions, this map may be highly unreliable, +leading the downstream robotics application to make incorrect but fully confident choices +based on the unreliable map. Future work should work to incorporate uncertainty into +the scene representation itself, in order to enable robot policies which act intelligently in +the face of uncertainty. When in a highly uncertain environment such a robot could, for +example, ask a human for help, or gather additional information about its environment +before taking action. The issue of uncertainty estimation is well studied in mobile robotics, +especially in the context of SLAM [94]. Recent approaches in novel view synthesis have +also enabled dense estimates for geometric and photometric uncertainty [95, 96], as shown +in fig. 7. Such approaches could be applied to robot manipulators, especially for problems +like collision avoidance. +Joint prediction of scene semantics and geometry. Present approaches which +infer scene semantics do so by taking some geometric representation of the scene as input, +and producing a semantic map based on that geometry. With this approach, scene geometry +informs scene semantics, but semantics has no influence on geometry. In reality, semantic +information about an object can provide queues about its likely geometry. If an object is +identified as a coffee mug, it probably has a handle on it. If an object is identified as a +12 + +Figure 7: Shen et. al. [95] builds a scene representation which includes dense uncertainty +estimates. A similar approach could prove useful to build scene representations for robot +manipulators. Image from Shen et. al. +baseball, it is probably spherical. Floors and tabletops tend to be flat and level with one +another. Future approaches could infer scene geometry and semantics jointly, by encoding +sensor data in a learned latent representation, which is used to inform both geometric +and semantic inference. +Joint inference approaches have been used for semantic SLAM +[97, 98, 99]. Implementing such an approach for robot manipulators would enable higher +fidelity predictive mapping and improved semantic understanding. +Sensing-first development and real-world benchmarking. +Many existing ap- +proaches, particularly for grasping and collision avoidance, are benchmarked entirely in +simulation. Real-world demonstrations are addressed after the development has been op- +timized for simulation, and often only occur under highly controlled conditions. Future +work should consider developing methods with sensing in mind from the beginning, and +aiming for high performance on real-world tasks with sensing, rather than in simulation. +This means building systems which are robust to sensor noise, and able to act on raw sensor +data in real time. One encouraging sign towards such a goal is the just announced RT-1 +project from Google [100], which is an embodied agent that is benchmarked entirely on +real-world tasks. +Representations for Alternative Sensing Modalities. All works addressed in this +survey utilize data from typical vision based sensors: RGB cameras, depth cameras, and/or +LiDaR. There are many other sensor types used in robotics, such as force torque sensors +and tactile sensors, which provide information about the environment. However, existing +approaches react immediately to information from these sensors, and do not build an inter- +mediate representation which persists over time. Can we build scene representations which +model the factors that these sensors measure? For example, the measurement from a tactile +sensor may be influenced by how hard the object is, how heavy it is, or even how conductive +it is, alongside the object’s geometry. Present scene representations do not encode all of this +information, making it very difficult to relate new observations from a tactile sensor to the +known scene. Creating scene representations which do encode information from alternative +sensors could enable applications such as SLAM or collision avoidance under low vision +conditions. +6. Conclusion +Building new scene representations for robot manipulation is an important step towards +creating fully autonomous embodied agents which can interact intelligently with the world. +13 + +Ground-Truth +Rendered novel view +RGB-Uncertainty +Depth +Depth-UncertaintyContinued advances in sensing and robotics will necessitate new representations which en- +code data from new sensors and support new robot form factors and applications. There +are two challenges which are present in all works covered in this review: building represen- +tations which can be constructed in real time, and which are robust to sensor noise. Given +the recent pace of advancement in robotics and computing, we are confident that these +challenges will be sufficiently overcome. +Application +Common +Representations +Less Common +Representations +Collision Avoidance +Signed distance fields +Convex hulls +Potential fields +Geometric primitives +Learned representations +Manipulation and Grasping +Direct action +Meshes +Point clouds +Voxel grids +Signed Distance Fields +Learned representations +Teleoperation +Point clouds +Meshes +Occupancy grids +Table 2: Scene representations covered in this review +14 + +References +[1] Z. Moore, C. Sifferman, S. Tullis, M. Ma, R. Proffitt, and M. Skubic, “Depth sensor- +based in-home daily activity recognition and assessment system for stroke rehabili- +tation,” in 2019 IEEE International Conference on Bioinformatics and Biomedicine +(BIBM), 2019, pp. 1051–1056. +[2] M. Kang, Y. Cho, and S.-E. 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Available: https://arxiv.org/abs/2212.06817 +23 + diff --git a/PNFIT4oBgHgl3EQfeiuo/content/tmp_files/load_file.txt b/PNFIT4oBgHgl3EQfeiuo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d140a38a9cbc728f489f6f23f13dbc359b87effd --- /dev/null +++ b/PNFIT4oBgHgl3EQfeiuo/content/tmp_files/load_file.txt @@ -0,0 +1,1326 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf,len=1325 +page_content='A Review of Scene Representations for Robot Manipulators Carter Sifferman sifferman@wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='edu Department of Computer Sciences University of Wisconsin - Madison 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Introduction For a robot to act intelligently, it needs to sense the world around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Increasingly, robots build an internal representation of the world from sensor readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This representation can then be used to inform downstream tasks, such as manipulation, collision avoidance, or human interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In practice, scene representations vary widely depending on the type of robot, the sensing modality, and the task that the robot is designed to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This review provides an overview of the scene representations used for robot manipulators (robot arms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' We focus primarily on representations which are built from real world sensing and are used to inform some downstream robotics task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Building an intermediate scene representation is not necessary for a robotics system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' It is completely possible for a robotics system to act directly on sensor data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' predict appropriate grasps directly from RGB images), and we will look at many such systems within this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' However, we argue that intermediate scene representations are beneficial for robot manipulators as they: act as spatial memory are efficient storage of past memories allow long-horizon planning can act as regularization and encode spatial priors for learning systems In this review, we organize scene representations into three categories depending on the task that the representation supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' These categories make up the sections of our review: collision avoidance (section 2), manipulation (section 3), and teleoperation (section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Within each section, we provide a review of the existing literature, summarize the challenges in the area, and consider directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In section 5, we look at scene representations for manipulators as a whole, and consider directions for future research which cut across our three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='1 Literature Survey Process Our literature survey consisted of two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In phase 1, we performed a broad search of existing reviews in order to gain context and understand the broader landscape of robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='11275v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='RO] 22 Dec 2022 Figure 1: This review focuses on robotics applications which build some intermediate scene representation between sensor data and a downstream robotics task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Image sources: [1, 2, 3] Category Snowball Seeds # Papers Found Collision avoidance [2, 18, 3] 26 Manipulation [19, 20, 21] 28 Teleoperation [22, 23, 14] 16 Table 1: The three sections that this review is organized into, and the “snowball seeds” which began the literature review process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' To find these reviews, we use Google Scholar search with the “Review Articles” filter en- abled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The result of this search is 13 reviews spanning a broad spectrum: manipulation and grasping [4, 5, 6, 7, 8, 9, 10], SLAM [11], human-robot interaction [12, 13], teleoperation [14], motion planning [15, 16] and inverse kinematics [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Knowledge gained from these reviews was used to determine our taxonomy of scene representations, and the scope of this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The goal of phase 2 of our survey process was to find directly relevant papers which employ a scene representation for robot manipulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' As the papers that we search span a wide range of topics, keyword searches did not prove effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Instead, we “snowballed” through references, beginning at seed papers which were found through keyword search, reviews, or consulting with colleagues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' These seeds are shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' We snowballed through references both through traditional reference chasing, and using Google Scholar’s “Cited By” page to find papers published after the seed paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' We also use the Abstract Viewer Project1, a system for finding related papers developed as a research project at UW- Madison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Abstract Viewer does not use citations to find related papers, instead relying on analysis of textual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This proved helpful for finding still-related papers when references ran dry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In total, 69 papers were collected in phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This review is not meant to act as a comprehensive survey of any one subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Instead, we hope to give a broad overview of the field and provide jumping-off points for further reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Articles were not selected to be a representative sample of the entire field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' papers which use an identical scene representation to existing work may be excluded while those with a unique scene representation are generally included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' https://pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='edu/AbstractsViewer/ 2 Sensing data Intermediate Downstream task Representation2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Collision Avoidance A central challenge in robotics is being able to avoid collisions, which can potentially be very costly with a powerful, fast, and expensive robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Some approaches for collision avoidance respond directly to measurements from robot-mounted sensors, without building any inter- mediate representation [24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A downside of this approach is that it has no memory: the robot can only act based on what it currently observes, and cannot plan based on its previous observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' As a result, these approaches are overly cautious and perform poorly in challenging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Modern approaches for collision avoidance can be broadly grouped into two categories: motion planning [27], in which the start and goal position of the robot end effector are known, and the goal is to find a viable, collision-free path between the two, and inverse kinematics, in which the end effector position is known, and the goal is to find a viable joint configuration of the robot which matches that end effector position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This distinction is unimportant for this review, as both of these approaches have the same requirements of their scene representations: fast collision checks and (sometimes) fast calculation of the distance to the nearest obstacle or direction to the nearest obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' As a result, scene representations for collision avoidance are similar whether the problem is formulated as inverse kinematics or motion planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Potential Fields– Early approaches to collision avoidance in a motion planning context represented the environment with a potential field, first proposed in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This potential field can be evaluated at any point to yield a scalar “potential” value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This value is determined by both the scene geometry (which have high potential around them) and the desired location (which has a low potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In order to move through space, the robot simply follows the negative gradient of this potential field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The potential field was heavily utilized in early motion planning work [29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' While effective for the time, potential fields suffer from a few problems: the potential function is prone to local minima, and can be very difficult and computationally intensive to construct [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Additionally, it is impractical to construct potential fields in real-time from sensor measurements, both because they are slow to construct, and because they require scene geometry to be described in a friendly closed-form, which sensors cannot provide natively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Later work [32] improved on the local minimum problem by taking into account the relative starting position as well as scene geometry during potential field construction, but nonetheless potential fields have fallen out of favor in collision avoidance applications since the early 2000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Signed Distance Fields– A signed distance field is a mapping between 3D points in space x and the scalar distance d to the nearest obstacle: SDF(x) = d The SDF has the nice property that taking −∇SDF(x) yields a vector pointing towards the nearest obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This representation has been used in computer graphics since at least 1998 [33, 34], and became popular for robot collision avoidance with the introduction of the highly influential CHOMP motion planner in 2009 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' CHOMP uses the signed distance field, along with pre-computed gradients to perform optimization over the robot configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Subsequently, the popular STOMP [35], TrajOpt [36], and ITOMP [37] motion planners also used a signed distance field to represent their environment, and used gradients in a 3 similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In each of these works, the signed distance field is computed at fixed points on a regular voxel grid as a pre-processing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' To do such a computation, a precise model of the underlying geometry is needed, typically in the form of a mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Similarly to potential fields, computing a signed distance field is computationally expensive, and generating it from noisy sensor data is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In practice, collision avoidance approaches which use an SDF are constrained to simulations, where the SDF can be pre-computed, or static environments in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Regardless, SDFs are the most popular scene representation for collision avoidance, largely because they are supported by popular and effective motion planners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' There exist libraries such as VoxBlox [38] and FIESTA [39] which efficiently compute and store discretely sampled SDFs for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Collections of Primitives– A less common method for representing geometry for col- lision avoidance is with a collection of primitive shapes, such as spheres, cylinders, and cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' These primitives are usually stored parametrically, so that collision checking can be done quickly and the objects represented natively in optimization solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Toussaint et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [40] proposed Logic-Geometric Programming, in which the scene is composed entirely of parametric cylinders, blocks, and planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This paper has been influential for its elegant optimization-centered formulation, but the scene representation used within has not been heavily utilized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' it serves more as a demonstration of the approach’s capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Similarly, Gaertner et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [41] considers collision avoidance with humanoid robots, and uses a col- lection of primitives to represent dynamic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Similarly to Toussaint, the collection of primitives is used primarily to demonstrate the capabilities of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In contrast, Zim- merman et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [42] proposes a method for using collections of primitives in gradient-based optimization methods such as TrajOpt [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They provide a unified method for dealing with many types of primitives, and a method for taking the derivative of the distance to the nearest primitive, making a collection of primitives a drop-in replacement for SDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' However, this approach has not seen widespread adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Collections of Convex Hulls– A collection of convex hulls is another less common way to represent scene geometry for collision avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Similarly to primitive shapes, convex hulls allow for fast collision checking and natural representation in optimization solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Convex hulls have the additional benefit that any shape can be broken down to a collection of convex hulls via an algorithm like QuickHull [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Schulman et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [44] uses a set of convex hulls to represent a scene, and outperforms the SDF-based motion planners of the time like CHOMP [3] and STOMP [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' CollisionIK [18] introduces an optimization-based method for inverse kinematics, which is able to operate in real-time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' for mimicry control) and avoid collisions with dynamic obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' CollisionIK mentions that point cloud objects could be broken down into convex hulls in real time, but does not demonstrate such a process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' To our knowledge, no existing approach constructs convex hulls in real time from sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Learned Representations– Within the last year, one approach has emerged which uses a learned environment representation to enable real-time collision avoidance with dynamic obstacles and real-world sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This paper is somewhat influenced by the growing lit- erature around learned representations in computer vision [45], graphics [46], and SLAM [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' RCIK [2] proposes a collision cost prediction network, a neural network which takes as input features extracted from an occupancy grid, as well as a 3D point in that grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' from this input the network predicts the collision cost, which is an approximation of the 4 Figure 2: The collision cost prediction network of RCIK [2] is trained on simulated data, and outputs a collsion cost Fcol which approximates the signed distance function to enable real-world real-time collision avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' SDF evaluated at that 3D point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The occupancy grid can be generated in real time with one or more depth cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The network is trained on one million simulated examples of random environments and joint configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' While this method does not have the col- lision avoidance guarantees provided in theory by other methods, it is the first method to perform collision-free inverse kinematics in real time with real sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This same approach was later evaluated under real-time control [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='1 Future Directions in Collision Avoidance Real-time generation of signed distance fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Signed distance fields have proven highly effective for enabling collision avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' However, SDFs are very costly to generate, and require a very accurate representation of the underlying environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Because of this, the vast majority of work on collision avoidance with robot manipulators does so only in simulation, or in manually recreated static environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A pressing challenge is finding ways to bring these methods to the real world by constructing an SDF in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Recent work on neural representations has enabled real-time construction of a neural SDF from depth imagery called iSDF [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Similarly to RCIK [2], the SDF produced by iSDF is only an approximation, however it is generated over time from multiple sensor observations, and does not rely on simulated pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Adapting a similar approach for static or manipulator-mounted depth cameras could enable real-world operation of the many collision avoidance approaches which rely on SDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Moving beyond signed distance fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Signed distance fields alone are great for collision avoidance, but offer some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' For example, not all collisions are equally costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Colliding with a pillow might be admissible if it means avoiding collision with a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Future representations, and algorithms which act on them, could store semantic information along with scene geometry, to enable such decisions to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Learning such semantic scene properties may be possible via extensive pre-training, or via interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 5 Joint configuration Joint configurations feature extractor col Regression module obs Obstacle Occupancy grid feature extractor3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Manipulation Arguably the most important task for a robot manipulator is to manipulate things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Manipu- lation can mean grasping with a simple one degree-of-freedom gripper, articulated grasping, or simple pushing and nudging of objects with any part of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Robot manipulation is a vast field, with approaches specialized for dealing with many specific challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' To keep the scope of this review reasonable, we focus on scene representations which are used for: Basic grasping with a 1DoF gripper Generalizable grasping Articulated grasping Predicting scene flow Direct Action on Images– While this review focuses on intermediate scene representa- tions, it is clear that, for the purposes of robot manipulation, an intermediate representation is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A seminal work in this area was Saxena et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [21] in 2006, which was the first work to directly predict grasping points from an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They use a neural network to, given an RGB image of an object to be grasped, predict pixels in the image at which the object is most suitable for grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' To train their neural network, they use supervised learning on a large fully synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Two years later, the same authors improved on the approach by using RGB-D imagery as input to the network [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Lenz et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [52] improved upon previous work by aiming to predict the single best grasp, rather than listing many viable ones, and predicting the orientation and extent of the grasp along with the location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Later, the “DexNet” series of papers offered iterative improvements by improving training data and tweaking the neural network outputs, as well as considering alternative grip types such as articulated hands and suction cups [53, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Other work aims to grasp objects given some semantic label, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' “coffee mug” or “plate”, this is sometimes called semantic grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Jang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [56] proposes a two-stream approach to semantic grasping, in which one stream identifies objects while another finds suitable grasps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Schwarz et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [57] demonstrates a semantic grasping pipeline which uses a suction cup gripper and works by simply segmenting out objects and finding their center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This approach per- formed well at the highly competitive Amazon Picking Challenge2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' These direct prediction approaches are highly effective for real world operation, however they are fairly limited in their potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' These approaches can only perform single-shot grasping, meaning they are unable to, for example, rotate an object, let go of it, and grab it again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They also have a limited ability to reason about novel shapes in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Lastly, these approaches typically rely on some synthetic training data, which must be generated via other 3D-aware methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Meshes– A number of works use a 3D triangular mesh to represent objects for manipu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The problem of finding suitable grasps given a 3D mesh is a long-standing problem with active research [58, 59, 60, 61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This paragraph focuses not on those algorithms, but on real-world systems for manipulation which represent objects as meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The first of such real-world systems was Berenson et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [63] in 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This work makes grasping 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' https://robohub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='org/amazon-picking-challenge/ 6 Figure 3: An early approach for real world grasping, Berenson et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [63], relied on a motion capture system and pre-defined meshes to perform grasping in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' possible in the real world by incorporating information about not only the object to be grasped, but also nearby obstacles, such as the table or other objects, into the grasp selec- tion algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In order to sense the positions of objects in real-world tests, this approach relies on motion capture markers being placed on each object, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Goldfeder et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [64] introduced a method for finding good grasps given a mesh, and tested their method by scanning real objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This approach was somewhat effective, but the conversion from scan to mesh does not happen in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Collet et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [65] circumvents the prob- lem of real-time mesh construction by modeling each object as a primitive, and fitting that primitive to a point cloud in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Later work by the same authors [66] extends this idea to arbitrary meshes by using a 6D pose recognition algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Assuming that a mesh of the object is known, this approach enables prediction of the mesh’s pose in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Papazov et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [67] takes a similar approach, but assumes that the object is represented with a set of points and surface normals, rather than an RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Varley et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [68] removes the requirement of a pre-made mesh by teaching a neural network to complete a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' From the completed point cloud, a mesh can be built and that mesh passed off to a mesh-based grasp planner in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The steps in Varley et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.’s pipeline are shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In each of these approaches, we see that the limiting factor is not the grasp planning algorithms themselves, but the generation of meshes in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Point Clouds– Point clouds have been used as the basis for grasp planning in a similar manner to meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In constrast to meshes, point clouds are closer to the native output of commonly used depth cameras, making them more practical for real world construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Approaches exist for grasp planning directly on point clouds [69, 70, 71, 72], although less numerous than those for meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Florence et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [73] (covered in more depth in the following paragraph) introduces a method for finding corresponding grasps between similar objects, and uses point clouds to perform the grasping in real-world examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Simeonov et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [74] considers the problem of manipulation planning, and is able to predict the movement of scene objects directly from their point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This prediction is used to plan for manipulations which move the scene towards a goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 7 2Figure 4: Varley et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [68] uses a neural network to enable shape completion on partial point cloud observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The completed shapes can then be transformed to a mesh and used in any mesh-based grasp planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Image from Varley et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Voxel Grids– Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [75] constructs a gripper-sized voxel grid from a point cloud and a given gripper position, with each voxel encoding whether the space is occupied by a point in the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This voxel grid can then be compared (via nearest neighbors) to previously simulated voxel grids to determine whether it corresponds to a good gripper position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Learned Representations– Dense object nets [73] are an approach for generalizable grasp- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' For a given input image, a neural network is trained to produce a dense pixel-level feature map which produces similar output feature vectors for semantically similar points in multiple images of similar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' For example, for any two images of a mug, the goal is that pixels corresponding to the same point on the mug handle will have the same feature vector in the output representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This representation is trained to be consistent across object instance, pose, and deformation, enabling generalizable grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They demonstrate that this approach is effective at generalizable grasping in the real world, using merged point cloud data from multiple depth cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Simeonov et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [19] expands upon this approach by taking a 3D point as input to the neural network, rather than a 2D pixel position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The object properties are encoded within the weights of a multi-layer perceptron, similarly to NeRF [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They demonstrate that this approach can be used to perform pick-and-place tasks of novel objects of a given class given fewer than 10 demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Additionally, the architecture of their MLP ensures that the descriptor fields are SE(3)-equivariant, making them robust to arbitrary object poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Van Der Merwe et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [76] uses a learned represen- tation for articulated grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A neural network is trained to, given a point cloud and a 3D query point, approximate the signed distance function at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The latent space of this network is concatenated with information about desired grasp qualities and robot 8 15 Kinect 430 98499824 26 (a) Image of Occluded Side (b) Point Cloud (c) Segmented and Meshed (d) CNN Input 403530252015105 15 25 30 10 15 20 (e) CNN Output (f) Fast Mesh (g) Detailed Mesh (h) Grasp Planningconfiguration, and fed into another network which predicts the success rate of the proposed grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Xu et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [20] builds a visual predictive model for robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Their goal is to, given some representation of the scene along with a robot action, predict the scene after the action has occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Rather than using a manually constructed scene representation, they allow the scene representation to be learned in an end-to-end manner, as a 128x128x48 feature grid with an 8-dimensional vector at each point in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Using this representation, they are able to predict 3D scene flow and plan manipulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='1 Future Directions in Manipulation Performing 3D grasp planning based on real-world sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' There is a dis- connect between the way scenes are represented for grasp planning (primarily meshes) and the way that the most effective robotics systems, such as those used in the Amazon Pick- ing Challenge, are operated (primarily direct prediction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' An important direction for future work is bridging this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Direct prediction methods are severely limited in their spatial rea- soning, while mesh-based methods are severely limited in their real-world operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Point cloud manipulation and learned representations may bridge the gap, but do not currently offer the high grasping accuracy of mesh based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Representing complex object properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Any object may have many properties which are relevant to manipulation: its center of mass, deformation properties, or con- straints on its motion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' hold a mug full of coffee upright, pull a drawer straight out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Learning-based approaches, such as Dense PhysNet [77] have shown promise towards being able to learn these properties autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A needed direction for future research is de- termining ways to learn these properties, generalize them to novel objects, and store these properties alongside their geometric representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Teleoperation In a teleoperation scenario, a human controls a robot remotely, and relies on an intermediate interface, such as a monitor or VR headset, to understand the robot’s surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Much recent work on teleoperation places the human operator in virtual reality (VR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' studies have shown increased task performance in virtual reality, due to the ease of controlling the user’s view and all six degrees-of-freedom of the robot [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' However, the scene representations used in these VR approaches can generally applied to any sort of teleoperation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='1 Human Control of Mobile Robots Mobile robots are much more likely than robot manipulators to venture into unknown environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Because of this, the foundational work in representing scenes to remote operators has taken place in mobile robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' For context, we offer a brief overview of such work here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Nielsen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [79] was an early attempt at incorporating data beyond RGB camera feeds into robot teleoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They create a dashboard which shows an RGB camera feed along with LiDaR scans and a rough map of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' While all of the information is presented to the user, it is not combined to create an intuitive display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Kelly et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [80] builds a 3D map of the environment by using LiDaR scans to create a 3D map, and coloring that map with data from RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A CAD model of the mobile robot is 9 (a) Dashboard-based interface from [82] (2004) (b) Augmented interface from [79] (2007) (c) 3D interface from [80] (2011) Figure 5: A history of the typical information displays used in mobile robot teleoperation placed in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This allows rendering arbitrary viewpoints, such as an overview of the entire scene, an overhead view, or a third person view, as would be seen in a racing video game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Stotko et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [81] builds a 3D mesh-based scene representation for a mobile manipulator, and displays the mesh to the user interactively in virtual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They find that users have fewer collisions, and report a greater level of immersion and awareness than with a 2D interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Livantino et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [23] augments robot-attached camera views with 3D data to overlay data such as desired path, destination, and label traversable terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' For mobile robot teleoperation, the trend has been towards a free floating, user controllable view and natural display of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' These same goals apply to robot manipulator teleoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='2 Human Control of Robot Manipulators No intermediate representation– A simple baseline for teleoperation is the use of one or more static cameras, which the user can either see all at once, or switch between manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' While simple to implement, a static camera approach is limiting: they present the user with limited geometric information, and don’t leverage computation to enhance human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Static cameras are also prone to being blocked by the robot manipulator itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' There are a number of approaches which improve on the static camera baseline without building an intermediate representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Murata et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [83] considers mobile manipulators (manipulators mounted on mobile robots), and renders a CAD model of the robot on top of a background made of stitched, observed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The CAD model is kept updated to represent the robot’s current state, and the user is able to position the camera to generate an arbitrary view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A different approach is taken by Rakita et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [84], in which a robot arm is used as a camera operator for another robot arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The camera operator’s pose is automatically optimized in real time to present a clear view of the manipulating robot’s end effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Point Clouds– Point clouds are a common choice for representing scenes for real time operation, because they are the native output of depth cameras, and can be displayed in real time with little processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Brizzi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [85] considers augmented reality for VR teleoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Operators see an RGB image of the scene, along with features, such as distance to the target, direction to the target, or gripper state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Point clouds from a depth 10 QuitProgram Select Camera Home Camere Zoom In ZoomOut X Feedbock Resistonce Limit H Velocity Limit Auto ON Robot: Sensorstatus Health Sensor Status Heoding Roll Anitude Sonar OK ojps 02LF Laser OK Pitch Camera 05DN OK Pressfor Sensor Tele Velocity Inertials OK Power Status 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='5Volts Inclinometer OK Update Turn n/ 2025 Compass Escape OK 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='0 m/s Thermometer OK Bump OK Motion Pursuit CommunicationsHealth OK Enobled FLIR OK StopVideo Map Map Robota 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='92kph 655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='1 m UnknownFigure 6: Kohn et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [86] uses meshes to represent known objects (table and robot) and point clouds to represent unknown objects (puzzle box) for teleoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Image from Kohn et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' camera are used to calculate these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Kohn et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [86] displays a combination of point clouds from RGBD cameras and meshes to a user in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Meshes are used for known objects (such as the floor, table, and robot) and point clouds are used for the unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The unknown objects are filtered in real time according to the meshes, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A similar approach is taken by Su et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Wei et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [22] similarly displays point clouds to users, but unlike previous approaches, they do not use meshes to represent known objects, aside from the robot gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They perform a user study, comparing the point cloud to multi-view RGB and to a point cloud projected onto an RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In their experiments, the hybrid point cloud and RGB view performs best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Meshes– While point clouds are native and fast to display, meshes may be preferred due to their potentially higher visual fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Wonsick et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [88] takes an approach similar to the mesh-based approaches in section 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' a point cloud of the scene is semantically segmented to identify its class amongst known objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A known 3D model is then fit to each point cloud segment, and that 3D model is displayed to the user in virtual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Models are able to be constructed in much higher fidelity, and the scene can be entirely rendered, enabling control of lighting, texture, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They run a user study and say that users find their approach more usable and less cognitive load than point-cloud streaming alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' However their approach relies on having known 3D models of objects in the environment, and has no fallback if such a model does not exist, as such it is not suitable to unknown environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Piyavichayanon et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [89] generates a “depth mesh” by combining the view of multiple depth cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This mesh is used to display augmented reality features, such as distance to a collision state, on a handheld smartphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Merwe et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [90] performs a user study to investigate how the type of information presented to the operator changes their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They compare a full information 3D model to a “representative” mesh based model, which only displays crucial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They find that task completion time is lower with the full model, but cognitive load is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Occupancy Grids– Omarali et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [91] considers multiple modes of visualization for VR teleoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Alongside the depth camera-based baselines, they present a hybrid view, which displays the current output from the depth camera, alongside a translucent occupancy map in the previously observed areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This occupancy map gives the user some context for the greater environment, while indicating that the context is not as reliable as the currently 11 observed region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' They find that users prefer the hybrid depth camera + occupancy map approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='3 Future Directions in Teleoperation Improving real-time visualization of the robot’s environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Existing approaches tend to rely on depth cameras, which can render the world in real time, but are low in detail and high in noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Approaches which attempt to process this data into something more appealing, like a mesh, require heavy computation and require meshes of potential objects to be known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Future work should consider ways to improve the fidelity and detail of reconstruction in real time and in unknown environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' To some extent, advances in imaging, such as high resolution depth sensors may improve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Another potential solution is dynamically directing sensing towards areas that need high detail, such as around the end effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The problem of representing the environment from a novel perspective is known as novel view synthesis and is well studied in the computer vision literature (see the recent review [92]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Incorporating techniques from novel view synthesis could allow a more complete environment representation to be displayed to the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Building representations suited to the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Currently, the vast majority of approaches focus on building representations which represent the environment faithfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' However, it has long been known that realistic displays are not always ideal, as they can make it difficult to parse what is relevant [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Some work covered in this review has considered building a sparse representation of only relevant features [90, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' However, these approaches require such a representation to be specified by hand prior to operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Future work should consider ways of filtering the scene representation to represent only what is important to the user in an unknown scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Future Directions Quantifying Uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Present systems for robot manipulation build a most likely map of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In poor sensing conditions, this map may be highly unreliable, leading the downstream robotics application to make incorrect but fully confident choices based on the unreliable map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Future work should work to incorporate uncertainty into the scene representation itself, in order to enable robot policies which act intelligently in the face of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' When in a highly uncertain environment such a robot could, for example, ask a human for help, or gather additional information about its environment before taking action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' The issue of uncertainty estimation is well studied in mobile robotics, especially in the context of SLAM [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Recent approaches in novel view synthesis have also enabled dense estimates for geometric and photometric uncertainty [95, 96], as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Such approaches could be applied to robot manipulators, especially for problems like collision avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Joint prediction of scene semantics and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Present approaches which infer scene semantics do so by taking some geometric representation of the scene as input, and producing a semantic map based on that geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' With this approach, scene geometry informs scene semantics, but semantics has no influence on geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' In reality, semantic information about an object can provide queues about its likely geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' If an object is identified as a coffee mug, it probably has a handle on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' If an object is identified as a 12 Figure 7: Shen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [95] builds a scene representation which includes dense uncertainty estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' A similar approach could prove useful to build scene representations for robot manipulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Image from Shen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' baseball, it is probably spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Floors and tabletops tend to be flat and level with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Future approaches could infer scene geometry and semantics jointly, by encoding sensor data in a learned latent representation, which is used to inform both geometric and semantic inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Joint inference approaches have been used for semantic SLAM [97, 98, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Implementing such an approach for robot manipulators would enable higher fidelity predictive mapping and improved semantic understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Sensing-first development and real-world benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Many existing ap- proaches, particularly for grasping and collision avoidance, are benchmarked entirely in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Real-world demonstrations are addressed after the development has been op- timized for simulation, and often only occur under highly controlled conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Future work should consider developing methods with sensing in mind from the beginning, and aiming for high performance on real-world tasks with sensing, rather than in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' This means building systems which are robust to sensor noise, and able to act on raw sensor data in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' One encouraging sign towards such a goal is the just announced RT-1 project from Google [100], which is an embodied agent that is benchmarked entirely on real-world tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Representations for Alternative Sensing Modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' All works addressed in this survey utilize data from typical vision based sensors: RGB cameras, depth cameras, and/or LiDaR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' There are many other sensor types used in robotics, such as force torque sensors and tactile sensors, which provide information about the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' However, existing approaches react immediately to information from these sensors, and do not build an inter- mediate representation which persists over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Can we build scene representations which model the factors that these sensors measure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' For example, the measurement from a tactile sensor may be influenced by how hard the object is, how heavy it is, or even how conductive it is, alongside the object’s geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Present scene representations do not encode all of this information, making it very difficult to relate new observations from a tactile sensor to the known scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Creating scene representations which do encode information from alternative sensors could enable applications such as SLAM or collision avoidance under low vision conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Conclusion Building new scene representations for robot manipulation is an important step towards creating fully autonomous embodied agents which can interact intelligently with the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' 13 Ground-Truth Rendered novel view RGB-Uncertainty Depth Depth-UncertaintyContinued advances in sensing and robotics will necessitate new representations which en- code data from new sensors and support new robot form factors and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' There are two challenges which are present in all works covered in this review: building represen- tations which can be constructed in real time, and which are robust to sensor noise.' 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“Rt-1: Robotics transformer for real-world control at scale,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='org/abs/2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} +page_content='06817 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFIT4oBgHgl3EQfeiuo/content/2301.11275v1.pdf'} diff --git a/Q9AyT4oBgHgl3EQfU_ew/content/2301.00136v1.pdf b/Q9AyT4oBgHgl3EQfU_ew/content/2301.00136v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f4774c63e0ce760b594f529cf2af2df31a9b9936 --- 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discrete symmetry can preserve black +hole information∗ +or +Turning a Black Hole Inside Out +Gerard ’t Hooft +Faculty of Science, Department of Physics +Institute for Theoretical Physics +Princetonplein 5, 3584 CC Utrecht +The Netherlands http://www.staff.science.uu.nl/˜hooft101 +Abstract +To apply the laws of General Relativity to quantum black holes, one first +needs to remove the horizon singularity by means of Kruskal-Szekeres coor- +dinates. This however doubles spacetime, which thereby is equipped with +an exact binary symmetry. All particles near a black hole share the same +symmetry, and conservation of this symmetry may completely remove the in- +formation paradox: the quantum black hole has no interior, or equivalently, +the black hole interior is a quantum clone of the exterior region. These obser- +vations, totally overlooked in most of the literature on quantum black holes, +resolve some issues concerning conservation of information. Some other prob- +lems do remain . +∗Presented at DICE2022, Castiglioncello, Italy, May 19-22, 2022. +1 +arXiv:2301.08708v1 [gr-qc] 20 Jan 2023 + +1 +Introduction +A silent revolution took place in the author’s understanding of the quantum mechanical +equations for black holes, as first mentioned in Ref. [1]. +In summary, what happens +is that the space-time singularity associated to the horizon of a black hole, can easily +be removed by an appropriate coordinate transformation, showing that a black hole +horizon is about as regular as the apparent singularity of planet Earth at its North and +South poles: it is a coordinate singularity. Removing this singularity should answer +all questions about the nature of space and time there, but it leads to a new question +instead. Apparently, beyond the coordinate singularity, space and time are extended into +a new copy of the existing space-time. Our new insight, about which we also reported +in Ref. [2], is that this is not a new space-time region at all, but merely an exact mirror +image of existing space-time. The black hole carries along a clone, not only of itself, but +also of the entire surrounding universe. +An observer1, moving from ordinary space-time to the cloned region will notice no +differences at all, but in fact space-time in this region is quite remarkable, because the +clone is connected to the original via time reversal. Clearly, this is not an ordinary +symmetry, but it does allow us to handle it as such. We can limit ourselves to states +in Hilbert space that are either symmetric or anti-symmetric2 under the switch between +the black hole and its clone. As we shall show briefly, this reduction has a big effect on +the expected properties of the black hole. Its entropy is now only half of what is usually +found, and consequently, the temperature of the emitted radiation doubles. +But we shall come to that; let us first return to some basic physical issues concerning +quantum gravity. Usually, our starting point is chosen to be Newton’s force law, +F = G m1 m2 +r2 +, +(1.1) +where one imposes the condition that this law needs to be made compatible with the +highly successful laws of special relativity. As is well-known, one then ends up with +General Relativity. It is found that space and time are curved, and the curvature is +everywhere in the vicinity of the source. Making the next step, needed to impose the +laws of quantum mechanics, is now very hard, and usually comes with new assumptions +and unprovable principles, such as string theory and holography. +There is nothing +wrong with such approaches, except that the results are generally uncertain. One has +to understand Nature’s (or God’s3) way; you have to be religious to understand that, +1The concept of an observer can perhaps better be replaced by the concept of observable variables, +so that some problems such as finite-size effects and finiteness of memory space, are more easy to avoid, +but clearly, utmost diligence would be asked for. +2symmetric for the real parts of wave functions, and antisymmetric for the imaginary parts, as will +be explained, see section 4. +3I should henceforth avoid using God’s name, as this apparently suggests religious feelings that I do +not have. +2 + +and if you are not religious, you just miss the boat. +An approach that requires (almost4) no religion is the path offered by black holes. If +we remove the coordinate singularities by a coordinate transformation, we only encounter +small amounts of local curvature. +However, the effects of gravity still diverge in a +remarkable way, which is the effects of the gravitational force acting between in- and +out-going matter. This diverges badly, in a process that explodes exponentially in time, +but in a way that triggered our interest: this divergence only occurs on the horizon. +Elsewhere in space-time, one may assume that standard knowledge of general relativity +and quantum mechanics offers sufficient clues as to what happens there, so, unintelligible +curvature does not occur everywhere, and we may attempt to couple understandable +features of regions of space and time, while reserving our attention to the tiny subset of +space-time points right at the horizon. This is where a black is connected to its clone at +the other side. +Instead of Newton’s law, Eq. (1.1) , we focus on the Shapiro effect. This effect brings +about gravitational lensing, as is well known in astronomy: +δu− = −8πG p− log |˜x1 − ˜x2| , +(1.2) +where p− is the momentum of the source, which we take as a vector in the light cone +minus-direction, while u− is the displacement it generates in a reference mass, also in the +minus-direction. ˜x1 and ˜x2 are the transverse components of the locations of source and +effect. The most noticeable distinction with Newton’s law is the dependence on these +coordinates. It goes as 1/r2 in the Newtonian case, while it is only logarithmic in the +case of Shapiro. The logarithm reflects the fact that the associated phenomenon takes +place in a two-dimensional subspace of space-time. Laplace’s field equation generates +logarithms in two dimensions. +The Shapiro equation (1.2) offers an easy way to handle the gravitational interactions +between in- and out-particles. One finds that, in the reference frame of the out-particles, +the momentum p− of the in-particles increases exponentially as a function of the time +separation between in and out. Remarkably, here momenta of the in-particles affect the +positions of the out-particles. By applying Fourier transformations, one derives directly +that (apart from a sign) the same relation holds if for both particles momenta and +positions are interchanged. This is time reflection symmetry, see Section 3. Readers +familiar with these issues might want to proceed directly to Section ??, where some +subtle novelties enter. +4A critical reader may observe that we do make use of a belief: the fact that black holes should in +no way differ from ordinary particles in the way they obey quantum mechanical laws. Yes, I admit that +this is a belief. +3 + +2 +Perturbations around the stationary black hole +Many researchers now continue with the black hole as it is being formed by some implo- +sion event.[3] It then seems as if the details of this implosion are intimately responsible +for what happens next. We claim however that this only holds for time spans shorter +than the scrambling time. This is the time it takes for a particle to reach the horizon +within a few Planck lengths, in natural units: +TS = O +� +MBH log(MBH/MPlanck) +� +. +(2.1) +Classically we have the no-hair theorem stating that the details of the implosion event +are immaterial. In practice one expects that only minute quantum details will continue +to depend on details of the original implosion, but this situation would be similar to what +happens to an atom after its formation at the beginning of the universe. Surely such +details will determine which of its quantum states it will be in right now, but as physicists +we should be primarily interested in a proper categorisation of all possible quantum +states, and the Schr¨odinger equation dictating how these evolve one into another. We +here take the same attitude regarding black holes. Its history before scrambling time, +or its evaporation process after scrambling time, should not be relevant for its physical +equations at present. +Soon after its formation, the black hole is almost stable, absorbing and emitting +particles every now and then, according to a Schr¨odinger equation and a Hamiltonian +connecting all microstates. It is our job to use basic principles for deriving the proper +classification of states and the fundamental Hamiltonian. At some decent distance from +the horizon, the categorisation is clear and the Hamiltonian is known, it is called the +Standard Model of the subatomic particles, which can be put in a gently curved space- +time. At all space-time points where the temperature dropped to below a few TeV, we +know how it generates the answers to our questions. +We now assume that the complete set of states is determined by the spectrum of +particles excited thermally at all distances beyond a few Planck lengths from the horizon, +where the exact numbers are still to be derived. Since the particles are thermal and the +local temperature is expected to have dropped well below the Planck temperature, the +effects of these particles on the black hole space-time metric must have dropped to values +small enough to neglect them for the time being. We plan to postpone these details to +being justified a posteriori (See Section 6). Right now we can observe that the sets +of states and operators used at time scales that may vary over more than one unit of +the scrambling time will definitely not commute with one another. The Schwarzschild +4 + +time ↑ +singularity +in +out +−2GM +0 +2GM +r +r = 2GM +IV +III +II +I +x +y +a) +b) +Figure 1: a) The Schwarzschild metric, in Schwarzschild coordinates. Vertical dashed +lines: the event horizon. Curved lines: light like radial geodesics: y = const (smooth), +and x = const (dashed). +Cones: the orientation of the local light cones. +Angular +coordinates (θ, ϕ) are not shown. +b) The same space-time in Tortoise coordinates +(x, y), showing regions I − IV and the orientations of the local light cones. Curves +show r = const. lines. +metric, in Schwarzschild’s coordinates, demonstrates clearly the time dilation invariance: +ds2 += +1 +1 − 2GM +r +dr2 − +� +1 − 2GM +r +� +dt2 + r2dΩ2 , +(2.2) +where +Ω +≡ +(θ, ϕ) , +dΩ +≡ +(dθ, sin θ dϕ) . +(2.3) +It is sketched in Figure 1a. The regular coordinate frame is the Kruskal [4] Szekeres [5] +coordinates, to be referred to as tortoise coordinates x and y, see Figure 1b. One defines +x y += +� +r +2GM − 1 +� +er/2GM ; +(2.4) +y/x += +et/2GM , +(2.5) +in terms of which, the spacetime described by Eq. (2.2) is regular at the horizon: +ds2 = 32(GM)3 +r +e−r/2GMdx dy + r2dΩ2 . +(2.6) +All of space-time as we know it, is covered by the set {x ≥ 0 , y ≥ 0} (region +I ). +As we mentioned in the Introduction, this new coordinate frame allows for an +5 + +analytic continuation towards another black hole space-time, defined by the new region +II : +{x ≤ 0 , y ≤ 0}. In contrast, the regions III where x ≤ 0 and y ≥ 0, and IV +where x ≥ 0, y ≤ 0, are situated beyond the infinite future or infinite past, and these +are physically less bothering. +Both coordinate frames depicted in Figure 1 are of importance, and we have to +understand better how they are related. The arrow of time is oriented upward in all +outside regions of the Schwarzschild frame Fig. 1a. In its inside regions it is pointing +inwards, and this means that a local observer will be grilled at the singularity soon after +having passed the horizon. But we are more interested in the way regions I and II are +linked together, in Fig. 1b. +The arrow of time can be read off from Eq. (2.5). It implies that, as a local observer +in the horizon region sees time always moving in the upwards direction inside its local +light cone, an outside observer using the growing time coordinate t, will be moving +downwards in region II . Wave functions in I will rotate as e−iEt, but in region II as +e+iEt. However, as soon as the Standard Model applies, such negative energy modes +are forbidden. +Either one can say that the Dirac ket states seen by the local observer relate to the bra +states of the global observer, or one can say that, in the static black hole approximation, +the states seen by the local observer in region II describe a universe that is almost +full rather than almost empty. Considering the fact that, as derived by Hawking [14], +quantum effects generate particles at the crossing points of the two horizons, which are +seen as particles in both regions, the idea that we have a space full of particles in region +II leads to a satisfactory image of a quantum mechanical leak at that point, producing +a steady shower of Hawking particles. +At first sight one might object that there is no such leak; a time translation for the +outside observer leads to a Lorentz transformation for the local observer, and it might +seem that no particles should leak out. Now of course quantum effects may generate +tunnelling, and there is a more calculable genuine effect that generates such tunnelling: +the Shapiro effect mentioned in the Introduction. +3 +The Shapiro effect +How to employ the Shapiro effect [6] in understanding the dynamics of the Hawking +effect, was elaborately explained in our previous publications [1, 7]. Eq. (1.2) is derived +by first considering the Schwarzschild metric of a particle at rest, and then subjecting +that to a Lorentz transformation with a large γ factor. The particle momentum is the +limit p− → γm for large γ, and small rest mass m. The result [8, 9] still describes flat +space-time, but General Relativity tells us that there is curvature, delta-peaked at the +location of the shock wave associated to the fast source particle. Next, one generalises +6 + +the equation for the case that the source mass goes inwards through an S2 surface, +the horizon, rather than a flat shock wave. The result [10] is almost the same, but the +transverse separation ˜x − ˜x′ is now described by an angular separation, the logarithm +turns into a Laplace function on the sphere. We see that Eq. (1.2) turns into +δu− = Gp−f(Ω, Ω′) +with +(1 − ∆Ω)f(Ω, Ω′) = 8πGδ2(Ω, Ω′) . +(3.1) +Here, Ω is the solid angle at which the out-particle leaves, and Ω′ is where the in-particle +entered the black hole horizon. +We add normalisation constants that turn the tortoise coordinate x into the lightcone +coordinate u− and y into u+. For large black holes, the metric in the u± coordinates +is almost flat. Henceforth, we ignore the slight curvature that remains. +With u± we indicate the positions at the future or past horizon, i.e., the time at +which the particles enter or leave. p± are their momenta. These equations are linear +in the position and momentum variables, and consequently, it is easy to generalise this +result to handle the case of many particles going in or out. These lead to momentum +and position distributions on the horizons: +p± → p±(Ω) , +u± → u±(Ω) . +(3.2) +And this allows us to use spherical wave expansions5 +u±(Ω) = +� +ℓ,m +u± +ℓ mYℓ m(Ω) , +p±(Ω) = +� +ℓ,m +p± +ℓ mYℓ m(Ω) . +(3.3) +Since, for each spherical harmonic, ∆ = −ℓ(ℓ + 1), we can write Eq. (3.1) as +δu− +ℓm = +8πG +ℓ2 + ℓ + 1p− +ℓm . +(3.4) +Now we can use the commutator equations for all in- and out-going particles i, j, +[u± +i , p∓ +j ] = iδi j , +(3.5) +to arrive at the algebra +[u+ +ℓ m, u− +ℓ′ m′] = iλℓ δmm′ δℓℓ′ ; +λℓ = +8πG +ℓ2 + ℓ + 1 . +(3.6) +A few remarks: +We dropped the symbol δ from δu±, which means that it is assumed that all displace- +ments δu added up give the positions u themselves, provided coordinates are chosen +appropriately. The same holds for the momenta. +5These distribution functions are to be defined accurately: the momenta emerge as sums of δ distri- +butions, while positions are represented as centre of mass positions. See for example Ref. [2], sections 3 +and 4. +7 + +Secondly, all different (ℓ, m) modes decouple, so we obtain new, one-dimensional +quantum systems that generate the total Hilbert space in a very simple manner. These +equations are highly trivial. +Furthermore, the Fourier transformation from momenta to positions and back is a +unitary transformation. Hence, the algebra (3.6) should act as a perfect, information +preserving, mapping from in- to out-particles (and back). +4 +Restoring unitarity +But then there is a more serious problem that has to be addressed. The Fourier trans- +forms arrived at in the previous section, are only unitary if the data stretch over the +entire real line both for the u+ and the u− variables. But the physical data in the +observable part of the universe are exclusively on the half-lines u+ > 0 and u− > 0. If +we drop the data on the negative half-lines, the Fourier transformation is not unitary at +all. +The cure to this problem may well come from the observation proposed in the Intro- +duction section: just choose region II to be a quantum clone of region I . In that case, +also the Fourier variables in region II are quantum clones of those in region I . Is this +an elegant solution to our problem? +Unfortunately no. If all wave functions were real, then indeed the equation ψ(u) = +ψ(−u) copies correctly in Fourier space as well: ˆψ(p) = ˆψ(−p). But wave functions are +not in general real. Can we impose reality as an extra condition? +In ordinary quantum mechanics, the condition that wave functions are real is not +impossible. Taking the Hamiltonian to be an imaginary, antisymmetric matrix would be +allowed in quantum mechanics, provided that we accept the corresponding symmetry in +the energy spectrum: every state with energy E in the energy spectrum, is associated +to a state with the opposite sign of the energy. Reality of the wave function implies a +symmetry +E ↔ −E +(4.1) +in the energy spectrum. Remember that, right at the beginning of our treatment, it +was imposed that for ordinary particles the effect of the particles to the total mass of +the black hole may be ignored, so indeed, we may assume the condition that the entire +energy spectrum of particles is shifted to obey the symmetry (4.1). +This line of arguments needs to be carefully checked to ascertain that it be realistic. +In previous work, it was proposed to combine the transformation u ↔ −u with the +antipodal mapping: +Ω = (θ, ϕ) ↔ −Ω ≡ (π − θ, ϕ + π) , +(4.2) +8 + +but this was found to lead to other difficulties: the antipodal mapping is the mapping +that replaces all spherical harmonic functions Yℓ m by (−1)ℓ Yℓ m. But since we already +had that the mapping I ↔ II corresponds to u± ↔ −u± , this would imply +u± +ℓ,m(Ω) = (−1)ℓ+1u± +ℓ,m(Ω) , +(4.3) +so that u± = 0 if ℓ is even, which cannot be squared with the commutation equation +saying that all u variables obey [u+, u−] = iλℓ, see Eq. (3.6). Therefore the transition +to antipodes is herewith dismissed. +An other possibility was proposed im 1984 by the author [11]: should we regard all +states in region II as the bra states ⟨ψ| associated to the ket states |ψ⟩ in region I ? The +elegance of this proposal is that, together, these structures form the density matrices of +the system in region I . Is it possible to accept the mathematics of the states in region +I and II taken together, as representing the quantum density states? Actually, this +proposal almost coincides with our attempt to impose the symmetry (4.1). It is not as +yet ruled out, and it seems to be the only possibility left. Note that we are forced to +supplant our complex wave functions by real ones. It is as if the use of complex global +U(1) symmetries is forbidden, which might enforce violation of exact, additive, infinite +conservation laws. These are replaced by a discrete Z2 symmetry that switches the signs +of the tortoise coordinates x and y. +And it does lead to a more general, really important observation: this proposal +replaces Hawking’s calculation for the temperature of a black hole by another one! +If we use density matrices to compute probabilities, we find that probabilities are +proportional to the matrix elements of the density matrix, whereas Hawking regards +these as quantum vectorstates, and consequently, he averages over the squares of the +states describing region II as these can’t be observed. +Thus, in the thermal Boltzmann factors, we now disagree. In our expressions, the +temperature of radiation emitted by a black hole, of a given mass, must be twice the +temperature found by Hawking. +5 +Black hole thermodynamics in a nut shell +In statistical physics and quantum field theory it is well-known that a unitary evolution +operator, e−iHt can be analytically continued towards imaginary time t → −iβ to +become the operator e−βE , which is exactly the operator that produces a thermal +probability distribution if the temperature T obeys β = 1/kT , with k being the +Boltzmann constant. [12, 13]. One also finds that the distribution of the energy levels is +given by the entropy S(E) if we enumerate the energy eigenstates. In a grand canonical +ensemble, one would write ϱ(E)dE for the number of energy levels between E and +9 + +E + dE, to define the free energy F(β) by +Z(β) = +� ∞ +0 +ϱ(E)dE e−βE = +� ∞ +0 +eS(E) − βEdE = e−βF , +(5.1) +where S(E) = log(ϱ(E) is the entropy of the system if E is handled as the expectation +value ⟨E⟩. The usual thermodynamical equations emerge if we assume ⟨E⟩ to be the +macroscopic energy, F is the free energy, and we use +⟨E⟩ = − ∂ +∂β log Z(β) . +(5.2) +In black holes however, one expects the integral (5.1) to diverge, as the entropy +increases as E2. This leads to a negative specific heat , which destabilises the black +hole, and therefore it is better to go to a micro-canonical ensemble, where we keep the +total energy fixed. In a micro-canonical sample, we define the entropy by ordering all +energy eigenstates, writing them as E(n), and we focus on a single value for n. For +large systems, such as large black holes, one can take the continuum limit. Then we +write +eS(E) = dn +dE , +dS(E) +dE += β(E) . +(5.3) +Here, one defines +log Z = −βF = E − S/β = E − TS +(5.4) +(in units where k = 1). While in the grand canonical ensemble, the temperature could +be chosen, here in the micro-canonical ensemble, β is a fixed function of the black hole +mass, being the total energy E. +The fact that β is fixed, originates from the fixed background configuration, being +the Schwarzschild metric. +What is the number β for a black hole with mass M ? Consider all states on a +trajectory in complex space-time. Choose the trajectory in tortoise coordinates to be +x = A eiϕ , +y = A e−iϕ , +xy = |x|2 = A2 +is fixed , +t = −4iGMϕ . +(5.5) +We are dealing with a quantum system where all states in the stretch at imaginary +time t ranging from zero to −iβ can be excited to arbitrary values. The total number +of energy eigenstates is then given by eS(E) = Tr +� +U(−iβ) +� +, where β is the total length +of the path. The length of this path is fixed if we assume the end point of the path is +the first point where x and y take on the values they had at the starting point. This is +at ϕ = 2π in Eq. (5.5). One thus finds that +kT = 1/β = 1/8πGM , +(5.6) +10 + +which is kT = +ℏc3 +8πG MBH when units are put back in. This is Hawking’s famous result [14]. +Now consider the theory where all states in region II are determined by a mapping +from region I , where the exact nature of the mapping is immaterial; it just simply +must be a unique mapping. Let us again follow our trajectory (5.5) in imaginary time. +At ϕ = π we are at t = −4πGM and we arrive at the spot where (x, y) take the +values (−x, −y), which is where the clone of the starting point begins. Therefore, the +trajectory closes at ϕ = π, which is at time t = −4πiGM , so that β = 4πGM , and +the temperature turns into +kT = +ℏc3 +4πGMBH +. +(5.7) +Inspecting Eq. (5.3), we find that not only β but also the total entropy takes on half +the value of Hawking’s calculation, Eq. (5.6). We can understand why this is so. In +terms of our local theory, the entropy is an integral over Euclidean spacetime. It counts +all independent quantum states in space-time, but the states localised after the point +(−x, −y) are not independent, they are the states in region II , which are the same +states as the ones seen before in region I . They should therefore not be counted again. +6 +Conclusions +Our revision of the black hole temperature, Eq. (5.7), is a direct consequence of the +fact that we identify the states in the second region of the Penrose diagram with the +ones in region I . We do not merely claim that the particles emerging there, carry all +information of the in-going particles, they are the in-going particles. This is easiest to +comprehend if we return to the Schwarzschild coordinates. There, one can argue that +the cusp singularity at the horizon is somewhat more delicate than just a coordinate +artefact. It asks us not to consider all states you get by filling up all of space-time there +with any states we like, but we have to fill in region II exactly as region I . Here, +the black hole we describe, fundamentally differs from Rindler space, Rindler space is +not just a big black hole because, from the outset, it contains different regions I and +II . One can’t create such configurations from the Schwarzschild metric; this is where +the Kruskal-Szekeres coordinates for a black hole are somewhat deceptive, suggesting a +world that does not exist. It does exist in Rindler spacetime. +Note that whenever we throw something in, it is seen as in going material both in +region I and in region II . For instance, when we throw in a dust shell of matter, both +regions I and II undergo modification. After a large amount of time (more than the +scrambling time), the dust shell should become invisible. This only happens if a dust +shell is assumed to enter also in region II . This does not only hold for dust shells but +for all events in the outside world. +11 + +At the beginning of our presentation, in Section 2, we mentioned some a posteriori +verifications. +In particular we claim that a stationary black hole is the appropriate +background for an accurate description of quantum black hole dynamics, as soon as it +is large enough compared to the Planck scale. To really appreciate our procedures, our +paper was not enough; one should do much more test calculations to appreciate how our +picture hangs together. We do hope that our general philosophy will be receiving more +attention than it had so-far. The fact that the Hawking temperature is modified by a +factor 2 indicates that more is going on than merely a change in semantics. +References +[1] G. ’t Hooft, Quantum clones inside black holes, Universe 2022, 8(10), 537; +https://doi.org/10.3390/universe8100537; http://arxiv.org/abs/2206.04608. +[2] G. ’t Hooft, How studying black hole theory may help us to quantise gravity. pre- +sented at the Conference on “Eternity between and Space and Time”, Padova, May +19-21, 2022, http://arXiv:2211.10723v2 [gr-qc]. +[3] A. Ashtekar and M. Bojowald, Black hole evaporation: +A paradigm, Class. +Quant.Grav. 22 (2005) 3349-3362, arXiv:gr-qc/0504029v2. +[4] M.D. Kruskal, Maximal extension of Schwarzschild metric, Phys. Rev. 119 (1960) +1743 . +[5] G. Szekeres, On the singularities of a Riemannian manifold, Publ. Math. Debrecen +7, 285 (1960) (now available at General Relativity and Gravitation 34 (2002) 2001). +[6] Irwin I. Shapiro (1964). Fourth Test of General Relativity. Phys. Rev. Letters. 13 +(26): 789-791. Bibcode:1964PhRvL..13..789S. doi:10.1103/PhysRevLett.13.789. +[7] The +Black +Hole +Firewall +Transformation +and +Realism +in +Quantum +Me- +chanics, +Universe +2021, +7, +298. +https://doi.org/10.3390/universe7080298. +http://arxiv.org/abs/2106.11152v2. +[8] W.B. Bonnor, The gravitational field of light, Commun. Math. Phys. 13 (1969) 163. +[9] P.C. Aichelburg and R.U. Sexl, On the Gravitational field of a massless particle, +Gen.Rel. and Gravitation 2 303-312 (1971); +[10] T. Dray and G. ’t Hooft, The gravitational shock wave of a massless particle, Nucl. +Phys. B253 (1985) 173. +[11] G. ’t Hooft, An ambiguity of the equivalence principle and Hawking’s temperature. +J. of Geometry and Physics 1 (1984) 45-52. +12 + +[12] K. Symanzik, Euclidean quantum field theory, I, Equations for a scalar model, J. +Math. Phys. 7, 510 (1966). +[13] K. Symanzik, Euclidean quantum field theory, in “Local Quantum Theory”, (ed. R. +Jost), Academic Press, New York, 1969. +[14] S.W. Hawking, +Particle creation by black holes, Commun. Math. Phys., 43(3) +(1975) 199. +13 + diff --git a/R9FAT4oBgHgl3EQf1R5m/content/tmp_files/load_file.txt b/R9FAT4oBgHgl3EQf1R5m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a02c8738f168c371ef3780fbc87d25d1f742b6f5 --- /dev/null +++ b/R9FAT4oBgHgl3EQf1R5m/content/tmp_files/load_file.txt @@ -0,0 +1,359 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf,len=358 +page_content='How an exact discrete symmetry can preserve black hole information∗ or Turning a Black Hole Inside Out Gerard ’t Hooft Faculty of Science, Department of Physics Institute for Theoretical Physics Princetonplein 5, 3584 CC Utrecht The Netherlands http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='nl/˜hooft101 Abstract To apply the laws of General Relativity to quantum black holes, one first needs to remove the horizon singularity by means of Kruskal-Szekeres coor- dinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This however doubles spacetime, which thereby is equipped with an exact binary symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' All particles near a black hole share the same symmetry, and conservation of this symmetry may completely remove the in- formation paradox: the quantum black hole has no interior, or equivalently, the black hole interior is a quantum clone of the exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' These obser- vations, totally overlooked in most of the literature on quantum black holes, resolve some issues concerning conservation of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Some other prob- lems do remain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' ∗Presented at DICE2022, Castiglioncello, Italy, May 19-22, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='08708v1 [gr-qc] 20 Jan 2023 1 Introduction A silent revolution took place in the author’s understanding of the quantum mechanical equations for black holes, as first mentioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In summary, what happens is that the space-time singularity associated to the horizon of a black hole, can easily be removed by an appropriate coordinate transformation, showing that a black hole horizon is about as regular as the apparent singularity of planet Earth at its North and South poles: it is a coordinate singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Removing this singularity should answer all questions about the nature of space and time there, but it leads to a new question instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Apparently, beyond the coordinate singularity, space and time are extended into a new copy of the existing space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Our new insight, about which we also reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' [2], is that this is not a new space-time region at all, but merely an exact mirror image of existing space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The black hole carries along a clone, not only of itself, but also of the entire surrounding universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' An observer1, moving from ordinary space-time to the cloned region will notice no differences at all, but in fact space-time in this region is quite remarkable, because the clone is connected to the original via time reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Clearly, this is not an ordinary symmetry, but it does allow us to handle it as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We can limit ourselves to states in Hilbert space that are either symmetric or anti-symmetric2 under the switch between the black hole and its clone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' As we shall show briefly, this reduction has a big effect on the expected properties of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Its entropy is now only half of what is usually found, and consequently, the temperature of the emitted radiation doubles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' But we shall come to that;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' let us first return to some basic physical issues concerning quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Usually, our starting point is chosen to be Newton’s force law, F = G m1 m2 r2 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1) where one imposes the condition that this law needs to be made compatible with the highly successful laws of special relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' As is well-known, one then ends up with General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' It is found that space and time are curved, and the curvature is everywhere in the vicinity of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Making the next step, needed to impose the laws of quantum mechanics, is now very hard, and usually comes with new assumptions and unprovable principles, such as string theory and holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' There is nothing wrong with such approaches, except that the results are generally uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' One has to understand Nature’s (or God’s3) way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' you have to be religious to understand that, 1The concept of an observer can perhaps better be replaced by the concept of observable variables, so that some problems such as finite-size effects and finiteness of memory space, are more easy to avoid, but clearly, utmost diligence would be asked for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 2symmetric for the real parts of wave functions, and antisymmetric for the imaginary parts, as will be explained, see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 3I should henceforth avoid using God’s name, as this apparently suggests religious feelings that I do not have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 2 and if you are not religious, you just miss the boat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' An approach that requires (almost4) no religion is the path offered by black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' If we remove the coordinate singularities by a coordinate transformation, we only encounter small amounts of local curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' However, the effects of gravity still diverge in a remarkable way, which is the effects of the gravitational force acting between in- and out-going matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This diverges badly, in a process that explodes exponentially in time, but in a way that triggered our interest: this divergence only occurs on the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Elsewhere in space-time, one may assume that standard knowledge of general relativity and quantum mechanics offers sufficient clues as to what happens there, so, unintelligible curvature does not occur everywhere, and we may attempt to couple understandable features of regions of space and time, while reserving our attention to the tiny subset of space-time points right at the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This is where a black is connected to its clone at the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Instead of Newton’s law, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1) , we focus on the Shapiro effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This effect brings about gravitational lensing, as is well known in astronomy: δu− = −8πG p− log |˜x1 − ˜x2| , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='2) where p− is the momentum of the source, which we take as a vector in the light cone minus-direction, while u− is the displacement it generates in a reference mass, also in the minus-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' ˜x1 and ˜x2 are the transverse components of the locations of source and effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The most noticeable distinction with Newton’s law is the dependence on these coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' It goes as 1/r2 in the Newtonian case, while it is only logarithmic in the case of Shapiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The logarithm reflects the fact that the associated phenomenon takes place in a two-dimensional subspace of space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Laplace’s field equation generates logarithms in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The Shapiro equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='2) offers an easy way to handle the gravitational interactions between in- and out-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' One finds that, in the reference frame of the out-particles, the momentum p− of the in-particles increases exponentially as a function of the time separation between in and out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Remarkably, here momenta of the in-particles affect the positions of the out-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' By applying Fourier transformations, one derives directly that (apart from a sign) the same relation holds if for both particles momenta and positions are interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This is time reflection symmetry, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Readers familiar with these issues might want to proceed directly to Section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=', where some subtle novelties enter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 4A critical reader may observe that we do make use of a belief: the fact that black holes should in no way differ from ordinary particles in the way they obey quantum mechanical laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Yes, I admit that this is a belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 3 2 Perturbations around the stationary black hole Many researchers now continue with the black hole as it is being formed by some implo- sion event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' [3] It then seems as if the details of this implosion are intimately responsible for what happens next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We claim however that this only holds for time spans shorter than the scrambling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This is the time it takes for a particle to reach the horizon within a few Planck lengths, in natural units: TS = O � MBH log(MBH/MPlanck) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1) Classically we have the no-hair theorem stating that the details of the implosion event are immaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In practice one expects that only minute quantum details will continue to depend on details of the original implosion, but this situation would be similar to what happens to an atom after its formation at the beginning of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Surely such details will determine which of its quantum states it will be in right now, but as physicists we should be primarily interested in a proper categorisation of all possible quantum states, and the Schr¨odinger equation dictating how these evolve one into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We here take the same attitude regarding black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Its history before scrambling time, or its evaporation process after scrambling time, should not be relevant for its physical equations at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Soon after its formation, the black hole is almost stable, absorbing and emitting particles every now and then, according to a Schr¨odinger equation and a Hamiltonian connecting all microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' It is our job to use basic principles for deriving the proper classification of states and the fundamental Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' At some decent distance from the horizon, the categorisation is clear and the Hamiltonian is known, it is called the Standard Model of the subatomic particles, which can be put in a gently curved space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' At all space-time points where the temperature dropped to below a few TeV, we know how it generates the answers to our questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We now assume that the complete set of states is determined by the spectrum of particles excited thermally at all distances beyond a few Planck lengths from the horizon, where the exact numbers are still to be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Since the particles are thermal and the local temperature is expected to have dropped well below the Planck temperature, the effects of these particles on the black hole space-time metric must have dropped to values small enough to neglect them for the time being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We plan to postpone these details to being justified a posteriori (See Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Right now we can observe that the sets of states and operators used at time scales that may vary over more than one unit of the scrambling time will definitely not commute with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The Schwarzschild 4 time ↑ singularity in out −2GM 0 2GM r r = 2GM IV III II I x y a) b) Figure 1: a) The Schwarzschild metric, in Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Vertical dashed lines: the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Curved lines: light like radial geodesics: y = const (smooth), and x = const (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Cones: the orientation of the local light cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Angular coordinates (θ, ϕ) are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' b) The same space-time in Tortoise coordinates (x, y), showing regions I − IV and the orientations of the local light cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Curves show r = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' metric, in Schwarzschild’s coordinates, demonstrates clearly the time dilation invariance: ds2 = 1 1 − 2GM r dr2 − � 1 − 2GM r � dt2 + r2dΩ2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='2) where Ω ≡ (θ, ϕ) , dΩ ≡ (dθ, sin θ dϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='3) It is sketched in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The regular coordinate frame is the Kruskal [4] Szekeres [5] coordinates, to be referred to as tortoise coordinates x and y, see Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' One defines x y = � r 2GM − 1 � er/2GM ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='4) y/x = et/2GM , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='5) in terms of which, the spacetime described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='2) is regular at the horizon: ds2 = 32(GM)3 r e−r/2GMdx dy + r2dΩ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='6) All of space-time as we know it, is covered by the set {x ≥ 0 , y ≥ 0} (region I ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' As we mentioned in the Introduction, this new coordinate frame allows for an 5 analytic continuation towards another black hole space-time, defined by the new region II : {x ≤ 0 , y ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In contrast, the regions III where x ≤ 0 and y ≥ 0, and IV where x ≥ 0, y ≤ 0, are situated beyond the infinite future or infinite past, and these are physically less bothering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Both coordinate frames depicted in Figure 1 are of importance, and we have to understand better how they are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The arrow of time is oriented upward in all outside regions of the Schwarzschild frame Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In its inside regions it is pointing inwards, and this means that a local observer will be grilled at the singularity soon after having passed the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' But we are more interested in the way regions I and II are linked together, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The arrow of time can be read off from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' It implies that, as a local observer in the horizon region sees time always moving in the upwards direction inside its local light cone, an outside observer using the growing time coordinate t, will be moving downwards in region II .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Wave functions in I will rotate as e−iEt, but in region II as e+iEt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' However, as soon as the Standard Model applies, such negative energy modes are forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Either one can say that the Dirac ket states seen by the local observer relate to the bra states of the global observer, or one can say that, in the static black hole approximation, the states seen by the local observer in region II describe a universe that is almost full rather than almost empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Considering the fact that, as derived by Hawking [14], quantum effects generate particles at the crossing points of the two horizons, which are seen as particles in both regions, the idea that we have a space full of particles in region II leads to a satisfactory image of a quantum mechanical leak at that point, producing a steady shower of Hawking particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' At first sight one might object that there is no such leak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' a time translation for the outside observer leads to a Lorentz transformation for the local observer, and it might seem that no particles should leak out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Now of course quantum effects may generate tunnelling, and there is a more calculable genuine effect that generates such tunnelling: the Shapiro effect mentioned in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 3 The Shapiro effect How to employ the Shapiro effect [6] in understanding the dynamics of the Hawking effect, was elaborately explained in our previous publications [1, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='2) is derived by first considering the Schwarzschild metric of a particle at rest, and then subjecting that to a Lorentz transformation with a large γ factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The particle momentum is the limit p− → γm for large γ, and small rest mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The result [8, 9] still describes flat space-time, but General Relativity tells us that there is curvature, delta-peaked at the location of the shock wave associated to the fast source particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Next, one generalises 6 the equation for the case that the source mass goes inwards through an S2 surface, the horizon, rather than a flat shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The result [10] is almost the same, but the transverse separation ˜x − ˜x′ is now described by an angular separation, the logarithm turns into a Laplace function on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We see that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='2) turns into δu− = Gp−f(Ω, Ω′) with (1 − ∆Ω)f(Ω, Ω′) = 8πGδ2(Ω, Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1) Here, Ω is the solid angle at which the out-particle leaves, and Ω′ is where the in-particle entered the black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We add normalisation constants that turn the tortoise coordinate x into the lightcone coordinate u− and y into u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' For large black holes, the metric in the u± coordinates is almost flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Henceforth, we ignore the slight curvature that remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' With u± we indicate the positions at the future or past horizon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=', the time at which the particles enter or leave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' p± are their momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' These equations are linear in the position and momentum variables, and consequently, it is easy to generalise this result to handle the case of many particles going in or out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' These lead to momentum and position distributions on the horizons: p± → p±(Ω) , u± → u±(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='2) And this allows us to use spherical wave expansions5 u±(Ω) = � ℓ,m u± ℓ mYℓ m(Ω) , p±(Ω) = � ℓ,m p± ℓ mYℓ m(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='3) Since, for each spherical harmonic, ∆ = −ℓ(ℓ + 1), we can write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1) as δu− ℓm = 8πG ℓ2 + ℓ + 1p− ℓm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='4) Now we can use the commutator equations for all in- and out-going particles i, j, [u± i , p∓ j ] = iδi j , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='5) to arrive at the algebra [u+ ℓ m, u− ℓ′ m′] = iλℓ δmm′ δℓℓ′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' λℓ = 8πG ℓ2 + ℓ + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='6) A few remarks: We dropped the symbol δ from δu±, which means that it is assumed that all displace- ments δu added up give the positions u themselves, provided coordinates are chosen appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The same holds for the momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 5These distribution functions are to be defined accurately: the momenta emerge as sums of δ distri- butions, while positions are represented as centre of mass positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' See for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' [2], sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 7 Secondly, all different (ℓ, m) modes decouple, so we obtain new, one-dimensional quantum systems that generate the total Hilbert space in a very simple manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' These equations are highly trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Furthermore, the Fourier transformation from momenta to positions and back is a unitary transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Hence, the algebra (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='6) should act as a perfect, information preserving, mapping from in- to out-particles (and back).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 4 Restoring unitarity But then there is a more serious problem that has to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The Fourier trans- forms arrived at in the previous section, are only unitary if the data stretch over the entire real line both for the u+ and the u− variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' But the physical data in the observable part of the universe are exclusively on the half-lines u+ > 0 and u− > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' If we drop the data on the negative half-lines, the Fourier transformation is not unitary at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The cure to this problem may well come from the observation proposed in the Intro- duction section: just choose region II to be a quantum clone of region I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In that case, also the Fourier variables in region II are quantum clones of those in region I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Is this an elegant solution to our problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Unfortunately no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' If all wave functions were real, then indeed the equation ψ(u) = ψ(−u) copies correctly in Fourier space as well: ˆψ(p) = ˆψ(−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' But wave functions are not in general real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Can we impose reality as an extra condition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In ordinary quantum mechanics, the condition that wave functions are real is not impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Taking the Hamiltonian to be an imaginary, antisymmetric matrix would be allowed in quantum mechanics, provided that we accept the corresponding symmetry in the energy spectrum: every state with energy E in the energy spectrum, is associated to a state with the opposite sign of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Reality of the wave function implies a symmetry E ↔ −E (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1) in the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Remember that, right at the beginning of our treatment, it was imposed that for ordinary particles the effect of the particles to the total mass of the black hole may be ignored, so indeed, we may assume the condition that the entire energy spectrum of particles is shifted to obey the symmetry (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This line of arguments needs to be carefully checked to ascertain that it be realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In previous work, it was proposed to combine the transformation u ↔ −u with the antipodal mapping: Ω = (θ, ϕ) ↔ −Ω ≡ (π − θ, ϕ + π) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='2) 8 but this was found to lead to other difficulties: the antipodal mapping is the mapping that replaces all spherical harmonic functions Yℓ m by (−1)ℓ Yℓ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' But since we already had that the mapping I ↔ II corresponds to u± ↔ −u± , this would imply u± ℓ,m(Ω) = (−1)ℓ+1u± ℓ,m(Ω) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='3) so that u± = 0 if ℓ is even, which cannot be squared with the commutation equation saying that all u variables obey [u+, u−] = iλℓ, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Therefore the transition to antipodes is herewith dismissed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' An other possibility was proposed im 1984 by the author [11]: should we regard all states in region II as the bra states ⟨ψ| associated to the ket states |ψ⟩ in region I ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The elegance of this proposal is that, together, these structures form the density matrices of the system in region I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Is it possible to accept the mathematics of the states in region I and II taken together, as representing the quantum density states?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Actually, this proposal almost coincides with our attempt to impose the symmetry (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' It is not as yet ruled out, and it seems to be the only possibility left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Note that we are forced to supplant our complex wave functions by real ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' It is as if the use of complex global U(1) symmetries is forbidden, which might enforce violation of exact, additive, infinite conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' These are replaced by a discrete Z2 symmetry that switches the signs of the tortoise coordinates x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' And it does lead to a more general, really important observation: this proposal replaces Hawking’s calculation for the temperature of a black hole by another one!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' If we use density matrices to compute probabilities, we find that probabilities are proportional to the matrix elements of the density matrix, whereas Hawking regards these as quantum vectorstates, and consequently, he averages over the squares of the states describing region II as these can’t be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Thus, in the thermal Boltzmann factors, we now disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In our expressions, the temperature of radiation emitted by a black hole, of a given mass, must be twice the temperature found by Hawking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 5 Black hole thermodynamics in a nut shell In statistical physics and quantum field theory it is well-known that a unitary evolution operator, e−iHt can be analytically continued towards imaginary time t → −iβ to become the operator e−βE , which is exactly the operator that produces a thermal probability distribution if the temperature T obeys β = 1/kT , with k being the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' One also finds that the distribution of the energy levels is given by the entropy S(E) if we enumerate the energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In a grand canonical ensemble, one would write ϱ(E)dE for the number of energy levels between E and 9 E + dE, to define the free energy F(β) by Z(β) = � ∞ 0 ϱ(E)dE e−βE = � ∞ 0 eS(E) − βEdE = e−βF , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1) where S(E) = log(ϱ(E) is the entropy of the system if E is handled as the expectation value ⟨E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The usual thermodynamical equations emerge if we assume ⟨E⟩ to be the macroscopic energy, F is the free energy, and we use ⟨E⟩ = − ∂ ∂β log Z(β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='2) In black holes however, one expects the integral (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='1) to diverge, as the entropy increases as E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This leads to a negative specific heat , which destabilises the black hole, and therefore it is better to go to a micro-canonical ensemble, where we keep the total energy fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In a micro-canonical sample, we define the entropy by ordering all energy eigenstates, writing them as E(n), and we focus on a single value for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' For large systems, such as large black holes, one can take the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Then we write eS(E) = dn dE , dS(E) dE = β(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='3) Here, one defines log Z = −βF = E − S/β = E − TS (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='4) (in units where k = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' While in the grand canonical ensemble, the temperature could be chosen, here in the micro-canonical ensemble, β is a fixed function of the black hole mass, being the total energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The fact that β is fixed, originates from the fixed background configuration, being the Schwarzschild metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' What is the number β for a black hole with mass M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Consider all states on a trajectory in complex space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Choose the trajectory in tortoise coordinates to be x = A eiϕ , y = A e−iϕ , xy = |x|2 = A2 is fixed , t = −4iGMϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='5) We are dealing with a quantum system where all states in the stretch at imaginary time t ranging from zero to −iβ can be excited to arbitrary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The total number of energy eigenstates is then given by eS(E) = Tr � U(−iβ) � , where β is the total length of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The length of this path is fixed if we assume the end point of the path is the first point where x and y take on the values they had at the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This is at ϕ = 2π in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' One thus finds that kT = 1/β = 1/8πGM , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='6) 10 which is kT = ℏc3 8πG MBH when units are put back in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This is Hawking’s famous result [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Now consider the theory where all states in region II are determined by a mapping from region I , where the exact nature of the mapping is immaterial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' it just simply must be a unique mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Let us again follow our trajectory (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='5) in imaginary time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' At ϕ = π we are at t = −4πGM and we arrive at the spot where (x, y) take the values (−x, −y), which is where the clone of the starting point begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Therefore, the trajectory closes at ϕ = π, which is at time t = −4πiGM , so that β = 4πGM , and the temperature turns into kT = ℏc3 4πGMBH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='7) Inspecting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='3), we find that not only β but also the total entropy takes on half the value of Hawking’s calculation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We can understand why this is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In terms of our local theory, the entropy is an integral over Euclidean spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' It counts all independent quantum states in space-time, but the states localised after the point (−x, −y) are not independent, they are the states in region II , which are the same states as the ones seen before in region I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' They should therefore not be counted again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 6 Conclusions Our revision of the black hole temperature, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='7), is a direct consequence of the fact that we identify the states in the second region of the Penrose diagram with the ones in region I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We do not merely claim that the particles emerging there, carry all information of the in-going particles, they are the in-going particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This is easiest to comprehend if we return to the Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' There, one can argue that the cusp singularity at the horizon is somewhat more delicate than just a coordinate artefact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' It asks us not to consider all states you get by filling up all of space-time there with any states we like, but we have to fill in region II exactly as region I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Here, the black hole we describe, fundamentally differs from Rindler space, Rindler space is not just a big black hole because, from the outset, it contains different regions I and II .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' One can’t create such configurations from the Schwarzschild metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' this is where the Kruskal-Szekeres coordinates for a black hole are somewhat deceptive, suggesting a world that does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' It does exist in Rindler spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Note that whenever we throw something in, it is seen as in going material both in region I and in region II .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' For instance, when we throw in a dust shell of matter, both regions I and II undergo modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' After a large amount of time (more than the scrambling time), the dust shell should become invisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This only happens if a dust shell is assumed to enter also in region II .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' This does not only hold for dust shells but for all events in the outside world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 11 At the beginning of our presentation, in Section 2, we mentioned some a posteriori verifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' In particular we claim that a stationary black hole is the appropriate background for an accurate description of quantum black hole dynamics, as soon as it is large enough compared to the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' To really appreciate our procedures, our paper was not enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' one should do much more test calculations to appreciate how our picture hangs together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' We do hope that our general philosophy will be receiving more attention than it had so-far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' The fact that the Hawking temperature is modified by a factor 2 indicates that more is going on than merely a change in semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' ’t Hooft, Quantum clones inside black holes, Universe 2022, 8(10), 537;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='3390/universe8100537;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' http://arxiv.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Kruskal, Maximal extension of Schwarzschild metric, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' 119 (1960) 1743 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FAT4oBgHgl3EQf1R5m/content/2301.08708v1.pdf'} 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a/SNAyT4oBgHgl3EQfVPeU/vector_store/index.faiss b/SNAyT4oBgHgl3EQfVPeU/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..90b340f9b5d8f3f6db350288f80195801d2508ae --- /dev/null +++ b/SNAyT4oBgHgl3EQfVPeU/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1f18033ca0ade2a7180ef706176aac6b434da203b2b8e3034d80221043c0d278 +size 9633837 diff --git a/WdAyT4oBgHgl3EQfhvjT/content/tmp_files/2301.00384v1.pdf.txt b/WdAyT4oBgHgl3EQfhvjT/content/tmp_files/2301.00384v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..75ccb44fed35344978ff10d160b17d19b5a98b26 --- /dev/null +++ b/WdAyT4oBgHgl3EQfhvjT/content/tmp_files/2301.00384v1.pdf.txt @@ -0,0 +1,1314 @@ +Correlation Clustering Algorithm for Dynamic Complete Signed +Graphs: An Index-based Approach +Ali Shakiba +Department of Computer Science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran. +ali.shakiba@vru.ac.ir;a.shakiba.iran@gmail.com +Abstract +In this paper, we reduce the complexity of approximating the correlation clustering problem from +O (m × (2 + α(G)) + n) to O (m + n) for any given value of ε for a complete signed graph with n vertices +and m positive edges where α(G) is the arboricity of the graph. Our approach gives the same output as +the original algorithm and makes it possible to implement the algorithm in a full dynamic setting where +edge sign flipping and vertex addition/removal are allowed. Constructing this index costs O (m) memory +and O (m × α(G)) time. We also studied the structural properties of the non-agreement measure used +in the approximation algorithm. The theoretical results are accompanied by a full set of experiments +concerning seven real-world graphs. These results shows superiority of our index-based algorithm to the +non-index one by a decrease of %34 in time on average. +Keywords: Correlation clustering · Dynamic graphs · Online Algorithms +1 +Introduction +Clustering is one of the most studied problems in machine learning with various applications in analyzing +and visualizing large datasets. There are various models and technique to obtain a partition of elements, +such that elements belonging to different partitions are dissimilar to each other and the elements in the same +partition are very similar to each other. +The problem of correlation clustering, introduced in [1], is known to be an NP-hard problem for the +disagree minimization. Therefore, several different approximation solutions based on its IP formulation exist +in the literature. Recently, the idea of a 2-approximation algorithm in [1] is extended in [4] for constructing +a O (1)-approximation algorithm. The experiments in [4] show acceptable performance for this algorithm in +practice, although its theoretical guarantee can be too high, e.g. 1 442 for β = λ = +1 +36. In [3], this algorithm +is extended to an online setting where just vertex additions are allowed, and whenever a new vertex is added, +it reveals all its positively signed edges. Shakiba in [12] studied the effect of vertex addition/removal and edge +sign flipping in the underlying graph to the final clustering result, in order to make the algorithm suitable for +dynamic graphs. However, one bottleneck in this way is computing the values of NonAgreement among +the edges and identifying the ε-lightness of vertices. The current paper proposes a novel indexing scheme +to remedy this and make the algorithm efficient, not just in terms of dynamic graphs, but for even dynamic +hyper-parameter ε. Our proposed method, in comparison with the online method of [3] is that we allow +a full dynamic setting, i.e. vertex addition/removal and edge’s sign flipping. It is known that any online +algorithm for the correlation clustering problem has at least Ω(n)-approximation ratio [10]. Note that the +underlying algorithm used in the current paper is consistent, as is shown via experimental results [3]. +The rest of the paper is organized as follows: In Section 1.1, we highlight our contributions. This is +followed by a reminding some basic algorithms and results in Section 2. +Then, we introduce the novel +indexing structure in Section 3.1 and show how it can be employed to enhance the running-time of the +approximate correlation clustering algorithm. Then, we show how to maintain the proposed indices in a +full dynamic settings in Section 3.2. In Section 4, we give an extensive experiments which accompanies the +1 +arXiv:2301.00384v1 [cs.DS] 1 Jan 2023 + +theoretical results and show the effectiveness of the proposed indexing structure. Finally, a conclusion is +drawn. +1.1 +Our Contribution +In this paper, we simply ask +“How can one reduce the time to approximate a correlation clustering of the input graph [4] for +varying values of ε?” +We also ask +“How can we make the solution to the first question an online solution for dynamic graphs?” +Our answer to the first question is devising a novel indexing-structure which is constructed based on the +structural properties of the approximation algorithm and its NonAgreement measure. As our experiments +in Section 4 show, the proposed method enhanced the total running-time of querying the clustering for about +%34 on average for seven real-world datasets. Then, we make this structure online to work with dynamic +graphs based on theoretical results in [12]. The construction of the index itself is highly parallelizable, up +to the number of the vertices in the input graph. The idea for parallelization is simple: construct each +NAO (v) in the NAO (G) with a separate parallel thread. +We also study the intrinsic structures in the +NonAgreement measure, to bake more efficient algorithms for index-maintenance due to updates to the +underlying graph. More precisely, we show that using the proposed index structure, we can find a correlation +clustering for a graph for any given value of ε in time O (m + n), compared to the O (m × (2 + α(G)) + n) +time for the CC. The pre-processing time of the ICC would be O (m × α(G)) with O (m) space complexity. +2 +Preliminaries +Let G = (V, E) be a complete undirected signed graph with |V | = n vertices. The set of edges E is naturally +partitioned into positive and negative signed edges, E+ and E−, respectively. Then, we use m to denote +|E+|. The correlation clustering problem is defined as +cost(C) = +� +{u,v}∈E+ +u∈Ci,v∈Cj,i̸=j +1 + +� +{u,v}∈E− +u,v∈Ci +1, +(1) +where C = {C1, . . . , Cℓ} is a clustering. Note that this is the min-disagree variant of the problem. +The constant factor approximation algorithm of [4] is based on two main quantities: (1) ε-agreement of +a positively signed edge {u, v}, i.e. u and v are in ε-agreement if and only if NonAgreementG (u, v) = +|NG(u)∆NG(v)| +max{|NG(u)|,|NG(v)|} < ε, and (2) ε-lightness, where a vertex u is said to be ε-light if AgreeCntG+(u) +|NG+(u)| +< ε where +AgreeCntG+(u) = |{w ∈ V |u and v are in ε-agreement}|. Note that a vertex which is not ε-light is called +ε-heavy. This is a 2 + 4 +ε + +1 +ε2 -approximation algorithm, as is shown in [3]. This algorithm is described in +Algorithm 1, which we will refer to the CC algorithm, for short. +Shakiba in [12] studied theoretical foundation of the CC algorithm in a full dynamic setting. The following +result is a summary of Table 1, Corollary 1, and Theorem 4 in [12]. +Theorem 1. Suppose the sign of an edge u = {u, v} is flipped. Then, the non-agreement and ε-lightness of +vertices other than the ones whose distance to either u and v is more than two would not change. +The arboricity of the graph G is the minimum number of edge-disjoint spanning forests into which G can +be decomposed. The following lemma for arboricity is useful in bounding the number of operations. +Lemma 1. Lemma 2 in [2] Suppose the graph G = (V, E) has n vertices with m edges. Then, +� +{u,v}∈E +min {degG(u), degG(v)} ≤ 2a(G) × m. +(2) +2 + +Algorithm 1 CorrelationClustering(G) [4] +1: procedure CorrelationClustering(G, ε) +2: +Let G+ = G[E+] where E+ is the set of edges whose sign is + +3: +Discard all edges whose endpoints are not in ε-agreement +4: +Discard all edges between two ε-light vertices +5: +Let � +G+ be the sparsified graph G+ after performing previous two operations +6: +Let C be the collection of connected components in � +G+ +7: +return C as the output clustering +8: end procedure +3 +Proposed Method +In this section, we describe our novel indexing structure. +This structure allows dynamic queries of the +correlation clustering with varying values of ε for dynamic graphs. The proposed algorithm which uses the +indexing structure would be called ICC, or indexed-based correlation clustering. +3.1 +Indexing structure +For an edge e = {u, v} with positive sign, we define its ε-agreement distance as NonAgreementG+ (u, v). +Intuitively, this is the infimum of the values ε which the nodes u and v are not in ε-agreement. Let define +the set E = {NonAgreementG+ (u, v) |e = {u, v} ∈ E+}. Without loss of generality, let E = {ε0, . . . , εℓ−1} +with the ordering min E = ε0 < ε1 < · · · < εℓ−1 = max E. For a fixed value of ε, let G+ +ε = (V, E+ +ε ) where +E+ +ε = {e = {u, v} ∈ E+|NonAgreementG+ (u, v) < ε}. +Observation 1. For all ε ≤ ε0, G+ +ε is the null graph, i.e. a graph on all nodes without any edges. Moreover, +for all ε > εℓ, G+ +ε = G+. +Next, we introduce the key ingredient to our indexing structure, called NAO. +Definition 1 (NonAgreement Node Ordering). The ε-agreement ordering for each node v ∈ V , denoted +by NAO (v), is defined as an ordered subset of vertices in G where: +1. node u ∈ V appears in the ordering NAO (v) if and only if e = {u, v} is a positive edge in G. +2. for each two distinct vertices u, w ∈ V which appear in NAO (v), +NonAgreementG+ (v, u) < NonAgreementG+ (v, w) , +(3) +implies u appears before w. +3. for each node u ∈ NAO (v), its ε-agreement distance is also stored with that node. +The NonAgreement node ordering of the graph G is defined as NAO (G) = {(v, NAO (v))|v ∈ V }. +In other words, the NAO (v) is a sorted array of neighboring nodes of v in G+ in their ε-agreement +distance value. An example NonAgreement node ordering for all vertices in a sample graph is illustrated +in Figure 1. The space and construction time complexities of the NAO(G) are investigated in the next two +lemmas. +Lemma 2. The NonAgreement node ordering for a graph G, NAO (G), can be represented in O (m) +memory. +Proof. The number of nodes inside NAO (v) equals to the degG+(v) + 1, for all vertices in NG+(v) as well +as the vertex v itself. Note that it is not required to explicitly store the vertex v itself in the ordering. +Cumulatively, the total size required for representing NAO (v) is 2 × m entries. +3 + +v0 +v1 +v2 +v3 +v4 +v5 +v1 +v5 +v3 +v2 +v0 +v1 +v3 +v0 +v0 +v4 +v3 +v0 +v0 +v1 +v2 +v3 +v4 +v5 +0 +0.35 +0.44 +0.72 +0 +0.2 +0.35 +0.46 +0 +0 +0 +0 +0.44 +0.46 +0.29 +0.29 +0.72 +0.2 +Figure 1: An illustrative example of NAO (v) for an example graph. +Lemma 3. The time complexity to construct the NAO (v) for all vertices v ∈ V is O (m × (α(G) + lg m)) +where α(G) is the arboricity of the graph G. +Proof. To compute the value of NonAgreementG+ (u, v) for all edges e = {u, v} ∈ E+, one needs to +compute |NG+(u)∆NG+(v)|. This requires O (degG+(u) + degG+(v)) operations to compute the symmetric +difference, given the adjacency lists of u and v are sorted. However, we can compute their intersection and +use it to compute the NonAgreement as follows +NonAgreementG+ (u, v) = degG+(u) + degG+(v) − 2 × |NG+(u) ∩ NG+(v)| +max {NG+(u), NG+(v)} +. +(4) +Hence, the total time required to compute the values of NonAgreement is equal to +� +{u,v}∈E+ +min {degG+(u), degG+(v)} , +which is known to be bounded by 2α(G) × m (Lemma 1). Moreover, each NAO (v) is of length 1 + degG+(v) +which requires sorting by the value of their NonAgreement in O (degG+(v) lg (degG+(v))), which accumu- +lates to O (m lg m). +The correlation clustering corresponding to a given value of ε and a graph G is the set of connected +components of the graph � +G+. Given access to NAO (G) = {NAO (v) |v ∈ V (G)}, one can respond to the +following queries: (1) Is the vertex v an ε-heavy vertex or an ε-light? (2) Are the endpoints of an edge {u, v} +in ε-agreement? As we will show in this section, both of these questions can be answered in O (1) time using +the NAO structure. For a vertex v ∈ V , its ε-light threshold is defined as LTHε (v) = ⌈ε degG+(v)⌉ + 1. +Lemma 4. For an ε and a vertex v ∈ V , v is ε-heavy if and only if the ε-agreement distance of the +LTHε (v)-th smallest vertex in NAO (v) is less than ε. Otherwise, it is ε-light. +Proof. The vertices u and v would be in ε-agreement if and only if NonAgreementG+ (u, v) < ε. Whenever +the ε-agreement distance of the LTHε (v)-th smallest vertex in NAO (v) is less than ε, it means that there +are at least ⌈ε degG+(v)⌉ + 1 vertices, including the v itself, in ε-agreement with v. This is equivalent the ε- +heavyness of the vertex v. On the other hand, if the ε-agreement distance of the LTHε (v)-th smallest vertex +in NAO (v) is greater than or equal to ε, this is equivalent to that the number of vertices in ε-agreement +with v is less than ε × degG+(v), i.e. v is ε-light. +Lemma 5. Given access to NAO (v), for all values of ε and any vertex v ∈ V , identifying the ε-lightness of +v can be accomplished in O (1). +4 + +Proof. This is implied by Lemma 4, assuming that the NAO (v) is implemented as an array. +Given access to the index NAO (G) = {(v, NAO (v))|v ∈ V } for a graph G, a query simply asks for a +clustering of the graph for a given value of ε. +Theorem 2. Given access to the index NAO (G), computing a clustering for a given parameter ε can be +accomplished in O (m (a(n) + 1)) amortized time, where a(n) is the slowly growing inverse of the single-valued +Ackermann’s function. +Proof. Intuitively, we are going to use NAO (G) to construct the graph � +G+ and compute its connected +components incrementally. More formally, we start by putting each vertex to its isolated set in a disjoin- +union structure. Then, we identify ε-heavy vertices H ⊆ V , which takes O (m). Next, consider NAO (v) for +v ∈ H. For each vertex u ∈ NAO (v) whose key is smaller than ε and u ∈ H, we call Union(u, v), which +merges the cluster which u and v belong. These operations takes O (m × a(n)) amortized time [5] using a +Disjoint-Set data structure. The correctness of the query algorithm is implied by the Definition 1 and the +Lemmas 4 and 5. +Before going any further, we need to state the following results, which intuitively investigate the behavior +of the ε-agreement and ε-light of edges and vertices through the E.Let +E = +� +NonAgreementG+ (u, v) | {u, v} ∈ E+� += {ε0, ε1, . . . , εℓ−1} , +(5) +where εi−1 < εi for i = 1, . . . , ℓ − 1. Moreover, assume that εℓ > max E. +Observation 2. For any ε ≤ ε0, the endpoints of all positive signed edges {u, v} are not in ε-agreement. +Observation 3. For any ε > εℓ−1, the endpoints of all positive signed edges {u, v} are in ε-agreement. +By the Observation 2, the correlation clustering output by the Algorithm 1 for all values of ε ≤ ε0 would +be the collection of singleton vertices, i.e. {{v} |v ∈ V }. However, we cannot say that for ε > εell−1, the +output is a single cluster, i.e. the set V . Why is that? As ε increases, the number of edges in ε-agreement +would increase, however, the number of ε-light vertices would increase, too (By Corollary 1, which follows +soon). Hence, there is a trade-off for choosing a suitable value of ε to get the minimum number of clusters. +We would discuss this issue further in Section 4 (Experiments). +Theorem 3. Let ε < ε′ and u, v ∈ V be two distinct vertices. If u and v are in ε-agreement, then they would +be in ε′-agreement, too. Also, if u and v are not in ε′-agreement, then they would not be in ε-agreement. +Proof. The proof is a direct implication of the NonAgreement definition. +Given that u and v are in +ε-agreement, we have NonAgreementG+ (u, v) < ε < ε′, i.e. u and v are in ε′-agreement, too. Similarly, +given that u and v are not in ε′-agreement, we have NonAgreementG+ (u, v) ≥ ε′ > ε, i.e. u and v are not +in ε-agreement, too. +Theorem 4. Let ε < ε′ and u ∈ V . If u is ε-light, then u would be ε′-light, too. Also, if u is ε′-heavy, then +it would be ε-heavy. +Proof. This is implied by the definition of ε-lightness and Theorem 3. +Two other important cases are not considered in Theorem 4, i.e. (1) Is it possible that a vertex u be +ε-heavy, but becomes ε′-light for some values ε < ε′?, and (2) Is it possible that a ε-light vertex becomes +ε′-heavy for some values of ε < ε′? The answer to both of these questions is affirmative. By Theorems 3 +and 4 and the previous discussion, we can state the following corollary. +Corollary 1. Let ε < ε′. +1. The number of positively signed edges whose endpoints are in ε′-agreement is greater than or equal +to the number of positive edges whose endpoints are in ε-agreement. In other words, the agreement +relation is monotone. +5 + +2. The number of vertices which are ε′-light can be either greater or less than or even equal to the number +of ε-light vertices. In other words, the lightness relation is not necessarily monotone. +The Baseline idea is to recompute the NonAgreement for each new value of ε, which takes O (m × α(G)), +as is described in the proof of Lemma 2, deciding on the ε-heaviness of the vertices in G in time O (n) and +computing the connected components of the graph � +G+ as the output in time O (m + n). Totally, the time +complexity of the Baseline is O (m × (2 + α(G)) + n). +To conclude, using the index structure we invest O (m) space (Lemma 2) and O (m × (α(G) + lg m)) +(Lemma 3) time to construct the NAO (G) which make it possible to answer each query with varying ε +values in time O (m(a(n) + 1)) (Theorem 2). Comparing this with O (m × (2 + α(G)) + n) time for the +Baseline reveals that our index-based structure makes query times faster for variable values of ε. +Given access to NAO, the algorithm 2 constructs the graph � +G+. Note that the output of the algorithms +1 and 2 are the same. As answering whether a vertex v is ε-heavy or the endpoints of an edge {u, v} are in ε- +agreement can be answered in O (1) time, the running-time of the algorithm 2 would be O (max {|V |, |E+|}) +for the for loop and O +� +|V� +G+| + |E+ +� +G+| +� +to compute the connected components of � +G+. Totally, the running- +time of the query algorithm in the ICC would be O (m + n), compared to the O (m × (2 + α(G)) + n) time +for the CC. +To model queries over a graph for various values of ε, we use a ε-Schedule defined as +ε-Schedule = {ε0 < ε1 < · · · < εℓ} ⊆ [0, 2]. +(6) +Note that we consider ε-Schedule as a set of strictly increasing real numbers, just for the sake of simplicity +in notation. In a real life scenario, any time one can ask for a clustering with any value of ε. +Algorithm 2 The index-based correlation clustering algorithm with NAO (G) structure. +1: procedure IndexCorrelationClustering(G, NAO (G) , ε) +2: +for all v ∈ V do +3: +Use the NAO (() v) to identify and remove all the edges {v, u} which are in ε-NonAgreement +4: +Use the NAO (v) to identify and remove edges {u, v} where u and v are both ε-light +5: +end for +6: +return The connected components of the remaining graph as the clustering output. +7: end procedure +3.2 +Maintaining the index +The index structure introduced in Section 3.1 is used to compute a clustering of a static graph for user- +defined dynamic values of ε. In this section, we revise this structure to make it suitable for computing a +clustering of a dynamic graph for dynamic values of ε. +There are three different operations applicable to the underlying graph of the CorrelationClustering: +(1) flipping the sign of an edge e = {u, v}, (2) adding a new vertex v, and (3) removing an existing vertex +v. These operations are considered in Lemmas 6, 7, and 8, respectively. +Shakiba in [12] has shown that flipping the sign of an edge {u, v}, the ε-agreement of edges whose both +endpoints are not in the union of the positive neighborhood of its endpoints does not change (Propositions +2 and 4 in [12]). +Therefore, we just need to compute the ε-agreement for positive edges {x, w} where +x, w ∈ (NG+(u) ∪ NG+(v)). +Lemma 6. Assuming the sign of an edge e = {u, v} is flipped. +1. Assuming the sign of edge was + prior to the flipping, then u and v would be no more in ε-agreement +since their edge is now negatively signed. The vertices v and u are removed from the arrays NAO (u) +and NAO (v), respectively. Moreover, we need to update the values of the non-agreement for vertices u +and v in NAO (w) for all vertices w ∈ NAO (u) ∪ NAO (v). +6 + +2. Assuming the sign of edge was − prior to the flipping, then their NonAgreementG+ (u, v) is computed +and the vertices v and u are added to proper sorted place in NAO (u) and NAO (v), respectively. +Moreover, we need to recompute the NonAgreementG+ (x, w) for all x, w ∈ NAO (u) ∪ NAO (v) and +update their ε-agreement values. +Applying these changes is applicable in O +�� +{x,w}∈E+∩X×X (log degG+(x) + log degG+(w)) +� +where X = +NAO (u) ∪ NAO (v). +Proof. The correctness of this lemma follows from the discussion just before it. Therefore, we would just give +the time analysis. In the first case, finding the vertices in each NAO is of O (log degG+(u) + log degG+(v)). +The removal would be O (1) amortized time. +The second case, i.e. flipping the sign from − to +, would cost more, as it may require changing all the pos- +itive edges with both endpoints the in the set X. In the worst case, one may assume that all the ε-agreement +between the edges with both endpoints inside X is changed. Therefore, we need to update each of them on +their corresponding NAO indices. We need to add vertices v and u to the NAO(u) and NAO(v), respectively, +based on NonAgreementG+ (u, v). +Finding their correct position inside sorted NAOs is possible with +O (log degG+(u) + log degG+(v)) comparisons. Again, adding them at their corresponding position would +take O (1) amortized time. For the other positive edges with both endpoints in the set X, such as {x, w}, +we might need to update their ε-agreement. Without loss of generality, assume that degG+(x) ≤ degG+(w). +Doing so requires re-computation of NonAgreementG+ (x, w), in O (min {degG+(x), degG+(w)}), querying +the NonAgreement inside NAO (x), in time O (log degG+(x)), and then possible updating its position, in +both NAO (x) and NAO (w) requires O (log degG+(x) + log degG+(w)). In the worst case, all the elements +in X may need update. Therefore, this case can be accomplished in +O +� +� +� +{x,w}∈E+∩X×X +(log degG+(x) + log degG+(w)) +� +� , +in the worst-case. +Using Lemma 6, we can state the NAO-FlipEdge(u, v) algorithm (Algorithm 3) which flips the sign +of the edge {u, v}. The correctness and the running-time of this algorithm follows directly from Lemma 6. +There are cases where a set of positive edges E+ +v are also given for a newly added vertex v. One can first add +Algorithm 3 The NAO-FlipEdge(u, v) algorithm to apply the effect of flipping the sign of the edge (u, v) +in NAO (G). +1: procedure NAO-FlipEdge(u, v) +2: +Let A ← NAO (u) ∪ NAO (v). +3: +if Sign of edge {u, v} is positive then +4: +Update the graph G+ by removing edge {u, v}. +5: +Remove the vertices u and v from NAO (v) and NAO (u), respectively. +6: +else +▷ Sign of edge {u, v} is negative +7: +Update the graph G+ by adding edge {u, v}. +8: +Compute NonAgreementG+ (u, v). +9: +Add the vertices u and v to NAO (v) and NAO (u), respectively. +10: +end if +11: +for all w ∈ A do +12: +Recompute NonAgreementG+ (u, w) and update NAO (u) and NAO (w). +13: +Recompute NonAgreementG+ (v, w) and update NAO (v) and NAO (w). +14: +end for +15: end procedure +7 + +the vertex with all negative edges, and afterwards, flip all the edges in E+ +v , so they would become positively +signed. However, a batch operation would give us higher performance in practice. +Lemma 7. Assume that a vertex v is added to graph G with new positive signed edges E+ +v . Then, +1. If E+ +v = ∅, then NAO (G) does not need any updates and we just add a new NAO (v) to NAO (G) with +v as its only element in constant-time. +2. Otherwise, let X = NG+(v)∪ +� +∪x∈NG+(v)NG+(x) +� +. We need to compute the NonAgreementG+ (x, w) +for each positive edges {x, w} with x, w ∈ X after adding all the new positive edges in E+ +v , and possibly +update NAO (x) and NAO (w). This would require +O +� +� +� +{x,w}∈(E+∪E+ +v )∩X×X +(log degG+(x) + log degG+(w)) +� +� , +(7) +operations in the worst case. +Proof. The correctness would follow immediately from the Lemma 6. Again, the first case can be accom- +plished in O (1) time. For the second case, we need to calculate |X| values of NonAgreement, and add +them or update them in their corresponding NAOs. This would require +O +� +� +� +{x,w}∈(E+∪E+ +v )∩X×X +(log degG+(x) + log degG+(w)) +� +� , +(8) +by exactly the same discussion as in Lemma 6. +Remark 1. In comparing the time required for Lemmas 6 and 7, please note that the size of the set X can +be much larger in Lemma 7 than the one in Lemma 6, depending on the size of E+ +v for the new vertex v. +The NAO-AddVertex(x, Nx) algorithm (Algorithm 4) adds a new vertex x to G+ and all its neighboring +positively signed edges Nx, and updates the NAO (G). The correctness and the running-time of this algorithm +follows directly from Lemma 7. For the last operation, removing an existing vertex v with a set of positive +Algorithm 4 The NAO-AddVertex(x, Nx) algorithm to add a new vertex x and all its positive neighbors +Nx to NAO (G). +1: procedure NAO-AddVertex(x, Nx) +2: +Update graph G+ by adding the new vertex x. +3: +Construct a new NAO (x) and add it to NAO (G). +4: +Let X ← Nx ∪ (∪z∈NxNG+(z)). +5: +for all y ∈ X do +6: +for all z ∈ X \ y do +7: +Update NAO (y) and NAO (z) if the NonAgreementG+ (y, z) changes. +8: +end for +9: +end for +10: end procedure +adjacent edges E+ +v , can be accomplished by first flipping the sign of all of its adjacent edges in E+ +v from + +to −, and afterwards, removing its NAO (v) from NAO (G). Similar to vertex addition, we can do it also in +batch-mode, hoping for better performance in practice. The algorithm for the NAO-RemoveVertex(x) is +similar to the NAO-AddVertex(x, Nx) algorithm (Algorithm 4). +Lemma 8. Assume that an existing vertex v is removed from the graph G with the set of adjacent positive +signed edges E+ +v . Then, +8 + +1. If E+ +v = ∅, then NAO (v) has a single element, itself, and can be easily removed from NAO (G). Nothing +else would require a change, so it is of constant-time complexity. +2. Otherwise, let X = NG+(v)∪ +� +∪x∈NG+(v)NG+(x) +� +. We need to compute the NonAgreementG+ (x, w) +for each positive edges {x, w} with x, w ∈ X after removing all the edges in E+ +v , and possibly update +NAO (x) and NAO (w). In the worst-case, this would require +O +� +� +� +{x,w}∈(E+∪E+ +v )∩X×X +(log degG+(x) + log degG+(w)) +� +� , +(9) +operations. +Proof. The correctness of this lemma is implied by the Lemma 6. The discussion of the running-time is +exactly the same as in the proof of Lemma 7. +4 +Experiments +To evaluate the proposed method, we used 7 graphs which are formed by user-user interactions. These +datasets are described in Table 1 and are accessible through the SNAP [8]. The smallest dataset consists of +22 470 nodes with 171 002 edges and the largest one consists of 317 080 nodes with 1 049 866 edges. In all +these graphs, we consider the existing edges as positively signed and non-existing edges as negatively signed. +The distribution of the NonAgreements for each dataset is illustrated in Figures 2a to 8a. The distri- +bution of NonAgreements of the edges in almost all the datasets obeys the normal distribution, except +small imperfections in Arxiv ASTRO-PH (Figure 2a), Arxiv COND-MAT (Figure 4a), and DBLP (Figure +8a). Moreover, more detailed statistics on these distributions are given in Tables 2. One single observation is +that the most frequent value of the NonAgreement in all the sample datasets is 1. Why is that? Without +loss of generality, assume that degG(u) ≥ degG(v). Then, |NG(v)| = 2 |NG(u) ∩ NG(v)|. By the assumption +|NG(u)| ≥ 2 |NG(u) ∩ NG(v)|, i.e. the intersection of the neighborhood of vertices u and v consists of at +most half of the neighborhood of u. Also, exactly half of the neighborhood of v falls at the intersection of +the neighborhood of u and v. Intuitively, the vertex u has clustering preference with extra vertices at least +the number of vertices which both u and v have clustering preference. Similarly, the vertex v has clustering +preference with exactly half extra vertices which both u and v have clustering preference. +For each dataset, we have used a different set of ε-Schedules, depending on the distribution of their +NonAgreements. More precisely: (1) we have sorted the values of non-agreements in each dataset in a +non-decreasing order, with repetitions. (2) Then, we have selected 21 distinct values equally spaces of these +values. (3) The ε-Schedule was set to the selected values in the second step, which appended the value +of 0 at the beginning and the value of 1.99 to the end if either does not exists. Totally, the number of +ε-Schedule for each dataset is either 21, 22 or 23. +An interesting observation is the number of clusters for ε ≈ 1− in Figures 2b to 8b. Note that we use 1− +to denote the interval [1 − ϵ, 1] for some non-zero constant ϵ > 0. When ε ≈ 1 but less than 1, the number +of clusters is minimum. As it gets closer to 1, the number of clusters increases with a much greater descent +than the decrease in the number of clusters as it gets close to 1 from 0. +As in Corollary 1, by increasing the ε, the number of vertices in ε-agreement is a non-decreasing function, +which is confirmed by the plots in Figures 2a to 8a as the number of vertices in ε-non-agreement is given +by a non-increasing function. By closer visual inspection of these figures, we can see that the shape of the +plot for the number of ε-non-agreement vertices in all these graphs is almost the same, with inflection point +around the value of ε ≈ 1. This is due to the intrinsic nature of the NonAgreementG (u, v). +Similarly, the non-monotonicity result stated in Corollary 1 is observed in the same figures for the number +of ε-light vertices. By a visual inspection, the trend of the number of ε-light vertices for almost all datasets, +except for the Arxiv HEP-TH (Figure 5a) and the EU-Email (Figure 7a), is the same: the number of ε-light +vertices increases as ε increases up to some point (first interval), then decreases slightly (second interval), and +9 + +finally increases and would be asymptotically equal to the number of vertices in the graphs (last interval). +For Arxiv HEP-TH and EU-Email, we have the same trend, however, the second interval is very small. +Table 1: Description of the datasets. +Dataset +Nodes +Edges +Arxiv ASTRO-PH [7] +18 772 +198 110 +MUSAE-Facebook [11] +22 470 +171 002 +Arxiv COND-MAT [7] +23 133 +93 497 +Arxiv HEP-TH [6] +27 770 +352 807 +Enron-Email [9] +36 692 +183 831 +EU-Email [7] +265 214 +420 045 +DBLP [13] +317 080 +1 049 866 +Table 2: Statistics of NonAgreement in each dataset. +Dataset +Distinct +Minimum +Maximum +Top 2 frequent values +Arxiv ASTRO-PH +16 436 +0.015 873 0 +1.967 21 +1 — 9 664 +0.5 — 2 992 +MUSAE-Facebook +12 988 +0.031 746 +1.978 02 +1 — 18 534 +1.25 — 2 346 +Arxiv COND-MAT +3 893 +0.044 444 4 +1.964 91 +1 — 12 018 +0.5 — 4 954 +Arxiv HEP-TH +23 285 +0.086 956 5 +1.976 19 +1 — 24 852 +1.333 33 — 4 910 +Enron-Email +20 273 +0.090 909 1 +1.954 55 +1 — 31 704 +0.5 — 6 796 +EU-Email +28 612 +0.117 647 +1.990 52 +1 — 441 520 +0.997 658 — 682 +DBLP +11 611 +0.013 793 1 +1.981 13 +1 — 167 590 +0.5 — 67 238 +All the algorithms for the naive and the proposed index-based correlation clustering algorithms are +implemented in C++1 without any parallelization and the experiments are done using an Ubuntu 22.04.1 +TLS with an Intel Core i7-10510U CPU @ 1.80GHz with 12 GB of RAM. The time for running the naive +correlation clustering algorithm (Algorithm 1), denoted here as CC, as well as the time for the index-based +correlation clustering algorithm denoted as ICC, is given in Figures 2d to 8d). Note that the time reported +for the ICC in these figures does not include the required time for constructing the NAOs, as they are +constructed once and used throughout the ε-Schedule. The running-time to read the graph as well as +constructing the NAOs is reported in Table 3 in milliseconds. The CC and ICC algorithms are the same, +except that in CC, the non-agreement values of the edges and the ε-lightness of the vertices are computed +for each given value of ε, however, in ICC these are computed and stored in the proposed NAOs structure +once and used for clustering with respect to different values of ε. As it can be observed in Figures 2d to 8d, +the running time for the ICC, excluding the time to construct the NAOs for once, is largely smaller than +the one for CC. On average, our approach for the described ε-Schedule lead to %25 decrease in clustering +time. This enhancement comes at the cost of pre-computing the NAOs, which costs on average %34 of the +time for a single run of CC, which is quite small and makes the ICC efficient in cases where one requires to +have multiple clustering for various values of ε. +5 +Conclusion +In this paper, we proposed a novel indexing structure to decrease the overall running-time of an approximation +algorithm for the correlation clustering problem. This structure can be constructed in O (m × α(G)) time +with O (m) memory. Then, we can output a correlation clustering for any value of ε in O (m + n), compared +with O (m × (2 + α(G)) + n) time complexity of the ordinary correlation clustering algorithm. Moreover, +1https://github.com/alishakiba/Correlation-Clustering-Algorithm-for-Dynamic-Complete-Signed-Graphs-An-Index-based- +Approach +10 + +Table 3: Time to construct the NAO for each dataset in milliseconds. +Dataset +Time to construct NAO (ms) +Time to read graph (ms) +Arxiv ASTRO-PH +1 677 +543 +MUSAE-Facebook +1 391 +427 +Arxiv COND-MAT +521 +346 +Arxiv HEP-TH +12 530 +725 +Enron-Email +2 498 +525 +EU-Email +4 479 +958 +DBLP +4 620 +2 167 +the proposed index can be efficiently maintained during updates to the underlying graph, including edge +sign flip, vertex addition and vertex deletion. The theoretical results are accompanied with practical results +in the experiments using seven real world graphs. The experimental results show about %34 decrease in the +running-time of queries. +A future research direction would be studying this algorithm in parallel frameworks such as Map-Reduce +and make it scalable to very Big graphs. Another research direction would be enhancing the approximation +guarantee of the algorithm, or devising more efficient algorithms in terms of approximation ratio. +References +[1] Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Machine learning, 56(1):89–113, +2004. +[2] Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on +computing, 14(1):210–223, 1985. +[3] Vincent Cohen-Addad, Silvio Lattanzi, Andreas Maggiori, and Nikos Parotsidis. Online and consistent +correlation clustering. +In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang +Niu, and Sivan Sabato, editors, Proceedings of the 39th International Conference on Machine Learning, +volume 162 of Proceedings of Machine Learning Research, pages 4157–4179. PMLR, 17–23 Jul 2022. +[4] Vincent Cohen-Addad, Silvio Lattanzi, Slobodan Mitrovi´c, Ashkan Norouzi-Fard, Nikos Parotsidis, and +Jakub Tarnawski. 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Community structure in +large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics, +6(1):29–123, 2009. +11 + +(a) NonAgreements +(b) Number of clusters +(c) The number of NonAgreement and ε-light edges +(d) Clustering time in milliseconds +Figure 2: The graph Arxiv ASTRO-PH. +[10] Claire Mathieu, Ocan Sankur, and Warren Schudy. Online correlation clustering. In 27th International +Symposium on Theoretical Aspects of Computer Science-STACS 2010, pages 573–584, 2010. +[11] Benedek Rozemberczki, Carl Allen, and Rik Sarkar. Multi-scale attributed node embedding, 2019. +[12] Ali Shakiba. +Online correlation clustering for dynamic complete signed graphs. +arXiv preprint +arXiv:2211.07000, 2022. +[13] Jaewon Yang and Jure Leskovec. Defining and evaluating network communities based on ground-truth. +In Proceedings of the ACM SIGKDD Workshop on Mining Data Semantics, pages 1–8, 2012. +12 + +Non-agreement distribution for Arxiv ASTRO-PH +17500 +15000 +12500 +edges +10000 +# +7500 +5000 +2500 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Value of edge non-agreementArxiv ASTRO-PH +18000 +16000 +14000 +lusters +of Cl +12000 +# +10000 +8000 +6000 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +wArxivASTRO-PHDataset +400000 +Non-agreement edges +-light edges +350000 +300000 +250000 +200000 +0 +150000 +100000 +50000 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EArxiyASTRO-PHDataset +5000 +CC Time (ms) +ICC Time (ms) +4500 +4000 +3500 +Time +3000 +2500 +2000 +1500 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00(a) NonAgreements +(b) Number of clusters +(c) The number of NonAgreement and ε-light edges +(d) Clustering time in milliseconds +Figure 3: The graph MUSAE-Facebook. +13 + +Non-agreement distribution for MUSAE-Facebook +30000 +25000 +20000 +↓0 # +15000 +10000 +5000 +00'0 +0.25 +0.500.751.001.25150 +1.75 +2.00 +Value of edqe non-agreementMUSAE-Facebook +22000 +21000 +of Clusters +20000 +# +19000 +18000 +17000 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EMUSAE-FacebookDataset +350000 +300000 +250000 +200000 +Non-agreement edges +-light edges +150000 +100000 +50000 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EMUSAE-FacebookDataset +7000 +CC Time (ms) +ICC Time (ms) +6500 +(sw) +6000 +Time +5500 +5000 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +E(a) NonAgreements +(b) Number of clusters +(c) The number of NonAgreement and ε-light edges +(d) Clustering time in milliseconds +Figure 4: The graph Arxiv COND-MAT. +14 + +Non-agreementdistributionforArxivCOND-MAT +12000 +10000 +of edges +8000 +# +6000 +4000 +2000 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Value of edge non-agreementArxiy COND-MAT +22500 +20000 +17500 +ofClusters +15000 +12500 +# +10000 +7500- +5000 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +wArxivCOND-MATDataset +175000 +150000 +125000 +100000 +Non-agreement edges +0 +-light edges +75000 +50000 +25000 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EArxivCOND-MATDataset +6000 +5000 +(sw) +4000 +Time +3000 +2000 +CC Time (ms) +ICC Time (ms) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +E(a) NonAgreements +(b) Number of clusters +(c) The number of NonAgreement and ε-light edges +(d) Clustering time in milliseconds +Figure 5: The graph Arxiv HEP-TH. +15 + +Non-aqreement distribution for Arxiv HEP-TH +60000 +50000 +E40000 +ebpa +0000E, +# +20000 +10000 +0 +00'0 +0.25 +0.500.751.001.251.50 +1.75 +2.00 +Value of edqe non-agreementArxiv HEP-TH +27500 +27000 +of Clusters +26500 +#26000 +25500 +25000 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EArxiyHEP-THDataset +700000 +600000 +500000 +400000 +Non-agreement edges +-light edges +300000 +200000 +100000 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EArxivHEP-THDataset +12000 +CC Time (ms) +ICC Time (ms) +11500 +11000 +(sw) +10500 +ne +wll +10000 +9500 +9000 +8500 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00(a) NonAgreements +(b) Number of clusters +(c) The number of NonAgreement and ε-light edges +(d) Clustering time in milliseconds +Figure 6: The graph Enron-Email. +16 + +Non-agreement distribution for Enron-Email +80000 +70000 +00009 +Dpe +50000 +40000 +# +30000 +20000 +10000 +0.00 +0.25 +0.500.75100125150 +0175 +2.00 +Value of edqe non-agreementEmail-Enron +37500 +35000 +32500 +30000 +27500 +of +# +25000 +22500 +20000 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EEmail-EnronDataset +350000 +300000 +250000 +200000 +Non-agreement edges +0 +-light edges +150000 +100000 +50000 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EEmail-EnronDataset +16000 +14000 +(sw) +Time +12000 +10000 +CC Time (ms) +8000 +ICC Time (ms) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00(a) NonAgreements +(b) Number of clusters +(c) The number of NonAgreement and ε-light edges +(d) Clustering time in milliseconds +Figure 7: The graph EU-Email. +17 + +Non-agreement distribution for EU-Emai +600000 +500000 +400000 +300000 +200000 +100000 +0 +0.00 +0.25 +0.500.751.001.251.50 +1.75 +2.00 +Value of edge non-agreementEmail-EU +265200 +265100 +265000 +264900 +264800 +264700 +264600 +264500 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EEU-EmailDataset +Non-agreement edges +700000 +-light edges +600000 +500000 +400000 +0 +300000 +200000 +100000 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +EEU-EmailDataset +810000 +CC Time (ms) +ICC Time (ms) +800000 +790000 +S +780000 +770000 +760000 +750000 +740000 +730000 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +E(a) NonAgreements +(b) Number of clusters +(c) The number of NonAgreement and ε-light edges +(d) Clustering time in milliseconds +Figure 8: The graph DBLP. +18 + +Non-agreement distribution for DBLP +175000 +150000 +125000 +100000 +75000 +50000 +25000 ++ 0 +00'0 +0.25 +0.500.75100125150 +175 +2.00 +Value of edqe non-agreementDBLP +300000 +250000 +200000 +150000 +100000 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +E1e6 +DBLPDataset +2.0 +1.5 +edges +Non-agreement edges +of +1.0 +-light edges +# +0.5 +0.0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +E1e6 +DBLPDataset +1.0 +0.8 +(sw) +0.6 +0.4 - +CC Time (ms) +0.2 +ICC Time (ms) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +E \ No newline at end of file diff --git a/WdAyT4oBgHgl3EQfhvjT/content/tmp_files/load_file.txt b/WdAyT4oBgHgl3EQfhvjT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d29ed89d5232b91820520913f13655ee710aa9c --- /dev/null +++ b/WdAyT4oBgHgl3EQfhvjT/content/tmp_files/load_file.txt @@ -0,0 +1,699 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf,len=698 +page_content='Correlation Clustering Algorithm for Dynamic Complete Signed Graphs: An Index-based Approach Ali Shakiba Department of Computer Science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='shakiba@vru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='ir;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='shakiba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='iran@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='com Abstract In this paper, we reduce the complexity of approximating the correlation clustering problem from O (m × (2 + α(G)) + n) to O (m + n) for any given value of ε for a complete signed graph with n vertices and m positive edges where α(G) is the arboricity of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Our approach gives the same output as the original algorithm and makes it possible to implement the algorithm in a full dynamic setting where edge sign flipping and vertex addition/removal are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Constructing this index costs O (m) memory and O (m × α(G)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' We also studied the structural properties of the non-agreement measure used in the approximation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The theoretical results are accompanied by a full set of experiments concerning seven real-world graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' These results shows superiority of our index-based algorithm to the non-index one by a decrease of %34 in time on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Keywords: Correlation clustering · Dynamic graphs · Online Algorithms 1 Introduction Clustering is one of the most studied problems in machine learning with various applications in analyzing and visualizing large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' There are various models and technique to obtain a partition of elements, such that elements belonging to different partitions are dissimilar to each other and the elements in the same partition are very similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The problem of correlation clustering, introduced in [1], is known to be an NP-hard problem for the disagree minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Therefore, several different approximation solutions based on its IP formulation exist in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Recently, the idea of a 2-approximation algorithm in [1] is extended in [4] for constructing a O (1)-approximation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The experiments in [4] show acceptable performance for this algorithm in practice, although its theoretical guarantee can be too high, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 1 442 for β = λ = 1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In [3], this algorithm is extended to an online setting where just vertex additions are allowed, and whenever a new vertex is added, it reveals all its positively signed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Shakiba in [12] studied the effect of vertex addition/removal and edge sign flipping in the underlying graph to the final clustering result, in order to make the algorithm suitable for dynamic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' However, one bottleneck in this way is computing the values of NonAgreement among the edges and identifying the ε-lightness of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The current paper proposes a novel indexing scheme to remedy this and make the algorithm efficient, not just in terms of dynamic graphs, but for even dynamic hyper-parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Our proposed method, in comparison with the online method of [3] is that we allow a full dynamic setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' vertex addition/removal and edge’s sign flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' It is known that any online algorithm for the correlation clustering problem has at least Ω(n)-approximation ratio [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Note that the underlying algorithm used in the current paper is consistent, as is shown via experimental results [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The rest of the paper is organized as follows: In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='1, we highlight our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This is followed by a reminding some basic algorithms and results in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, we introduce the novel indexing structure in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='1 and show how it can be employed to enhance the running-time of the approximate correlation clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, we show how to maintain the proposed indices in a full dynamic settings in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In Section 4, we give an extensive experiments which accompanies the 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00384v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='DS] 1 Jan 2023 theoretical results and show the effectiveness of the proposed indexing structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Finally, a conclusion is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='1 Our Contribution In this paper, we simply ask “How can one reduce the time to approximate a correlation clustering of the input graph [4] for varying values of ε?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' We also ask “How can we make the solution to the first question an online solution for dynamic graphs?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Our answer to the first question is devising a novel indexing-structure which is constructed based on the structural properties of the approximation algorithm and its NonAgreement measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' As our experiments in Section 4 show, the proposed method enhanced the total running-time of querying the clustering for about %34 on average for seven real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, we make this structure online to work with dynamic graphs based on theoretical results in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The construction of the index itself is highly parallelizable, up to the number of the vertices in the input graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The idea for parallelization is simple: construct each NAO (v) in the NAO (G) with a separate parallel thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' We also study the intrinsic structures in the NonAgreement measure, to bake more efficient algorithms for index-maintenance due to updates to the underlying graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' More precisely, we show that using the proposed index structure, we can find a correlation clustering for a graph for any given value of ε in time O (m + n), compared to the O (m × (2 + α(G)) + n) time for the CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The pre-processing time of the ICC would be O (m × α(G)) with O (m) space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 2 Preliminaries Let G = (V, E) be a complete undirected signed graph with |V | = n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The set of edges E is naturally partitioned into positive and negative signed edges, E+ and E−, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, we use m to denote |E+|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The correlation clustering problem is defined as cost(C) = � {u,v}∈E+ u∈Ci,v∈Cj,i̸=j 1 + � {u,v}∈E− u,v∈Ci 1, (1) where C = {C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' , Cℓ} is a clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Note that this is the min-disagree variant of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The constant factor approximation algorithm of [4] is based on two main quantities: (1) ε-agreement of a positively signed edge {u, v}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' u and v are in ε-agreement if and only if NonAgreementG (u, v) = |NG(u)∆NG(v)| max{|NG(u)|,|NG(v)|} < ε, and (2) ε-lightness, where a vertex u is said to be ε-light if AgreeCntG+(u) |NG+(u)| < ε where AgreeCntG+(u) = |{w ∈ V |u and v are in ε-agreement}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Note that a vertex which is not ε-light is called ε-heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This is a 2 + 4 ε + 1 ε2 -approximation algorithm, as is shown in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This algorithm is described in Algorithm 1, which we will refer to the CC algorithm, for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Shakiba in [12] studied theoretical foundation of the CC algorithm in a full dynamic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The following result is a summary of Table 1, Corollary 1, and Theorem 4 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Suppose the sign of an edge u = {u, v} is flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, the non-agreement and ε-lightness of vertices other than the ones whose distance to either u and v is more than two would not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The arboricity of the graph G is the minimum number of edge-disjoint spanning forests into which G can be decomposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The following lemma for arboricity is useful in bounding the number of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Lemma 2 in [2] Suppose the graph G = (V, E) has n vertices with m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, � {u,v}∈E min {degG(u), degG(v)} ≤ 2a(G) × m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' (2) 2 Algorithm 1 CorrelationClustering(G) [4] 1: procedure CorrelationClustering(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' ε) 2: Let G+ = G[E+] where E+ is the set of edges whose sign is + 3: Discard all edges whose endpoints are not in ε-agreement 4: Discard all edges between two ε-light vertices 5: Let � G+ be the sparsified graph G+ after performing previous two operations 6: Let C be the collection of connected components in � G+ 7: return C as the output clustering 8: end procedure 3 Proposed Method In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' we describe our novel indexing structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This structure allows dynamic queries of the correlation clustering with varying values of ε for dynamic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The proposed algorithm which uses the indexing structure would be called ICC, or indexed-based correlation clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='1 Indexing structure For an edge e = {u, v} with positive sign, we define its ε-agreement distance as NonAgreementG+ (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Intuitively, this is the infimum of the values ε which the nodes u and v are not in ε-agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Let define the set E = {NonAgreementG+ (u, v) |e = {u, v} ∈ E+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Without loss of generality, let E = {ε0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' , εℓ−1} with the ordering min E = ε0 < ε1 < · · · < εℓ−1 = max E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For a fixed value of ε, let G+ ε = (V, E+ ε ) where E+ ε = {e = {u, v} ∈ E+|NonAgreementG+ (u, v) < ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For all ε ≤ ε0, G+ ε is the null graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' a graph on all nodes without any edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Moreover, for all ε > εℓ, G+ ε = G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Next, we introduce the key ingredient to our indexing structure, called NAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Definition 1 (NonAgreement Node Ordering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The ε-agreement ordering for each node v ∈ V , denoted by NAO (v), is defined as an ordered subset of vertices in G where: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' node u ∈ V appears in the ordering NAO (v) if and only if e = {u, v} is a positive edge in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' for each two distinct vertices u, w ∈ V which appear in NAO (v), NonAgreementG+ (v, u) < NonAgreementG+ (v, w) , (3) implies u appears before w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' for each node u ∈ NAO (v), its ε-agreement distance is also stored with that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The NonAgreement node ordering of the graph G is defined as NAO (G) = {(v, NAO (v))|v ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In other words, the NAO (v) is a sorted array of neighboring nodes of v in G+ in their ε-agreement distance value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' An example NonAgreement node ordering for all vertices in a sample graph is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The space and construction time complexities of the NAO(G) are investigated in the next two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The NonAgreement node ordering for a graph G, NAO (G), can be represented in O (m) memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The number of nodes inside NAO (v) equals to the degG+(v) + 1, for all vertices in NG+(v) as well as the vertex v itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Note that it is not required to explicitly store the vertex v itself in the ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Cumulatively, the total size required for representing NAO (v) is 2 × m entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 3 v0 v1 v2 v3 v4 v5 v1 v5 v3 v2 v0 v1 v3 v0 v0 v4 v3 v0 v0 v1 v2 v3 v4 v5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='72 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='46 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='2 Figure 1: An illustrative example of NAO (v) for an example graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The time complexity to construct the NAO (v) for all vertices v ∈ V is O (m × (α(G) + lg m)) where α(G) is the arboricity of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' To compute the value of NonAgreementG+ (u, v) for all edges e = {u, v} ∈ E+, one needs to compute |NG+(u)∆NG+(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This requires O (degG+(u) + degG+(v)) operations to compute the symmetric difference, given the adjacency lists of u and v are sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' However, we can compute their intersection and use it to compute the NonAgreement as follows NonAgreementG+ (u, v) = degG+(u) + degG+(v) − 2 × |NG+(u) ∩ NG+(v)| max {NG+(u), NG+(v)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' (4) Hence, the total time required to compute the values of NonAgreement is equal to � {u,v}∈E+ min {degG+(u), degG+(v)} , which is known to be bounded by 2α(G) × m (Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Moreover, each NAO (v) is of length 1 + degG+(v) which requires sorting by the value of their NonAgreement in O (degG+(v) lg (degG+(v))), which accumu- lates to O (m lg m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The correlation clustering corresponding to a given value of ε and a graph G is the set of connected components of the graph � G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Given access to NAO (G) = {NAO (v) |v ∈ V (G)}, one can respond to the following queries: (1) Is the vertex v an ε-heavy vertex or an ε-light?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' (2) Are the endpoints of an edge {u, v} in ε-agreement?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' As we will show in this section, both of these questions can be answered in O (1) time using the NAO structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For a vertex v ∈ V , its ε-light threshold is defined as LTHε (v) = ⌈ε degG+(v)⌉ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For an ε and a vertex v ∈ V , v is ε-heavy if and only if the ε-agreement distance of the LTHε (v)-th smallest vertex in NAO (v) is less than ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Otherwise, it is ε-light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The vertices u and v would be in ε-agreement if and only if NonAgreementG+ (u, v) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Whenever the ε-agreement distance of the LTHε (v)-th smallest vertex in NAO (v) is less than ε, it means that there are at least ⌈ε degG+(v)⌉ + 1 vertices, including the v itself, in ε-agreement with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This is equivalent the ε- heavyness of the vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' On the other hand, if the ε-agreement distance of the LTHε (v)-th smallest vertex in NAO (v) is greater than or equal to ε, this is equivalent to that the number of vertices in ε-agreement with v is less than ε × degG+(v), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' v is ε-light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Given access to NAO (v), for all values of ε and any vertex v ∈ V , identifying the ε-lightness of v can be accomplished in O (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This is implied by Lemma 4, assuming that the NAO (v) is implemented as an array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Given access to the index NAO (G) = {(v, NAO (v))|v ∈ V } for a graph G, a query simply asks for a clustering of the graph for a given value of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Given access to the index NAO (G), computing a clustering for a given parameter ε can be accomplished in O (m (a(n) + 1)) amortized time, where a(n) is the slowly growing inverse of the single-valued Ackermann’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Intuitively, we are going to use NAO (G) to construct the graph � G+ and compute its connected components incrementally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' More formally, we start by putting each vertex to its isolated set in a disjoin- union structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, we identify ε-heavy vertices H ⊆ V , which takes O (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Next, consider NAO (v) for v ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For each vertex u ∈ NAO (v) whose key is smaller than ε and u ∈ H, we call Union(u, v), which merges the cluster which u and v belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' These operations takes O (m × a(n)) amortized time [5] using a Disjoint-Set data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The correctness of the query algorithm is implied by the Definition 1 and the Lemmas 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Before going any further, we need to state the following results, which intuitively investigate the behavior of the ε-agreement and ε-light of edges and vertices through the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='Let E = � NonAgreementG+ (u, v) | {u, v} ∈ E+� = {ε0, ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' , εℓ−1} , (5) where εi−1 < εi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' , ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Moreover, assume that εℓ > max E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For any ε ≤ ε0, the endpoints of all positive signed edges {u, v} are not in ε-agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For any ε > εℓ−1, the endpoints of all positive signed edges {u, v} are in ε-agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' By the Observation 2, the correlation clustering output by the Algorithm 1 for all values of ε ≤ ε0 would be the collection of singleton vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' {{v} |v ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' However, we cannot say that for ε > εell−1, the output is a single cluster, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' the set V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Why is that?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' As ε increases, the number of edges in ε-agreement would increase, however, the number of ε-light vertices would increase, too (By Corollary 1, which follows soon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Hence, there is a trade-off for choosing a suitable value of ε to get the minimum number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' We would discuss this issue further in Section 4 (Experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Let ε < ε′ and u, v ∈ V be two distinct vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' If u and v are in ε-agreement, then they would be in ε′-agreement, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Also, if u and v are not in ε′-agreement, then they would not be in ε-agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The proof is a direct implication of the NonAgreement definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Given that u and v are in ε-agreement, we have NonAgreementG+ (u, v) < ε < ε′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' u and v are in ε′-agreement, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Similarly, given that u and v are not in ε′-agreement, we have NonAgreementG+ (u, v) ≥ ε′ > ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' u and v are not in ε-agreement, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Let ε < ε′ and u ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' If u is ε-light, then u would be ε′-light, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Also, if u is ε′-heavy, then it would be ε-heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This is implied by the definition of ε-lightness and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Two other important cases are not considered in Theorem 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' (1) Is it possible that a vertex u be ε-heavy, but becomes ε′-light for some values ε < ε′?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=', and (2) Is it possible that a ε-light vertex becomes ε′-heavy for some values of ε < ε′?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The answer to both of these questions is affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' By Theorems 3 and 4 and the previous discussion, we can state the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Let ε < ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The number of positively signed edges whose endpoints are in ε′-agreement is greater than or equal to the number of positive edges whose endpoints are in ε-agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In other words, the agreement relation is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The number of vertices which are ε′-light can be either greater or less than or even equal to the number of ε-light vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In other words, the lightness relation is not necessarily monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The Baseline idea is to recompute the NonAgreement for each new value of ε, which takes O (m × α(G)), as is described in the proof of Lemma 2, deciding on the ε-heaviness of the vertices in G in time O (n) and computing the connected components of the graph � G+ as the output in time O (m + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Totally, the time complexity of the Baseline is O (m × (2 + α(G)) + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' To conclude, using the index structure we invest O (m) space (Lemma 2) and O (m × (α(G) + lg m)) (Lemma 3) time to construct the NAO (G) which make it possible to answer each query with varying ε values in time O (m(a(n) + 1)) (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Comparing this with O (m × (2 + α(G)) + n) time for the Baseline reveals that our index-based structure makes query times faster for variable values of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Given access to NAO, the algorithm 2 constructs the graph � G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Note that the output of the algorithms 1 and 2 are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' As answering whether a vertex v is ε-heavy or the endpoints of an edge {u, v} are in ε- agreement can be answered in O (1) time, the running-time of the algorithm 2 would be O (max {|V |, |E+|}) for the for loop and O � |V� G+| + |E+ � G+| � to compute the connected components of � G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Totally, the running- time of the query algorithm in the ICC would be O (m + n), compared to the O (m × (2 + α(G)) + n) time for the CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' To model queries over a graph for various values of ε, we use a ε-Schedule defined as ε-Schedule = {ε0 < ε1 < · · · < εℓ} ⊆ [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' (6) Note that we consider ε-Schedule as a set of strictly increasing real numbers, just for the sake of simplicity in notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In a real life scenario, any time one can ask for a clustering with any value of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Algorithm 2 The index-based correlation clustering algorithm with NAO (G) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 1: procedure IndexCorrelationClustering(G, NAO (G) , ε) 2: for all v ∈ V do 3: Use the NAO (() v) to identify and remove all the edges {v, u} which are in ε-NonAgreement 4: Use the NAO (v) to identify and remove edges {u, v} where u and v are both ε-light 5: end for 6: return The connected components of the remaining graph as the clustering output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 7: end procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='2 Maintaining the index The index structure introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='1 is used to compute a clustering of a static graph for user- defined dynamic values of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In this section, we revise this structure to make it suitable for computing a clustering of a dynamic graph for dynamic values of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' There are three different operations applicable to the underlying graph of the CorrelationClustering: (1) flipping the sign of an edge e = {u, v}, (2) adding a new vertex v, and (3) removing an existing vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' These operations are considered in Lemmas 6, 7, and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Shakiba in [12] has shown that flipping the sign of an edge {u, v}, the ε-agreement of edges whose both endpoints are not in the union of the positive neighborhood of its endpoints does not change (Propositions 2 and 4 in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Therefore, we just need to compute the ε-agreement for positive edges {x, w} where x, w ∈ (NG+(u) ∪ NG+(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Assuming the sign of an edge e = {u, v} is flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Assuming the sign of edge was + prior to the flipping, then u and v would be no more in ε-agreement since their edge is now negatively signed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The vertices v and u are removed from the arrays NAO (u) and NAO (v), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Moreover, we need to update the values of the non-agreement for vertices u and v in NAO (w) for all vertices w ∈ NAO (u) ∪ NAO (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Assuming the sign of edge was − prior to the flipping, then their NonAgreementG+ (u, v) is computed and the vertices v and u are added to proper sorted place in NAO (u) and NAO (v), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Moreover, we need to recompute the NonAgreementG+ (x, w) for all x, w ∈ NAO (u) ∪ NAO (v) and update their ε-agreement values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Applying these changes is applicable in O �� {x,w}∈E+∩X×X (log degG+(x) + log degG+(w)) � where X = NAO (u) ∪ NAO (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The correctness of this lemma follows from the discussion just before it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Therefore, we would just give the time analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In the first case, finding the vertices in each NAO is of O (log degG+(u) + log degG+(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The removal would be O (1) amortized time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The second case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' flipping the sign from − to +, would cost more, as it may require changing all the pos- itive edges with both endpoints the in the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In the worst case, one may assume that all the ε-agreement between the edges with both endpoints inside X is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Therefore, we need to update each of them on their corresponding NAO indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' We need to add vertices v and u to the NAO(u) and NAO(v), respectively, based on NonAgreementG+ (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Finding their correct position inside sorted NAOs is possible with O (log degG+(u) + log degG+(v)) comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Again, adding them at their corresponding position would take O (1) amortized time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For the other positive edges with both endpoints in the set X, such as {x, w}, we might need to update their ε-agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Without loss of generality, assume that degG+(x) ≤ degG+(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Doing so requires re-computation of NonAgreementG+ (x, w), in O (min {degG+(x), degG+(w)}), querying the NonAgreement inside NAO (x), in time O (log degG+(x)), and then possible updating its position, in both NAO (x) and NAO (w) requires O (log degG+(x) + log degG+(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In the worst case, all the elements in X may need update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Therefore, this case can be accomplished in O � � � {x,w}∈E+∩X×X (log degG+(x) + log degG+(w)) � � , in the worst-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Using Lemma 6, we can state the NAO-FlipEdge(u, v) algorithm (Algorithm 3) which flips the sign of the edge {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The correctness and the running-time of this algorithm follows directly from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' There are cases where a set of positive edges E+ v are also given for a newly added vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' One can first add Algorithm 3 The NAO-FlipEdge(u, v) algorithm to apply the effect of flipping the sign of the edge (u, v) in NAO (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 1: procedure NAO-FlipEdge(u, v) 2: Let A ← NAO (u) ∪ NAO (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 3: if Sign of edge {u, v} is positive then 4: Update the graph G+ by removing edge {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 5: Remove the vertices u and v from NAO (v) and NAO (u), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 6: else ▷ Sign of edge {u, v} is negative 7: Update the graph G+ by adding edge {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 8: Compute NonAgreementG+ (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 9: Add the vertices u and v to NAO (v) and NAO (u), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 10: end if 11: for all w ∈ A do 12: Recompute NonAgreementG+ (u, w) and update NAO (u) and NAO (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 13: Recompute NonAgreementG+ (v, w) and update NAO (v) and NAO (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 14: end for 15: end procedure 7 the vertex with all negative edges, and afterwards, flip all the edges in E+ v , so they would become positively signed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' However, a batch operation would give us higher performance in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Assume that a vertex v is added to graph G with new positive signed edges E+ v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' If E+ v = ∅, then NAO (G) does not need any updates and we just add a new NAO (v) to NAO (G) with v as its only element in constant-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Otherwise, let X = NG+(v)∪ � ∪x∈NG+(v)NG+(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' We need to compute the NonAgreementG+ (x, w) for each positive edges {x, w} with x, w ∈ X after adding all the new positive edges in E+ v , and possibly update NAO (x) and NAO (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This would require O � � � {x,w}∈(E+∪E+ v )∩X×X (log degG+(x) + log degG+(w)) � � , (7) operations in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The correctness would follow immediately from the Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Again, the first case can be accom- plished in O (1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For the second case, we need to calculate |X| values of NonAgreement, and add them or update them in their corresponding NAOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This would require O � � � {x,w}∈(E+∪E+ v )∩X×X (log degG+(x) + log degG+(w)) � � , (8) by exactly the same discussion as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In comparing the time required for Lemmas 6 and 7, please note that the size of the set X can be much larger in Lemma 7 than the one in Lemma 6, depending on the size of E+ v for the new vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The NAO-AddVertex(x, Nx) algorithm (Algorithm 4) adds a new vertex x to G+ and all its neighboring positively signed edges Nx, and updates the NAO (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The correctness and the running-time of this algorithm follows directly from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For the last operation, removing an existing vertex v with a set of positive Algorithm 4 The NAO-AddVertex(x, Nx) algorithm to add a new vertex x and all its positive neighbors Nx to NAO (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 1: procedure NAO-AddVertex(x, Nx) 2: Update graph G+ by adding the new vertex x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 3: Construct a new NAO (x) and add it to NAO (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 4: Let X ← Nx ∪ (∪z∈NxNG+(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 5: for all y ∈ X do 6: for all z ∈ X \\ y do 7: Update NAO (y) and NAO (z) if the NonAgreementG+ (y, z) changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 8: end for 9: end for 10: end procedure adjacent edges E+ v , can be accomplished by first flipping the sign of all of its adjacent edges in E+ v from + to −, and afterwards, removing its NAO (v) from NAO (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Similar to vertex addition, we can do it also in batch-mode, hoping for better performance in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The algorithm for the NAO-RemoveVertex(x) is similar to the NAO-AddVertex(x, Nx) algorithm (Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Assume that an existing vertex v is removed from the graph G with the set of adjacent positive signed edges E+ v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' If E+ v = ∅, then NAO (v) has a single element, itself, and can be easily removed from NAO (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Nothing else would require a change, so it is of constant-time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Otherwise, let X = NG+(v)∪ � ∪x∈NG+(v)NG+(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' We need to compute the NonAgreementG+ (x, w) for each positive edges {x, w} with x, w ∈ X after removing all the edges in E+ v , and possibly update NAO (x) and NAO (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In the worst-case, this would require O � � � {x,w}∈(E+∪E+ v )∩X×X (log degG+(x) + log degG+(w)) � � , (9) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The correctness of this lemma is implied by the Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The discussion of the running-time is exactly the same as in the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 4 Experiments To evaluate the proposed method, we used 7 graphs which are formed by user-user interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' These datasets are described in Table 1 and are accessible through the SNAP [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The smallest dataset consists of 22 470 nodes with 171 002 edges and the largest one consists of 317 080 nodes with 1 049 866 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In all these graphs, we consider the existing edges as positively signed and non-existing edges as negatively signed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The distribution of the NonAgreements for each dataset is illustrated in Figures 2a to 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The distri- bution of NonAgreements of the edges in almost all the datasets obeys the normal distribution, except small imperfections in Arxiv ASTRO-PH (Figure 2a), Arxiv COND-MAT (Figure 4a), and DBLP (Figure 8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Moreover, more detailed statistics on these distributions are given in Tables 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' One single observation is that the most frequent value of the NonAgreement in all the sample datasets is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Why is that?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Without loss of generality, assume that degG(u) ≥ degG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, |NG(v)| = 2 |NG(u) ∩ NG(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' By the assumption |NG(u)| ≥ 2 |NG(u) ∩ NG(v)|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' the intersection of the neighborhood of vertices u and v consists of at most half of the neighborhood of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Also, exactly half of the neighborhood of v falls at the intersection of the neighborhood of u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Intuitively, the vertex u has clustering preference with extra vertices at least the number of vertices which both u and v have clustering preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Similarly, the vertex v has clustering preference with exactly half extra vertices which both u and v have clustering preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For each dataset, we have used a different set of ε-Schedules, depending on the distribution of their NonAgreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' More precisely: (1) we have sorted the values of non-agreements in each dataset in a non-decreasing order, with repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' (2) Then, we have selected 21 distinct values equally spaces of these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' (3) The ε-Schedule was set to the selected values in the second step, which appended the value of 0 at the beginning and the value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='99 to the end if either does not exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Totally, the number of ε-Schedule for each dataset is either 21, 22 or 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' An interesting observation is the number of clusters for ε ≈ 1− in Figures 2b to 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Note that we use 1− to denote the interval [1 − ϵ, 1] for some non-zero constant ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' When ε ≈ 1 but less than 1, the number of clusters is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' As it gets closer to 1, the number of clusters increases with a much greater descent than the decrease in the number of clusters as it gets close to 1 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' As in Corollary 1, by increasing the ε, the number of vertices in ε-agreement is a non-decreasing function, which is confirmed by the plots in Figures 2a to 8a as the number of vertices in ε-non-agreement is given by a non-increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' By closer visual inspection of these figures, we can see that the shape of the plot for the number of ε-non-agreement vertices in all these graphs is almost the same, with inflection point around the value of ε ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This is due to the intrinsic nature of the NonAgreementG (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Similarly, the non-monotonicity result stated in Corollary 1 is observed in the same figures for the number of ε-light vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' By a visual inspection, the trend of the number of ε-light vertices for almost all datasets, except for the Arxiv HEP-TH (Figure 5a) and the EU-Email (Figure 7a), is the same: the number of ε-light vertices increases as ε increases up to some point (first interval), then decreases slightly (second interval), and 9 finally increases and would be asymptotically equal to the number of vertices in the graphs (last interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' For Arxiv HEP-TH and EU-Email, we have the same trend, however, the second interval is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Table 1: Description of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Dataset Nodes Edges Arxiv ASTRO-PH [7] 18 772 198 110 MUSAE-Facebook [11] 22 470 171 002 Arxiv COND-MAT [7] 23 133 93 497 Arxiv HEP-TH [6] 27 770 352 807 Enron-Email [9] 36 692 183 831 EU-Email [7] 265 214 420 045 DBLP [13] 317 080 1 049 866 Table 2: Statistics of NonAgreement in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Dataset Distinct Minimum Maximum Top 2 frequent values Arxiv ASTRO-PH 16 436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='015 873 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='967 21 1 — 9 664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='5 — 2 992 MUSAE-Facebook 12 988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='031 746 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='978 02 1 — 18 534 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 — 2 346 Arxiv COND-MAT 3 893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='044 444 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='964 91 1 — 12 018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='5 — 4 954 Arxiv HEP-TH 23 285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='086 956 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='976 19 1 — 24 852 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='333 33 — 4 910 Enron-Email 20 273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='090 909 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='954 55 1 — 31 704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='5 — 6 796 EU-Email 28 612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='117 647 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='990 52 1 — 441 520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='997 658 — 682 DBLP 11 611 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='013 793 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='981 13 1 — 167 590 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='5 — 67 238 All the algorithms for the naive and the proposed index-based correlation clustering algorithms are implemented in C++1 without any parallelization and the experiments are done using an Ubuntu 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='1 TLS with an Intel Core i7-10510U CPU @ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='80GHz with 12 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The time for running the naive correlation clustering algorithm (Algorithm 1), denoted here as CC, as well as the time for the index-based correlation clustering algorithm denoted as ICC, is given in Figures 2d to 8d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Note that the time reported for the ICC in these figures does not include the required time for constructing the NAOs, as they are constructed once and used throughout the ε-Schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The running-time to read the graph as well as constructing the NAOs is reported in Table 3 in milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The CC and ICC algorithms are the same, except that in CC, the non-agreement values of the edges and the ε-lightness of the vertices are computed for each given value of ε, however, in ICC these are computed and stored in the proposed NAOs structure once and used for clustering with respect to different values of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' As it can be observed in Figures 2d to 8d, the running time for the ICC, excluding the time to construct the NAOs for once, is largely smaller than the one for CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' On average, our approach for the described ε-Schedule lead to %25 decrease in clustering time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This enhancement comes at the cost of pre-computing the NAOs, which costs on average %34 of the time for a single run of CC, which is quite small and makes the ICC efficient in cases where one requires to have multiple clustering for various values of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 5 Conclusion In this paper, we proposed a novel indexing structure to decrease the overall running-time of an approximation algorithm for the correlation clustering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' This structure can be constructed in O (m × α(G)) time with O (m) memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Then, we can output a correlation clustering for any value of ε in O (m + n), compared with O (m × (2 + α(G)) + n) time complexity of the ordinary correlation clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Moreover, 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='com/alishakiba/Correlation-Clustering-Algorithm-for-Dynamic-Complete-Signed-Graphs-An-Index-based- Approach 10 Table 3: Time to construct the NAO for each dataset in milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Dataset Time to construct NAO (ms) Time to read graph (ms) Arxiv ASTRO-PH 1 677 543 MUSAE-Facebook 1 391 427 Arxiv COND-MAT 521 346 Arxiv HEP-TH 12 530 725 Enron-Email 2 498 525 EU-Email 4 479 958 DBLP 4 620 2 167 the proposed index can be efficiently maintained during updates to the underlying graph, including edge sign flip, vertex addition and vertex deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The theoretical results are accompanied with practical results in the experiments using seven real world graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' The experimental results show about %34 decrease in the running-time of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' A future research direction would be studying this algorithm in parallel frameworks such as Map-Reduce and make it scalable to very Big graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Another research direction would be enhancing the approximation guarantee of the algorithm, or devising more efficient algorithms in terms of approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' References [1] Nikhil Bansal, Avrim Blum, and Shuchi Chawla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Correlation clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Machine learning, 56(1):89–113, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' [2] Norishige Chiba and Takao 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densification laws, shrink- ing diameters and possible explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining, pages 177–187, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' [7] Jure Leskovec, Jon Kleinberg, and Christos Faloutsos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Graph evolution: Densification and shrinking diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' ACM transactions on Knowledge Discovery from Data (TKDD), 1(1):2–es, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' [8] Jure Leskovec and Andrej Krevl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' SNAP Datasets: Stanford large network dataset collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' http: //snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='edu/data, June 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' [9] Jure Leskovec, Kevin J Lang, Anirban Dasgupta, and Michael W Mahoney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Internet Mathematics, 6(1):29–123, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 11 (a) NonAgreements (b) Number of clusters (c) The number of NonAgreement and ε-light edges (d) Clustering time in milliseconds Figure 2: The graph Arxiv ASTRO-PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' [10] Claire Mathieu, Ocan Sankur, and Warren Schudy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Online correlation clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In 27th International Symposium on Theoretical Aspects of Computer Science-STACS 2010, pages 573–584, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' [11] Benedek Rozemberczki, Carl Allen, and Rik Sarkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Multi-scale attributed node embedding, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' [12] Ali Shakiba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Online correlation clustering for dynamic complete signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='07000, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' [13] Jaewon Yang and Jure Leskovec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' Defining and evaluating network communities based on ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' In Proceedings of the ACM SIGKDD Workshop on Mining Data Semantics, pages 1–8, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 12 Non-agreement distribution for Arxiv ASTRO-PH 17500 15000 12500 edges 10000 # 7500 5000 2500 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 Value of edge non-agreementArxiv ASTRO-PH 18000 16000 14000 lusters of Cl 12000 # 10000 8000 6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 wArxivASTRO-PHDataset 400000 Non-agreement edges light edges 350000 300000 250000 200000 0 150000 100000 50000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 EArxiyASTRO-PHDataset 5000 CC Time (ms) ICC Time (ms) 4500 4000 3500 Time 3000 2500 2000 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00(a) NonAgreements (b) Number of clusters (c) The number of NonAgreement and ε-light edges (d) Clustering time in milliseconds Figure 3: The graph MUSAE-Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=" 13 Non-agreement distribution for MUSAE-Facebook 30000 25000 20000 ↓0 # 15000 10000 5000 00'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 Value of edqe non-agreementMUSAE-Facebook 22000 21000 of Clusters 20000 # 19000 18000 17000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 EMUSAE-FacebookDataset 350000 300000 250000 200000 Non-agreement edges light edges 150000 100000 50000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 EMUSAE-FacebookDataset 7000 CC Time (ms) ICC Time (ms) 6500 (sw) 6000 Time 5500 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 E(a) NonAgreements (b) Number of clusters (c) The number of NonAgreement and ε-light edges (d) Clustering time in milliseconds Figure 4: The graph Arxiv COND-MAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 14 Non-agreementdistributionforArxivCOND-MAT 12000 10000 of edges 8000 # 6000 4000 2000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 Value of edge non-agreementArxiy COND-MAT 22500 20000 17500 ofClusters 15000 12500 # 10000 7500- 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 wArxivCOND-MATDataset 175000 150000 125000 100000 Non-agreement edges 0 light edges 75000 50000 25000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 EArxivCOND-MATDataset 6000 5000 (sw) 4000 Time 3000 2000 CC Time (ms) ICC Time (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 E(a) NonAgreements (b) Number of clusters (c) The number of NonAgreement and ε-light edges (d) Clustering time in milliseconds Figure 5: The graph Arxiv HEP-TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=" 15 Non-aqreement distribution for Arxiv HEP-TH 60000 50000 E40000 ebpa 0000E, # 20000 10000 0 00'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 Value of edqe non-agreementArxiv HEP-TH 27500 27000 of Clusters 26500 #26000 25500 25000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 EArxiyHEP-THDataset 700000 600000 500000 400000 Non-agreement edges light edges 300000 200000 100000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 EArxivHEP-THDataset 12000 CC Time (ms) ICC Time (ms) 11500 11000 (sw) 10500 ne wll 10000 9500 9000 8500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00(a) NonAgreements (b) Number of clusters (c) The number of NonAgreement and ε-light edges (d) Clustering time in milliseconds Figure 6: The graph Enron-Email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 16 Non-agreement distribution for Enron-Email 80000 70000 00009 Dpe 50000 40000 # 30000 20000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75100125150 0175 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 Value of edqe non-agreementEmail-Enron 37500 35000 32500 30000 27500 of # 25000 22500 20000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 EEmail-EnronDataset 350000 300000 250000 200000 Non-agreement edges 0 light edges 150000 100000 50000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 EEmail-EnronDataset 16000 14000 (sw) Time 12000 10000 CC Time (ms) 8000 ICC Time (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00(a) NonAgreements (b) Number of clusters (c) The number of NonAgreement and ε-light edges (d) Clustering time in milliseconds Figure 7: The graph EU-Email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content=' 17 Non-agreement distribution for EU-Emai 600000 500000 400000 300000 200000 100000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 Value of edge non-agreementEmail-EU 265200 265100 265000 264900 264800 264700 264600 264500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='25 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} +page_content='00 E' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQfhvjT/content/2301.00384v1.pdf'} diff --git a/YdE0T4oBgHgl3EQf3wJf/content/tmp_files/2301.02729v1.pdf.txt b/YdE0T4oBgHgl3EQf3wJf/content/tmp_files/2301.02729v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..46b042d27cb38df29dd02216a393cd0c13ff61a3 --- /dev/null +++ b/YdE0T4oBgHgl3EQf3wJf/content/tmp_files/2301.02729v1.pdf.txt @@ -0,0 +1,2360 @@ +arXiv:2301.02729v1 [cs.LG] 6 Jan 2023 +A Characterization of Multilabel Learnability +Vinod Raman, Unique Subedi, Ambuj Tewari +Abstract +We consider the problem of multilabel classification and investigate learnability in batch and +online settings. In both settings, we show that a multilabel function class is learnable if and only +if each single-label restriction of the function class is learnable. As extensions, we also study mul- +tioutput regression in the batch setting and bandit feedback in the online setting. For the former, +we characterize learnability w.r.t. Lp losses. For the latter, we show a similar characterization as +in the full-feedback setting. +1 +Introduction +Multilabel classification is a learning problem where multiple labels can be assigned to an instance. This +is a generalization of multiclass classification, where an instance is classified into one of the multiple +possible labels. Multilabel classification has enjoyed a wide range of practical applications like image +tagging, document categorization, and recommender systems, to name a few. This widespread appli- +cability has motivated the development of several practical methods [KVJ12, YWJZ20, NLMKF17], as +well as theoretical analysis [KNRD15, LT15]. However, the most fundamental question of learnability +in a multilabel setting remains unanswered. In this work, we address this question by characterizing +multilabel learnability in two major learning settings: batch and online learning. +Characterizing learnability is the foundational step toward understanding any statistical learning +problem. +The fundamental theorem of statistical learning characterizes the learnability of binary +function class in terms of the finiteness of a combinatorial quantity called the Vapnik-Chervonenkis +(VC) dimension [VC71, VC74]. Extending VC theory, [Nat89] proposed and studied the Natarajan +dimension, which was later shown by [BDCBL92] to characterize learnability in multiclass settings +with a finite number of labels. Recent work by [BCD+22] shows that Daniely-Shwartz (DS) dimension, +originally proposed by [DSS14], characterizes multiclass learnability in infinite labels setting. Similarly, +in an online setting, the Littlestone dimension [Lit87] characterizes the online learnability of binary +function class. And a generalization of the Littlestone dimension [DSBDSS11] characterizes online +learnability in a multiclass setting with finite labels. Nevertheless, to our best knowledge, no such +characterization of the learnability of multilabel function classes exists in the literature. +To make our problem precise, let X be the instance space and Y = {−1, 1}K the target space for +some K ∈ N. The main question we study is: what characterizes the learnability of the function class +F ⊂ YX ? For each k ∈ {1, 2, . . ., K}, define a scalar-valued function class Fk = {fk | (f1, . . . , fK) ∈ F} +by restricting each function in F to its kth coordinate output. The following informal version of our +main result characterizes the learnability of F in terms of the learnability of each component class Fk. +Theorem. (Informal) The function class F ⊂ YX is learnable if and only if each of Fk ⊂ YX +k +is +learnable. +We prove a version of this result in both the batch and online settings. As extensions, we also prove +a version of this result for batch regression and bandit online settings, considering the appropriate +definition of learnability. +A unifying theme throughout all four learning settings is our ability to +constructively convert a learning algorithm A for F, into a learning algorithm Ak for Fk for each +k ∈ {1, ..., K} and vice versa. Thus, we do not study combinatorial dimensions, but rather take a more +direct, algorithmic approach to address the learnability of F. +1 + +2 +Preliminaries +Let X denote the instance space and Y = {−1, +1}K be the target space for some K ∈ N, unless +otherwise stated. +Consider a vector-valued function class F ⊂ YX , where YX denotes set of all +functions from X to Y. +Define a scalar-valued function class Fk = {fk | (f1, . . . , fK) ∈ F} by +restricting each function in F to its kth coordinate output. Here, each Fk ⊂ YX +k , where Yk denotes +the restriction of the target space to its kth component. Conveniently, we will write F = (F1, . . . , FK) +and Y = (Y1, . . . , YK). We denote ℓ : Y × Y → R≥0 to be some bounded, non-negative loss function. +For a function f ∈ F, we will use fk(x) to denote the kth coordinate output of f(x). On the other +hand, we will use yk to denote kth coordinate of y ∈ Y. As usual, [N] is used to denote a set of natural +numbers up to N, that is {1, 2, . . . , N}. Throughout the paper we will focus on loss functions that +also satisfy the identity of indiscernibles, defined formally below. +Definition 1 (Identity of Indiscernibles). A loss function ℓ : Y × Y → R≥0 satisfies the identity of +indiscernibles if ℓ(y1, y2) = 0 if and only if y1 = y2. +At a high-level, the identity of indiscernibles guarantees that the loss functions we consider are zero- +aligned. That is, if ℓ1 and ℓ2 are two multilabel loss functions that satisfy the identity of indiscernibles, +then ℓ1(y1, y2) = 0 if and only if ℓ2(y1, y2) = 0. +2.1 +Batch Setting +In the batch setting, we are interested in characterizing the learnability of F under the classical PAC +model [Val84]. +Definition 2 (Agnostic PAC Learnability). A function class F is agnostic PAC learnable w.r.t. loss +ℓ : Y×Y → R≥0, if there exists a function m : (0, 1)2 → N and a learning algorithm A : (X ×Y)∗ → YX +with the following property: for every ǫ, δ ∈ (0, 1) and for every distribution D on X × Y, running +algorithm A on n ≥ m(ǫ, δ) iid samples from D outputs a predictor g = A(S) such that with probability +at least 1 − δ over S ∼ Dn, +ED[ℓ(g(x), y)] ≤ inf +f∈F ED[ℓ(f(x), y)] + ǫ. +Note that we do not require the output predictor A(S) to be in F, but only require A(S) to compete +with the best predictor in F. If we restrict D to a class of distributions such that inff∈F ED[ℓ(f(x), y)] = +0 and, then we get realizable PAC learnability. +Definition 3 (Realizable PAC Learnability). A function class F is realizable PAC learnable w.r.t. loss +ℓ : Y×Y → R≥0, if there exists a function m : (0, 1)2 → N and a learning algorithm A : (X ×Y)∗ → YX +with the following property: for every ǫ, δ ∈ (0, 1) and for every distribution D on X × Y where +inff∈F ED[ℓ(f(x), y)] = 0 , running algorithm A on n ≥ m(ǫ, δ) iid samples from D outputs a predictor +g = A(S) such that with probability at least 1 − δ over S ∼ Dn, +ED[ℓ(g(x), y)] ≤ ǫ. +It is well known that for binary function classes with 0-1 loss, realizable learnability and agnostic +learnability are equivalent [SSBD14, Theorem 6.7]. For equivalence between realizable and agnostic +learnability for a more general class of loss function and target spaces, we recall the following recent +result. +Lemma 1 ([HKLM22]). Let F be a function class from instance space X to a finite target space +Y. Consider a general loss function ℓ : Y × Y → R≥0 that satisfies identity of indiscernible, that is +ℓ(y1, y2) = 0 if and only if y1 = y2. Then, the following are equivalent: +(1) F is realizable PAC learnable w.r.t. ℓ. +(2) F is agnostic PAC learnable w.r.t. ℓ. +2 + +2.2 +Online Setting +In the online setting, we place no distributional assumptions on the set of labeled instances that the +learner may observe. Instead, an adversary plays a sequential game with the learner over T rounds. +In each round t ∈ [T ], an adversary selects a labeled instance (xt, yt) ∈ X × Y and reveals xt to +the learner. The learner makes a (potentially randomized) prediction ˆyt ∈ Y. Finally, the adversary +reveals the true label yt, and the learner suffers the loss ℓ(yt, ˆyt), where ℓ is some pre-specified loss +function. Given a function class F ⊆ YX , the goal of the learner is to output predictions ˆyt such +that it’s cumulative loss is close to the best possible cumulative loss over functions in F. A function +class is online learnable if there exists an algorithm such that for any sequence of labeled examples +(x1, y1), ..., (xT , yT ), the difference in cumulative loss between its predictions and the predictions of +the best possible function in F is small. Definition 4 makes this precise and formally defines online +learnability. +Definition 4 (Online Agnostic Learnability). A function class F is online learnable w.r.t. loss ℓ, if +there exists an (potentially randomized) algorithm A such that for any adaptively chosen sequence of +labelled examples (xt, yt) ∈ X ×Y, the algorithm outputs A(xt) ∈ Y at every iteration t ∈ [T ] such that +E +� T +� +t=1 +ℓ(A(xt), yt) − inf +f∈F +T +� +t=1 +ℓ(f(xt), yt) +� +≤ R(T ) +where the expectation is taken w.r.t. the randomness of A and that of the possibly adaptive adversary, +and R(T ) : N → R+ is the additive regret: a non-decreasing, sub-linear function of T . +If for the sequence of labelled examples, there exists a f ∈ F s.t for all t ∈ [T ], f(xt) = yt, then +we say that the sequence is realizable. If it is further guaranteed that the learner always observes a +realizable sequence, then we say we are in the realizable setting. Definition 5 then provides a formal +statement on what it means for a function class to be online learnable under realizability. +Definition 5 (Online Realizable Learnability). A function class F is online learnable w.r.t. loss ℓ +under realizability, if there exists an (potentially randomized) algorithm A such that for any realizable +sequence of labelled examples (xt, yt) ∈ X × Y, the algorithm outputs A(xt) ∈ Y at every iteration +t ∈ [T ] such that +E +� T +� +t=1 +ℓ(A(xt), yt) +� +≤ R(T ) +where the expectation is taken w.r.t. the randomness of A, and R(T ) : N → R+ is the additive regret: +a non-decreasing, sub-linear function of T . If A is deterministic and R(T ) = M for some M ∈ N, +then we say A is a mistake-bound online learner. +In many situations, however, the true label yt is not revealed to the learner in each round t ∈ [T ]. +Instead, the adversary only reveals the loss ℓ(ˆyt, yt) that the learner has incurred in this round. If ℓ is +the 0-1 loss, then this means that the learner does not get to observe where it made a mistake, only +that it made a mistake. This type of partial feedback is commonly referred to as bandit information. +Definition 6 defines online learnability under bandit feedback. +Definition 6 (Bandit Online Learnability). A function class F is bandit online learnable w.r.t. loss +ℓ, if there exists a randomized algorithm A such that for any adaptively chosen sequence of labelled +examples (xt, yt) ∈ X × Y, under bandit feedback, the algorithm outputs A(xt) ∈ Y at every iteration +t ∈ [T ] such that +E +� T +� +t=1 +ℓ(A(xt), yt) − inf +f∈F +T +� +t=1 +ℓ(f(xt), yt) +� +≤ R(T ) +where the expectation is taken w.r.t. the randomness of A and that of the possibly adaptive adversary, +and R(T ) : N → R+ is the additive regret: a non-decreasing, sub-linear function of T . +3 + +3 +Batch Multilabel Classification +In this section, we study learnability in batch multilabel classification settings. First, we consider +learnability with respect to a natural decomposable loss. Then, we extend the result to more general +non-decomposable losses. +3.1 +Characterizing Batch Learnability for the Hamming Loss +A canonical and natural loss function for multilabel classification is the Hamming loss, defined as +ℓH(f(x), y) := +K +� +i=1 +1 +� +fi(x) ̸= yi� +, +where f(x) = (f1(x), . . . , fK(x)) and y = (y1, . . . , yK). The following result establishes an equivalence +between the learnability of F w.r.t. Hamming loss and the learnability of each Fk w.r.t. 0-1 loss. +Theorem 2. The function class F ⊂ YX is agnostic PAC learnable w.r.t. the Hamming loss ℓH if +and only if each of Fk ⊂ YX +k is agnostic PAC learnable w.r.t. the 0-1 loss. +Proof. (of sufficiency in Theorem 2) We will first prove that the agnostic PAC learnability of each Fk +is sufficient for agnostic PAC learnability of F. Our proof will be constructive: given oracle access to +agnostic PAC learners Ak for each Fk w.r.t. 0-1 loss, we will construct an agnostic PAC learner A for +F w.r.t. ℓH. +Algorithm 1 Agnostic PAC Learner for F w.r.t. ℓH +Input: Agnostic PAC learners {Ak}K +k=1 for Fk’s and samples S = {(xi, yi)}n +i=1 ∼ Dn on X × Y. +1 Construct marginal Sk = {(xi, yk +i )}n +i=1 for all k ∈ [K] with scalar-valued targets for Ak’s. +2 Get hypothesis hk = Ak(Sk) for all k ∈ [K]. +3 Output h = (h1, . . . , hK). +Note that Algorithm 1 could be improper as the predictor h may not necessarily be in F. In fact, +each of the algorithms Ak could be improper. Next, we will show that Algorithm 1 is an agnostic PAC +learner for F w.r.t. ℓH. Denote Dk to be the marginal distribution of D restricted to X × Yk. Let +mk(ǫ, δ) denote the sample complexity of Ak. Since Ak is an agnostic PAC learner for Fk’s, we have +that for n ≥ maxk mk( ǫ +K , δ +K ), with probability at least 1 − δ/K over samples Sk ∼ Dn +k , +EDk +� +1 +� +hk(x) ̸= yk�� +≤ +inf +fk∈Fk EDk +� +1 +� +fk(x) ̸= yk�� ++ ǫ +K . +Summing these risk bounds over all coordinates k and using union bounds over probabilities, we get +that with probability at least 1 − δ over samples S ∼ Dn, +K +� +k=1 +EDk +� +1 +� +hk(x) ̸= yk�� +≤ +K +� +k=1 +inf +fk∈Fk EDk +� +1 +� +fk(x) ̸= yk�� ++ ǫ. +Now using the fact that the sum of infimums is at most the infimum of sums followed by the +linearity of expectation gives +ED +� K +� +k=1 +1 +� +hk(x) ̸= yk� +� +≤ inf +f∈F ED +� K +� +k=1 +1 +� +fk(x) ̸= yk� +� ++ ǫ. +This completes our proof of sufficiency as it shows that Algorithm 1 is an agnostic PAC learner for F +w.r.t. ℓH with sample complexity at most maxk mk(ǫ/K, δ/K). +■ +Proof. (of necessity in Theorem 2) Next, we will show that if F is learnable w.r.t. ℓH, then each Fk +is PAC learnable w.r.t. the 0-1 loss. Our proof is again based on reduction: given oracle access to +4 + +Algorithm 2 Agnostic PAC learner for F1 w.r.t. 0-1 loss +Input: Agnostic PAC learner A for F and samples S = {(xi, yi1)}n +i=1 ∼ Dn +1 +1 Augment the sample S to create a K-variate target, where the augmented sample is +�S += +{(xi, (y1 +i , . . . , yK +i ))}n +i=1 such that yik ∼ {−1, 1} each with probability 1/2 for all i ∈ [n] and +k ∈ {2, . . . , K} +2 Output h1 = A1(�S), the restriction of A(�S) to its first output coordinate. +agnostic PAC learner A for F, we will construct agnostic PAC learners A1, stated as Algorithm 2, for +F1. By symmetry, similar construction can be used for all other Fk’s. +Let us consider a distribution �D on X × Y such that a sample (x, (y1, . . . , yK)) from �D is obtained +by first sampling (x, y1) ∼ D1 and appending yk’s sampled independently from uniform distribution +on {−1, 1} for each k ∈ {2, . . ., K}. +For the sake of analysis, let us also denote hk = Ak(�S) to +be restrictions of A(�S) from Algorithm 2 to its kth coordinate. Let m(ǫ, δ, K) denote the sample +complexity of A. Since A is an agnostic PAC learner for F, for n ≥ m(ǫ, δ, K), with probability at +least 1 − δ, we have +E � +D +� K +� +k=1 +1 +� +hk(x) ̸= yk� +� +≤ inf +f∈F E � +D +� K +� +k=1 +1 +� +fk(x) ̸= yk� +� ++ ǫ. +For k ≥ 2, since the target is chosen uniformly at random from {−1, 1}, the 0-1 risk of any predictor +is 1/2. Therefore, the expression above can be written as +ED1 +� +1 +� +h1(x) ̸= y1�� ++ +K +� +k=2 +1/2 ≤ inf +f∈F +� +ED1 +� +1 +� +f1(x) ̸= y1�� ++ +K +� +k=2 +1/2 +� ++ ǫ, +which upon cancellation of constant factors reduces to +ED1[1 +� +h1(x) ̸= y1� +] ≤ +inf +f1∈F1 ED1[1 +� +f1(x) ̸= y1� +] + ǫ. +Therefore, Algorithm 2 is an agnostic PAC learner for F1 w.r.t ℓ0-1 with sample complexity at most +m(ǫ, δ, K). +■ +3.2 +Characterizing Batch Learnability for General Losses +In Theorem 2, we characterized the learnability of a multilabel classifier w.r.t. the Hamming loss. Now, +we characterize the learnability of general multilabel losses that satisfy the identity of indiscernibles, +namely ℓ(y1, y2) = 0 if and only if y1 = y2. +Lemma 3. Let ℓ be a multilabel loss that satisfies the identity of indiscernibles. A function class +F ⊂ YX is agnostic PAC learnable w.r.t. ℓ if and only if F ⊂ YX is agnostic PAC learnable w.r.t. +Hamming loss ℓH. +Proof. (of sufficiency in Lemma 3) We will show that if F is learnable w.r.t. Hamming loss ℓH, then +F is learnable w.r.t. ℓ. First, we show this for any realizable distribution D w.r.t. ℓ. Since ℓ = 0 +if and only if ℓH = 0, the distribution D is also realizable w.r.t. Hamming loss. Furthermore, since +there are at most 22K distinct possible inputs to ℓ(·, ·), the loss function can only take a finite number +of values. So, for any ℓ, we can always find universal constants a and b such that aℓH ≤ ℓ ≤ bℓH. +Since F is learnable w.r.t ℓH, there exists a learning algorithm A with the following property: for any +ǫ, δ > 0, for a sufficiently large S ∼ Dn, the algorithm outputs a predictor h = A(S) such that, with +probability 1 − δ over S ∼ Dn, +ED[ℓH(h(x), y)] ≤ ǫ +b. +This inequality upon using the fact that ℓ(h(x), y) ≤ bℓH(h(x), y) pointwise reduces to ED[ℓ(h(x), y)] ≤ +ǫ. Therefore, any realizable learner A for ℓH is also a realizable learner for ℓ. Since ℓ satisfies the +identity of indiscernible, Lemma 1 guarantees the existence of agnostic PAC learner B for F w.r.t. +5 + +Algorithm 3 Realizable to agnostic reduction for F w.r.t. ℓ +Input: Realizable PAC learner A for F, unlabeled samples SU, labeled samples SL +1 Run A over all possible labelings of SU by F to get a concept class +C(SU) = {A(SU, f(SU)) | f ∈ F|SU }. +2 Return a predictor ˆh ∈ C(SU) with lowest empirical error on SL, +ˆh = arg min +h∈C(SU) +1 +|SL| +� +(x,y)∈SL +ℓ(h(x), y). +ℓ. In particular, the agnostic PAC learner B is Algorithm 1 in [HKLM22], which we restate below in +Algorithm 3 for completion. +We refer the reader to [HKLM22] for the complete analysis of the Algorithm 3. That said, we +now give a high-level idea of why it works. Suppose A is a realizable PAC learner for F w.r.t. ℓ +with sample complexity mA(ǫ, δ, K). Then, for any function f ∈ F, given an unlabeled sample SU of +size mA(ǫ/2, δ/2, K), running A on the labeled sample (SU, f(SU)) guarantees that, with probability +1 − δ/2 over SU ∼ DX , +EDX [ℓ(f(x), f ′(x))] ≤ ǫ/2, +where f ′ = A +� +(SU, f(SU)) +� +. This subsequently guarantees that running A over all possible labelings +generates a function class C(SU) with the property that for every f ∈ F, with high probability, there +exists f ′ ∈ C(SU) such that the risk of f and f ′ are close under D. So, with high probability, C(SU) +must contain a function ˜f whose risk is close to that of the optimal predictor f ⋆ in F for D. Since +C(SU) is a finite function class, running ERM over it with sufficiently large labeled samples SL should +return a predictor h whose risk is arbitrarily close to that of ˜f, and thus close to the optimal predictor. +One can formalize this argument using Hoeffding’s bound to show that for |SL| ≥ O +� +log (|C(SU)|/δ) +ǫ2 +� +, +we obtain with probability 1 − δ, +ED +� +ℓ(ˆh(x), y) +� +≤ inf +f∈F ED [ℓ(f(x), y)] + ǫ. +Together, the sample complexity of algorithm B, denoted as mB(ǫ, δ, K), is the sample complexity +of A plus the number of samples required for empirical risk minimizer of step 2 to generalize well. +That is, +mB(ǫ, δ, K) ≤ mA(ǫ/2, δ/2, K) + O +� 1 +ǫ2 log |C(SU)| +δ +� +≤ mA(ǫ/2, δ/2, K) + O +�mA(ǫ/2, δ/2, K) K + log 1 +δ +ǫ2 +� +where the last step follows upon using the fact that |C(SU)| ≤ 2mA(ǫ/2,δ/2,K) K. Therefore, we first +showed that an agnostic PAC learner A of F w.r.t. ℓH is a realizable PAC learner for F w.r.t. ℓ, and +then provided a black-box reduction of A to B, an agnostic PAC learner for F w.r.t. ℓ. +■ +Proof. (of necessity in Lemma 3) Next, we will show that if F is learnable w.r.t. ℓ, then F is learnable +w.r.t. Hamming ℓH. Our proof will follow a similar route as in the proof for sufficiency. First, we +show this for any realizable distribution D w.r.t. ℓH. Due to the alignment of 0, the distribution +D is also realizable w.r.t. ℓ. As both ℓH and ℓ take at most finite number of values, we can find +universal constants a and b such that aℓH ≤ ℓ ≤ bℓH. Since F is learnable w.r.t ℓ, there exists a +learning algorithm A with the following property: for any ǫ, δ > 0, for a sufficiently large S ∼ Dn, the +algorithm outputs a predictor h = A(S) such that, with probability 1 − δ over S ∼ Dn, +ED[ℓ(h, (x, y))] ≤ aǫ. +6 + +This inequality upon using the fact that aℓH(h, (x, y)) ≤ ℓ(h, (x, y)) pointwise yields ED[ℓH(h, (x, y))] ≤ +ǫ. Therefore, any realizable learner A for ℓ is also a realizable learner for ℓH. Since ℓH satisfies the +identity of indiscernibles, Lemma 1 guarantees the existence of agnostic PAC learner B for F w.r.t. +ℓH. In fact, the algorithm B is Algorithm 3 after replacing ℓ with ℓH. If mA(ǫ, δ, K) is the sample +complexity of A, then a similar argument as in the sample complexity analysis of the sufficiency +direction gives that the sample complexity of B is +mB(ǫ, δ, K) ≤ mA(ǫ/2, δ/2, K) + O +�mA(ǫ/2, δ/2, K) K + log 1 +δ +ǫ2 +� +. +■ +As an immediate consequence of Lemma 3 and Theorem 2, we can deduce the following result. +Theorem 4. Let ℓ be a multilabel loss function satisfying the identity of indiscernibles. A function +class F ⊂ YX is agnostic learnable w.r.t. ℓ if and only if each restriction Fk ⊂ YX +k is agnostic PAC +learnable w.r.t. the 0-1 loss. +4 +Online Multilabel Classification +In this section, we provide analogs of Theorem 2 and 4 in the online setting. Like before, we begin +by characterizing the learnability of the Hamming loss and then move to give a characterization of +learnability for all losses satisfying the identity of indiscernibles. Throughout this section, we give +regret bounds assuming an oblivious adversary. A standard reduction (see Chapter 4 in [CBL06]) +allows us to convert oblivious regret bounds to adaptive regret bounds. +4.1 +Characterizing Online Learnability for the Hamming Loss +Theorem 5 presents the main result of this subsection. +Theorem 5. A function class F ⊂ YX is online learnable w.r.t. the Hamming loss if and only if each +restriction Fk ⊂ YX +k is online learnable w.r.t. the 0-1 loss. +Proof. (of sufficiency for Theorem 2) We first prove that online learnability of each restriction is +sufficient for online learnability of ℓH. Our proof is based on a reduction: given oracle access to online +learners for {Fk}K +k=1 w.r.t. ℓ0-1, we will construct an online learner A for F w.r.t. ℓH. +Algorithm 4 Online Learner A for F w.r.t. ℓH +Input: Online learners {Ak}K +k=1 for Fk’s +1 for t = 1, ..., T do +2 +Receive example xt +3 +Predict ˆyt = (A1(xt), ..., AK(xt)) +4 +Receive true label yt = (y1 +t , ..., yK +t ) and suffer loss ℓM(ˆyt, yt) +5 +Update Ak by passing (xt, yk +t ) for k ∈ [K] +6 end +It suffices to show that the expected regret of A is sublinear in T w.r.t. ℓH. By Definition 4, we +have that for all k ∈ [K], +E +� T +� +t=1 +1{Ak(xt) ̸= yk +t } − +inf +fk∈Fk +T +� +t=1 +1{fk(xt) ̸= yk +t } +� +≤ Rk(T ) +where Rk(T ) is some sublinear function in T . Summing the regret bounds across all k ∈ [K] splitting +up the expectations, and using linearity of expectation, we get that +E +� K +� +k=1 +T +� +t=1 +1{Ak(xt) ̸= yk +t } +� +− E +� K +� +k=1 +inf +fk∈Fk +T +� +t=1 +1{fk(xt) ̸= yk +t } +� +≤ +K +� +k=1 +Rk(T ). +7 + +Noting that �K +k=1 inffk∈Fk +�T +t=1 +1{fk(xt) ̸= yk +t } ≤ inff∈F +�K +k=1 +�T +t=1 +1{fk(xt) ̸= yk +t }, the bound +above reduces to +E +� K +� +k=1 +T +� +t=1 +1{Ak(xt) ̸= yk +t } +� +− E +� +inf +f∈F +� K +� +k=1 +T +� +t=1 +1{fk(xt) ̸= yk +t } +�� +≤ +K +� +k=1 +Rk(T ). +Swapping the order of summations, noting that A(xt) = (A1(xt), ..., AK(xt)), and using the definition +of Hamming loss we have that, +E +� T +� +t=1 +ℓH(A(xt), yt) +� +− E +� +inf +f∈F +T +� +t=1 +ℓH(f(xt), yt) +� +≤ +K +� +k=1 +Rk(T ). +This concludes the proof of this direction since �K +k=1 Rk(T ) is still a sublinear function in T . +■ +Proof. (of necessity for Theorem 2) Next, we prove that online learnability of each restriction is nec- +essary for online learnability of ℓH. Our proof again is based on a reduction: given oracle access to an +online learner for F w.r.t. ℓH, we will construct online learners for {Fk}K +k=1 w.r.t. ℓ0-1. The Algorithm +below makes this precise by constructing an online learner for F1. +Algorithm 5 Online Learner for F1 w.r.t. ℓ0-1 +Input: Online learner B for F w.r.t. ℓH +1 for t = 1, ..., T do +2 +Receive example xt +3 +Predict ˆy1 +t = B1(xt) +4 +Receive true label y1 +t and suffer loss +1{ˆy1 +t ̸= y1 +t )} +5 +Update B by passing (xt, (y1 +t , y2 +t , ..., yK +t )), where yk +t ∼ {−1, 1}, each with probability 1/2, for +k ∈ {2, ..., K} +6 end +It suffices to show that the expected regret of A1 is sublinear in T . As previously mentioned, we +assume that the sequence (x1, y1 +1), ..., (xT , y1 +T) is chosen by an oblivious adversary, and thus is not +random. Let yt = (y1 +t , y2 +t , ..., yK +t ). By Definition 4, we have that, +E +� T +� +t=1 +ℓH(B(xt), yt) − inf +f∈F +T +� +t=1 +ℓH(f(xt), yt) +� +≤ R(T, K) +where the expectation is over both the randomness of B(xt) and (y2 +t , ..., yK +t ) and R(T, K) is a sub- +linear function of T . Splitting up the expectation, using the definition of the Hamming loss, and by +the linearity of expectation, we have that +T +� +t=1 +K +� +k=1 +E +� +1{Bk(xt) ̸= yk +t } +� +− E +� +inf +f∈F +T +� +t=1 +K +� +k=1 +1{fk(xt) ̸= yk +t } +� +≤ R(T, K). +Since E +� +inff∈F +�T +t=1 +�K +k=1 +1{fk(xt) ̸= yk +t } +� +≤ inff∈F E +��T +t=1 +�K +k=1 +1{fk(xt) ̸= yk +t } +� +, we get that +T +� +t=1 +K +� +k=1 +E +� +1{Bk(xt) ̸= yk +t } +� +− inf +f∈F +T +� +t=1 +K +� +k=1 +E +� +1{fk(xt) ̸= yk +t } +� +≤ R(T, K). +Next, observe that for every t ∈ [T ], for every k ∈ {2, ..., K}, and for any predictor fk, by the +randomness of yk +t , we have that E +� +1{Bk(xt) ̸= yk +t } +� += E +� +1{f(xt) ̸= yk +t } +� += 1 +2. Thus, we get that, +T +� +t=1 +E +� +1{B1(xt) ̸= y1 +t } + K − 1 +2 +� +− inf +f∈F +T +� +t=1 +E +� +1{f1(xt) ̸= y1 +t } + K − 1 +2 +� +≤ R(T, K). +Canceling constant factors we get that, +E +� T +� +t=1 +1{B1(xt) ̸= y1 +t } +� +− inf +f1∈F1 +T +� +t=1 +1{f1(xt) ̸= y1 +t } ≤ R(T, K), +8 + +showcasing that Algorithm 5 is an online agnostic learner for F1 with respect to ℓ0-1. Finally, by +symmetry, a similar reduction can be used to construct online learners for each restriction Fk. This +completes the proof of both directions. +■ +Before we move on to characterize the online learnability of general losses, we first need an online +analog of the realizable-to-agnostic conversion in the batch setting. The next subsection covers this +result. +4.2 +Realizable-to-Agnostic Online Conversion +Our strategy for constructively characterizing the learnability of general multilabel losses in the batch +setting required the ability to convert a realizable learner to an agnostic learner in a black-box fashion. +In this section, we provide an analog of this conversion for the online setting. More specifically, we +focus on the multiclass classification setting with the 0-1 loss where |Y| = K. +Then, we give an +algorithm that takes as input a (potentially randomized) realizable (multiclass) online learner for ℓ0-1 +and outputs an agnostic (multiclass) online learner for ℓ0-1. Theorem 6 formalizes the main result of +this subsection. +Theorem 6. Let A be a realizable (multiclass) online learner for ℓ0-1 with expected mistake-bound +R(T, K). Then, there is an agnostic (multiclass) online learner for ℓ0-1 with expected regret bound +O( +� +T R(T, K) ln(KT )). +Proof. As previously mentioned, we will assume the adversary is oblivious. Accordingly, let (x1, y1), ..., (xT , yT ) +denote the sequence of labelled instances to be observed by the online learner. Let H ⊂ YX be a hy- +pothesis class and A be a (potentially randomized) online realizable learner for H w.r.t. ℓ0-1. By +definition, this means that for any realizable sequence (x1, h(x1)), ..., (xT , h(xT )), we have that +E +� T +� +t=1 +1{A(xt) ̸= h(xt)} +� +≤ R(T, K) +where R(T, K) is a sub-linear function of T . +We now use A to construct an agnostic online learner w.r.t. ℓ0-1. The high-level idea is very similar +to the conversion of the Standard Optimal Algorithm (SOA) in the multiclass classification setting into +an agnostic online learner (see [DSBDSS11]). The only wrinkle we will need to deal with is the fact that +A could be a potentially randomized algorithm. To this end, we will construct a finite set of experts E +such that for every h ∈ H, with high probability, there exists a E ∈ E s.t. for all t ∈ [T ], h(xt) = E(xt). +In particular, this will also be true for the optimal hypothesis h∗ = arg minh∈H +�T +t=1 ℓ0-1(h(xt), yt). +Then, we can run the celebrated Randomized Exponential Weights Algorithm (REWA) using E as the +set of experts and ℓ0-1 as the loss function. Since, with high probability, the behavior of h∗ is exactly +captured by an expert in E, the best possible loss over the experts infE∈E +�T +t=1 ℓ0-1(E(xt), yt) is at +most the best possible loss over H, infh∈H +�T +t=1 ℓ0-1(h(xt), yt), with high probability. +We now formally begin this construction. Recall, that we want to construct a finite set of experts +E such that for all hypothesis h ∈ H, with high probability, there exists a E ∈ E s.t. for all t ∈ [T ], +h(xt) = E(xt). Fix a h ∈ H. Given time horizon T , let LT = {L ⊂ [T ]; |L| ≤ 2R(T, K)}. For every +L ∈ LT , let ΦL = YL be the set of all possible functions φ : L → Y mapping time points in L to +labels in Y. Note that |ΦL| = K|L|. For every L ∈ LT , further denote φh +L ∈ ΦL as the mapping that +exactly corresponds to h, that is φh +L(t) = h(xt) for all t ∈ L. Finally, for every L ∈ LT and every +φ ∈ ΦL we define an expert EL,φ. The expert EL,φ simulates a game between A and the environment +on the sequence of instances x1, ..., xT assuming that A makes a mistake (w.r.t. h) precisely on the +rounds in L and the true labels of these mistaken examples are determined by φ. The algorithm below +formalizes this idea. +Using the procedure above, we have constructed N = �2R(T,K) +j=0 +�T +j +� +Kj ≤ (T K)2R(T,K) experts, +each of them using an independent copy of A. Next, we construct M such independent batches of +N experts. This will allow us to amplify the probability that there exists at least one expert whose +behavior matches h. That is, each of the M batches will contain N = �2R(T,K) +j=0 +�T +j +� +Kj ≤ (KT )2R(T,K) +experts constructed exactly in the same way as described above. Because of this, we let Ei +L,φ denote +the expert EL,φ in the i’th batch. E = �M +i=1 +� +L∈LT +� +φ∈ΦL{Ei +L,φ} denotes the set of all experts across +9 + +Algorithm 6 Expert(L, φ) +Input: Independent copy of A +1 for t = 1, ..., T do +2 +Receive example xt +3 +Let ˜yt = A(xt) +4 +if t ∈ L then +5 +Predict ˆyt = φ(t) +6 +else +7 +Predict ˆyt = ˜yt +8 +end +9 +Update A by passing (xt, ˆyt) +10 end +all batches, and lastly, Eh ⊂ E denotes those set of experts parameterized by functions φh. That is, +Eh = �M +i=1 +� +L∈LT {Ei +L,φh +L}. +Since we are in the oblivious setting, the true sequence (x1, yt), ..., (xT , yT ) is not random and +therefore the sequence (x1, h(xt)), ..., (xT , h(xT )) is not random. But, since A is a potentially random- +ized algorithm, the time points it makes mistakes when fed (x1, h(xt)), ..., (xT , h(xT )) are random. In +this sense, A induces a distribution over the subsets of indices in [T ] where it makes mistakes on the +sequence (x1, h(xt)), ..., (xT , h(xT )). Notice that in the event that exists an expert Ei +L,φh +L ∈ Eh whose +indices L exactly match up with the time points where its copy of A makes mistakes on the sequence +(x1, h(x1), ..., (xT , h(xT )), then Ei +L,φh +L feeds its copy of A exactly the sequence of instances labelled by +h, and therefore predicts exactly h(xt) in each round t ∈ [T ]. Thus, this is precisely the event that we +want to show occurs with high probability. +To that end, let Ai +L,φ be the copy of A associated with expert Ei +L,φ and Ih(Ai +L,φ) be the ran- +dom variable denoting the set of time points where Ai +L,φ makes mistakes when run on the sequence +(x1, h(x1)), ..., (xT , h(xT )). Then, we have +P +� +∃Ei +L,φh +L ∈ Eh s.t. Ih(Ai +L,φh +L) = L +� += 1 − P +� +∀Ei +L,φh +L ∈ Eh, Ih(Ai +L,φh +L) ̸= L +� += 1 − ΠM +i=1ΠL∈LT P +� +I(Ai +L,φh +L) ̸= L +� += 1 − +� +ΠL∈LT P +� +Ih(A1 +L,φh +L) ̸= L +��M += 1 − +� +ΠL∈LT (1 − P +� +Ih(A1 +L,φh +L) = L +� +) +�M += 1 − (ΠL∈LT (1 − P [Ih(A) = L]))M , +where the last equality follows from the fact that {A1 +L,φh +L}L∈LT are independent copies of A. +Let Bh denote the event that A makes at most 2R(T, K) mistakes when run on the sequence +(x1, h(x1)), ..., (xT , h(xT )). Then, +P [Ih(A) = L] ≥ P [Ih(A) = L|Bh] P [Bh] +≥ P [Ih(A) = L|Bh] +2 +, +where the last inequality follows from the fact that A has an expected mistake bound of R(T, K) and +therefore by Markov’s inequality, with probability at least 1 +2, �T +t=1 +1{A(xt) ̸= h(xt)} ≤ 2R(T, K). +Putting things together, we have that, +P +� +∃Ei +L,φh +L ∈ Eh s.t. Ih(Ai +L,φh +L) = L +� +≥ 1 − +� +ΠL∈LT +� +1 − P [Ih(A) = L|Bh] +2 +��M +. +Next, note that � +L∈LT P [Ih(A) = L|Bh] = 1. +This is because the support of the conditional +distribution over the set of time points that A makes mistakes, given it makes at most 2R(T, K) +10 + +mistakes, is exactly LT . Therefore, we have that ΠL∈LT +� +1 − P[Ih(A)=L|Bh] +2 +� +≤ e− 1 +2 using the fact that +1−x ≤ e−x. Putting things together, we get that P +� +∃Ei +L,φh +L ∈ Eh s.t. Ih(Ai +L,φh +L) = L +� +≥ 1−e− M +2 . Since +Eh ⊂ E, this further implies that P +� +∃Ei +L,φ ∈ E s.t. Ih(Ai +L,φ) = L +� +≥ 1 − e− M +2 . Setting M = 2 ln( 2 +δ ), +we get that with probability at least 1 − δ +2, ∃Ei +L,φ ∈ E s.t. Ih(Ai +L,φ) = L. This means that we have +successfully constructed a set E of MN ≤ 2 ln( 2 +δ )(KT )2R(T,K) experts such that with high probability, +there exists an expert who exactly matches the behavior of h on the sequence of instances. Since h is +arbitrary, this must be true for any hypothesis and in particular, for the optimal hypothesis h∗. +Next, we run REWA over E on the stream (x1, y1), ..., (xT , yT ) using ℓ0-1. A nice property about +the REWA is that for any finite set of experts G, for ℓ0-1, and for learning rate η = +� +8 ln(|G|) +T +, with +probability at least 1 − δ +2, +T +� +t=1 +ℓ0-1(ˆyt, yt) − inf +g∈G +� T +� +t=1 +ℓ0-1(g(xt), yt) +� +≤ +� +T +2 ln(|G|) + +� +T +2 ln +�2 +δ +� +where ˆyt is the prediction of REWA in the t’th round. Taking G to be E, and union bounding over the +event that there exists an expert E ∈ E that exactly matches the behavior of h∗ and the event that +REWA succeeds, we get that with probability 1 − δ, +T +� +t=1 +ℓ0-1(ˆyt, yt) − inf +h∈H +� T +� +t=1 +ℓ0-1(h(xt), yt) +� +≤ +� +T +2 ln(MN) + +� +T +2 ln +�2 +δ +� +≤ +� +T +2 ln(M(KT )2R(T,K)) + +� +T +2 ln +�2 +δ +� += +� +T +2 (ln(M) + 2R(T, K) ln(KT )) + +� +T +2 ln +�2 +δ +� +≤ +� +T +2 ln(2 ln(2 +δ )) + +� +T R(T, K) ln(KT )) + +� +T +2 ln +�2 +δ +� += O +� +� +T R(T, K) ln(KT ) + +� +T ln(1 +δ ) +� +. +Picking δ = 1 +T , a standard argument yields +E +� T +� +t=1 +ℓ0-1(ˆyt, yt) − inf +h∈H +T +� +t=1 +ℓ0-1(h(xt), yt) +� +≤ O +�� +T R(T, K) ln(KT ) +� +as desired. Thus, running REWA over of set up experts E gives an online agnostic learner for H w.r.t. +ℓ0-1. +■ +The reduction from realizable to agnostic online learning can be extended beyond the multiclass +classification setting and the 0-1 loss to the multilabel classification setting and any multilabel loss +function satisfying the identity of indiscernibles. Corollary 7 formalizes this idea. +Corollary 7. Let ℓ be any multilabel loss function satisfying the identity of indiscernibles. If A is +a realizable (multilabel) online learner for ℓ with expected loss bound R(T, K), then there exists an +agnostic (multilabel) online learner for ℓ with expected regret bound O +� +b +� +KT R(T,K) +a +ln(T ) +� +, where a +and b are universal constants that depend on ℓ. +Proof. Let F ⊂ YX be a multilabel function class with target space Y = {−1, 1}K for some K ∈ N. +Let ℓ : Y ×Y → R be any non-negative multilabel loss function satisfying the identity of indiscernibles. +Then, since there are at most 2K possible values for ℓ, there must exist constants a and b s.t. for +11 + +all y1, y2 ∈ Y, aℓ0-1(y1, y2) ≤ ℓ(y1, y2) ≤ bℓ0-1(y1, y2). Let A be an online realizable learner w.r.t. ℓ. +Then, by definition, for any sequence of instances (x1, y1), ..., (xT , yT) realized by F, +E +� T +� +t=1 +ℓ(A(xt), yt) +� +≤ R(T, K). +Furthermore, since aℓ0-1 ≤ ℓ, we can write +E +� T +� +t=1 +1{A(xt) ̸= yt} +� +≤ R(T, K) +a +. +Therefore, A is also a multiclass online realizable learner w.r.t. +ℓ0-1 and therefore a valid input +realizable online learner for the construction in the proof of Theorem 6. +To that end, we will follow the same construction in the proof of Theorem 6 using A as the +input realizable online learner. +However, we will need a slight modification to accommodate the +fact that we are interested in the multilabel loss ℓ. +Notice that one can run REWA used in the +construction of the agnostic learner in the proof of Theorem 6 with any loss function ℓ ∈ [0, 1]. Thus, +instead of running the REWA using ℓ0-1, we can run REWA using ℓ +b ∈ [0, 1], since b trivially upper +bounds ℓ. Moreover, since our set of experts E enjoys the property that for any f ∈ F, with high +probability, there exists an expert in E ∈ E that exactly matches the behavior of f, we have that +infE∈E +��T +t=1 +ℓ(E(xt),yt) +b +� +≤ inff∈F +��T +t=1 +ℓ(f(xt),yt) +b +� +with high probability for the loss function ℓ +b as +well. Let ˜ +A be the result of following the same construction used in the proof of Theorem 6, but using +A as the online realizable learner and ℓ +b as the loss function in REWA. Then, by Theorem 6 and the +observations above, we have that +E +� T +� +t=1 +ℓ( ˜ +A(xt), yt) − inf +f∈F +T +� +t=1 +ℓ(f(xt), yt) +� +≤ O +� +b +� +T R(T, K) +a +ln(2KT ) +� +≤ O +� +b +� +KT R(T, K) +a +ln(T ) +� +. +This completes the proof as we have shown that ˜ +A is an online agnostic learner for F w.r.t. ℓ. +■ +4.3 +Characterizing Online Learnability for General Losses +Using Theorem 5 and Corollary 7, we now characterize the learnability of arbitrary multilabel loss +functions ℓ as long as they satisfy the identity of indiscernibles. The key idea is that since there are +only finite number of possible inputs to ℓ, for any ℓ satisfying the identity of indiscernibles, there must +exist universal constants a and b s.t. aℓH(y1, y2) ≤ ℓ(y1, y2) ≤ bℓH(y1, y2). Then, we can characterize +the learnability of ℓ by relating it to the learnability of ℓH. +Lemma 8. Let ℓ be any loss function satisfying the identity of indiscernibles. A function class F ⊂ YX +is online learnable w.r.t. ℓ if and only if F is online learnable w.r.t. ℓH. +Proof. Let a and b be the universal constants s.t. +for all y1, y2 ∈ Y, aℓH(y1, y2) ≤ ℓ(y1, y2) ≤ +bℓH(y1, y2). We will first show that if F is online learnable w.r.t. ℓH, then F is online learnable +w.r.t. ℓ. By Corollary 7, it suffices to give a realizable online learner for F w.r.t. ℓ. Since F is online +learnable w.r.t. ℓH, there exists an algorithm A s.t. for any sequence (x1, y1), ..., (xT , yT ), we have +E +� T +� +t=1 +ℓH(A(xt), yt) − inf +f∈F +T +� +t=1 +ℓH(f(xt), yt) +� +≤ R(T, K) +where R(T, K) is a sublinear function of T . In the realizable setting, we are guaranteed that for any +sequence (x1, y1), ..., (xT , yT ) that the online learner may observe, there exists a f ∈ F s.t f(xt) = yt +for all t ∈ [T ]. Since ℓH satisfies the identity of indiscernibles, we have that for any realizable sequence +(x1, y1), ..., (xT , yT ), inff∈F +�T +t=1 ℓH(f(xt), yt) = 0. Thus, we have that E +��T +t=1 ℓH(A(xt), yt) +� +≤ +12 + +R(T, K). Noting that ℓH(A(xt), yt) ≥ ℓ(A(xt),yt) +b +completes this portion of the proof as it implies that +E +��T +t=1 ℓ(A(xt), yt) +� +≤ bR(T, K), showcasing that A is also a realizable online learner for F w.r.t. ℓ. +The construction in Corollary 7 can then be used to convert A into an agnostic online learner for F +w.r.t. ℓ. +Next, we show the reverse direction - if F is online learnable w.r.t. ℓ, then F is online learnable +w.r.t. ℓH. Again, by Corollary 7 it suffices to construct an online learner for F w.r.t. ℓH in the +realizable setting. We can repeat the exact same procedure above. +Since F is online learnable w.r.t. ℓ, there exists an algorithm A s.t. for any sequence (x1, y1), ..., (xT , yT ), +we have +E +� T +� +t=1 +ℓ(A(xt), yt) − inf +f∈F +T +� +t=1 +ℓ(f(xt), yt) +� +≤ R(T, K) +where R(T, K) is a sublinear function of T . In the realizable setting, we are guaranteed that for any +sequence (x1, y1), ..., (xT , yT ) that the online learner may observe, there exists a f ∈ F s.t f(xt) = yt +for all t ∈ [T ]. +Since ℓ satisfies the identity of indiscernibles, we have that for any realizable se- +quence (x1, y1), ..., (xT , yT ), inff∈F +�T +t=1 ℓ(f(xt), yt) = 0. Thus, we have that, E +��T +t=1 ℓ(A(xt), yt) +� +≤ +R(T, K). Noting that ℓ(A(xt), yt) ≥ aℓH(A(xt), yt) completes this portion of the proof as it implies +that E +��T +t=1 ℓH(A(xt), yt) +� +≤ R(T,K) +a +, showcasing that A is also a realizable online learner for F w.r.t. +ℓH. The construction in Corollary 7 can then be used to convert A into an agnostic online learner for +F w.r.t. ℓH. +■ +As an immediate consequence of Lemma 8 and Theorem 5, we get the following Theorem charac- +terizing the online learnability of general multilabel losses. +Theorem 9. Let ℓ be any multilabel loss function that satisfies the identity of indiscernibles. +A +function class F ⊂ YX is online learnable w.r.t. ℓ if and only if each restriction Fk ⊂ YX +k is online +learnable w.r.t. the 0-1 loss. +4.4 +Bandit Online Multilabel Classification +We extend the results in the previous subsection to the online setting where the learner only observes +bandit feedback in each round. Theorem 10 gives a characterization of bandit online learnability of a +function class F in terms of the online learnability of each restriction. +Theorem 10. Let ℓ be any loss function that satisfies the identity of indiscernibles. A function class +F ⊂ YX is bandit online learnable w.r.t. ℓ if and only if each restriction Fk ⊂ YX +k is online learnable +w.r.t. the 0-1 loss. +Proof. Let b be such that for all y1, y2 ∈ Y, ℓ(y1, y2) ≤ b. We first show necessity - if F is bandit +online learnable w.r.t. ℓ, then each restriction Fk is online learnable w.r.t. ℓ0-1. This follows trivially +from the fact that if A is a bandit online learner for F, then A is also an online learner for F under +full-feedback. Thus, by Theorem 9, online learnability of F w.r.t. ℓ implies online learnability of +restriction Fk w.r.t. the 0-1 loss. +We now focus on showing sufficiency - if every restriction {Fk}K +k=1 is online learnable w.r.t. 0-1 +loss, then F is bandit online learnable w.r.t. loss ℓ. At a high level, the proof of this direction is very +similar to the proof of Theorem 10 and the proof of Theorem 2.3 in [DH13]. We first construct a finite +set of experts E such that with high probability there exists an expert E ∈ E that exactly matches the +behavior of the best function f ∗ with high probability. Then, we run EXP4.IX from [Neu15] over this +set of experts using the scaled loss ℓ +b, which gives a high probability regret bound. Union bounding +with the event that there exists an expert that exactly matches the behavior of the optimal function +completes the proof. +We now formalize the sketch above. By Theorem 9, if all restrictions Fk are online learnable, then +F is online agnostic learnable w.r.t. ℓ0-1 and therefore online realizable learnable w.r.t. ℓ0-1. This +means that there exists an online learner A s.t. for any sequence (x1, y1), ..., (xT , yT ) realized by F, +we have +13 + +E +� T +� +t=1 +1 {A(xt) ̸= yt} +� +≤ R(T, K). +Note that this means that A is also a realizable multiclass online learner for F w.r.t. ℓ0-1. Therefore, +we can use A to construct the same set of experts E as in the proof of Theorem 6. Namely, by using A as +the realizable multiclass online learner, we can construct a finite set of experts of size M(T 2K)2R(T,K) +such that with probability at least 1 − e +−M +2 , there exists an expert E ∈ E that exactly matches the +behavior of the optimal function in hindsight f ∗. If we set M = ln( 2 +δ ), then with probability 1 − δ +2, +there exists an expert E ∈ E that exactly matches the behavior of f ∗. Note that in each round t ∈ [T ], +every expert E ∈ E outputs an element of Y, which can also be thought of as a distribution over Y +that places all mass on one particular y ∈ Y. Thus, for every round t ∈ [T ], we can view each expert +as outputting a distribution over the label (action) space Y. +From this perspective, we can run the bandit algorithm EXP4.IX from [Neu15] over our set of +experts E using the scaled loss ℓ +b ∈ [0, 1]. For an appropriately chosen learning rate η, this guarantees +that with probability 1 − δ +2, +T +� +t=1 +ℓ(ˆyt, yt) − inf +E∈E +T +� +t=1 +ℓ(E(xt), yt) ≤ O +� +b +� +|Y|T ln(|E|) + b +� +|Y|T ln(4 +δ ) +� +where ˆyt is the prediction of EXP4.IX in the t’th round. Union bounding with the event that there +exists an expert that matches the behavior of the optimal function f ∗, we get that with probability +1 − δ, +T +� +t=1 +ℓ(ˆyt, yt) − inf +f∈F +T +� +t=1 +ℓ(f(xt), yt) ≤ O +� +b +� +2KT ln(ln(2 +δ )(T 2K)2R(T,K)) + b +√ +2KT ln(4 +δ ) +� +≤ O +� +b +� +K2KR(T, K)T ln(T ) + b +√ +2KT ln(4 +δ ) +� +Taking, δ = 1 +T , we have that with probability 1 − 1 +T , +T +� +t=1 +ℓ(ˆyt, yt) − inf +f∈F +T +� +t=1 +ℓ(f(xt), yt) ≤ O +� +b +� +K2KR(T, K)T ln(T ) + b +√ +2KT ln(T ) +� +, +from which we can conclude that +E +� T +� +t=1 +ℓ(ˆyt, yt) +� +− inf +f∈F +T +� +t=1 +ℓ(f(xt), yt) ≤ O +� +b +� +K2KR(T, K)T ln(T ) +� +. +This concludes the proof as we have shown that EXP4.IX run over E is a bandit online learner for +ℓ. +■ +5 +Batch Multioutput Regression +In this section, we consider the case when Y = [0, 1]K ⊂ RK for K ∈ N. This target space is without +loss of generality because one can always normalize each Yk to [0,1] by subtracting the lower bound +and dividing by the upper bound of Yk. Following our outline in classification, we will first study +learnability under decomposable losses and then study a non-decomposable loss. +5.1 +Characterizing Learnability for Lp norms +A canonical loss function for multioutput regression is the p-norm, defined as +ℓp(f(x), y) = +K +� +k=1 +|fk(x) − yk|p, +14 + +for 1 ≤ p < ∞ and f(x), y ∈ Y. The p-norm loss is a natural multivariate extension of the metric +dp(fk(x), yk) := |fk(x) − yk|p, +which is generally taken as a loss function in a real-valued regression setting. The following result +establishes an equivalence between the learnability of F ⊂ YX w.r.t. the p-norm and the learnability +of each Fk w.r.t. the dp-metric. +Theorem 11. The function class F ⊂ YX is agnostic learnable w.r.t. +ℓp if and only if each of +Fk ⊂ YX +k is agnostic learnable w.r.t. dp. +Proof. (of sufficiency in Theorem 11) We will first prove that the agnostic learnability of each Fk is +sufficient for the agnostic learnability of F. As in the classification setting, the proof here is based +on reduction. That is, given oracle access to agnostic learners Ak for each Fk w.r.t. dp loss, we will +construct an agnostic learner A for F w.r.t. ℓp loss. +Algorithm 7 Agnostic Learner for F w.r.t. ℓp +Input: Agnostic learners {Ak}K +k=1 for Fk’s and samples S = {(xi, yi)}n +i=1 ∼ Dn on X × Y. +1 Construct samples Sk = {xi, yk +i }n +i=1 with scalar-valued targets for all k ∈ [K] +2 Get predictors gk = Ak(Sk) for all k ∈ [K]. +3 Output g = (g1, . . . , gk). +We will show that Algorithm 7 is an agnostic learner for F w.r.t. ℓp. Denote Dk to be the marginal +distribution of D restricted to X ×Yk. Let us use mk(ǫ, δ) to denote the sample complexity of Ak. Since +Ak is an agnostic learners for Fk, we have that for sample size n ≥ maxk mk( ǫ +K , δ +K ), with probability +at least 1 − δ/K over samples Sk ∼ Dn +k , +EDk[|gk(x) − yk|p] ≤ +inf +fk∈Fk EDk[|fk(x) − yk|p] + ǫ +K . +Summing these risk bounds over all k coordinates and union bounding over the success probabilities, +we get that with probability at least 1 − δ over samples S ∼ Dn, +K +� +k=1 +EDk[|gk(x) − yk|p] ≤ +K +� +k=1 +inf +fk∈Fk EDk[|fk(x) − yk|p] + ǫ. +Using the fact that the sum of infimums over individual coordinates is at most the overall infimum of +sums followed by the linearity of expectation, we can write the expression above as +ED +� K +� +k=1 +|gk(x) − yk|p +� +≤ inf +f∈F ED +� K +� +k=1 +|fk(x) − yk|p +� ++ ǫ, +showcasing that Algorithm 7 is an agnostic learner for F w.r.t. ℓp with sample complexity at most +maxk mk(ǫ/K, δ/K). This completes our proof of sufficiency. +■ +Proof. (of necessity in Theorem 11) Next, we will show that if F is agnostic learnable w.r.t. ℓp, then +each Fk is agnostic learnable w.r.t. +dp. +Given oracle access to agnostic learner A for F, we will +construct agnostic learners A1 for F1. By symmetry, a similar reduction can then be used to construct +an agnostic learner for each component Fk. +Since we will be given a sample with a single variate target, the main question is to find the right +way to augment samples to a K-variate target. In the proof of Theorem 2, we showed that randomly +choosing yik ∼ Uniform({−1, 1}) for k ≥ 2 results in all predictors having a constant 1/2 risk–leaving +only the risk of the first component on both sides. Unfortunately, for regression settings, no single +augmentation works for every distribution on X. Thus, we will augment the samples by considering +all possible behaviors of (F2, . . . , FK) on the sample. Since the function class maps to a potentially +uncountably infinite space, we will first discretize each component of the function class and consider +all possible labelings over the discretized space. Fix 1 > α > 0. For k ≥ 2, define the discretization +f α +k (x) = +�f(x) +α +� +α +15 + +for every fk ∈ Fk and the discretized component class Fα +k = {f α +k |fk ∈ Fk}. Note that a function +in Fk maps to {0, α, 2α, . . . , ⌊1/α⌋α} and the size of the range of the discretized function class Fα +k is +1 + ⌊1/α⌋ ≤ (α + 1)/α ≤ 2/α. For the convenience of exposition, let us define Fα +2:K to be Fα without +the first component, and we will denote f α +2:K to be an element of Fα +2:K. +Algorithm 8 Agnostic learner for F1 w.r.t. dp +Input: Agnostic learner A for F samples S = (x1:n, y1 +1:n) ∼ Dn +1 and another independent samples �S +from D1 +1 Define Saug = {(x1:n, y1 +1:n, f α +2:K(x1:n) | f2:K ∈ Fα +2:K}, all possible augmentations of S. +2 Run A over all possible augmentations to get +C(S) := +� +A +� +Sa +� +| Sa ∈ Saug +� +3 Define C1(S) = {g1 | (g1, . . . , gk) ∈ C(S)}, a restriction of C(S) to its first coordinate output. +4 Return ˆg1, the predictor in C1(S) with the lowest empirical error over �S w.r.t. dp. +We will now show that Algorithm 8 is an agnostic learner for F1. First, let us define +f ⋆ +1 := arg min +f1∈F1 +ED1[|f1(x) − y1|p], +to be optimal predictor in F1 w.r.t. D1. By definition of F1, there must exist f ⋆ +2:K ∈ F2:K such that +(f ⋆ +1 , f ⋆ +2:K) ∈ F. We note that f ⋆ +k need not be optimal predictors in Fk for k ≥ 2, but we use the ⋆ +notation just to associate these component functions with the first component function f ⋆ +1 . Define +f ⋆,α +2:K ∈ Fα +2:K to be the corresponding discretization of f ⋆ +2:K. +Suppose g = A((x1:n, y1 +1:n, f ⋆,α +2:K(x1:n)) is the predictor obtained by running A on the sample aug- +mented by f ⋆,α +2:K. Note that g ∈ C(S) by definition. Let mA(ǫ, δ, K) be the sample complexity of A. +Since A is an agnostic learner for F w.r.t ℓp, we have that for n ≥ mA(ǫ/4, δ/2, K), with probability +at least 1 − δ/2, +ED1 +� +|g1(x) − y1|p� ++ +K +� +k=2 +EDX +� +|gk(x) − f ⋆,α +k +(x)|p� +≤ inf +f∈F +� +ED1 +� +|f1(x) − y1|p� ++ +K +� +k=2 +EDX +� +|fk(x) − f ⋆,α +k +(x)|p� +� ++ ǫ +4 +Note that the quantity on the left is trivially lower bounded by the risk of the first component +and the optimal risk on the right-hand side is trivially upper bounded by the risk of (f ⋆ +1 , f ⋆ +2:K). In +particular, we have +ED1 +� +|g1(x) − y1|p� +≤ ED1 +� +|f ⋆ +1 (x) − y1|p� ++ +K +� +k=2 +EDX +� +|f ⋆ +k(x) − f ⋆,α +k +(x)|p� ++ ǫ +4 +≤ ED1 +� +|f ⋆ +1 (x) − y1|p� ++ Kαp + ǫ +4, +where the last inequality follows upon using the fact that |f ⋆ +k(x) − f ⋆,α +k +(x)| ≤ α for all x and k ≥ 2. +Picking α = (ǫ/4K)1/p and using the definition of f ⋆ +1 , we obtain +ED1 +� +|g1(x) − y1|p� +≤ +inf +f1∈F1 ED1[|f1(x) − y1|p] + ǫ +2. +Therefore, we have shown the existence of one predictor g ∈ C(S) such that its restriction to the +first component, g1, obtains the agnostic bound. +Recall that by Hoeffding’s Inequality and union +bound, with probability at least 1 − δ/2, the empirical risk of every hypothesis in C1(S) on a sample +of size ≥ O +� +1 +ǫ2 log |C1(S)| +δ +� +is at most ǫ/4 away from its true error. So, if |�S| ≥ O +� +1 +ǫ2 log |C1(S)| +δ +� +, then +with probability at least 1 − δ/2, we have +16 + +1 +|�S| +� +(x,y1)∈ � +S +|g1(x) − y1|p ≤ ED1 +� +|g1(x) − y1|p� ++ ǫ +4 ≤ +inf +f1∈F1 ED1[|f1(x) − y1|p] + 3ǫ +4 . +Since ˆg1 is the ERM on �S over C1(S), its empirical risk can be at most inff1∈F1 ED1[|f1(x)−y1|p]+ +3ǫ +4 . Given that the population risk of ˆg1 is at most ǫ/4 away from its empirical risk, we have that +ED1[|ˆg1(x) − y1|p] ≤ +inf +f1∈F1 ED1[|f1(x) − y1|p] + ǫ. +Applying union bounds, the entire process succeeds with probability 1 − δ. Letting m1(ǫ, δ) denote +the sample complexity of Algorithm 8, we have that m1(ǫ, δ) is at most the sample complexity of A +plus the number of samples required for ERM in step 4 to succeed. Thus, +m1(ǫ, δ) ≤ mA(ǫ/4, δ/2, K) + O +� 1 +ǫ2 log |C1(S)| +δ +� +≤ mA(ǫ/4, δ/2, K) + O +�mA(ǫ/4, δ/2, K) K log 2 +α + log 1 +δ +ǫ2 +� +≤ mA(ǫ/4, δ/2, K) + O + + +mA(ǫ/4, δ/2, K) K +� +1 + 1 +p log K +ǫ +� ++ log 1 +δ +ǫ2 + + , +where the second inequality follows due to |C1(S)| ≤ (2/α)mA(ǫ/4,δ/2,K) Kand the last equality follows +due to our choice of α = (ǫ/4K)1/p. This completes the proof as it shows that Algorithm 8 is an +agnostic learner for F1 w.r.t. dp. +■ +5.2 +Characterizing Learnability for the Max Loss +Next, we will study the learnability of function class F w.r.t. a non-decomposable loss. In the regression +setting, the natural non-decomposable loss to consider is ℓ∞, which is defined as +ℓ∞(f(x), y) := max +k∈[K] |fk(x) − yk|. +The following result characterizes the agnostic learnability of F w.r.t. ℓ∞. +Theorem 12. The function class F ⊂ YX is agnostic learnable w.r.t. +ℓ∞ if and only if each of +Fk ⊂ YX +k is agnostic learnable w.r.t. the absolute value loss, d1. +Proof. (of sufficiency of Theorem 12) +We will first prove that agnostic learnability of each Fk w.r.t. d1 is sufficient for agnostic learnability +of F w.r.t. ℓ∞. For this direction, we will first discretize the function class F. Then, we will show +how agnostic learners Ak’s for Fk’s can be used to construct a realizable learner for the discretization +of F w.r.t. ℓ∞. Since the discretized function class maps to finite label space, we can do the standard +realizable-to-agnostic reduction. +Finally, we will show that an agnostic learner for the discretized +function class is an agnostic learner for the original function class F. To that end, for 0 < α < 1 and +for each f ∈ F, define its corresponding discretization as +f α(x) := +��f1(x) +α +� +α, . . . , +�fK(x) +α +� +α +� +, +where f(x) = (f1(x), . . . , fK(x)). Next, define the discretized function class, Fα = {f α | f ∈ F}. +First, we will show that A constructed in step 1 of Algorithm 9 is a realizable learner for Fα. +Consider a distribution D that is realizable by Fα w.r.t. ℓ∞, that is inff α∈F α ED[maxk |f α +k (x)−yk|] = 0. +Since maxk |f α +k (x) − yk| ≥ |f α +k (x) − yk| for each k ∈ [K], we have that +inf +f α +k ∈F α +k +EDk[|f α +k (x) − yk|] = 0. +17 + +Algorithm 9 Agnostic learner for F w.r.t. ℓ∞ +Input: Agnostic learner Ak’s for Fk’s, unlabed samples SU ∼ DX , and labeled samples SL ∼ D +1 For any S ∼ D and its restriction Sk ∼ Dk, define algorithm A(S) := (A1(S1), . . . , AK(SK)). +2 Discretize F to get Fα. +3 Run A over all possible labelings of SU by Fα to get +C(SU) := +� +A +� +SU, f α(SU) +� +| f α ∈ Fα +|SU +� +4 Return ˆg ∈ C(SU) with the lowest empirical error over SL w.r.t. ℓ∞. +The triangle inequality gives us |fk(x) − yk| ≤ |f α +k (x) − yk| + |fk(x) − f α +k (x)| ≤ |f α +k (x) − yk| + α, +which we can use to obtain +inf +fk∈Fk EDk +� +|fk(x) − yk| +� +≤ +inf +f α +k ∈F α +k +EDk +� +|f α +k (x) − yk| +� ++ α = α, +for each k ∈ [K]. Suppose S = {(xi, yi)}n +i=1 ∼ Dn for n ≥ maxk mk(ǫ/2K, δ/K), where mk(ǫ, δ) is the +sample complexity of Ak. Let Sk = {(xi, y1 +i )}n +i=1 ∼ Dn +1 and gk = Ak(Sk) be the predictor that Ak +outputs when trained on Sk. Then, with probability 1 − δ/K over samples Sk ∼ Dn +k, we have +EDk +� +|gk(x) − yk| +� +≤ +inf +fk∈Fk EDk +� +|fk(x) − yk| +� ++ +ǫ +2K ≤ α + +ǫ +2K , +upon using the inequality established above. Union bounding over these events, we obtain with prob- +ability 1 − δ, +K +� +k=1 +EDk +� +|gk(x) − yk| +� +≤ Kα + ǫ +2. +Since maxk |gk(x) − yk| ≤ � +k |gk(x) − yk|, the bound above reduces to +ED +� +max +k∈[K] |gk(x) − yk| +� +≤ Kα + ǫ +2 ≤ ǫ, +where the last inequality follows from picking α = ǫ/4K. Therefore, the algorithm A is a realizable +learner for Fα with respect to ℓ∞ with sample complexity mA(ǫ, δ, K) = maxk mk(ǫ/2K, δ/K). +Next, we will show that Algorithm 9 is an agnostic learner for Fα. The argument here is similar +to the one used for realizable to agnostic reduction in [HKLM22]. Let D be an arbitrary distribution +over X × Y. Define +f ⋆,α := +inf +f α∈F α ED +� +max +k∈[K] |f α +k (x) − yk| +� +, +to be the optimal predictor in Fα. Note that we can do this reduction because Fα maps to finite +label space of size ≤ (2/α)K, and thus it makes sense to consider all possible labeling by Fα over the +unlabeled samples. Let mA(ǫ, δ, K) be the sample complexity of the realizable learner A. Since C(SU) +contains a predictor returned by A when trained on all possible labelings of SU by Fα, it must contain +a predictor ˜g = A( ˜S), where ˜S = (SU, f ⋆,α(SU)) is labeled by f ⋆,α. Since A is a realizable learner +with respect to Fα, for |SU| ≥ mA(ǫ/4, δ/2, K), with probability 1 − δ/2, we have +EDX +� +max +k∈[K] |˜g(x) − f ⋆,α(x)| +� +≤ ǫ +4. +Using the triangle inequality, this can be further reduced to +ED +� +max +k∈[K] |˜g(x) − yk| +� +≤ ED +� +max +k∈[K] |f ⋆,α(x) − yk| +� ++ EDX +� +max +k∈[K] |˜g(x) − f ⋆,α(x)| +� +≤ ED +� +max +k∈[K] |f ⋆,α(x) − yk| +� ++ ǫ +4. +Thus, we have shown that there exists one predictor ˜g ∈ C(SU) that has good population risk. +Recall that by Hoeffding’s inequality and union bound, with probability at least 1 − δ/2, the empirical +18 + +risk of every hypothesis in C(SU) on a sample of size ≥ +8 +ǫ2 log 4|C(SU)| +δ +is at most ǫ/4 away from its +population risk. So, if |SL| ≥ +8 +ǫ2 log 4|C(SU)| +δ +, then with probability at least 1 − δ/2, we have +1 +|SL| +� +(x,y)∈SL +max +k∈[K] |˜gk(x) − yk| ≤ ED +� +max +k∈[K] |˜g(x) − yk| +� ++ ǫ +4 ≤ +inf +f α∈F α ED[max +k∈[K] |f α +k (x) − yk|] + ǫ +2. +Next, consider the predictor ˆg returned by Algorithm 9. +Then, its empirical risk can be at most +inff α∈F α ED[maxk |f α +k (x)−yk|]+ ǫ +2. Given that the population risk of ˆg can be at most ǫ/4 away from +its empirical risk, we have that +ED +� +max +k∈[K] |ˆgk(x) − yk| +� +≤ +inf +f α∈F α ED +� +max +k∈[K] |f α +k (x) − yk| +� ++ 3ǫ +4 = inf +f∈F ED +� +max +k∈[K] |f α +k (x) − yk| +� ++ 3ǫ +4 , +where the last step follows because taking infimum over Fα and F are equivalent if the quantity inside +expectation is defined with the discretized version of the function. Applying union bounds, the entire +process succeeds with probability 1−δ. Next, we note that |f α +k (x)−yk| ≤ |fk(x)−yk|+|f α +k (x)−fk(x)| ≤ +|fk(x) − yk| + α. Using this inequality, we have +ED +� +max +k∈[K] |ˆgk(x) − yk| +� +≤ inf +f∈F ED +� +max +k∈[K] |fk(x) − yk| +� ++ α + 3ǫ +4 . +Note that we have already picked α = ǫ/4K and the same choice of α yields +ED +� +max +k∈[K] |ˆgk(x) − yk| +� +≤ inf +f∈F ED +� +max +k∈[K] |fk(x) − yk| +� ++ ǫ. +We now upper bound the sample complexity of Algorithm 9, denoted m(ǫ, δ, K) hereinafter. Note +that m(ǫ, δ, K) is at most the number of unlabeled samples required for the realizable algorithm A to +succeed plus the number of labeled samples for the ERM step to succeed. Thus, +m(ǫ, δ, K) ≤ mA(ǫ/4, δ/2, K) + O +� 1 +ǫ2 log |C(SU)| +δ +� +≤ mA(ǫ/4, δ/2, K) + O +� +mA(ǫ/4, δ/2, K) K log K +ǫ + log 1 +δ +ǫ2 +� +, +where the second inequality follows due to |C(SU)| ≤ (2/α)mA(ǫ/4,δ/2,K) Kand our choice of α = +ǫ/4K. Since the realizable algorithm A is constructed using individual algorithms Ak’s, we can re- +late its sample complexity to the sample complexity of Ak’s. +Using the fact that mA(ǫ, δ, K) = +maxk mk(ǫ/2K, δ/K), the sample complexity above can be rewritten as +m(ǫ, δ, K) ≤ max +k +mk(ǫ/8K, δ/2K) + O +� +K log K +ǫ maxk mk(ǫ/8K, δ/2K) + log 1 +δ +ǫ2 +� +. +This completes our proof of sufficiency as we have shown that Algorithm 9 is also an agnostic learner +for F w.r.t ℓ∞. +■ +Proof. (of necessity of Theorem 12) +Next, we will show that if F is agnostic learnable w.r.t. ℓ∞, then each restriction Fk is agnostic +learnable w.r.t. d1. Our proof will be constructive: given oracle access to an agnostic learner A for F +w.r.t. ℓ∞, we will construct agnostic learners Ak for each Fk w.r.t. d1. In particular, we will show that +Algorithm 8, given as input our agnostic learner A for ℓ∞, is an agnostic learner for F1 w.r.t. d1. By +symmetry, a similar reduction can then be used to construct an agnostic learner for each component +Fk. +Our proof here is essentially the same as the proof of necessity in Theorem 11. So, in the same +spirit, fix α > 0 and for k ≥ 2, define the discretization +f α +k (x) = +�f(x) +α +� +α +19 + +for every fk ∈ Fk. Define the discretized component class Fα +k = {f α +k |fk ∈ Fk} and define Fα +2:K to be +Fα without the first component. We will denote f α +2:K to be an element of Fα +2:K. Let us define +f ⋆ +1 := arg min +f1∈F1 +ED1[|f1(x) − y1|], +to be optimal predictor in F1 w.r.t. D1. By definition of F1, there must exist f ⋆ +2:K ∈ F2:K such that +(f ⋆ +1 , f ⋆ +2:K) ∈ F. We note that f ⋆ +k need not be optimal predictors in Fk for k ≥ 2, but we use the ⋆ +notation just to associate these component functions with the first component function f ⋆ +1 . Define +f ⋆,α +2:K ∈ Fα +2:K to be the corresponding discretization of f ⋆ +2:K. +Suppose g = A((x1:n, y1 +1:n, f ⋆,α +2:K(x1:n)) is the predictor obtained by running A on the sample aug- +mented by f ⋆,α +2:K. Let mA(ǫ, δ, K) denote the sample complexity of A. Since A is an agnostic learner +for F, we have that for n ≥ mA(ǫ/4, δ/2, K) , with probability at least 1 − δ/2, +E(x,y1)∼D1 +� +max +k∈[K] |gk(x) − yk| +� +≤ inf +f∈F E(x,y1)∼D1 +� +max +k∈[K] |fk(x) − yk| +� ++ ǫ +4, +where yk = f ⋆,α +k +(x) for k ≥ 2. Note that the quantity on the left is trivially lower bounded by the risk +of the first component and the optimal risk on the right-hand side is trivially upper bounded by the +risk of (f ⋆ +1 , f ⋆ +2:K). In particular, we have +ED1 +� +|g1(x) − y1| +� +≤ E(x,y1)∼D1 +� +max +k∈[K] |f ⋆ +k(x) − yk| +� ++ ǫ +4 +≤ ED1 +� +|f ⋆ +1 (x) − y1| +� ++ +K +� +k=2 +EDX +� +|f ⋆ +k(x) − f ⋆,α +k +(x)| +� ++ ǫ +4 +≤ ED1 +� +|f ⋆ +1 (x) − y1| +� ++ Kα + ǫ +4, +where the second inequality follows upon using the fact that the sum of positive real numbers is greater +than their maximum. The last inequality uses the fact that f ⋆,α is the α discretization of f ⋆. Picking +α = ǫ/4K and using the definition of f ⋆ +1 , we obtain +ED1 +� +|g1(x) − y1| +� +≤ +inf +f1∈F1 ED1[|f1(x) − y1|] + ǫ +2. +Therefore, we have shown the existence of one predictor g ∈ C(S) such that its restriction to +the first component, g1, obtains the agnostic bound. Recall that by Hoeffding’s inequality and union +bound, with probability at least 1 − δ/2, the empirical risk of every hypothesis in C1(S) on a sample +of size ≥ +8 +ǫ2 log 4|C1(S)| +δ +is at most ǫ/4 away from its true error. So, if |�S| ≥ +8 +ǫ2 log 4|C1(S)| +δ +, then with +probability at least 1 − δ/2, we have +1 +|�S| +� +(x,y1)∈ � +S +|g1(x) − y1| ≤ ED1 +� +|g1(x) − y1| +� ++ ǫ +4 ≤ +inf +f1∈F1 ED1[|f1(x) − y1|] + 3ǫ +4 . +Since ˆg1 is the ERM on �S over C1(S), its empirical risk can be at most inff1∈F1 ED1[|f1(x) − y1|] + 3ǫ +4 . +Given that the population risk of ˆg1 can be at most ǫ/4 away from its empirical risk, we have that +ED1[|ˆg1(x) − y1|] ≤ +inf +f1∈F1 ED1[|f1(x) − y1|] + ǫ. +Applying union bounds, the entire process succeeds with probability 1−δ. Like before, we can compute +the upper bound on the sample complexity of Algorithm 8, denoted m1(ǫ, δ), as +m1(ǫ, δ) ≤ mA(ǫ/4, δ/2, K) + O +� 1 +ǫ2 log |C1(S)| +δ +� +≤ mA(ǫ/4, δ/2, K) + O +� +mA(ǫ/4, δ/2, K) K log K +ǫ + log 1 +δ +ǫ2 +� +, +where we use C1(S) ≤ (2/α)mA(ǫ/4,δ/2,K) K and our choice of α = ǫ/4K. This completes the proof as +it shows that Algorithm 8, given as input an agnostic learner A for F w.r.t. ℓ∞, outputs an agnostic +learner for F1 w.r.t. d1. +■ +20 + +As a final remark, we want to distinguish between the role of discretization in Algorithms 8 and +9. In Algorithm 8, we only discretize the components F2:K to augment the input sample containing a +single-variate target to a K-variate target in all possible ways. Since all possible augmentations of the +sample by F2:K could potentially be of infinite size, we first discretized F2:K so that we can construct +a finite cover of all possible augmentations. However, the role of discretization is more fundamental +in Algorithm 9. In this case, we first use learners Ak for Fk to construct an algorithm, which we +showed to be a realizable learner for the discretized function class Fα. Then, step 3 and step 4 are +standard realizable to agnostic reduction for Fα. 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In +I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and +R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. +Curran Associates, Inc., 2017. +[SSBD14] +Shai Shalev-Shwartz and Shai Ben-David. Understanding Machine Learning: From The- +ory to Algorithms. Cambridge University Press, USA, 2014. +[Val84] +Leslie G. Valiant. A theory of the learnable. In Symposium on the Theory of Computing, +1984. +[VC71] +Vladimir Naumovich Vapnik and Aleksei Yakovlevich Chervonenkis. On uniform con- +vergence of the frequencies of events to their probabilities. Teoriya Veroyatnostei i ee +Primeneniya, 16(2):264–279, 1971. +[VC74] +V. Vapnik and A. Chervonenkis. Theory of Pattern Recognition [in Russian]. 1974. +[YWJZ20] +Liang Yang, Xi-Zhu Wu, Yuan Jiang, and Zhi-Hua Zhou. Multi-label learning with deep +forest. In ECAI, volume 325 of Frontiers in Artificial Intelligence and Applications, pages +1634–1641. IOS Press, 2020. +22 + diff --git a/YdE0T4oBgHgl3EQf3wJf/content/tmp_files/load_file.txt b/YdE0T4oBgHgl3EQf3wJf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ad0c5bdef6f893057603d636b34e7e001f8db8c --- /dev/null +++ b/YdE0T4oBgHgl3EQf3wJf/content/tmp_files/load_file.txt @@ -0,0 +1,1154 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf,len=1153 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='02729v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='LG] 6 Jan 2023 A Characterization of Multilabel Learnability Vinod Raman, Unique Subedi, Ambuj Tewari Abstract We consider the problem of multilabel classification and investigate learnability in batch and online settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In both settings, we show that a multilabel function class is learnable if and only if each single-label restriction of the function class is learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' As extensions, we also study mul- tioutput regression in the batch setting and bandit feedback in the online setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For the former, we characterize learnability w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Lp losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For the latter, we show a similar characterization as in the full-feedback setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 1 Introduction Multilabel classification is a learning problem where multiple labels can be assigned to an instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This is a generalization of multiclass classification, where an instance is classified into one of the multiple possible labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Multilabel classification has enjoyed a wide range of practical applications like image tagging, document categorization, and recommender systems, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This widespread appli- cability has motivated the development of several practical methods [KVJ12, YWJZ20, NLMKF17], as well as theoretical analysis [KNRD15, LT15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' However, the most fundamental question of learnability in a multilabel setting remains unanswered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In this work, we address this question by characterizing multilabel learnability in two major learning settings: batch and online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Characterizing learnability is the foundational step toward understanding any statistical learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The fundamental theorem of statistical learning characterizes the learnability of binary function class in terms of the finiteness of a combinatorial quantity called the Vapnik-Chervonenkis (VC) dimension [VC71, VC74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Extending VC theory, [Nat89] proposed and studied the Natarajan dimension, which was later shown by [BDCBL92] to characterize learnability in multiclass settings with a finite number of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Recent work by [BCD+22] shows that Daniely-Shwartz (DS) dimension, originally proposed by [DSS14], characterizes multiclass learnability in infinite labels setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Similarly, in an online setting, the Littlestone dimension [Lit87] characterizes the online learnability of binary function class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' And a generalization of the Littlestone dimension [DSBDSS11] characterizes online learnability in a multiclass setting with finite labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Nevertheless, to our best knowledge, no such characterization of the learnability of multilabel function classes exists in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' To make our problem precise, let X be the instance space and Y = {−1, 1}K the target space for some K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The main question we study is: what characterizes the learnability of the function class F ⊂ YX ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', K}, define a scalar-valued function class Fk = {fk | (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , fK) ∈ F} by restricting each function in F to its kth coordinate output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The following informal version of our main result characterizes the learnability of F in terms of the learnability of each component class Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (Informal) The function class F ⊂ YX is learnable if and only if each of Fk ⊂ YX k is learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We prove a version of this result in both the batch and online settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' As extensions, we also prove a version of this result for batch regression and bandit online settings, considering the appropriate definition of learnability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A unifying theme throughout all four learning settings is our ability to constructively convert a learning algorithm A for F, into a learning algorithm Ak for Fk for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', K} and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, we do not study combinatorial dimensions, but rather take a more direct, algorithmic approach to address the learnability of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 1 2 Preliminaries Let X denote the instance space and Y = {−1, +1}K be the target space for some K ∈ N, unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Consider a vector-valued function class F ⊂ YX , where YX denotes set of all functions from X to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Define a scalar-valued function class Fk = {fk | (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , fK) ∈ F} by restricting each function in F to its kth coordinate output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Here, each Fk ⊂ YX k , where Yk denotes the restriction of the target space to its kth component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Conveniently, we will write F = (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , FK) and Y = (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , YK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We denote ℓ : Y × Y → R≥0 to be some bounded, non-negative loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For a function f ∈ F, we will use fk(x) to denote the kth coordinate output of f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' On the other hand, we will use yk to denote kth coordinate of y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' As usual, [N] is used to denote a set of natural numbers up to N, that is {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Throughout the paper we will focus on loss functions that also satisfy the identity of indiscernibles, defined formally below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Definition 1 (Identity of Indiscernibles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A loss function ℓ : Y × Y → R≥0 satisfies the identity of indiscernibles if ℓ(y1, y2) = 0 if and only if y1 = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' At a high-level, the identity of indiscernibles guarantees that the loss functions we consider are zero- aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' That is, if ℓ1 and ℓ2 are two multilabel loss functions that satisfy the identity of indiscernibles, then ℓ1(y1, y2) = 0 if and only if ℓ2(y1, y2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='1 Batch Setting In the batch setting, we are interested in characterizing the learnability of F under the classical PAC model [Val84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Definition 2 (Agnostic PAC Learnability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F is agnostic PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' loss ℓ : Y×Y → R≥0, if there exists a function m : (0, 1)2 → N and a learning algorithm A : (X ×Y)∗ → YX with the following property: for every ǫ, δ ∈ (0, 1) and for every distribution D on X × Y, running algorithm A on n ≥ m(ǫ, δ) iid samples from D outputs a predictor g = A(S) such that with probability at least 1 − δ over S ∼ Dn, ED[ℓ(g(x), y)] ≤ inf f∈F ED[ℓ(f(x), y)] + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that we do not require the output predictor A(S) to be in F, but only require A(S) to compete with the best predictor in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' If we restrict D to a class of distributions such that inff∈F ED[ℓ(f(x), y)] = 0 and, then we get realizable PAC learnability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Definition 3 (Realizable PAC Learnability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F is realizable PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' loss ℓ : Y×Y → R≥0, if there exists a function m : (0, 1)2 → N and a learning algorithm A : (X ×Y)∗ → YX with the following property: for every ǫ, δ ∈ (0, 1) and for every distribution D on X × Y where inff∈F ED[ℓ(f(x), y)] = 0 , running algorithm A on n ≥ m(ǫ, δ) iid samples from D outputs a predictor g = A(S) such that with probability at least 1 − δ over S ∼ Dn, ED[ℓ(g(x), y)] ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' It is well known that for binary function classes with 0-1 loss, realizable learnability and agnostic learnability are equivalent [SSBD14, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For equivalence between realizable and agnostic learnability for a more general class of loss function and target spaces, we recall the following recent result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Lemma 1 ([HKLM22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let F be a function class from instance space X to a finite target space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Consider a general loss function ℓ : Y × Y → R≥0 that satisfies identity of indiscernible, that is ℓ(y1, y2) = 0 if and only if y1 = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, the following are equivalent: (1) F is realizable PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (2) F is agnostic PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='2 Online Setting In the online setting, we place no distributional assumptions on the set of labeled instances that the learner may observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Instead, an adversary plays a sequential game with the learner over T rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In each round t ∈ [T ], an adversary selects a labeled instance (xt, yt) ∈ X × Y and reveals xt to the learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The learner makes a (potentially randomized) prediction ˆyt ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Finally, the adversary reveals the true label yt, and the learner suffers the loss ℓ(yt, ˆyt), where ℓ is some pre-specified loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Given a function class F ⊆ YX , the goal of the learner is to output predictions ˆyt such that it’s cumulative loss is close to the best possible cumulative loss over functions in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class is online learnable if there exists an algorithm such that for any sequence of labeled examples (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ), the difference in cumulative loss between its predictions and the predictions of the best possible function in F is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Definition 4 makes this precise and formally defines online learnability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Definition 4 (Online Agnostic Learnability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' loss ℓ, if there exists an (potentially randomized) algorithm A such that for any adaptively chosen sequence of labelled examples (xt, yt) ∈ X ×Y, the algorithm outputs A(xt) ∈ Y at every iteration t ∈ [T ] such that E � T � t=1 ℓ(A(xt), yt) − inf f∈F T � t=1 ℓ(f(xt), yt) � ≤ R(T ) where the expectation is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the randomness of A and that of the possibly adaptive adversary, and R(T ) : N → R+ is the additive regret: a non-decreasing, sub-linear function of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' If for the sequence of labelled examples, there exists a f ∈ F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t for all t ∈ [T ], f(xt) = yt, then we say that the sequence is realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' If it is further guaranteed that the learner always observes a realizable sequence, then we say we are in the realizable setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Definition 5 then provides a formal statement on what it means for a function class to be online learnable under realizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Definition 5 (Online Realizable Learnability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' loss ℓ under realizability, if there exists an (potentially randomized) algorithm A such that for any realizable sequence of labelled examples (xt, yt) ∈ X × Y, the algorithm outputs A(xt) ∈ Y at every iteration t ∈ [T ] such that E � T � t=1 ℓ(A(xt), yt) � ≤ R(T ) where the expectation is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the randomness of A, and R(T ) : N → R+ is the additive regret: a non-decreasing, sub-linear function of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' If A is deterministic and R(T ) = M for some M ∈ N, then we say A is a mistake-bound online learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In many situations, however, the true label yt is not revealed to the learner in each round t ∈ [T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Instead, the adversary only reveals the loss ℓ(ˆyt, yt) that the learner has incurred in this round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' If ℓ is the 0-1 loss, then this means that the learner does not get to observe where it made a mistake, only that it made a mistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This type of partial feedback is commonly referred to as bandit information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Definition 6 defines online learnability under bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Definition 6 (Bandit Online Learnability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F is bandit online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' loss ℓ, if there exists a randomized algorithm A such that for any adaptively chosen sequence of labelled examples (xt, yt) ∈ X × Y, under bandit feedback, the algorithm outputs A(xt) ∈ Y at every iteration t ∈ [T ] such that E � T � t=1 ℓ(A(xt), yt) − inf f∈F T � t=1 ℓ(f(xt), yt) � ≤ R(T ) where the expectation is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the randomness of A and that of the possibly adaptive adversary, and R(T ) : N → R+ is the additive regret: a non-decreasing, sub-linear function of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 3 3 Batch Multilabel Classification In this section, we study learnability in batch multilabel classification settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' First, we consider learnability with respect to a natural decomposable loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, we extend the result to more general non-decomposable losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='1 Characterizing Batch Learnability for the Hamming Loss A canonical and natural loss function for multilabel classification is the Hamming loss, defined as ℓH(f(x), y) := K � i=1 1 � fi(x) ̸= yi� , where f(x) = (f1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , fK(x)) and y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , yK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The following result establishes an equivalence between the learnability of F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Hamming loss and the learnability of each Fk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The function class F ⊂ YX is agnostic PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the Hamming loss ℓH if and only if each of Fk ⊂ YX k is agnostic PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of sufficiency in Theorem 2) We will first prove that the agnostic PAC learnability of each Fk is sufficient for agnostic PAC learnability of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Our proof will be constructive: given oracle access to agnostic PAC learners Ak for each Fk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 0-1 loss, we will construct an agnostic PAC learner A for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Algorithm 1 Agnostic PAC Learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH Input: Agnostic PAC learners {Ak}K k=1 for Fk’s and samples S = {(xi, yi)}n i=1 ∼ Dn on X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 1 Construct marginal Sk = {(xi, yk i )}n i=1 for all k ∈ [K] with scalar-valued targets for Ak’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 2 Get hypothesis hk = Ak(Sk) for all k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 3 Output h = (h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , hK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that Algorithm 1 could be improper as the predictor h may not necessarily be in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In fact, each of the algorithms Ak could be improper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, we will show that Algorithm 1 is an agnostic PAC learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Denote Dk to be the marginal distribution of D restricted to X × Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let mk(ǫ, δ) denote the sample complexity of Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since Ak is an agnostic PAC learner for Fk’s, we have that for n ≥ maxk mk( ǫ K , δ K ), with probability at least 1 − δ/K over samples Sk ∼ Dn k , EDk � 1 � hk(x) ̸= yk�� ≤ inf fk∈Fk EDk � 1 � fk(x) ̸= yk�� + ǫ K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Summing these risk bounds over all coordinates k and using union bounds over probabilities, we get that with probability at least 1 − δ over samples S ∼ Dn, K � k=1 EDk � 1 � hk(x) ̸= yk�� ≤ K � k=1 inf fk∈Fk EDk � 1 � fk(x) ̸= yk�� + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Now using the fact that the sum of infimums is at most the infimum of sums followed by the linearity of expectation gives ED � K � k=1 1 � hk(x) ̸= yk� � ≤ inf f∈F ED � K � k=1 1 � fk(x) ̸= yk� � + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This completes our proof of sufficiency as it shows that Algorithm 1 is an agnostic PAC learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH with sample complexity at most maxk mk(ǫ/K, δ/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of necessity in Theorem 2) Next, we will show that if F is learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH, then each Fk is PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Our proof is again based on reduction: given oracle access to 4 Algorithm 2 Agnostic PAC learner for F1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 0-1 loss Input: Agnostic PAC learner A for F and samples S = {(xi, yi1)}n i=1 ∼ Dn 1 1 Augment the sample S to create a K-variate target, where the augmented sample is �S = {(xi, (y1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , yK i ))}n i=1 such that yik ∼ {−1, 1} each with probability 1/2 for all i ∈ [n] and k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , K} 2 Output h1 = A1(�S), the restriction of A(�S) to its first output coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' agnostic PAC learner A for F, we will construct agnostic PAC learners A1, stated as Algorithm 2, for F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By symmetry, similar construction can be used for all other Fk’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let us consider a distribution �D on X × Y such that a sample (x, (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , yK)) from �D is obtained by first sampling (x, y1) ∼ D1 and appending yk’s sampled independently from uniform distribution on {−1, 1} for each k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For the sake of analysis, let us also denote hk = Ak(�S) to be restrictions of A(�S) from Algorithm 2 to its kth coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let m(ǫ, δ, K) denote the sample complexity of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since A is an agnostic PAC learner for F, for n ≥ m(ǫ, δ, K), with probability at least 1 − δ, we have E � D � K � k=1 1 � hk(x) ̸= yk� � ≤ inf f∈F E � D � K � k=1 1 � fk(x) ̸= yk� � + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For k ≥ 2, since the target is chosen uniformly at random from {−1, 1}, the 0-1 risk of any predictor is 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, the expression above can be written as ED1 � 1 � h1(x) ̸= y1�� + K � k=2 1/2 ≤ inf f∈F � ED1 � 1 � f1(x) ̸= y1�� + K � k=2 1/2 � + ǫ, which upon cancellation of constant factors reduces to ED1[1 � h1(x) ̸= y1� ] ≤ inf f1∈F1 ED1[1 � f1(x) ̸= y1� ] + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, Algorithm 2 is an agnostic PAC learner for F1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t ℓ0-1 with sample complexity at most m(ǫ, δ, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='2 Characterizing Batch Learnability for General Losses In Theorem 2, we characterized the learnability of a multilabel classifier w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the Hamming loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Now, we characterize the learnability of general multilabel losses that satisfy the identity of indiscernibles, namely ℓ(y1, y2) = 0 if and only if y1 = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let ℓ be a multilabel loss that satisfies the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F ⊂ YX is agnostic PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ if and only if F ⊂ YX is agnostic PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Hamming loss ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of sufficiency in Lemma 3) We will show that if F is learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Hamming loss ℓH, then F is learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' First, we show this for any realizable distribution D w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since ℓ = 0 if and only if ℓH = 0, the distribution D is also realizable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Hamming loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Furthermore, since there are at most 22K distinct possible inputs to ℓ(·, ·), the loss function can only take a finite number of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' So, for any ℓ, we can always find universal constants a and b such that aℓH ≤ ℓ ≤ bℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since F is learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t ℓH, there exists a learning algorithm A with the following property: for any ǫ, δ > 0, for a sufficiently large S ∼ Dn, the algorithm outputs a predictor h = A(S) such that, with probability 1 − δ over S ∼ Dn, ED[ℓH(h(x), y)] ≤ ǫ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This inequality upon using the fact that ℓ(h(x), y) ≤ bℓH(h(x), y) pointwise reduces to ED[ℓ(h(x), y)] ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, any realizable learner A for ℓH is also a realizable learner for ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since ℓ satisfies the identity of indiscernible, Lemma 1 guarantees the existence of agnostic PAC learner B for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 5 Algorithm 3 Realizable to agnostic reduction for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ Input: Realizable PAC learner A for F, unlabeled samples SU, labeled samples SL 1 Run A over all possible labelings of SU by F to get a concept class C(SU) = {A(SU, f(SU)) | f ∈ F|SU }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 2 Return a predictor ˆh ∈ C(SU) with lowest empirical error on SL, ˆh = arg min h∈C(SU) 1 |SL| � (x,y)∈SL ℓ(h(x), y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In particular, the agnostic PAC learner B is Algorithm 1 in [HKLM22], which we restate below in Algorithm 3 for completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We refer the reader to [HKLM22] for the complete analysis of the Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' That said, we now give a high-level idea of why it works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Suppose A is a realizable PAC learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ with sample complexity mA(ǫ, δ, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, for any function f ∈ F, given an unlabeled sample SU of size mA(ǫ/2, δ/2, K), running A on the labeled sample (SU, f(SU)) guarantees that, with probability 1 − δ/2 over SU ∼ DX , EDX [ℓ(f(x), f ′(x))] ≤ ǫ/2, where f ′ = A � (SU, f(SU)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This subsequently guarantees that running A over all possible labelings generates a function class C(SU) with the property that for every f ∈ F, with high probability, there exists f ′ ∈ C(SU) such that the risk of f and f ′ are close under D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' So, with high probability, C(SU) must contain a function ˜f whose risk is close to that of the optimal predictor f ⋆ in F for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since C(SU) is a finite function class, running ERM over it with sufficiently large labeled samples SL should return a predictor h whose risk is arbitrarily close to that of ˜f, and thus close to the optimal predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' One can formalize this argument using Hoeffding’s bound to show that for |SL| ≥ O � log (|C(SU)|/δ) ǫ2 � , we obtain with probability 1 − δ, ED � ℓ(ˆh(x), y) � ≤ inf f∈F ED [ℓ(f(x), y)] + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Together, the sample complexity of algorithm B, denoted as mB(ǫ, δ, K), is the sample complexity of A plus the number of samples required for empirical risk minimizer of step 2 to generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' That is, mB(ǫ, δ, K) ≤ mA(ǫ/2, δ/2, K) + O � 1 ǫ2 log |C(SU)| δ � ≤ mA(ǫ/2, δ/2, K) + O �mA(ǫ/2, δ/2, K) K + log 1 δ ǫ2 � where the last step follows upon using the fact that |C(SU)| ≤ 2mA(ǫ/2,δ/2,K) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, we first showed that an agnostic PAC learner A of F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH is a realizable PAC learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ, and then provided a black-box reduction of A to B, an agnostic PAC learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of necessity in Lemma 3) Next, we will show that if F is learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ, then F is learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Hamming ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Our proof will follow a similar route as in the proof for sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' First, we show this for any realizable distribution D w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Due to the alignment of 0, the distribution D is also realizable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' As both ℓH and ℓ take at most finite number of values, we can find universal constants a and b such that aℓH ≤ ℓ ≤ bℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since F is learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t ℓ, there exists a learning algorithm A with the following property: for any ǫ, δ > 0, for a sufficiently large S ∼ Dn, the algorithm outputs a predictor h = A(S) such that, with probability 1 − δ over S ∼ Dn, ED[ℓ(h, (x, y))] ≤ aǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 6 This inequality upon using the fact that aℓH(h, (x, y)) ≤ ℓ(h, (x, y)) pointwise yields ED[ℓH(h, (x, y))] ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, any realizable learner A for ℓ is also a realizable learner for ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since ℓH satisfies the identity of indiscernibles, Lemma 1 guarantees the existence of agnostic PAC learner B for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In fact, the algorithm B is Algorithm 3 after replacing ℓ with ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' If mA(ǫ, δ, K) is the sample complexity of A, then a similar argument as in the sample complexity analysis of the sufficiency direction gives that the sample complexity of B is mB(ǫ, δ, K) ≤ mA(ǫ/2, δ/2, K) + O �mA(ǫ/2, δ/2, K) K + log 1 δ ǫ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ As an immediate consequence of Lemma 3 and Theorem 2, we can deduce the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let ℓ be a multilabel loss function satisfying the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F ⊂ YX is agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ if and only if each restriction Fk ⊂ YX k is agnostic PAC learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 4 Online Multilabel Classification In this section, we provide analogs of Theorem 2 and 4 in the online setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Like before, we begin by characterizing the learnability of the Hamming loss and then move to give a characterization of learnability for all losses satisfying the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Throughout this section, we give regret bounds assuming an oblivious adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A standard reduction (see Chapter 4 in [CBL06]) allows us to convert oblivious regret bounds to adaptive regret bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='1 Characterizing Online Learnability for the Hamming Loss Theorem 5 presents the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F ⊂ YX is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the Hamming loss if and only if each restriction Fk ⊂ YX k is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of sufficiency for Theorem 2) We first prove that online learnability of each restriction is sufficient for online learnability of ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Our proof is based on a reduction: given oracle access to online learners for {Fk}K k=1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1, we will construct an online learner A for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Algorithm 4 Online Learner A for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH Input: Online learners {Ak}K k=1 for Fk’s 1 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', T do 2 Receive example xt 3 Predict ˆyt = (A1(xt), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', AK(xt)) 4 Receive true label yt = (y1 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', yK t ) and suffer loss ℓM(ˆyt, yt) 5 Update Ak by passing (xt, yk t ) for k ∈ [K] 6 end It suffices to show that the expected regret of A is sublinear in T w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By Definition 4, we have that for all k ∈ [K], E � T � t=1 1{Ak(xt) ̸= yk t } − inf fk∈Fk T � t=1 1{fk(xt) ̸= yk t } � ≤ Rk(T ) where Rk(T ) is some sublinear function in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Summing the regret bounds across all k ∈ [K] splitting up the expectations, and using linearity of expectation, we get that E � K � k=1 T � t=1 1{Ak(xt) ̸= yk t } � − E � K � k=1 inf fk∈Fk T � t=1 1{fk(xt) ̸= yk t } � ≤ K � k=1 Rk(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 7 Noting that �K k=1 inffk∈Fk �T t=1 1{fk(xt) ̸= yk t } ≤ inff∈F �K k=1 �T t=1 1{fk(xt) ̸= yk t }, the bound above reduces to E � K � k=1 T � t=1 1{Ak(xt) ̸= yk t } � − E � inf f∈F � K � k=1 T � t=1 1{fk(xt) ̸= yk t } �� ≤ K � k=1 Rk(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Swapping the order of summations, noting that A(xt) = (A1(xt), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', AK(xt)), and using the definition of Hamming loss we have that, E � T � t=1 ℓH(A(xt), yt) � − E � inf f∈F T � t=1 ℓH(f(xt), yt) � ≤ K � k=1 Rk(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This concludes the proof of this direction since �K k=1 Rk(T ) is still a sublinear function in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of necessity for Theorem 2) Next, we prove that online learnability of each restriction is nec- essary for online learnability of ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Our proof again is based on a reduction: given oracle access to an online learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH, we will construct online learners for {Fk}K k=1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The Algorithm below makes this precise by constructing an online learner for F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Algorithm 5 Online Learner for F1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1 Input: Online learner B for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH 1 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', T do 2 Receive example xt 3 Predict ˆy1 t = B1(xt) 4 Receive true label y1 t and suffer loss 1{ˆy1 t ̸= y1 t )} 5 Update B by passing (xt, (y1 t , y2 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', yK t )), where yk t ∼ {−1, 1}, each with probability 1/2, for k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', K} 6 end It suffices to show that the expected regret of A1 is sublinear in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' As previously mentioned, we assume that the sequence (x1, y1 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , y1 T) is chosen by an oblivious adversary, and thus is not random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let yt = (y1 t , y2 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', yK t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By Definition 4, we have that, E � T � t=1 ℓH(B(xt), yt) − inf f∈F T � t=1 ℓH(f(xt), yt) � ≤ R(T, K) where the expectation is over both the randomness of B(xt) and (y2 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', yK t ) and R(T, K) is a sub- linear function of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Splitting up the expectation, using the definition of the Hamming loss, and by the linearity of expectation, we have that T � t=1 K � k=1 E � 1{Bk(xt) ̸= yk t } � − E � inf f∈F T � t=1 K � k=1 1{fk(xt) ̸= yk t } � ≤ R(T, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since E � inff∈F �T t=1 �K k=1 1{fk(xt) ̸= yk t } � ≤ inff∈F E ��T t=1 �K k=1 1{fk(xt) ̸= yk t } � , we get that T � t=1 K � k=1 E � 1{Bk(xt) ̸= yk t } � − inf f∈F T � t=1 K � k=1 E � 1{fk(xt) ̸= yk t } � ≤ R(T, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, observe that for every t ∈ [T ], for every k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', K}, and for any predictor fk, by the randomness of yk t , we have that E � 1{Bk(xt) ̸= yk t } � = E � 1{f(xt) ̸= yk t } � = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, we get that, T � t=1 E � 1{B1(xt) ̸= y1 t } + K − 1 2 � − inf f∈F T � t=1 E � 1{f1(xt) ̸= y1 t } + K − 1 2 � ≤ R(T, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Canceling constant factors we get that, E � T � t=1 1{B1(xt) ̸= y1 t } � − inf f1∈F1 T � t=1 1{f1(xt) ̸= y1 t } ≤ R(T, K), 8 showcasing that Algorithm 5 is an online agnostic learner for F1 with respect to ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Finally, by symmetry, a similar reduction can be used to construct online learners for each restriction Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This completes the proof of both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ Before we move on to characterize the online learnability of general losses, we first need an online analog of the realizable-to-agnostic conversion in the batch setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The next subsection covers this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='2 Realizable-to-Agnostic Online Conversion Our strategy for constructively characterizing the learnability of general multilabel losses in the batch setting required the ability to convert a realizable learner to an agnostic learner in a black-box fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In this section, we provide an analog of this conversion for the online setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' More specifically, we focus on the multiclass classification setting with the 0-1 loss where |Y| = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, we give an algorithm that takes as input a (potentially randomized) realizable (multiclass) online learner for ℓ0-1 and outputs an agnostic (multiclass) online learner for ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 6 formalizes the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let A be a realizable (multiclass) online learner for ℓ0-1 with expected mistake-bound R(T, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, there is an agnostic (multiclass) online learner for ℓ0-1 with expected regret bound O( � T R(T, K) ln(KT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' As previously mentioned, we will assume the adversary is oblivious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Accordingly, let (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ) denote the sequence of labelled instances to be observed by the online learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let H ⊂ YX be a hy- pothesis class and A be a (potentially randomized) online realizable learner for H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By definition, this means that for any realizable sequence (x1, h(x1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , h(xT )), we have that E � T � t=1 1{A(xt) ̸= h(xt)} � ≤ R(T, K) where R(T, K) is a sub-linear function of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We now use A to construct an agnostic online learner w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The high-level idea is very similar to the conversion of the Standard Optimal Algorithm (SOA) in the multiclass classification setting into an agnostic online learner (see [DSBDSS11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The only wrinkle we will need to deal with is the fact that A could be a potentially randomized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' To this end, we will construct a finite set of experts E such that for every h ∈ H, with high probability, there exists a E ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' for all t ∈ [T ], h(xt) = E(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In particular, this will also be true for the optimal hypothesis h∗ = arg minh∈H �T t=1 ℓ0-1(h(xt), yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, we can run the celebrated Randomized Exponential Weights Algorithm (REWA) using E as the set of experts and ℓ0-1 as the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since, with high probability, the behavior of h∗ is exactly captured by an expert in E, the best possible loss over the experts infE∈E �T t=1 ℓ0-1(E(xt), yt) is at most the best possible loss over H, infh∈H �T t=1 ℓ0-1(h(xt), yt), with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We now formally begin this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Recall, that we want to construct a finite set of experts E such that for all hypothesis h ∈ H, with high probability, there exists a E ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' for all t ∈ [T ], h(xt) = E(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Fix a h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Given time horizon T , let LT = {L ⊂ [T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' |L| ≤ 2R(T, K)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For every L ∈ LT , let ΦL = YL be the set of all possible functions φ : L → Y mapping time points in L to labels in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that |ΦL| = K|L|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For every L ∈ LT , further denote φh L ∈ ΦL as the mapping that exactly corresponds to h, that is φh L(t) = h(xt) for all t ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Finally, for every L ∈ LT and every φ ∈ ΦL we define an expert EL,φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The expert EL,φ simulates a game between A and the environment on the sequence of instances x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', xT assuming that A makes a mistake (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' h) precisely on the rounds in L and the true labels of these mistaken examples are determined by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The algorithm below formalizes this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Using the procedure above, we have constructed N = �2R(T,K) j=0 �T j � Kj ≤ (T K)2R(T,K) experts, each of them using an independent copy of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, we construct M such independent batches of N experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This will allow us to amplify the probability that there exists at least one expert whose behavior matches h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' That is, each of the M batches will contain N = �2R(T,K) j=0 �T j � Kj ≤ (KT )2R(T,K) experts constructed exactly in the same way as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Because of this, we let Ei L,φ denote the expert EL,φ in the i’th batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' E = �M i=1 � L∈LT � φ∈ΦL{Ei L,φ} denotes the set of all experts across 9 Algorithm 6 Expert(L, φ) Input: Independent copy of A 1 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', T do 2 Receive example xt 3 Let ˜yt = A(xt) 4 if t ∈ L then 5 Predict ˆyt = φ(t) 6 else 7 Predict ˆyt = ˜yt 8 end 9 Update A by passing (xt, ˆyt) 10 end all batches, and lastly, Eh ⊂ E denotes those set of experts parameterized by functions φh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' That is, Eh = �M i=1 � L∈LT {Ei L,φh L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since we are in the oblivious setting, the true sequence (x1, yt), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ) is not random and therefore the sequence (x1, h(xt)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , h(xT )) is not random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' But, since A is a potentially random- ized algorithm, the time points it makes mistakes when fed (x1, h(xt)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , h(xT )) are random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In this sense, A induces a distribution over the subsets of indices in [T ] where it makes mistakes on the sequence (x1, h(xt)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , h(xT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Notice that in the event that exists an expert Ei L,φh L ∈ Eh whose indices L exactly match up with the time points where its copy of A makes mistakes on the sequence (x1, h(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , h(xT )), then Ei L,φh L feeds its copy of A exactly the sequence of instances labelled by h, and therefore predicts exactly h(xt) in each round t ∈ [T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, this is precisely the event that we want to show occurs with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' To that end, let Ai L,φ be the copy of A associated with expert Ei L,φ and Ih(Ai L,φ) be the ran- dom variable denoting the set of time points where Ai L,φ makes mistakes when run on the sequence (x1, h(x1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , h(xT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, we have P � ∃Ei L,φh L ∈ Eh s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Ih(Ai L,φh L) = L � = 1 − P � ∀Ei L,φh L ∈ Eh, Ih(Ai L,φh L) ̸= L � = 1 − ΠM i=1ΠL∈LT P � I(Ai L,φh L) ̸= L � = 1 − � ΠL∈LT P � Ih(A1 L,φh L) ̸= L ��M = 1 − � ΠL∈LT (1 − P � Ih(A1 L,φh L) = L � ) �M = 1 − (ΠL∈LT (1 − P [Ih(A) = L]))M , where the last equality follows from the fact that {A1 L,φh L}L∈LT are independent copies of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let Bh denote the event that A makes at most 2R(T, K) mistakes when run on the sequence (x1, h(x1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , h(xT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, P [Ih(A) = L] ≥ P [Ih(A) = L|Bh] P [Bh] ≥ P [Ih(A) = L|Bh] 2 , where the last inequality follows from the fact that A has an expected mistake bound of R(T, K) and therefore by Markov’s inequality, with probability at least 1 2, �T t=1 1{A(xt) ̸= h(xt)} ≤ 2R(T, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Putting things together, we have that, P � ∃Ei L,φh L ∈ Eh s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Ih(Ai L,φh L) = L � ≥ 1 − � ΠL∈LT � 1 − P [Ih(A) = L|Bh] 2 ��M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, note that � L∈LT P [Ih(A) = L|Bh] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This is because the support of the conditional distribution over the set of time points that A makes mistakes, given it makes at most 2R(T, K) 10 mistakes, is exactly LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, we have that ΠL∈LT � 1 − P[Ih(A)=L|Bh] 2 � ≤ e− 1 2 using the fact that 1−x ≤ e−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Putting things together, we get that P � ∃Ei L,φh L ∈ Eh s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Ih(Ai L,φh L) = L � ≥ 1−e− M 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since Eh ⊂ E, this further implies that P � ∃Ei L,φ ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Ih(Ai L,φ) = L � ≥ 1 − e− M 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Setting M = 2 ln( 2 δ ), we get that with probability at least 1 − δ 2, ∃Ei L,φ ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Ih(Ai L,φ) = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This means that we have successfully constructed a set E of MN ≤ 2 ln( 2 δ )(KT )2R(T,K) experts such that with high probability, there exists an expert who exactly matches the behavior of h on the sequence of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since h is arbitrary, this must be true for any hypothesis and in particular, for the optimal hypothesis h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, we run REWA over E on the stream (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ) using ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A nice property about the REWA is that for any finite set of experts G, for ℓ0-1, and for learning rate η = � 8 ln(|G|) T , with probability at least 1 − δ 2, T � t=1 ℓ0-1(ˆyt, yt) − inf g∈G � T � t=1 ℓ0-1(g(xt), yt) � ≤ � T 2 ln(|G|) + � T 2 ln �2 δ � where ˆyt is the prediction of REWA in the t’th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Taking G to be E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' and union bounding over the event that there exists an expert E ∈ E that exactly matches the behavior of h∗ and the event that REWA succeeds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' we get that with probability 1 − δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' T � t=1 ℓ0-1(ˆyt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' yt) − inf h∈H � T � t=1 ℓ0-1(h(xt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' yt) � ≤ � T 2 ln(MN) + � T 2 ln �2 δ � ≤ � T 2 ln(M(KT )2R(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='K)) + � T 2 ln �2 δ � = � T 2 (ln(M) + 2R(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' K) ln(KT )) + � T 2 ln �2 δ � ≤ � T 2 ln(2 ln(2 δ )) + � T R(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' K) ln(KT )) + � T 2 ln �2 δ � = O � � T R(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' K) ln(KT ) + � T ln(1 δ ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Picking δ = 1 T , a standard argument yields E � T � t=1 ℓ0-1(ˆyt, yt) − inf h∈H T � t=1 ℓ0-1(h(xt), yt) � ≤ O �� T R(T, K) ln(KT ) � as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, running REWA over of set up experts E gives an online agnostic learner for H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ The reduction from realizable to agnostic online learning can be extended beyond the multiclass classification setting and the 0-1 loss to the multilabel classification setting and any multilabel loss function satisfying the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Corollary 7 formalizes this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let ℓ be any multilabel loss function satisfying the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' If A is a realizable (multilabel) online learner for ℓ with expected loss bound R(T, K), then there exists an agnostic (multilabel) online learner for ℓ with expected regret bound O � b � KT R(T,K) a ln(T ) � , where a and b are universal constants that depend on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let F ⊂ YX be a multilabel function class with target space Y = {−1, 1}K for some K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let ℓ : Y ×Y → R be any non-negative multilabel loss function satisfying the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, since there are at most 2K possible values for ℓ, there must exist constants a and b s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' for 11 all y1, y2 ∈ Y, aℓ0-1(y1, y2) ≤ ℓ(y1, y2) ≤ bℓ0-1(y1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let A be an online realizable learner w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, by definition, for any sequence of instances (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT) realized by F, E � T � t=1 ℓ(A(xt), yt) � ≤ R(T, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Furthermore, since aℓ0-1 ≤ ℓ, we can write E � T � t=1 1{A(xt) ̸= yt} � ≤ R(T, K) a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, A is also a multiclass online realizable learner w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1 and therefore a valid input realizable online learner for the construction in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' To that end, we will follow the same construction in the proof of Theorem 6 using A as the input realizable online learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' However, we will need a slight modification to accommodate the fact that we are interested in the multilabel loss ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Notice that one can run REWA used in the construction of the agnostic learner in the proof of Theorem 6 with any loss function ℓ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, instead of running the REWA using ℓ0-1, we can run REWA using ℓ b ∈ [0, 1], since b trivially upper bounds ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Moreover, since our set of experts E enjoys the property that for any f ∈ F, with high probability, there exists an expert in E ∈ E that exactly matches the behavior of f, we have that infE∈E ��T t=1 ℓ(E(xt),yt) b � ≤ inff∈F ��T t=1 ℓ(f(xt),yt) b � with high probability for the loss function ℓ b as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let ˜ A be the result of following the same construction used in the proof of Theorem 6, but using A as the online realizable learner and ℓ b as the loss function in REWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, by Theorem 6 and the observations above, we have that E � T � t=1 ℓ( ˜ A(xt), yt) − inf f∈F T � t=1 ℓ(f(xt), yt) � ≤ O � b � T R(T, K) a ln(2KT ) � ≤ O � b � KT R(T, K) a ln(T ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This completes the proof as we have shown that ˜ A is an online agnostic learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='3 Characterizing Online Learnability for General Losses Using Theorem 5 and Corollary 7, we now characterize the learnability of arbitrary multilabel loss functions ℓ as long as they satisfy the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The key idea is that since there are only finite number of possible inputs to ℓ, for any ℓ satisfying the identity of indiscernibles, there must exist universal constants a and b s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' aℓH(y1, y2) ≤ ℓ(y1, y2) ≤ bℓH(y1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, we can characterize the learnability of ℓ by relating it to the learnability of ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let ℓ be any loss function satisfying the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F ⊂ YX is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ if and only if F is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let a and b be the universal constants s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' for all y1, y2 ∈ Y, aℓH(y1, y2) ≤ ℓ(y1, y2) ≤ bℓH(y1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We will first show that if F is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH, then F is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By Corollary 7, it suffices to give a realizable online learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since F is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH, there exists an algorithm A s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' for any sequence (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ), we have E � T � t=1 ℓH(A(xt), yt) − inf f∈F T � t=1 ℓH(f(xt), yt) � ≤ R(T, K) where R(T, K) is a sublinear function of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In the realizable setting, we are guaranteed that for any sequence (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ) that the online learner may observe, there exists a f ∈ F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t f(xt) = yt for all t ∈ [T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since ℓH satisfies the identity of indiscernibles, we have that for any realizable sequence (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ), inff∈F �T t=1 ℓH(f(xt), yt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, we have that E ��T t=1 ℓH(A(xt), yt) � ≤ 12 R(T, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Noting that ℓH(A(xt), yt) ≥ ℓ(A(xt),yt) b completes this portion of the proof as it implies that E ��T t=1 ℓ(A(xt), yt) � ≤ bR(T, K), showcasing that A is also a realizable online learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The construction in Corollary 7 can then be used to convert A into an agnostic online learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, we show the reverse direction - if F is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ, then F is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Again, by Corollary 7 it suffices to construct an online learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH in the realizable setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We can repeat the exact same procedure above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since F is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ, there exists an algorithm A s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' for any sequence (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ), we have E � T � t=1 ℓ(A(xt), yt) − inf f∈F T � t=1 ℓ(f(xt), yt) � ≤ R(T, K) where R(T, K) is a sublinear function of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In the realizable setting, we are guaranteed that for any sequence (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ) that the online learner may observe, there exists a f ∈ F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t f(xt) = yt for all t ∈ [T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since ℓ satisfies the identity of indiscernibles, we have that for any realizable se- quence (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ), inff∈F �T t=1 ℓ(f(xt), yt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, we have that, E ��T t=1 ℓ(A(xt), yt) � ≤ R(T, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Noting that ℓ(A(xt), yt) ≥ aℓH(A(xt), yt) completes this portion of the proof as it implies that E ��T t=1 ℓH(A(xt), yt) � ≤ R(T,K) a , showcasing that A is also a realizable online learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The construction in Corollary 7 can then be used to convert A into an agnostic online learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ As an immediate consequence of Lemma 8 and Theorem 5, we get the following Theorem charac- terizing the online learnability of general multilabel losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let ℓ be any multilabel loss function that satisfies the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F ⊂ YX is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ if and only if each restriction Fk ⊂ YX k is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='4 Bandit Online Multilabel Classification We extend the results in the previous subsection to the online setting where the learner only observes bandit feedback in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 10 gives a characterization of bandit online learnability of a function class F in terms of the online learnability of each restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let ℓ be any loss function that satisfies the identity of indiscernibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' A function class F ⊂ YX is bandit online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ if and only if each restriction Fk ⊂ YX k is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let b be such that for all y1, y2 ∈ Y, ℓ(y1, y2) ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We first show necessity - if F is bandit online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ, then each restriction Fk is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This follows trivially from the fact that if A is a bandit online learner for F, then A is also an online learner for F under full-feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, by Theorem 9, online learnability of F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ implies online learnability of restriction Fk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We now focus on showing sufficiency - if every restriction {Fk}K k=1 is online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 0-1 loss, then F is bandit online learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' loss ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' At a high level, the proof of this direction is very similar to the proof of Theorem 10 and the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='3 in [DH13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We first construct a finite set of experts E such that with high probability there exists an expert E ∈ E that exactly matches the behavior of the best function f ∗ with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, we run EXP4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='IX from [Neu15] over this set of experts using the scaled loss ℓ b, which gives a high probability regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Union bounding with the event that there exists an expert that exactly matches the behavior of the optimal function completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We now formalize the sketch above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By Theorem 9, if all restrictions Fk are online learnable, then F is online agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1 and therefore online realizable learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This means that there exists an online learner A s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' for any sequence (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=', (xT , yT ) realized by F, we have 13 E � T � t=1 1 {A(xt) ̸= yt} � ≤ R(T, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that this means that A is also a realizable multiclass online learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, we can use A to construct the same set of experts E as in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Namely, by using A as the realizable multiclass online learner, we can construct a finite set of experts of size M(T 2K)2R(T,K) such that with probability at least 1 − e −M 2 , there exists an expert E ∈ E that exactly matches the behavior of the optimal function in hindsight f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' If we set M = ln( 2 δ ), then with probability 1 − δ 2, there exists an expert E ∈ E that exactly matches the behavior of f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that in each round t ∈ [T ], every expert E ∈ E outputs an element of Y, which can also be thought of as a distribution over Y that places all mass on one particular y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, for every round t ∈ [T ], we can view each expert as outputting a distribution over the label (action) space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' From this perspective, we can run the bandit algorithm EXP4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='IX from [Neu15] over our set of experts E using the scaled loss ℓ b ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For an appropriately chosen learning rate η, this guarantees that with probability 1 − δ 2, T � t=1 ℓ(ˆyt, yt) − inf E∈E T � t=1 ℓ(E(xt), yt) ≤ O � b � |Y|T ln(|E|) + b � |Y|T ln(4 δ ) � where ˆyt is the prediction of EXP4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='IX in the t’th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Union bounding with the event that there exists an expert that matches the behavior of the optimal function f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' we get that with probability 1 − δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' T � t=1 ℓ(ˆyt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' yt) − inf f∈F T � t=1 ℓ(f(xt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' yt) ≤ O � b � 2KT ln(ln(2 δ )(T 2K)2R(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='K)) + b √ 2KT ln(4 δ ) � ≤ O � b � K2KR(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' K)T ln(T ) + b √ 2KT ln(4 δ ) � Taking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' δ = 1 T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' we have that with probability 1 − 1 T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' T � t=1 ℓ(ˆyt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' yt) − inf f∈F T � t=1 ℓ(f(xt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' yt) ≤ O � b � K2KR(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' K)T ln(T ) + b √ 2KT ln(T ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' from which we can conclude that E � T � t=1 ℓ(ˆyt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' yt) � − inf f∈F T � t=1 ℓ(f(xt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' yt) ≤ O � b � K2KR(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' K)T ln(T ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This concludes the proof as we have shown that EXP4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='IX run over E is a bandit online learner for ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ 5 Batch Multioutput Regression In this section, we consider the case when Y = [0, 1]K ⊂ RK for K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This target space is without loss of generality because one can always normalize each Yk to [0,1] by subtracting the lower bound and dividing by the upper bound of Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Following our outline in classification, we will first study learnability under decomposable losses and then study a non-decomposable loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='1 Characterizing Learnability for Lp norms A canonical loss function for multioutput regression is the p-norm, defined as ℓp(f(x), y) = K � k=1 |fk(x) − yk|p, 14 for 1 ≤ p < ∞ and f(x), y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The p-norm loss is a natural multivariate extension of the metric dp(fk(x), yk) := |fk(x) − yk|p, which is generally taken as a loss function in a real-valued regression setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The following result establishes an equivalence between the learnability of F ⊂ YX w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the p-norm and the learnability of each Fk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the dp-metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The function class F ⊂ YX is agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓp if and only if each of Fk ⊂ YX k is agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of sufficiency in Theorem 11) We will first prove that the agnostic learnability of each Fk is sufficient for the agnostic learnability of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' As in the classification setting, the proof here is based on reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' That is, given oracle access to agnostic learners Ak for each Fk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' dp loss, we will construct an agnostic learner A for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓp loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Algorithm 7 Agnostic Learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓp Input: Agnostic learners {Ak}K k=1 for Fk’s and samples S = {(xi, yi)}n i=1 ∼ Dn on X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 1 Construct samples Sk = {xi, yk i }n i=1 with scalar-valued targets for all k ∈ [K] 2 Get predictors gk = Ak(Sk) for all k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 3 Output g = (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , gk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We will show that Algorithm 7 is an agnostic learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Denote Dk to be the marginal distribution of D restricted to X ×Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let us use mk(ǫ, δ) to denote the sample complexity of Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since Ak is an agnostic learners for Fk, we have that for sample size n ≥ maxk mk( ǫ K , δ K ), with probability at least 1 − δ/K over samples Sk ∼ Dn k , EDk[|gk(x) − yk|p] ≤ inf fk∈Fk EDk[|fk(x) − yk|p] + ǫ K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Summing these risk bounds over all k coordinates and union bounding over the success probabilities, we get that with probability at least 1 − δ over samples S ∼ Dn, K � k=1 EDk[|gk(x) − yk|p] ≤ K � k=1 inf fk∈Fk EDk[|fk(x) − yk|p] + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Using the fact that the sum of infimums over individual coordinates is at most the overall infimum of sums followed by the linearity of expectation, we can write the expression above as ED � K � k=1 |gk(x) − yk|p � ≤ inf f∈F ED � K � k=1 |fk(x) − yk|p � + ǫ, showcasing that Algorithm 7 is an agnostic learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓp with sample complexity at most maxk mk(ǫ/K, δ/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This completes our proof of sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of necessity in Theorem 11) Next, we will show that if F is agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓp, then each Fk is agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Given oracle access to agnostic learner A for F, we will construct agnostic learners A1 for F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By symmetry, a similar reduction can then be used to construct an agnostic learner for each component Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since we will be given a sample with a single variate target, the main question is to find the right way to augment samples to a K-variate target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In the proof of Theorem 2, we showed that randomly choosing yik ∼ Uniform({−1, 1}) for k ≥ 2 results in all predictors having a constant 1/2 risk–leaving only the risk of the first component on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Unfortunately, for regression settings, no single augmentation works for every distribution on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, we will augment the samples by considering all possible behaviors of (F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , FK) on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since the function class maps to a potentially uncountably infinite space, we will first discretize each component of the function class and consider all possible labelings over the discretized space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Fix 1 > α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For k ≥ 2, define the discretization f α k (x) = �f(x) α � α 15 for every fk ∈ Fk and the discretized component class Fα k = {f α k |fk ∈ Fk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that a function in Fk maps to {0, α, 2α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , ⌊1/α⌋α} and the size of the range of the discretized function class Fα k is 1 + ⌊1/α⌋ ≤ (α + 1)/α ≤ 2/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For the convenience of exposition, let us define Fα 2:K to be Fα without the first component, and we will denote f α 2:K to be an element of Fα 2:K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Algorithm 8 Agnostic learner for F1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' dp Input: Agnostic learner A for F samples S = (x1:n, y1 1:n) ∼ Dn 1 and another independent samples �S from D1 1 Define Saug = {(x1:n, y1 1:n, f α 2:K(x1:n) | f2:K ∈ Fα 2:K}, all possible augmentations of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 2 Run A over all possible augmentations to get C(S) := � A � Sa � | Sa ∈ Saug � 3 Define C1(S) = {g1 | (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , gk) ∈ C(S)}, a restriction of C(S) to its first coordinate output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 4 Return ˆg1, the predictor in C1(S) with the lowest empirical error over �S w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We will now show that Algorithm 8 is an agnostic learner for F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' First, let us define f ⋆ 1 := arg min f1∈F1 ED1[|f1(x) − y1|p], to be optimal predictor in F1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By definition of F1, there must exist f ⋆ 2:K ∈ F2:K such that (f ⋆ 1 , f ⋆ 2:K) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We note that f ⋆ k need not be optimal predictors in Fk for k ≥ 2, but we use the ⋆ notation just to associate these component functions with the first component function f ⋆ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Define f ⋆,α 2:K ∈ Fα 2:K to be the corresponding discretization of f ⋆ 2:K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Suppose g = A((x1:n, y1 1:n, f ⋆,α 2:K(x1:n)) is the predictor obtained by running A on the sample aug- mented by f ⋆,α 2:K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that g ∈ C(S) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let mA(ǫ, δ, K) be the sample complexity of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since A is an agnostic learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t ℓp, we have that for n ≥ mA(ǫ/4, δ/2, K), with probability at least 1 − δ/2, ED1 � |g1(x) − y1|p� + K � k=2 EDX � |gk(x) − f ⋆,α k (x)|p� ≤ inf f∈F � ED1 � |f1(x) − y1|p� + K � k=2 EDX � |fk(x) − f ⋆,α k (x)|p� � + ǫ 4 Note that the quantity on the left is trivially lower bounded by the risk of the first component and the optimal risk on the right-hand side is trivially upper bounded by the risk of (f ⋆ 1 , f ⋆ 2:K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In particular, we have ED1 � |g1(x) − y1|p� ≤ ED1 � |f ⋆ 1 (x) − y1|p� + K � k=2 EDX � |f ⋆ k(x) − f ⋆,α k (x)|p� + ǫ 4 ≤ ED1 � |f ⋆ 1 (x) − y1|p� + Kαp + ǫ 4, where the last inequality follows upon using the fact that |f ⋆ k(x) − f ⋆,α k (x)| ≤ α for all x and k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Picking α = (ǫ/4K)1/p and using the definition of f ⋆ 1 , we obtain ED1 � |g1(x) − y1|p� ≤ inf f1∈F1 ED1[|f1(x) − y1|p] + ǫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, we have shown the existence of one predictor g ∈ C(S) such that its restriction to the first component, g1, obtains the agnostic bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Recall that by Hoeffding’s Inequality and union bound, with probability at least 1 − δ/2, the empirical risk of every hypothesis in C1(S) on a sample of size ≥ O � 1 ǫ2 log |C1(S)| δ � is at most ǫ/4 away from its true error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' So, if |�S| ≥ O � 1 ǫ2 log |C1(S)| δ � , then with probability at least 1 − δ/2, we have 16 1 |�S| � (x,y1)∈ � S |g1(x) − y1|p ≤ ED1 � |g1(x) − y1|p� + ǫ 4 ≤ inf f1∈F1 ED1[|f1(x) − y1|p] + 3ǫ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since ˆg1 is the ERM on �S over C1(S), its empirical risk can be at most inff1∈F1 ED1[|f1(x)−y1|p]+ 3ǫ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Given that the population risk of ˆg1 is at most ǫ/4 away from its empirical risk, we have that ED1[|ˆg1(x) − y1|p] ≤ inf f1∈F1 ED1[|f1(x) − y1|p] + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Applying union bounds, the entire process succeeds with probability 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Letting m1(ǫ, δ) denote the sample complexity of Algorithm 8, we have that m1(ǫ, δ) is at most the sample complexity of A plus the number of samples required for ERM in step 4 to succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, m1(ǫ, δ) ≤ mA(ǫ/4, δ/2, K) + O � 1 ǫ2 log |C1(S)| δ � ≤ mA(ǫ/4, δ/2, K) + O �mA(ǫ/4, δ/2, K) K log 2 α + log 1 δ ǫ2 � ≤ mA(ǫ/4, δ/2, K) + O \uf8eb \uf8ed mA(ǫ/4, δ/2, K) K � 1 + 1 p log K ǫ � + log 1 δ ǫ2 \uf8f6 \uf8f8 , where the second inequality follows due to |C1(S)| ≤ (2/α)mA(ǫ/4,δ/2,K) Kand the last equality follows due to our choice of α = (ǫ/4K)1/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This completes the proof as it shows that Algorithm 8 is an agnostic learner for F1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='2 Characterizing Learnability for the Max Loss Next, we will study the learnability of function class F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' a non-decomposable loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In the regression setting, the natural non-decomposable loss to consider is ℓ∞, which is defined as ℓ∞(f(x), y) := max k∈[K] |fk(x) − yk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The following result characterizes the agnostic learnability of F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The function class F ⊂ YX is agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞ if and only if each of Fk ⊂ YX k is agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' the absolute value loss, d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of sufficiency of Theorem 12) We will first prove that agnostic learnability of each Fk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' d1 is sufficient for agnostic learnability of F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' For this direction, we will first discretize the function class F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, we will show how agnostic learners Ak’s for Fk’s can be used to construct a realizable learner for the discretization of F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since the discretized function class maps to finite label space, we can do the standard realizable-to-agnostic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Finally, we will show that an agnostic learner for the discretized function class is an agnostic learner for the original function class F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' To that end, for 0 < α < 1 and for each f ∈ F, define its corresponding discretization as f α(x) := ��f1(x) α � α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , �fK(x) α � α � , where f(x) = (f1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , fK(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, define the discretized function class, Fα = {f α | f ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' First, we will show that A constructed in step 1 of Algorithm 9 is a realizable learner for Fα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Consider a distribution D that is realizable by Fα w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞, that is inff α∈F α ED[maxk |f α k (x)−yk|] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since maxk |f α k (x) − yk| ≥ |f α k (x) − yk| for each k ∈ [K], we have that inf f α k ∈F α k EDk[|f α k (x) − yk|] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 17 Algorithm 9 Agnostic learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞ Input: Agnostic learner Ak’s for Fk’s, unlabed samples SU ∼ DX , and labeled samples SL ∼ D 1 For any S ∼ D and its restriction Sk ∼ Dk, define algorithm A(S) := (A1(S1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' , AK(SK)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 2 Discretize F to get Fα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 3 Run A over all possible labelings of SU by Fα to get C(SU) := � A � SU, f α(SU) � | f α ∈ Fα |SU � 4 Return ˆg ∈ C(SU) with the lowest empirical error over SL w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The triangle inequality gives us |fk(x) − yk| ≤ |f α k (x) − yk| + |fk(x) − f α k (x)| ≤ |f α k (x) − yk| + α, which we can use to obtain inf fk∈Fk EDk � |fk(x) − yk| � ≤ inf f α k ∈F α k EDk � |f α k (x) − yk| � + α = α, for each k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Suppose S = {(xi, yi)}n i=1 ∼ Dn for n ≥ maxk mk(ǫ/2K, δ/K), where mk(ǫ, δ) is the sample complexity of Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let Sk = {(xi, y1 i )}n i=1 ∼ Dn 1 and gk = Ak(Sk) be the predictor that Ak outputs when trained on Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, with probability 1 − δ/K over samples Sk ∼ Dn k, we have EDk � |gk(x) − yk| � ≤ inf fk∈Fk EDk � |fk(x) − yk| � + ǫ 2K ≤ α + ǫ 2K , upon using the inequality established above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Union bounding over these events, we obtain with prob- ability 1 − δ, K � k=1 EDk � |gk(x) − yk| � ≤ Kα + ǫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since maxk |gk(x) − yk| ≤ � k |gk(x) − yk|, the bound above reduces to ED � max k∈[K] |gk(x) − yk| � ≤ Kα + ǫ 2 ≤ ǫ, where the last inequality follows from picking α = ǫ/4K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, the algorithm A is a realizable learner for Fα with respect to ℓ∞ with sample complexity mA(ǫ, δ, K) = maxk mk(ǫ/2K, δ/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, we will show that Algorithm 9 is an agnostic learner for Fα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The argument here is similar to the one used for realizable to agnostic reduction in [HKLM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let D be an arbitrary distribution over X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Define f ⋆,α := inf f α∈F α ED � max k∈[K] |f α k (x) − yk| � , to be the optimal predictor in Fα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that we can do this reduction because Fα maps to finite label space of size ≤ (2/α)K, and thus it makes sense to consider all possible labeling by Fα over the unlabeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let mA(ǫ, δ, K) be the sample complexity of the realizable learner A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since C(SU) contains a predictor returned by A when trained on all possible labelings of SU by Fα, it must contain a predictor ˜g = A( ˜S), where ˜S = (SU, f ⋆,α(SU)) is labeled by f ⋆,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since A is a realizable learner with respect to Fα, for |SU| ≥ mA(ǫ/4, δ/2, K), with probability 1 − δ/2, we have EDX � max k∈[K] |˜g(x) − f ⋆,α(x)| � ≤ ǫ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Using the triangle inequality, this can be further reduced to ED � max k∈[K] |˜g(x) − yk| � ≤ ED � max k∈[K] |f ⋆,α(x) − yk| � + EDX � max k∈[K] |˜g(x) − f ⋆,α(x)| � ≤ ED � max k∈[K] |f ⋆,α(x) − yk| � + ǫ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, we have shown that there exists one predictor ˜g ∈ C(SU) that has good population risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Recall that by Hoeffding’s inequality and union bound, with probability at least 1 − δ/2, the empirical 18 risk of every hypothesis in C(SU) on a sample of size ≥ 8 ǫ2 log 4|C(SU)| δ is at most ǫ/4 away from its population risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' So, if |SL| ≥ 8 ǫ2 log 4|C(SU)| δ , then with probability at least 1 − δ/2, we have 1 |SL| � (x,y)∈SL max k∈[K] |˜gk(x) − yk| ≤ ED � max k∈[K] |˜g(x) − yk| � + ǫ 4 ≤ inf f α∈F α ED[max k∈[K] |f α k (x) − yk|] + ǫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, consider the predictor ˆg returned by Algorithm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, its empirical risk can be at most inff α∈F α ED[maxk |f α k (x)−yk|]+ ǫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Given that the population risk of ˆg can be at most ǫ/4 away from its empirical risk, we have that ED � max k∈[K] |ˆgk(x) − yk| � ≤ inf f α∈F α ED � max k∈[K] |f α k (x) − yk| � + 3ǫ 4 = inf f∈F ED � max k∈[K] |f α k (x) − yk| � + 3ǫ 4 , where the last step follows because taking infimum over Fα and F are equivalent if the quantity inside expectation is defined with the discretized version of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Applying union bounds, the entire process succeeds with probability 1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Next, we note that |f α k (x)−yk| ≤ |fk(x)−yk|+|f α k (x)−fk(x)| ≤ |fk(x) − yk| + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Using this inequality, we have ED � max k∈[K] |ˆgk(x) − yk| � ≤ inf f∈F ED � max k∈[K] |fk(x) − yk| � + α + 3ǫ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that we have already picked α = ǫ/4K and the same choice of α yields ED � max k∈[K] |ˆgk(x) − yk| � ≤ inf f∈F ED � max k∈[K] |fk(x) − yk| � + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We now upper bound the sample complexity of Algorithm 9, denoted m(ǫ, δ, K) hereinafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that m(ǫ, δ, K) is at most the number of unlabeled samples required for the realizable algorithm A to succeed plus the number of labeled samples for the ERM step to succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Thus, m(ǫ, δ, K) ≤ mA(ǫ/4, δ/2, K) + O � 1 ǫ2 log |C(SU)| δ � ≤ mA(ǫ/4, δ/2, K) + O � mA(ǫ/4, δ/2, K) K log K ǫ + log 1 δ ǫ2 � , where the second inequality follows due to |C(SU)| ≤ (2/α)mA(ǫ/4,δ/2,K) Kand our choice of α = ǫ/4K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since the realizable algorithm A is constructed using individual algorithms Ak’s, we can re- late its sample complexity to the sample complexity of Ak’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Using the fact that mA(ǫ, δ, K) = maxk mk(ǫ/2K, δ/K), the sample complexity above can be rewritten as m(ǫ, δ, K) ≤ max k mk(ǫ/8K, δ/2K) + O � K log K ǫ maxk mk(ǫ/8K, δ/2K) + log 1 δ ǫ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This completes our proof of sufficiency as we have shown that Algorithm 9 is also an agnostic learner for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' (of necessity of Theorem 12) Next, we will show that if F is agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞, then each restriction Fk is agnostic learnable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Our proof will be constructive: given oracle access to an agnostic learner A for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞, we will construct agnostic learners Ak for each Fk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In particular, we will show that Algorithm 8, given as input our agnostic learner A for ℓ∞, is an agnostic learner for F1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By symmetry, a similar reduction can then be used to construct an agnostic learner for each component Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Our proof here is essentially the same as the proof of necessity in Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' So, in the same spirit, fix α > 0 and for k ≥ 2, define the discretization f α k (x) = �f(x) α � α 19 for every fk ∈ Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Define the discretized component class Fα k = {f α k |fk ∈ Fk} and define Fα 2:K to be Fα without the first component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We will denote f α 2:K to be an element of Fα 2:K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let us define f ⋆ 1 := arg min f1∈F1 ED1[|f1(x) − y1|], to be optimal predictor in F1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' By definition of F1, there must exist f ⋆ 2:K ∈ F2:K such that (f ⋆ 1 , f ⋆ 2:K) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' We note that f ⋆ k need not be optimal predictors in Fk for k ≥ 2, but we use the ⋆ notation just to associate these component functions with the first component function f ⋆ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Define f ⋆,α 2:K ∈ Fα 2:K to be the corresponding discretization of f ⋆ 2:K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Suppose g = A((x1:n, y1 1:n, f ⋆,α 2:K(x1:n)) is the predictor obtained by running A on the sample aug- mented by f ⋆,α 2:K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Let mA(ǫ, δ, K) denote the sample complexity of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since A is an agnostic learner for F, we have that for n ≥ mA(ǫ/4, δ/2, K) , with probability at least 1 − δ/2, E(x,y1)∼D1 � max k∈[K] |gk(x) − yk| � ≤ inf f∈F E(x,y1)∼D1 � max k∈[K] |fk(x) − yk| � + ǫ 4, where yk = f ⋆,α k (x) for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Note that the quantity on the left is trivially lower bounded by the risk of the first component and the optimal risk on the right-hand side is trivially upper bounded by the risk of (f ⋆ 1 , f ⋆ 2:K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In particular, we have ED1 � |g1(x) − y1| � ≤ E(x,y1)∼D1 � max k∈[K] |f ⋆ k(x) − yk| � + ǫ 4 ≤ ED1 � |f ⋆ 1 (x) − y1| � + K � k=2 EDX � |f ⋆ k(x) − f ⋆,α k (x)| � + ǫ 4 ≤ ED1 � |f ⋆ 1 (x) − y1| � + Kα + ǫ 4, where the second inequality follows upon using the fact that the sum of positive real numbers is greater than their maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' The last inequality uses the fact that f ⋆,α is the α discretization of f ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Picking α = ǫ/4K and using the definition of f ⋆ 1 , we obtain ED1 � |g1(x) − y1| � ≤ inf f1∈F1 ED1[|f1(x) − y1|] + ǫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Therefore, we have shown the existence of one predictor g ∈ C(S) such that its restriction to the first component, g1, obtains the agnostic bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Recall that by Hoeffding’s inequality and union bound, with probability at least 1 − δ/2, the empirical risk of every hypothesis in C1(S) on a sample of size ≥ 8 ǫ2 log 4|C1(S)| δ is at most ǫ/4 away from its true error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' So, if |�S| ≥ 8 ǫ2 log 4|C1(S)| δ , then with probability at least 1 − δ/2, we have 1 |�S| � (x,y1)∈ � S |g1(x) − y1| ≤ ED1 � |g1(x) − y1| � + ǫ 4 ≤ inf f1∈F1 ED1[|f1(x) − y1|] + 3ǫ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since ˆg1 is the ERM on �S over C1(S), its empirical risk can be at most inff1∈F1 ED1[|f1(x) − y1|] + 3ǫ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Given that the population risk of ˆg1 can be at most ǫ/4 away from its empirical risk, we have that ED1[|ˆg1(x) − y1|] ≤ inf f1∈F1 ED1[|f1(x) − y1|] + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Applying union bounds, the entire process succeeds with probability 1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Like before, we can compute the upper bound on the sample complexity of Algorithm 8, denoted m1(ǫ, δ), as m1(ǫ, δ) ≤ mA(ǫ/4, δ/2, K) + O � 1 ǫ2 log |C1(S)| δ � ≤ mA(ǫ/4, δ/2, K) + O � mA(ǫ/4, δ/2, K) K log K ǫ + log 1 δ ǫ2 � , where we use C1(S) ≤ (2/α)mA(ǫ/4,δ/2,K) K and our choice of α = ǫ/4K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' This completes the proof as it shows that Algorithm 8, given as input an agnostic learner A for F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ℓ∞, outputs an agnostic learner for F1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' ■ 20 As a final remark, we want to distinguish between the role of discretization in Algorithms 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In Algorithm 8, we only discretize the components F2:K to augment the input sample containing a single-variate target to a K-variate target in all possible ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Since all possible augmentations of the sample by F2:K could potentially be of infinite size, we first discretized F2:K so that we can construct a finite cover of all possible augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' However, the role of discretization is more fundamental in Algorithm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In this case, we first use learners Ak for Fk to construct an algorithm, which we showed to be a realizable learner for the discretized function class Fα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Then, step 3 and step 4 are standard realizable to agnostic reduction for Fα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Finally, we argued that an agnostic learner for the discretized function class is also an agnostic learner for the original function class F for a small enough discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' 6 Discussion In this work, we give a characterization of multilabel learnability for batch, online, and bandit settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In all three settings, we show that a multilabel function class is learnable if and only if each restriction is learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' As future work, we hope to characterize learnability when the number of tasks K is countably infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} 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Natarajan, Pradeep K Ravikumar, and Inderjit S Dhillon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Consistent multilabel classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Cortes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Lawrence, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Sugiyama, and R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Pereira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Burges, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Bottou, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Wein- berger, editors, Advances in 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+page_content=' Explore no more: Improved high-probability regret bounds for non- stochastic bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 28, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' [NLMKF17] Jinseok Nam, Eneldo Loza Menc´ıa, Hyunwoo J Kim, and Johannes F¨urnkranz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' Maxi- mizing subset accuracy with recurrent neural networks in multi-label classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} +page_content=' In I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE0T4oBgHgl3EQf3wJf/content/2301.02729v1.pdf'} 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We are moti- +vated by the fact that most works that analyze this data con- +sider a specific conversion mechanism, such as the MSW +effect, although flavor conversion is still an open question +in supernovae due to the presence of neutrino-neutrino in- +teractions. In our analysis, instead of considering a specific +conversion mechanism, we let the electron antineutrino sur- +vival probability be a free parameter. We fit the data from +Kamiokande-II, Baksan, and IMB detected spectrum with +two classes of models: time-integrated and time-dependent. +For the time-integrated model, it is not possible to put limits +above 1σ on the survival probability. The same happens for +the time-dependent model when cooling is the only mecha- +nism of antineutrino emission. However, for models consid- +ering an accretion phase, Pee ∼ 0 is strongly rejected, show- +ing a preference for the existence of an accretion component +in the detected antineutrino flux, and a preference for normal +mass ordering when only the MSW is present. +1 Introduction +The detection of antineutrinos coming from the SN1987A +supernova, the first and only detection of supernova neutri- +nos up to this date, was a big event for particle and astro- +physics. The events were observed by the underground neu- +trino experiments Kamiokande-II (KII) [1, 2], IMB [3, 4] +and Baksan [5]. Since then, many works were produced to +analyze and understand this data [6–11], which gave us in- +formation to put bound in supernova models and neutrino +properties. However, some conditions used in previous works +ae-mail: dedin@ifi.unicamp.br +be-mail: mvsantos@ifi.unicamp.br +ce-mail: holanda@ifi.unicamp.br +de-mail: kemp@ifi.unicamp.br +do not fit well in the picture that we have today. In this con- +text, this paper is intended to be complementary to [6, 7]. +One of the main questions regarding supernova neutri- +nos today is the flavor conversion mechanism. It is expected +for the supernova neutrinos to suffer MSW conversion [12– +14] and a substantial number of works were done consider- +ing this as the only conversion mechanism in action, includ- +ing the ones that analyze the SN1987A data [6, 7]. How- +ever, today it is expected that neutrino-neutrino interactions +(forward scattering) become relevant in a supernova envi- +ronment leading the neutrinos to a non-linear collective evo- +lution [15]. Due to the complications that emerge from this +type of evolution, there is not a conclusive picture of neu- +trino conversion in the supernova environment. +Nevertheless, given the equal amount of non-electron +antineutrinos νx = (νµ,ντ) emitted from the supernova, it +is possible to write the flavor conversion in terms of only the +electron antineutrino survival probability Pee. Therefore, we +treat this probability as a free parameter to see how SN1987A +data can constrain it. Something similar was done by F. Vis- +sani in [16]. However, it seems that the influence of the sur- +vival probability is analyzed only for the MSW normal hi- +erarchy scenario (Pee = 0.64) against the no oscillation one +(Pee = 0). Here we take a more complete analysis for Pee, +allowing it to range from 0 to 1. +In section 2 we describe our model for the detected event +rate in each detector (KII,IMB, Baksan) based on two differ- +ent neutrino emission models, the flavor conversion mecha- +nism, and the detection properties. In section 3 we describe +our statistical analysis of the SN1987A data. In section 4 we +show our results and discuss them, and finally, in section 5 +we present our conclusions. +arXiv:2301.11407v1 [hep-ph] 26 Jan 2023 + +2 +2 Model for the neutrino signal +In this section, we describe the model for the expected +neutrino event rate in each of the detectors, which is used +to fit the SN1987A data. First, we describe the two neutrino +emission models considered in this paper: a time-dependent +and a time-integrated. In sequence, we describe the flavor +conversion in the flux, which depends only on Pee, and, in +the end, we discuss the detection features of this analysis. +Given that the most relevant cross-section for the consid- +ered detectors is the IBD, we will restrict our model to the +antineutrino sector (¯νe, ¯νµ, ¯ντ) +2.1 Neutrino Emission +Based on previous SN1987A neutrino data analysis [6– +10], we use two distinct models for the neutrino emission: +time-integrated and time-dependent ones. +Time-dependent Given that the neutrino emission evolves +in time, a time-dependent model should be at least consid- +ered in data analysis. This approach can be found in the fa- +mous paper of Lamb and Loredo [6] and some other works +[7]. In this approach, the antineutrino emission can be di- +vided into two phases: the accretion and cooling phases. +Here we will follow the path of [6, 7] and model each phase +by its most relevant mechanism of emission. +In this case, the accretion phase can be modeled as a +positron thermal flux with temperature Ta incident in a neu- +tron target, that composes the mass in accretion in the proto- +neutron star. Therefore, as in [6, 7], we consider that only +electron antineutrinos are emitted in this phase and the flux +is given by: +φ 0 +a,¯νe(Eν,t) = 8πc +(hc)3 [Nn(t)σe+n(Eν)ge+(Ee+,Ta)], +(1) +with +N(t) = Yn +mn +×Ma × +jk(t) +1+t/0.5s, +ge+(Ee+,Ta) = +E2 +e+ +1+exp[Ee+/Ta], +(2) +where Nn(t) is the number of neutrons as a function of the +time, σe+n(Eν) the positron-neutron cross-section, and +ge+(Ee+,Ta) the thermal distribution of positrons with en- +ergy Ee+ in a temperature Ta. The number of neutrons is +given by the initial accreting mass Ma with a fraction of +neutrons Yn, and its time behavior is given by the factor +jk(t) = exp +� +−(t/τa)k� +, with τa being the characteristic time +of the accretion phase and the parameter k = 2 following the +parametrization in [7]1. The denominator 1 +t/0.5s, as in +1In [6] it is used k = 10, however, as discussed in [7] k = 2 adjust better +to supernova simulations. +[6, 7], is used to mimic the behavior from supernova simu- +lations, where we have a constant flux within the first 0.5s +followed by a fast decrease. +The cooling phase, which is dominated by neutrinos emit- +ted by the cooling neutron star, is modeled by a thermal dis- +tribution of fermions with temperature Tc(t), with character- +istic time τc, emitted from a sphere with fixed radius Rc and +is given by +φ 0 +c (E,t) = +πc +(hc)3 4πR2 +c +E2 +1+exp[E/Tc(t)], +(3) +with the cooling temperature being a function of time +Tc(t) = Tc exp[−t/(4τc)]. +(4) +To combine the fluxes of both phases of emission, we +follow [7] where the cooling phase starts after the accretion +one. As argued in the cited work, if the accretion and cool- +ing phases were contemporaneous the first seconds would be +composed of two different spectra, given the different tem- +peratures of each of these phases. As numerical simulations +of supernovae do not show this feature, we assume that the +different emission phases are separated in time. We do this +using the following parameterization: +φ 0 +¯ν(t) = φ 0 +a (t)+(1− jk(t))φ 0 +c (t −τa), +(5) +where we have to remind that the accretion flux is consid- +ered only for the electron antineutrinos. +Time-integrated In this model, we consider that the time- +integrated flux can be described by the following pinched +spectrum [17]: +φ 0 +β(E) = Lβ +E0β +1 +(αβ +1)−(αβ +1)Γ (αβ +1)E0β +× +� E +E0 +�αβ +e−(αβ +1)E/E0β , +(6) +where, for a specific neutrino flavor β, Lβ is the total energy +(time-integrated luminosity), E0β the mean energy, and αβ +the pinching parameter. We are mainly motivated to use this +model due to a collection of works which only use the en- +ergy information from the SN1987A [8–10]. Although the +time data could bring new information, it is interesting to +check if the energy alone can say something about the flavor +conversion. +2.2 Flavor Conversion +From emission until detection, the neutrino may suffer +flavor conversion. It is still an open question for supernova +neutrinos which is the complete mechanism of flavor con- +version, given the complications that arise with neutrino- +neutrino interactions. However, due to unitarity and the equal + +3 +initial flux of non-electron antineutrinos φ 0 +νµ = φ 0 +ντ = φ 0 +νx, +the equations for flavor conversion can be simplified so that +it will only depend on the electron antineutrino survival prob- +ability Pee and initial fluxes [18], such that +φνe = φ 0 +νe −(1−Pee)(φ 0 +νe −φ 0 +νx), +(7a) +2φνx = 2φ 0 +νx +(1−Pee)(φ 0 +νe −φ 0 +νx). +(7b) +Therefore, we can explore the survival probability Pee as +a free parameter representing the flavor conversion occur- +ring during the neutrino propagation. In this paper, we want +to see how strong the SN1987A data can constrain Pee in the +fitted models, given that the flavor conversion mechanism is +still an open question in a supernova environment. Although +this probability may be time and/or energy-dependent, we +will consider it independent of these variables, given that +we do not want to use a specific model. +We will also consider the MSW-only conversion sce- +nario in order to compare it to our free Pee model. In this +scenario, the electron antineutrino is created as a ¯ν1 for nor- +mal mass hierarchy (NH) and ¯ν3 for inverted mass hierar- +chy (IH). Therefore, the survival probability for each mass +ordering can be written as follows +PNH +ee += U2 +e1, +(8a) +PIH +ee = (1−Pf (E))U2 +e3 +Pf (E)U2 +e1, +(8b) +Pf (E) = exp +� +− +U2 +e3 +3.5×10−5 +�20MeV +E +�2/3� +, +(9) +where we have considered an adiabatic evolution, except on +the high-density resonance of the IH, where the flip prob- +ability from ¯ν3 to ¯ν1 is parameterized by Pf (E) which de- +pends on the energy. This picture is similar to the MSW ef- +fect considered in previous works [7–10]. The energy depen- +dency in the MSW effect may appear when considering pos- +sible non-adiabaticity in the high-density resonance layer +[7, 14]. However, for the usual parameterization (equation +(9)), the conversion probability in the resonance is negligi- +ble. Then, the constant and energy-independent Pee is a good +representation of what has been done in SN1987A analyses +until now. +Although this energy dependence of Pee is negligible in +the standard MSW effect, other possible effects associated +with collective effects, such as spectral split among different +neutrino flavors lead to a strong energy dependency, chang- +ing drastically this scenario [15]. However, given the un- +knowns associated with such collective effects nowadays, +we limit our analysis to consider a Pee that is uniform in en- +ergy, leaving the spectral split analysis for a future work. +2.3 Detection +In the case of the SN1987A, we have data from three +detectors: Kamiokande-II, IMB, and Baksan. In all of them, +the dominant channel for electron antineutrino detection is +the Inverse Beta-decay (IBD), which is the only one that we +will consider. Therefore, the event rate RIBD +¯νe +as a function +of the positron measured energy Ee+, the angle between the +incoming neutrino and the scattered positron θ and time (for +the time-dependent model) can be calculated as follows +RIBD +¯νe (Ee+,t,cosθ) = Np ×φ¯νe(Eν,t) +× dσIBD +¯νe +d cosθ (Eν)×ηd(Ee+), +(10) +where Np is the number of free protons, φ¯νe(Eν,t) the elec- +tron antineutrino flux at the detector, dσIBD +¯νe (Eν)/d cosθ the +differential cross-section for IBD, and ηd(Ee+) the detec- +tor efficiency. For the IBD, the incoming neutrino energy +Eν is related to the created positron energy by Ee+ ≈ Eν − +1.293MeV, due to the mass difference between the initial +proton and the final neutron. The energy threshold for the +IBD is Eth +¯ν = 1.806MeV [19]. +2.4 Efficiency +Instead of assuming the procedure followed in [7], where +authors performed a Monte Carlo simulation to calculate an +average efficiency, we decided to adopt the functions from +[5], that simply fit the efficiency points reported from the +three collaborations. These functions are shown in Figure +12. +2.5 Cross-section +The exclusive interaction considered in the analysis was +the inverse beta decay, given the high cross-section com- +pared to other possible channels of KII, IMB, and Baksan. +We adopted the differential cross section (in the scattering +angle) calculated by Vogel and Beacom in [20]. +2.6 Off-set time +Another thing that we have to be careful of is to not con- +fuse the time of the first detected neutrino t1 with the time +t0 = t = 0 which indicates the time that the first neutrino ar- +rives at the detector, even if it was not detected. Not consid- +ering this may force that the first detected neutrino is origi- +nated from the initial accretion phase, which may not be the +case. As we will discuss later, for the MSW conversion in +the inverted mass hierarchy scenario (IH), the initial ¯νe flux +contributes only to 2% of the detected flux, which makes it + +4 +probable that the first detected neutrino came from the cool- +ing phase and then t1 ̸= t0. To get around this problem, it is +usual to introduce an offset time td +off = t1 −t0 between the +first detected neutrino and the time of arrival of the first neu- +trino, which may be different for each detector given that +they do not have an equal absolute time. +2.7 Background Modeling +In a realistic approach, we have to consider that detected +events may come from background sources. The background +rate is considered to be constant over the time of exposure, +and also uniform over space, i.e., it depends only on the +positron energy of the event B = B(Ei) = d2NB/dtdE. The +independence regarding the spatial position is an approxi- +mation, given that there is more background at the wall of +the detector, due to the surrounding material. +The background can be measured and it is published +by the collaborations. In our case, we use the background +rate from [21] for Kamiokande-II and [6] for Baksan. The +background is irrelevant for the IMB detector. In the case +of the Time-Integrated analysis, we have to integrate the +background rate in time to get the event rate per energy +B = B(Ei) = dNB/dE. The integration has to be done on the +time of exposure to the supernova signal, i.e., the data-taking +duration (∼ 30s). +3 Statistical Analysis +For the statistical analysis, we use the method of maxi- +mum unbinned likelihood, due to the low number of events. +Our expression for the likelihood is the same as in [7] +L = e−fd +� R(t)dt +N +∏ +i=1 +eR(ti)τd +× +�Bi +2 + +� +R(ti,Ee,i,cosθi)Li(Ee)dEe +� +. +(11) +Here we made implicitly the dependency of L in the +parameters of our models. In this equation, i is the index +of each event, R(t,E,cosθ) is the expected event rate from +equation (10), R(t) the event rate integrated in the angle and +energy, and B the background rate2 discussed in section 2.7. +The integration in the positron energy Ee is made consid- +ering a Gaussian distribution Li(Ee) around the measured +value Ee,i with standard deviation given by the measurement +uncertainty. As in [7], we consider that the time and angle +uncertainties are irrelevant. We also consider the dead time +τd for each detector (d = K,B,I), where fd is the live-time +2The factor of 1/2 in the background rate term comes from its angular +dependency in cosθ, which we consider to be uniform. +fraction [7]. In the case of the time-independent model, we +only have to consider a time integration in the event rate for +the signal R(ti,Ee,i,cosθi) and for the background B(Ei). +To find the set of parameters that best adjusts our model +to the data, we only have to maximize the likelihood L or +minimize −2log(L ). The last one is useful because it trans- +forms multiplication into a sum and has a straightforward +connection to confidence intervals. Given that we have a set +of parameters ⃗θ, taking their the best-fit ˆ⃗θ we can define the +likelihood ratio as follows. +λ(⃗θ) ≡ L (⃗θ)/L ( ˆ⃗θ) +(12) +so that −2logλ(⃗θ) follows a χ2 distribution in the asymp- +totic limit of large samples N → ∞, with m degrees of free- +dom representing the number of parameters not constrained +to be in its best-fit value. With this procedure, we can esti- +mate the best-fit values for the parameters and their confi- +dence interval, given a confidence level. However, we have +to note that our data is not a large sample so our confi- +dence level is an approximation. In any case, in this paper, +we consider that it is an acceptable approximation given the +allowed region for the astrophysical parameters to be com- +parable to previous works [6] that use other approaches to +set the confidence levels, as we discuss in Appendix A. +4 Results and Discussion +4.1 Time-dependent model +For the time-dependent model, following the references +[6, 7], we consider two possible cases, one with just cooling +emission and the other with an initial accretion phase. For +the cooling component, we have four astrophysical param- +eters, the initial cooling temperature Tc, the time constant +of the phase τc, the radius of the neutrinosphere Rc, and the +ratio between the initial temperatures of the electronic and +non-electronic antineutrinos τ = T¯νx/T¯νe. Previous works [7] +fix this temperature ratio based on supernova simulations. +Here, we check the impact of changing this ratio given that it +has strong implications in how similar the initial spectra are, +which reflects how well we can identify flavor conversion +in the detected spectrum. Nevertheless, we limit ourselves +to the range of temperature ratio expected from supernova +simulations [17]. When considering the accretion phase, we +introduce three new astrophysical parameters: the initial ac- +cretion temperature Ta, the time constant of the phase τa, +and the accretion mass Ma. In addition to the astrophysical +parameters, there is the offset time for each detector and the +survival probability, resulting in a total of 8 parameters for +the cooling model and 11 for the cooling plus accretion. +To analyze how the SN1987A data can put limits on Pee, +we can do a marginal analysis, as described in section 3. + +5 +Figures 1 and 2 show the marginal plot of Pee for the models +with only cooling component and for the one with cooling +and accretion, respectively. For the model with just cooling, +we can see that it is not possible to put limits on Pee up to the +1σ for τ values considered. This probably happens because +both initial fluxes φ 0 +νe and φ 0 +νx come from the same mecha- +nism, resulting in almost indistinguishable spectra, even al- +lowing the temperatures to be different. +When we consider the accretion phase, we have a dif- +ferent scenario, where Pee ∼ 0 is strongly rejected, as we +can see in Figure 2. This stronger constraint in Pee happens +because in the accretion mechanism only electrons antineu- +trinos are emitted, making their initial flux φ 0 +νe more distin- +guishable from the non-electronic one φ 0 +νx, which in turns +facilitates the identification of flavor conversion. Given that, +the excluded region of Pee ∼ 0 corresponds to the case where +the detected flux is composed only by the initial φ 0 +νx, i.e., a +flux with no accretion component. This shows us that the +detected electron antineutrinos are better described by a flux +with an accretion component coming from φ 0 +νe, as already +found by [6]. However, in [6] they do not consider the role +of flavor conversion, while here we can see that the exis- +tence of an accretion component has strong implications on +the conversion mechanism. If we consider only the MSW ef- +fect with adiabatic propagation, this implies that the normal +hierarchy scenario is favored over the inverted. Comparing +them with the best-fit of free Pee, the normal hierarchy sce- +nario is not significantly rejected, while the inverted one is +rejected by ∼ 3σ of significance. +We have also tested the implications of considering the +cooling and accretion components as contemporaneous. As +argued by [7], there is no evidence of a composed spectrum +in supernova simulations, so the two mechanisms with dif- +ferent mean energies should occur at different times. How- +ever, from supernovae physics, we may expect that the PNS +starts to cool down by neutrino emission soon after its for- +mation, simultaneously with the accretion mechanism [22]. +Therefore, we decide to test the implications of that hypoth- +esis in our analysis. As we can see in Figure 3 there is no +significant modification on Pee limits. The only modification +appears on the best-fit of tIMB +off , which can be seen in Ap- +pendix A. +4.2 Time-integrated model +For the time-integrated model, we considered a Fermi- +Dirac emission (ανe = ανx = 2.3), a choice that does not +have big impact in the fitting for 2.3 < α < 4 3. We also con- +sider a hierarchy for the mean energy Eνx > Eνe, which is +3By letting ανe and ανx run free in this interval, the variation of the +likelihood ratio L /Lmax was not above 1σ (C.L. ≈ 68%). +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pee +0 +2 +4 +6 +8 +10 +∆χ2 +|Ue1|2 +|Ue3|2 +MSW (NH) +MSW (IH) +τ = Tx/Te = 1.00 +τ = Tx/Te = 1.10 +τ = Tx/Te = 1.20 +τ = Tx/Te = 1.30 +τ = Tx/Te = 1.40 +Fig. 1 Pee likelihood ratio (∆χ2 = −2logL /Lmax) for the SN1987A +data considering the time-dependent model with only the cooling com- +ponent. The horizontal dashed lines corresponds to 1, 2 and 3σ of C.L. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pee +0 +2 +4 +6 +8 +10 +∆χ2 +τ = Tx/Te = 1.00 +τ = Tx/Te = 1.10 +τ = Tx/Te = 1.20 +τ = Tx/Te = 1.30 +τ = Tx/Te = 1.40 +Fig. 2 Same as Fig. 1 with two components: accretion and cooling. In +this case, the two phases are considered to be separated in time. The +horizontal dashed lines corresponds to 1, 2 and 3σ of C.L. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pee +0 +2 +4 +6 +8 +10 +∆χ2 +τ = Tx/Te = 1.00 +τ = Tx/Te = 1.10 +τ = Tx/Te = 1.20 +τ = Tx/Te = 1.30 +τ = Tx/Te = 1.40 +Fig. 3 Same as Fig. 1 with two components: accretion and cooling. In +this case, the two phases are considered to be contemporaneous. The +horizontal dashed lines corresponds to 1, 2 and 3σ of C.L. + +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pee +0 +2 +4 +6 +8 +10 +∆χ2 +Ta < 0.6Tc +Ta unconstrained +Fig. 4 Pee likelihood ratio comparing the scenario with (solid) and +without (dashed) the assumption of Ta < 0.6Tc. The vertical lines cor- +respond to MSW-LMA solution to an adiabatic neutrino propagation. +The horizontal grey lines correspond to 1, 2 and 3σ of C.L. +physically motivated given that non-electron neutrinos inter- +act less (lack of τ and µ leptons in the environment) and then +escape from deeper regions in the supernova with higher +temperatures. The best-fit values for the astrophysical pa- +rameters are shown in Table 3 considering the 3 different +conversion scenarios. As we can see, there is a preference +for a detected spectrum φνe to be composed mostly by the +initial non-electron neutrino spectrum φ 0 +νx, given that there is +basically no constraint for the total energy ενe, the same be- +havior was also found in [10]. Even in the MSW mechanism +with inverted mass hierarchy, where the composition of φ 0 +νx +in the final flux is small (Pee ≈ 67.8%), the flavor conversion +is compensated by a higher total energy ενx. This preference +is a combination of the imposed energy hierarchy Eνx > Eνe +and the low detection efficiency for lower energies, where +the low energy events can be as well described as coming +from the background. However, we did not investigate this +preference deeply4. As we are interested in the flavor con- +version parameter Pee, we leave the Appendix A to compare +our marginal and contour plots with previous analyses to +show the consistency of our method, at least regarding the +astrophysical parameters. +For the flavor conversion analysis, we again fix the ini- +tial temperature ratio (more precisely the mean energy ratio +τ = Eνx/Eνe = Tνx/Tνe) and let the other parameters run +freely over the allowed range (Table 3). Figure 5 shows the +marginal plot of Pee minimizing over the other model param- +eters. Again, there is no constraint on the survival probabil- +ity above 68% of confidence, even for spectra with higher +mean energy differences such as τ = 1.4. +4We only tested a scenario with relaxed bound conditions for the pa- +rameters. However, we obtained nonsensical values for the electron +antineutrino total energy, such as ενe ∼ 1055ergs for the inverted mass +hierarchy. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pee +0 +2 +4 +6 +8 +10 +∆χ2 +τ = Tx/Te = 1.00 +τ = Tx/Te = 1.10 +τ = Tx/Te = 1.20 +τ = Tx/Te = 1.30 +τ = Tx/Te = 1.40 +Fig. 5 Pee likelihood ratio for the SN1987A data considering the time- +integrated model. +4.3 Problems with fitting the data with some models +In our numerical implementation, we found some diffi- +culties in working with the two-component model (accretion ++ cooling). The main one is the existence of different local +minima, which make the minimizer algorithm give different +best fits depending on the initial conditions. To get around +this problem, we used two methods to find the global min- +imum. In the first method we fit this model multiple times +(≈ 1000) fluctuating the initial conditions of parameters uni- +formly in the ranges shown in Table 2, and taking the min- +imum value of −2logL as the initial condition to find the +global best-fit. The second method was based on using dif- +ferent minimizers (MINOS, scipy, simplex)5 to see if this +dependency on the initial conditions was algorithm depen- +dent. In the end, we found that all the different minimizers +obtained the same best fit given initial conditions around it, +and in agreement with the first method. Given the concor- +dance between the two methods and algorithms, we have +confidence that the best fit obtained is the most probable one +inside the allowed parameter space. +5 Conclusion +In this paper, we have explored the role of flavor conver- +sion in the SN1987A neutrino data, and how it can impose +limits on the flavor conversion mechanism. We found that +the time-integrated model, which uses only the energy infor- +mation, could not put any limit on the electron antineutrino +survival probability Pee. The same happens for the time- +dependent models that consider antineutrino emission only +from the cooling mechanism. However, with the existence of +an accretion emission of electron antineutrinos, strong limits +are imposed on low values of Pee. This is impressive given +the low statistics of the SN1987A neutrino data and it is in +5All of them implemented in the iminuit library [23]. + +7 +agreement with the previous work of Lamb and Loredo [6] +in which the data shows a strong preference for the existence +of an accretion component. Our results may contradict the +conclusions of Vissani in [16], where it is placed that flavor +conversion play no significant role in the analysis. Neverthe- +less, from his paper description, it seems that he uses only +one spectrum for each flavor instead of two (cooling and +accretion), which is equivalent to our results on the model +with only a cooling component. Moreover, his affirmation is +regarding the implications of flavor conversion on the astro- +physical parameters, which seems indeed negligible. +Despite [7] claim a stability of best-fit values of astro- +physical parameters, we found a high dependency on initial +conditions in the frequentist approach of maximum likeli- +hood estimation in equation (11), where multiple local min- +ima could easily be interpreted as the global best-fit. +As we discussed, our analysis does not consider any time +or energy dependency on Pee, which may happen when we +consider collective effects due to neutrino-neutrino forward +scattering. We leave the study of time and energy depen- +dency for a future paper. In any case, our results can still +be used to constrain conversion models that result in a fixed +value for Pee. +Acknowledgments +This work was supported by São Paulo Research Foun- +dation (FAPESP) grants no. 2019/08956-2, no. 14/19164-6, +and no. 2022/01568-0 and also financed in part by the Coor- +denação de Aperfeiçoamento de Pessoal de Nível Superior +– Brasil (CAPES) – Finance Code 001. +References +1. K. Hirata et al. Observation of a Neutrino Burst from the +Supernova SN 1987a. Phys. Rev. Lett., 58:1490–1493, +1987. +2. K. S. Hirata et al. +Observation in the Kamiokande- +II Detector of the Neutrino Burst from Supernova SN +1987a. Phys. Rev. D, 38:448–458, 1988. +3. R. M. Bionta et al. Observation of a Neutrino Burst +in Coincidence with Supernova SN 1987a in the Large +Magellanic Cloud. Phys. Rev. Lett., 58:1494, 1987. +4. C. B. Bratton et al. Angular Distribution of Events From +Sn1987a. Phys. Rev. D, 37:3361, 1988. +5. E. N. Alekseev, L. N. Alekseeva, I. V. 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Dec 2020. + +8 +0 +2 +4 +6 +8 +10 +Tc (MeV) +0 +2 +4 +6 +8 +10 +∆χ2 +NH +IH +Pee +0 +5 +10 +15 +20 +τc (s) +0 +20 +40 +60 +80 +100 +Rc (km) +0 +1 +2 +3 +4 +5 +6 +tK +off (s) +0 +2 +4 +6 +8 +10 +∆χ2 +0 +1 +2 +3 +4 +5 +tI +off (s) +0 +1 +2 +3 +4 +5 +tB +off (s) +Astrophysical Parameters Profile (Only Cooling) +Fig. 6 Marginal plots for the astrophysical parameters Tc,τc,Rc and +detection off-set times tKII +off ,tIMB +off ,tBak +off for the only cooling model. +Appendix A: Comparing results with other works +Here we show our results for the astrophysical parame- +ters fit in the format of marginalized/profile plots for each +individual parameter and contour plots for some key combi- +nation of parameters. +Appendix A.1: Time-Dependent +The results of the time-dependent analysis are very com- +parable to Loredo and Lamb [6] and Pagliaroli et al. [7] +work. Both authors also analyzed SN1987A data to respect +to the same time-dependent model used here. In figure 8, we +show the statistical limits on Tc × Rc. Our bounds overlap +with both works but it is not identical to them. We attribute +this difference to our use of detector efficiencies reported +by the original collaborations and shown in Figure 12 and +up-to-date neutrino mixing parameters. For a more detailed +view of our analysis, we also show ∆χ2 profiles of parame- +ters for the only Cooling and Cooling + Accretion models in +Figures 6 and 7 respectively, as well as the best values found +and intervals used shown in Tables 1 and 2. +Table 1 Range and best-fit (BF) for all parameters in the time- +dependent model Only Cooling. We show the best-fit for three flavor +conversion scenarios: MSW with NH, MSW with IH, and a model- +independent free Pee. +Parameter +NH BF +IH BF +Free Pee BF +Range +T0,c [MeV] +3.66+0.56 +−0.37 +3.5+0.4 +−0.4 +3.66+0.024 +−0.024 +1-10 +τc [s] +4.1+1.1 +−0.9 +4.1+1.2 +−0.8 +4.1+1.1 +−0.8 +1-40 +Rc [km] +36+15 +−13 +30+15 +−9 +34+16 +−13 +1-100 +tKII +off [s] +0.0+0.19 +−0 +0.0+0.19 +−0 +0.0+0.19 +−0 +0-6 +tIMB +off +[s] +0.0+0.12 +−0 +0.0+0.12 +−0 +0.0+0.12 +−0 +0-6 +tBak +off [s] +0.0+0.41 +−0 +0.0+0.42 +−0 +0.0+0.41 +−0 +0-6 +Table 2 Range and best-fit (BF) for all parameters in the time- +dependent model Cooling+Accretion. We show the best-fit for three +flavor conversion scenarios: MSW with NH, MSW with IH, and a +model-independent free Pee. +Parameter +NH BF +IH BF +Free Pee BF +Range +T0,c [MeV] +4.81+0.72 +−0.78 +3.975+0.02 +−0.02 +5.37+0.78 +−0.06 +1-10 +τc [s] +4.1+1.5 +−1.0 +4.6+1.6 +−1.2 +4.1+1.5 +−1.0 +1-40 +Rc [km] +12.3+8.7 +−4.2 +14.4+9.2 +−7.7 +11.3+7.8 +−3.8 +1-100 +T0,a [MeV] +2.00+0.13 +−0.13 +3.12+0.19 +−0.18 +1.91+0.17 +−0.12 +0.1-10 +τa [s] +0.57+0.38 +−0.20 +0.7+0.38 +−0.2 +0.58+0.37 +−0.20 +0.3-3.5 +Ma [M⊙] +0.6+0 +−0.46 +0.6+0 +−0.17 +0.6+0 +−0.43 +0-0.6 +tKII +off [s] +0.0+0.03 +−0 +0.0+0.05 +−0 +0.0+0.03 +−0 +0-6 +tIMB +off +[s] +0.4+0.4 +−0.4 +0.0+0.08 +−0 +0.45+0.42 +−0.45 +0-6 +tBak +off [s] +0.0+0.1 +−0 +0.0+0.11 +−0 +0.0+0.09 +−0 +0-6 +2 +4 +6 +8 +10 +Tc (MeV) +0 +2 +4 +6 +8 +10 +∆χ2 +NH +IH +Pee +5 +10 +15 +20 +τc (s) +20 +40 +60 +80 +100 +Rc (km) +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Ta (MeV) +0 +2 +4 +6 +8 +10 +∆χ2 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +τa (s) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ma (M⊙) +0 +1 +2 +3 +4 +5 +6 +tK +off (s) +0 +2 +4 +6 +8 +10 +∆χ2 +0 +1 +2 +3 +4 +5 +6 +tI +off (s) +0 +1 +2 +3 +4 +5 +6 +tB +off (s) +Astrophysical Parameters Profile (Cooling + Accretion) +Fig. +7 Marginal +plots +for +the +astrophysical +parameters +Tc,τc,Rc,Ta,τa,Ma and detection off-set times tKII +off ,tIMB +off ,tBak +off +for +the cooling plus accretion model. +Appendix A.2: Time-Integrated +For the time-integrated model, we use the work of C. Lu- +nardini [10] for comparison. Figure 9 shows the marginal- +ized plots of each of the four parameters ¯Ee,εe, ¯Ex,εx for +the three flavor conversion scenario. As already discussed in +the paper, there is a preference for φ¯νe ≈ φ 0 +¯νx, with almost +no bound on εe and only a hard upper bound in ¯Ee due to +the imposed hierarchy in the mean energy. This is consistent +with the results shown in Table 1 of [10]. +A more direct comparison can be done with the contour +plots of ¯Ee× ¯Ex and ¯Ex×εx, which are explicitly shown [10]. +The contour plot for ¯Ee × ¯Ex is shown in figure 10. The ob- +tained bounds are similar to the one from [10], where we +get stronger bounds in ¯Ex in the flavor conversion with fixed +Pee, i.e., the MSW scenario with fixed mass hierarchy (NH +or IH). This is expected given that sinθ13 is treated as a free +parameter in [10], which results in a free Pee within a specif + +9 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +Tc (MeV) +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Rc (km) +Only Cooling Model +Cool. - NH (C.L.=68%) +Cool. - NH (C.L.=90%) +LL (C.L.=68%) +LL (C.L.=90%) +Pagliaroli et al. (C.L.=68%) +Pagliaroli et al. (C.L.=90%) +LL - B.F. +Pagliaroli - B.F. +Fig. 8 Tc,0 vs Rc contour plots comparing our results with previous +ones [6, 7] +. +range6, given a bound similar to our free Pee ∈ [0,1]. A sim- +ilar behavior is found for the bounds on the ¯Ex ×εx contour +plot, where the results of [10] are somewhere between our +fixed (NH or IH) and free Pee scenarios, where in the last sce- +nario no bound is found for εe. With this picture in mind, we +can conclude that our analysis of the time-integrated model +is in relatively good agreement with previous works, given +the peculiarities discussed above. +Table 3 Range and best-fit (BF) for all parameters in the time- +integrated model. We show the best-fit for three flavor conversion sce- +narios: MSW with NH, MSW with IH, and a model-independent free +Pee. +Parameter +NH BF +IH BF +Free Pee BF +Range +[10] +¯Ee [MeV] +5.7+13.7 +−2.7 +5.8+5.8 +−2.8 +5.5+4.6 +−2.5 +3−30 +εe [1052ergs] +1.518+43.48 +−0.02 +43.9+1.1 +−23.3 +1.5+43.5 +−0 +1.5−45 +¯Ex [MeV] +11.50+0.03 +−0.03 +11.50+0.06 +−0.06 +11.50+0.03 +−0.32 +3−30 +εx [1052ergs] +6.3+3.5 +−2.5 +2.1+0.6 +−0.6 +8+36 +−7 +1.5−45 +Appendix B: Detection information +In this appendix, the reader can found information about +the detection properties and data used in this work. In Table +4 we show the detectors properties and in Figure 12 the con- +sidered efficiency function. By last, we show the neutrino +data form Kamiokande-II, IMB, and Baksan in Tables 5, 6, +and 7 respectively. +6The range used in [10] correspond to the interval 10−7 < sin2 θ13 < +10−2, which is smaller than our range [0,1]. +0 +5 +10 +15 +20 +25 +30 +Ee [MeV] +0 +2 +4 +6 +8 +10 +∆χ2 +NH +IH +Free Pee +5 +10 +15 +20 +25 +Ex [MeV] +0 +2 +4 +6 +8 +10 +Profile over E with Energy Hierarchy Ex > Ee +εe, εx = [1.5-45.0] 1052 ergs, Ee, Ex = [3.0-30.0] MeV, αe = αx = 2.3 +0 +10 +20 +30 +40 +εe [1052 ergs] +0 +2 +4 +6 +8 +10 +∆χ2 +NH +IH +Free Pee +0 +10 +20 +30 +40 +εx [1052 ergs] +0 +2 +4 +6 +8 +10 +Profile over ε with Energy Hierarchy Ex > Ee +εe, εx = [1.5-45.0] 1052 ergs, Ee, Ex = [3.0-30.0] MeV, αe = αx = 2.3 +Fig. 9 Marginal plots for the astrophysical parameters ¯E¯νe, ¯E¯νx,ε¯νe,ε¯νx +(bottom) in the time-integrated model. Here we consider a Fermi-Dirac +emission αe = αx = 2.3 and spectral energy hierarchy ¯Ex > ¯Ee. +5 +10 +15 +20 +25 +30 +Ee [MeV] +5 +10 +15 +20 +25 +30 +Ex [MeV] +Normal Hierarchy +Best Fit +66% C.L. +95% C.L. +99.7% C.L. +5 +10 +15 +20 +25 +30 +Ee [MeV] +5 +10 +15 +20 +25 +30 +Inverted Hierarchy +5 +10 +15 +20 +25 +30 +Ee [MeV] +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +22.5 +25.0 +Pee Free +Ee vs Ex with Energy Hierarchy Ex > Ee +εe, εx = 1.5 − 45.0 × 1052ergs, Ee, Ex = 3.0 − 30.0MeV , αe = αx = 2.3 +Fig. 10 Contour plot ¯Ee × ¯Ex for the time-integrated model. Here we +consider a Fermi-Dirac emission αe = αx = 2.3 and spectral energy hi- +erarchy ¯Ex > ¯Ee. The bottom plot was taken from [10] for comparison. +Table 4 Characteristics of each detector +Detector +Fiducial Mass [kton] +Free Protons +Composition +[kton] +[1032] +Kamiokande-II +2.14 +1.43 +H2O +IMB +6.80 +4.54 +H2O +Baksan +0.20 +0.19 +C9H2O + +MeV +30 +Hierarchy of energy +25 +20 +15 +10 +5 +10 +20 +30 +E.-/Mev10 +5 +10 +15 +20 +25 +30 +Ex [MeV] +5 +10 +15 +20 +25 +30 +35 +40 +45 +εx [1052 ergs] +Normal Hierarchy +Best Fit +66% C.L. +95% C.L. +99.7% C.L. +5 +10 +15 +20 +25 +30 +Ex [MeV] +5 +10 +15 +20 +25 +30 +35 +40 +45 +Inverted Hierarchy +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +22.5 +25.0 +Ex [MeV] +5 +10 +15 +20 +25 +30 +35 +40 +45 +Free Pee +Ex vs εx with Energy Hierarchy Ex > Ee +εe, εx = 1.5 − 45.0 × 1052ergs, Ee, Ex = 3.0 − 30.0MeV , αe = αx = 2.3 +Fig. 11 Contour plot ¯Ex × εx for the time-integrated model. Here we +consider a Fermi-Dirac emission αe = αx = 2.3 and spectral energy hi- +erarchy ¯Ex > ¯Ee. The bottom plot was taken from [10] for comparison. +5 +10 +15 +20 +25 +30 +35 +40 +Eν[MeV ] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +η(Eν) +Kamiokande II +IMB +Baksan +Fig. 12 Detectors efficiency [5] +Table 5 SN1987A data from Kamiokande-II. +Kamiokande-II +Event +Time +Energy +Angle +Background [21] +[s] +[MeV] +[Degree] +[MeV−1.s−1] +1 +0 +20±2,9 +18±18 +1.6×10−5 +2 +0,107 +13,5±3,2 +40± 27 +1.9×10−3 +3 +0,303 +7,5± +108±32 +2.9×10−2 +4 +0,324 +9,2±2,7 +70±30 +1.2×10−2 +5 +0,507 +12,8±2,9 +135±23 +2.1×10−3 +6 +0,686 +6,3±1,7 +68±77 +3.7×10−2 +7 +1,541 +35,4±8 +32±16 +4.5×10−5 +8 +1,728 +21±4,2 +30±18 +8.2×10−5 +9 +1,915 +19,8±3,2 +38±22 +1.5×10−5 +10 +9,219 +8,6±2,7 +122±30 +1.5×10−2 +11 +10,433 +13±2,6 +49±26 +1.9×10−3 +12 +12,439 +8,9±1,9 +91±39 +1.6×10−2 +13 +17,641 +6,5 ±1,6 +— +3.8×10−2 +14 +20,257 +5,4±1,4 +— +2.9×10−2 +15 +21,355 +4,6± 1,3 +— +2.8×10−2 +16 +23,814 +6,5±1,6 +— +3.8×10−2 +Table 6 SN1987A data from IMB. +IMB +Event +Time +Energy +Angle +Background +[s] +[MeV] +[Degree] +[MeV−1.s−1] +1 +0 +38±7 +80±10 +0 +2 +0,412 +37±7 +44±15 +0 +3 +0,65 +28±6 +56 ±20 +0 +4 +1,141 +39±7 +65±20 +0 +5 +1,562 +36±9 +33±5 +0 +6 +2,684 +36±6 +52±0 +0 +7 +5,01 +19±5 +42±20 +0 +8 +5,582 +22±5 +104±20 +0 +Table 7 SN1987A data from Baksan. +Baksan +Event +Time +Energy +Angle +Background [6] +[s] +[MeV] +[Degree] +[MeV−1.s−1] +1 +0 +12±2,4 +— +8.4×10−4 +2 +0,435 +17,9±3,6 +— +1.3×10−3 +3 +1,71 +23,5±4,7 +— +1.2×10−3 +4 +7,687 +17,6±3,5 +— +1.3×10−3 +5 +9,099 +20,3±4,1 +— +1.3×10−3 + +Hierarchy of energy +2. +5 +￥2 +1.5 +1 +0.5 +0 +10 +20 +30 +Eox/MeV \ No newline at end of file diff --git a/_NFIT4oBgHgl3EQf9ivV/content/tmp_files/load_file.txt b/_NFIT4oBgHgl3EQf9ivV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9572126bb37f28a1718ec2c96e7ed0afa3cedad1 --- /dev/null +++ b/_NFIT4oBgHgl3EQf9ivV/content/tmp_files/load_file.txt @@ -0,0 +1,787 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf,len=786 +page_content='SN1987A neutrino burst: limits on flavor conversion Pedro Dedin Netoa,1, Marcos V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' dos Santosb,1, Pedro Cunha de Holandac,1, Ernesto Kempd,1 1Instituto de Física Gleb Wataghin, UNICAMP, Rua Sérgio Buarque de Holanda 777, Campinas-SP, Brazil Abstract In this paper, we revisit the SN1987A neutrino data to see its constraints on flavor conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We are moti- vated by the fact that most works that analyze this data con- sider a specific conversion mechanism, such as the MSW effect, although flavor conversion is still an open question in supernovae due to the presence of neutrino-neutrino in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In our analysis, instead of considering a specific conversion mechanism, we let the electron antineutrino sur- vival probability be a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We fit the data from Kamiokande-II, Baksan, and IMB detected spectrum with two classes of models: time-integrated and time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' For the time-integrated model, it is not possible to put limits above 1σ on the survival probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The same happens for the time-dependent model when cooling is the only mecha- nism of antineutrino emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, for models consid- ering an accretion phase, Pee ∼ 0 is strongly rejected, show- ing a preference for the existence of an accretion component in the detected antineutrino flux, and a preference for normal mass ordering when only the MSW is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 1 Introduction The detection of antineutrinos coming from the SN1987A supernova, the first and only detection of supernova neutri- nos up to this date, was a big event for particle and astro- physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The events were observed by the underground neu- trino experiments Kamiokande-II (KII) [1, 2], IMB [3, 4] and Baksan [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Since then, many works were produced to analyze and understand this data [6–11], which gave us in- formation to put bound in supernova models and neutrino properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, some conditions used in previous works ae-mail: dedin@ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='br be-mail: mvsantos@ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='br ce-mail: holanda@ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='br de-mail: kemp@ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='br do not fit well in the picture that we have today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In this con- text, this paper is intended to be complementary to [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' One of the main questions regarding supernova neutri- nos today is the flavor conversion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' It is expected for the supernova neutrinos to suffer MSW conversion [12– 14] and a substantial number of works were done consider- ing this as the only conversion mechanism in action, includ- ing the ones that analyze the SN1987A data [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' How- ever, today it is expected that neutrino-neutrino interactions (forward scattering) become relevant in a supernova envi- ronment leading the neutrinos to a non-linear collective evo- lution [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Due to the complications that emerge from this type of evolution, there is not a conclusive picture of neu- trino conversion in the supernova environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Nevertheless, given the equal amount of non-electron antineutrinos νx = (νµ,ντ) emitted from the supernova, it is possible to write the flavor conversion in terms of only the electron antineutrino survival probability Pee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Therefore, we treat this probability as a free parameter to see how SN1987A data can constrain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Something similar was done by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Vis- sani in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, it seems that the influence of the sur- vival probability is analyzed only for the MSW normal hi- erarchy scenario (Pee = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='64) against the no oscillation one (Pee = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Here we take a more complete analysis for Pee, allowing it to range from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In section 2 we describe our model for the detected event rate in each detector (KII,IMB, Baksan) based on two differ- ent neutrino emission models, the flavor conversion mecha- nism, and the detection properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In section 3 we describe our statistical analysis of the SN1987A data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In section 4 we show our results and discuss them, and finally, in section 5 we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='11407v1 [hep-ph] 26 Jan 2023 2 2 Model for the neutrino signal In this section, we describe the model for the expected neutrino event rate in each of the detectors, which is used to fit the SN1987A data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' First, we describe the two neutrino emission models considered in this paper: a time-dependent and a time-integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In sequence, we describe the flavor conversion in the flux, which depends only on Pee, and, in the end, we discuss the detection features of this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Given that the most relevant cross-section for the consid- ered detectors is the IBD, we will restrict our model to the antineutrino sector (¯νe, ¯νµ, ¯ντ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1 Neutrino Emission Based on previous SN1987A neutrino data analysis [6– 10], we use two distinct models for the neutrino emission: time-integrated and time-dependent ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Time-dependent Given that the neutrino emission evolves in time, a time-dependent model should be at least consid- ered in data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' This approach can be found in the fa- mous paper of Lamb and Loredo [6] and some other works [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In this approach, the antineutrino emission can be di- vided into two phases: the accretion and cooling phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Here we will follow the path of [6, 7] and model each phase by its most relevant mechanism of emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In this case, the accretion phase can be modeled as a positron thermal flux with temperature Ta incident in a neu- tron target, that composes the mass in accretion in the proto- neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Therefore, as in [6, 7], we consider that only electron antineutrinos are emitted in this phase and the flux is given by: φ 0 a,¯νe(Eν,t) = 8πc (hc)3 [Nn(t)σe+n(Eν)ge+(Ee+,Ta)], (1) with N(t) = Yn mn ×Ma × jk(t) 1+t/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5s, ge+(Ee+,Ta) = E2 e+ 1+exp[Ee+/Ta], (2) where Nn(t) is the number of neutrons as a function of the time, σe+n(Eν) the positron-neutron cross-section, and ge+(Ee+,Ta) the thermal distribution of positrons with en- ergy Ee+ in a temperature Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The number of neutrons is given by the initial accreting mass Ma with a fraction of neutrons Yn, and its time behavior is given by the factor jk(t) = exp � −(t/τa)k� , with τa being the characteristic time of the accretion phase and the parameter k = 2 following the parametrization in [7]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The denominator 1 +t/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5s, as in 1In [6] it is used k = 10, however, as discussed in [7] k = 2 adjust better to supernova simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' [6, 7], is used to mimic the behavior from supernova simu- lations, where we have a constant flux within the first 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5s followed by a fast decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The cooling phase, which is dominated by neutrinos emit- ted by the cooling neutron star, is modeled by a thermal dis- tribution of fermions with temperature Tc(t), with character- istic time τc, emitted from a sphere with fixed radius Rc and is given by φ 0 c (E,t) = πc (hc)3 4πR2 c E2 1+exp[E/Tc(t)], (3) with the cooling temperature being a function of time Tc(t) = Tc exp[−t/(4τc)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' (4) To combine the fluxes of both phases of emission, we follow [7] where the cooling phase starts after the accretion one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As argued in the cited work, if the accretion and cool- ing phases were contemporaneous the first seconds would be composed of two different spectra, given the different tem- peratures of each of these phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As numerical simulations of supernovae do not show this feature, we assume that the different emission phases are separated in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We do this using the following parameterization: φ 0 ¯ν(t) = φ 0 a (t)+(1− jk(t))φ 0 c (t −τa), (5) where we have to remind that the accretion flux is consid- ered only for the electron antineutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Time-integrated In this model, we consider that the time- integrated flux can be described by the following pinched spectrum [17]: φ 0 β(E) = Lβ E0β 1 (αβ +1)−(αβ +1)Γ (αβ +1)E0β × � E E0 �αβ e−(αβ +1)E/E0β , (6) where, for a specific neutrino flavor β, Lβ is the total energy (time-integrated luminosity), E0β the mean energy, and αβ the pinching parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We are mainly motivated to use this model due to a collection of works which only use the en- ergy information from the SN1987A [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Although the time data could bring new information, it is interesting to check if the energy alone can say something about the flavor conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 Flavor Conversion From emission until detection, the neutrino may suffer flavor conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' It is still an open question for supernova neutrinos which is the complete mechanism of flavor con- version, given the complications that arise with neutrino- neutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, due to unitarity and the equal 3 initial flux of non-electron antineutrinos φ 0 νµ = φ 0 ντ = φ 0 νx, the equations for flavor conversion can be simplified so that it will only depend on the electron antineutrino survival prob- ability Pee and initial fluxes [18], such that φνe = φ 0 νe −(1−Pee)(φ 0 νe −φ 0 νx), (7a) 2φνx = 2φ 0 νx +(1−Pee)(φ 0 νe −φ 0 νx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' (7b) Therefore, we can explore the survival probability Pee as a free parameter representing the flavor conversion occur- ring during the neutrino propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In this paper, we want to see how strong the SN1987A data can constrain Pee in the fitted models, given that the flavor conversion mechanism is still an open question in a supernova environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Although this probability may be time and/or energy-dependent, we will consider it independent of these variables, given that we do not want to use a specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We will also consider the MSW-only conversion sce- nario in order to compare it to our free Pee model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In this scenario, the electron antineutrino is created as a ¯ν1 for nor- mal mass hierarchy (NH) and ¯ν3 for inverted mass hierar- chy (IH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Therefore, the survival probability for each mass ordering can be written as follows PNH ee = U2 e1, (8a) PIH ee = (1−Pf (E))U2 e3 +Pf (E)U2 e1, (8b) Pf (E) = exp � − U2 e3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5×10−5 �20MeV E �2/3� , (9) where we have considered an adiabatic evolution, except on the high-density resonance of the IH, where the flip prob- ability from ¯ν3 to ¯ν1 is parameterized by Pf (E) which de- pends on the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' This picture is similar to the MSW ef- fect considered in previous works [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The energy depen- dency in the MSW effect may appear when considering pos- sible non-adiabaticity in the high-density resonance layer [7, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, for the usual parameterization (equation (9)), the conversion probability in the resonance is negligi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Then, the constant and energy-independent Pee is a good representation of what has been done in SN1987A analyses until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Although this energy dependence of Pee is negligible in the standard MSW effect, other possible effects associated with collective effects, such as spectral split among different neutrino flavors lead to a strong energy dependency, chang- ing drastically this scenario [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, given the un- knowns associated with such collective effects nowadays, we limit our analysis to consider a Pee that is uniform in en- ergy, leaving the spectral split analysis for a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 Detection In the case of the SN1987A, we have data from three detectors: Kamiokande-II, IMB, and Baksan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In all of them, the dominant channel for electron antineutrino detection is the Inverse Beta-decay (IBD), which is the only one that we will consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' the event rate RIBD ¯νe as a function of the positron measured energy Ee+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' the angle between the incoming neutrino and the scattered positron θ and time (for the time-dependent model) can be calculated as follows RIBD ¯νe (Ee+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='cosθ) = Np ×φ¯νe(Eν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='t) × dσIBD ¯νe d cosθ (Eν)×ηd(Ee+),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' (10) where Np is the number of free protons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' φ¯νe(Eν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='t) the elec- tron antineutrino flux at the detector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' dσIBD ¯νe (Eν)/d cosθ the differential cross-section for IBD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' and ηd(Ee+) the detec- tor efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' For the IBD, the incoming neutrino energy Eν is related to the created positron energy by Ee+ ≈ Eν − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='293MeV, due to the mass difference between the initial proton and the final neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The energy threshold for the IBD is Eth ¯ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='806MeV [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 Efficiency Instead of assuming the procedure followed in [7], where authors performed a Monte Carlo simulation to calculate an average efficiency, we decided to adopt the functions from [5], that simply fit the efficiency points reported from the three collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' These functions are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 Cross-section The exclusive interaction considered in the analysis was the inverse beta decay, given the high cross-section com- pared to other possible channels of KII, IMB, and Baksan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We adopted the differential cross section (in the scattering angle) calculated by Vogel and Beacom in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 Off-set time Another thing that we have to be careful of is to not con- fuse the time of the first detected neutrino t1 with the time t0 = t = 0 which indicates the time that the first neutrino ar- rives at the detector, even if it was not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Not consid- ering this may force that the first detected neutrino is origi- nated from the initial accretion phase, which may not be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As we will discuss later, for the MSW conversion in the inverted mass hierarchy scenario (IH), the initial ¯νe flux contributes only to 2% of the detected flux, which makes it 4 probable that the first detected neutrino came from the cool- ing phase and then t1 ̸= t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' To get around this problem, it is usual to introduce an offset time td off = t1 −t0 between the first detected neutrino and the time of arrival of the first neu- trino, which may be different for each detector given that they do not have an equal absolute time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7 Background Modeling In a realistic approach, we have to consider that detected events may come from background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The background rate is considered to be constant over the time of exposure, and also uniform over space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=', it depends only on the positron energy of the event B = B(Ei) = d2NB/dtdE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The independence regarding the spatial position is an approxi- mation, given that there is more background at the wall of the detector, due to the surrounding material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The background can be measured and it is published by the collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In our case, we use the background rate from [21] for Kamiokande-II and [6] for Baksan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The background is irrelevant for the IMB detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In the case of the Time-Integrated analysis, we have to integrate the background rate in time to get the event rate per energy B = B(Ei) = dNB/dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The integration has to be done on the time of exposure to the supernova signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=', the data-taking duration (∼ 30s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 3 Statistical Analysis For the statistical analysis, we use the method of maxi- mum unbinned likelihood, due to the low number of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Our expression for the likelihood is the same as in [7] L = e−fd � R(t)dt N ∏ i=1 eR(ti)τd × �Bi 2 + � R(ti,Ee,i,cosθi)Li(Ee)dEe � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' (11) Here we made implicitly the dependency of L in the parameters of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In this equation, i is the index of each event, R(t,E,cosθ) is the expected event rate from equation (10), R(t) the event rate integrated in the angle and energy, and B the background rate2 discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The integration in the positron energy Ee is made consid- ering a Gaussian distribution Li(Ee) around the measured value Ee,i with standard deviation given by the measurement uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As in [7], we consider that the time and angle uncertainties are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We also consider the dead time τd for each detector (d = K,B,I), where fd is the live-time 2The factor of 1/2 in the background rate term comes from its angular dependency in cosθ, which we consider to be uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' fraction [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In the case of the time-independent model, we only have to consider a time integration in the event rate for the signal R(ti,Ee,i,cosθi) and for the background B(Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' To find the set of parameters that best adjusts our model to the data, we only have to maximize the likelihood L or minimize −2log(L ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The last one is useful because it trans- forms multiplication into a sum and has a straightforward connection to confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Given that we have a set of parameters ⃗θ, taking their the best-fit ˆ⃗θ we can define the likelihood ratio as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' λ(⃗θ) ≡ L (⃗θ)/L ( ˆ⃗θ) (12) so that −2logλ(⃗θ) follows a χ2 distribution in the asymp- totic limit of large samples N → ∞, with m degrees of free- dom representing the number of parameters not constrained to be in its best-fit value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' With this procedure, we can esti- mate the best-fit values for the parameters and their confi- dence interval, given a confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, we have to note that our data is not a large sample so our confi- dence level is an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In any case, in this paper, we consider that it is an acceptable approximation given the allowed region for the astrophysical parameters to be com- parable to previous works [6] that use other approaches to set the confidence levels, as we discuss in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 4 Results and Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1 Time-dependent model For the time-dependent model, following the references [6, 7], we consider two possible cases, one with just cooling emission and the other with an initial accretion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' For the cooling component, we have four astrophysical param- eters, the initial cooling temperature Tc, the time constant of the phase τc, the radius of the neutrinosphere Rc, and the ratio between the initial temperatures of the electronic and non-electronic antineutrinos τ = T¯νx/T¯νe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Previous works [7] fix this temperature ratio based on supernova simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Here, we check the impact of changing this ratio given that it has strong implications in how similar the initial spectra are, which reflects how well we can identify flavor conversion in the detected spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Nevertheless, we limit ourselves to the range of temperature ratio expected from supernova simulations [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' When considering the accretion phase, we introduce three new astrophysical parameters: the initial ac- cretion temperature Ta, the time constant of the phase τa, and the accretion mass Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In addition to the astrophysical parameters, there is the offset time for each detector and the survival probability, resulting in a total of 8 parameters for the cooling model and 11 for the cooling plus accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' To analyze how the SN1987A data can put limits on Pee, we can do a marginal analysis, as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 5 Figures 1 and 2 show the marginal plot of Pee for the models with only cooling component and for the one with cooling and accretion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' For the model with just cooling, we can see that it is not possible to put limits on Pee up to the 1σ for τ values considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' This probably happens because both initial fluxes φ 0 νe and φ 0 νx come from the same mecha- nism, resulting in almost indistinguishable spectra, even al- lowing the temperatures to be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' When we consider the accretion phase, we have a dif- ferent scenario, where Pee ∼ 0 is strongly rejected, as we can see in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' This stronger constraint in Pee happens because in the accretion mechanism only electrons antineu- trinos are emitted, making their initial flux φ 0 νe more distin- guishable from the non-electronic one φ 0 νx, which in turns facilitates the identification of flavor conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Given that, the excluded region of Pee ∼ 0 corresponds to the case where the detected flux is composed only by the initial φ 0 νx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=', a flux with no accretion component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' This shows us that the detected electron antineutrinos are better described by a flux with an accretion component coming from φ 0 νe, as already found by [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, in [6] they do not consider the role of flavor conversion, while here we can see that the exis- tence of an accretion component has strong implications on the conversion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' If we consider only the MSW ef- fect with adiabatic propagation, this implies that the normal hierarchy scenario is favored over the inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Comparing them with the best-fit of free Pee, the normal hierarchy sce- nario is not significantly rejected, while the inverted one is rejected by ∼ 3σ of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We have also tested the implications of considering the cooling and accretion components as contemporaneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As argued by [7], there is no evidence of a composed spectrum in supernova simulations, so the two mechanisms with dif- ferent mean energies should occur at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' How- ever, from supernovae physics, we may expect that the PNS starts to cool down by neutrino emission soon after its for- mation, simultaneously with the accretion mechanism [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Therefore, we decide to test the implications of that hypoth- esis in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As we can see in Figure 3 there is no significant modification on Pee limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The only modification appears on the best-fit of tIMB off , which can be seen in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 Time-integrated model For the time-integrated model, we considered a Fermi- Dirac emission (ανe = ανx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3), a choice that does not have big impact in the fitting for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 < α < 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We also con- sider a hierarchy for the mean energy Eνx > Eνe, which is 3By letting ανe and ανx run free in this interval, the variation of the likelihood ratio L /Lmax was not above 1σ (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' ≈ 68%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Pee 0 2 4 6 8 10 ∆χ2 |Ue1|2 |Ue3|2 MSW (NH) MSW (IH) τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='00 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='10 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='20 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='30 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 1 Pee likelihood ratio (∆χ2 = −2logL /Lmax) for the SN1987A data considering the time-dependent model with only the cooling com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The horizontal dashed lines corresponds to 1, 2 and 3σ of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Pee 0 2 4 6 8 10 ∆χ2 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='00 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='10 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='20 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='30 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 2 Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 1 with two components: accretion and cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In this case, the two phases are considered to be separated in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The horizontal dashed lines corresponds to 1, 2 and 3σ of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Pee 0 2 4 6 8 10 ∆χ2 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='00 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='10 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='20 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='30 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 3 Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 1 with two components: accretion and cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In this case, the two phases are considered to be contemporaneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The horizontal dashed lines corresponds to 1, 2 and 3σ of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Pee 0 2 4 6 8 10 ∆χ2 Ta < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6Tc Ta unconstrained Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 4 Pee likelihood ratio comparing the scenario with (solid) and without (dashed) the assumption of Ta < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The vertical lines cor- respond to MSW-LMA solution to an adiabatic neutrino propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The horizontal grey lines correspond to 1, 2 and 3σ of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' physically motivated given that non-electron neutrinos inter- act less (lack of τ and µ leptons in the environment) and then escape from deeper regions in the supernova with higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The best-fit values for the astrophysical pa- rameters are shown in Table 3 considering the 3 different conversion scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As we can see, there is a preference for a detected spectrum φνe to be composed mostly by the initial non-electron neutrino spectrum φ 0 νx, given that there is basically no constraint for the total energy ενe, the same be- havior was also found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Even in the MSW mechanism with inverted mass hierarchy, where the composition of φ 0 νx in the final flux is small (Pee ≈ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8%), the flavor conversion is compensated by a higher total energy ενx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' This preference is a combination of the imposed energy hierarchy Eνx > Eνe and the low detection efficiency for lower energies, where the low energy events can be as well described as coming from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, we did not investigate this preference deeply4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As we are interested in the flavor con- version parameter Pee, we leave the Appendix A to compare our marginal and contour plots with previous analyses to show the consistency of our method, at least regarding the astrophysical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' For the flavor conversion analysis, we again fix the ini- tial temperature ratio (more precisely the mean energy ratio τ = Eνx/Eνe = Tνx/Tνe) and let the other parameters run freely over the allowed range (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Figure 5 shows the marginal plot of Pee minimizing over the other model param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Again, there is no constraint on the survival probabil- ity above 68% of confidence, even for spectra with higher mean energy differences such as τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 4We only tested a scenario with relaxed bound conditions for the pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, we obtained nonsensical values for the electron antineutrino total energy, such as ενe ∼ 1055ergs for the inverted mass hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Pee 0 2 4 6 8 10 ∆χ2 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='00 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='10 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='20 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='30 τ = Tx/Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 5 Pee likelihood ratio for the SN1987A data considering the time- integrated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 Problems with fitting the data with some models In our numerical implementation, we found some diffi- culties in working with the two-component model (accretion + cooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The main one is the existence of different local minima, which make the minimizer algorithm give different best fits depending on the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' To get around this problem, we used two methods to find the global min- imum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In the first method we fit this model multiple times (≈ 1000) fluctuating the initial conditions of parameters uni- formly in the ranges shown in Table 2, and taking the min- imum value of −2logL as the initial condition to find the global best-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The second method was based on using dif- ferent minimizers (MINOS, scipy, simplex)5 to see if this dependency on the initial conditions was algorithm depen- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In the end, we found that all the different minimizers obtained the same best fit given initial conditions around it, and in agreement with the first method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Given the concor- dance between the two methods and algorithms, we have confidence that the best fit obtained is the most probable one inside the allowed parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 5 Conclusion In this paper, we have explored the role of flavor conver- sion in the SN1987A neutrino data, and how it can impose limits on the flavor conversion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We found that the time-integrated model, which uses only the energy infor- mation, could not put any limit on the electron antineutrino survival probability Pee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The same happens for the time- dependent models that consider antineutrino emission only from the cooling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' However, with the existence of an accretion emission of electron antineutrinos, strong limits are imposed on low values of Pee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' This is impressive given the low statistics of the SN1987A neutrino data and it is in 5All of them implemented in the iminuit library [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 7 agreement with the previous work of Lamb and Loredo [6] in which the data shows a strong preference for the existence of an accretion component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Our results may contradict the conclusions of Vissani in [16], where it is placed that flavor conversion play no significant role in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Neverthe- less, from his paper description, it seems that he uses only one spectrum for each flavor instead of two (cooling and accretion), which is equivalent to our results on the model with only a cooling component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Moreover, his affirmation is regarding the implications of flavor conversion on the astro- physical parameters, which seems indeed negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Despite [7] claim a stability of best-fit values of astro- physical parameters, we found a high dependency on initial conditions in the frequentist approach of maximum likeli- hood estimation in equation (11), where multiple local min- ima could easily be interpreted as the global best-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As we discussed, our analysis does not consider any time or energy dependency on Pee, which may happen when we consider collective effects due to neutrino-neutrino forward scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We leave the study of time and energy depen- dency for a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In any case, our results can still be used to constrain conversion models that result in a fixed value for Pee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Acknowledgments This work was supported by São Paulo Research Foun- dation (FAPESP) grants no.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Ianni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Improved analysis of SN1987A antineutrino events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As- tropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=', 31:163–176, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Lunardini and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Smirnov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Neutrinos from SN1987A, earth matter effects and the LMA solution of the solar neutrino problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Phys.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Maria Laura Costantini, Aldo Ianni, Giulia Pagliaroli, and Francesco Vissani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Is there a problem with low en- ergy SN1987A neutrinos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' JCAP, 05:014, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Jackson Olsen and Yong-Zhong Qian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Prospects for dis- tinguishing supernova models using a future neutrino signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' D, 105(8):083017, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Hans Dembinski and Piti Ongmongkolkul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' scikit- hep/iminuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Dec 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 8 0 2 4 6 8 10 Tc (MeV) 0 2 4 6 8 10 ∆χ2 NH IH Pee 0 5 10 15 20 τc (s) 0 20 40 60 80 100 Rc (km) 0 1 2 3 4 5 6 tK off (s) 0 2 4 6 8 10 ∆χ2 0 1 2 3 4 5 tI off (s) 0 1 2 3 4 5 tB off (s) Astrophysical Parameters Profile (Only Cooling) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 6 Marginal plots for the astrophysical parameters Tc,τc,Rc and detection off-set times tKII off ,tIMB off ,tBak off for the only cooling model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Appendix A: Comparing results with other works Here we show our results for the astrophysical parame- ters fit in the format of marginalized/profile plots for each individual parameter and contour plots for some key combi- nation of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1: Time-Dependent The results of the time-dependent analysis are very com- parable to Loredo and Lamb [6] and Pagliaroli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' [7] work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Both authors also analyzed SN1987A data to respect to the same time-dependent model used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In figure 8, we show the statistical limits on Tc × Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Our bounds overlap with both works but it is not identical to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We attribute this difference to our use of detector efficiencies reported by the original collaborations and shown in Figure 12 and up-to-date neutrino mixing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' For a more detailed view of our analysis, we also show ∆χ2 profiles of parame- ters for the only Cooling and Cooling + Accretion models in Figures 6 and 7 respectively, as well as the best values found and intervals used shown in Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Table 1 Range and best-fit (BF) for all parameters in the time- dependent model Only Cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We show the best-fit for three flavor conversion scenarios: MSW with NH, MSW with IH, and a model- independent free Pee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Parameter NH BF IH BF Free Pee BF Range T0,c [MeV] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='024 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='024 1-10 τc [s] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 1-40 Rc [km] 36+15 −13 30+15 −9 34+16 −13 1-100 tKII off [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='19 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='19 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='19 −0 0-6 tIMB off [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='12 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='12 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='12 −0 0-6 tBak off [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='41 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='42 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='41 −0 0-6 Table 2 Range and best-fit (BF) for all parameters in the time- dependent model Cooling+Accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We show the best-fit for three flavor conversion scenarios: MSW with NH, MSW with IH, and a model-independent free Pee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Parameter NH BF IH BF Free Pee BF Range T0,c [MeV] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='78 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='975+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='06 1-10 τc [s] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 1-40 Rc [km] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 1-100 T0,a [MeV] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1-10 τa [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='57+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='58+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 Ma [M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6+0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6+0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6+0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='43 0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 tKII off [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='03 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='05 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='03 −0 0-6 tIMB off [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='08 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='45+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='45 0-6 tBak off [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='11 −0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='09 −0 0-6 2 4 6 8 10 Tc (MeV) 0 2 4 6 8 10 ∆χ2 NH IH Pee 5 10 15 20 τc (s) 20 40 60 80 100 Rc (km) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Ta (MeV) 0 2 4 6 8 10 ∆χ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 τa (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Ma (M⊙) 0 1 2 3 4 5 6 tK off (s) 0 2 4 6 8 10 ∆χ2 0 1 2 3 4 5 6 tI off (s) 0 1 2 3 4 5 6 tB off (s) Astrophysical Parameters Profile (Cooling + Accretion) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 7 Marginal plots for the astrophysical parameters Tc,τc,Rc,Ta,τa,Ma and detection off-set times tKII off ,tIMB off ,tBak off for the cooling plus accretion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2: Time-Integrated For the time-integrated model, we use the work of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Lu- nardini [10] for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Figure 9 shows the marginal- ized plots of each of the four parameters ¯Ee,εe, ¯Ex,εx for the three flavor conversion scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' As already discussed in the paper, there is a preference for φ¯νe ≈ φ 0 ¯νx, with almost no bound on εe and only a hard upper bound in ¯Ee due to the imposed hierarchy in the mean energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' This is consistent with the results shown in Table 1 of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' A more direct comparison can be done with the contour plots of ¯Ee× ¯Ex and ¯Ex×εx, which are explicitly shown [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The contour plot for ¯Ee × ¯Ex is shown in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The ob- tained bounds are similar to the one from [10], where we get stronger bounds in ¯Ex in the flavor conversion with fixed Pee, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=', the MSW scenario with fixed mass hierarchy (NH or IH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' This is expected given that sinθ13 is treated as a free parameter in [10], which results in a free Pee within a specif 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Tc (MeV) 10 20 30 40 50 60 70 80 90 100 Rc (km) Only Cooling Model Cool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' - NH (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='=68%) Cool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' - NH (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='=90%) LL (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='=68%) LL (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='=90%) Pagliaroli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='=68%) Pagliaroli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='=90%) LL - B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Pagliaroli - B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 8 Tc,0 vs Rc contour plots comparing our results with previous ones [6, 7] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' range6, given a bound similar to our free Pee ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' A sim- ilar behavior is found for the bounds on the ¯Ex ×εx contour plot, where the results of [10] are somewhere between our fixed (NH or IH) and free Pee scenarios, where in the last sce- nario no bound is found for εe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' With this picture in mind, we can conclude that our analysis of the time-integrated model is in relatively good agreement with previous works, given the peculiarities discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Table 3 Range and best-fit (BF) for all parameters in the time- integrated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' We show the best-fit for three flavor conversion sce- narios: MSW with NH, MSW with IH, and a model-independent free Pee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Parameter NH BF IH BF Free Pee BF Range [10] ¯Ee [MeV] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7+13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 3−30 εe [1052ergs] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='518+43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='02 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='9+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1 −23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5+43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 −0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5−45 ¯Ex [MeV] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='50+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='03 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='50+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='06 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='50+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='32 3−30 εx [1052ergs] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 8+36 −7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5−45 Appendix B: Detection information In this appendix, the reader can found information about the detection properties and data used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' In Table 4 we show the detectors properties and in Figure 12 the con- sidered efficiency function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' By last, we show the neutrino data form Kamiokande-II, IMB, and Baksan in Tables 5, 6, and 7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 6The range used in [10] correspond to the interval 10−7 < sin2 θ13 < 10−2, which is smaller than our range [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 0 5 10 15 20 25 30 Ee [MeV] 0 2 4 6 8 10 ∆χ2 NH IH Free Pee 5 10 15 20 25 Ex [MeV] 0 2 4 6 8 10 Profile over E with Energy Hierarchy Ex > Ee εe, εx = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5-45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0] 1052 ergs, Ee, Ex = [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0] MeV, αe = αx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 0 10 20 30 40 εe [1052 ergs] 0 2 4 6 8 10 ∆χ2 NH IH Free Pee 0 10 20 30 40 εx [1052 ergs] 0 2 4 6 8 10 Profile over ε with Energy Hierarchy Ex > Ee εe, εx = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5-45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0] 1052 ergs, Ee, Ex = [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0] MeV, αe = αx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 9 Marginal plots for the astrophysical parameters ¯E¯νe, ¯E¯νx,ε¯νe,ε¯νx (bottom) in the time-integrated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Here we consider a Fermi-Dirac emission αe = αx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 and spectral energy hierarchy ¯Ex > ¯Ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 5 10 15 20 25 30 Ee [MeV] 5 10 15 20 25 30 Ex [MeV] Normal Hierarchy Best Fit 66% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 5 10 15 20 25 30 Ee [MeV] 5 10 15 20 25 30 Inverted Hierarchy 5 10 15 20 25 30 Ee [MeV] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Pee Free Ee vs Ex with Energy Hierarchy Ex > Ee εe, εx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 − 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 × 1052ergs, Ee, Ex = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 − 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0MeV , αe = αx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 10 Contour plot ¯Ee × ¯Ex for the time-integrated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Here we consider a Fermi-Dirac emission αe = αx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 and spectral energy hi- erarchy ¯Ex > ¯Ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The bottom plot was taken from [10] for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Table 4 Characteristics of each detector Detector Fiducial Mass [kton] Free Protons Composition [kton] [1032] Kamiokande-II 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='43 H2O IMB 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='54 H2O Baksan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='19 C9H2O MeV 30 Hierarchy of energy 25 20 15 10 5 10 20 30 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='-/Mev10 5 10 15 20 25 30 Ex [MeV] 5 10 15 20 25 30 35 40 45 εx [1052 ergs] Normal Hierarchy Best Fit 66% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 5 10 15 20 25 30 Ex [MeV] 5 10 15 20 25 30 35 40 45 Inverted Hierarchy 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 Ex [MeV] 5 10 15 20 25 30 35 40 45 Free Pee Ex vs εx with Energy Hierarchy Ex > Ee εe, εx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5 − 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 × 1052ergs, Ee, Ex = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 − 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0MeV , αe = αx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 11 Contour plot ¯Ex × εx for the time-integrated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Here we consider a Fermi-Dirac emission αe = αx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3 and spectral energy hi- erarchy ¯Ex > ¯Ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' The bottom plot was taken from [10] for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 5 10 15 20 25 30 35 40 Eν[MeV ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='0 η(Eν) Kamiokande II IMB Baksan Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' 12 Detectors efficiency [5] Table 5 SN1987A data from Kamiokande-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Kamiokande-II Event Time Energy Angle Background [21] [s] [MeV] [Degree] [MeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='s−1] 1 0 20±2,9 18±18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6×10−5 2 0,107 13,5±3,2 40± 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='9×10−3 3 0,303 7,5± 108±32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='9×10−2 4 0,324 9,2±2,7 70±30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2×10−2 5 0,507 12,8±2,9 135±23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='1×10−3 6 0,686 6,3±1,7 68±77 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='7×10−2 7 1,541 35,4±8 32±16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5×10−5 8 1,728 21±4,2 30±18 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2×10−5 9 1,915 19,8±3,2 38±22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5×10−5 10 9,219 8,6±2,7 122±30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='5×10−2 11 10,433 13±2,6 49±26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='9×10−3 12 12,439 8,9±1,9 91±39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='6×10−2 13 17,641 6,5 ±1,6 — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8×10−2 14 20,257 5,4±1,4 — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='9×10−2 15 21,355 4,6± 1,3 — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8×10−2 16 23,814 6,5±1,6 — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='8×10−2 Table 6 SN1987A data from IMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' IMB Event Time Energy Angle Background [s] [MeV] [Degree] [MeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='s−1] 1 0 38±7 80±10 0 2 0,412 37±7 44±15 0 3 0,65 28±6 56 ±20 0 4 1,141 39±7 65±20 0 5 1,562 36±9 33±5 0 6 2,684 36±6 52±0 0 7 5,01 19±5 42±20 0 8 5,582 22±5 104±20 0 Table 7 SN1987A data from Baksan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content=' Baksan Event Time Energy Angle Background [6] [s] [MeV] [Degree] [MeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='s−1] 1 0 12±2,4 — 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='4×10−4 2 0,435 17,9±3,6 — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3×10−3 3 1,71 23,5±4,7 — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='2×10−3 4 7,687 17,6±3,5 — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3×10−3 5 9,099 20,3±4,1 — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFIT4oBgHgl3EQf9ivV/content/2301.11407v1.pdf'} +page_content='3×10−3 Hierarchy of energy 2.' metadata={'source': 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Magdeburg, Universitätsplatz 2, +Magdeburg, 39106, Germany. +*Corresponding author(s). E-mail(s): blind@iag.uni-stuttgart.de; +Contributing authors: kleinert@iag.uni-stuttgart.de; +lutz@iag.uni-stuttgart.de; beck@iag.uni-stuttgart.de; +Abstract +Generating turbulent inflow data is a challenging task in zonal Large +Eddy Simulation (zLES) and often relies on predefined DNS data to gen- +erate synthetic turbulence with the correct statistics. The more accurate, +but more involved alternative is to use instantaneous data from a precur- +sor simulation. Using instantaneous data as an inflow condition allows +to conduct high fidelity simulations of subdomains of e.g. an aircraft +including all non-stationary or rare events. In this paper we introduce +a tool-chain that is capable of interchanging highly resolved spatial and +temporal data between flow solvers with different discretization schemes. +To accomplish this, we use interpolation algorithms suitable for scattered +data in order to interpolate spatially. In time we use one-dimensional +interpolation schemes for each degree of freedom. The results show that +we can get stable simulations that map all flow features from the source +data into a new target domain. Thus, the coupling is capable of mapping +arbitrary data distributions and formats into a new domain while also +recovering and conserving turbulent structures and scales. The necessary +time and space resolution requirements can be defined knowing the reso- +lution requirements of the used numerical scheme in the target domain. +Keywords: DGSEM, instantaneous inflow condition, coupling, zonal LES +1 +arXiv:2301.03192v1 [physics.flu-dyn] 9 Jan 2023 + +Springer Nature 2021 LATEX template +2 +A Time-Accurate Inflow Coupling for Zonal LES +1 Introduction +In modern Computational Fluid Dynamics (CFD) research Large Eddy Sim- +ulation (LES) is becoming more popular due to increased computation +performance [1]. However, many practical applications still are out of range +for a detailed investigation using this technique. Therefore, many simulations +run today are based on so called zonal approaches that often depend on pre- +defined DNS data to generate synthetic turbulence with the correct statistics. +By zonal we mean that only a small subset of a domain is simulated with +the LES method in order to decrease computational cost, while the majority +of the domain is for example solved by a much cheaper Reynolds-Averaged +Navier-Stokes (RANS) method, or the subset is equipped with suitable bound- +ary conditions, in particular scale-resolving inflow data. This can be of special +interest in the simulation of turbulent boundary layers, where we do not want +to simulate the initial transition process, but are just interested in the fully +developed boundary layer as a starting point. To achieve this, there have +been developed many approaches, such as the synthetic eddy method [2, 3] +or the recycling-rescaling approach [4, 5] which allow for significantly smaller +domains. A practical example is the simulation of acoustic noise at the trail- +ing edge of an airfoil where a detailed simulation of only a part of the airfoil is +needed [6]. One similarity of the applications just described is their dependency +on boundary layer properties and therefore reference data from literature. +Another slightly different example is the investigation of the interaction of +an incoming turbulent wake with the boundary layer. For example this can +be applied to the interaction of a turbulent wing wake with the horizontal +stabilizer of an aircraft (cf. Fig. 1). The described scenario poses a challenge, +since fully scale resolving codes often can not afford to compute the whole +aircraft and codes that are capable of running a simulation of a whole aircraft +are often not able to run high fidelity simulation of parts of it. Therefore, there +is the need to map the results in a time-accurate manner from one simulation +to an inflow/initial condition on a detailed simulation with a smaller domain +and to impose them as inflow conditions. This approach allows for refined +simulation of areas of special interest. Also such a coupling between these +codes enables a way to further investigate the interaction between turbulence +of different physical scales very efficiently. Therefore, zonal simulations of high +Reynolds number flow become feasible. +When trying to couple different numerical codes we encounter several +problems on how to approach this: +1. Is a two-way coupling necessary or is one-way sufficient? +2. Do we couple the codes during runtime? +3. Are the underlying numerics compatible? +The first question is - in context of the scenario described above - easy to +answer. Since we only are interested in the effects of an incoming flow to the +target domain, a one-way coupling is sufficient. We note that, depending on +the equation system, a one-way coupling via a prescribed Dirichlet boundary + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +3 +is prone to errors, since information transport is limited to one direction. How- +ever, we can justify this simplification by assuming that we e.g. extract the +flow in a wake region with no proximity to a wall in case of the compressible +Navier-Stokes equations. Additionally, the simplification removes a lot of com- +plexity and thus enables efficient coupling of different solvers and experiments +which would not be possible in a two-way coupled way. +Thus, we can directly answer the second question. Having both codes run +separately allows us to perform the mapping in a preprocessing step for the +zonal simulation and thus removes a bottleneck during runtime. In addition, +it avoids the complexities of having to solve the High Performance Comput- +ing (HPC) problem of having to run two possibly very heterogeneous codes +synchronously and establish efficient parallel communication patterns. As men- +tioned before the coupling is designed to perform detailed simulations of a +subdomain, meaning that the area of special interest is also contained in the +full domain simulation and therefore is assumed to be represented in a suffi- +cient way to capture all the necessary physics. Also we have to consider that +the incoming physics can be truncated. We thus investigate the influence of +different resolution combinations in order to quantify this error. +The third question is harder to answer since we not only have to take +the spatial discretization like finite volume, finite difference and finite element +methods into account, but also take care of the temporal discretization, which +in most applications is either implicit or explicit. In general, two choices for +mapping the solutions between two heterogeneous representations are possi- +ble: A projection approach, and an interpolation method. While the former is +(approximately) conservative, ensuring this property on arbitrary meshes in +space and time is cumbersome, expensive (as it requires non-local operations) +and not very flexible. The interpolation approach relaxes this condition, mak- +ing the mapping process very general and allows for extending the algorithms +to work with arbitrary (x, y, z) data as an input and map it to a compatible +data format. The mapping algorithms thus have to be able to capture reso- +lution differences from both grid spacing and numerical efficiency per DOF, +arbitrary points and inconsistent time steps. Hence, interpolation algorithms +seem to be a good choice for achieving these properties in space and time. +Another problem we have to tackle is how to deal with large data sets. +Although we opt for an offline coupling which avoids having to transfer the +data in situ, time resolved surface or volume data is very memory intensive. +Thus, memory management algorithms have to be taken into account, includ- +ing parallelization, load balancing and data reduction in order to keep compute +costs low. +In this paper we want to show that mapping instantaneous data from the +DLR finite volume code TAU to the high-order spectral element code FLEXI +is possible and allows for detailed simulation of subdomains. Thus, we want to +answer the following research questions: +1. Is it possible to get a mapping for the data which allows for a numerically +stable coupled simulation? + +Springer Nature 2021 LATEX template +4 +A Time-Accurate Inflow Coupling for Zonal LES +2. Which temporal resolution is needed? +3. How does the difference in spatial resolution (mesh and numerical efficiency +per DOF) affect the mapping error? +2 Numerical Methods +An important aspect for the mapping algorithm is the knowledge of the under- +lying numerics and the associated scale resolving properties which we are going +to assess in more detail in the following section. +2.1 Code Frameworks +The target numerical scheme for which the data has to be prepared is the open- +source discontinuous Galerkin framework FLEXI, developed at the University +of Stuttgart [7]. In this scheme the domain is partitioned into non-overlapping +unstructured hexahedral elements. We can choose an arbitrary polynomial +degree for the elements. This also means that the solution in each element is +represented as a polynomial. +In contrast to the spectral element code, the source data is typically point- +wise data resulting from an experiment or finite volume code. The presented +mapping algorithms are designed and implemented to be generally applicable, +but are optimized to work with the DLR finite volume code TAU. In Fig. 1 a +typical application of TAU is visualized. TAU uses hybrid RANS/LES methods +e.g. for cases with separated flows, where attached boundary layers are treated +in RANS mode and detached wake regions are resolved in LES mode. Thus the +effect of the wing wake on the boundary layer of the HTP is hard to investigate +within TAU. FLEXI on the other hand resolves the boundary layer and thus +is capable of quantifying the influence of the wing wake. Thus, the region of +interest that can be simulated within FLEXI is marked in Fig. 1. TAU will +also be used to validate the results in Sec. 4 later on. +2.1.1 TAU +The finite volume solver TAU is developed by the German Aerospace Center +(DLR) and widely used among the aviation industry [8]. It solves the Euler +or RANS equations both on structured and unstructured grids. Several one- +and two-equation models and Reynolds stress models are implemented for +turbulence modeling. Additionally, LES or hybrid RANS/LES simulations can +also be performed. Hexahedra, tetrahedra, triangular prisms and pyramids +are supported elements for the cells of the primary grid. For the computation +of the numerical fluxes at the cell boundaries, different upwind schemes and +central approximations are available. Both explicit and implicit schemes can +be chosen for the integration in time. The resulting linear system is solved with +SGS or LUSGS schemes. For convergence acceleration, local time stepping, +residual smoothing and multigrid methods are used. Parallelization is achieved +by domain decomposition, with communication through the message passing +interface (MPI). + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +5 +TAU +FLEXI +Fig. 1: Tandem wing configuration with visualized wing wake interacting with +the HTP simulated in TAU. Region of interest for detailed simulation in FLEXI +is highlighted. +2.1.2 FLEXI +FLEXI is a high-performance open-source CFD solver based on the Discon- +tinuous Galerkin Spectral Element Method (DGSEM). It utilizes hexahedral +tensor product elements with an arbitrary polynomial degree in each element. +Since DG methods are hybrid schemes combining finite element and finite vol- +ume methods, we use a Roe Riemann solver with minimum dissipation for +the fluxes between the elements [9]. Additionally, we represent the polyno- +mial solution on a non-equispaced Legendre-Gauss or Legendre-Gauss-Lobatto +point set. Since FLEXI acts as a framework there are multiple equation sys- +tems implemented. In this paper we mainly use the compressible Navier-Stokes +equations. For validation the linear scalar advection system is used. The results +in the application section are created using the compressible Navier-Stokes +equations, which are implemented as skew-symmetric split form approxima- +tions to minimize aliasing instabilities [10]. The boundary conditions generally +are imposed weakly. This means, that we do not prescribe the state at the +corresponding solution point in the element, but rather prescribe the numer- +ical flux. FLEXI is parallelized using MPI and was successfully tested on up +to O(105) cores [11]. +2.1.3 Comparison of the Code Frameworks +Discontinuous Galerkin methods are commonly used high-order schemes. +Finite volume methods in contrast are for unstructured meshes often limited to +second order. It is well known that for the same number of degrees of freedom +high-order methods can achieve lower error and need fewer solution points to +resolve the same structures. This is known as numbers per wavelength nPPW +criteria [1, 12]. High-order methods can achieve nP P W ≈ 4, while second finite + +Springer Nature 2021 LATEX template +6 +A Time-Accurate Inflow Coupling for Zonal LES +101.6 +101.8 +102 +102.2 +102.4 +10−11 +10−6 +10−1 +1 +-2 +1 +-6 +h +L2-error of ρ +TAU RANS mode +TAU LES mode +FLEXI N = 1 +FLEXI N = 5 +Fig. 2: Comparison of the convergence behavior of TAU and FLEXI for +different settings and meshes. +volume method is often limited to nPPW ≈ 20. This means that for resolv- +ing a structure of a given wavelength, high-order methods need 5 times fewer +resolution points. However, this property is heavily dependent on the used +polynomial degree N as shown in Fig. 2. Thus, on the same mesh the simu- +lation with FLEXI N = 5 not only has a faster decreasing error but also has +the smaller error for a given amount of degrees of freedom. This becomes obvi- +ous since FLEXI N = 1 or TAU on the h = 102.4 grid correspond in terms +of amount of degree of freedom with FLEXI N = 5 at h = 101.6. The Shu- +vortex test case [13] is utilized to conduct this study. All simulations are run +independently and thus without mapping. +The results denoted as “TAU RANS mode” are obtained with numerical set- +tings commonly used for the RANS zones of hybrid RANS/LES simulations in +TAU. A second order central flux approximation is used as Riemann solver sta- +bilized by artificial dissipation, derived from the scheme after James, Schmidt +and Turkel [14]. Applying a skew-symmetric scheme with matrix dissipation +[15] already reduces the dissipation level compared to the TAU-default aver- +age of flux scheme with scalar dissipation. The simulations denoted with “TAU +LES mode” additionally utilize a reconstruction of the convective fluxes using +a linear gradient extrapolation at the cell faces, in a way to reduce the numer- +ical dispersion of the skew-symmetric scheme [16]. Moreover, the coefficient of +artificial dissipation is lowered by a factor of 16. These settings are suitable +for the LES zones of a hybrid simulation. In the FLEXI runs a Roe Riemann +solver is used [9]. The FLEXI simulation for N = 1 shows a result similar to +the TAU runs with the same order of convergence. However N = 1 is typically +not used in practical application, since the advantages of high-order schemes +are not visible for such low polynomial degree. The runs with N = 5 represents +a more realistic scenario and show the advantage of high-order polynomials. + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +7 +2.2 Workflow +Before presenting the mapping routines, we first discuss the general workflow +of how to run a simulation with time resolved input data. The general workflow +is visualized in Fig. 3. +First, the source data has to be provided. Generally this can be in the form +of point-wise scattered data. Since in this paper we focus on the procedure for +mapping TAU data to FLEXI we assume to get either volume snapshots or +surface data from TAU. +Second, we process the data by choosing an appropriate spatial mapping +mechanism. Also we have to decide if we only want to map surface data for +an instantaneous boundary condition, or if we also want to get the volume +information to e.g. generate a restart file for FLEXI. The tool creates an inter- +polated file for each input file. The results are saved in a HDF5® format that +uses a polynomial structure closely related to FLEXI. To ensure compatibility +with FLEXI, the output polynomial degree is identical to the degree of the +simulation we want to run afterwards. +Higher-order representations are prone to aliasing and oscillations in gen- +eral and the quality of the results heavily relies on the used point set and +polynomial degree we perform the interpolation on. Since the output degree +and point set is defined by the simulation we want to perform eventually, we +can not use these parameters for mitigating errors. Thus, we super-sample +the target point representation which helps avoiding errors due to oscilla- +tions resulting from the non-polynomials character of the source points. We +then map the source data to the super-sampled target data points. We found +that using M ≈ [1.5, 2] · (N + 1) target points for super-sampling yields good +agreement and mitigates oscillations significantly. This is in line with the lit- +erature values for overintegration of turbulent data, which is commonly used +in the DG community for dealiasing [17]. After interpolating the source data +to super-sampled target points, we project the solution to the original basis +N < M which removes the high modal information that is especially affected +by aliasing. +Third, we interpolate the results from the second step temporally. The +mapped volume or surface files created in step two get converted into a dataset +containing the temporal interpolator for each solution point. The interpolator +consists of the coefficients of the polynomial, which are dependent on the +evaluation time. Additionally, we partition the data into a user-defined number +of subsets to limit the amount of data of each interpolator and avoid memory +overflow during FLEXI runtime. +Fourth, we provide FLEXI with the resulting file. FLEXI then evaluates +the interpolant in each time step and sets the according boundary condition +to the interpolated values. + +Springer Nature 2021 LATEX template +8 +A Time-Accurate Inflow Coupling for Zonal LES +Generate Data +Spatial Inter- +polation +Temporal +Interpolation +Run FLEXI +Generate the source data and +provide it as a point wise array +Run the spatial interpolation algorithms and +save the mapped solution for every timestep +Interpolate the mapped data temporally +and partition the data to fit into memory +Provide FLEXI with the interpolator +created before and run the simulation +Simulation +Tools +Explanation +Fig. 3: Workflow of imposing a time resolved boundary condition. +2.2.1 Some Remarks on Surface Data +The mapping algorithms we present can be applied to volume as well as sur- +face data. Depending on the provided data the spatial mapping algorithms +will either use the volume solution to extract the target boundary or use the +provided surface plane directly. +TAU is able to write 2D data from a user-defined plane, onto which the flow +variables are interpolated internally using algorithms of the chimera technique +[18]. This interpolation will also of course introduce an error that leads to a +mismatch between the volume solution of TAU and the 2D data on the plane. +The resulting chimera plane can be read in separately. Hence, there can be +made significant simplifications in term of area reduction which reduces the +overall cost of the mapping algorithms. +2.3 Spatial Interpolation +An important step to achieve the coupling of TAU with FLEXI is the spatial +mapping. Since ultimately we want to create a instantaneous boundary condi- +tion we have to map surface data only. To keep the interpolation more general +we implement a three-dimensional method to also allow volume interpolation +and arbitrary oriented surface planes in the source domain. +A major challenge in creating a mapping between TAU and FLEXI are +the differences in spatial discretization. Industrial finite volume codes rely +often on tetrahedral meshes. Therefore, a direct interpolation is difficult, since +mesh data structures for hexahedral only codes are dissimilar. In contrast to +FLEXI, TAU stores its solution as low order point wise data. This results +in arrays containing the coordinates (x, y, z) and the solution ⃗U [19]. FLEXI +however stores its solution as polynomial data for each element [7, 20]. The + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +9 +difficulty now is to consistently map the point-wise TAU data to the polynomial +coefficients needed for the solution polynomials in FLEXI. Since FLEXI only +uses unstructured hexahedral elements and TAU is in contrast also based on +tetrahedrals we can not use the TAU mesh information natively. An example +of this incompatibility including a schematic of the interpolated solution is +visualized in Fig. 4. +TAU +FLEXI +Fig. 4: Visualization of the TAU-FLEXI interface including different point- +sets. Light green curve shows an approximate solution after interpolating the +TAU solution to the FLEXI point-set. +The interpolation algorithms thus have to work with scattered data. This +means we have a cloud of points with a submerged target mesh. Therefore, +interpolation from the TAU source data to the FLEXI target data has to be +done using unstructured interpolation algorithms. There are several algorithms +available that are suitable for such tasks. +Scattered data interpolation generally can achieve good accuracy and per- +formance but is highly dependent on the distribution of the source points [21]. +On the other hand implementing scattered data interpolation routines enable +us to gain a more universal interface, since other codes and solution formats +can be implemented easily. +Since the solution data of the source solver is provided beforehand, we do +not have to map the data during run time, which saves a lot of computation +time. In addition, the meshes used by both codes are known a priori (and +remain constant during the computation). Still, good performance is crucial. +Therefore, we implement the interface framework with MPI parallelization. +This is done by reading in the mesh and distributing the elements equally +between each processor using MPI. The elements are sorted along a Hilbert +curve [20] which minimizes the communication interfaces between the individ- +ual processors cf. red curve in Fig. 5a. This approach however can only be +done for the target FLEXI mesh, because we need the mesh information to +distribute the data directly. For the source data we only read in data points. + +Springer Nature 2021 LATEX template +10 +A Time-Accurate Inflow Coupling for Zonal LES +Hence, a simplification and distribution of the load between the processors is +not possible in a first step. Thus, we read in the source data into shared memory +[11, 22]. Each processor then accesses the shared memory source data and sim- +plifies it by only selecting the data which is necessary to spatially interpolate +in its individual domain. With this method we can save a lot of computational +time and decrease our memory footprint without sacrificing accuracy. +If we use surface data as an input to the spatial interpolation algorithms +we have to consider load balancing in more detail. The distribution of volume +elements that has just been described, does not take surfaces into account. +Thus, for the surface mapping we can not ensure that every part of the decom- +posed domain contains surface elements that have to be mapped (cf. Fig. 5a). +Therefore, especially for small domains with many processors, it is possible +that not all processors are working on the task resulting in a inefficient map- +ping. Hence, we have to redistribute the load between the processors according +to the number of surface elements (cf. Fig. 5b). This can be done by assigning +each surface element a high weight for domain decomposition. This weight is +chosen by counting the number of boundary sides that have to be mapped per +element and it generally reduces the computational load on MPI ranks that +contain such a coupling interface. To do so we first have to read in the mesh +file normally, then apply the surface weighting and finally reinitialize the mesh +reading process [23]. With this approach we can gain performance improve- +ments while sacrificing a few seconds in the initialization process due to the +necessary reinitialization of the mesh. The overall cost of surface interpola- +tion will be lower than using volume data. The difference between volume and +surface distribution is visualized in Fig. 5. +Additionally, we have to ensure to provide a buffer region around every +individual MPI domain in order to establish the interpolation stencils for each +point. The buffer area is estimated by taking the size of the largest element +in the complete domain of the source points into account. Since the largest +element is not known directly, we take the distances between the scattered +points into account and use the largest distance for that matter. +2.3.1 Nearest neighbor interpolation +The nearest neighbor interpolation checks the source data coordinates and +finds the closest point to the desired FLEXI target point by point-wise com- +parison. The values U of the source data are then directly stored as a nodal +coefficient in FLEXI. This type of interpolation yields a piecewise constant +solution. Another requirement is an evenly distributed source mesh. If these +requirements can not be met, there is risk of bad results. This does not auto- +matically mean the the results are not physical, but rather that the resulting +interpolation polynomial inside a DG cell is ill conditioned and can, due to the +massive jumps, result in an unnaturally oscillating mapped solution. Another +phenomenon one can observe is the possibility to get jumps on the element +boundaries of the target mesh, if the boundary nodes are not included. On the + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +11 +Volume Mapping +(a) Target domain is fully submerged in +the source data (gray). Domain decom- +position does not have to take surface +elements into account. +Surface Mapping +(b) Boundary is aligned with surface +source data (gray). MPI distribution has +to be adapted in order for all three proces- +sors to contain mapped surface elements. +Fig. 5: Differences between MPI domain decomposition for volume and surface +mapping on three MPI processors. Same colors correspond with the same MPI +domain. Space filling curve is visualized in red. +other side this interpolation technique yields fast and good results if the source +and target data are well aligned or if the meshes coincide at the interface. +2.3.2 Inverse Distance Weighting +A more general approach is available using inverse distance weighting [24]. The +target solution is calculated using a weighted average of the the source value +u(⃗x)target = +�Nsource +i=i +ωi(⃗x)ui,source +�Nsource +i=i +ωi(⃗x) +(1) +with Nsource denoting the number of source points in the whole domain. In +contrast to the nearest neighbor approach we not only take one point into +account, but all in the source area. The weights ω(⃗x) are depending on the +distance between the source points and the target solution point +ωi = +1 +� +∥⃗x − ⃗xi∥L2 +�p +(2) + +Springer Nature 2021 LATEX template +12 +A Time-Accurate Inflow Coupling for Zonal LES +and a weighting exponent p. For p ⇒ ∞ the inverse distance weighting +approach resembles the nearest neighbor method. A modification to the gen- +eral inverse distance weighting was introduced by Shepard, who proposed to +only take the source points into account that are within a predefined radius R +around the target point [25]. This reads as +ωi = +�max(0, R − ∥⃗x − ⃗xi∥L2) +R ∥⃗x − ⃗xi∥2 +�2 +(3) +with R denoting a predefined search radius. +2.3.3 Radial Basis Functions +A third option to consider for unstructured interpolation are radial basis func- +tions ϕ [26, 27]. These methods allow for high-order accurate interpolants s +of unstructured data. The interpolant consists of the weighted sum of radial +basis functions. In contrast to the other methods introduced earlier we have +to solve a linear equation system to invert the Vandermonde and to determine +the weights ω satisfying +s(⃗x) = +Nsource +� +i=1 +ωiϕ(∥⃗x − ⃗xi∥L2) +(4) +and therefore +uj,source = +Nsource +� +i=1 +ωiϕ(rji) +(5) +with rki = ∥⃗xk − ⃗xi∥L2. We rewrite the interpolation condition in matrix +notation +� +���� +ϕ(r11) ϕ(r12) · · · ϕ(r1N) +ϕ(r21) ϕ(r22) · · · ϕ(r2N) +... +... +... +... +ϕ(rN1) ϕ(rN2) · · · ϕ(rNN) +� +���� +� +���� +ω1 +ω2 +... +ωN +� +���� = +� +���� +u1,source +u2,source +... +uN,source +� +���� . +(6) +This can be rewritten in matrix form as Φij⃗ωi = ⃗uj,source using index nota- +tion. Since we have to invert the matrix Φ for interpolation, the radial basis +approach is the most expensive of the introduced methods. +We evaluate the interpolant +u(⃗x) ≈ +N +� +i=1 +ωiϕ(∥⃗x − ⃗xi∥L2) +(7) +and get the value at an arbitrary point in the computational domain. + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +13 +101 +102 +103 +10−17 +10−12 +10−7 +10−2 +1 +-2 +1 +-5 +gridsrc +L2-error +Nearest Neighbor +Modified Shepard +RBF (thin plate) +Fig. 6: Convergence of the L2-error of an interpolated one-dimensional sine +function for different interpolation methods. +Typical radial basis functions for interpolation are multiquadratic ϕ(r) = +� +1 + (εr)2, inverse multiquadratic ϕ(r) = +1 +√ +1+(εr)2 , Gaussian ϕ(r) = e−(εr)2 +and thin plate spline ϕ(r) = r2 ln(r) functions with r = ∥⃗xj − ⃗xi∥L2. The +parameter ε defines the shape of the function and is used for scaling. The +multiquadratic and the thin plate spline have shown to be the most reliable +radial basis functions for this use case. Since the thin plate spline does not +require any additional user parameter ε we use this function for all further +investigations. +During implementation of the algorithms above some observations were +made. First, none of the scattered interpolation method is designed in a +way to be conservative. Thus, we interpolate the primitive variables and, for +consistency reasons, convert to conservative variables after mapping. +2.3.4 Comparison of the Spatial Interpolation Methods +Before assessing the performance of the spatial interpolation routines in con- +text of the mapping routines, we investigate the convergence behavior in an +isolated test case. Thus, we calculate the L2-error of a simple one-dimensional +interpolation of a sine function f(x) = sin(2πx). +We plot the error in Fig. 6 over the sampling resolution of the source +data. We can see that radial basis function interpolation clearly yields the +best results with lower errors and a better convergence rate EOC = 5 than +nearest neighbor and inverse distance weighting interpolation with EOC = 2. +We assess the accuracy and the differences between the spatial interpolation +schemes in more detail in Sec. 3. + +Springer Nature 2021 LATEX template +14 +A Time-Accurate Inflow Coupling for Zonal LES +2.4 Temporal Interpolation +In addition to the spatial interpolation we also have to interpolate temporally +in order to account for the different time stepping schemes in the source and +target codes. FLEXI e.g. uses an explicit low storage Runge-Kutta method to +advance the equation systems in time. TAU on the other hand uses an implicit +time discretization to accomplish that. However, in addition to the different +time stepping schemes, the time step and output rate of the simulation data can +change between different simulations. For using the data as an instantaneous +boundary condition we have to ensure that we can provide the target solver +with the correct inflow data at an arbitrary point of time. Thus, it is crucial to +interpolate the results of the spatial interpolation in time to get a continuous +temporal interpolator. +In contrast to the volume and surface mapping the temporal interpola- +tion consists of purely one-dimensional problems. For one-dimensional data +there are vast numbers of different interpolation techniques. In this work we +use polynomial interpolation in combination with a Lagrange basis and spline +interpolation. +We use the Lagrange interpolation basis since coefficient and solution values +coincide [28]. Also the evaluation can be easily done with the tools already built +into FLEXI, since the solution in each element consists of the tensorproduct +of three one-dimensional nodal Lagrange functions. +Furthermore two different variants of spline interpolation are implemented. +A common open spline as well as the Akima spline [29]. In contrast to a typical +spline an Akima spline does not take the second derivative into account. This +leads to a more evenly distributed solution and fewer overshoots compared to +the open spline. The problem of overshoots can also be found in polynomial +interpolation of degree p ≥ 2. This becomes especially important if an implicit +source method is paired with an explicit target solver. In this case the temporal +interpolation has to come up for a huge number of time steps since the time +step in an explicit scheme is typically much smaller than implicit time steps. +Thus, overshoots can play a significant role for the overall mapping quality. +If the time steps of source and target method are similar, the effect of the +temporal interpolation becomes smaller. However, it should be noted that even +in this case, overshoots can be generated. Generally, for these reasons it is +recommended to either use linear interpolation or the Akima interpolation for +the most reliable results. We will show this in Sec. 3. +Hence, the resulting quality of the interpolation depends on multiple fac- +tors. First, on the chosen interpolation method. Second, on the sample rate of +the provided state or boundary source files. Thus, a general prediction of the +error resulting from temporal interpolation is difficult. +The interpolation is done in a separate tool and is not only limited to sur- +face data, but can also be done with restart files of any FLEXI simulation. The +result is processed and saved in an HDF5® file which includes the coefficients +for every polynomial at every temporal sample point. + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +15 +The resulting files of the temporal interpolation routine can either be +directly used in FLEXI for evaluation of the interpolant or even be used to +generate a restart file to continue simulation at an arbitrary point of time. +The temporal interpolator generated contains the resulting polynomial/s- +pline at each degree of freedom. Thus, the overall size of the interpolator array +has more dimension (polynomial coefficients and time) than the solution array. +With increasing dimensionality the memory requirements of the array also +increase. Depending on the amount of source data available it might be neces- +sary to partition the resulting temporal interpolant in order to avoid memory +overflow during simulation of the target domain. Therefore, a maximal size for +the interpolant array has to be provided by the user. The interpolation algo- +rithm will then partition the data into equally sized datasets, each containing +a period of time which results from the user parameter. FLEXI then only reads +in the dataset that contains the temporal information of the current FLEXI +time step. Thus, during runtime of FLEXI the saved interpolant is only eval- +uated, allowing for obtaining an interpolated solution at an arbitrary point of +time. +3 Validation of the Interface +In this section we start validating the mapping algorithms. We chose a gradual +approach and start by showing a proof of concept, followed by the tempo- +ral algorithms and in the end assess the convergence behavior of the spatial +interpolation algorithms of the interface. +We start to evaluate the algorithms by applying them to very simple test +cases. Thus, we chose the linear scalar advection equation +ut + ∇ · (au) = 0 +with +a ∈ R +(8) +due to its simplicity and a priori known exact solution for given initial con- +ditions. The transport velocity is set to a = 1. We vary the initial conditions +between the tested scenarios and describe them in the corresponding sections +in more detail. The source domain Ω ∈ [−1, 1]3 and the target domain +Ω ∈ [1, 3] × [−1, 1]2 are designed to have a shared interface at x = 1. The +source data as well as target data for these test cases are fully generated using +FLEXI. Triple-periodic boundary conditions are used for the source mesh and +the target mesh is designed to have periodic boundary conditions in y and z. +In x-direction we have the instantaneous interface condition at x = 1 and a +outflow at x = 3. +3.1 Proof of Concept +We start the validation of the interface by applying it to a very basic sine test +case with +u(x, t) = sin(π(x − at)). +(9) + +Springer Nature 2021 LATEX template +16 +A Time-Accurate Inflow Coupling for Zonal LES +−1 +−0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +−1 +−0.5 +0 +0.5 +1 +x +u +t = 0.0 +t = 0.8 +t = 1.6 +Interface +Fig. 7: Overview of the spatial mapping process for a traveling sine wave. +The initial conditions are set to u0(x, t = 0). The exact solution u(x, t) is +purely x dependent and thus the values at the interface plane u(x = 1, t) do +not vary in y and z. The target domain is initialized with a constant solution +u0 = const. = 0. +For this first test we match the surface elements at the interface and thus +can use nearest neighbor interpolation without sacrificing accuracy (source +and target points coincide). In this case the nearest neighbor algorithm will +just copy the data from the source to the target domain. Source and target +mesh are only offset in x-direction by the length of the domain. In Fig. 7 the +initial condition is depicted in light green. One can also see the solution of (8) +after t = 0.8 and t = 1.6. The vertical gray dashed grid lines depict the mesh +of the simulation grid and the red dotted line visualizes the interface between +source and target domain. The graphs are extracted from the center line in +x-direction of the equispaced Cartesian cubes, which each have a resolution +of 16 × 16 × 16 using N = 4 polynomials in each element. Legendre-Gauss- +Lobatto points are used for the source and target simulation. Additionally, we +avoid temporal interpolation by sampling the interface data at every physical +time step dt. +In Fig. 7 we can see that the general workflow presented performs as +expected and the information gets propagated over the interface with a = 1. +Since we do not interpolate the data in any way in this test case we expect +the overall errors between source sine and target sine wave to be comparable. +In the source domain we have an L2-error of 9.6506E−7 after t = 2. The L2- +error in the target domain after t = 2 is evaluated in the same way and is +9.6859E−7. This successfully proves that the workflow is capable of mapping +the data without any information loss. We stress that the full framework is +working as if we were coupling between two heterogeneous solvers, with the +exception that the source and target points coincide. +Note that we used continuous initial conditions between source and target +domain. We found that one should avoid having jumps or large discrepancies + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +17 +of the initial conditions between the source and target domain due to nonphys- +ical disturbances created at the inlet of the target domain which are further +propagated. This however, is not due to the interface mapping algorithms but +rather due to the nature of the high-order scheme. In practical applications, +especially for transient simulations, this does not pose a problem since all +structures starting from free-stream, will be mapped into the target domain. +3.2 Assessing the Temporal Interpolation and Sampling +Next, we evaluate the effect of temporal interpolation/sampling on the inter- +face mapping process. Thus, we investigate the effects of different temporal +interpolation schemes and sampling rates on the incoming solution, which we +map via the instantaneous boundary condition. For this test case we chose dif- +ferent exact solution and initial conditions for the linear advection equation 8. +To evaluate the effect of the sampling we chose a initial condition that includes +a discontinuity in order to visualize the information loss. Thus, we use +u(x, t) = +� +1. +if +− 0.5 < x − at < 0.5 +0. +else +. +(10) +We use Legendre-Gauss-Lobatto nodes with N = 4 on a 256×1×1 source and +target domain. The surface elements at the interface are again matched. Thus, +we can use nearest neighbor interpolation to interpolate the surface data in +space, without sacrificing accuracy (copy values from source to target). Fig. 8a +depicts the simulation at two discrete points in time evaluated with different +∆t-interpolants. The interface is located at x = 1 and the discontinuities travel +into the target domain on the right side of the red dotted interface with a = 1. +Already in the overview graph in Fig. 8a we can see substantial differences +between the two interpolated jumps. For this test we chose in total three +sampling rates. A fine sampling rate at ∆t ≈ 10dt which is approximately +ten times the explicit FLEXI time step dt and two coarser sampling rates at +∆t ≈ 50dt and ∆t ≈ 100dt. In Fig. 8a only the finest and the coarsest sampling +rate are visualized. Fig. 8b shows the jumps of all three sampling rates at the +same evaluation time t in more detail. Since the x-axis is scaled identically +one can see that the influence of the temporal mapping on the target FLEXI +simulation is very high. There are two main effects visible: +1. The temporal distance between two samples effects the slope of the jump +and +2. the temporal interpolation method has an effect on the quality of the jump +representation. +The first observation has to only result from the temporal interpolation since +the slope of the shock has been steeper in the source domain. Additionally we +see that lowering ∆t increases the slope again. Thus, ∆t has to be chosen in +a way that the steepest gradient in data can be represented sufficiently. This +however is very much problem depending and requires knowledge of the data. + +Springer Nature 2021 LATEX template +18 +A Time-Accurate Inflow Coupling for Zonal LES +−1 +−0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +0 +0.5 +1 +x +u +t = 1.5 & ∆t = 10dt +t = 2.0 & ∆t = 100dt +Interface +(a) Overview over the simulation domain of the jump test case using spline +interpolation. The jumps are shown in more detail in Fig. 8b. +1.4 +1.5 +1.6 +0 +0.5 +1 +∆t = 10dt +x +u +1.4 +1.5 +1.6 +0 +0.5 +1 +∆t = 50dt +x +1.4 +1.5 +1.6 +0 +0.5 +1 +∆t = 100dt +x +Linear +Quadratic +Spline +Akima +(b) Detailed view of the jumps containing the interpolation techniques visualized in +8a. +Fig. 8: Overview and detailed plots of the jump test case for different ∆t and +temporal interpolation algorithms. +In Sec. 4 we assess an approach on how to determine this in the context of +turbulent eddies. +For the second point a more general statement can be made, since this +observation is nearly independent of ∆t and only becomes more prominent if +∆t becomes sufficiently large. Higher order polynomial approximations tend +to oscillate, especially for equispaced point distribution which is the case for +temporal sampling. Therefore, in Fig. 8 polynomial interpolation is only shown +up to second degree. Also the well known Spline interpolation tends to oscilla- +tions for high ∆t. Most favorable thus is linear or Akima interpolation, which +represent the vertical jump best and recover the steepest gradients. Higher +order polynomial and spline interpolation in this case fail mainly because the + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +19 +physical time steps dt at which we sample are roughly equispaced. Thus, we +see the so-called Runge’s phenomenon for interpolation using an equispaced +point set in time. This is a crucial point since the TAU output frequency is +only determined by its implicit timestep. The target solver FLEXI thus has +to recover the data in every explicit time step. However, having a fixed source +sampling rate and decreasing the target time step, will not further increase +the error since the interpolant is only determined by the sampling rate of the +source data and only is evaluated during FLEXI runtime. +Since Akima interpolation yields slightly smoother results in combination +with steeper gradients, we use Akima interpolation for all following test cases. +3.3 Convergence of the Spatial Mapping +Another important aspect one has to consider is the convergence behavior of +the mapping process. To measure the effect of the interpolation routines we +decided to calculate the error norm of the whole mapping and simulation pro- +cess. Thus, the error includes spatial and temporal interpolation error as well +as the error associated with imposing the instantaneous boundary condition +in FLEXI (e.g. discretization error). +To run the convergence test, we modify the initial conditions from the one- +dimensional sine in Fig. 7. We add a y and z dependency to the exact solution +u in order to have varying u values on the interface plane. Thus, we get +u(x, y, z, t) = sin(π(x − at)) + sin(πy) + sin(πz). +(11) +For the sine wave and the linear transport we have seen earlier that we can +recover the exact solution on the target domain and that the information is +propagated correctly via the instantaneous boundary condition if there is no +spatial and temporal interpolation involved (just copy the values from source +to target). Thus, we want to investigate the effect of different non-matching +interfaces (point sets and resolution) on the error of the simulation and there- +fore have to combine spatial and temporal interpolation techniques for the first +time. We again use Legendre-Gauss-Lobatto points with N = 5 in the source +and target domain. Additionally, we use super-sampling with N = 8. For the +linear transport this is not necessary, since in contrast to the Navier-Stokes +equations we do not see aliasing here. However, we want to assess the con- +vergence as close to the later application as possible and additionally avoid +matching all the degrees of freedom in any case (Nsrc = 5 ̸= 8 = Ntar,super). +In Fig. 9 we see two different testing scenarios. The first in Fig. 9a shows +the L2-error for increasing target resolution and a fixed source mesh. The +second scenario in Fig. 9b depicts the error for an increasing source resolution +and a fixed target mesh. With “grid” we mean the number of elements in each +spatial direction of the Cartesian cube. The sampling timestep is defined by +the physical timestep dt of the source data. Thus, for e.g. the source grid +“1” we extract the interface data at every physical time step and use Akima + +Springer Nature 2021 LATEX template +20 +A Time-Accurate Inflow Coupling for Zonal LES +1 +2 +4 +8 +16 +32 +10−6 +10−4 +10−2 +gridsrc = 8 × 8 × 8 +1 +-1 +1 +-4 +gridtar +L2-error +1 +2 +4 +8 +16 +32 +10−6 +10−4 +10−2 +gridsrc = 32 × 32 × 32 +1 +-1 +1 +-5 +gridtar +L2-error +(a) Convergence behavior for the linear scalar advection equation system for two +different source meshes. +1 +2 +4 +8 +16 +32 +10−6 +10−4 +10−2 +gridtar = 8 × 8 × 8 +1 +-1.5 +1 +-3 +gridsrc +L2-error +1 +2 +4 +8 +16 +32 +10−6 +10−4 +10−2 +gridtar = 32 × 32 × 32 +1 +-1.5 +1 +-3 +gridsrc +L2-error +(b) Convergence behavior for the linear scalar advection equation system for two +different target meshes. +Nearest Neighbor +Modified Shepard +RBF (thin plate) +Fig. 9: Comparison of the convergence behavior of the entire mapping pro- +cedure including spatial, temporal mapping and the instantaneous boundary +condition. +interpolation to interpolate it to the physical time step of the “32” target grid +that is 32 times smaller. +In Fig. 9a we assess the effect of varying target mesh resolutions on the +overall error. The resolution of the source mesh is fixed at 8 × 8 × 8 and +32 × 32 × 32 respectively. We expect the overall error to converge, since the +error can not be mitigated any further if it is dominated by the source data. +Thus, we can see the influence of the source data on the target domain. For +the 8 × 8 × 8 source mesh we can observe this behavior very well. Starting + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +21 +at gridtar ≈ 4 we see that for all interpolation algorithms there is no further +improvement. For the finer source mesh we can observe a similar behavior, +however the overall error is lower and the error is converged later. Due to the +error introduced with the spatial interpolation we can not see a declining error +until the source mesh resolution. +In Fig. 9b we investigate the effect of a varying source mesh on a fixed target +mesh. This can be interpreted as increased input quality for the mapping for +a given target domain. In this test case we again assessed the influence for two +fixed target resolutions at 8×8×8 and 32×32×32. Nearest neighbor, Shepard +as well as RBF interpolation show declining errors for increasing source grid +resolution. This time nearly linear decaying errors can be seen up to the reso- +lution of the target mesh. However, especially for the gridtar = 8 × 8 × 8 case, +we can see that RBF interpolation is capable of recovering information from +source grids with finer resolution than the target mesh. Shepard and nearest +neighbor show clearly weaker performance here and have changing slopes of +the error in this source grid regime. +Overall we can rank the performance of the three tested spatial interpola- +tion techniques. Nearest neighbor interpolation shows as expected the weakest +performance with an experimental order of convergence of EOC ≈ 1.2 in +Fig. 9b. Shepard interpolation shows overall lower errors at roughly the same +order of convergence EOC ≈ 1.5. However, Shepard interpolation is capable +of retaining the error even for source resolutions higher than the target resolu- +tion in Fig. 9b where nearest neighbor interpolation show inconsistent results. +Finally, radial basis function interpolation clearly yields the best results with +an order of convergence of EOC ≈ 2.8. Thus, using RBF interpolation is rec- +ommended. Overall the results in Fig. 9b underline the observations made in +Fig. 6. However, the convergence rates in Fig. 9b are lower for all interpolation +methods. The qualitative observations however are identical and the losses in +EOC are equivalent for all interpolation techniques. One should note, that the +test case in Fig. 9b has an increased complexity, since it is two-dimensional +and we evaluate the error over the whole mapping process compared to an +isolated one-dimensioal interpolation test in Fig. 6. +For large source datasets one should keep in mind, that solving the equation +system necessary to the get the interpolation coefficients for the radial basis +functions gets very expensive and RBF interpolation even in this simple test +case was noticeably (approx. up to an order of magnitude) slower than nearest +neighbor and Shepard interpolation. +4 Results: Cylinder Flow +In this section we investigate the flow around a cylinder at a Reynolds number +of Rec = 3900 [30, 31]. The diameter of the cylinder is defined as c and is used +as the characteristic length in this investigation. The domain has a spanwise +extension of c. For the first time we now map actual TAU surface data into a +FLEXI domain. + +Springer Nature 2021 LATEX template +22 +A Time-Accurate Inflow Coupling for Zonal LES +In Fig. 10 the simulation setup is depicted. Note that the size of the inter- +face planes in Fig. 10 at xI does not match the size in the actual simulation. +In the setup the interface planes are designed in a way that all the turbulent +wake structures are captured by the planes and all vortical structures of the +wake are fully contained in the interface planes. +The most important aspects for assessing the performance of the interface +is to define the interface locations and to define record points (also known as +probe points). We decide to place two interface planes at position xI in the +wake as well as two record points xP . +We use the same number of degrees of freedom in the TAU source and the +appended FLEXI target mesh. Thus, the target mesh resolution including the +interface has to be divided by a factor of eight in order to accommodate for the +higher polynomial degree of FLEXI N = 7. With this approach we minimize +the resulting errors (cf. Fig. 9a). We will show later that this resolution is +sufficient to map all physical structures occurring in this test case. The target +mesh in this case is a simple box that has the same y and z dimensions as the +interface and a sufficiently long x extension for the turbulent wake to develop +and travel. +u∞ +x0 +xP1 xP2 +xI1 +xI2 +x +z +Fig. 10: Cylinder at Rec = 3900 test case definition containing interface planes +and probe points for evaluation. +4.1 Simulation Setup +Next, we discuss the simulation setup of the cylinder test case. We describe +the setup for FLEXI as well as TAU. +The main reason we chose the cylinder flow as the main test case is the +fact, that we can afford to run the whole domain in FLEXI and in TAU. Thus, +we not only can compare the mapped results against the TAU solution but +also against the reference DNS created with FLEXI. Additionally, the cylinder +is a well known geometry in the fluid mechanics community and has been +investigated in detail before. +We use the same base mesh setup for the TAU and FLEXI DNS reference +simulation. The only difference is again that the FLEXI case is coarser by a +factor of eight to consider the high-order polynomials that are used in each +element. Thus, if FLEXI is run with N = 7 we have the same number of + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +23 +degrees of freedom as TAU. We also investigate the solution quality of lower +order polynomials later. For the simulation in FLEXI we use N ∈ [3, 5, 7] +polynomials. +4.2 Sensitivity on Resolution +First, we assess the sensitivity of the test case on the resolution. We conduct +this study in FLEXI since we are interested if the chosen resolution is sufficient +for a DNS. This test case was specifically chosen since it allows to conduct a +fully resolved simulation in FLEXI and in TAU. For typical applications of the +inflow condition this will not be possible. +In Fig. 11 the spectra of the turbulent kinetic energy are shown at three +discrete points in the wake of the cylinder. Each figure contains the spectrum +for three simulations. Each with different polynomial degree. +100 +102 +10−7 +10−5 +10−3 +10−1 +101 +log(k) +log(E(k)) +Spectrum x = 1.75c +100 +102 +10−7 +10−5 +10−3 +10−1 +101 +log(k) +Spectrum x = 4.25c +100 +102 +10−7 +10−5 +10−3 +10−1 +101 +log(k) +Spectrum x = 6.75c +N = 3 +N = 5 +N = 7 +Fig. 11: Comparison of the turbulent kinetic energy of different polynomial +degrees N. +The N = 3 spectrum in Fig. 11 shows a deviation from the N = 5 and N = +7 curves at all evaluation locations. Thus, we can assume that the resolution for +N = 3 is not sufficient for a DNS and does not yield enough dissipation. Since +the turbulent kinetic energy spectra for N = 5 and N = 7 coincide, we can +assume that we are converged at this resolution and thus N = 5 is sufficient +for running a DNS. The expected Strouhal frequency of the cylinder is clearly +visible as a distinct peak in the spectrum [31] for all polynomial degrees. +The simulation with N = 7 (same amount of degrees of freedom as TAU +mesh) is too fine for a typical LES/DNS since the mesh was originally created + +Springer Nature 2021 LATEX template +24 +A Time-Accurate Inflow Coupling for Zonal LES +−0.8 −0.6 −0.4 −0.2 +−1 +0 +1 +Source +−0.8 −0.6 −0.4 −0.2 +−1 +0 +1 +Mapped +1.99 +2 +2.01 +·10−3 +ρ +Fig. 12: Comparison of the instantaneous flow fields of a cylinder wake state. +to be suitable for a hybrid RANS/DNS and a FV code. Evaluating the viscous +wall spacing in FLEXI yields y+ ≈ 0.01 which is more than sufficient, even +for a DNS. This however underlines the benefits of a high-order scheme when +resolving turbulent eddies. +As already shown in Fig. 2 using the same amount of DOFs in FLEXI and +in TAU with a high polynomial degree in FLEXI shows the strength of the +high-order scheme, since this resolution is sufficient in FLEXI to run a DNS. +4.3 Interpolation Error +Next, we assess the interpolation error resulting from interpolating a wake +plane as defined in Fig. 10 onto the FLEXI boundary condition (gridtar = +32×8) using the modified Shepard method. We investigate the error for plane +xI1, which is the one that is located closest to the cylinder. The error is assessed +by using the TAU source data as reference data. To get the same point set for +TAU and FLEXI we evaluate the polynomials of the mapped FLEXI solution +on the TAU solution points. The results are visualized in Fig. 12. +By eye norm the results in Fig. 12 look very convincing. The structures of +the source data are all represented in the mapped solution. Taking a closer look +one can see small overshoots of the mapped solution at the element boundaries. +This effect has already been mitigated by using a super-sampling as dealiasing +technique. To quantify the error we look at the difference between the source +and the mapped data visualized in Fig. 13. +Thus, we evaluate the error of the density for three resolutions gridtar = +32 × 8, gridtar = 16 × 4 and gridtar = 8 × 2. The plots show the relative devia- +tion of the interpolated target data based on the source data. The density plot +confirms the observations we made in Fig. 12 and shows a very small error in + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +25 +−0.8 −0.5 −0.2 +−1 +0 +1 +gridtar = 32 × 8 +−0.8 −0.5 −0.2 +−1 +0 +1 +gridtar = 16 × 4 +−0.8 −0.5 −0.2 +−1 +0 +1 +gridtar = 8 × 2 +0 +0.2 +0.4 +0.6 +0.8 +1 +·10−3 +|∆ρu|/ρusrc +Fig. 13: Comparison of the interpolation error of a cylinder wake state. +Table 1: Minimum, maximum and integral mean values of the primitive +variables. +ρ +u +v +w +p +Mean +Source +2.014E-03 +2.806E+01 +2.001E-01 +-6.915E-01 +1.578E+02 +Mapped +2.014E-03 +2.803E+01 +2.013E-01 +-6.888E-01 +1.578E+02 +Min. +Source +1.986E-03 +-2.975E+01 +-3.402E+01 +-2.510E+01 +1.553E+02 +Mapped +1.986E-03 +-2.978E+01 +-3.333E+01 +-2.508E+01 +1.553E+02 +Max. +Source +2.020E-03 +4.800E+01 +3.178E+01 +2.967E+01 +1.583E+02 +Mapped +2.020E-03 +4.787E+01 +3.153E+01 +2.932E+01 +1.583E+02 +the whole domain. The error increases as espected for coarser resolutions. Cal- +culating the L2-errors for all three meshes yields L2-error(32 × 8) = 5.89E−7, +L2-error(16 × 4) = 3.03E−7 and L2-error(8 × 2) = 9.65E−8 yielding an con- +vergence rate of EOC = 1.31 which is in line with our findings from Fig. 9b +for Shepard’s method. Especially in the part containing eddies in the middle +of the interface planes we see large errors at the eddy boundaries. +In Tab. 1 we can see the minimal and maximal values of the primitive +variables for source and mapped data. Additionally, the integral mean value +is listed. One can note that the mapping yields very good results for density +and pressure. In contrast especially for the velocities we see small deviations +from source data. This error is based on the fact that the interpolation is +not conservative. This gets especially pronounced for the velocity components +due to changing signs. By applying the conversion of primitive to conservative +variables after interpolation we ensure that - despite the interpolation not +being conservative - we get consistent conservative variables. + +Springer Nature 2021 LATEX template +26 +A Time-Accurate Inflow Coupling for Zonal LES +Hence, due to flexibility of the scheme and the generally very small effect +on the mapped results we can neglect the effects of the non-conservativity (cf. +Tab. 1) and directly use the mapped plane as an inflow condition. +4.4 Influence of the Sampling Rate +Now we assess the effect of the sampling rate of the source data on the quality +of the solution in the target domain, which is a very important user parameter +that has to be considered when creating a coupled simulation. We do so by +investigating the effect on the contribution of the incoming turbulence on the +turbulent kinetic energy. +In Fig. 14 the turbulent kinetic energy spectra at two distinct probe points +are visualized. The different colors depict different temporal sampling. With +NSkip we mean how many TAU snapshots are skipped in time. NSkip = 1 +means that every temporal snapshot is used. The physical TAU sampling rate +is ∼ 150 snapshots per characteristic time. The characteristic time is defined as +the time it takes the fluid to cover the distance of the diameter of the cylinder. +For NSkip = 2 we only use every second snapshot. The lighter the color gets +the fewer snapshots are used to recover the TAU solution in FLEXI. +100 +101 +102 +10−6 +10−3 +100 +103 +log(k) +log(E(k)) +x = 4.75c +100 +101 +102 +10−6 +10−3 +100 +103 +log(k) +x = 5.25c +NSkip = 1 +NSkip = 64 +NSkip = 256 +NSkip = 512 +NSkip = 1024 +Reference +Fig. 14: Turbulent kinetic energy at two distinct probe points in the wake of +the cylinder with varying sampling rate. +Fig. 14 shows that the results are heavily dependent on the sampling rate. +This seams reasonable since the sampling rate determines which structures are +mapped via the instantaneous boundary condition. According to the Nyquist +criterion there is a value for NSkip for which the solution is not represented +anymore. In this case for NSkip ≥ 512 we no longer see agreement with the + +Springer Nature 2021 LATEX template +A Time-Accurate Inflow Coupling for Zonal LES +27 +reference solution. For smaller NSkip there is better agreement with the black +reference solution. Hence, two major observations can be made. First, for high +NSkip the major flow structures can not be recovered and even the Strouhal +frequency is not represented correctly. Additionally, after some development in +the target domain at x = 5.25c, we can see the there is a lot of disagreement +even for low k. Second, we can observe that the energy does not adapt and we +loose energy in high modes for large NSkip. +From these observations we can conclude that the sampling frequency is +dependent on the structures that have to be mapped to the new domain. Thus, +we define a measure to quantify the “eddy size - sampling rate” relation which is +closely related to the underlying spatial discretization scheme. From literature +(e.g. [1, 10]) we know that there is a similar criteria for spatial discretization, +which uses the parameter numbers-per-wavelength nPPW to quantify the prop- +erty of a spatial discretization scheme in resolving multi-scale structures. For +DGSEM it is known that nPPW,DGSEM ≈ 4. +In this case we take two sizes as reference. First, according to [32] the large +structures are of the size of the cylinder which corresponds to L = 1c. From +the simulation setup and the properties of the DG scheme we estimate the +smallest structures according to +l = +Ldomain +#DOF · nPPW,DGSEM +≈ 0.06c. +(12) +with Ldomain denoting the size of the domain and #DOF the number of DOFs +used to discretize the domain. Taking u∞ into account, we get an approxi- +mation for how long it takes an eddy to be advected over the interface plane, +assuming Taylor’s hypothesis [33]. Taking the sampling frequency into account +we can estimate that for the smallest structures we need NSkip ≈ 4 and for the +large structures NSkip ≈ 64 is sufficient. This behavior for L = c is also under- +lined in Fig. 14. Only using every 64th sample NSkip = 64 still provides us with +the main structures and correct amplitudes, while NSkip > 64 shows signs of +underresolution. Using these information we can approximate a criterion on +how many points we need per structure/eddy that has to be transported over +the interface. It turns out that for both large and small eddies we need approx- +imately 2.3 samples per eddy. As one would expect, we can conclude that +spatial and temporal discretization requirements are similar for the interface. +We repeated this evaluation for both interface planes xI1 and xI2. Both +showed qualitatively identical results. +5 Summary +In this work we introduced a method to generate an instantaneous boundary +condition relying on a precursor simulation. We presented the numerical meth- +ods necessary to handle differences in spatial and temporal discretization via +interpolation as well as validated the scheme for simple test cases and a more +complex cylinder wake. + +Springer Nature 2021 LATEX template +28 +A Time-Accurate Inflow Coupling for Zonal LES +We have shown how to generate numerically stable inflow and initial condi- +tions with the methods described in this paper that are universally applicable +also to other solvers than TAU and even experimental data. +The requirements regarding sampling rate are similar to those of the +spatial discretization and thus need approximately four sampling points per +wavelength, depending on the temporal interpolation scheme used. +We implemented several mapping techniques and showed the differences +in interpolation quality and additionally demonstrated their capabilities of +reconstructing scattered source data. In addition we utilized super-sampling +of the interpolation to increase the overall accuracy and to mitigate the errors +due to aliasing and numerical incompatibilities. +In terms of spatial resolution difference at the interface we observed that +increasing the resolution of the source data never posed a problem. How- +ever, coarsening the data too much can produce large aliasing errors which +cause trouble for the high-order scheme. Thus, we recommend using a similar +resolution on both sides of the interface. +The introduced interface now has to be applied to more complex scenarios. +The tool-chain introduced in this paper is already designed to handle these +kind of challenging simulations. +Acknowledgments. +The authors gratefully acknowledge the Deutsche +Forschungsgemeinschaft DFG (German Research Foundation) for funding this +work in the framework of the research unit FOR2895. We also thank the +Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this +project (GCS-lesdg) by providing computing time on the GCS Supercomputer +HAWK at Höchstleistungsrechenzentrum Stuttgart (www.hlrs.de). +References +[1] Flad, D., Beck, A., Guthke, P.: A large eddy simulation method +for DGSEM using non-linearly optimized relaxation filters. Journal of +Computational Physics 408 (2020). https://doi.org/10.1016/j.jcp.2020. +109303 +[2] Jarrin, N., Benhamadouche, S., Laurence, D., Prosser, R.: A synthetic- +eddy-method for generating inflow conditions for large-eddy simulations. +International Journal of Heat and Fluid Flow 27(4), 585–593 (2006). +https://doi.org/10.1016/j.ijheatfluidflow.2006.02.006 +[3] Jarrin, N., Prosser, R., Uribe, J.C., Benhamadouche, S., Laurence, +D.: Reconstruction of turbulent fluctuations for hybrid RANS/LES +simulations using a Synthetic-Eddy Method. 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Journal of Fluid Mechanics 640, +1–4 (2009). https://doi.org/10.1017/S0022112009992126 + diff --git a/bNE1T4oBgHgl3EQfdQQZ/content/tmp_files/load_file.txt b/bNE1T4oBgHgl3EQfdQQZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..97a84969dfe7e2765f546b855393c46a9ff49a38 --- /dev/null +++ b/bNE1T4oBgHgl3EQfdQQZ/content/tmp_files/load_file.txt @@ -0,0 +1,1085 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf,len=1084 +page_content='Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES Marcel Blind1*, Johannes Kleinert1, Thorsten Lutz1 and Andrea Beck2,1 1Institute of Aerodynamics and Gas Dynamics, University of Stuttgart, Pfaffenwaldring 21, Stuttgart, 70569, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2Institute of Fluid Dynamics and Thermodynamics, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, Magdeburg, 39106, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' E-mail(s): blind@iag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Contributing authors: kleinert@iag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' lutz@iag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' beck@iag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Abstract Generating turbulent inflow data is a challenging task in zonal Large Eddy Simulation (zLES) and often relies on predefined DNS data to gen- erate synthetic turbulence with the correct statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The more accurate, but more involved alternative is to use instantaneous data from a precur- sor simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Using instantaneous data as an inflow condition allows to conduct high fidelity simulations of subdomains of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' an aircraft including all non-stationary or rare events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this paper we introduce a tool-chain that is capable of interchanging highly resolved spatial and temporal data between flow solvers with different discretization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To accomplish this, we use interpolation algorithms suitable for scattered data in order to interpolate spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In time we use one-dimensional interpolation schemes for each degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The results show that we can get stable simulations that map all flow features from the source data into a new target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, the coupling is capable of mapping arbitrary data distributions and formats into a new domain while also recovering and conserving turbulent structures and scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The necessary time and space resolution requirements can be defined knowing the reso- lution requirements of the used numerical scheme in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Keywords: DGSEM, instantaneous inflow condition, coupling, zonal LES 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='03192v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='flu-dyn] 9 Jan 2023 Springer Nature 2021 LATEX template 2 A Time-Accurate Inflow Coupling for Zonal LES 1 Introduction In modern Computational Fluid Dynamics (CFD) research Large Eddy Sim- ulation (LES) is becoming more popular due to increased computation performance [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However, many practical applications still are out of range for a detailed investigation using this technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Therefore, many simulations run today are based on so called zonal approaches that often depend on pre- defined DNS data to generate synthetic turbulence with the correct statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' By zonal we mean that only a small subset of a domain is simulated with the LES method in order to decrease computational cost, while the majority of the domain is for example solved by a much cheaper Reynolds-Averaged Navier-Stokes (RANS) method, or the subset is equipped with suitable bound- ary conditions, in particular scale-resolving inflow data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This can be of special interest in the simulation of turbulent boundary layers, where we do not want to simulate the initial transition process, but are just interested in the fully developed boundary layer as a starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To achieve this, there have been developed many approaches, such as the synthetic eddy method [2, 3] or the recycling-rescaling approach [4, 5] which allow for significantly smaller domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' A practical example is the simulation of acoustic noise at the trail- ing edge of an airfoil where a detailed simulation of only a part of the airfoil is needed [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' One similarity of the applications just described is their dependency on boundary layer properties and therefore reference data from literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Another slightly different example is the investigation of the interaction of an incoming turbulent wake with the boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For example this can be applied to the interaction of a turbulent wing wake with the horizontal stabilizer of an aircraft (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The described scenario poses a challenge, since fully scale resolving codes often can not afford to compute the whole aircraft and codes that are capable of running a simulation of a whole aircraft are often not able to run high fidelity simulation of parts of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Therefore, there is the need to map the results in a time-accurate manner from one simulation to an inflow/initial condition on a detailed simulation with a smaller domain and to impose them as inflow conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This approach allows for refined simulation of areas of special interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Also such a coupling between these codes enables a way to further investigate the interaction between turbulence of different physical scales very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Therefore, zonal simulations of high Reynolds number flow become feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' When trying to couple different numerical codes we encounter several problems on how to approach this: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Is a two-way coupling necessary or is one-way sufficient?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Do we couple the codes during runtime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Are the underlying numerics compatible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The first question is - in context of the scenario described above - easy to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since we only are interested in the effects of an incoming flow to the target domain, a one-way coupling is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We note that, depending on the equation system, a one-way coupling via a prescribed Dirichlet boundary Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 3 is prone to errors, since information transport is limited to one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' How- ever, we can justify this simplification by assuming that we e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' extract the flow in a wake region with no proximity to a wall in case of the compressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, the simplification removes a lot of com- plexity and thus enables efficient coupling of different solvers and experiments which would not be possible in a two-way coupled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we can directly answer the second question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Having both codes run separately allows us to perform the mapping in a preprocessing step for the zonal simulation and thus removes a bottleneck during runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In addition, it avoids the complexities of having to solve the High Performance Comput- ing (HPC) problem of having to run two possibly very heterogeneous codes synchronously and establish efficient parallel communication patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' As men- tioned before the coupling is designed to perform detailed simulations of a subdomain, meaning that the area of special interest is also contained in the full domain simulation and therefore is assumed to be represented in a suffi- cient way to capture all the necessary physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Also we have to consider that the incoming physics can be truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We thus investigate the influence of different resolution combinations in order to quantify this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The third question is harder to answer since we not only have to take the spatial discretization like finite volume, finite difference and finite element methods into account, but also take care of the temporal discretization, which in most applications is either implicit or explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In general, two choices for mapping the solutions between two heterogeneous representations are possi- ble: A projection approach, and an interpolation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' While the former is (approximately) conservative, ensuring this property on arbitrary meshes in space and time is cumbersome, expensive (as it requires non-local operations) and not very flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The interpolation approach relaxes this condition, mak- ing the mapping process very general and allows for extending the algorithms to work with arbitrary (x, y, z) data as an input and map it to a compatible data format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The mapping algorithms thus have to be able to capture reso- lution differences from both grid spacing and numerical efficiency per DOF, arbitrary points and inconsistent time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Hence, interpolation algorithms seem to be a good choice for achieving these properties in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Another problem we have to tackle is how to deal with large data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Although we opt for an offline coupling which avoids having to transfer the data in situ, time resolved surface or volume data is very memory intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, memory management algorithms have to be taken into account, includ- ing parallelization, load balancing and data reduction in order to keep compute costs low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this paper we want to show that mapping instantaneous data from the DLR finite volume code TAU to the high-order spectral element code FLEXI is possible and allows for detailed simulation of subdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we want to answer the following research questions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Is it possible to get a mapping for the data which allows for a numerically stable coupled simulation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 A Time-Accurate Inflow Coupling for Zonal LES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Which temporal resolution is needed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' How does the difference in spatial resolution (mesh and numerical efficiency per DOF) affect the mapping error?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2 Numerical Methods An important aspect for the mapping algorithm is the knowledge of the under- lying numerics and the associated scale resolving properties which we are going to assess in more detail in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1 Code Frameworks The target numerical scheme for which the data has to be prepared is the open- source discontinuous Galerkin framework FLEXI, developed at the University of Stuttgart [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this scheme the domain is partitioned into non-overlapping unstructured hexahedral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We can choose an arbitrary polynomial degree for the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This also means that the solution in each element is represented as a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In contrast to the spectral element code, the source data is typically point- wise data resulting from an experiment or finite volume code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The presented mapping algorithms are designed and implemented to be generally applicable, but are optimized to work with the DLR finite volume code TAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 1 a typical application of TAU is visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' TAU uses hybrid RANS/LES methods e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' for cases with separated flows, where attached boundary layers are treated in RANS mode and detached wake regions are resolved in LES mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus the effect of the wing wake on the boundary layer of the HTP is hard to investigate within TAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' FLEXI on the other hand resolves the boundary layer and thus is capable of quantifying the influence of the wing wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, the region of interest that can be simulated within FLEXI is marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' TAU will also be used to validate the results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 4 later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1 TAU The finite volume solver TAU is developed by the German Aerospace Center (DLR) and widely used among the aviation industry [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' It solves the Euler or RANS equations both on structured and unstructured grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Several one- and two-equation models and Reynolds stress models are implemented for turbulence modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, LES or hybrid RANS/LES simulations can also be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Hexahedra, tetrahedra, triangular prisms and pyramids are supported elements for the cells of the primary grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For the computation of the numerical fluxes at the cell boundaries, different upwind schemes and central approximations are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Both explicit and implicit schemes can be chosen for the integration in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The resulting linear system is solved with SGS or LUSGS schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For convergence acceleration, local time stepping, residual smoothing and multigrid methods are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Parallelization is achieved by domain decomposition, with communication through the message passing interface (MPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 5 TAU FLEXI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 1: Tandem wing configuration with visualized wing wake interacting with the HTP simulated in TAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Region of interest for detailed simulation in FLEXI is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 FLEXI FLEXI is a high-performance open-source CFD solver based on the Discon- tinuous Galerkin Spectral Element Method (DGSEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' It utilizes hexahedral tensor product elements with an arbitrary polynomial degree in each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since DG methods are hybrid schemes combining finite element and finite vol- ume methods, we use a Roe Riemann solver with minimum dissipation for the fluxes between the elements [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, we represent the polyno- mial solution on a non-equispaced Legendre-Gauss or Legendre-Gauss-Lobatto point set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since FLEXI acts as a framework there are multiple equation sys- tems implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this paper we mainly use the compressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For validation the linear scalar advection system is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The results in the application section are created using the compressible Navier-Stokes equations, which are implemented as skew-symmetric split form approxima- tions to minimize aliasing instabilities [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The boundary conditions generally are imposed weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This means, that we do not prescribe the state at the corresponding solution point in the element, but rather prescribe the numer- ical flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' FLEXI is parallelized using MPI and was successfully tested on up to O(105) cores [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3 Comparison of the Code Frameworks Discontinuous Galerkin methods are commonly used high-order schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Finite volume methods in contrast are for unstructured meshes often limited to second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' It is well known that for the same number of degrees of freedom high-order methods can achieve lower error and need fewer solution points to resolve the same structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This is known as numbers per wavelength nPPW criteria [1, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' High-order methods can achieve nP P W ≈ 4, while second finite Springer Nature 2021 LATEX template 6 A Time-Accurate Inflow Coupling for Zonal LES 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 102 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 10−11 10−6 10−1 1 2 1 6 h L2-error of ρ TAU RANS mode TAU LES mode FLEXI N = 1 FLEXI N = 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2: Comparison of the convergence behavior of TAU and FLEXI for different settings and meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' volume method is often limited to nPPW ≈ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This means that for resolv- ing a structure of a given wavelength, high-order methods need 5 times fewer resolution points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However, this property is heavily dependent on the used polynomial degree N as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, on the same mesh the simu- lation with FLEXI N = 5 not only has a faster decreasing error but also has the smaller error for a given amount of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This becomes obvi- ous since FLEXI N = 1 or TAU on the h = 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 grid correspond in terms of amount of degree of freedom with FLEXI N = 5 at h = 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The Shu- vortex test case [13] is utilized to conduct this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' All simulations are run independently and thus without mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The results denoted as “TAU RANS mode” are obtained with numerical set- tings commonly used for the RANS zones of hybrid RANS/LES simulations in TAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' A second order central flux approximation is used as Riemann solver sta- bilized by artificial dissipation, derived from the scheme after James, Schmidt and Turkel [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Applying a skew-symmetric scheme with matrix dissipation [15] already reduces the dissipation level compared to the TAU-default aver- age of flux scheme with scalar dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The simulations denoted with “TAU LES mode” additionally utilize a reconstruction of the convective fluxes using a linear gradient extrapolation at the cell faces, in a way to reduce the numer- ical dispersion of the skew-symmetric scheme [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Moreover, the coefficient of artificial dissipation is lowered by a factor of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' These settings are suitable for the LES zones of a hybrid simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In the FLEXI runs a Roe Riemann solver is used [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The FLEXI simulation for N = 1 shows a result similar to the TAU runs with the same order of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However N = 1 is typically not used in practical application, since the advantages of high-order schemes are not visible for such low polynomial degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The runs with N = 5 represents a more realistic scenario and show the advantage of high-order polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 Workflow Before presenting the mapping routines, we first discuss the general workflow of how to run a simulation with time resolved input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The general workflow is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' First, the source data has to be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Generally this can be in the form of point-wise scattered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since in this paper we focus on the procedure for mapping TAU data to FLEXI we assume to get either volume snapshots or surface data from TAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Second, we process the data by choosing an appropriate spatial mapping mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Also we have to decide if we only want to map surface data for an instantaneous boundary condition, or if we also want to get the volume information to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' generate a restart file for FLEXI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The tool creates an inter- polated file for each input file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The results are saved in a HDF5® format that uses a polynomial structure closely related to FLEXI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To ensure compatibility with FLEXI, the output polynomial degree is identical to the degree of the simulation we want to run afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Higher-order representations are prone to aliasing and oscillations in gen- eral and the quality of the results heavily relies on the used point set and polynomial degree we perform the interpolation on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since the output degree and point set is defined by the simulation we want to perform eventually, we can not use these parameters for mitigating errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we super-sample the target point representation which helps avoiding errors due to oscilla- tions resulting from the non-polynomials character of the source points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We then map the source data to the super-sampled target data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We found that using M ≈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5, 2] · (N + 1) target points for super-sampling yields good agreement and mitigates oscillations significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This is in line with the lit- erature values for overintegration of turbulent data, which is commonly used in the DG community for dealiasing [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' After interpolating the source data to super-sampled target points, we project the solution to the original basis N < M which removes the high modal information that is especially affected by aliasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Third, we interpolate the results from the second step temporally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The mapped volume or surface files created in step two get converted into a dataset containing the temporal interpolator for each solution point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The interpolator consists of the coefficients of the polynomial, which are dependent on the evaluation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, we partition the data into a user-defined number of subsets to limit the amount of data of each interpolator and avoid memory overflow during FLEXI runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fourth, we provide FLEXI with the resulting file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' FLEXI then evaluates the interpolant in each time step and sets the according boundary condition to the interpolated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Springer Nature 2021 LATEX template ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='A Time-Accurate Inflow Coupling for Zonal LES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Generate Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Spatial Inter- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='polation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Interpolation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Run FLEXI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Generate the source data and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='provide it as a point wise array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Run the spatial interpolation algorithms and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='save the mapped solution for every timestep ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Interpolate the mapped data temporally ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='and partition the data to fit into memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Provide FLEXI with the interpolator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='created before and run the simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Tools ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Explanation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3: Workflow of imposing a time resolved boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1 Some Remarks on Surface Data The mapping algorithms we present can be applied to volume as well as sur- face data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Depending on the provided data the spatial mapping algorithms will either use the volume solution to extract the target boundary or use the provided surface plane directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' TAU is able to write 2D data from a user-defined plane, onto which the flow variables are interpolated internally using algorithms of the chimera technique [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This interpolation will also of course introduce an error that leads to a mismatch between the volume solution of TAU and the 2D data on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The resulting chimera plane can be read in separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Hence, there can be made significant simplifications in term of area reduction which reduces the overall cost of the mapping algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3 Spatial Interpolation An important step to achieve the coupling of TAU with FLEXI is the spatial mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since ultimately we want to create a instantaneous boundary condi- tion we have to map surface data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To keep the interpolation more general we implement a three-dimensional method to also allow volume interpolation and arbitrary oriented surface planes in the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' A major challenge in creating a mapping between TAU and FLEXI are the differences in spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Industrial finite volume codes rely often on tetrahedral meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Therefore, a direct interpolation is difficult, since mesh data structures for hexahedral only codes are dissimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In contrast to FLEXI, TAU stores its solution as low order point wise data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This results in arrays containing the coordinates (x, y, z) and the solution ⃗U [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' FLEXI however stores its solution as polynomial data for each element [7, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 9 difficulty now is to consistently map the point-wise TAU data to the polynomial coefficients needed for the solution polynomials in FLEXI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since FLEXI only uses unstructured hexahedral elements and TAU is in contrast also based on tetrahedrals we can not use the TAU mesh information natively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' An example of this incompatibility including a schematic of the interpolated solution is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' TAU FLEXI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 4: Visualization of the TAU-FLEXI interface including different point- sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Light green curve shows an approximate solution after interpolating the TAU solution to the FLEXI point-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The interpolation algorithms thus have to work with scattered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This means we have a cloud of points with a submerged target mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Therefore, interpolation from the TAU source data to the FLEXI target data has to be done using unstructured interpolation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' There are several algorithms available that are suitable for such tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Scattered data interpolation generally can achieve good accuracy and per- formance but is highly dependent on the distribution of the source points [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' On the other hand implementing scattered data interpolation routines enable us to gain a more universal interface, since other codes and solution formats can be implemented easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since the solution data of the source solver is provided beforehand, we do not have to map the data during run time, which saves a lot of computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In addition, the meshes used by both codes are known a priori (and remain constant during the computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Still, good performance is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Therefore, we implement the interface framework with MPI parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This is done by reading in the mesh and distributing the elements equally between each processor using MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The elements are sorted along a Hilbert curve [20] which minimizes the communication interfaces between the individ- ual processors cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This approach however can only be done for the target FLEXI mesh, because we need the mesh information to distribute the data directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For the source data we only read in data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 10 A Time-Accurate Inflow Coupling for Zonal LES Hence, a simplification and distribution of the load between the processors is not possible in a first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we read in the source data into shared memory [11, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Each processor then accesses the shared memory source data and sim- plifies it by only selecting the data which is necessary to spatially interpolate in its individual domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' With this method we can save a lot of computational time and decrease our memory footprint without sacrificing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' If we use surface data as an input to the spatial interpolation algorithms we have to consider load balancing in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The distribution of volume elements that has just been described, does not take surfaces into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, for the surface mapping we can not ensure that every part of the decom- posed domain contains surface elements that have to be mapped (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Therefore, especially for small domains with many processors, it is possible that not all processors are working on the task resulting in a inefficient map- ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Hence, we have to redistribute the load between the processors according to the number of surface elements (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This can be done by assigning each surface element a high weight for domain decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This weight is chosen by counting the number of boundary sides that have to be mapped per element and it generally reduces the computational load on MPI ranks that contain such a coupling interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To do so we first have to read in the mesh file normally, then apply the surface weighting and finally reinitialize the mesh reading process [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' With this approach we can gain performance improve- ments while sacrificing a few seconds in the initialization process due to the necessary reinitialization of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The overall cost of surface interpola- tion will be lower than using volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The difference between volume and surface distribution is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, we have to ensure to provide a buffer region around every individual MPI domain in order to establish the interpolation stencils for each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The buffer area is estimated by taking the size of the largest element in the complete domain of the source points into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since the largest element is not known directly, we take the distances between the scattered points into account and use the largest distance for that matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1 Nearest neighbor interpolation The nearest neighbor interpolation checks the source data coordinates and finds the closest point to the desired FLEXI target point by point-wise com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The values U of the source data are then directly stored as a nodal coefficient in FLEXI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This type of interpolation yields a piecewise constant solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Another requirement is an evenly distributed source mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' If these requirements can not be met, there is risk of bad results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This does not auto- matically mean the the results are not physical, but rather that the resulting interpolation polynomial inside a DG cell is ill conditioned and can, due to the massive jumps, result in an unnaturally oscillating mapped solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Another phenomenon one can observe is the possibility to get jumps on the element boundaries of the target mesh, if the boundary nodes are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' On the Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 11 Volume Mapping (a) Target domain is fully submerged in the source data (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Domain decom- position does not have to take surface elements into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Surface Mapping (b) Boundary is aligned with surface source data (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' MPI distribution has to be adapted in order for all three proces- sors to contain mapped surface elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 5: Differences between MPI domain decomposition for volume and surface mapping on three MPI processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Same colors correspond with the same MPI domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Space filling curve is visualized in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' other side this interpolation technique yields fast and good results if the source and target data are well aligned or if the meshes coincide at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 Inverse Distance Weighting A more general approach is available using inverse distance weighting [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The target solution is calculated using a weighted average of the the source value u(⃗x)target = �Nsource i=i ωi(⃗x)ui,source �Nsource i=i ωi(⃗x) (1) with Nsource denoting the number of source points in the whole domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In contrast to the nearest neighbor approach we not only take one point into account, but all in the source area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The weights ω(⃗x) are depending on the distance between the source points and the target solution point ωi = 1 � ∥⃗x − ⃗xi∥L2 �p (2) Springer Nature 2021 LATEX template 12 A Time-Accurate Inflow Coupling for Zonal LES and a weighting exponent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For p ⇒ ∞ the inverse distance weighting approach resembles the nearest neighbor method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' A modification to the gen- eral inverse distance weighting was introduced by Shepard, who proposed to only take the source points into account that are within a predefined radius R around the target point [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This reads as ωi = �max(0, R − ∥⃗x − ⃗xi∥L2) R ∥⃗x − ⃗xi∥2 �2 (3) with R denoting a predefined search radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3 Radial Basis Functions A third option to consider for unstructured interpolation are radial basis func- tions ϕ [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' These methods allow for high-order accurate interpolants s of unstructured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The interpolant consists of the weighted sum of radial basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In contrast to the other methods introduced earlier we have to solve a linear equation system to invert the Vandermonde and to determine the weights ω satisfying s(⃗x) = Nsource � i=1 ωiϕ(∥⃗x − ⃗xi∥L2) (4) and therefore uj,source = Nsource � i=1 ωiϕ(rji) (5) with rki = ∥⃗xk − ⃗xi∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We rewrite the interpolation condition in matrix notation � ���� ϕ(r11) ϕ(r12) · · · ϕ(r1N) ϕ(r21) ϕ(r22) · · · ϕ(r2N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' ϕ(rN1) ϕ(rN2) · · · ϕ(rNN) � ���� � ���� ω1 ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' ωN � ���� = � ���� u1,source u2,source .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' uN,source � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' (6) This can be rewritten in matrix form as Φij⃗ωi = ⃗uj,source using index nota- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since we have to invert the matrix Φ for interpolation, the radial basis approach is the most expensive of the introduced methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We evaluate the interpolant u(⃗x) ≈ N � i=1 ωiϕ(∥⃗x − ⃗xi∥L2) (7) and get the value at an arbitrary point in the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 13 101 102 103 10−17 10−12 10−7 10−2 1 2 1 5 gridsrc L2-error Nearest Neighbor Modified Shepard RBF (thin plate) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 6: Convergence of the L2-error of an interpolated one-dimensional sine function for different interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Typical radial basis functions for interpolation are multiquadratic ϕ(r) = � 1 + (εr)2, inverse multiquadratic ϕ(r) = 1 √ 1+(εr)2 , Gaussian ϕ(r) = e−(εr)2 and thin plate spline ϕ(r) = r2 ln(r) functions with r = ∥⃗xj − ⃗xi∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The parameter ε defines the shape of the function and is used for scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The multiquadratic and the thin plate spline have shown to be the most reliable radial basis functions for this use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since the thin plate spline does not require any additional user parameter ε we use this function for all further investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' During implementation of the algorithms above some observations were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' First, none of the scattered interpolation method is designed in a way to be conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we interpolate the primitive variables and, for consistency reasons, convert to conservative variables after mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 Comparison of the Spatial Interpolation Methods Before assessing the performance of the spatial interpolation routines in con- text of the mapping routines, we investigate the convergence behavior in an isolated test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we calculate the L2-error of a simple one-dimensional interpolation of a sine function f(x) = sin(2πx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We plot the error in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 6 over the sampling resolution of the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We can see that radial basis function interpolation clearly yields the best results with lower errors and a better convergence rate EOC = 5 than nearest neighbor and inverse distance weighting interpolation with EOC = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We assess the accuracy and the differences between the spatial interpolation schemes in more detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 14 A Time-Accurate Inflow Coupling for Zonal LES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 Temporal Interpolation In addition to the spatial interpolation we also have to interpolate temporally in order to account for the different time stepping schemes in the source and target codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' FLEXI e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' uses an explicit low storage Runge-Kutta method to advance the equation systems in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' TAU on the other hand uses an implicit time discretization to accomplish that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However, in addition to the different time stepping schemes, the time step and output rate of the simulation data can change between different simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For using the data as an instantaneous boundary condition we have to ensure that we can provide the target solver with the correct inflow data at an arbitrary point of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, it is crucial to interpolate the results of the spatial interpolation in time to get a continuous temporal interpolator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In contrast to the volume and surface mapping the temporal interpola- tion consists of purely one-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For one-dimensional data there are vast numbers of different interpolation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this work we use polynomial interpolation in combination with a Lagrange basis and spline interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We use the Lagrange interpolation basis since coefficient and solution values coincide [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Also the evaluation can be easily done with the tools already built into FLEXI, since the solution in each element consists of the tensorproduct of three one-dimensional nodal Lagrange functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Furthermore two different variants of spline interpolation are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' A common open spline as well as the Akima spline [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In contrast to a typical spline an Akima spline does not take the second derivative into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This leads to a more evenly distributed solution and fewer overshoots compared to the open spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The problem of overshoots can also be found in polynomial interpolation of degree p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This becomes especially important if an implicit source method is paired with an explicit target solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this case the temporal interpolation has to come up for a huge number of time steps since the time step in an explicit scheme is typically much smaller than implicit time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, overshoots can play a significant role for the overall mapping quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' If the time steps of source and target method are similar, the effect of the temporal interpolation becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However, it should be noted that even in this case, overshoots can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Generally, for these reasons it is recommended to either use linear interpolation or the Akima interpolation for the most reliable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We will show this in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Hence, the resulting quality of the interpolation depends on multiple fac- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' First, on the chosen interpolation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Second, on the sample rate of the provided state or boundary source files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, a general prediction of the error resulting from temporal interpolation is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The interpolation is done in a separate tool and is not only limited to sur- face data, but can also be done with restart files of any FLEXI simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The result is processed and saved in an HDF5® file which includes the coefficients for every polynomial at every temporal sample point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 15 The resulting files of the temporal interpolation routine can either be directly used in FLEXI for evaluation of the interpolant or even be used to generate a restart file to continue simulation at an arbitrary point of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The temporal interpolator generated contains the resulting polynomial/s- pline at each degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, the overall size of the interpolator array has more dimension (polynomial coefficients and time) than the solution array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' With increasing dimensionality the memory requirements of the array also increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Depending on the amount of source data available it might be neces- sary to partition the resulting temporal interpolant in order to avoid memory overflow during simulation of the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Therefore, a maximal size for the interpolant array has to be provided by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The interpolation algo- rithm will then partition the data into equally sized datasets, each containing a period of time which results from the user parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' FLEXI then only reads in the dataset that contains the temporal information of the current FLEXI time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, during runtime of FLEXI the saved interpolant is only eval- uated, allowing for obtaining an interpolated solution at an arbitrary point of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3 Validation of the Interface In this section we start validating the mapping algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We chose a gradual approach and start by showing a proof of concept, followed by the tempo- ral algorithms and in the end assess the convergence behavior of the spatial interpolation algorithms of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We start to evaluate the algorithms by applying them to very simple test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we chose the linear scalar advection equation ut + ∇ · (au) = 0 with a ∈ R (8) due to its simplicity and a priori known exact solution for given initial con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The transport velocity is set to a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We vary the initial conditions between the tested scenarios and describe them in the corresponding sections in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The source domain Ω ∈ [−1, 1]3 and the target domain Ω ∈ [1, 3] × [−1, 1]2 are designed to have a shared interface at x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The source data as well as target data for these test cases are fully generated using FLEXI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Triple-periodic boundary conditions are used for the source mesh and the target mesh is designed to have periodic boundary conditions in y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In x-direction we have the instantaneous interface condition at x = 1 and a outflow at x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1 Proof of Concept We start the validation of the interface by applying it to a very basic sine test case with u(x, t) = sin(π(x − at)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' (9) Springer Nature 2021 LATEX template 16 A Time-Accurate Inflow Coupling for Zonal LES −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 3 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1 x u t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='0 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6 Interface Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 7: Overview of the spatial mapping process for a traveling sine wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The initial conditions are set to u0(x, t = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The exact solution u(x, t) is purely x dependent and thus the values at the interface plane u(x = 1, t) do not vary in y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The target domain is initialized with a constant solution u0 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For this first test we match the surface elements at the interface and thus can use nearest neighbor interpolation without sacrificing accuracy (source and target points coincide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this case the nearest neighbor algorithm will just copy the data from the source to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Source and target mesh are only offset in x-direction by the length of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 7 the initial condition is depicted in light green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' One can also see the solution of (8) after t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 and t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The vertical gray dashed grid lines depict the mesh of the simulation grid and the red dotted line visualizes the interface between source and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The graphs are extracted from the center line in x-direction of the equispaced Cartesian cubes, which each have a resolution of 16 × 16 × 16 using N = 4 polynomials in each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Legendre-Gauss- Lobatto points are used for the source and target simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, we avoid temporal interpolation by sampling the interface data at every physical time step dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 7 we can see that the general workflow presented performs as expected and the information gets propagated over the interface with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since we do not interpolate the data in any way in this test case we expect the overall errors between source sine and target sine wave to be comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In the source domain we have an L2-error of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6506E−7 after t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The L2- error in the target domain after t = 2 is evaluated in the same way and is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6859E−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This successfully proves that the workflow is capable of mapping the data without any information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We stress that the full framework is working as if we were coupling between two heterogeneous solvers, with the exception that the source and target points coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Note that we used continuous initial conditions between source and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We found that one should avoid having jumps or large discrepancies Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 17 of the initial conditions between the source and target domain due to nonphys- ical disturbances created at the inlet of the target domain which are further propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This however, is not due to the interface mapping algorithms but rather due to the nature of the high-order scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In practical applications, especially for transient simulations, this does not pose a problem since all structures starting from free-stream, will be mapped into the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 Assessing the Temporal Interpolation and Sampling Next, we evaluate the effect of temporal interpolation/sampling on the inter- face mapping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we investigate the effects of different temporal interpolation schemes and sampling rates on the incoming solution, which we map via the instantaneous boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For this test case we chose dif- ferent exact solution and initial conditions for the linear advection equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To evaluate the effect of the sampling we chose a initial condition that includes a discontinuity in order to visualize the information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we use u(x, t) = � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' if − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 < x − at < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' (10) We use Legendre-Gauss-Lobatto nodes with N = 4 on a 256×1×1 source and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The surface elements at the interface are again matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we can use nearest neighbor interpolation to interpolate the surface data in space, without sacrificing accuracy (copy values from source to target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 8a depicts the simulation at two discrete points in time evaluated with different ∆t-interpolants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The interface is located at x = 1 and the discontinuities travel into the target domain on the right side of the red dotted interface with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Already in the overview graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 8a we can see substantial differences between the two interpolated jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For this test we chose in total three sampling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' A fine sampling rate at ∆t ≈ 10dt which is approximately ten times the explicit FLEXI time step dt and two coarser sampling rates at ∆t ≈ 50dt and ∆t ≈ 100dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 8a only the finest and the coarsest sampling rate are visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 8b shows the jumps of all three sampling rates at the same evaluation time t in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since the x-axis is scaled identically one can see that the influence of the temporal mapping on the target FLEXI simulation is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' There are two main effects visible: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The temporal distance between two samples effects the slope of the jump and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' the temporal interpolation method has an effect on the quality of the jump representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The first observation has to only result from the temporal interpolation since the slope of the shock has been steeper in the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally we see that lowering ∆t increases the slope again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, ∆t has to be chosen in a way that the steepest gradient in data can be represented sufficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This however is very much problem depending and requires knowledge of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 18 A Time-Accurate Inflow Coupling for Zonal LES −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1 x u t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 & ∆t = 10dt t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='0 & ∆t = 100dt Interface (a) Overview over the simulation domain of the jump test case using spline interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The jumps are shown in more detail in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1 ∆t = 10dt x u 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1 ∆t = 50dt x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1 ∆t = 100dt x Linear Quadratic Spline Akima (b) Detailed view of the jumps containing the interpolation techniques visualized in 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 8: Overview and detailed plots of the jump test case for different ∆t and temporal interpolation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 4 we assess an approach on how to determine this in the context of turbulent eddies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For the second point a more general statement can be made, since this observation is nearly independent of ∆t and only becomes more prominent if ∆t becomes sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Higher order polynomial approximations tend to oscillate, especially for equispaced point distribution which is the case for temporal sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Therefore, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 8 polynomial interpolation is only shown up to second degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Also the well known Spline interpolation tends to oscilla- tions for high ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Most favorable thus is linear or Akima interpolation, which represent the vertical jump best and recover the steepest gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Higher order polynomial and spline interpolation in this case fail mainly because the Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 19 physical time steps dt at which we sample are roughly equispaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we see the so-called Runge’s phenomenon for interpolation using an equispaced point set in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This is a crucial point since the TAU output frequency is only determined by its implicit timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The target solver FLEXI thus has to recover the data in every explicit time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However, having a fixed source sampling rate and decreasing the target time step, will not further increase the error since the interpolant is only determined by the sampling rate of the source data and only is evaluated during FLEXI runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since Akima interpolation yields slightly smoother results in combination with steeper gradients, we use Akima interpolation for all following test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3 Convergence of the Spatial Mapping Another important aspect one has to consider is the convergence behavior of the mapping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To measure the effect of the interpolation routines we decided to calculate the error norm of the whole mapping and simulation pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, the error includes spatial and temporal interpolation error as well as the error associated with imposing the instantaneous boundary condition in FLEXI (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' discretization error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To run the convergence test, we modify the initial conditions from the one- dimensional sine in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We add a y and z dependency to the exact solution u in order to have varying u values on the interface plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we get u(x, y, z, t) = sin(π(x − at)) + sin(πy) + sin(πz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' (11) For the sine wave and the linear transport we have seen earlier that we can recover the exact solution on the target domain and that the information is propagated correctly via the instantaneous boundary condition if there is no spatial and temporal interpolation involved (just copy the values from source to target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we want to investigate the effect of different non-matching interfaces (point sets and resolution) on the error of the simulation and there- fore have to combine spatial and temporal interpolation techniques for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We again use Legendre-Gauss-Lobatto points with N = 5 in the source and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, we use super-sampling with N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For the linear transport this is not necessary, since in contrast to the Navier-Stokes equations we do not see aliasing here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However, we want to assess the con- vergence as close to the later application as possible and additionally avoid matching all the degrees of freedom in any case (Nsrc = 5 ̸= 8 = Ntar,super).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9 we see two different testing scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The first in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9a shows the L2-error for increasing target resolution and a fixed source mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The second scenario in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9b depicts the error for an increasing source resolution and a fixed target mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' With “grid” we mean the number of elements in each spatial direction of the Cartesian cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The sampling timestep is defined by the physical timestep dt of the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' the source grid “1” we extract the interface data at every physical time step and use Akima Springer Nature 2021 LATEX template 20 A Time-Accurate Inflow Coupling for Zonal LES 1 2 4 8 16 32 10−6 10−4 10−2 gridsrc = 8 × 8 × 8 1 1 1 4 gridtar L2-error 1 2 4 8 16 32 10−6 10−4 10−2 gridsrc = 32 × 32 × 32 1 1 1 5 gridtar L2-error (a) Convergence behavior for the linear scalar advection equation system for two different source meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 1 2 4 8 16 32 10−6 10−4 10−2 gridtar = 8 × 8 × 8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1 3 gridsrc L2-error 1 2 4 8 16 32 10−6 10−4 10−2 gridtar = 32 × 32 × 32 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 1 3 gridsrc L2-error (b) Convergence behavior for the linear scalar advection equation system for two different target meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Nearest Neighbor Modified Shepard RBF (thin plate) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9: Comparison of the convergence behavior of the entire mapping pro- cedure including spatial, temporal mapping and the instantaneous boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' interpolation to interpolate it to the physical time step of the “32” target grid that is 32 times smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9a we assess the effect of varying target mesh resolutions on the overall error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The resolution of the source mesh is fixed at 8 × 8 × 8 and 32 × 32 × 32 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We expect the overall error to converge, since the error can not be mitigated any further if it is dominated by the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we can see the influence of the source data on the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For the 8 × 8 × 8 source mesh we can observe this behavior very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Starting Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 21 at gridtar ≈ 4 we see that for all interpolation algorithms there is no further improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For the finer source mesh we can observe a similar behavior, however the overall error is lower and the error is converged later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Due to the error introduced with the spatial interpolation we can not see a declining error until the source mesh resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9b we investigate the effect of a varying source mesh on a fixed target mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This can be interpreted as increased input quality for the mapping for a given target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this test case we again assessed the influence for two fixed target resolutions at 8×8×8 and 32×32×32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Nearest neighbor, Shepard as well as RBF interpolation show declining errors for increasing source grid resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This time nearly linear decaying errors can be seen up to the reso- lution of the target mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However, especially for the gridtar = 8 × 8 × 8 case, we can see that RBF interpolation is capable of recovering information from source grids with finer resolution than the target mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Shepard and nearest neighbor show clearly weaker performance here and have changing slopes of the error in this source grid regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Overall we can rank the performance of the three tested spatial interpola- tion techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Nearest neighbor interpolation shows as expected the weakest performance with an experimental order of convergence of EOC ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Shepard interpolation shows overall lower errors at roughly the same order of convergence EOC ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However, Shepard interpolation is capable of retaining the error even for source resolutions higher than the target resolu- tion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9b where nearest neighbor interpolation show inconsistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Finally, radial basis function interpolation clearly yields the best results with an order of convergence of EOC ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, using RBF interpolation is rec- ommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Overall the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9b underline the observations made in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' However, the convergence rates in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9b are lower for all interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The qualitative observations however are identical and the losses in EOC are equivalent for all interpolation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' One should note, that the test case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9b has an increased complexity, since it is two-dimensional and we evaluate the error over the whole mapping process compared to an isolated one-dimensioal interpolation test in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For large source datasets one should keep in mind, that solving the equation system necessary to the get the interpolation coefficients for the radial basis functions gets very expensive and RBF interpolation even in this simple test case was noticeably (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' up to an order of magnitude) slower than nearest neighbor and Shepard interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 4 Results: Cylinder Flow In this section we investigate the flow around a cylinder at a Reynolds number of Rec = 3900 [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The diameter of the cylinder is defined as c and is used as the characteristic length in this investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The domain has a spanwise extension of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For the first time we now map actual TAU surface data into a FLEXI domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 22 A Time-Accurate Inflow Coupling for Zonal LES In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 10 the simulation setup is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Note that the size of the inter- face planes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 10 at xI does not match the size in the actual simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In the setup the interface planes are designed in a way that all the turbulent wake structures are captured by the planes and all vortical structures of the wake are fully contained in the interface planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The most important aspects for assessing the performance of the interface is to define the interface locations and to define record points (also known as probe points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We decide to place two interface planes at position xI in the wake as well as two record points xP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We use the same number of degrees of freedom in the TAU source and the appended FLEXI target mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, the target mesh resolution including the interface has to be divided by a factor of eight in order to accommodate for the higher polynomial degree of FLEXI N = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' With this approach we minimize the resulting errors (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We will show later that this resolution is sufficient to map all physical structures occurring in this test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The target mesh in this case is a simple box that has the same y and z dimensions as the interface and a sufficiently long x extension for the turbulent wake to develop and travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' u∞ x0 xP1 xP2 xI1 xI2 x z Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 10: Cylinder at Rec = 3900 test case definition containing interface planes and probe points for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1 Simulation Setup Next, we discuss the simulation setup of the cylinder test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We describe the setup for FLEXI as well as TAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The main reason we chose the cylinder flow as the main test case is the fact, that we can afford to run the whole domain in FLEXI and in TAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we not only can compare the mapped results against the TAU solution but also against the reference DNS created with FLEXI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, the cylinder is a well known geometry in the fluid mechanics community and has been investigated in detail before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We use the same base mesh setup for the TAU and FLEXI DNS reference simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The only difference is again that the FLEXI case is coarser by a factor of eight to consider the high-order polynomials that are used in each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, if FLEXI is run with N = 7 we have the same number of Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 23 degrees of freedom as TAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We also investigate the solution quality of lower order polynomials later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For the simulation in FLEXI we use N ∈ [3, 5, 7] polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 Sensitivity on Resolution First, we assess the sensitivity of the test case on the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We conduct this study in FLEXI since we are interested if the chosen resolution is sufficient for a DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This test case was specifically chosen since it allows to conduct a fully resolved simulation in FLEXI and in TAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For typical applications of the inflow condition this will not be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 11 the spectra of the turbulent kinetic energy are shown at three discrete points in the wake of the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Each figure contains the spectrum for three simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Each with different polynomial degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 100 102 10−7 10−5 10−3 10−1 101 log(k) log(E(k)) Spectrum x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='75c 100 102 10−7 10−5 10−3 10−1 101 log(k) Spectrum x = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='25c 100 102 10−7 10−5 10−3 10−1 101 log(k) Spectrum x = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='75c N = 3 N = 5 N = 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 11: Comparison of the turbulent kinetic energy of different polynomial degrees N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The N = 3 spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 11 shows a deviation from the N = 5 and N = 7 curves at all evaluation locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we can assume that the resolution for N = 3 is not sufficient for a DNS and does not yield enough dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Since the turbulent kinetic energy spectra for N = 5 and N = 7 coincide, we can assume that we are converged at this resolution and thus N = 5 is sufficient for running a DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The expected Strouhal frequency of the cylinder is clearly visible as a distinct peak in the spectrum [31] for all polynomial degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The simulation with N = 7 (same amount of degrees of freedom as TAU mesh) is too fine for a typical LES/DNS since the mesh was originally created Springer Nature 2021 LATEX template 24 A Time-Accurate Inflow Coupling for Zonal LES −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 −1 0 1 Source −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 −1 0 1 Mapped 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='99 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='01 10−3 ρ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 12: Comparison of the instantaneous flow fields of a cylinder wake state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' to be suitable for a hybrid RANS/DNS and a FV code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Evaluating the viscous wall spacing in FLEXI yields y+ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='01 which is more than sufficient, even for a DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This however underlines the benefits of a high-order scheme when resolving turbulent eddies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' As already shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 2 using the same amount of DOFs in FLEXI and in TAU with a high polynomial degree in FLEXI shows the strength of the high-order scheme, since this resolution is sufficient in FLEXI to run a DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3 Interpolation Error Next, we assess the interpolation error resulting from interpolating a wake plane as defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 10 onto the FLEXI boundary condition (gridtar = 32×8) using the modified Shepard method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We investigate the error for plane xI1, which is the one that is located closest to the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The error is assessed by using the TAU source data as reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To get the same point set for TAU and FLEXI we evaluate the polynomials of the mapped FLEXI solution on the TAU solution points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The results are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' By eye norm the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 12 look very convincing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The structures of the source data are all represented in the mapped solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Taking a closer look one can see small overshoots of the mapped solution at the element boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This effect has already been mitigated by using a super-sampling as dealiasing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' To quantify the error we look at the difference between the source and the mapped data visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we evaluate the error of the density for three resolutions gridtar = 32 × 8, gridtar = 16 × 4 and gridtar = 8 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The plots show the relative devia- tion of the interpolated target data based on the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The density plot confirms the observations we made in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 12 and shows a very small error in Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 −1 0 1 gridtar = 32 × 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 −1 0 1 gridtar = 16 × 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 −1 0 1 gridtar = 8 × 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='8 1 10−3 |∆ρu|/ρusrc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 13: Comparison of the interpolation error of a cylinder wake state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Table 1: Minimum, maximum and integral mean values of the primitive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' ρ u v w p Mean Source 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='014E-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='806E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='001E-01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='915E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='578E+02 Mapped 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='014E-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='803E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='013E-01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='888E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='578E+02 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Source 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='986E-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='975E+01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='402E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='510E+01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='553E+02 Mapped 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='986E-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='978E+01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='333E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='508E+01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='553E+02 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Source 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='020E-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='800E+01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='178E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='967E+01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='583E+02 Mapped 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='020E-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='787E+01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='153E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='932E+01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='583E+02 the whole domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The error increases as espected for coarser resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Cal- culating the L2-errors for all three meshes yields L2-error(32 × 8) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='89E−7, L2-error(16 × 4) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='03E−7 and L2-error(8 × 2) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='65E−8 yielding an con- vergence rate of EOC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='31 which is in line with our findings from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 9b for Shepard’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Especially in the part containing eddies in the middle of the interface planes we see large errors at the eddy boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 1 we can see the minimal and maximal values of the primitive variables for source and mapped data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, the integral mean value is listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' One can note that the mapping yields very good results for density and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In contrast especially for the velocities we see small deviations from source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This error is based on the fact that the interpolation is not conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This gets especially pronounced for the velocity components due to changing signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' By applying the conversion of primitive to conservative variables after interpolation we ensure that - despite the interpolation not being conservative - we get consistent conservative variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 26 A Time-Accurate Inflow Coupling for Zonal LES Hence, due to flexibility of the scheme and the generally very small effect on the mapped results we can neglect the effects of the non-conservativity (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 1) and directly use the mapped plane as an inflow condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='4 Influence of the Sampling Rate Now we assess the effect of the sampling rate of the source data on the quality of the solution in the target domain, which is a very important user parameter that has to be considered when creating a coupled simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We do so by investigating the effect on the contribution of the incoming turbulence on the turbulent kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 14 the turbulent kinetic energy spectra at two distinct probe points are visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The different colors depict different temporal sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' With NSkip we mean how many TAU snapshots are skipped in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' NSkip = 1 means that every temporal snapshot is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The physical TAU sampling rate is ∼ 150 snapshots per characteristic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The characteristic time is defined as the time it takes the fluid to cover the distance of the diameter of the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For NSkip = 2 we only use every second snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The lighter the color gets the fewer snapshots are used to recover the TAU solution in FLEXI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 100 101 102 10−6 10−3 100 103 log(k) log(E(k)) x = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='75c 100 101 102 10−6 10−3 100 103 log(k) x = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='25c NSkip = 1 NSkip = 64 NSkip = 256 NSkip = 512 NSkip = 1024 Reference Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 14: Turbulent kinetic energy at two distinct probe points in the wake of the cylinder with varying sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 14 shows that the results are heavily dependent on the sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This seams reasonable since the sampling rate determines which structures are mapped via the instantaneous boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' According to the Nyquist criterion there is a value for NSkip for which the solution is not represented anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this case for NSkip ≥ 512 we no longer see agreement with the Springer Nature 2021 LATEX template A Time-Accurate Inflow Coupling for Zonal LES 27 reference solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For smaller NSkip there is better agreement with the black reference solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Hence, two major observations can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' First, for high NSkip the major flow structures can not be recovered and even the Strouhal frequency is not represented correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Additionally, after some development in the target domain at x = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='25c, we can see the there is a lot of disagreement even for low k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Second, we can observe that the energy does not adapt and we loose energy in high modes for large NSkip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' From these observations we can conclude that the sampling frequency is dependent on the structures that have to be mapped to the new domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we define a measure to quantify the “eddy size - sampling rate” relation which is closely related to the underlying spatial discretization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' From literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' [1, 10]) we know that there is a similar criteria for spatial discretization, which uses the parameter numbers-per-wavelength nPPW to quantify the prop- erty of a spatial discretization scheme in resolving multi-scale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' For DGSEM it is known that nPPW,DGSEM ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In this case we take two sizes as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' First, according to [32] the large structures are of the size of the cylinder which corresponds to L = 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' From the simulation setup and the properties of the DG scheme we estimate the smallest structures according to l = Ldomain #DOF · nPPW,DGSEM ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='06c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' (12) with Ldomain denoting the size of the domain and #DOF the number of DOFs used to discretize the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Taking u∞ into account, we get an approxi- mation for how long it takes an eddy to be advected over the interface plane, assuming Taylor’s hypothesis [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Taking the sampling frequency into account we can estimate that for the smallest structures we need NSkip ≈ 4 and for the large structures NSkip ≈ 64 is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' This behavior for L = c is also under- lined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Only using every 64th sample NSkip = 64 still provides us with the main structures and correct amplitudes, while NSkip > 64 shows signs of underresolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Using these information we can approximate a criterion on how many points we need per structure/eddy that has to be transported over the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' It turns out that for both large and small eddies we need approx- imately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='3 samples per eddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' As one would expect, we can conclude that spatial and temporal discretization requirements are similar for the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We repeated this evaluation for both interface planes xI1 and xI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Both showed qualitatively identical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' 5 Summary In this work we introduced a method to generate an instantaneous boundary condition relying on a precursor simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We presented the numerical meth- ods necessary to handle differences in spatial and temporal discretization via interpolation as well as validated the scheme for simple test cases and a more complex cylinder wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 28 A Time-Accurate Inflow Coupling for Zonal LES We have shown how to generate numerically stable inflow and initial condi- tions with the methods described in this paper that are universally applicable also to other solvers than TAU and even experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The requirements regarding sampling rate are similar to those of the spatial discretization and thus need approximately four sampling points per wavelength, depending on the temporal interpolation scheme used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We implemented several mapping techniques and showed the differences in interpolation quality and additionally demonstrated their capabilities of reconstructing scattered source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In addition we utilized super-sampling of the interpolation to increase the overall accuracy and to mitigate the errors due to aliasing and numerical incompatibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' In terms of spatial resolution difference at the interface we observed that increasing the resolution of the source data never posed a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' How- ever, coarsening the data too much can produce large aliasing errors which cause trouble for the high-order scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Thus, we recommend using a similar resolution on both sides of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The introduced interface now has to be applied to more complex scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The tool-chain introduced in this paper is already designed to handle these kind of challenging simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' The authors gratefully acknowledge the Deutsche Forschungsgemeinschaft DFG (German Research Foundation) for funding this work in the framework of the research unit FOR2895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' We also thank the Gauss Centre for Supercomputing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='gauss-centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='eu) for funding this project (GCS-lesdg) by providing computing time on the GCS Supercomputer HAWK at Höchstleistungsrechenzentrum Stuttgart (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='hlrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' References [1] Flad, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=', Beck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=', Guthke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=': A large eddy simulation method for DGSEM using non-linearly optimized relaxation filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Journal of Computational Physics 408 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1016/j.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='2957018 [32] Pope, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' : Turbulent Flows, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' print edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Press, Cambridge (2015) [33] Moin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=': Revisiting Taylor’s hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' Journal of Fluid Mechanics 640, 1–4 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} +page_content='1017/S0022112009992126' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE1T4oBgHgl3EQfdQQZ/content/2301.03192v1.pdf'} diff --git a/bdA0T4oBgHgl3EQfGf9T/vector_store/index.faiss b/bdA0T4oBgHgl3EQfGf9T/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..50fe081b6caf93e6e992bb8a3aabb5b491b379ce --- /dev/null +++ b/bdA0T4oBgHgl3EQfGf9T/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:66af8c32b59390438492424103f1091ed261d4876294af5d678f01e3a302ceea +size 7274541 diff --git a/bdE5T4oBgHgl3EQffA9R/content/tmp_files/2301.05623v1.pdf.txt b/bdE5T4oBgHgl3EQffA9R/content/tmp_files/2301.05623v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6ccbe460941232b13d0468d913e8b5b6b9646a4 --- /dev/null +++ b/bdE5T4oBgHgl3EQffA9R/content/tmp_files/2301.05623v1.pdf.txt @@ -0,0 +1,3317 @@ +arXiv:2301.05623v1 [cs.CV] 13 Jan 2023 +Reworking geometric morphometrics +into a methodology of transformation grids +Fred L. Bookstein +University of Vienna, University of Washington +ORCID: 0000-0003-2716-8471 +fred.bookstein@univie.ac.at, flbookst@uw.edu +This is an update of a manuscript of the same name and authorship that was +submitted to Evolutionary Biology on January 2, 2023. +1/16/2023 +1:30 +1 + +Abstract. Today’s typical application of geometric morphometrics to a quantita- +tive comparison of organismal anatomies begins by standardizing samples of homologously +labelled point configurations for location, orientation, and scale, and then renders the en- +suing comparisons graphically by thin-plate spline as applied to group averages, principal +components, regression predictions, or canonical variates. The scale-standardization step +has recently come under criticism as unnecessary and indeed inappropriate, at least for +growth studies. This essay argues for a similar rethinking of the centering and rotation, +and then the replacement of the thin-plate spline interpolant of the resulting configurations +by a different strategy that leaves unexplained residuals at every landmark individually in +order to simplify the interpretation of the displayed grid as a whole, the “transformation +grid” that has been highlighted as the true underlying topic ever since D’Arcy Thompson’s +celebrated exposition of 1917. For analyses of comparisons involving gradients at large ge- +ometric scale, this paper argues for replacement of all three of the Procrustes conventions +by a version of my two-point registration of 1986 (originally Francis Galton’s of 1907). +The choice of the two points interacts with another non-Procrustes concern, interpretabil- +ity of the grid lines of a coordinate system deformed according to a fitted polynomial +trend rather than an interpolating thin-plate spline. The paper works two examples using +previously published midsagittal cranial data; there result new findings pertinent to the +interpretation of both of these classic data sets. A concluding discussion suggests that the +current toolkit of geometric morphometrics, centered on Procrustes shape coordinates and +thin-plate splines, is too restricted to suit many of the interpretive purposes of evolutionary +and developmental biology. +KEYWORDS: Procrustes analysis, thin-plate spline, geometric morphometrics, Vil- +mann neurocranial octagons, anthropoid midsagittal crania, transformation grids, quadratic +fits, bilinear maps, cubic fits, two-point shape coordinates, modularity, baseline registra- +tion, D’Arcy Thompson. +1/16/2023 +1:30 +2 + +I. Introduction +Figure 1 here arose simply as free play with the tools of geometric morphometrics +(GMM). The data set comprises the familiar “Vilmann octagons” tracing around the mid- +sagittal neurocrania of close-bred laboratory rats radiographed in the 1960’s by the Danish +anatomist Henning Vilmann at eight ages between 7 days and 150 days and digitized some +years later by the New York craniofacial biologist Melvin Moss. This version of the data +is the one explored in my textbook of 2018: the subset of 18 animals with complete data +(all eight landmarks) at all eight ages. The concern of Figure 1 is the contrast of the +Procrustes-averaged shapes for the age-7 and age-150 animals (only the averages, no con- +sideration of covariances). The heavy lines are for the age-150 data subset, the light lines, +the data from the animals at age 7 days (a configuration this paper will occasionally refer +to as the “template”). All panels of the figure complicate the usual Procrustes plot of shape +coordinate pairs by all or some of the segments connecting these coordinate pairs. In the +figure’s left column, all 8·7/2 = 28 of the interlandmark segments have been drawn; in the +right column, only the subset that are the reason for calling your attention to this figure. +In the top row, those average locations correspond to the usual Procrustes-registered shape +coordinates, partialling out only centering, size, and rotation. Panel (b) is limited just to +the nine (out of 28) interlandmark segments from panel (a) that rotated either way by at +least 8.6 degrees (the figure label expresses this as “0.15 radians”) 1 between age 7 and 150 +days. A pretty graphic, but it features too much overlay of signals to qualify as a legible +pattern analysis. +However, most of the clutter is due to the substantial change of aspect ratio (height-to- +width ratio, obvious in the left column) that rotated both of the longer diagonals (Basion +to Bregma, Lambda to SES) of the template. Fortunately, we already know how to remove +this unwanted uniformity of relative vertical compression from our comparison: recourse +to the “nonuniform” component of Procrustes shape space, complement to the subspace of +uniform transformations (those that take all rectangles into parallelograms). The resulting +plots are the pair in the bottom row. +It is no surprise that the diagram at lower left, panel (c), looks even more cluttered +than panel (a), because now the calvarial roof, not just the cranial base, overlaps between +the ages. But there is also a new signal once the diagram is edited to suppress all the +segments that didn’t rotate much, a signal that seems not to have been anticipated in +previously published analyses of these data. As panel (d) shows, six of the 28 possible +segments rotate by more than 0.15 radian after standardizing this uniform aspect of the +young-to-old comparison. And now the pattern is obvious. The five landmarks at left +(anatomical posterior, SOS around to Lam) are rotating clockwise (in this projection) +over growth, while the three at the right, located anatomically anteriorly, are rotating +counterclockwise, all this to a longitudinal arrangement (think of the centroid of the set of +five, versus the centroid of the frontmost three) that isn’t rotating either way. This paper +will refer to the segmented polygon SOS-Bas-Opi-IPS-Lam, the set of five landmarks at +1 One radian is the mathematician’s natural metric of angle, the angle (about 57◦) at +which the extent of a circular arc is equal to the radius of the circle. +1/16/2023 +1:30 +3 + +left in Figure 1(d), as the “posterior pentagon” and the remaining three, Brg-SES-ISS, as +the “anterior triangle.” +The opposition of rotations in panel (d) is consistent with a report using an alternative +arithmetic of intersegment length-ratios. There is evidently shortening of upper calvarial +anteroposterior length, Lam to Brg, relative to the central segment of the cranial base from +ISS to SOS. Now there is no need to state that this pattern is “relative to the sequestering +of the uniform term,” as uniform transformations do not alter ratios of distances in the +same direction, whether concurrent or parallel. +This relative rotation, including that contrast of vertically aligned horizontal growth +rates, the central cranial base versus the calvarial roof above it, is surely a feature of the +143-day change of form here. But where is it to be found in the GMM toolkit? Figure +2 recovers exactly the same report from a quantitative style dating back more than 80 +years prior to GMM, analysis via the coordinates Francis Galton introduced in 1907 for +“classification of portraits.”2 Here I have diagrammed every possible two-point registra- +tion of these octagons (quantified only by their average coordinates as Moss originally +digitized them). For each alternative baseline, the original Cartesian coordinate average +configuration has been separately rotated and scaled so that the first baseline point is at +(0, 0) of a new coordinate system and the second is at (1, 0) in the same system (the two +points circled in every panel of the figure). We have thereby altered every single step of +the Procrustes toolkit — the centering, the rotating, the scaling — while eschewing any +recourse to the thin-plate spline for separating out that uniform term. And yet ten of the +panels clearly show the same phenomenon, the relative rotation between the anatomically +posterior pentagon of landmarks and the anterior triangle. Whenever both ends of the +baseline are in the same sector (here numbered [8,1,2,3,4] versus [5,6,7]), the rotation is +clear in the behavior of the complementary sector. +This is particularly evident in the +analysis to baselines 5-6 (row 4 column 5), 5-7 (row 4 column 6), or 3-4 (row 3 column 2), +where, regardless of any overall change of aspect ratio, the border of the octagon opposite +the baseline appears to have radically shifted by a rotation with respect to that baseline. +The disparity between ratios of change of length for segments ISS-SOS and Lam-Brg is +clearest, perhaps, in the panel for that ISS-SOS baseline, fifth row, fourth column. +Such an analysis, both elegant and elementary, shares no arithmetic with the standard +GMM toolkit of Procrustes registration and thin-plate splines. (For a good overview of +computational aspects of that standard toolkit in a format suitable for routine biometric +applications, see Claude 2008.) It is far older than that morphometric synthesis of the +2 The GMM literature usually refers to these as “two-point coordinates” or an “edge +registration,” while the statistical literature (Stuart and Ord 1994:279) calls them “Book- +stein coordinates” in keeping with Stigler’s Law, which states that usually innovations are +named after the second person to stumble across them. Ignoring the scaling aspect of this +tool, the centering and orientation here was already explicit in Boas (1905) and probably +can be traced back all the way to the German anthropologists’ adoption of the celebrated +“Frankfurt Horizontal” in 1882 (see Garson, 1885 [which, remarkably enough, is available +from JSTOR] — Orbital set to (0, 0), Porion along the positive x-axis: the Ohr-Augen +Horizontale of Martin 1914). For a contemporary critique of this specific convention of +1882, see Bookstein, 2016. +1/16/2023 +1:30 +4 + +1990’s, older even than analysis by triangles (“tensor biometrics,” Bookstein, 1984) or by +biorthogonal grids (Bookstein, 1978). Both of these versions involve attention to short +or long transects of the form that intersect internally, where, by analogy with the change +of form from a square to a rectangle, for one particular pair of directions (sides of the +square) the ratios of change of distance are greatest or least and the angle of intersection +is invariant at 90◦, while the ratio of change of the two distances at 45◦ to these directions +(diagonals of the square) is unity and it is the change of their angle that is maximized. A +closer inspection of the interlandmark-distance interpretation of Figure 1(d) instead makes +reference to distances that are parallel at some spacing (upper calvarial width versus lower), +a change visible equally in the Procrustes fits and in the two-point versions, especially +versions 7-8 (row 5 column 4) and 3-5 (row 3 column 3). The idea of examining ratios of +parallel distances like these is already present in some much earlier applied treatises, such +as Martin 1914. +For an intuitive understanding of what is going on here, turn back to the earliest +textbook introduction of the thin-plate spline, Bookstein 1991, where analyses like these, +restricted to just a quadrilateral of landmarks, exemplify what I called “purely inhomoge- +neous transformations” there, meaning, transformations without any uniform component. +Figure 7.3.6 of that book displays, within the limits of the software tools of the time, the +effect of rotating the starting grid on the graphs of this purely inhomogeneous component +(here, the sole nonlinear component) of the deformations of a square that minimize net +bending energy — the now-ubiquitous thin-plate spline. +Figure 3 is a modification of that textbook figure intended to clarify the contrast of the +different types of salience (length-ratios and rotations) for the pairs of segments of interest. +Its four columns prototype different types of the transformations, each of a starting square +of landmarks. In the top row are the starting squares, twice in Cartesian alignment with +the page and twice at 45◦. Below are the corresponding analyses, enhanced by ordinary +thin-plate splines that are not actually part of the arithmetical report. (But that spline +has nothing to do with the analysis here, which deals only with the landmark positions +per se, not any interstitial tissue. The quadratic extension to the interstitial rendering +in Figures 4 through 11 requires a minimum of six landmarks, not just these four; the +further extension to a cubic fit in Figures 17 and 19 requires at least ten.) In column +(a) the square is transformed to a rhombus by rotating two of its edges without change +in length. What change are the angles between the concurrent edges. In column (b) the +same transformation is applied to the square of landmarks at 45◦ (in other words, the grid +has rotated with respect to the landmarks, the configuration of which has not changed in +either row). Now the report is reversed: the greatest change is in the ratio of lengths of +diagonals, while the angle between them is left invariant at 90◦. This was also the case for +the configuration in column 1, where it was confounded by the inconvenient orientation of +the grid lines. +The situation in Figure 1(d) or Figure 2 corresponds instead to the prototype in +columns (c) or (d) of Figure 3. The starting configuration is still the same square. But +in column (c) the transformation changes the ratio of lengths of two edges that are par- +allel (horizontal in the figure), not perpendicular as in columns (a) or (b), while leaving +unchanged the ratio of the other two edge lengths (the other pair of parallels in panel c1) +1/16/2023 +1:30 +5 + +while radically altering their angle. This is a transformation from a square to an isosceles +trapezoid. The complementary transformation in column (d), which the geometer would +call square-to-kite, leaves the diagonals unchanged in length and in angle while altering +the relation between their midpoints. Now it is a different pair of paired edges whose +length-ratio has not changed — the top and bottom V ’s — and while the angles at the +end of the horizontal diagonal are hardly altered, those at the ends of the vertical diagonal +are greatly changed, one increased and the other decreased. To repeat, these reports rely +not at all on any GMM technology, neither Procrustes nor thin-plate spline. +The aim of this paper is to push this insight as far as it can go while remaining +elementary in its biomathematics. (For instance, its multivariate analysis is limited to +the familiar setting of multiple regression.) While the idea of two-point coordinates was +originally Galton’s in 1907, the idea at which the analysis here is aimed, the quadratic +growth-gradient, is only half as old: it is present in embryo in Peter Sneath’s underrated +paper of 1967 on trend-surface analysis of D’Arcy Thompson’s transformation grids. The +core of the argument inheres in any of the next eight figures, which selected eight interesting +baselines from the 28 in Figure 2 for expansion of the analysis to include an explicit +quadratic regression of the averaged age-150 Cartesian coordinates against the same from +the age-7 octagons. These analyses completely ignore the tools of standard GMM — there +is no Procrustes centering, no scaling or reorientation beyond the (arbitrary) choice of +baseline, and thin-plate splines are drawn only to be dismissed — while what results, you +will see, is a coherent summary of this particular change of neurocranial form. A combined +Figure 12 arrays the eight separate summaries for a synthesis of their information content +abstracted in Figures 13 and 14. Following this exploration, a further analysis of some +data from a study of cranial hominization my Vienna group published twenty years ago +will consider some extensions of this approach, and a concluding Discussion will reflect on +some implications of this seeming irrelevance of today’s conventional GMM toolkit for the +explanatory purposes of evolutionary or developmental morphology. +II. Vilmann 7-to-150-day growth analyzed without Procrustes GMM +The recommended alternate analysis of the Vilmann growth gradient in Figure 1 may +be narrated by an extraction of common findings from a suite of separate analyses to my +selection of baselines, some transects of the octagon and others circumferential to it. The +analyses to be synthesized are laid out in Figures 4 through 11. +Each of these eight composites offers four panels. At upper left will be the conventional +thin-plate spline of the averaged octagon of Cartesian coordinates of the age-7-day animals +as warped into the analogous average at age 150 days. Analysis is to the baseline of the pair +of landmarks indicated in larger circles, which specifies the orientation of the thin-plate +spline grid in each example. To its right will be a gridded version (in this orientation) +of the “growth fit” likewise displayed first for comparison as a thin-plate spline. +Here +the x-coordinate of the deformed grid is the predicted x-coordinate from the regression of +the baseline-standardized octagon vertices of the 150-day average on both coordinates of +the age-7 average, and also their squares and their crossproduct (i.e., a regression of each +x150 on (x7, y7, x 2 +7 , y 2 +7 , x7y7)) and likewise the y150-coordinates. Each of these regressions +1/16/2023 +1:30 +6 + +involves five predictors, plus a constant, for only eight “cases” (the relevant coordinate, +x or y, of the eight landmarks), and so has only two degrees of freedom for error; they +are not really regressions, but rather almost interpolations, when the landmark count is so +small. At left in the lower row will be a more appropriate representation of the quadratic +fit splined at above right: actual transforms of the grid lines of the starting form to this +baseline, with the regressions’ “dependent variable” the x− and y−coordinates of the filled +dots corresponding to the fitted locations at the open circles nearby. Finally, the panel at +lower right will restrict this grid to just the interior of the age-7 octagon — this portion +of the graphic deserves attention first, before any extensions to the exterior. +Consider, then, the first figure in this series, Figure 4, which is the analysis for a +baseline from Basion to Opisthion — the shortest interlandmark segment in the template, +but one contained entirely within the posterior pentagonal compartment of Figure 1(d) +and hence one that might enlighten us as to the rotation archived there. That rotation +between anterior landmark triangle and posterior landmark pentagon is essentially the +same that is displayed in Figure 1 for the conventional GMM approach and in Figure 2 for +the panel corresponding to this baseline (there, the panel in row 1, column 1) — indeed +it will be the same in all eight of the series of figures. Here in Figure 4, for the baseline +Basion to Opisthion (axis of the midsagittal foramen magnum), the orientation of the form +is rotated about 130◦ from the Procrustes convention in Figure 1. The thin-plate spline +(upper left panel) is interesting in that inside the posterior five-landmark component, SOS +around to Lam, the interior as rendered by the spline appears to be nearly affine (all grid +cells the same size and shape) except near IPS, and likewise nearly affine for the anterior +component Bas-SES-ISS. The growth fit (upper right panel) apparently has pulled IPS to +the left in this diagram. As Figure 1 shows, and as has been exposited in earlier papers +(e.g., Bookstein 2017), this point participates in a specific focal process displacing it upward +in the more realistic anatomical setting of Figure 1. Thus the fit in this upper right panel +of Figure 4 does not show the deviation of change at IPS from change at its neighbors that +is present in the actual data. +Either panel of the lower row shows how closely the fitted landmarks (open circles) +track the averaged 150-day locations observed (the solid circles). The horizontal grid lines +in the interior of the form (lower right panel) are mainly straight, while their orientation +on this diagram is graded from top to bottom more smoothly than one would infer from +the analogous diagram at upper left (the thin-plate spline based on the fully detailed data +record, which, by design, is not conducive to any lower-dimensional summary). The steady +rotation of this imputed grid line direction is complemented by a gentle curvature of the +other grid line direction, a curvature that is not so apparent in the explicit thin-plate +spline at upper left — the transformation that appears segmented there, one nearly linear +system for the posterior pentagon and another for the anterior triangle, is smoothed by +the quadratic regression into a continuous gradient from end to end of the template (top +to bottom of the grid, in this coordinate system). Note that the prediction error of the +quadratic fit (lower row) specifically implicates the length of the chosen baseline, at both +ends. +Figure 5 shows the same analysis for a different baseline, Basion to Interparietal suture, +from the same posterior pentagon. Again the quadratic fit (lower row) shows a substantial +1/16/2023 +1:30 +7 + +residual, this time at only one of the baseline points (IPS). In the deformed grid, both +systems of lines are curved, a feature that makes interpretation more difficult. +Figure 6 is the first to involve a cross-component baseline, Basion (from the posterior +pentagon) to Bregma (from the anterior triangle). The starting grid has rotated about +80◦ from its position in the first of this series (i.e., the angle between segments Basion- +Opisthion and Basion-Bregma in the age-7 average is about 80◦). Again the panels in +the lower row inform us that the initially vertical grid lines (lines along Lambda-ISS or +IPS-SOS) are transformed by the quadratic fit into a pencil of nearly straight lines at +varying orientations, while the lines of the originally orthogonal system are gently curved +in a manner that will concern us in detail in Figure 13. At neither of the baseline points is +there any substantial fitting error of the quadratic regressions. The approximate uniformity +of cell sizes across the trimmed grid at lower right here and in every other figure of this +series assures us that the recourse to distances from the centroid in models of centric +allometry, such as Bookstein 2021a, is a reasonable default. Indeed the separation between +the actual age-150 centroid and the quadratic trend transform of the age-7 centroid is a +mere 0.054 units in the scale of this figure. +Figure 7, to a baseline from IPS to SOS, is very nearly the same analysis as in Figure +6 inasmuch as the two baselines, IPS-SOS and Bas-Brg, are nearly at 90◦ in the age-7 +template. The main difference is the substantial increase in fitting error, owing to the fact +that landmark 3, IPS, is known to be strongly loaded on a special factor not shared with +the rest of the configuration. Nevertheless, the grids of the lower row still greatly resemble +those of the preceding figure, for the baseline at 90◦ to this one: lines parallel to IPS-ISS +(here, the baseline) remain straight but rotate from end to end, while the orthogonals are +gently curved. +Let us move more quickly through the remaining versions of this four-panel scheme. +In Figure 8, baseline Lambda-SOS, both systems of grid lines are gently curved (although +the rotation from end to end of the original octagon is as clear as if they had remained +straight). The errors of fit at the baseline points are moderate in magnitude, partly because +the fit at Lambda is distorted by the need to accommodate the deviation at IPS. +Figure 9, for a baseline Bregma-SES within the anterior component of Figure 1, dis- +plays gentle curves in both grid systems. Errors of the quadratic fit are again moderate, +and the rotation so evident in Figure 1(d) is very clear in spite of the curvature of these +deformed grid lines. The baseline in Figure 10, Bregma-ISS, has similar errors of fit and +similar curving of the grid lines. Finally, Figure 11, for an ISS-Bas baseline, is roughtly +the 90◦ rotation of the analysis in Figure 5, whose baseline (Bas-IPS) is roughly at 90◦ to +the baseline here. +Figure 12 summarizes all eight of these analyses in a way that permits some criteria of +interpretability to emerge regarding replacement of the Procrustes rotation by a protocol +more conducive to reportage: a protocol that associates the reorientation of specimens +to the ultimate simplification of their deformation by reference to the specific coordinate +lines as deformed from the template’s square grid. We have seen that baseline analyses +can sometime come in pairs if the corresponding interlandmark segments themselves lie at +approximately 90◦, and it is better if they run close to the centroid of the octagon. More +subtly, morphological comparisons that can result in reports of relative rotations of parts +1/16/2023 +1:30 +8 + +of a landmark configuration may be diagrammed best not by a thin-plate spline but by a +choice of a specific baseline that highlights the rotation in question, like Figure 6 or Figure +7 here, by leaving one set of grid lines straight lines even as they are rotated. (For instance, +in Figure 6, the panel at lower right is more interpretable than the panel at upper left, +even though the information content is effectively the same.) The thin-plate renderings in +Figure 12, columns 1 and 3, all confirm the relative rotation detectable already in Figure +1, but do not otherwise appear to offer much intuitive accessibility. By comparison, the +quadratic-fit displays, columns 2 and 4, vary enough in their legibility that some are truly +insightful. Those that seem most helpful are the pair of analyses in the second row, to +baseline Bas-Lam or IPS-SOS (two directions that happen to be nearly perpendicular) +— these seem to be considerably better than the standard GMM analysis at showing a +potentially meaningful gradient for the growth process being visualized here. +The analysis in Figure 7 suggested a scenario I have highlighted in Figure 13 by the +simple trick of extending the domain of the quadratic fit beyond the bounds of the land- +mark locations being fitted. The diagram here extends the earlier gridded transformation +merely by evaluating it on the new real estate to the left in the same template coordinate +system, i.e. into the empty space some distance above the foramen magnum of these an- +imals, where the horizontal grid lines of Figure 7 appear to be converging. We see that +the near-linearity of the transformation along the baseline and all the grid lines parallel +to that direction persists quite far beyond the actual anatomical limits of the comparison, +resulting in the strong impression of some sort of descriptive center at an unphysiologi- +cal distance outside the actual calva. The apparent rotation suggested in Figure 1(d) is +embedded here in a larger system of reorientations that might be viewed as continuous +rather than segmented, or, in a more suggestive language, graded rather than modular. +The suggestion is strong, for instance, that this grading ought to be checked for extending +further anteriorly to the facial skeleton (a description that will be tied to the classic inter- +pretation of “orthocephalization” by a footnote in Section IV) or other features outside of +this particular neurocranial data set. +Figure 14 sketches two geometrical interpretations of Figure 13, one more familiar to +the applied mathematician and the other less so. Each is an alternative to the thin-plate +spline of column (d) in Figure 3; one will prove more realistic than the other for this +paper’s examples. The more familiar map is the projection, central panel, that takes every +straight line onto another straight line. But this mapping substantially alters the spacing +of the points where these deformed grid lines meet the bounding kite. No such respacing +appears in the extended quadratic fit itself, Figure 13. An alternative better matching that +observed quadratic fit is the family prototyped in the right-hand panel of the figure, the +bilinear mapa that I discussed in considerable detail in Bookstein 1985. Bilinear maps3 +take one quadrilateral onto another as follows. Every point (x, y) in the interior of the +template quadrilateral is the intersection of two lines connecting opposite edges that divide +those edges in the same ratio. The map takes (x, y) to the intersection of the two lines that +divide the homologous pair of edges in the target in the same ratio. The bilinear map of +square onto kite can be written (x, y) → (x, y)+a(1+xy, 1+xy) for some a. The projection +3 In finite-element analysis, these are often called isoparametric coordinates of the +quadrilateral. +1/16/2023 +1:30 +9 + +in Figure 14 required the upper isosceles triangle of the template to be mapped into the +space above the horizontal diagonal of the kite, entailing a considerable compression of +its vertical coordinate; the bilinear transformation enforces much less compression here, +at the cost of bending that horizontal diagonal over the course of the deformation. This +attenuation of the variability of those ratios of area change seems to match the graphics +of all the quadratic fits in Figures 4 through 11 after an appropriate rotation. +Returning one final time to the scheme in Figure 1(d), the decomposition of the +neurocranial octagon into two nonoverlapping components, we see that the figure has +indeed oversimplified the situation there. That the rotation of all edges of the posterior +pentagon leaps to the viewer’s eye obscures the fact that all but one of these segments have +changed their length. And likewise the anterior “triangle,” Brg-SES-ISS, does not rotate +rigidly — its edge from SES to ISS shortens and also does not rotate as far as the other two. +Any report focusing on the two “components” is deficient in failing to refer to the coordinate +space in-between them, where the unconformity between anteroposterior changes of length +along the cranial base versus along the calvarial roof seems better captured by the rotating +lines of the Bas-Brg baseline and IPS-SOS baseline analyses (Figure 12, row 2, columns +2 and 4) than by the irregularities of the corresponding thin-plate splines (columns 1 and +3). +III. An example from hominization of the skull +The Vilmann analysis of Section II exploited the best study design that experimental +zoomorphology has to offer: a sample of close-bred animals imaged by identical machinery +at a fixed sequence of developmental ages. (The identification of this research design as +the summum bonum of laboratory evo-devo research is a century old — it dates from no +later than Przibram 1922.) Most of the data structures to which GMM has been applied +are not so elegantly designed. This paper’s final example is a pair of comparisons, each +much more typical in its design, that share one 20-landmark configuration scheme. The +data are a selection from the 29 forms analyzed in Chapter 4 of Weber and Bookstein +(2011) that originated in computed midsagittal sections of a larger sample of CT scans +digitized by Philipp Gunz for the growth analysis in Bookstein et al. (2003). That original +analysis explicitly relied upon the same GMM toolkit that is most commonly invoked +today: Procrustes analysis, principal components of the resulting shape coordinates, and +visualizations by thin-plate spline.4 +Of the specimens homologously digitized in 2003, most are Homo sapiens, while four +are named specimens of H. neanderthalensis (Atapuerca, Kabwe, Guattari, and Petralona), +and two are specimens of Pan, one of each sex. For the present reanalysis I have averaged +the 18 adult sapiens (one of which, Mladeˇc, is an archaic specimen) and, as a separate +group, the four neanderthals. As a third “group” (present for a didactic purpose, a com- +4 In one version or another these data have already been used for demonstrations of +GMM in textbooks three different times: not only Weber and Bookstein (2011) but also +Bookstein (2014, 2018). The approach circumventing those typical GMM maneuvers is +new to the present paper. +1/16/2023 +1:30 +10 + +parison of comparisons) I selected the female adult chimpanzee, because the adult male +shows even more of the heterochrony that will render my final figure so extreme in cer- +tain aspects of its geometry. Of course these samples are far more limited than any data +resources that would be brought to bear on the same comparisons today. It would be +unreasonable to claim that the computations to be reported presently are valid empirical +findings; my purpose is instead to demonstrate a methodological alternative to Procrustes- +and spline-based GMM. +The left panel of Figure 15 names these twenty landmarks at their positions in the +average of the H. sapiens sample in the original CT coordinates, which were not far from +a Sella-Nasion orientation. In the right panel this configuration is supplemented by the +configurations of the same twenty points for the female chimpanzee and also for the nean- +derthal average, all after the two-point transformation (Bookstein coordinates) that put +all three ANS’s (of which two are group averages) at (0, 0) and all three internal Lambda’s +at (1, 0). Evidently this coordinate system has been rotated, translated, and scaled from +the panel at its left, but none of these steps proceeded by the Procrustes method. +Consider first the analysis in Figure 16, which in its design echoes three of the four +panels of the Vilmann series, Figures 4 through 11, but in this case only for one selected +baseline, from ANS to LaI, as in the right panel of Figure 15. (Analysis to a roughly +perpendicular baseline, Opi–BrI, results in essentially the same diagrams.) The comparison +in Figure 16 is from the averaged points for H. sapiens in Figure 15 to the averaged +points for neanderthalensis. In both of these Homo averages (and also in the single female +adult Pan specimen to come) the baseline crosses the cranial base near Sella roughly +halfway along its length. The thin-plate spline deformation from the average of the eighteen +humans to the average of the four neanderthals, upper left in the figure, shows the expected +contrast of shrinking neurocranium and expanding splanchnocranium, particularly along +the palate; the cranial base interposes itself as the so-called “hafting zone.” As the upper- +right panel shows, this grid is tracked to some extent by the analogous grid for the fitted +values of the same neanderthalis landmarks from the quadratic regression on the sapiens +coordinates, That quadratic regression, already demonstrated many times in the Vilmann +example preceding, shows most of its failure of fit (discrepancies between the open circles +and their filled neighbors in the lower-left panel) along that central separatrix, with a +possible exception at lower right where the pairings of the two inions are rearranged in +both separation and orientation. As Figure 15 hinted, this rearrangement is due mainly +to excessive variation at InE, external inion. +The final quadratic trend grid, at lower left in Figure 16, is strikingly different from +the thin-plate spline of the same point loci (upper right). Indeed this grid for the fit looks +remarkably like a rotation of the grid at right in Figure 14, the bilinear transformation +leaving two specific families of straight lines straight after the deformation, while their +orientations rotate across. the diagram. At this large scale, the comparison of midsagittal +crania of these sister species is largely smooth — the points in the hafting zone differ +hardly at all from their predicted locations under the quadratic analysis. In particular, +the implication of modularity in the upper right panel is completely effaced in the actual +quadratic fit grid at lower left, indicating instead an approximating spatial process that is +homogeneously graded with no natural boundaries embryological or otherwise. The grading +1/16/2023 +1:30 +11 + +is consistent with the observation that relative to the face the neanderthal neurocranium +is smaller than that of sapiens with some relative rotation as well. +Figure 17 analyzes the same comparison by a cubic fit instead of the quadratic fit +in Figure 16. (Specifically, this fit models each of the twenty x−coordinates of the H. +neanderthalensis average and then each of its twenty y−coordinates as a linear combina- +tion of nine terms xsap, ysap, x2 +sap, y2 +sap, xsapysap, x3 +sap, y3 +sap, x2 +sapysap, and xsapy2 +sap. The +quadratic regressions used only the first five of these predictors.) These cubic grids show +bizarre behavior outside the limits of their driving data (the strange cusps already clear +in Sneath’s examples of 1967), so as in the Vilmann exposition of Section II I extended +the figure by one more panel, lower right, that trims the grid to just the interior of the +region occupied by the actual target configuration (here the H. neanderthalensis average). +The straight lines of the rendering in Figure 16 now appear as S-curves across that same +hafting zone, and of course the new fit, a regression on nine predictors, has to be closer +than that in Figure 16 based on only five of the nine. But the change of size-ratios between +neurocranium and splanchnocranium remains clear, as does the directional extension along +the palate and the relative rotations from anterior to posterior and from caudal to cranial. +The situation is quite different for the comparison of the H. sapiens average to our more +distant relative, the female chimpanzee. The quadratic analysis analogous to Figure 16 +can be found in Figure 18, but it no longer appears to look entirely like the bilinear map of +Figure 14. Instead we encounter a strong local feature of the transformation, the apparent +flattening of the parietal region, that is seen in both of the thin-plate spline renderings of +the top row (at left, for the actual shape coordinates; at right, for the quadratic fit) and +likewise in the gridded representation of that quadratic fit at lower left. Strikingly, the +residuals of this analysis seem no greater than those of the comparison of the sapiens sample +with the neanderthals, Figure 16, yet the flattening of the splines is clearly detected by +this quadratic fit as well, which has so many fewer coefficients (and also a matrix inversion +step of much lower rank, 5 × 5 instead of 23 × 23). The bidirectional linearity of the lower +left panel in Figure 16 has certainly ceased to apply globally, while the hafting zone here +seems still to be no sort of natural boundary between multiple modules. The deformation +remains smoothly graded except locally, in the parietal region. +Yet when we switch the algorithm from the quadratic (five-term) fit to the cubic (nine- +term) fit, Figure 19, nothing essential changes in the analysis as a result of these additional +four degrees of freedom per coordinate. The thin-plate spline of the fitted points (upper +right panel) is not much altered from that in the previous figure except in that same +nonconforming parietal region, and while the cubic fit here leads to pathologies of the +extrapolated grid at every corner of the original scheme (lower left panel), its restriction +to the interior of the actual anatomy, lower right in the figure, shows grid lines that, +ignoring their curvature, are actually well-aligned with those of the lower left panel in +Figure 16, the comparison from sapiens to neanderthalensis. We have thereby confirmed +graphically that the shape difference in the parietal region is indeed local. Put this another +way: the quadratic fit (Figure 18) and the cubic fit (Figure 19) convey the same message, +a relatively continuous gradient of deformation right across the hafting zone. And they +agree, too, that the situation at the parietal (landmarks Opi through LaE) is not coherent +with this large-scale gradient. From Bregma forward, the lower right panels in Figures 17 +1/16/2023 +1:30 +12 + +and 19 differ mainly in the intensity of rotation of these gridline segments; but posterior +to that arbitrary boundary the parietal landmarks participate in a reorganization that is +incommensurate between the two comparisons. +Thus we see again that, just as in the Vilmann growth example, an approach that +eschews all of the standard Procrustes steps and also the usual thin-plate spline is capa- +ble of generating the same understanding of a morphological phenomenon, in this case a +somewhat more complicated one. +IV. Discussion +A. The main concern of GMM ought to be the transformation grid per se. This was +already clear from the earliest formal appearance of the concept in D’Arcy Thompson’s +On Growth and Form (Thompson, 1917), where the review literature usually begins (even +though portrait artists like Albrecht D¨urer had thought about this much earlier). The +endpoint of the method ought to be not statistical but graphical, and the derived report +should be geometrical, not statistical, en route to an ultimately biophysical or otherwise +morphogenesis-informed endpoint. The main dilemmas in this tradition were already well- +critiqued over the first six decades of its development as I reviewed them in Chapter 5 +of Bookstein, 1978. No matter how clearly defined the positions of individual landmark +points might be, there was no complementary rhetoric for reporting meaningful features of +the transformation grid that expressed comparisons of their configurations over meaningful +biological contrasts. The best exposition of this problem remains Sneath, 1967, a paper +that struggled, ultimately unsuccessfully, to bring the algebra of landmark analysis (in +that pre-spline era) into alignment with the reasoning of numerical taxonomy. Yet D’Arcy +Thompson would have been delighted with the grid in Figure 13, while presentations of the +same information in Procrustes style, Figure 1a, or spline-style, panels 4(a) through 11(a), +would have been of no use to him at all. A more contemporary and quite distinct tradition +of transformation studies approaches the problem via a calculus of diffeomorphisms (see, for +example, Grenander and Miller, 2007), which makes no essential reference to landmarks +at all, instead basing its computations on the full field of image contents, gray-scale or +even colored, spanning the organ(s) of interest. The approach seems particularly helpful +in neurological applications to imagery of the human brain. +This contrasting method, +however, is beyond the scope of my Procrustes critique here. +The analysis in Figure 7 suggests renewing Thompson’s original concern in this do- +main, the interpretation of grids per se, via injecting a new theme into the discussion, an +anatomical basis for orienting the starting grid on the template, that more intensively ex- +ploits the interaction between deformation graphics and the investigator’s prior awareness +of how coordinate systems themselves can vary in their visually dominant features. The +biomathematics ought to begin, then, with a confluence of two insights: one, that some +morphological domains might be amenable to some kind of functionally interpretable large- +scale pattern analysis, and the other, an intuition about the geometrical language by which +the pattern of interest might be quantified. For Henning Vilmann, this translation began +with the knowledge that growth of rodent neurocrania is a plausible domain for morpho- +metric exploration and that its midsagittal aspect bears enough information about growth +1/16/2023 +1:30 +13 + +and function to be worthy of geometrization not only in his own measurements of extent, +nor the numerous intermediate multivariate investigations of this same data set (including +several of my own), but also in the novelties of Section II. But given these two axioms, an +applied study would culminate in an exploration not of alternative statistics but of alter- +native graphics: a survey not of diverse linear combinations but of diverse grid renderings. +Information about absolute scale change, where relevant (as in biomechanical aspects of +interpretation), can be embedded in any of these grid figures by a simple magnification over +the course of printing, or can be inscribed on interlandmark segments or the line-elements +of a transformation grid by overprinting. In this context of large-scale comparison, rotation +is a tool of rendering clarification, not a nuisance variable of digitizing. +The quadratic regressions in Figures 4 through 11 all used the same list of five predic- +tors x, y, x2, y2, xy. This consistency lets the renderings here, unlike the approach in the +lower row of Figure 1, preserve the uniform component of the transformation grid, where +we can see how it interacts with these gradients of large but finite scale. But the directions +corresponding to those two axes x and y vary from baseline to baseline, and the baseline +points are not privileged by the regressions. Consequently the coordinates pinned by the +two-point registration are not quite pinned by the regression — they are permitted to shift +to some extent from solid to fitted circles in the grid figures here. +The resulting dataflow sheds new light on what we mean by “the best rotation” +when, as in both of this paper’s examples, different parts of an organ appear to rotate +relative to one another over a comparison of interest. The role of the multiple two-point +registrations that this paper recommends as a substitute for the Procrustes algorithm is +not itself a “finding” of any sort but merely a convenience, a simple way of regularizing +the landmarks’ Cartesian coordinates in order that a selection of reasonable polynomial +trends can be fitted, each in a reasonably equably weighted way. Its advantage is that +unlike the case for the Procrustes method, there is more than one of them. The Procrustes +approach optimizes a quantity (sums of squares of landmark shifts) that is irrelevant to +the ultimate purpose of an evolutionary or developmental GMM analysis, which is not a +minimized sum of squares or a singular-value decomposition or a classification but rather +a plausible biological hypothesis for the observed form-differences, their causes, or their +consequences for the organism. +Then the logic of the inference engine we need is not the operationalized Procrustes +arithmetic itself, the least-squares fit to what is almost always a completely wrong model +(the null model, a pure similarity transformation). Instead we need the logic of E. T. +Jaynes’s approach to numerical inference (e.g., Jaynes 2003): the explicit acknowledge- +ment of what we do not know — what is missing from the list of data-driven constraints +on some quantitative empirical inference. (I have recently reviewed this logic in the rather +different context of paleoseismology, which is the history of great earthquakes — see Book- +stein 2021b.) What is missing from a Procrustes analysis is, among other things, the ac- +knowledgement that choice of an orientation constraint affects the resulting report: what +we seek is the orientation that will best clarify the final published diagram. Furthermore, +regardless of this issue of orientation, in every GMM context we already know there is no +“correct” registration, because there is no “correct” list of landmarks — in the presence +of any regional rotation or rescaling, different lists of landmarks or semilandmarks lead +1/16/2023 +1:30 +14 + +to different Procrustes registrations, and the empirical report of a shape comparison must +accommodate that specific form of ignorance. That is the whole purpose of the grids — to +free our attention from the landmark data per se to the space in-between where biological +processes actually take place. +The particular protocol dictating the selection of orientations to be considered may +be irrelevant to the quantitative morphological inference under study. (Recall that in this +paper the two points fixed in the baseline registration are not fixed by the fitted trend — +the registration is not an inferential component of the grid report at all.) Orientation may +be specified as any interlandmark segment from the available pairings, or any homologous +boundary alignment, or even a specific force vector such as a muscle load or gravitational +vertical — or possibly all of these. Whatever the choices of orientation, the investigator +of a global deformation is led to the approach here, which is the selection of at least one +satisfactory such orientation as judged by the ultimate diagram at the end of the workflow. +In 3D, one could proceed via an assortment of large landmark triangles passing near the +centroid, similarly searching for clarity and redundancy. But in other contexts that issue of +orientation may be quite relevant to the interpretation. The examples here have all dealt +with global trends, but Figures 18 and 19 hinted at a need for a deformation tool suitable +for local features as well. Such a tool would likewise entail a rotation of the Cartesian +coordinate system prior to grid computation, but in general a different one — see, for +example, the model of the crease in Bookstein 2000 or Bookstein 2014, Figure 7.19. +B. We need to broaden the range of ideas we borrow from geometry. A combination +of two branches of geometry led us to the bilinear interpretation in Figure 14 of the +grid in Figure 13, but this other toolkit is not among those currently being taught to +biomathematicians. The kernel r2 log r of the thin-plate spline doesn’t much resemble +the biological processes we are trying to understand, but the algebra of polynomial fits +(here, mainly the specific appearance of bilinear maps leaving both pencils of coordinate +lines almost straight and almost evenly spaced after deformation) does pick up much of the +classic appearance of growth-gradients as laid out for analysis from Thompson on. More +important than the extension of the idea of a coordinate system, though, is an extension +of the domain of morphometric data to include empirical entities other than landmark +points. The description of the grid in Figure 13 makes no essential mention of any of the +landmarks — the simple exegesis here (bilinear reorganization of that particular family of +grid lines while remaining lines) pertains much more to the interior of this octagon (the +directions of those transects across it, or, if you will, the pairing of points across the left +and right sides of the outline in this orientation) than to any of its boundary delineation +detail, even though that boundary is the sole data source for the example. Thus at root +the finding exemplifies a language of intraorganismal matching, the pairing of points along +a shared curve bounding some anatomical entity in section. Pairings like these are not like +landmarks in any formal aspect. +So even though this paper’s first example argument began from a playful GMM- +derived diagram, Figure 1d, it ends up formalized in the rhetoric of a spatial extension +(Figure 13) unknown to GMM but comprehensible by every reader of Thompson’s chapter, +as interpreted in Figure 14 via a similar-looking figure from a subchapter of college geom- +1/16/2023 +1:30 +15 + +etry. This logical sequence can be reversed: beginning from those same textbooks, to try +finding biological examples that illustrate them. We are used to polar coordinates, for ex- +ample (most recently in the study of centric allometry, Bookstein 2021a), but what about +bipolar coordinates or confocal coordinates (Bookstein 1981, 1985) and other schemes that +(literally) co-ordinate position with respect to two origins or two axial systems at the same +time? The range of coordinate systems is vastly broader than the Cartesian on which +today’s GMM automatically relies. My biorthogonal grids (Bookstein 1978) already went +beyond this possibility, though not in a statistically feasible way, via their formalism of +one-axis and three-axis singularities corresponding to the “lemon” and “star” umbilics that +are the topic of advanced treatises such as Koenderink (1990). From the earliest years of +the twentieth century the mathematics of geometry has permitted us to talk about coor- +dinates of many different extended structures: not just points, but lines, planes, circles, +and many other formalisms. See, at first, Hilbert and Cohn-Vossen, 1931/1952, and then, +among the more contemporary surveys, Porteous 2001 or Glaeser 2012. +Thus the word “geometric” in the phrase “geometric morphometrics” needs to have +its meaning broadened beyond the current focus on the Procrustes component of GMM or +indeed any version based on analysis of landmark points as logically separate data elements. +“Procrustes distance” between specimens, when computed as a minimizing sum of squared +Cartesian coordinate differences, is just a theory-free proxy for the far more subtle and +multifarious concept the biologist knows as the opposite of “similarity,” and today’s GMM +treats Procrustes shape coordinates as just a list of Cartesian pairs (or triples) in their +own coordinate space of position, without reference to any explicit features for describing +how their interrelationships (e.g. the interlandmark segments of Figure 1) actually change +across a comparison of configurations. D’Arcy Thompson got this correct back in 1917: +“The deformation of a complicated figure,” he wrote (Thompson 1961:271), “may be a +phenomenon easy of comprehension, though the figure itself have to be left unanalyzed +and undefined. This process of comparison, recognizing in one form a definite permutation +or deformation of another, apart altogether from a precise and adequate understanding +of the original ‘type’ or standard of comparison, lies within the immediate province of +mathematics.” +That geometry of “recognizing deformation” is not limited to the geometry of points +referred individually to Cartesian axes. Thompson himself referred explicitly to the ap- +pearance of the deformed grid lines in his drawings. +For the comparison to Mola, for +instance, he wrote, “I have deformed [Diodon’s] vertical coordinates into a system of con- +centric circles, and its horizontal coordinates into a system of curves which, approximately +and provisionally, are made to resemble a system of hyperbolas” (Thompson 1961:300). It +is the configuration of these curves, not the landmarks on them, that is the bridge from +arithmetic to understanding. In other words, the elementary language of deformation, the +language by which we report morphological comparisons as deformations, must be based +in a glossary of multiple elementary types of deformable image components, not disartic- +ulated landmarks. The roster of these is broad indeed, including, among other options, +the changes of point-pairs to other point-pairs at a different distance or direction that we +already saw in Figure 1, but also changes of triangles to other triangles, squares to any +quadrilateral whether rectangle, parallelogram, trapezoid, or some other form, displace- +1/16/2023 +1:30 +16 + +ment of interior points with respect to an unchanging boundary, circles to ellipses, ellipses +to any other simple closed curve, straight lines to other straight lines, lines to any other +open curve, line-elements having an orientation in the small as well as a location (for a +spline cognizant of this structure, see Bookstein and Green, 1993), or nearby pairs of par- +allel lines to any bent ribbon tracing the sequence of changes all along their shared length. +All of these have appeared in biometric examples; each requires a different geometric gram- +mar for its reporting. For instance (in another acknowledgement of our sister discipline of +neuromorphometrics), line elements per se summarize image data for the method known +as diffusion tensor analysis that traces and summarizes patterns of wiring in the human +brain. +As I hope you have already come to suspect from the figures in this paper, the thin- +plate spline is not designed to be of any particular help in this matter. Its functional form +is mainly a sum of terms r2 log r, where r is the distance from each grid point to each +landmark of the template in turn, and so it has no machinery for collecting references to +two or more landmarks at the same time, but must revert to the nonbiological symmetries +of linear multivariate statistics for this purpose (so that the partial warps, for instance, +are just a (2k − 4)−dimensional rotation of its Cartesian coordinates however they were +arrived at to that point, while the relative warps are just a different (2k − 4)−dimensional +rotation of the same coordinates). No, the elements of a quantitative morphometric com- +parison in terms of deformation must be the whole coordinate systems of our deformation +diagrams, and the features we extract must be features that refer to those deformed lines +and areas, whether end to end or truncated to the vicinity of specific landmark subsets. +Any geometric report qualified to drive a programme like Thompson’s aimed at simple +descriptions of relationships among individually complicated specimens must begin with +more complicated elementary entities than positions of discrete landmark points. A search +for such explananda, beginning from the paired interlandmark segments in Figure 1, leads +immediately to the elementary aspects of this paper’s two examples, which make no refer- +ence to the formula r2 log r nor indeed any quantification beyond the squaring or cubing of +coordinates and products of those powers that allows us to parameterize families of nearly +parallel curves that began as parallel lines. +C. The exterior of an organ or an organism is a useful domain for communication of +findings even in the absence of tissue. This comment has real bite for a GMM that depends +on the conventional thin-plate spline, which does not understand exteriors at all. So the +usual interpolating spline is precisely the wrong tool for detecting large-scale gradients that, +like the one summarizing the Vilmann comparison, are not affine — are not conducive to +descriptions emphasizing some pair of directions at 90◦ bearing the maximum ratio of rates +of change. Because the conventional thin-plate spline relaxes to uniform at great distances, +it is not a helpful component of answers to any question about large-scale organization of +a form-comparison, the question asked by most morphologists (and dysmorphologists, and +paleontologists) ever since Thompson’s time. To quantify the cunning hint from Figure 1d, +I needed the tool of a quadratic trend surface (i.e., a fit, not an interpolation), and when +the graphic of that fit proved intriguing, a suitable summary arose only when the rendering +was extended (Figure 13) far enough beyond the actual convex hull of the landmarks that +1/16/2023 +1:30 +17 + +Figure 14 could show us how to report its structure. However vague the language might +be for a discussion of Figure 7 by itself, the reworking that is Figure 13 makes the implicit +explicit — the extended grid now is exactly the report we seek, no actual words required +except the legend explaining how the graphic was produced. But such a graphic no longer +resembles any sort of conventional GMM output. +Because the interior of any non-nested module is at the same time a part of the exterior +of every other module, one sees from the hominization example that the morphometric +aspect of “modularity,” whatever its exact morphogenetic definition, is a matter not of +landmark coordinates but of what happens to coordinate grid lines. Figures 17 and 19 +confirm that, within the limits of these data resources (adult forms only, no growth series, +a mere 20 landmarks), there is no graphical evidence for the cranial base as a separatrix +between braincase and face, in spite of their obvious differences in function, but strong +evidence for a separation of the whole anterior two-thirds of this landmark scheme from +the five parietal landmarks, Opi through LaI and LaE, that so clearly seize control of the +lower-right corner of the grids for either the quadratic fit (Figure 18 lower left) or the cubic +fit (Figure 19 lower right) to the comparison across genera. While the empirical import +of this second data example is obsolete, owing to advances in the accrual of samples of +all these species, the practice whereby consideration of the transformation grids per se +might shape inferences from landmark data about morphogenetic control processes ought +to be transferred from the current GMM toolkit to these more integrated investigative +tools along the lines of the examples here. +D. The implications of a diminished role for the existing core of geometric morpho- +metrics in quantitative morphology are liberating. Via a new toolbox that intentionally +discards Procrustes centering, Procrustes scaling, and Procrustes orientation, and that +downplays the role of thin-plate splines — the whole core of today’s GMM — we may be +able to better achieve GMM’s principal declared purpose, the quantitative understanding +of morphological variation and its causes or effects, by recourse to more diverse geometrical +formalisms, some ancient and some relatively novel. This methodological possibility has +several implications, some for actual analysis of morphologies and others for the method- +ological component of graduate curricula in the evo-devo sciences. The aspects of geometry +that GMM is accustomed to borrowing for its tools concentrate much too heavily on ma- +trix algebra and linear multivariate analysis. As Peter Sneath suspected so long ago in +his paper on trend-surface analysis, there are other geometric entities, such as those here +dealing with quadratic bivariate polynomials, that speak more clearly to the investigator’s +visual instincts, especially as regards phenomena of orientation. (Examine, for instance, +panel 1d of Sneath 1967,5 which shows a relative rotation between face and braincase in +the comparison of Homo to Pan similar to the one in Figure 18 here, without, however, the +optimization of coordinates that Section III exploited.) And far more objects can be as- +signed coordinates than discrete points (or semilandmarks) alone: grid lines, for instance, +deserve coordinates of their own (Figures 4 through 11) and also interlandmark segments +5 According to Biegert 1957, the orientation is along the central plane of the sphenoid +(in Latinate German, “Planum-sphenoideum-Ebene”) to suit the needs of a much broader +study of the midsagittal skull across the order Primates. +1/16/2023 +1:30 +18 + +(Figure 1). +Similarly, the way GMM relies on thin-plate splines for its published renderings ex- +aggerates their importance for organismal biology. The spline is an interpolating map, +whereas, in view of how arbitrary our landmark lists actually are, biological interpreta- +tion often goes deeper and better via approximating maps instead. The actual role of +interpolating splines in the research cycle, then, might be shifted well earlier, all the way +back to before the final rendering style is chosen, in order to supply guidance about which +geometrical languages should be exploited for the most effective dissemination. At that +early stage, interpolating splines are good aids to the search for component processes that +are primarily local, but are poor at the analogous global reports, which, as Sneath already +knew in 1967, do better with polynomial analyses. Both possibilities should be checked, +and perhaps both preserved in the final analysis, the way Figures 4 ff. show both the +thin-plate spline, which reveals the local change at IPS, and the quadratic grid, which +summarizes the overall change of form so much better (in both contexts ignoring the Pro- +crustes side of GMM in favor of the different optimization of orientation recommended +here). +The finding in Figure 1d should not have been new to this paper. In the many previ- +ous GMM investigations of the Vilmann data there should long since have been mention +of rotations of subanatomies, a rhetoric that has been suppressed, perhaps unintentionally, +by virtue of our current traditions of overly symmetric data summaries like Procrustes +distance, principal component analysis and interpolating splines.6 It is time for the mor- +phological side of biomathematics to return to its roots in biological geometry sensu lato +— what might the organism’s function space “know” about its own form? — in order to +rebuild the interplay between data and explanation using a much broader range of geomet- +ric formalisms than just “points” (or their “modules”) and “deformations.” The method +of cubic regression, Figures 17 and 19, is likewise not new; I copied it straight from Sneath +(1967). The particularly careless way the Procrustes method dismisses orientation as just +a nuisance variable has blinded our field to the possibility that relative intraspecimen ori- +entations can be just as informative a channel of insight and explanation as relative extents +(proportions). To restore and then extend this symmetry we need to abandon the stan- +dard Procrustes tool in favor of explorations that explicitly consider multiple orientations +at the same time, just as studies of allometry have been considering multiple size measures +since at least Blackith and Reyment (1971). More generally, to understand transformation +grids we must extend our understanding of the sort of entities that can have coordinates +from points to more extended structures. Only then can we trust our diagrams to provide +straightforward practical summaries of the “blooming, buzzing confusion” (W. James) that +is the spectrum of Darwinian phenomena we call evo-devo. +Acknowledgements. I thank Jim Rohlf, Stony Brook University, for thoughtful +6 In an ironic exception, a non-Procrustes analysis in my 1991 textbook refers to this +rotation as an epiphenomenon (a side-effect) of orthocephalization, the usual name for the +process by which the anterior cranial base thrusts under the facial skeleton — but the verb +“rotate” itself is in scare quotes! See Bookstein (1991:312). +1/16/2023 +1:30 +19 + +commentary on the basic thrust of this manuscript at several earlier stages. It was Joe +Felsenstein, University of Washington, who first alerted me to foundational problems in +the way GMM handles the concept of “rotation.” +Competing interests and funding. There has been no support from any external +funding source, and no conflicts of interest thereby. +1/16/2023 +1:30 +20 + +Literature Cited +Biegert, J. Der Formwandel des Primatensch¨adels. Gegenbaurs morphologisches +Jahrbuch 98:77–199, 1957. +Blackith, R. E., and R. A. Reyment Multivariate Morphometrics. Academic Press, +1971. +Boas, F. The horizontal plane of the skull and the general problem of the comparison +of variable forms. Science 21:862–863, 1905. +Bookstein, F. L. The Measurement of Biological Shape and Shape Change. Lecture +Notes in Biomathematics, vol. 24. Springer-Verlag, 1978. +Bookstein, F. L. Coordinate systems and morphogenesis. In Morphogenesis and Pat- +tern Formation, ed. T. G. Connelly, L. Brinkley, and B. Carlson. Raven Press, 1981, pp. +265–282. +Bookstein, F. L. Tensor biometrics for changes in cranial shape. Annals of Human +Biology 11:413–437, 1984. +Bookstein, F. L. Transformations of quadrilaterals, tensor fields, and morphogenesis. +In Mathematical Essays on Growth and the Emergence of Form, ed. +P. L. Antonelli. +University of Alberta Press, 1985, pp. 221–265. +Bookstein, F. L. Size and shape spaces for landmark data in two dimensions. (With +Discussion and Rejoinder.) Statistical Science 1:181–242, 1986. +Bookstein, F. L. 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Reconsidering “The inappropriateness of conventional cephalomet- +rics.” The American Journal of Orthodontics 149:784–797, 2016. +Bookstein, F. L. A method for factor analysis of shape coordinates. American Journal +of Physical Anthropology 64:221–245, 2017. +Bookstein, F. L. A Course of Morphometrics for Biologists. Cambridge University +Press, 2018. +Bookstein, F. L. Centric allometry: Studying growth using landmark data. Evolu- +1/16/2023 +1:30 +21 + +tionary Biology doi:/10.1007/s11692-020-09530-w, 48:129–159, 2021a. +Bookstein, F. L. Estimating earthquake probabilities by Jaynes’s method of maximum +entropy. Bulletin of the Seismological Society of America 111:2846–2861, 2021b. +Claude, J. Morphometrics with R. Springer, 2008. +Galton, F. Classification of portraits. Nature 76:617–618, 1907. +Garson, J. G. The Frankfort craniometric agreement, with critical remarks thereon. +Journal of the the Anthropological Institute of Great Britain and Ireland, 14:64–83, 1885. +Glaeser, G. Geometry and its Applications in Arts, Nature, and Technology. (English +edition, modified from the German original.) Springer, 2012. +Grenander, U., and M. Miller. Pattern Theory: from Representation to Inference. +Oxford University Press, 2007. +Hilbert, D., and D. Cohn-Vossen. Anschauliche Geometry. 1931. Tr. by P. Nemenyi +as Geometry and the Imagination, Chelsea Pub. Co., 1952. +Jaynes, E. T. Probability Theory: the Logic of Science. +(Ed. +G. L. Bretthorst.) +Cambridge University Press, 2003. +Koenderink, J. Solid Shape. MIT Press, 1990. +Martin R. Lehrbuch der Anthropologie in systematischer Darstellung. Jena: Gustav +Fischer, 1914. +Porteous, I. R. Geometric Differentiation for the intelligence of Curves and Surfaces, +2nd ed. Cambridge, 2001. +Przibram, H. Form und Formel im Tierreiche. Beitr¨age zu einer quantitativen Biologie +I–XX. Franz Deuticke, 1922. +Sneath, P. H. A. Trend-surface analysis of transformation grids. Journal of Zoology, +London 151:65–122, 1967. +Stuart, A., and K. Ord. Kendall’s Advanced Theory of Statistics. Volume 1, Distri- +bution Theory. Wiley, 1994. +Thompson, D’A. W. On Growth and Form. Macmillan, 1917. Abridged edition, ed. +J. T. Bonner, Cambridge University Press, 1961. +Weber, G. W., and F. L. Bookstein. Virtual Anthropology: a Guide to a New Inter- +disciplinary Field. Springer Verlag, 2011. +Captions for figures +Figure 1. Unexpected pattern in the much-analyzed Vilmann data set of neurocranial +octagons for growing laboratory rats. (upper left) Saturated network of interlandmark seg- +ments, Procrustes average shapes of the octagons at ages 7 days (light lines) and 150 days +(heavy lines). Landmarks: Bas, Basion; Opi, Opisthion; IPS, Interparietal suture; Lam, +Lambda; Brg, Bregma; SES, Spheno¨ethmoid synchondrosis; ISS, Intersphenoidal synchon- +drosis; SOS, Spheno¨occipital synchondrosis. (upper right) Subnetwork of segments rotating +by at least 0.15 radians (8.6◦) over this age comparison. (lower left) The same saturated +network for the nonaffine component only of the same Procrustes shape coordinates, with +1/16/2023 +1:30 +22 + +landmark numbers. (lower right) Now tht the uniform component of this shape coordinate +space has been partialled out, there emerges a considerably simpler subnetwork, explicitly +displaying the relative rotation of the anteriormost three landmarks with respect to the +other five. +Figure 2. Two-point superpositions (Bookstein coordinates) of the Vilmann age-7 +and age-150 average octagons for every possible baseline. Landmarks are numbered as in +Figure 1. Circled landmarks: ends of the baseline as registered to (0, 0) and (1, 0). Light +lines, age-7 average; heavy lines, age-150 average. +Figure 3. Contrasting morphometric renderings for diverse transformations by thin- +plate spline (lower row) of a variously oriented square (upper row). (a) Square to paral- +lelogram, grid aligned with the edges of the square. (b) The same, grid now aligned with +the square’s diagonals. (c) Square to trapezoid. (d) Almost the same, grid rotated 45◦: +square to kite. Adapted from Bookstein, 1991, Figure 7.3.6. +Figure 4. This is the first of eight figures that all have the same four-panel format +as applied to one of eight selected baselines from the array of 28 offered in Figure 2. Top- +row panels, left to right: actual change of averaged Cartesian coordinates, with thin-plate +spline oriented to selected baseline; ordinary thin-plate spline of the quadratic fit to the +age-150 average as regressed on first and second powers of the x− and y− coordinates of +the template and also their product xy. Bottom row, left, the quadratic fit (not a spline) +as a grid of its own. Solid circles, the observed data; open circles, predictions from this +regression. Bottom right: restriction of the display list of grid vertices to the interior of +the age-7 octagon as explained in the text. The baseline of this figure runs from Basion to +Opisthion (landmark 1 to landmark 2). +Figure 5. The same as Figure 4 for a baseline from Basion to Interparietal suture, +landmark 1 to landmark 3. +Figure 6. The same for a baseline from Basion to Lambda, landmark 1 to landmark +5. +Figure 7. +The same for a baseline from Interparietal suture to Spheno¨occipital +synchondrosis, landmark 3 to landmark 8. Each panel is roughly a 90◦ rotation of the +corresponding panel in Figure 6, having a baseline at about 90◦ to this one. +Figure 8. The same for a baseline from Lambda to Spheno¨occipital synchondrosis, +landmark 4 to landmark 8. +Figure 9. The same for a baseline from Bregma to Spheno¨ethmoid synchondrosis, +landmark 5 to landmark 6. +Figure 10. The same for a baseline from Bregma to Intersphenoidal suture, landmark +5 to landmark 7. +Figure 11. The same for a baseline from Spheno¨ethmoid synchondrosis to Basion, +1/16/2023 +1:30 +23 + +landmark 6 to landmark 1. +Figure 12. Synthesis of upper left and lower right panels of Figures 4 through 11, +analyses to eight of the 28 possible two-point baselines. Clearly some of these choices lead +to simpler reports than others do. As the thin-plate spline is covariant with similarity +transformations of its target, all the splines here (columns 1 and 3) are the same except for +grid orientation and spacing. But the regressions associated with columns 2 and 4 weight +different landmarks differently (in particular, weighting the two ends of the baseline not +at all), so these grids can vary in more aspects than the baseline orientation per se. +Figure 13. Graphical extension of the quadratic fit to the IPS-SOS baseline yields +a striking reinterpretation of the phenomenon. Left, grid extended to the left over the +baseline-registered template; right, corresponding version of the fitted quadratic trend from +Figure 7. Filled dots, observed average configurations after the two-point registration (left, +age 7 days; right, age 150 days). Open dots, fitted values of the quadratic regression as in +the earlier figures. +Figure 14. +Two alternatives for column (d) of Figure 3. The map in Figure 13 +more closely resembles the bilinear map (far right panel) than the projection map (central +panel). The projection map sends all straight lines to other straight lines; the bilinear map, +in general, only the lines that join matched proportional aliquots from opposite edges. +In both deformations the dashed line delineates the effect of the map on the horizontal +diameter of the starting diamond shape. The projection takes this curve to a straight line, +the bilinear map, to a parabola engendering a less extreme reduction of the template cells’ +areas above this diameter. Owing to the shared symmetry axis of square and kite there is +another set of straight lines within the grid in the rightmost panel — the verticals — but +this third set is not present in the general case, hence the “bi” of “bilinear,” and so I have +not drawn them here. +Figure 15. Landmark configurations for the hominization example, Section III. (left) +Abbreviated names of the twenty landmarks printed at the raw digitized coordinate aver- +ages of the adult sapiens subsample of Bookstein et al. (2003). Alv, alveolare, inferior tip +of the bony septum between the two maxillary central incisors; ANS, anterior nasal spine, +top of the spina nasalis anterior; Bas, basion, midsagittal point on the anterior margin of +the foramen magnum; BrE, BrI, external and internal bregma, outermost and innermost +innermost intersections of sagittal and lambdoidal sutures; CaO, canalis opticus intersec- +tion, intersection point of a chord connecting the two canalis opticus landmarks with the +midsagittal plane; CrG, crista galli, point at the posterior base of the crista galli; FCe, +foramen caecum, anterior margin of foramen caecum in the midsagittal plane; FoI, fossa +incisiva, midsagittal point on the posterior margin of the fossa incisiva; Gla, glabella, most +anterior point of the frontal in the midsagittal; InE, InI, external and internal inion, most +prominent projections of the occipital bone in the midsagittal; LaE, LaI, external and in- +ternal lambda, outermost and innermost intersections of sagittal and lambdoidal sutures; +Nas, nasion, highest point on the nasal bones in the midsagittal plane; Opi, opisthion, +midsagittal point on the posterior margin of the foramen magnum; PNS, posterior nasal +spine, most posterior point of the spina nasalis; Rhi, rhinion, lowest point of the internasal +1/16/2023 +1:30 +24 + +suture in the midsagittal plane; Sel, sella turcica, top of dorsum sellae; Vmr, vomer, sphe- +nobasilar suture in the midsagittal plane. (right) Bookstein coordinates to an ANS-LaI +baseline for the averaged adult H. sapiens and H. neanderthalensis samples and the single +adult female chimpanzee. +Figure 16. Three grid diagrams for the comparison of the averaged H. sapiens and +H. neanderthalensis twenty-landmark configurations, to an ANS-LaI baseline. (upper left) +Conventional thin-plate spline grid deforming the sapiens average to the neanderthalensis. +(upper right) Thin-plate spline rendering of the deformation from the same averaged sapi- +ens to the quadratic regression fits (regressions on first and second powers of the x− and +y−coordinates and also their product xy) of the neanderthalensis configuration. (lower left) +Explicit grid of that quadratic regression. Solid circles, observed averaged neanderthalensis +two-point coordinates; open circles, fitted locations. +Figure 17. The same for a cubic regression of the neanderthalensis coordinates, nine +predictors instead of five. Upper left, upper right, and lower left panels as in Figure 16. +At lower right, an enlarged version of the fitted grid (lower left) as trimmed to the interior +of the actual neanderthalensis average. +Figure 18. The same as Figure 16 for the comparison of the averaged H. sapiens to +the single female chimpanzee in the data base of Bookstein et al. 2003. +Figure 19. The same as Figure 18 for the comparison of H. sapiens to the female +Pan using these tools. The grid at lower left, for the cubic fit, is correctly drawn even +though it looks like a whale. +1/16/2023 +1:30 +25 + +−0.5 +0.0 +0.5 +−0.5 +0.0 +0.5 +1.0 + + + + + + + + + + + + +(a) average Vilmann octagons, 7 and 150 days, + Procrustes superposition +Bas +Opi +IPP +Lam +Brg +SES +ISS +SOS +−0.4 +0.0 +0.2 +0.4 +−0.2 +0.2 +0.4 +0.6 + + + + + + + + + + + + + + +(b) cut at rotation 0.15 radians +−0.5 +0.0 +0.5 +0.0 +0.5 +1.0 + + + + + + + + + + + +(c) the same, nonaffine component only +1 +2 +3 +4 +5 +6 +7 +8 +−0.4 +0.0 +0.2 +0.4 +−0.2 0.0 +0.2 +0.4 +0.6 +• +• +•• +•• +•• +• +• +•• +•• +•• +(d) the same, cut at 0.15 radians +Figure 1. +1/16/2023 +1:30 +26 + +−1 0 +1 +2 +3 +−4 −3 −2 −1 0 + + + + + + + + + + + + + + + + + baseline 1−2 +0.0 +1.0 +2.0 +−2.5 −1.5 −0.5 + + + + + + + + + + + + + + + + + baseline 1−3 +0.0 +1.0 +2.0 +−1.5 +−0.5 +0.5 + + + + + + + + + + + + + + + + + baseline 1−4 +0.0 +0.4 +0.8 +1.2 +−0.6 +0.0 +0.4 + + + + + + + + + + + + + + + + + baseline 1−5 +−0.2 +0.4 +0.8 +0.0 +0.4 +0.8 +1.2 + + + + + + + + + + + + + + + + + baseline 1−6 +0.0 +1.0 +0.0 +1.0 + + + + + + + + + + + + + + + + + baseline 1−7 +0 +1 +2 +3 +0 +1 +2 +3 +4 + + + + + + + + + + + + + + + + + baseline 1−8 +−1 0 +1 +2 +3 +4 +−5 +−3 +−1 0 + + + + + + + + + + + + + + + + + baseline 2−3 +0.0 +1.0 +2.0 +−2.0 +−1.0 +0.0 + + + + + + + + + + + + + + + + + baseline 2−4 +0.0 +0.4 +0.8 +1.2 +−0.6 +0.0 +0.4 + + + + + + + + + + + + + + + + + baseline 2−5 +0.0 +0.4 +0.8 +−0.2 +0.2 +0.6 + + + + + + + + + + + + + + + + + baseline 2−6 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 + + + + + + + + + + + + + + + + + baseline 2−7 +0.0 +1.0 +2.0 +0.0 +1.0 +2.0 + + + + + + + + + + + + + + + + + baseline 2−8 +0 +1 +2 +3 +4 +−2 −1 0 +1 +2 + + + + + + + + + + + + + + + + baseline 3−4 +0.0 0.5 1.0 1.5 +−0.5 +0.5 1.0 + + + + + + + + + + + + + + + + + baseline 3−5 +0.0 +0.4 +0.8 +−0.4 +0.0 +0.4 + + + + + + + + + + + + + + + + + baseline 3−6 +0.0 0.4 0.8 1.2 +−0.4 +0.2 0.6 + + + + + + + + + + + + + + + + + baseline 3−7 +0.0 +1.0 +−0.5 +0.5 + + + + + + + + + + + + + + + + + baseline 3−8 +−1.0 +0.0 +1.0 +2.0 +−1.5 −0.5 +0.5 + + + + + + + + + + + + + + + + + baseline 4−5 +−0.2 +0.4 0.8 +−0.8 +−0.2 +0.4 + + + + + + + + + + + + + + + + + baseline 4−6 +0.0 0.5 1.0 1.5 +−0.5 +0.5 + + + + + + + + + + + + + + + + + baseline 4−7 +0.0 +1.0 +2.0 +−0.5 +0.5 +1.5 + + + + + + + + + + + + + + + + + baseline 4−8 +−0.5 +0.5 +−1.5 +−0.5 + + + + + + + + + + + + + + + + + baseline 5−6 +0.0 +1.0 +2.0 +−2.0 −1.0 +0.0 + + + + + + + + + + + + + + + + + baseline 5−7 +0.0 +1.0 +−1.0 +0.0 + + + + + + + + + + + + + + + + + baseline 5−8 +0 +1 +2 +3 +4 +−2 −1 +0 +1 + + + + + + + + + + + + + + + + + baseline 6−7 +0.0 +1.0 +−1.0 +0.0 + + + + + + + + + + + + + + + + + baseline 6−8 +−1 +0 +1 +2 +−2 +−1 +0 +1 + + + + + + + + + + + + + + + + + baseline 7−8 +Figure 2. +1/16/2023 +1:30 +27 + +(a1) +(a2) +(b1) +(b2) +(c1) +(c2) +(d1) +(d2) +Figure 3. +1/16/2023 +1:30 +28 + +tps of actual growth, 7 days to 150, + baseline 1 to 2 +Vil 7−da to 150−da, baseline 1 to 2 + tps of growth fit +−2 +0 +2 +4 +−4 +−2 +0 +.............. . . +............... . +.............. . . +.............. . . +............... . +................ +................ +................ +................ +................ +................ +................ +................ +................ +................ +................ +................ +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 1 to 2 + quad trend fit +−1 +0 +1 +2 +−4 +−3 +−2 +−1 +0 +. . +. . . . . . +. . . . . . . . . . +. . . . . . . . . +. . . . . . . . . . +. . . . . . . . . +. . . . . . . . . . +. . . . . . . . . +. . . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . +. . . . . +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 1 to 2 + quad trend fit, interior +Figure 4. +1/16/2023 +1:30 +29 + +tps of actual growth, 7 days to 150, + baseline 1 to 3 +Vil 7−da to 150−da, baseline 1 to 3 + tps of growth fit +0 +1 +2 +3 +−2 +−1 +0 +1 +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +. +. . +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 1 to 3 + quad trend fit +0.0 0.5 1.0 1.5 2.0 2.5 +−2.0 +−1.0 +0.0 +. .. . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . +. . . .. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 1 to 3 + quad trend fit, interior +Figure 5. +1/16/2023 +1:30 +30 + +tps of actual growth, 7 days to 150, + baseline 1 to 5 +Vil 7−da to 150−da, baseline 1 to 5 + tps of growth fit +0.0 +0.5 +1.0 +1.5 +−0.5 +0.0 +0.5 +1.0 +. . . . . . . . . . . . . .... +. . . . . . . . . . . . . .... +. . . . . . . . . . . . ..... +. . . . . . . . . . . ...... +. . . . . . . . . . ....... +. . . . . . . . . . ....... +. . . . . . . . . ........ +. . . . . . . . ......... +. . . . . . . .......... +. . . . . . ........... +. . . . . . ........... +. . . . . ............ +. . . . . ............ +. . . . ............. +. . . .............. +. . . .............. +. . ............... +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 1 to 5 + quad trend fit +0.0 0.2 0.4 0.6 0.8 1.0 +−0.6 +−0.2 +0.2 0.4 +. . +. . . +. . . . . . +. . . . . . . +. . . . . . . . . +. . . . . . . . . . . +. . . . . . . . . . . .. +. . . . . . . . . . .. +. . . . . . . . . . . +. . . . . . . . . +. . . . . . . . +. . . . +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 1 to 5 + quad trend fit, interior +Figure 6. +1/16/2023 +1:30 +31 + +tps of actual growth, 7 days to 150, + baseline 3 to 8 +Vil 7−da to 150−da, baseline 3 to 8 + tps of growth fit +0.0 0.5 1.0 1.5 2.0 +−0.5 +0.5 1.0 1.5 +. ............... +................ +. ............... +. . .............. +. . .............. +. . . ............. +. . . . ............ +. . . . ............ +. . . . . . .......... +. . . . . ........... +. . . . . . .......... +. . . . . . . ......... +. . . . . . . . ........ +. . . . . . . . ........ +. . . . . . . . . ....... +. . . . . . . . . . ...... +. . . . . . . . . . ...... +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 3 to 8 + quad trend fit +0.0 +0.5 +1.0 +1.5 +−0.5 +0.0 +0.5 +1.0 +. . . . +. . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . . +. . . . . . . . . +. . . . . . . . . . +. . . . . . . . . . +. . . . . . . . . . +. . . . . . . . . . +. . . . . . . . +. . . . +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 3 to 8 + quad trend fit, interior +Figure 7. +1/16/2023 +1:30 +32 + +tps of actual growth, 7 days to 150, + baseline 4 to 8 +Vil 7−da to 150−da, baseline 4 to 8 + tps of growth fit +−0.5 +0.5 +1.5 +2.5 +−1.5 +−0.5 +0.5 +1.5 +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +............. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 4 to 8 + quad trend fit +0.0 0.5 1.0 1.5 2.0 +−0.5 0.0 0.5 1.0 +. . . . +. . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 4 to 8 + quad trend fit, interior +Figure 8. +1/16/2023 +1:30 +33 + +tps of actual growth, 7 days to 150, + baseline 5 to 6 +Vil 7−da to 150−da, baseline 5 to 6 + tps of growth fit +−1.0 +0.0 0.5 1.0 +−2.5 +−1.5 +−0.5 +.............. +.............. +.............. +. ............. +. ............. +. . ............ +. . ............ +. . . ........... +. . . . .......... +. . . . .......... +. . . . .......... +. . . . . ......... +. . . . . ......... +. . . . . . ........ +. . . . . . . ....... +. . . . . . . ....... +. . . . . . . ....... +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 5 to 6 + quad trend fit +−0.5 +0.0 +0.5 +1.0 +−1.5 −1.0 −0.5 0.0 +. . +. . . . .. +. . . . . . . +. . . . . . . +. . . . . . . . +. . . . . . . . . +. . . . . . . . . +. . . . . . . . . +. . . . . . . . +. . . . . . . . . +. . . . . . . . . +. . . . . . . . +. . . . . . . . +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 5 to 6 + quad trend fit, interior +Figure 9. +1/16/2023 +1:30 +34 + +tps of actual growth, 7 days to 150, + baseline 5 to 7 +Vil 7−da to 150−da, baseline 5 to 7 + tps of growth fit +0 +1 +2 +3 +−2 +−1 +0 +1 +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 5 to 7 + quad trend fit +0.0 0.5 1.0 1.5 2.0 2.5 +−1.5 +−0.5 +0.5 +. . . +. . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . +. . . . . . . . +. . . . . . . . +. . . . . . . +. . . . . +. . . +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 5 to 7 + quad trend fit, interior +Figure 10. +1/16/2023 +1:30 +35 + +tps of actual growth, 7 days to 150, + baseline 6 to 1 +Vil 7−da to 150−da, baseline 6 to 1 + tps of growth fit +0.0 +0.5 +1.0 +1.5 +−0.5 +0.0 +0.5 +................. +................. +................ . +........... .. . . . . +......... . . . . . . . . +..... . . . . . . . . . . . . +.. . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 6 to 1 + quad trend fit +0.0 +0.4 +0.8 +−0.4 +0.0 +0.4 +. . . . . . . . +. . . . . . . . . . +. . . . . . . . . . . . +. . . . . . . . . . . . +. . . . . . . . . . . . +. . . . . . . . . . . . +. . . . . . . . . . . +. . . . . . . . . . .. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 6 to 1 + quad trend fit, interior +Figure 11. +1/16/2023 +1:30 +36 + +Vil 7−da to 150−da, baseline 1 to 2 + tps of growth +−2 +0 +2 +4 +−4 +−2 +0 +...... . . . . . . . . . . +...... . . . . . . . . . . +...... . . . . . . . . . . +...... . . . . . . . . . . +...... . . . . . . . . . . +....... . . . . . . . . . +....... . . . . . . . . . +........ . . . . . . . . +........ . . . . . . . . +....... .. . . . . . . . +........ . . . . . . . . +....... .. . . . . . . . +........ .. . . . . . . +........ .. . . . . . . +......... . . . . . . . +......... . . . . . . . +.......... . . . . . . +Vil 7−da to 150−da, baseline 1 to 2 + quad trend fit +Vil 7−da to 150−da, baseline 1 to 3 + tps of growth +0 +1 +2 +3 +−2 +−1 +0 +1 +............. +............. +............. +............. +............. +............. +......... ... . +..... .... .. .. +... ... .. .. .. . +. ... .. .. . . .. . +.. .. . .. . . . . . . +.. . .. . . . . . . . . +. .. . . . . . . . . . . +. . . . . . . . . . . . . +. . . . . . . . . . . . . +. . . . . . . . . . . . . +. . . . . . . . . . . . . +Vil 7−da to 150−da, baseline 1 to 3 + quad trend fit +Vil 7−da to 150−da, baseline 1 to 5 + tps of growth +0.0 +0.5 +1.0 +1.5 +−0.5 +0.0 +0.5 +1.0 +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . .. +. . . . . . . . . . . . . . ... +. . . . . . . . . . . . . . ... +. . . . . . . . . . . . ..... +. . . . . . . . . . . . ..... +. . . . . . . . . . . ...... +. . . . . . . . . . . ...... +. . . . . . . . . . ....... +. . . . . . . . . ........ +. . . . . . . . . ........ +. . . . . . . . ......... +. . . . . . . .......... +. . . . . . . .......... +. . . . . . ........... +. . . . . ............ +Vil 7−da to 150−da, baseline 1 to 5 + quad trend fit +Vil 7−da to 150−da, baseline 3 to 8 + tps of growth +0.0 0.5 1.0 1.5 2.0 +−0.5 +0.5 1.0 1.5 +. . . . . . . .. ....... +. . . . . . . . ........ +. . . . . . . . . ....... +. . . . . . . . . ....... +. . . . . . . . . . ...... +. . . . . . . . . . .. .... +. . . . . . . . . . . ..... +. . . . . . . . . . . . .... +. . . . . . . . . . . . . ... +. . . . . . . . . . . . . ... +. . . . . . . . . . . . . ... +. . . . . . . . . . . . . . .. +. . . . . . . . . . . . . . .. +. . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . +Vil 7−da to 150−da, baseline 3 to 8 + quad trend fit +Vil 7−da to 150−da, baseline 4 to 8 + tps of growth +−0.5 +0.5 +1.5 +2.5 +−1.5 +−0.5 +0.5 +1.5 +. . . . . . . . . .. .. +. . . . . . . .. . ... +. . . . . . . . .. ... +. . . . . . . .. .... +. . . . . . . . ... .. +. . . . . . . . ..... +. . . . . . . .. .... +. . . . . . . .. .... +. . . . . . . ...... +. . . . . . ... .... +. . . . . . ... .... +. . . . . .. ...... +. . . . .. ....... +. . . . .. ....... +. . . .. .. ...... +. . . . .. ....... +. . . . ......... +Vil 7−da to 150−da, baseline 4 to 8 + quad trend fit +Vil 7−da to 150−da, baseline 5 to 6 + tps of growth +−1.0 +0.0 0.5 1.0 +−2.5 +−1.5 +−0.5 +. . . . .......... +. . . . .......... +. . . . . ......... +. . . . . ......... +. . . . . . ........ +. . . . . . . ....... +. . . . . . . ....... +. . . . . . . . ...... +. . . . . . . . ...... +. . . . . . . . ...... +. . . . . . . . . ..... +. . . . . . . . . . .... +. . . . . . . . . .. ... +. . . . . . . . . . .... +. . . . . . . . . . . ... +. . . . . . . . . . . . .. +. . . . . . . . . . . . .. +Vil 7−da to 150−da, baseline 5 to 6 + quad trend fit +Vil 7−da to 150−da, baseline 5 to 7 + tps of growth +0 +1 +2 +3 +−2 +−1 +0 +1 +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +.............. +Vil 7−da to 150−da, baseline 5 to 7 + quad trend fit +Vil 7−da to 150−da, baseline 6 to 1 + tps of growth +0.0 +0.5 +1.0 +1.5 +−0.5 +0.0 +0.5 +...... .. . . . . . . . . . +.... . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . +Vil 7−da to 150−da, baseline 6 to 1 + quad trend fit +Figure 12. +1/16/2023 +1:30 +37 + +−2 +−1 +0 +1 +2 +0 +1 +2 +3 +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. +. +. +. +. +. +. +. +−3 +−2 +−1 +0 +1 +2 +−1 +0 +1 +2 +3 +4 +. . . . . . . . . . . . . . . . ..... +. . . . . . . . . . . . . . . . . .... +. . . . . . . . . . . . . . . . . . ... +. . . . . . . . . . . . . . . . . .... +. . . . . . . . . . . . . . . . . . ... +. . . . . . . . . . . . . . . . . . . .. +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . +. +. +. +. +. +. +. +. +Vil 7−da to 150−da, baseline 3 to 8 + quad trend extrapolation +Figure 13. +1/16/2023 +1:30 +38 + +square in diamond orientation + +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +projection map onto kite + +bilinear map onto kite + +Figure 14. +1/16/2023 +1:30 +39 + +200 +400 +600 +200 +400 +600 + + + + + + + + + + + + + + + + + + + + +twenty landmarks on the anthropoid midsagittal +Alv +ANS +Rhi +Nas +Gla +BrE +LaE +InE +Opi +InI +LaI +BrI +FCe +CrGCaO +Bas +PNS +Sel +Vmr +FoI +0.0 +0.2 +0.4 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H.neand, baseline 2 to 11 + conventional thin−plate spline +H.sap to H.neand, baseline 2 to 11 + tps of cubic fit +−0.5 0.0 0.5 1.0 1.5 +−0.5 0.0 0.5 1.0 1.5 +. . . ......... . . . . . +. . . ........... . . . +. . ............. . . +. . . ............. . +. . . .............. +. . . .............. +. . . .............. +. . . . ............. +. . . . ............. +. . . . . ............ +. . . . . . ........... +. . . . . . ........... +. . . . . . . .......... +..... +. +. +. +. . +. +. +... +. +. .. +...... +. +. +. +. . +. +. +... +. +. .. +. +H.sap to H.neand, baseline 2 to 11 + cubic trend fit +0.0 +0.4 +0.8 +−0.2 +0.2 +0.6 +. . . . . . +. . . . . . . . . . +. . . . . . . . . . . . +. . . . . . . . . . . . +. . . . . . . . . . . +. . . . . . . . . +. . . . . . . +. . . .. +. +. +. .. +. +. +. +. +. +. +. +. +. . +. +. +. +. +. +. +. +. .. +. +. +. +. +. +. +. +.. . +. +. +.. +. +H.sap to H.neand, baseline 2 to 11 + cubic trend fit, trimmed view +Figure 17. +1/16/2023 +1:30 +42 + +H.sap. to Pan.f, baseline 2 to 11 + conventional thin−plate spline +H.sap to Pan.f, baseline 2 to 11 + tps of quadratic fit +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +. . . . . . . . . . . ...... +. . . . . . . . . . . ...... +. . . . . . . . . . ....... +. . . . . . . . . . ....... +. . . . . . . . . ........ +. . . . . . . . . ........ +. . . . . . . . ......... +. . . . . . . .......... +. . . . . . . .......... +. . . . . . ........... +. . . . . . ........... +. . . . . ............ +. . . . ............. +. . +. .. +. +. +. +. . +. +. +.. . +. +. +.. +. +. . ... +. +.. +. . +. +. +.. . +. +. +.. +. +H.sap to Pan.f, baseline 2 to 11 + quad trend fit +Figure 18. +1/16/2023 +1:30 +43 + +H.sap. to Pan.f, baseline 2 to 11 + conventional thin−plate spline +H.sap to Pan.f, baseline 2 to 11 + tps of cubic fit +−0.5 +0.5 1.0 1.5 2.0 +0.0 0.5 1.0 1.5 2.0 +. . . . . . . .......... +. . . . . . ........... +. . . . . ............ +. . . . ............. +. . . .............. +. . ............... +. ................ +................. +................ . +.............. . . . +............ . . . . . +........... . . . . . . +......... . . . . . . . . +. . ... +. +.. +... +. +... +. +. .. +.. . ... +. +.. +... +. +.. . +. +. .. +. +H.sap to Pan.f, baseline 2 to 11 + cubic trend fit +0.0 +0.4 +0.8 +−0.2 +0.2 +0.6 +. . . . . . +. . . . . . . . .. +. . . . . . . . . . . . +. . . . . . . . . . . . +. . . . . . . . . . . +. . . . . . . . . +. . . . . . . +. . . .. +. +. +. .. +. +. +. +. +. +. +. +. +. . +. +. +.. +. +. +. . .. +. +.. +. . +. +. +.. . +. +. +. +. +. +H.sap to Pan.f, baseline 2 to 11 + cubic trend fit, trimmed view +Figure 19. +1/16/2023 +1:30 +44 + diff --git a/bdE5T4oBgHgl3EQffA9R/content/tmp_files/load_file.txt b/bdE5T4oBgHgl3EQffA9R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..85b41eaabba7cf48dd6120d6375f10e23f1ab70c --- /dev/null +++ b/bdE5T4oBgHgl3EQffA9R/content/tmp_files/load_file.txt @@ -0,0 +1,6344 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf,len=6343 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='05623v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='CV] 13 Jan 2023 Reworking geometric morphometrics into a methodology of transformation grids Fred L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Bookstein University of Vienna, University of Washington ORCID: 0000-0003-2716-8471 fred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='bookstein@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='at, flbookst@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='edu This is an update of a manuscript of the same name and authorship that was submitted to Evolutionary Biology on January 2, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 1 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Today’s typical application of geometric morphometrics to a quantita- tive comparison of organismal anatomies begins by standardizing samples of homologously labelled point configurations for location, orientation, and scale, and then renders the en- suing comparisons graphically by thin-plate spline as applied to group averages, principal components, regression predictions, or canonical variates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The scale-standardization step has recently come under criticism as unnecessary and indeed inappropriate, at least for growth studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This essay argues for a similar rethinking of the centering and rotation, and then the replacement of the thin-plate spline interpolant of the resulting configurations by a different strategy that leaves unexplained residuals at every landmark individually in order to simplify the interpretation of the displayed grid as a whole, the “transformation grid” that has been highlighted as the true underlying topic ever since D’Arcy Thompson’s celebrated exposition of 1917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' For analyses of comparisons involving gradients at large ge- ometric scale, this paper argues for replacement of all three of the Procrustes conventions by a version of my two-point registration of 1986 (originally Francis Galton’s of 1907).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The choice of the two points interacts with another non-Procrustes concern, interpretabil- ity of the grid lines of a coordinate system deformed according to a fitted polynomial trend rather than an interpolating thin-plate spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The paper works two examples using previously published midsagittal cranial data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' there result new findings pertinent to the interpretation of both of these classic data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' A concluding discussion suggests that the current toolkit of geometric morphometrics, centered on Procrustes shape coordinates and thin-plate splines, is too restricted to suit many of the interpretive purposes of evolutionary and developmental biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' KEYWORDS: Procrustes analysis, thin-plate spline, geometric morphometrics, Vil- mann neurocranial octagons, anthropoid midsagittal crania, transformation grids, quadratic fits, bilinear maps, cubic fits, two-point shape coordinates, modularity, baseline registra- tion, D’Arcy Thompson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Introduction Figure 1 here arose simply as free play with the tools of geometric morphometrics (GMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The data set comprises the familiar “Vilmann octagons” tracing around the mid- sagittal neurocrania of close-bred laboratory rats radiographed in the 1960’s by the Danish anatomist Henning Vilmann at eight ages between 7 days and 150 days and digitized some years later by the New York craniofacial biologist Melvin Moss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This version of the data is the one explored in my textbook of 2018: the subset of 18 animals with complete data (all eight landmarks) at all eight ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The concern of Figure 1 is the contrast of the Procrustes-averaged shapes for the age-7 and age-150 animals (only the averages, no con- sideration of covariances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The heavy lines are for the age-150 data subset, the light lines, the data from the animals at age 7 days (a configuration this paper will occasionally refer to as the “template”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' All panels of the figure complicate the usual Procrustes plot of shape coordinate pairs by all or some of the segments connecting these coordinate pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In the figure’s left column, all 8·7/2 = 28 of the interlandmark segments have been drawn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' in the right column, only the subset that are the reason for calling your attention to this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In the top row, those average locations correspond to the usual Procrustes-registered shape coordinates, partialling out only centering, size, and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Panel (b) is limited just to the nine (out of 28) interlandmark segments from panel (a) that rotated either way by at least 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='6 degrees (the figure label expresses this as “0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='15 radians”) 1 between age 7 and 150 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' A pretty graphic, but it features too much overlay of signals to qualify as a legible pattern analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' However, most of the clutter is due to the substantial change of aspect ratio (height-to- width ratio, obvious in the left column) that rotated both of the longer diagonals (Basion to Bregma, Lambda to SES) of the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Fortunately, we already know how to remove this unwanted uniformity of relative vertical compression from our comparison: recourse to the “nonuniform” component of Procrustes shape space, complement to the subspace of uniform transformations (those that take all rectangles into parallelograms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The resulting plots are the pair in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' It is no surprise that the diagram at lower left, panel (c), looks even more cluttered than panel (a), because now the calvarial roof, not just the cranial base, overlaps between the ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But there is also a new signal once the diagram is edited to suppress all the segments that didn’t rotate much, a signal that seems not to have been anticipated in previously published analyses of these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' As panel (d) shows, six of the 28 possible segments rotate by more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='15 radian after standardizing this uniform aspect of the young-to-old comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' And now the pattern is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The five landmarks at left (anatomical posterior, SOS around to Lam) are rotating clockwise (in this projection) over growth, while the three at the right, located anatomically anteriorly, are rotating counterclockwise, all this to a longitudinal arrangement (think of the centroid of the set of five, versus the centroid of the frontmost three) that isn’t rotating either way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This paper will refer to the segmented polygon SOS-Bas-Opi-IPS-Lam, the set of five landmarks at 1 One radian is the mathematician’s natural metric of angle, the angle (about 57◦) at which the extent of a circular arc is equal to the radius of the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 3 left in Figure 1(d), as the “posterior pentagon” and the remaining three, Brg-SES-ISS, as the “anterior triangle.” The opposition of rotations in panel (d) is consistent with a report using an alternative arithmetic of intersegment length-ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' There is evidently shortening of upper calvarial anteroposterior length, Lam to Brg, relative to the central segment of the cranial base from ISS to SOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Now there is no need to state that this pattern is “relative to the sequestering of the uniform term,” as uniform transformations do not alter ratios of distances in the same direction, whether concurrent or parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This relative rotation, including that contrast of vertically aligned horizontal growth rates, the central cranial base versus the calvarial roof above it, is surely a feature of the 143-day change of form here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But where is it to be found in the GMM toolkit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 2 recovers exactly the same report from a quantitative style dating back more than 80 years prior to GMM, analysis via the coordinates Francis Galton introduced in 1907 for “classification of portraits.”2 Here I have diagrammed every possible two-point registra- tion of these octagons (quantified only by their average coordinates as Moss originally digitized them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' For each alternative baseline, the original Cartesian coordinate average configuration has been separately rotated and scaled so that the first baseline point is at (0, 0) of a new coordinate system and the second is at (1, 0) in the same system (the two points circled in every panel of the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' We have thereby altered every single step of the Procrustes toolkit — the centering, the rotating, the scaling — while eschewing any recourse to the thin-plate spline for separating out that uniform term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' And yet ten of the panels clearly show the same phenomenon, the relative rotation between the anatomically posterior pentagon of landmarks and the anterior triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Whenever both ends of the baseline are in the same sector (here numbered [8,1,2,3,4] versus [5,6,7]), the rotation is clear in the behavior of the complementary sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This is particularly evident in the analysis to baselines 5-6 (row 4 column 5), 5-7 (row 4 column 6), or 3-4 (row 3 column 2), where, regardless of any overall change of aspect ratio, the border of the octagon opposite the baseline appears to have radically shifted by a rotation with respect to that baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The disparity between ratios of change of length for segments ISS-SOS and Lam-Brg is clearest, perhaps, in the panel for that ISS-SOS baseline, fifth row, fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Such an analysis, both elegant and elementary, shares no arithmetic with the standard GMM toolkit of Procrustes registration and thin-plate splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (For a good overview of computational aspects of that standard toolkit in a format suitable for routine biometric applications, see Claude 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') It is far older than that morphometric synthesis of the 2 The GMM literature usually refers to these as “two-point coordinates” or an “edge registration,” while the statistical literature (Stuart and Ord 1994:279) calls them “Book- stein coordinates” in keeping with Stigler’s Law, which states that usually innovations are named after the second person to stumble across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Ignoring the scaling aspect of this tool, the centering and orientation here was already explicit in Boas (1905) and probably can be traced back all the way to the German anthropologists’ adoption of the celebrated “Frankfurt Horizontal” in 1882 (see Garson, 1885 [which, remarkably enough, is available from JSTOR] — Orbital set to (0, 0), Porion along the positive x-axis: the Ohr-Augen Horizontale of Martin 1914).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' For a contemporary critique of this specific convention of 1882, see Bookstein, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 4 1990’s, older even than analysis by triangles (“tensor biometrics,” Bookstein, 1984) or by biorthogonal grids (Bookstein, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Both of these versions involve attention to short or long transects of the form that intersect internally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' by analogy with the change of form from a square to a rectangle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' for one particular pair of directions (sides of the square) the ratios of change of distance are greatest or least and the angle of intersection is invariant at 90◦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' while the ratio of change of the two distances at 45◦ to these directions (diagonals of the square) is unity and it is the change of their angle that is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' A closer inspection of the interlandmark-distance interpretation of Figure 1(d) instead makes reference to distances that are parallel at some spacing (upper calvarial width versus lower), a change visible equally in the Procrustes fits and in the two-point versions, especially versions 7-8 (row 5 column 4) and 3-5 (row 3 column 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The idea of examining ratios of parallel distances like these is already present in some much earlier applied treatises, such as Martin 1914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' For an intuitive understanding of what is going on here, turn back to the earliest textbook introduction of the thin-plate spline, Bookstein 1991, where analyses like these, restricted to just a quadrilateral of landmarks, exemplify what I called “purely inhomoge- neous transformations” there, meaning, transformations without any uniform component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='6 of that book displays, within the limits of the software tools of the time, the effect of rotating the starting grid on the graphs of this purely inhomogeneous component (here, the sole nonlinear component) of the deformations of a square that minimize net bending energy — the now-ubiquitous thin-plate spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 3 is a modification of that textbook figure intended to clarify the contrast of the different types of salience (length-ratios and rotations) for the pairs of segments of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Its four columns prototype different types of the transformations, each of a starting square of landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In the top row are the starting squares, twice in Cartesian alignment with the page and twice at 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Below are the corresponding analyses, enhanced by ordinary thin-plate splines that are not actually part of the arithmetical report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (But that spline has nothing to do with the analysis here, which deals only with the landmark positions per se, not any interstitial tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The quadratic extension to the interstitial rendering in Figures 4 through 11 requires a minimum of six landmarks, not just these four;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' the further extension to a cubic fit in Figures 17 and 19 requires at least ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') In column (a) the square is transformed to a rhombus by rotating two of its edges without change in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' What change are the angles between the concurrent edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In column (b) the same transformation is applied to the square of landmarks at 45◦ (in other words, the grid has rotated with respect to the landmarks, the configuration of which has not changed in either row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Now the report is reversed: the greatest change is in the ratio of lengths of diagonals, while the angle between them is left invariant at 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This was also the case for the configuration in column 1, where it was confounded by the inconvenient orientation of the grid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The situation in Figure 1(d) or Figure 2 corresponds instead to the prototype in columns (c) or (d) of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The starting configuration is still the same square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But in column (c) the transformation changes the ratio of lengths of two edges that are par- allel (horizontal in the figure), not perpendicular as in columns (a) or (b), while leaving unchanged the ratio of the other two edge lengths (the other pair of parallels in panel c1) 1/16/2023 1:30 5 while radically altering their angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This is a transformation from a square to an isosceles trapezoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The complementary transformation in column (d), which the geometer would call square-to-kite, leaves the diagonals unchanged in length and in angle while altering the relation between their midpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Now it is a different pair of paired edges whose length-ratio has not changed — the top and bottom V ’s — and while the angles at the end of the horizontal diagonal are hardly altered, those at the ends of the vertical diagonal are greatly changed, one increased and the other decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' To repeat, these reports rely not at all on any GMM technology, neither Procrustes nor thin-plate spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The aim of this paper is to push this insight as far as it can go while remaining elementary in its biomathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (For instance, its multivariate analysis is limited to the familiar setting of multiple regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') While the idea of two-point coordinates was originally Galton’s in 1907, the idea at which the analysis here is aimed, the quadratic growth-gradient, is only half as old: it is present in embryo in Peter Sneath’s underrated paper of 1967 on trend-surface analysis of D’Arcy Thompson’s transformation grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The core of the argument inheres in any of the next eight figures, which selected eight interesting baselines from the 28 in Figure 2 for expansion of the analysis to include an explicit quadratic regression of the averaged age-150 Cartesian coordinates against the same from the age-7 octagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' These analyses completely ignore the tools of standard GMM — there is no Procrustes centering, no scaling or reorientation beyond the (arbitrary) choice of baseline, and thin-plate splines are drawn only to be dismissed — while what results, you will see, is a coherent summary of this particular change of neurocranial form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' A combined Figure 12 arrays the eight separate summaries for a synthesis of their information content abstracted in Figures 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Following this exploration, a further analysis of some data from a study of cranial hominization my Vienna group published twenty years ago will consider some extensions of this approach, and a concluding Discussion will reflect on some implications of this seeming irrelevance of today’s conventional GMM toolkit for the explanatory purposes of evolutionary or developmental morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Vilmann 7-to-150-day growth analyzed without Procrustes GMM The recommended alternate analysis of the Vilmann growth gradient in Figure 1 may be narrated by an extraction of common findings from a suite of separate analyses to my selection of baselines, some transects of the octagon and others circumferential to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The analyses to be synthesized are laid out in Figures 4 through 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Each of these eight composites offers four panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' At upper left will be the conventional thin-plate spline of the averaged octagon of Cartesian coordinates of the age-7-day animals as warped into the analogous average at age 150 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Analysis is to the baseline of the pair of landmarks indicated in larger circles, which specifies the orientation of the thin-plate spline grid in each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' To its right will be a gridded version (in this orientation) of the “growth fit” likewise displayed first for comparison as a thin-plate spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Here the x-coordinate of the deformed grid is the predicted x-coordinate from the regression of the baseline-standardized octagon vertices of the 150-day average on both coordinates of the age-7 average, and also their squares and their crossproduct (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=', a regression of each x150 on (x7, y7, x 2 7 , y 2 7 , x7y7)) and likewise the y150-coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Each of these regressions 1/16/2023 1:30 6 involves five predictors, plus a constant, for only eight “cases” (the relevant coordinate, x or y, of the eight landmarks), and so has only two degrees of freedom for error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' they are not really regressions, but rather almost interpolations, when the landmark count is so small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' At left in the lower row will be a more appropriate representation of the quadratic fit splined at above right: actual transforms of the grid lines of the starting form to this baseline, with the regressions’ “dependent variable” the x− and y−coordinates of the filled dots corresponding to the fitted locations at the open circles nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Finally, the panel at lower right will restrict this grid to just the interior of the age-7 octagon — this portion of the graphic deserves attention first, before any extensions to the exterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Consider, then, the first figure in this series, Figure 4, which is the analysis for a baseline from Basion to Opisthion — the shortest interlandmark segment in the template, but one contained entirely within the posterior pentagonal compartment of Figure 1(d) and hence one that might enlighten us as to the rotation archived there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' That rotation between anterior landmark triangle and posterior landmark pentagon is essentially the same that is displayed in Figure 1 for the conventional GMM approach and in Figure 2 for the panel corresponding to this baseline (there, the panel in row 1, column 1) — indeed it will be the same in all eight of the series of figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Here in Figure 4, for the baseline Basion to Opisthion (axis of the midsagittal foramen magnum), the orientation of the form is rotated about 130◦ from the Procrustes convention in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The thin-plate spline (upper left panel) is interesting in that inside the posterior five-landmark component, SOS around to Lam, the interior as rendered by the spline appears to be nearly affine (all grid cells the same size and shape) except near IPS, and likewise nearly affine for the anterior component Bas-SES-ISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The growth fit (upper right panel) apparently has pulled IPS to the left in this diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' As Figure 1 shows, and as has been exposited in earlier papers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=', Bookstein 2017), this point participates in a specific focal process displacing it upward in the more realistic anatomical setting of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Thus the fit in this upper right panel of Figure 4 does not show the deviation of change at IPS from change at its neighbors that is present in the actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Either panel of the lower row shows how closely the fitted landmarks (open circles) track the averaged 150-day locations observed (the solid circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The horizontal grid lines in the interior of the form (lower right panel) are mainly straight, while their orientation on this diagram is graded from top to bottom more smoothly than one would infer from the analogous diagram at upper left (the thin-plate spline based on the fully detailed data record, which, by design, is not conducive to any lower-dimensional summary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The steady rotation of this imputed grid line direction is complemented by a gentle curvature of the other grid line direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' a curvature that is not so apparent in the explicit thin-plate spline at upper left — the transformation that appears segmented there,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' one nearly linear system for the posterior pentagon and another for the anterior triangle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' is smoothed by the quadratic regression into a continuous gradient from end to end of the template (top to bottom of the grid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' in this coordinate system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Note that the prediction error of the quadratic fit (lower row) specifically implicates the length of the chosen baseline, at both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 5 shows the same analysis for a different baseline, Basion to Interparietal suture, from the same posterior pentagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Again the quadratic fit (lower row) shows a substantial 1/16/2023 1:30 7 residual, this time at only one of the baseline points (IPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In the deformed grid, both systems of lines are curved, a feature that makes interpretation more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 6 is the first to involve a cross-component baseline, Basion (from the posterior pentagon) to Bregma (from the anterior triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The starting grid has rotated about 80◦ from its position in the first of this series (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=', the angle between segments Basion- Opisthion and Basion-Bregma in the age-7 average is about 80◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Again the panels in the lower row inform us that the initially vertical grid lines (lines along Lambda-ISS or IPS-SOS) are transformed by the quadratic fit into a pencil of nearly straight lines at varying orientations, while the lines of the originally orthogonal system are gently curved in a manner that will concern us in detail in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' At neither of the baseline points is there any substantial fitting error of the quadratic regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The approximate uniformity of cell sizes across the trimmed grid at lower right here and in every other figure of this series assures us that the recourse to distances from the centroid in models of centric allometry, such as Bookstein 2021a, is a reasonable default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Indeed the separation between the actual age-150 centroid and the quadratic trend transform of the age-7 centroid is a mere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='054 units in the scale of this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 7, to a baseline from IPS to SOS, is very nearly the same analysis as in Figure 6 inasmuch as the two baselines, IPS-SOS and Bas-Brg, are nearly at 90◦ in the age-7 template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The main difference is the substantial increase in fitting error, owing to the fact that landmark 3, IPS, is known to be strongly loaded on a special factor not shared with the rest of the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Nevertheless, the grids of the lower row still greatly resemble those of the preceding figure, for the baseline at 90◦ to this one: lines parallel to IPS-ISS (here, the baseline) remain straight but rotate from end to end, while the orthogonals are gently curved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Let us move more quickly through the remaining versions of this four-panel scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In Figure 8, baseline Lambda-SOS, both systems of grid lines are gently curved (although the rotation from end to end of the original octagon is as clear as if they had remained straight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The errors of fit at the baseline points are moderate in magnitude, partly because the fit at Lambda is distorted by the need to accommodate the deviation at IPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 9, for a baseline Bregma-SES within the anterior component of Figure 1, dis- plays gentle curves in both grid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Errors of the quadratic fit are again moderate, and the rotation so evident in Figure 1(d) is very clear in spite of the curvature of these deformed grid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The baseline in Figure 10, Bregma-ISS, has similar errors of fit and similar curving of the grid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Finally, Figure 11, for an ISS-Bas baseline, is roughtly the 90◦ rotation of the analysis in Figure 5, whose baseline (Bas-IPS) is roughly at 90◦ to the baseline here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 12 summarizes all eight of these analyses in a way that permits some criteria of interpretability to emerge regarding replacement of the Procrustes rotation by a protocol more conducive to reportage: a protocol that associates the reorientation of specimens to the ultimate simplification of their deformation by reference to the specific coordinate lines as deformed from the template’s square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' We have seen that baseline analyses can sometime come in pairs if the corresponding interlandmark segments themselves lie at approximately 90◦, and it is better if they run close to the centroid of the octagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' More subtly, morphological comparisons that can result in reports of relative rotations of parts 1/16/2023 1:30 8 of a landmark configuration may be diagrammed best not by a thin-plate spline but by a choice of a specific baseline that highlights the rotation in question, like Figure 6 or Figure 7 here, by leaving one set of grid lines straight lines even as they are rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (For instance, in Figure 6, the panel at lower right is more interpretable than the panel at upper left, even though the information content is effectively the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') The thin-plate renderings in Figure 12, columns 1 and 3, all confirm the relative rotation detectable already in Figure 1, but do not otherwise appear to offer much intuitive accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' By comparison, the quadratic-fit displays, columns 2 and 4, vary enough in their legibility that some are truly insightful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Those that seem most helpful are the pair of analyses in the second row, to baseline Bas-Lam or IPS-SOS (two directions that happen to be nearly perpendicular) — these seem to be considerably better than the standard GMM analysis at showing a potentially meaningful gradient for the growth process being visualized here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The analysis in Figure 7 suggested a scenario I have highlighted in Figure 13 by the simple trick of extending the domain of the quadratic fit beyond the bounds of the land- mark locations being fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The diagram here extends the earlier gridded transformation merely by evaluating it on the new real estate to the left in the same template coordinate system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' into the empty space some distance above the foramen magnum of these an- imals, where the horizontal grid lines of Figure 7 appear to be converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' We see that the near-linearity of the transformation along the baseline and all the grid lines parallel to that direction persists quite far beyond the actual anatomical limits of the comparison, resulting in the strong impression of some sort of descriptive center at an unphysiologi- cal distance outside the actual calva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The apparent rotation suggested in Figure 1(d) is embedded here in a larger system of reorientations that might be viewed as continuous rather than segmented, or, in a more suggestive language, graded rather than modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The suggestion is strong, for instance, that this grading ought to be checked for extending further anteriorly to the facial skeleton (a description that will be tied to the classic inter- pretation of “orthocephalization” by a footnote in Section IV) or other features outside of this particular neurocranial data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 14 sketches two geometrical interpretations of Figure 13, one more familiar to the applied mathematician and the other less so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Each is an alternative to the thin-plate spline of column (d) in Figure 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' one will prove more realistic than the other for this paper’s examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The more familiar map is the projection, central panel, that takes every straight line onto another straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But this mapping substantially alters the spacing of the points where these deformed grid lines meet the bounding kite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' No such respacing appears in the extended quadratic fit itself, Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' An alternative better matching that observed quadratic fit is the family prototyped in the right-hand panel of the figure, the bilinear mapa that I discussed in considerable detail in Bookstein 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Bilinear maps3 take one quadrilateral onto another as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Every point (x, y) in the interior of the template quadrilateral is the intersection of two lines connecting opposite edges that divide those edges in the same ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The map takes (x, y) to the intersection of the two lines that divide the homologous pair of edges in the target in the same ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The bilinear map of square onto kite can be written (x, y) → (x, y)+a(1+xy, 1+xy) for some a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The projection 3 In finite-element analysis, these are often called isoparametric coordinates of the quadrilateral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 9 in Figure 14 required the upper isosceles triangle of the template to be mapped into the space above the horizontal diagonal of the kite, entailing a considerable compression of its vertical coordinate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' the bilinear transformation enforces much less compression here, at the cost of bending that horizontal diagonal over the course of the deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This attenuation of the variability of those ratios of area change seems to match the graphics of all the quadratic fits in Figures 4 through 11 after an appropriate rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Returning one final time to the scheme in Figure 1(d), the decomposition of the neurocranial octagon into two nonoverlapping components, we see that the figure has indeed oversimplified the situation there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' That the rotation of all edges of the posterior pentagon leaps to the viewer’s eye obscures the fact that all but one of these segments have changed their length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' And likewise the anterior “triangle,” Brg-SES-ISS, does not rotate rigidly — its edge from SES to ISS shortens and also does not rotate as far as the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Any report focusing on the two “components” is deficient in failing to refer to the coordinate space in-between them, where the unconformity between anteroposterior changes of length along the cranial base versus along the calvarial roof seems better captured by the rotating lines of the Bas-Brg baseline and IPS-SOS baseline analyses (Figure 12, row 2, columns 2 and 4) than by the irregularities of the corresponding thin-plate splines (columns 1 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' An example from hominization of the skull The Vilmann analysis of Section II exploited the best study design that experimental zoomorphology has to offer: a sample of close-bred animals imaged by identical machinery at a fixed sequence of developmental ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (The identification of this research design as the summum bonum of laboratory evo-devo research is a century old — it dates from no later than Przibram 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') Most of the data structures to which GMM has been applied are not so elegantly designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This paper’s final example is a pair of comparisons, each much more typical in its design, that share one 20-landmark configuration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The data are a selection from the 29 forms analyzed in Chapter 4 of Weber and Bookstein (2011) that originated in computed midsagittal sections of a larger sample of CT scans digitized by Philipp Gunz for the growth analysis in Bookstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' That original analysis explicitly relied upon the same GMM toolkit that is most commonly invoked today: Procrustes analysis, principal components of the resulting shape coordinates, and visualizations by thin-plate spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 Of the specimens homologously digitized in 2003, most are Homo sapiens, while four are named specimens of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' neanderthalensis (Atapuerca, Kabwe, Guattari, and Petralona), and two are specimens of Pan, one of each sex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' For the present reanalysis I have averaged the 18 adult sapiens (one of which, Mladeˇc, is an archaic specimen) and, as a separate group, the four neanderthals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' As a third “group” (present for a didactic purpose, a com- 4 In one version or another these data have already been used for demonstrations of GMM in textbooks three different times: not only Weber and Bookstein (2011) but also Bookstein (2014, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The approach circumventing those typical GMM maneuvers is new to the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 10 parison of comparisons) I selected the female adult chimpanzee, because the adult male shows even more of the heterochrony that will render my final figure so extreme in cer- tain aspects of its geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Of course these samples are far more limited than any data resources that would be brought to bear on the same comparisons today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' It would be unreasonable to claim that the computations to be reported presently are valid empirical findings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' my purpose is instead to demonstrate a methodological alternative to Procrustes- and spline-based GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The left panel of Figure 15 names these twenty landmarks at their positions in the average of the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' sapiens sample in the original CT coordinates, which were not far from a Sella-Nasion orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In the right panel this configuration is supplemented by the configurations of the same twenty points for the female chimpanzee and also for the nean- derthal average, all after the two-point transformation (Bookstein coordinates) that put all three ANS’s (of which two are group averages) at (0, 0) and all three internal Lambda’s at (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Evidently this coordinate system has been rotated, translated, and scaled from the panel at its left, but none of these steps proceeded by the Procrustes method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Consider first the analysis in Figure 16, which in its design echoes three of the four panels of the Vilmann series, Figures 4 through 11, but in this case only for one selected baseline, from ANS to LaI, as in the right panel of Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (Analysis to a roughly perpendicular baseline, Opi–BrI, results in essentially the same diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') The comparison in Figure 16 is from the averaged points for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' sapiens in Figure 15 to the averaged points for neanderthalensis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In both of these Homo averages (and also in the single female adult Pan specimen to come) the baseline crosses the cranial base near Sella roughly halfway along its length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The thin-plate spline deformation from the average of the eighteen humans to the average of the four neanderthals, upper left in the figure, shows the expected contrast of shrinking neurocranium and expanding splanchnocranium, particularly along the palate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' the cranial base interposes itself as the so-called “hafting zone.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' As the upper- right panel shows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' this grid is tracked to some extent by the analogous grid for the fitted values of the same neanderthalis landmarks from the quadratic regression on the sapiens coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' That quadratic regression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' already demonstrated many times in the Vilmann example preceding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' shows most of its failure of fit (discrepancies between the open circles and their filled neighbors in the lower-left panel) along that central separatrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' with a possible exception at lower right where the pairings of the two inions are rearranged in both separation and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' As Figure 15 hinted, this rearrangement is due mainly to excessive variation at InE, external inion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The final quadratic trend grid, at lower left in Figure 16, is strikingly different from the thin-plate spline of the same point loci (upper right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Indeed this grid for the fit looks remarkably like a rotation of the grid at right in Figure 14, the bilinear transformation leaving two specific families of straight lines straight after the deformation, while their orientations rotate across.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' At this large scale, the comparison of midsagittal crania of these sister species is largely smooth — the points in the hafting zone differ hardly at all from their predicted locations under the quadratic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In particular, the implication of modularity in the upper right panel is completely effaced in the actual quadratic fit grid at lower left, indicating instead an approximating spatial process that is homogeneously graded with no natural boundaries embryological or otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The grading 1/16/2023 1:30 11 is consistent with the observation that relative to the face the neanderthal neurocranium is smaller than that of sapiens with some relative rotation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 17 analyzes the same comparison by a cubic fit instead of the quadratic fit in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (Specifically, this fit models each of the twenty x−coordinates of the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' neanderthalensis average and then each of its twenty y−coordinates as a linear combina- tion of nine terms xsap, ysap, x2 sap, y2 sap, xsapysap, x3 sap, y3 sap, x2 sapysap, and xsapy2 sap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The quadratic regressions used only the first five of these predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') These cubic grids show bizarre behavior outside the limits of their driving data (the strange cusps already clear in Sneath’s examples of 1967), so as in the Vilmann exposition of Section II I extended the figure by one more panel, lower right, that trims the grid to just the interior of the region occupied by the actual target configuration (here the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' neanderthalensis average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The straight lines of the rendering in Figure 16 now appear as S-curves across that same hafting zone, and of course the new fit, a regression on nine predictors, has to be closer than that in Figure 16 based on only five of the nine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But the change of size-ratios between neurocranium and splanchnocranium remains clear, as does the directional extension along the palate and the relative rotations from anterior to posterior and from caudal to cranial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The situation is quite different for the comparison of the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' sapiens average to our more distant relative, the female chimpanzee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The quadratic analysis analogous to Figure 16 can be found in Figure 18, but it no longer appears to look entirely like the bilinear map of Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Instead we encounter a strong local feature of the transformation, the apparent flattening of the parietal region, that is seen in both of the thin-plate spline renderings of the top row (at left, for the actual shape coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' at right, for the quadratic fit) and likewise in the gridded representation of that quadratic fit at lower left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Strikingly, the residuals of this analysis seem no greater than those of the comparison of the sapiens sample with the neanderthals, Figure 16, yet the flattening of the splines is clearly detected by this quadratic fit as well, which has so many fewer coefficients (and also a matrix inversion step of much lower rank, 5 × 5 instead of 23 × 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The bidirectional linearity of the lower left panel in Figure 16 has certainly ceased to apply globally, while the hafting zone here seems still to be no sort of natural boundary between multiple modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The deformation remains smoothly graded except locally, in the parietal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Yet when we switch the algorithm from the quadratic (five-term) fit to the cubic (nine- term) fit, Figure 19, nothing essential changes in the analysis as a result of these additional four degrees of freedom per coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The thin-plate spline of the fitted points (upper right panel) is not much altered from that in the previous figure except in that same nonconforming parietal region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' and while the cubic fit here leads to pathologies of the extrapolated grid at every corner of the original scheme (lower left panel),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' its restriction to the interior of the actual anatomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' lower right in the figure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' shows grid lines that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' ignoring their curvature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' are actually well-aligned with those of the lower left panel in Figure 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' the comparison from sapiens to neanderthalensis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' We have thereby confirmed graphically that the shape difference in the parietal region is indeed local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Put this another way: the quadratic fit (Figure 18) and the cubic fit (Figure 19) convey the same message, a relatively continuous gradient of deformation right across the hafting zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' And they agree, too, that the situation at the parietal (landmarks Opi through LaE) is not coherent with this large-scale gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' From Bregma forward, the lower right panels in Figures 17 1/16/2023 1:30 12 and 19 differ mainly in the intensity of rotation of these gridline segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' but posterior to that arbitrary boundary the parietal landmarks participate in a reorganization that is incommensurate between the two comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Thus we see again that, just as in the Vilmann growth example, an approach that eschews all of the standard Procrustes steps and also the usual thin-plate spline is capa- ble of generating the same understanding of a morphological phenomenon, in this case a somewhat more complicated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Discussion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The main concern of GMM ought to be the transformation grid per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This was already clear from the earliest formal appearance of the concept in D’Arcy Thompson’s On Growth and Form (Thompson, 1917), where the review literature usually begins (even though portrait artists like Albrecht D¨urer had thought about this much earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The endpoint of the method ought to be not statistical but graphical, and the derived report should be geometrical, not statistical, en route to an ultimately biophysical or otherwise morphogenesis-informed endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The main dilemmas in this tradition were already well- critiqued over the first six decades of its development as I reviewed them in Chapter 5 of Bookstein, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' No matter how clearly defined the positions of individual landmark points might be, there was no complementary rhetoric for reporting meaningful features of the transformation grid that expressed comparisons of their configurations over meaningful biological contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The best exposition of this problem remains Sneath, 1967, a paper that struggled, ultimately unsuccessfully, to bring the algebra of landmark analysis (in that pre-spline era) into alignment with the reasoning of numerical taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Yet D’Arcy Thompson would have been delighted with the grid in Figure 13, while presentations of the same information in Procrustes style, Figure 1a, or spline-style, panels 4(a) through 11(a), would have been of no use to him at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' A more contemporary and quite distinct tradition of transformation studies approaches the problem via a calculus of diffeomorphisms (see, for example, Grenander and Miller, 2007), which makes no essential reference to landmarks at all, instead basing its computations on the full field of image contents, gray-scale or even colored, spanning the organ(s) of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The approach seems particularly helpful in neurological applications to imagery of the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This contrasting method, however, is beyond the scope of my Procrustes critique here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The analysis in Figure 7 suggests renewing Thompson’s original concern in this do- main, the interpretation of grids per se, via injecting a new theme into the discussion, an anatomical basis for orienting the starting grid on the template, that more intensively ex- ploits the interaction between deformation graphics and the investigator’s prior awareness of how coordinate systems themselves can vary in their visually dominant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The biomathematics ought to begin, then, with a confluence of two insights: one, that some morphological domains might be amenable to some kind of functionally interpretable large- scale pattern analysis, and the other, an intuition about the geometrical language by which the pattern of interest might be quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' For Henning Vilmann, this translation began with the knowledge that growth of rodent neurocrania is a plausible domain for morpho- metric exploration and that its midsagittal aspect bears enough information about growth 1/16/2023 1:30 13 and function to be worthy of geometrization not only in his own measurements of extent, nor the numerous intermediate multivariate investigations of this same data set (including several of my own), but also in the novelties of Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But given these two axioms, an applied study would culminate in an exploration not of alternative statistics but of alter- native graphics: a survey not of diverse linear combinations but of diverse grid renderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Information about absolute scale change, where relevant (as in biomechanical aspects of interpretation), can be embedded in any of these grid figures by a simple magnification over the course of printing, or can be inscribed on interlandmark segments or the line-elements of a transformation grid by overprinting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In this context of large-scale comparison, rotation is a tool of rendering clarification, not a nuisance variable of digitizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The quadratic regressions in Figures 4 through 11 all used the same list of five predic- tors x, y, x2, y2, xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This consistency lets the renderings here, unlike the approach in the lower row of Figure 1, preserve the uniform component of the transformation grid, where we can see how it interacts with these gradients of large but finite scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But the directions corresponding to those two axes x and y vary from baseline to baseline, and the baseline points are not privileged by the regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Consequently the coordinates pinned by the two-point registration are not quite pinned by the regression — they are permitted to shift to some extent from solid to fitted circles in the grid figures here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The resulting dataflow sheds new light on what we mean by “the best rotation” when, as in both of this paper’s examples, different parts of an organ appear to rotate relative to one another over a comparison of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The role of the multiple two-point registrations that this paper recommends as a substitute for the Procrustes algorithm is not itself a “finding” of any sort but merely a convenience, a simple way of regularizing the landmarks’ Cartesian coordinates in order that a selection of reasonable polynomial trends can be fitted, each in a reasonably equably weighted way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Its advantage is that unlike the case for the Procrustes method, there is more than one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The Procrustes approach optimizes a quantity (sums of squares of landmark shifts) that is irrelevant to the ultimate purpose of an evolutionary or developmental GMM analysis, which is not a minimized sum of squares or a singular-value decomposition or a classification but rather a plausible biological hypothesis for the observed form-differences, their causes, or their consequences for the organism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Then the logic of the inference engine we need is not the operationalized Procrustes arithmetic itself, the least-squares fit to what is almost always a completely wrong model (the null model, a pure similarity transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Instead we need the logic of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Jaynes’s approach to numerical inference (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=', Jaynes 2003): the explicit acknowledge- ment of what we do not know — what is missing from the list of data-driven constraints on some quantitative empirical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (I have recently reviewed this logic in the rather different context of paleoseismology, which is the history of great earthquakes — see Book- stein 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') What is missing from a Procrustes analysis is, among other things, the ac- knowledgement that choice of an orientation constraint affects the resulting report: what we seek is the orientation that will best clarify the final published diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Furthermore, regardless of this issue of orientation, in every GMM context we already know there is no “correct” registration, because there is no “correct” list of landmarks — in the presence of any regional rotation or rescaling, different lists of landmarks or semilandmarks lead 1/16/2023 1:30 14 to different Procrustes registrations, and the empirical report of a shape comparison must accommodate that specific form of ignorance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' That is the whole purpose of the grids — to free our attention from the landmark data per se to the space in-between where biological processes actually take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The particular protocol dictating the selection of orientations to be considered may be irrelevant to the quantitative morphological inference under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (Recall that in this paper the two points fixed in the baseline registration are not fixed by the fitted trend — the registration is not an inferential component of the grid report at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') Orientation may be specified as any interlandmark segment from the available pairings, or any homologous boundary alignment, or even a specific force vector such as a muscle load or gravitational vertical — or possibly all of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Whatever the choices of orientation, the investigator of a global deformation is led to the approach here, which is the selection of at least one satisfactory such orientation as judged by the ultimate diagram at the end of the workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In 3D, one could proceed via an assortment of large landmark triangles passing near the centroid, similarly searching for clarity and redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But in other contexts that issue of orientation may be quite relevant to the interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The examples here have all dealt with global trends, but Figures 18 and 19 hinted at a need for a deformation tool suitable for local features as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Such a tool would likewise entail a rotation of the Cartesian coordinate system prior to grid computation, but in general a different one — see, for example, the model of the crease in Bookstein 2000 or Bookstein 2014, Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' We need to broaden the range of ideas we borrow from geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' A combination of two branches of geometry led us to the bilinear interpretation in Figure 14 of the grid in Figure 13, but this other toolkit is not among those currently being taught to biomathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The kernel r2 log r of the thin-plate spline doesn’t much resemble the biological processes we are trying to understand, but the algebra of polynomial fits (here, mainly the specific appearance of bilinear maps leaving both pencils of coordinate lines almost straight and almost evenly spaced after deformation) does pick up much of the classic appearance of growth-gradients as laid out for analysis from Thompson on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' More important than the extension of the idea of a coordinate system, though, is an extension of the domain of morphometric data to include empirical entities other than landmark points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The description of the grid in Figure 13 makes no essential mention of any of the landmarks — the simple exegesis here (bilinear reorganization of that particular family of grid lines while remaining lines) pertains much more to the interior of this octagon (the directions of those transects across it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' if you will,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' the pairing of points across the left and right sides of the outline in this orientation) than to any of its boundary delineation detail,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' even though that boundary is the sole data source for the example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Thus at root the finding exemplifies a language of intraorganismal matching, the pairing of points along a shared curve bounding some anatomical entity in section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Pairings like these are not like landmarks in any formal aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' So even though this paper’s first example argument began from a playful GMM- derived diagram, Figure 1d, it ends up formalized in the rhetoric of a spatial extension (Figure 13) unknown to GMM but comprehensible by every reader of Thompson’s chapter, as interpreted in Figure 14 via a similar-looking figure from a subchapter of college geom- 1/16/2023 1:30 15 etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This logical sequence can be reversed: beginning from those same textbooks, to try finding biological examples that illustrate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' We are used to polar coordinates, for ex- ample (most recently in the study of centric allometry, Bookstein 2021a), but what about bipolar coordinates or confocal coordinates (Bookstein 1981, 1985) and other schemes that (literally) co-ordinate position with respect to two origins or two axial systems at the same time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The range of coordinate systems is vastly broader than the Cartesian on which today’s GMM automatically relies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' My biorthogonal grids (Bookstein 1978) already went beyond this possibility, though not in a statistically feasible way, via their formalism of one-axis and three-axis singularities corresponding to the “lemon” and “star” umbilics that are the topic of advanced treatises such as Koenderink (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' From the earliest years of the twentieth century the mathematics of geometry has permitted us to talk about coor- dinates of many different extended structures: not just points, but lines, planes, circles, and many other formalisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' See, at first, Hilbert and Cohn-Vossen, 1931/1952, and then, among the more contemporary surveys, Porteous 2001 or Glaeser 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Thus the word “geometric” in the phrase “geometric morphometrics” needs to have its meaning broadened beyond the current focus on the Procrustes component of GMM or indeed any version based on analysis of landmark points as logically separate data elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' “Procrustes distance” between specimens, when computed as a minimizing sum of squared Cartesian coordinate differences, is just a theory-free proxy for the far more subtle and multifarious concept the biologist knows as the opposite of “similarity,” and today’s GMM treats Procrustes shape coordinates as just a list of Cartesian pairs (or triples) in their own coordinate space of position, without reference to any explicit features for describing how their interrelationships (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' the interlandmark segments of Figure 1) actually change across a comparison of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' D’Arcy Thompson got this correct back in 1917: “The deformation of a complicated figure,” he wrote (Thompson 1961:271), “may be a phenomenon easy of comprehension, though the figure itself have to be left unanalyzed and undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This process of comparison, recognizing in one form a definite permutation or deformation of another, apart altogether from a precise and adequate understanding of the original ‘type’ or standard of comparison, lies within the immediate province of mathematics.” That geometry of “recognizing deformation” is not limited to the geometry of points referred individually to Cartesian axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Thompson himself referred explicitly to the ap- pearance of the deformed grid lines in his drawings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' For the comparison to Mola, for instance, he wrote, “I have deformed [Diodon’s] vertical coordinates into a system of con- centric circles, and its horizontal coordinates into a system of curves which, approximately and provisionally, are made to resemble a system of hyperbolas” (Thompson 1961:300).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' It is the configuration of these curves, not the landmarks on them, that is the bridge from arithmetic to understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In other words, the elementary language of deformation, the language by which we report morphological comparisons as deformations, must be based in a glossary of multiple elementary types of deformable image components, not disartic- ulated landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The roster of these is broad indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' including,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' among other options,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' the changes of point-pairs to other point-pairs at a different distance or direction that we already saw in Figure 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' but also changes of triangles to other triangles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' squares to any quadrilateral whether rectangle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' parallelogram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' trapezoid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' or some other form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' displace- 1/16/2023 1:30 16 ment of interior points with respect to an unchanging boundary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' circles to ellipses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' ellipses to any other simple closed curve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' straight lines to other straight lines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' lines to any other open curve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' line-elements having an orientation in the small as well as a location (for a spline cognizant of this structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' see Bookstein and Green,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1993),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' or nearby pairs of par- allel lines to any bent ribbon tracing the sequence of changes all along their shared length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' All of these have appeared in biometric examples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' each requires a different geometric gram- mar for its reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' For instance (in another acknowledgement of our sister discipline of neuromorphometrics), line elements per se summarize image data for the method known as diffusion tensor analysis that traces and summarizes patterns of wiring in the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' As I hope you have already come to suspect from the figures in this paper, the thin- plate spline is not designed to be of any particular help in this matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Its functional form is mainly a sum of terms r2 log r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' where r is the distance from each grid point to each landmark of the template in turn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' and so it has no machinery for collecting references to two or more landmarks at the same time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' but must revert to the nonbiological symmetries of linear multivariate statistics for this purpose (so that the partial warps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' for instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' are just a (2k − 4)−dimensional rotation of its Cartesian coordinates however they were arrived at to that point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' while the relative warps are just a different (2k − 4)−dimensional rotation of the same coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' No, the elements of a quantitative morphometric com- parison in terms of deformation must be the whole coordinate systems of our deformation diagrams, and the features we extract must be features that refer to those deformed lines and areas, whether end to end or truncated to the vicinity of specific landmark subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Any geometric report qualified to drive a programme like Thompson’s aimed at simple descriptions of relationships among individually complicated specimens must begin with more complicated elementary entities than positions of discrete landmark points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' A search for such explananda, beginning from the paired interlandmark segments in Figure 1, leads immediately to the elementary aspects of this paper’s two examples, which make no refer- ence to the formula r2 log r nor indeed any quantification beyond the squaring or cubing of coordinates and products of those powers that allows us to parameterize families of nearly parallel curves that began as parallel lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The exterior of an organ or an organism is a useful domain for communication of findings even in the absence of tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This comment has real bite for a GMM that depends on the conventional thin-plate spline, which does not understand exteriors at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' So the usual interpolating spline is precisely the wrong tool for detecting large-scale gradients that, like the one summarizing the Vilmann comparison, are not affine — are not conducive to descriptions emphasizing some pair of directions at 90◦ bearing the maximum ratio of rates of change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Because the conventional thin-plate spline relaxes to uniform at great distances, it is not a helpful component of answers to any question about large-scale organization of a form-comparison, the question asked by most morphologists (and dysmorphologists, and paleontologists) ever since Thompson’s time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' To quantify the cunning hint from Figure 1d, I needed the tool of a quadratic trend surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=', a fit, not an interpolation), and when the graphic of that fit proved intriguing, a suitable summary arose only when the rendering was extended (Figure 13) far enough beyond the actual convex hull of the landmarks that 1/16/2023 1:30 17 Figure 14 could show us how to report its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' However vague the language might be for a discussion of Figure 7 by itself, the reworking that is Figure 13 makes the implicit explicit — the extended grid now is exactly the report we seek, no actual words required except the legend explaining how the graphic was produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But such a graphic no longer resembles any sort of conventional GMM output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Because the interior of any non-nested module is at the same time a part of the exterior of every other module, one sees from the hominization example that the morphometric aspect of “modularity,” whatever its exact morphogenetic definition, is a matter not of landmark coordinates but of what happens to coordinate grid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figures 17 and 19 confirm that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' within the limits of these data resources (adult forms only,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' no growth series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' a mere 20 landmarks),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' there is no graphical evidence for the cranial base as a separatrix between braincase and face,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' in spite of their obvious differences in function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' but strong evidence for a separation of the whole anterior two-thirds of this landmark scheme from the five parietal landmarks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Opi through LaI and LaE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' that so clearly seize control of the lower-right corner of the grids for either the quadratic fit (Figure 18 lower left) or the cubic fit (Figure 19 lower right) to the comparison across genera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' While the empirical import of this second data example is obsolete, owing to advances in the accrual of samples of all these species, the practice whereby consideration of the transformation grids per se might shape inferences from landmark data about morphogenetic control processes ought to be transferred from the current GMM toolkit to these more integrated investigative tools along the lines of the examples here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The implications of a diminished role for the existing core of geometric morpho- metrics in quantitative morphology are liberating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Via a new toolbox that intentionally discards Procrustes centering, Procrustes scaling, and Procrustes orientation, and that downplays the role of thin-plate splines — the whole core of today’s GMM — we may be able to better achieve GMM’s principal declared purpose, the quantitative understanding of morphological variation and its causes or effects, by recourse to more diverse geometrical formalisms, some ancient and some relatively novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This methodological possibility has several implications, some for actual analysis of morphologies and others for the method- ological component of graduate curricula in the evo-devo sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The aspects of geometry that GMM is accustomed to borrowing for its tools concentrate much too heavily on ma- trix algebra and linear multivariate analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' As Peter Sneath suspected so long ago in his paper on trend-surface analysis, there are other geometric entities, such as those here dealing with quadratic bivariate polynomials, that speak more clearly to the investigator’s visual instincts, especially as regards phenomena of orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (Examine, for instance, panel 1d of Sneath 1967,5 which shows a relative rotation between face and braincase in the comparison of Homo to Pan similar to the one in Figure 18 here, without, however, the optimization of coordinates that Section III exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=') And far more objects can be as- signed coordinates than discrete points (or semilandmarks) alone: grid lines, for instance, deserve coordinates of their own (Figures 4 through 11) and also interlandmark segments 5 According to Biegert 1957, the orientation is along the central plane of the sphenoid (in Latinate German, “Planum-sphenoideum-Ebene”) to suit the needs of a much broader study of the midsagittal skull across the order Primates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 18 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Similarly, the way GMM relies on thin-plate splines for its published renderings ex- aggerates their importance for organismal biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The spline is an interpolating map, whereas, in view of how arbitrary our landmark lists actually are, biological interpreta- tion often goes deeper and better via approximating maps instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The actual role of interpolating splines in the research cycle, then, might be shifted well earlier, all the way back to before the final rendering style is chosen, in order to supply guidance about which geometrical languages should be exploited for the most effective dissemination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' At that early stage, interpolating splines are good aids to the search for component processes that are primarily local, but are poor at the analogous global reports, which, as Sneath already knew in 1967, do better with polynomial analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Both possibilities should be checked, and perhaps both preserved in the final analysis, the way Figures 4 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' show both the thin-plate spline, which reveals the local change at IPS, and the quadratic grid, which summarizes the overall change of form so much better (in both contexts ignoring the Pro- crustes side of GMM in favor of the different optimization of orientation recommended here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The finding in Figure 1d should not have been new to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In the many previ- ous GMM investigations of the Vilmann data there should long since have been mention of rotations of subanatomies, a rhetoric that has been suppressed, perhaps unintentionally, by virtue of our current traditions of overly symmetric data summaries like Procrustes distance, principal component analysis and interpolating splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='6 It is time for the mor- phological side of biomathematics to return to its roots in biological geometry sensu lato — what might the organism’s function space “know” about its own form?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' — in order to rebuild the interplay between data and explanation using a much broader range of geomet- ric formalisms than just “points” (or their “modules”) and “deformations.” The method of cubic regression, Figures 17 and 19, is likewise not new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' I copied it straight from Sneath (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The particularly careless way the Procrustes method dismisses orientation as just a nuisance variable has blinded our field to the possibility that relative intraspecimen ori- entations can be just as informative a channel of insight and explanation as relative extents (proportions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' To restore and then extend this symmetry we need to abandon the stan- dard Procrustes tool in favor of explorations that explicitly consider multiple orientations at the same time, just as studies of allometry have been considering multiple size measures since at least Blackith and Reyment (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' More generally, to understand transformation grids we must extend our understanding of the sort of entities that can have coordinates from points to more extended structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Only then can we trust our diagrams to provide straightforward practical summaries of the “blooming, buzzing confusion” (W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' James) that is the spectrum of Darwinian phenomena we call evo-devo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' I thank Jim Rohlf, Stony Brook University, for thoughtful 6 In an ironic exception, a non-Procrustes analysis in my 1991 textbook refers to this rotation as an epiphenomenon (a side-effect) of orthocephalization, the usual name for the process by which the anterior cranial base thrusts under the facial skeleton — but the verb “rotate” itself is in scare quotes!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' See Bookstein (1991:312).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 19 commentary on the basic thrust of this manuscript at several earlier stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' It was Joe Felsenstein, University of Washington, who first alerted me to foundational problems in the way GMM handles the concept of “rotation.” Competing interests and funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' There has been no support from any external funding source, and no conflicts of interest thereby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 20 Literature Cited Biegert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Der Formwandel des Primatensch¨adels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Gegenbaurs morphologisches Jahrbuch 98:77–199, 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Blackith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=', and R.' 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In Mathematical Essays on Growth and the Emergence of Form, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Antonelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' University of Alberta Press, 1985, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 221–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Bookstein, F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Beitr¨age zu einer quantitativen Biologie I–XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Franz Deuticke, 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Sneath, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Trend-surface analysis of transformation grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Journal of Zoology, London 151:65–122, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Stuart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=', and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Ord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Kendall’s Advanced Theory of Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Volume 1, Distri- bution Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Wiley, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Thompson, D’A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' On Growth and Form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Macmillan, 1917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Abridged edition, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Bonner, Cambridge University Press, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Weber, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=', and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Bookstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Virtual Anthropology: a Guide to a New Inter- disciplinary Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Springer Verlag, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Captions for figures Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Unexpected pattern in the much-analyzed Vilmann data set of neurocranial octagons for growing laboratory rats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (upper left) Saturated network of interlandmark seg- ments, Procrustes average shapes of the octagons at ages 7 days (light lines) and 150 days (heavy lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Landmarks: Bas, Basion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Opi, Opisthion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' IPS, Interparietal suture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Lam, Lambda;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Brg, Bregma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' SES, Spheno¨ethmoid synchondrosis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' ISS, Intersphenoidal synchon- drosis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' SOS, Spheno¨occipital synchondrosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (upper right) Subnetwork of segments rotating by at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='15 radians (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='6◦) over this age comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (lower left) The same saturated network for the nonaffine component only of the same Procrustes shape coordinates, with 1/16/2023 1:30 22 landmark numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (lower right) Now tht the uniform component of this shape coordinate space has been partialled out, there emerges a considerably simpler subnetwork, explicitly displaying the relative rotation of the anteriormost three landmarks with respect to the other five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Two-point superpositions (Bookstein coordinates) of the Vilmann age-7 and age-150 average octagons for every possible baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Landmarks are numbered as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Circled landmarks: ends of the baseline as registered to (0, 0) and (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Light lines, age-7 average;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' heavy lines, age-150 average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Contrasting morphometric renderings for diverse transformations by thin- plate spline (lower row) of a variously oriented square (upper row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (a) Square to paral- lelogram, grid aligned with the edges of the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (b) The same, grid now aligned with the square’s diagonals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (c) Square to trapezoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (d) Almost the same, grid rotated 45◦: square to kite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Adapted from Bookstein, 1991, Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' This is the first of eight figures that all have the same four-panel format as applied to one of eight selected baselines from the array of 28 offered in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Top- row panels, left to right: actual change of averaged Cartesian coordinates, with thin-plate spline oriented to selected baseline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' ordinary thin-plate spline of the quadratic fit to the age-150 average as regressed on first and second powers of the x− and y− coordinates of the template and also their product xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Bottom row, left, the quadratic fit (not a spline) as a grid of its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Solid circles, the observed data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' open circles, predictions from this regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Bottom right: restriction of the display list of grid vertices to the interior of the age-7 octagon as explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The baseline of this figure runs from Basion to Opisthion (landmark 1 to landmark 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same as Figure 4 for a baseline from Basion to Interparietal suture, landmark 1 to landmark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same for a baseline from Basion to Lambda, landmark 1 to landmark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same for a baseline from Interparietal suture to Spheno¨occipital synchondrosis, landmark 3 to landmark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Each panel is roughly a 90◦ rotation of the corresponding panel in Figure 6, having a baseline at about 90◦ to this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same for a baseline from Lambda to Spheno¨occipital synchondrosis, landmark 4 to landmark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same for a baseline from Bregma to Spheno¨ethmoid synchondrosis, landmark 5 to landmark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same for a baseline from Bregma to Intersphenoidal suture, landmark 5 to landmark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same for a baseline from Spheno¨ethmoid synchondrosis to Basion, 1/16/2023 1:30 23 landmark 6 to landmark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Synthesis of upper left and lower right panels of Figures 4 through 11, analyses to eight of the 28 possible two-point baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Clearly some of these choices lead to simpler reports than others do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' As the thin-plate spline is covariant with similarity transformations of its target, all the splines here (columns 1 and 3) are the same except for grid orientation and spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' But the regressions associated with columns 2 and 4 weight different landmarks differently (in particular, weighting the two ends of the baseline not at all), so these grids can vary in more aspects than the baseline orientation per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Graphical extension of the quadratic fit to the IPS-SOS baseline yields a striking reinterpretation of the phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Left, grid extended to the left over the baseline-registered template;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' right, corresponding version of the fitted quadratic trend from Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Filled dots, observed average configurations after the two-point registration (left, age 7 days;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' right, age 150 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Open dots, fitted values of the quadratic regression as in the earlier figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Two alternatives for column (d) of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The map in Figure 13 more closely resembles the bilinear map (far right panel) than the projection map (central panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The projection map sends all straight lines to other straight lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' the bilinear map, in general, only the lines that join matched proportional aliquots from opposite edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' In both deformations the dashed line delineates the effect of the map on the horizontal diameter of the starting diamond shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The projection takes this curve to a straight line, the bilinear map, to a parabola engendering a less extreme reduction of the template cells’ areas above this diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Owing to the shared symmetry axis of square and kite there is another set of straight lines within the grid in the rightmost panel — the verticals — but this third set is not present in the general case, hence the “bi” of “bilinear,” and so I have not drawn them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Landmark configurations for the hominization example, Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (left) Abbreviated names of the twenty landmarks printed at the raw digitized coordinate aver- ages of the adult sapiens subsample of Bookstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Alv, alveolare, inferior tip of the bony septum between the two maxillary central incisors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' ANS, anterior nasal spine, top of the spina nasalis anterior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Bas, basion, midsagittal point on the anterior margin of the foramen magnum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' BrE, BrI, external and internal bregma, outermost and innermost innermost intersections of sagittal and lambdoidal sutures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' CaO, canalis opticus intersec- tion, intersection point of a chord connecting the two canalis opticus landmarks with the midsagittal plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' CrG, crista galli, point at the posterior base of the crista galli;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' FCe, foramen caecum, anterior margin of foramen caecum in the midsagittal plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' FoI, fossa incisiva, midsagittal point on the posterior margin of the fossa incisiva;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Gla, glabella, most anterior point of the frontal in the midsagittal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' InE, InI, external and internal inion, most prominent projections of the occipital bone in the midsagittal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' LaE, LaI, external and in- ternal lambda, outermost and innermost intersections of sagittal and lambdoidal sutures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Nas, nasion, highest point on the nasal bones in the midsagittal plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Opi, opisthion, midsagittal point on the posterior margin of the foramen magnum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' PNS, posterior nasal spine, most posterior point of the spina nasalis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Rhi, rhinion, lowest point of the internasal 1/16/2023 1:30 24 suture in the midsagittal plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Sel, sella turcica, top of dorsum sellae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Vmr, vomer, sphe- nobasilar suture in the midsagittal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (right) Bookstein coordinates to an ANS-LaI baseline for the averaged adult H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' sapiens and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' neanderthalensis samples and the single adult female chimpanzee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Three grid diagrams for the comparison of the averaged H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' sapiens and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' neanderthalensis twenty-landmark configurations, to an ANS-LaI baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (upper left) Conventional thin-plate spline grid deforming the sapiens average to the neanderthalensis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (upper right) Thin-plate spline rendering of the deformation from the same averaged sapi- ens to the quadratic regression fits (regressions on first and second powers of the x− and y−coordinates and also their product xy) of the neanderthalensis configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' (lower left) Explicit grid of that quadratic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Solid circles, observed averaged neanderthalensis two-point coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' open circles, fitted locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same for a cubic regression of the neanderthalensis coordinates, nine predictors instead of five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Upper left, upper right, and lower left panels as in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' At lower right, an enlarged version of the fitted grid (lower left) as trimmed to the interior of the actual neanderthalensis average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same as Figure 16 for the comparison of the averaged H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' sapiens to the single female chimpanzee in the data base of Bookstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The same as Figure 18 for the comparison of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' sapiens to the female Pan using these tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' The grid at lower left, for the cubic fit, is correctly drawn even though it looks like a whale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 (a) average Vilmann octagons, 7 and 150 days, Procrustes superposition Bas Opi IPP Lam Brg SES ISS SOS −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='6 (b) cut at rotation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='15 radians −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 (c) the same, nonaffine component only 1 2 3 4 5 6 7 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='6 • • •• •• •• • • •• •• •• (d) the same, cut at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='15 radians Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 26 −1 0 1 2 3 −4 −3 −2 −1 0 baseline 1−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 baseline 1−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 baseline 1−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 baseline 1−5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 baseline 1−7 0 1 2 3 0 1 2 3 4 baseline 1−8 −1 0 1 2 3 4 −5 −3 −1 0 baseline 2−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 −1.' metadata={'source': 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6−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='0 baseline 6−8 −1 0 1 2 −2 −1 0 1 baseline 7−8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 27 (a1) (a2) (b1) (b2) (c1) (c2) (d1) (d2) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 28 tps of actual growth, 7 days to 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' Vil 7−da to 150−da, baseline 3 to 8 quad trend fit, interior Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 32 tps of actual growth, 7 days to 150, baseline 4 to 8 Vil 7−da to 150−da, baseline 4 to 8 tps of growth fit −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='5 1.' 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baseline from ANS to LaI Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' 1/16/2023 1:30 40 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='sap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content=' to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='neand, baseline 2 to 11 conventional thin−plate spline H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='sap to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE5T4oBgHgl3EQffA9R/content/2301.05623v1.pdf'} +page_content='neand, baseline 2 to 11 tps of quadratic fit 0.' 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AHMED†, Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Labora- +tory, USA +This paper describes the first version (v1.0) of PyOED, a highly extensible scientific package that enables developing and testing +model-constrained optimal experimental design (OED) for inverse problems. Specifically, PyOED aims to be a comprehensive Python +toolkit for model-constrained OED. The package targets scientists and researchers interested in understanding the details of OED +formulations and approaches. It is also meant to enable researchers to experiment with standard and innovative OED technologies +with a wide range of test problems (e.g., simulation models). OED, inverse problems (e.g., Bayesian inversion), and data assimilation +(DA) are closely related research fields, and their formulations overlap significantly. Thus, PyOED is continuously being expanded +with a plethora of Bayesian inversion, DA, and OED methods as well as new scientific simulation models, observation error models, +and observation operators. These pieces are added such that they can be permuted to enable testing OED methods in various settings +of varying complexities. The PyOED core is completely written in Python and utilizes the inherent object-oriented capabilities; +however, the current version of PyOED is meant to be extensible rather than scalable. Specifically, PyOED is developed to “enable +rapid development and benchmarking of OED methods with minimal coding effort and to maximize code reutilization.” PyOED will be +continuously expanded with a plethora of Bayesian inversion, DA, and OED methods as well as new scientific simulation models, +observation error models, and observation operators. This paper provides a brief description of the PyOED layout and philosophy and +provides a set of exemplary test cases and tutorials to demonstrate how the package can be utilized. +Additional Key Words and Phrases: Optimal experimental design, OED, inverse problems, data assimilation +1 +INTRODUCTION +Recently, interest has increased in developing scalable data assimilation (DA) and uncertainty quantification methodolo- +gies for solving large-scale inverse problems. An inverse problem refers to the retrieval of a quantity of interest (QoI) +associated with or stemming from a physical phenomenon underlying partial noisy experimental or observational data +of that physical system [7, 44, 51]. The QoI could be, for example, the model state, initial condition, or other physics +quantity. Inverse problems are prominent in a wide spectrum of applications including power grids and atmospheric +numerical weather prediction [26, 43]. In these problems, the prediction of the physical phenomena is often formulated +as an initial value problem, while the initial condition of the simulator is corrected by fusing all available information. +Algorithmic approaches for solving inverse problems seek either a single-point estimate of the target QoI or a full +probabilistic description of the knowledge about the QoI given all available information. The underlying principle of +these methods is that information collected from observational systems is fused into computational models, along with +associated uncertainties, to produce an accurate estimate of the ground truth of the physical phenomena of interest. +∗The first author led the development of PyOED and this article. +†The second author developed a set of simulation models for initial testing of PyOED, and contributed to the introduction and revision of this article. +Authors’ addresses: Ahmed Attia, Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, Illinois, USA, 60349, aattia@anl.gov; +Shady E. Ahmed, Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Laboratory, Richland, Washington, USA, 99352, +shady.ahmed@pnnl.gov. +1 +arXiv:2301.08336v1 [cs.MS] 19 Jan 2023 + +2 +Ahmed Attia and Shady E. Ahmed +In the former approach seeking a single QoI estimate, the solution of an inverse problem is obtained by solving an +optimization problem with an objective to minimize the mismatch between observational data and model simulations, +possibly regularized by prior knowledge and uncertainty models. The latter approach, commonly known as Bayesian +inversion, seeks to characterize the probability distribution of the QoI through the posterior formulated by applying +Bayes’ rule, that is, the probability distribution of the QoI conditioned by all available information. +DA methods [12–16, 18, 20, 26, 35, 43] aim to solve large- to extreme-scale inverse problems. They work by fusing +information obtained from multiple sources, such as the dynamical model, prior knowledge, noisy and incomplete +measurements, and error models, in order to better estimate the state and parameters of the physical system. This +estimate improves the predictability of the simulation systems developed to make future predictions about the physical +phenomena of interest. +The quality of DA systems, and hence the accuracy of their predictions, is heavily influenced by the extent to which +the mathematical assumptions reflect reality and depends on the quality of the collected measurements. Optimal data +acquisition is the problem of determining the optimal observational strategy, for example, from a large set of candidate +observational schemes. This problem is widely formulated as an optimal experimental design (OED) problem [22, 39], +where the design parameterizes and thus determines the observational configuration. In an OED problem, a design +is defined to characterize a candidate configuration or a control, and the quality of the design is quantified by using +a utility function. The optimal design is then defined as the one that maximizes this utility function or, equivalently, +minimizes some OED criterion [9]. Since the aim of Bayesian inference is to estimate the QoI posterior, Bayesian OED +seeks an observational configuration that, when combined with the underlying dynamics, would maximize information +gain from the data or minimize the posterior uncertainty. Thus, an optimal design is found by solving an optimization +problem with the objective to maximize a utility function that quantifies the quality of the design and its influence +on the solution of the inverse problem. OED for inverse problems has experienced a recent surge in interest by the +scientific computing community; see, for example, [2] and references therein. +Numerical testing and experiments are critical for developing efficient OED formulations and algorithms. This +process is elementary for successful scientific research in general. Although statisticians have developed a plethora of +mathematical formulations and algorithmic approaches for general-purpose OED algorithms, most of the available and +publicly accessible OED software tools are limited to idealized formulations and specific applications such as finding +optimal collocation points for regression problems or designing clinical experiments. In addition, they are written +in the R programming language or MATLAB, thus limiting code reutilization and accessibility by a wider audience; +see, for example, [24, 40, 41, 46, 52]. Unfortunately, these tools do not align well with the interests of the increasingly +large computational science community for developing new OED formulations and algorithmic approaches for inverse +problems, DA, and model-constrained OED [4, 8, 19, 23, 27, 28, 30, 37, 39]. As a first step in alleviating this limitation, we +present and describe PyOED, a highly extensible open-source Python package written mainly to enable computational +scientists to formulate and rapidly test OED—as well as DA—formulations and algorithmic approaches. +PyOED is unique in several ways. First, to the best of our knowledge, it is the first open-source package for scientific +computing that allows implementing and testing the individual components of DA as well as OED systems in a +unified and streamlined environment. Second, it is written in Python, which is arguably the most popular and adopted +programming language for recent algorithmic developments in the computational science disciplines. It has a huge +user-support community, and the learning curve is relatively smoother than that of other lower-level programming + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +3 +languages such as C/C++. Third, PyOED is designed in an object-oriented programming (OOP) fashion, which enables +practitioners to reconfigure and reuse the individual building blocks. Moreover, it is easy to combine PyOED with +other user-defined routines, such as numerical integration of simulation models and optimization routines, which +makes PyOED highly extensible and adaptable to a wide range of applications. Fourth, PyOED is not limited to a +specific inverse problem formulation. Thus new DA and OED methods can be implemented and even interface with +other inversion packages, for example, hIPPYlib [50]. Fifth, PyOED leverages best practices in software development, +including detailed documentation with hands-on examples and robust unit-testing techniques. +The rest of this paper is organized as follows. Section 2 provides the mathematical formalism of inverse problems, +DA, and OED. Section 3 describes the structure and the philosophy of the PyOED package. In Section 4 we provide a +list of numerical experiments to demonstrate the general workflow and usage of PyOED. Concluding remarks are given +in Section 5. +2 +MATHEMATICAL BACKGROUND +In this section we provide a brief overview of the mathematical background of the PyOED core, which is important for +approaching the OED problem for Bayesian inversion. In this presentation we focus on sensor placement for Bayesian +inversion as a modal OED formulation. We start by discussing the forward problem in 2.1; then we introduce the inverse +problem in 2.2 and the OED formalism in 2.3. +2.1 +The forward problem +The forward problem maps the model parameters (e.g., the initial condition) onto the observation space. Consider the +forward problem described by +y = F (𝜃) + 𝛿 , +(1a) +where 𝜃 is the model parameter of interest, y ∈ RNobs is the observation, and 𝛿 ∈ RNobs is a noise term that accounts for +the inaccuracy of the observational system. The forward operator F is occasionally referred to as the “parameter-to- +observable” map and generally represents a composition of a simulation/solution model S and an observation operator +O. The simulation model S describes the evolution of the physical phenomena, for example, space-time advection and +diffusion of a contaminant simulated over a predefined model grid. The observation operator O projects the simulated +state onto the observational grid, for example, by interpolation or restriction to the observation grid. Thus, the forward +problem (1a) can be rewritten as +y = O ◦ S(𝜃) + 𝛿 , +(1b) +where ◦ is the composition operator, that is, O ◦ S(𝜃) ≡ O (S(𝜃)). +It is impossible to find a simple unique formalism of the simulation model that accurately represents all possible +dynamical systems. In this work and in the implementation of PyOED, we differentiate two types of simulation models: +time-independent and time-dependent simulations. A wide range of time-dependent simulation models for dynamical +systems governed by partial differential equations (PDEs) can be described as +𝜕𝒖 +𝜕𝑡 = 𝒇 (𝒖(𝒙,𝑡, 𝜇)), +(2) + +4 +Ahmed Attia and Shady E. Ahmed +where 𝒖 represents the prognostic variable(s) (e.g., physics), 𝒙 denotes the spatial coordinates, and 𝜇 defines the physics +parameters of the model. To numerically solve the simulation model S equations, we utilize spatial discretization and +temporal integration routines. If we follow a discretize-then-optimize approach, we can rewrite (2) in the form +u𝑛 := u(𝑡𝑛) = S𝑡𝑛−1→𝑡𝑛 (u0, 𝜇), +(3) +where u𝑛 := u(𝑡𝑛) ≡ u(𝑡𝑛, 𝒙, 𝜇) is the model state at time instance 𝑡𝑛 and S𝑡→𝑡+Δ𝑡 is a one-time step transition mapping +that results from the application of a standard spatial discretization method (e.g., finite difference, finite volume, or +finite element) and time integration scheme (e.g., Runge–Kutta routine) with a step size Δ𝑡. Thus, the prediction at time +𝑡𝑛 can be related to the initial condition u(𝑡0) and the model parameters using the recursive application of the mapping +S over a time interval [𝑡0,𝑡𝑛] as +u𝑛 = S𝑡𝑛−1→𝑡𝑛 ◦ · · · ◦ S𝑡0→𝑡1 (u0, 𝜇) . +(4) +In space-time formulations, one can stack the model state in one long vector u := +� +uT𝑛, . . . , uT +0 +�T +and define the +solution operator S as a block operator (e.g., a block matrix) that operates recursively on the the components of u. This +would enable unifying the formulation—to some extent—of the forward problem into the form (1a). +Since the observational measurements (e.g., sensory data) are not necessarily the same as the model state, we define +the state-to-observable mapping O𝑛(·) to map the state u𝑛 onto the observation space, for example, by restricting that +state onto the observational grid points, as +y(𝑡𝑛) = O𝑛(u(𝑡𝑛)) , +(5) +which then can be used to construct a general observational vector y ∈ RNobs representing spatiotemporal data, for +example, by stacking observations at multiple time points. +Note that while we focused the discussion above on time-dependent problems, the case of time-independent +simulations can be thought of as a special case of (4) where S maps, for example, the physics parameter 𝜇 to a model +state u and the time index is dropped. In most inverse problem formulations, based on the application of interest, the +inversion parameter 𝜃 stated in (1a) stands for the model physics 𝜇, the model initial condition u0, or both. Note also +that we have omitted details including adaptive time stepping where Δ𝑡 is adaptively adjusted to guarantee stability +and accuracy of the time integration methodology. We intentionally remove such details from the discussion here to +simplify the presentation and focus more on OED. In fact, the OED routines in PyOED are designed to solve OED +problems where the utility function is regarded as a black box, thus abstracting the OED capabilities from the inverse +problem definition. For these reasons, we take (1) to be an acceptable simplification that describes a forward problem +setup that is general enough for our purposes in this paper. +Both the simulation model S and the observation operator O are imperfect and generally include sensory noise and +representativeness errors characterizing imperfection of the map between the model space and observation space. The +fact that model observations F (𝜃) are not perfectly aligned with observational data (y) is modeled—assuming additive +noise—by adding the noise term 𝛿 to the simulated observations F (𝜃). In most applications, the observational noise +follows a Gaussian distribution 𝛿 ∼ N (0, Γnoise), where Γnoise is the observation error covariance matrix that captures +uncertainty stemming from sensory noise and representativeness errors. In this case, the data likelihood is +P (y|𝜃) ∝ exp +� +−1 +2 ∥F (𝜃) − y∥2 +Γ−1 +noise +� +, +(6) + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +5 +where the matrix-weighted norm in (6) is defined as ∥x∥2 +A = xTAx for a vector x and a square symmetric matrix A of +conformable sizes. +2.2 +The inverse problem +An inverse problem refers to the retrieval of the model parameter 𝜃 from noisy observation y, conditioned by the +model dynamics. This can be achieved by finding a point estimate or by building a complete probabilistic description as +discussed in Section 1. In the former, an optimization problem is solved to minimize the mismatch between the expected +observations (through simulation models) and real data. This is typically employed in variational DA methods where the +estimate of the true parameter is obtained by minimizing a regularized log-likelihood objective, where regularization is +employed to enforce smoothness or background information on the parameter. In this case, a point estimate of the true +𝜃 is obtained by solving +arg min +𝜃 +J (𝜃) := 1 +2 ∥F (𝜃) − y∥2 +Γ−1 +noise + 1 +2 +��𝜃 − 𝜃pr +��2 +Γ−1 +pr , +(7) +where 𝜃pr is an initial guess of the unknown true value of 𝜃. In general, the second term is added to enforce regularization +or prior knowledge on the solution, for example, if the solution is assumed a priori to follow a Gaussian distribution +𝜃 ∼ N �𝜃pr, Γpr +�. +Uncertainty envelopes around the single-point estimate obtained by solving (7) can be developed, for example, by +using Laplacian approximation [47] where the posterior is approximated by a Gaussian distribution. This approach +has been successfully employed in infinite-dimensional Bayesian inversion problems [44]. A fully Bayesian approach, +on the other hand, aims to provide a consistent probabilistic description of the unknown parameter along with the +associated uncertainties and is not limited to Gaussian distributions. This is achieved by describing the posterior, that +is, the probability distribution of the model parameter 𝜃 conditioned by the available simulations and noisy data y, and +is obtained by applying a form of Bayes’ theorem +P (𝜃|y) ∝ P (y|𝜃) P(𝜃) , +(8) +where P(𝜃) is the prior, P (y|𝜃) is the data likelihood, and ∝ indicates removal of a normalizing constant in the right-hand +side of (8). For further details on Bayesian inversion see, for example„ [43, 44]. Given the posterior (8), one can use +the maximum a posteriori (MAP) point as an estimate of the true unknown QoI or follow a Monte Carlo approach to +sample the posterior, thus building a complete probabilistic picture; see, for example, [16]. Both the variational and the +Bayesian inference approaches provide a plethora of techniques for statistical data analysis in general, and specifically +for solving inverse problems. +The Bayesian perspective provides a formal mathematical ground for estimating the physical QoI, for example, the +model parameter 𝜃, along with the associated uncertainties given the available sources of information. In many cases, +however, this inversion is an intermediate step, and the goal QoI is a function of the model parameter, that is, 𝜌 := P(𝜃). +A goal-oriented approach is followed in this case where one aims to inspect the posterior of the QoI conditioned by the +available data [32, 33]. + +6 +Ahmed Attia and Shady E. Ahmed +The ideal case: linear Gaussian problems. If the forward operator F is linear (or linearized), and assuming Gaussian +observational noise and a Gaussian prior N �𝜃pr, Γpr +�, then the posterior is Gaussian N +� +𝜃y +post, Γpost +� +with +Γpost = +� +F∗Γ−1 +noiseF + Γ−1 +pr +�−1 +, +𝜃y +post = Γpost +� +Γ−1 +pr𝜃pr + F∗Γ−1 +noise y +� +, +(9) +where F ≡ F is the forward model and F∗ is the associated adjoint. Despite being simple, this setup (9) is of utmost +importance in the Bayesian inversion and OED literature and is elementary for testing implementations of new DA and +OED approaches, mainly because the posterior can be formulated exactly. Moreover, in many large-scale applications, +the posterior can be approximated, to an acceptable degree, by a Gaussian distribution obtained by linearizing the +nonlinear operator F around the MAP estimate. The linearized model is also known as the tangent linear model (TLM) +obtained by differentiating F . +As mentioned earlier, inversion for the parameter 𝜃 is often an intermediate stage, and the end-goal is to describe the +posterior of a general QoI that is not the model parameter 𝜃 but rather a goal quantity 𝜌 that depends on the inversion +parameter 𝜃. Specifically, goal-oriented inversion seeks the posterior P (𝜌|y) ∝ L(y|𝜌,𝜃)P(𝜌) , where P(𝜌) is a prior on +the goal QoI and L(y|𝜃) = L(y|𝜌,𝜃) is the data-likelihood (6), where 𝜌 is determined completely based on 𝜃. We focus +the discussion here on the case of linear prediction operators P. That is, we consider prediction quantities of the form +𝜌 = P𝜃, +(10) +where P is a linear prediction operator. Within the Gaussian linear setting, the prior of the goal QoI 𝜌 is N �𝜌pr, Σpr +� +with +𝜌pr = P𝜃pr, +Σpr = PΓprP∗, +(11) +where P∗ is the adjoint of the prediction operator P. The posterior distribution of the prediction 𝜌, conditioned by the +observations y, is also Gaussian and is given by N �𝜌post, Σpost +�, where +𝜌post = P𝜃y +post , +Σpost = PΓpostP∗ = P +� +F∗Γ−1 +noiseF + Γ−1 +pr +� +P∗. +(12) +Note that the goal-oriented Bayesian inversion falls back to the standard formulation of a Bayesian inverse problem +if the prediction operator P is an identity operator. +2.3 +Optimal experimental design +Here we outline the basics of an OED problem for Bayesian inversion. An excellent review of recent advances on +model-constrained OED can be found in [1]. An OED optimization problem takes the general form +𝜻opt = arg max +𝜻 +U(𝜻) , +(13) +where U is a predefined utility function that quantifies the quality of the design 𝜻. The nature of 𝜻 depends on the +application at hand, and the utility function U is chosen to enable defining what an “optimal” design is. The optimization +problem (15) is often associated with an auxiliary sparsity-enforcing term −𝛼Φ(𝜻) to prevent dense designs and to +reduce the cost associated with deploying observational sensors. The utility function can be rewritten as +U(𝜻) = Ψ(𝜻) − 𝛼Φ(𝜻) , +(14) + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +7 +where Ψ(·) is an OED optimality objective, referred to hereafter as the “optimality criterion,” which is defined based on +the inverse problem at hand and on a chosen criterion (e.g., from the well-known OED alphabetic criteria). The function +Φ(𝜻) asserts regularization or sparsity on the design. For example, this could be a resource constraint: �𝑛s +𝑖 𝜻𝑖 = ∥𝜻 ∥ ≤ 𝑘 , +or �𝑛s +𝑖 𝜻𝑖 = ∥𝜻 ∥ = 𝑘 ;𝑘 ∈ Z+ , for example, an upper bound (or exact budget) on the number of sensors (in OED). It also +could be a sparsifying (possibly nondifferentiable) function, for example, ∥𝜻 ∥0, ∥𝜻 ∥1. +Generally speaking, we seek a design that maximizes the utility function U. Other formulations, however, involve +minimization of an OED optimality criterion; see, for example, [8]. Both formulations are adopted in the OED literature +and in PyOED and often are even equivalent, as explained below. In inverse problems, a design can be associated with +the observational configuration and thus can be used to optimally select an optimal observational policy. For example, +a design can be defined to select sensor location or temporal observation frequency that can help provide accurate +prediction with minimum uncertainty, or it can be defined to select an observational configuration that guarantees +maximum information gain from the data. +OED for sensor placement. In sensor placement we associate a binary design variable 𝜻𝑖 with the 𝑖th candidate sensor +location with 1 meaning activating the sensor and deactivating it otherwise. This defines the design as a binary vector +𝜻 ≡ 𝜻b ∈ {0, 1}𝑛s, which collectively define the observational configuration. In this case, the OED problem (13) takes +the form +𝜻opt = arg max +𝜻 ∈{0,1}𝑛s +U(𝜻) := Ψ(𝜻) − 𝛼Φ(𝜻) . +(15) +In Bayesian OED for sensor placement, the observation covariance Γnoise is replaced with a weighted version WΓ(𝜻), +resulting in the weighted data-likelihood +L(y|𝜃; 𝜻) ∝ exp +� +−1 +2 ∥F(𝜃) − y∥2 +WΓ (𝜻) +� +, +(16a) +where the weighted observation error covariance matrix takes the form +WΓ(𝜻) := +� +W(𝜻)ΓnoiseW(𝜻) +�† += LT(𝜻) +� +L(𝜻) +� +W(𝜻)ΓnoiseW(𝜻) +� +LT(𝜻) +�−1 +L(𝜻) , +(16b) +where † denotes the Moore–Penrose (pseudo) inverse and W(𝜻) := diag (𝜻) is a diagonal matrix with the binary design +𝜻 ∈ {0, 1}𝑛s on its diagonal. L(𝜻) is a sparse matrix that extracts nonzero rows/columns from the design matrix WΓ; +see [9] for further details. +The utility function . In the case of linear Bayesian inversion, the posterior is Gaussian with the covariance being +independent from the actual realizations of the data, as shown by (9) and (12). This fact enables designing observational +policies before actually deploying the observational sensors. Specifically, in linear Bayesian OED, we set the objective to +minimize a scalar summary of the posterior uncertainty, that is, the posterior covariance matrix. This is the underlying +principle of the alphabetical criteria [38]. For example, an A-optimal design is the one that minimizes the trace of the +posterior covariance matrix, and a D-optimal design is the one that minimizes its determinant (or equivalently the +log-determinant). Note that in the case of a linear model F, the Fisher information matrix FIM is equal to the inverse +of the posterior covariance matrix, that is, FIM = Γ−1 +post(𝜻) = F∗WΓ(𝜻)F + Γ−1 +pr . Thus, in this case the utility function— +discarding the penalty term—is set to U(𝜻) := Tr (FIM(𝜻)) for A-optimal designs and U(𝜻) := log det (FIM(𝜻)) for +D-optimal designs, and then the utility function is maximized. + +8 +Ahmed Attia and Shady E. Ahmed +When the model F is nonlinear, the FIM requires evaluating the TLM at the true parameter, that is, F = 𝜕F |𝜃=𝜃true. +Thus, to obtain an optimal design, one can iterate over finding the MAP estimate of 𝜃 and solving an OED problem +with Gaussian approximation around that estimate. Other utility functions employed in nonlinear OED problems or +non-Gaussian distributions include the Kullback–Leibler divergence between the posterior and the prior [29]. +Popular solution approaches. The OED problem (15) can be viewed as a mixed-integer program and can be solved by +using branch-and-bound [25, 31]. However, this type of research has not yet been applied to model-constrained-OED. A +common approach to solving (15) is to replace the binary optimization with the following relaxation: +𝜻opt = arg max +𝜻 ∈[0,1]𝑛s +U(𝜻) , +(17) +where the design variables are relaxed to take any values in the interval [0, 1] rather than only the binary values +{0, 1}. This approach, if carried out properly, has the effect of generating a continuous relaxation surface that connects +the values of the objective evaluated at the binary designs; see [9] for further details. A gradient-based optimization +approach is generally used to solve (17), which requires developing the gradient of both the optimality criterion Ψ and +the penalty Φ with respect to the design 𝜻. +As mentioned earlier, the penalty term is generally chosen to promote sparsity on the design. A popular penalty +function is based on the ℓ0 norm to promote design sparsity and is thus nonsmooth and consequently is nondifferentiable. +This difficulty can be alleviated, for example, by approximating the effect of ℓ0 with a sequence of differentiable functions +that converge in effect to the ℓ0; see, for example, [3]. In order to guarantee continuity of the relaxation surface, the +weighted precision matrix is defined in the general form +WΓ(𝜻) := +� +W(𝜻) ⊙ Γnoise +�† +, +W𝑖,𝑗 (𝜻) := + + +𝜔𝑖 𝜔𝑗 +; 𝑖 ≠ 𝑗 + + +0 +; 𝜔𝑖 = 0 +1 +𝜔2 +𝑖 +;𝜔𝑖 ≠ 0 +; 𝑖 = 𝑗 +; +𝑖, 𝑗 = 1, 2, . . . ,𝑛s , +𝑚,𝑛 = 1, 2, . . . ,𝑛𝑡 , +(18) +where ⊙ is the Hadamard (Schur) product of matrices and 𝜔𝑖 ∈ [0, 1] is a weight calculated by using 𝜻𝑖, for example, 𝜔𝑖 := +𝜻𝑖; see [9] for additional details. The formulation (18) guarantees that for 𝜻𝑖 ∈ [0, 1]𝑛s it holds that lim𝜻→𝜻 b WΓ(𝜻) = +WΓ(𝜻b) for a binary design 𝜻b ∈ {0, 1}𝑛s and thus guarantees continuity of the relaxation surface. Thus, the solution of +the relaxed OED optimization problem (17) is guaranteed to match the solution of the original binary OED optimization +problem. +The A- and D-optimal design relaxed optimization problems (17) take the following respective forms: +𝜻A−opt = arg max +𝜻 ∈[0,1]𝑛s +:= Tr +� +P +� +F∗� +Γnoise ⊙ W(𝜻) +�† +F + Γ−1 +pr +�−1 +P∗ +�−1 +− 𝛼Φ(𝜻) , +(19a) +𝜻D−opt = arg max +𝜻 ∈[0,1]𝑛s +:= log det +� +P +� +F∗� +Γnoise ⊙ W(𝜻) +�† +F + Γ−1 +pr +�−1 +P∗ +�−1 +− 𝛼Φ(𝜻) . +(19b) +The most important piece of information for solving the relaxation (17) is the gradient of the utility function; see, for +example, [9] for a detailed derivation of the gradients of the objective functions in (19). Gradient formulation, however, + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +9 +is mathematically involved and can be extremely computationally demanding because it requires numerous evaluations +of the forward operator, the goal operator, and the corresponding adjoints. Moreover, the penalty function Φ(·) is +required to be differentiable. +A stochastic learning approach to binary OED has been recently presented in [10], to solve the binary optimization +problem (15) without the need for relaxation. This approach does not require differentiability of the utility function U. +In this approach the optimal design is defined as +popt = arg max +p∈[0,1]𝑛s +E𝜻∼P(𝜻 |p) +� +U(𝜻) − 𝑏 +� +, +(20) +where P (𝜻 |p) is a multivariate Bernoulli distribution with parameter p specifying probabilities of success/activation of +each entry of 𝜻, that is, p𝑖 ∈ [0, 1]. Here 𝑏 is a constant “baseline” used to minimize variability of the stochastic estimate +of the gradient; see [10] for further details. Algorithm 1 summarizes the procedure followed to solve (20). +Algorithm 1 Stochastic optimization for binary OED with the optimal baseline. +Input: Initial distribution parameter p(0), step size schedule 𝜂 (𝑛), sample sizes Nens, 𝑚, baseline batch size 𝑏𝑚 +Output: 𝜻opt +1: initialize 𝑛 = 0 +2: while Not Converged do +3: +Update 𝑛 ← 𝑛 + 1 +4: +Sample {𝜻 [𝑗]; 𝑗 = 1, 2, . . . , Nens} ∼ P +� +𝜻 |p(𝑛)� +5: +Calculate 𝑏 = OptimalBaseline(p(𝑛), Nens, 𝑏𝑚) +6: +Calculate g(𝑛) = +1 +Nens +�Nens +𝑗=1 (J (𝜻 [𝑗] − 𝑏)) �𝑛s +𝑖=1 +� 𝜻𝑖 [𝑗 ] +p𝑖 ++ 𝜻 [𝑗 ]𝑖−1 +1−p𝑖 +� +e𝑖 +7: +Update p(𝑛+1) = L +� +p(𝑛) − 𝜂 (𝑛)𝑔(𝑛)� +8: end while +9: Set popt = p(𝑛) +10: +Sample {𝜻 [𝑗]; 𝑗 = 1, 2, . . . ,𝑚} ∼ P �𝜻 |popt�, and calculate J (𝜻 [𝑗]) +11: return 𝜻opt: the design 𝜻 with smallest value of J in the sample. +12: function OptimalBaseline(𝜃, Nens, 𝑏𝑚) +13: +Initialize 𝑏 ← 0 +14: +for 𝑒 ← 1 to 𝑏𝑚 do +15: +for 𝑗 ← 1 to Nens do +16: +Sample 𝜻 [𝑗] ∼ P (𝜻 |p) +17: +Calculate r[𝑗] = �𝑛s +𝑖=1 +� 𝜻𝑖 [𝑗 ] +p𝑖 ++ 𝜻 [𝑗 ]𝑖−1 +1−p𝑖 +� +e𝑖 +18: +end for +19: +Calculate d[𝑒] = +1 +Nens +�Nens +𝑗=1 r[𝑗] , and g[𝑒] = +1 +Nens +�Nens +𝑗=1 J (𝜻 [𝑗]) r[𝑗] +20: +Update 𝑏 ← 𝑏 + (g[𝑒])T d[𝑒] +21: +end for +22: +Update 𝑏 ← 𝑏 Nens / +� +𝑏𝑚 +�𝑛s +𝑖=1 +1 +p𝑖−p2 +𝑖 +� +23: +return 𝑏 +24: end function +Note that we do not provide an exclusive set of formulations or solution approaches in this study. We provide here +only an exemplary set of formulations and algorithms used to inspire the development of PyOED, which itself can be +used to test further formulations and algorithmic approaches. + +10 +Ahmed Attia and Shady E. Ahmed +3 +PYOED: STRUCTURE AND PHILOSOPHY +PyOED aims to provide a unified platform for implementing and testing model-constrained OED algorithmic approaches +including formalisms (15), (17), (19), and (20). Solving model-constrained OED and inverse problems requires proper +understanding and formulation of the underlying dynamical system, observational configuration, uncertainty models, +DA and inversion algorithms, and the OED objective [48] and the selected utility function. PyOED is a stand-alone, yet +extensible, Python package that provides users and researchers in the computational science and engineering disciplines +with a testing suite that effectively glues these components in an OOP fashion. For example, PyOED provides a variety +of time-dependent and time-independent simulation models. These include systems governed by linear algebraic +equations, ordinary differential equations, and PDEs. PyOED is also equipped with a set of classes implementing various +observational operators, probabilistic uncertainty models, and DA and OED methods. A high-level overview of the +PyOED major components and their coupling for solving DA and OED problems is provided in Figure 1. In Section 3.1 +we briefly describe the main components of PyOED and outline the functionality they provide in correspondence with +the diagram 1. +Fig. 1. High-level overview of the main components of PyOED. +3.1 +Code structure +Figure 2 shows the main subpackages (ordered alphabetically) shipped with the current version of PyOED (v1.0). The +rest of this section 3.1 provides a high-level description of the packages/subpackages of PyOED as displayed in Figure 2. +pyoed.models. Following the convention in the DATeS package [15], we use the word “model” to refer to three +entities: the simulation model, the observation model (or operator), and the error models. +The simulation model provides a prediction about the behavioral pattern of the physical phenomena of concern. In the +current version of PyOED (v1.0) we provide various simulation models under the pyoed.models.simulation_models, +including several versions of the Lorenz system [34] and advection-diffusion models. The structure of these prototyp- +ical simulation models should provide clear guidelines to practitioners willing to adopt PyOED for their particular +applications. + +Assimilation/lnversion +OED +Bayesian inversion, 3D-Var, 4D-Var, Kalman filtering, etc +Optimize +Simulation +Observation +Noisy Data +Acquisition +Model +Operator +Forward, adjoint, etc +Forward, adjoint, etc. +applylobserve, Jacobian, etc. +Data Sensors +Uncertainty/Error Model +Satellites, Doppler Lidar, etc. +prior, observation noise, etc.PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +11 +Fig. 2. Main subpackages (ordered alphabetically) available in the current version of PyOED (v1.0). +The observation operator maps the model state onto the observation grid, thus providing a functional mapping +between the model state and observational data. Two of the most prominent observation operators in experimen- +tal settings are the identity operator and an interpolator. PyOED provides implementations of several observa- +tion operators including these two, with an observational design properly incorporated to enable altering obser- +vational configurations at any point in the DA or OED solution process. Observation operators are provided in the +pyoed.models.observation_operators subpackage. +The error models quantify the uncertainty associated with the model parameter, model state, and observational data. +An experimental design can be associated with any of these pieces. For example, in sensor placement, an experimental +design is associated with the observational grid; thus, modifying the observational design affects the observational error +model. For example, in the relaxation approach (19), the design weights scale the entries of the covariance matrix, and +the stochastic approach (20) works by removing rows/columns of the observation error covariance matrix corresponding +to zero design variables. PyOED provides various implementations of error models suitable for modeling priors, as well +as observational errors in Bayesian inversion, where a design variable is consistently implemented to enable modifying +the experimental design during any step of the DA and/or OED solution process. The current version of PyOED (v1.0) +provides various error model implementations through the subpackage pyoed.models.error_models, including a +Gaussian model and Laplacian model. +pyoed.assimilation. PyOED provides a set of DA tools that include algorithms for “filtering” and “smoothing.” +These two terms are widely used in the DA literature. The former algorithm solves inverse problems that involve +time-independent or time-dependent simulation models, while the latter algorithm is restricted to time-dependent +models. Filtering involves prediction (of observation) using the parameter-to-observable map, followed by a correction + +filtering +assimilation +smoothing +examples +ml +error_models +models +observation operators +pyoed +oed +simulation models +optimization +stats +tutorials +utility12 +Ahmed Attia and Shady E. Ahmed +procedure to correct knowledge of the QoI given the observational data. In filtering for time-dependent simulations, the +observational data is assimilated sequentially, with one observation time per assimilation window/cycle. Examples of +filtering DA methods include three-dimensional variational DA, and Kalman filtering [6, 21]. Smoothing, on the other +hand, is concerned with history matching; these algorithms try to find the QoI that best matches multiple spatiotemporal +observations (a trajectory) and is usually defined as an initial value problem. Examples include space-time Bayesian +inversion [45] and four-dimensional variational DA [6], for which vanilla implementations are provided in PyOED. +Implementations of filtering DA algorithms are provided through pyoed.assimilation.filtering, and smoothing +algorithms are provided in pyoed.assimilation.smoothing. +pyoed.optimization. Numerical optimization routines are elementary for solving OED optimization problems, as +well as the variational approaches for solving DA problems. A variety of optimization software packages can be used +for solving numerical optimization problems including those described in this work. PyOED enables using external +optimization packages, including Python’s Scipy package, to solve DA and OED optimization problems. PyOED, +however, provides specific implementations of optimization procedures not available in popular optimization packages, +such as the stochastic algorithm described by Algorithm 1, various versions of the stochastic average approximation +(SAA) algorithm, and robust optimization [11]. +pyoed.ml. This subpackage is intended to provide implementations of machine learning algorithms useful for DA +and OED applications. For example, the stochastic learning approach to OED (20) can be seen as a reinforcement +learning (RL) approach to solving the OED problem. The module pyoed.ml.reinforcement_learning under this +package provides implementation of RL components, including an agent, a policy, transition probability, actions, and +utility functions. +pyoed.stats. This package aims to collect statistical procedures used by other parts of the package, such as sampling +routines, and implementation of random variables and their probabilistic utility functions including density evaluation +and log-probabilities. This version of PyOED (v1.0) provides an exemplary implementation of a multivariate Bernoulli +distribution required by the RL algorithm 1. Since statistical tools are crucial for various DA and OED algorithms, we +chose to keep the subpackage pyoed.stats rather than moving these implementations to other parts of the package. +This approach is advantageous because we continuously extend the package with various statistical tools, for example, +for randomized approximation methods for Bayesian inversion. +pyoed.oed. OED is the main component of PyOED that provides implementations of various algorithmic approaches +for solving OED problems, including relaxation (19) and stochastic learning (20), as well as recent developments +including robust OED [11]. Most implementations in this package take an inverse problem (DA object) as input and use +it to access all the underlying components, thus gaining access to the simulation model, error models, and observation +operator as well as the experimental design. This approach enables the user to modify an experimental design, solve +the DA problem if needed, and solve the underlying OED optimization problem. The core OED functionalities in most +PyOED routines, however, can be used with black-box utility functions, waiving the need for an inverse problem if +needed. +pyoed.utility. This subpackage aims to collect implementations of general-purpose functionality, such as file I/O +and visualization, as well as general mathematical and statistical procedures. The subpackage includes matrix-free + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +13 +implementations of expensive operations such as evaluating the trace and log-determinant of a matrix. It also provides +routines to approximate matrix trace using statistical randomization [5, 42]. +pyoed.examples. This subpackage provides various example scripts that users can follow to learn how to effectively +use various pieces of the package. The modules in this subpackage explain how to load all pieces of the subpackage +independently and explain how to properly coordinate these components to design a consistent DA and/or OED +experiment. +pyoed.tutorials. Given the popularity of Jupyter Notebooks in the computational science community, we converted +some of the examples in the subpackage pyoed.examples to Jupyter Notebooks and provided them in this subpackage +pyoed.tutorials. We employ them in the test cases presented in Section 4. We can add more tutorials on reasonable +demand. +3.2 +PyOED utilization workflow +While the components of DA and OED problems can be used independently of each other, some level of ordering is +mandatory for proper utilization. For example, an inverse problem (DA) object cannot be instantiated before a simulation +model, an observation operator, and error model objects. Similarly, an OED problem for Bayesian inversion cannot +be solved before creating an inverse problem. The general workflow for utilizing PyOED components is displayed in +Figure 3. A practical guide that illustrates how to follow this simple workflow is described in Section 4. +Fig. 3. Workflow describing initialization order and access level of PyOED components. +3.3 +Extending and contributing to PyOED +PyOED is meant to be extensible. Thus, we continuously interface other software tools that provide efficient imple- +mentations of the components of DA and OED problems. For example, hIPPYlib [50] is a software package for solving +high-dimensional inverse problems following an optimize-then-discretize approach. It has been employed to empirically +verify several inversion and OED algorithmic developments recently. Instead of rebuilding the functionality of hIPPYlib +and similar packages, we have interfaced with some of its components to show how easily and efficiently PyOED +can extend other successful packages. For example, PyOED interfaces with finite-element (FE) implementations of + +Models +Simulation model +Observation operator +Error models +Inverse problem (DA) +Data +Design +Prior +Observation noise +IP & OED +Inversion (analysis/posterior) +OED (optimal design)14 +Ahmed Attia and Shady E. Ahmed +advection-diffusion and Poisson models from hIPPYlib as well-as point observation operators. However, such extension +does not hinder the functionality or limit the extensibility of PyOED. Specifically, interfacing with such external +packages is optional and is not provided in the core of PyOED, mainly because the backbone of these packages is not +guaranteed not to be quickly outdated or be unmaintained. Thus, these extensions (e.g., interfacing with hIPPYlib) +are made optional during the import process of PyOED subpackages, and dependent functionality is used only when +properly installed and available on the current architecture. +3.4 +Code availability +The development version of PyOED is available from https://gitlab.com/ahmedattia/pyoed. +4 +TEST CASES +PyOED comes with a set of prototypical test problems with increasing complexity for both DA and OED. An ideal case +typically used in scientific publications is the linear Gaussian setup, where the simulation model and the observation +operator are both linear and the error models (observation noise and the prior) are both Gaussian; see Section 2. In +this case, the posterior is also a Gaussian; the solution of the inverse problem is unique—the posterior mean and mode +(MAP) are identical; and posterior moments (mean and covariance) both have closed forms that can be obtained by +applying the Kalman filter theory. Such a simplified setup can be used for testing new formulations in both DA and +OED, and thus it is provided in PyOED. We discuss this formulation and in Section 4.1 show how it can be utilized. In +Section 4.2 we discuss in further detail a standard experiment widely used in OED scientific research and offered by +PyOED. +4.1 +An ideal setup: linear Gaussian toy problem +Consider a time-dependent forward problem defined at time instances 𝑡0 + 𝑖Δ𝑡,𝑖 = 0, 1, . . . ,𝑛𝑡, for a fixed step size Δ𝑡, +as follows: +u𝑛 = A u𝑛−1 , +y𝑛 = Iu𝑛 + 𝛿 , +𝑛 = 1, 2, . . . , +(21) +where u𝑛 ∈ RNstate is the discrete model state at time instance 𝑡𝑛, A ∈ RNstate×Nstate is a matrix representing model +evolution over time interval [𝑡𝑛−1,𝑡𝑛], and I is the identity observation operator/matrix. If we assume 𝛿 ∼ N (0, R) and +assume a Gaussian prior u0 ∼ N +� +upr +0 , Γpr +� +, then the posterior is Gaussian N +� +upost +0 +, Γpost +� +with +Γpost = +� +ATR−1A + Γpr−1�−1 +, +upost +0 += Γpost +� +Γpr−1upr +0 + +𝑛𝑡 +∑︁ +𝑖=1 +ATR−1 y +� +. +(22) +Since (22) is a closed form of the posterior, we can use it to test and debug new DA and OED implementations. +This fact is highly utilized in the unit tests developed in PyOED. To create a proper experiment, we will follow the +workflow described by Figure 3. In the rest of this section (4.1) we describe how to initialize an inverse problem in +PyOED with the settings (22), and we provide a simple scheme that can be followed to initialize other experiments. +The code summarized here is provided in the pyoed.examples.fourDVar_driver module with additional comments, +details, and capabilities that can help the user understand the workflow for creating and solving an inverse problem. A + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +15 +Jupyter Notebook pyoed.tutorials.toy_linear is also available and can be used to regenerate the numerical results +presented in this section (4.1). +Creating the models. Assuming PyOED is already in the Python path, the first step is to import/load the simulation +model (that describes A), the observation operator (here an identity operator), and the error models to create the prior +and the observation error model. This can be done as described in the code snippet 1. +1 from pyoed.models.simulation_models.toy_linear import ToyLinearTimeDependent +2 from pyoed.models.observation_operators.identity import Identity +3 from pyoed.models.error_models.Gaussian import GaussianErrorModel +Snippet 1. Import essential modules to create the simulation model object, the observation operator, the prior, and the observation +error model. +Note that we have imported only the classes we need in this example. However, PyOED provides several other +implementations of the simulation models, observation operators, and error models. In order to create a simulation +model, an object of ToyLinearTimeDepndent is instantiated as described by snippet 2. This generates an internal +two-dimensional array of size 5 × 5 that represents the forward model A, which integrates the model state forward +by a timestep 𝑑𝑡 = 0.1. That internal array can be reproduced by setting the random_seed parameter in the passed +configurations dictionary. +1 model = ToyLinearTimeDependent(configs ={'nx':5, 'dt':0.1, 'random_seed ':123}) +Snippet 2. Instantiate the simulation model object. +Since each simulation model has its own configurations, many of which are assigned default values, we follow the +strategy of DATeS [15] and use dictionaries to pass model arguments. PyOED aggregates and validates the passed +dictionary against the default values and initiates the model accordingly. For example, in snippet 2 we specify a +random_seed argument that guarantees reproducibility of any randomly generated data inside the model object. +This is done by keeping an internal random state inside the model object that is independent from other objects +and is initialized to the passed random seed. Thus, if no random seed is passed, each time the same model object +is instantiated, completely random sequences will be generated if requested. Implementations of simulations model +must provide an implementation of a class-level method get_default_configs that returns a dictionary with all +default values used if not passed upon instantiation. In order to ensure that, all PyOED simulation models inherit +the class pyoed.models.simulation_models.SimulationModel that guarantees enforcing the implementation of +mandatory methods required for the seamless integration of various components in PyOED. The final configurations +of a simulation model is a combination of those in the passed configurations dictionary and the default values, with +precedence given to the passed configurations. The simulation model’s configurations (a copy of it, in fact) can be +accessed through the attribute configurations. We generally choose to return a copy to guarantee that all settings +are validated before modification. For example, one cannot modify the time step 𝑑𝑡 without verifying whether the +timestepping implementation is tied to that time step or not and updating dependencies accordingly. +The observation operator. Similar to simulation models, an observation operator is created by passing the settings in +the configurations dictionary to the observation operator class constructor as shown in snippet 3. Thus, the observation +operator has access to the model grid and other useful attributes to create and manipulate data structures (such as the + +16 +Ahmed Attia and Shady E. Ahmed +model state) without having to provide any new implementations. This is mainly because observations in this case are +the same as the corresponding model states (discarding observation noise). +1 +obs_oper = Identity(configs ={'model ':model }) +Snippet 3. Create an identity observation operator. +The prior and the data noise models. The error models (prior and observation noise) are created in this example as +shown in snippet 4. Note that the random_seed configurations variable is used to set the random number generator for +reproducibility. This enables us regenerate a set of experiments and generate proper benchmarks for a fair comparison +between various implementations. As mentioned earlier, if no random seed is passed, each instance of the error model +is assigned a randomly generated state that guarantees that each instance has its own different sequence of random +numbers/vectors realizations. +1 prior = GaussianErrorModel(configs ={'size':model.state_size , 'mean':1, 'variance ':1, 'random_seed ':1}) +2 +obs_noise = GaussianErrorModel(configs ={'size':obs_oper.shape [0], 'variance ':0.01 , 'random_seed ':1}) +Snippet 4. Create the prior and the observation error model. +The inverse problem (DA) object. The next step is to put these models together in action and use them to create an +inverse problem. We illustrate the utilization of a DA object to solve the inverse problem following a 4DVar formulation. +The literature provides a plethora of variants of the general 4DVar scheme. PyOED provides a few implementations; +however, the most basic (vanilla) implementation is used here for illustration. Two approaches are followed in PyOED +for instantiating a DA object. The first is to pass all configurations (upon initialization) in the configurations dictionary +configs similar to the case of simulation models, error models, and observation operators. The second approach is +to use the proper registration methods associated with the created object after instantiation. The latter can be also +used to update components of the DA object after initialization. For example, one might want to change the settings of +the assimilation time window, register new observations or remove the old ones, or modify or even change the prior. +Since the first approach has already been explained with the simulation and error models, we demonstrate the second +approach here. Specifically, a 4DVar assimilation object is created as in snippet 5. +1 +inverse_problem = VanillaFourDVar () +2 +inverse_problem.register_model(model) +3 +inverse_problem.register_observation_operator(obs_oper) +4 +inverse_problem.register_prior_model(prior) +5 +inverse_problem.register_observation_error_model(obs_noise) +Snippet 5. Create the inverse problem object with default settings, and then add (register) all the pieces created above, that is, the +simulation model, the observation operator, the prior, and the observation error model. +The next step is to register observational data (along with observation times). A standard strategy in experimentation +is to create synthetic data from a ground truth (known as a twin experiment). This is explained by snippet 6, where we +define the assimilation timespan (window) to be the interval [0, 0.3], and the observations are taken at 3 time instances +0.1, 0.2, 0.3. The observational data is mimicked by adding random noise (using the observation error model) to the +observed ground truth at the corresponding observation time instance. + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +17 +1 # Set the assimilation/simulation time window +2 tspan = (0, 0.3) +3 +inverse_problem.register_assimilation_window(tspan) +4 +5 # Create truth (true initial state and trajectory) +6 +true_IC = model.create_initial_condition () +7 +checkpoints = [0.1, 0.2, 0.3] +8 _, true_traject = model.integrate_state(true_IC , tspan=tspan , checkpoints=checkpoints) +9 +10 # Create synthetic data (perturbation to truth) and register them +11 for t, state in zip(checkpoints , true_traject): +12 +obs = obs_noise.add_noise(obs_oper(state)) +13 +inverse_problem.register_observation(t=t, observation=obs) +Snippet 6. Create synthetic noisy observations, and register all observation time points and observational data to the inverse problem +object. +The final step in the DA procedure is to solve the inverse problem and assess the quality of the solution. For this setup, +we know the ground truth, and thus one can evaluate the root mean squared error (RMSE), which is a standard error +metric in statistics in the DA literature. In order to solve the inverse problem, the solve_inverse_problem method of +the 4DVar DA object is called (snippet 7). This method will raise an instructive error if any of the essential elements, for +example, the simulation model, are not registered properly. Note that this function is flexible and allows the posterior +covariance to be constructed if needed. It also allows waiving finding the MAP estimate, which can be advantageous +in OED applications due to associated computational savings. For example, one might want to estimate the posterior +covariance in the linear Gaussian case without evaluating the MAP. +1 +inverse_problem.solve_inverse_problem(init_guess=prior.mean , update_posterior=True) +Snippet 7. Solve the inverse problem. The optimization initial point is set by default to the prior mean; however, it can be modified if +a better initial guess is known. Here, the posterior covariance is evaluated, and consequently the posterior is updated with both the +mean and the covariance matrix. +PyOED provides several utility functions to evaluate statistics, such as the RMSE, which can be used to quantify the +accuracy of the inverse problem solution. Snippet 8 shows how to call the utility function calculate_rmse and use it +to evaluate the prior and the analysis (posterior) RMSE, which are then printed. +1 from pyoed import utility +2 +prior_rmse = utility.calculate_rmse(true_IC , inverse_problem.prior.mean) +3 +posterior_rmse = utility.calculate_rmse(true_IC , inverse_problem.posterior.mean) +4 print(f"Prior RMSE: {prior_rmse}") +5 print(f"Posterrior RMSE: {posterior_rmse}") +Snippet 8. Calculate and print the RMSE values associated with the prior mean (initial guess here) and the posterior mean. +The same procedure can be easily followed to inspect the RMSE results over the whole assimilation timespan as +described by Snippet 9 with results plotted in Figure 4. +1 checkpoints , true_traject = model.integrate_state(true_IC , tspan=tspan) +2 _, prior_traject = model.integrate_state(inverse_problem.prior.mean , tspan=tspan) +3 _, posterior_traject = model.integrate_state(inverse_problem.posterior.mean , tspan=tspan) +4 +prior_rmse = [utility.calculate_rmse(xp, xt) for xp , xt in zip(prior_traject , true_traject)] + +18 +Ahmed Attia and Shady E. Ahmed +5 +posterior_rmse = [utility.calculate_rmse(xp , xt) for xp , xt in zip(posterior_traject , true_traject)] +Snippet 9. Generate RMSE over the whole assimilation window. +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Time +0 +10 +20 +30 +40 +50 +60 +RMSE +Prior +Posterior +Fig. 4. RMSE results of the solution of the inverse problem presented in Section 4.1. The RMSE results of both the prior and the +posterior trajectories plotted here are obtained by running the code in snippet 9. +One can also analyze the posterior covariances, for example, by generating and plotting the posterior covariance +matrix. Given the linear Gaussian settings in the present setup, one can validate the generated posterior covariance +matrix against the exact formula (22). One way to construct the posterior covariance matrix is to invoke the posterior +model as shown in snippet 10. +1 +post_cov = inverse_problem.posterior.covariance_matrix () +Snippet 10. Construct and retrieve the posterior covariance matrix +Note, however, that one should avoid constructing the covariance matrix for high-dimensional error models. Alterna- +tively, matrix-free implementations of covariance (and precision) matrix-vector product should be used. For example, to +multiply the prior covariance by state, one should call prior.covariance_matvec(state). The error models provide +many attributes to efficiently access the statistics of the model, such as covariance diagonal, and trace. The posterior +covariance matrix, constructed by employing the posterior functionality as in snippet 10 and the covariance matrix +evaluated by applying (22) along with the mismatch errors are plotted in Figure 5. +0.0 +2.5 +0.0 +2.5 +0.0 +2.5 +0.0 +2.5 +0.0 +2.5 +0.0 +2.5 +−0.05 +0.00 +0.05 +−0.05 +0.00 +0.05 +1 +2 +3 +×10−12 +Fig. 5. Entries of the posterior covariance matrix and the associated errors. Left: the posterior covariance matrix generated by +snippet 10. Middle: the closed-form posterior covariance matrix given by (22). Right: RMSE obtained by pointwise comparison of the +covariance matrices obtained by solving the inverse problem (left) and by using the closed form (middle). + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +19 +An OED experiment. The simulation model is instantiated with a model grid of size nx=5, and the observation operator +copies the model state. In fact, one can inspect the model array representation A for this toy linear model by calling +model.get_model_array(). Both the model state and the observation vector sizes here are 5. Thus, there are actually +5 candidate sensor locations (observation gridpoints), and one can try to find the optimal subset of sensors by using an +OED implementation. +Here we briefly illustrate utilizing an OED object to find the A-optimal design for the toy linear example discussed +above; see snippet 11. First, the proper OED module (here following [9]) is imported and is used to create the oed_problem +instance. The A-optimality criterion is registered (which can be changed later by registering a proper OED criterion), +and the OED problem is then solved. The results of solving the OED problem, for example, the optimal observational +design, are then stored in oed_result, which is an instance of (or derived from) the pyoed.oed.OEDResults class. +This class provides access to various attributes of the OED problem and the solution process (such as the optimization +trajectory and brute force solution if requested). This gives a taste of how simple it is to create and test DA and OED +problems in PyOED. Further details on OED implementations in PyOED are discussed in the following section (4.2). +1 from pyoed.oed.relaxed_oed import RelaxedOED +2 +oed_problem = RelaxedOED(inverse_problem=inverse_problem , problem_is_linear=True) +3 +oed_problem.register_optimality_criterion('A-opt') +4 +oed_results = oed_problem.solve_oed_problem () +Snippet 11. Create an OED object, and solve the OED optimization problem for the toy linear model. +4.2 +A standard model-constrained OED experiment +Parameter identification for an advection-diffusion (AD) model is the foundation of an experiment widely used in the +model-constrained OED literature for validating theoretical developments; see, for example, [4, 8, 19, 30]. Comparing +independent scientific OED developments is admittedly hard, mainly because of the lack of availability of open software +packages developed for OED. This is one of the main goals and features of PyOED. Specifically, PyOED will enable OED +researchers to compare the performance of new OED algorithmic approaches with other methods. Moreover, it enables +comparison with solution by brute force search for small- to moderate-dimensional problems. In this section 4.2 we +describe in detail the steps required to construct and solve an OED problem in PyOED with an AD simulation model. +This problem has been utilized independently in several OED developments; see, for example, [4, 8–10, 19]. Here, we +show how PyOED can be used to solve and benchmark this OED problem, thus providing a starting point for utilizing +and developing multiple approaches for solving OED problems in general in PyOED. Following the same approach as +in 4.1, we start by describing the components of the inverse problem and briefly show how they are initialized in PyOED; +then we initialize and solve the OED problem using the efficient stochastic approach summarized by Algorithm 1. +Additionally, we discuss the steps that should be modified to utilize other solution formulations and methods such as +the relaxation approach (17). +The code summarized here is provided in the pyoed.examples.OED_AD_FE module with additional comments, details, +and capabilities. A Jupyter Notebook pyoed.tutorials.OED_AD_FE is also available and can be used to regenerate the +numerical results presented in this section (4.1). + +20 +Ahmed Attia and Shady E. Ahmed +The simulation model. The governing equation of the contaminant field 𝑢 = 𝑢(x,𝑡) is modeled by the following AD +model equations with the associated boundary conditions: +𝑢𝑡 − 𝜅Δ𝑢 + v · ∇𝑢 = 0 +in D × [0,𝑇], +𝑢(𝑥, 0) = 𝜃 +in D, +𝜅∇𝑢 · n = 0 +on 𝜕D × [0,𝑇], +(23) +where 𝜅 > 0 is the diffusivity, 𝑇 is the simulation final time and v is the velocity field. The domain is D := (0, 1) × (0, 1) +with two rectangular regions modeling two buildings inside the domain. The velocity field v is known exactly and is +obtained by solving a steady Navier–Stokes equation, with the side walls driving the flow, as detailed in [9, 36]. +To create a simulation model object implementing (23), ground truth of the initial condition, and plot the domain +(with finite elements discretization) as well as the velocity field, one can use Snippet 12; the output is shown in Figure 6. +1 # Create the simulation model (AD with FE discretization) +2 from pyoed.models.simulation_models import fenics_models +3 +model_timestep = 1.0 +4 model = fenics_models.create_AdvectionDiffusion2D_model(dt=model_timestep) +5 +6 # Ground truth of the inversion parameter (initial condition here) +7 +true_model_state = model.create_initial_condition () +8 +9 # Plot the domain +10 model.plot_domain () +11 model.plot_velocity_field () +Snippet 12. Create an object representing the simulation model (23). +Fig. 6. Left: finite elements discretization of the domain D of the AD problem (23). Right: the velocity field v. +The prior. In this setup, following [8, 36, 49], we choose a Laplacian prior of the parameter 𝜃 is N �𝜃pr, Γpr +�, with Γpr +being a discretization of A−2, where A is a Laplacian operator. In PyOED, a Laplacian prior can be created as described +in snippet 13. +1 from pyoed.models.error_models import Laplacian +2 +configs = dict(Vh=model.parameter_dof , +3 +mean=model.parameter_vector(init_val =0.5) , + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +21 +4 +gamma =1.0, +5 +delta=16, +6 +random_seed =123, ) +7 prior = Laplacian.DolfinBiLaplacianErrorModel(configs) +Snippet 13. Create a Laplacian prior. +The observation operator. A common observational configuration is to consider uniformly distributed candidate +sensor locations and solve an OED problem to choose the optimal subset of candidate sensor locations. A uniform +observation operator can be created and incorporated in this problem as described in snippet 14. +1 from pyoed.models.observation_operators.fenics_observation_operators import( +2 +create_pointwise_observation_operator , +3 ) +4 +num_candidate_sensors = 10 +5 +obs_oper = create_pointwise_observation_operator(model=model , num_obs_points=num_candidate_sensors , ) +Snippet 14. Create a uniform observation operator with 10 candidate locations. +Assuming Gaussian observational noise model, a Gaussian observation error model is created as described in 15 +1 from pyoed.models.error_models.Gaussian import GaussianErrorModel +2 +obs_noise = GaussianErrorModel( +3 +configs ={'size':obs_oper.shape[0], 'variance ':0.1, 'random_seed ':2345} , +4 ) +Snippet 15. Create a Gaussian noise model. +The inverse problem: 4DVar. As with the case of the toy linear problem described above in Section 4.1, the elements +of the inverse problem here can be created as described by snippet 16. Similarly, synthetic observations (data) can be +created as in snippet 17. Note that all steps followed so far in this example are similar to those followed in the case of +the toy linear model discussed in Section 4.1. +1 +import numpy as np +2 from pyoed.assimilation.smoothing import fourDVar +3 +checkpoints += np.arange(0, model_timestep *(5.5) , model_timestep) +4 +DA_configs += dict(assimilation_window =( checkpoints [0], checkpoints [-1]), +5 +model=model , +6 +prior_model=prior , +7 +observation_operator=obs_oper , +8 +observation_error_model=obs_noise , ) +9 +inverse_problem = fourDVar.VanillaFourDVar(configs=DA_configs) +Snippet 16. Create the DA object. +1 # Create and register observations (perturb observation from true model trajectory) +2 obs_times , true_obs = model.integrate_state(true_model_state , +3 +tspan =( checkpoints [0], checkpoints [-1]), +4 +checkpoints=checkpoints [1: ], +5 +) +6 # Perturb with noise and register with the inverse problem +7 for t, y in zip(obs_times , true_obs): + +22 +Ahmed Attia and Shady E. Ahmed +8 +y += obs_oper.apply(y) +9 +yobs = obs_noise.add_noise(y) +10 +inverse_problem.register_observation(t=t, observation=yobs) +Snippet 17. Create synthetic observations, and associate them to the inverse problem object. +The OED problem. The discussion above is valid for all model-constrained OED approaches in PyOED. In what +follows, we describe the steps needed to create an OED object that follows the stochastic approach (20). An OED object +is created in PyOED by following the steps in snippet 18. Here, we seek to activate only 4 sensors out of the candidate +10, and we use the trace of the Fisher information matrix (FIM) as the utility function. To enforce the budget, we use an +ℓ0 penalty term as detailed in [10]. +1 # Create OED problem (stochastic formulation) +2 from pyoed.oed.binary_oed import BinaryOED +3 +oed_problem = BinaryOED(inverse_problem=inverse_problem , problem_is_linear=True ,) +4 +5 # Register the utility function: trace of the FIM (A-optimality) +6 +oed_problem.register_optimality_criterion('A-opt') +7 +8 # Register penalty/regularization term (with desired budge of only 4 active sensor) +9 +penalty_f = lambda design: np.abs(np.count_nonzero(design) - 4) +10 +oed_problem.register_penalty_term( +11 +penalty_function=penalty_f , +12 +penalty_weight =-1e+8, +# Negative as the objective (trace of the FIM below) +13 ) +Snippet 18. Create an OED object implementing the stochastic approach 1. +The OED problem can be solved as described by snippet 19. +1 +oed_results = oed_problem.solve_oed_problem( +2 +oed_evaluation_method='randomized ', +# randomized approximation of FIM trace +3 +learning_rate =1e-10, +4 +batch_size =32, +5 +bruteforce=True , +# To compare the solution to search by enumeration (bruteforce search) +6 ) +Snippet 19. Solve the OED problem. +The resulting oed_results object can be used to generate several analysis plots as described by the code in snippet 20. +This generates multiple standard plots, including the performance of the optimization algorithm over consecutive +iterations in Figure 7(left), the optimal sensor locations generated by the optimization algorithm in Figure 7(middle), +and comparison of the quality of the solution with respect to brute force search shown in Figure 7(right). +1 # Create standard plots of the OED results +2 +oed_problem.plot_results(oed_results) +Snippet 20. Create standard plots for assessing the performance of the optimization routine and the quality of the generated design. +To use other OED formulations to solve the same problem, the user only needs to update the code in the snippet 18 +with the proper OED implementation. For example, the relaxation approach (17) can be used as illustrated in the case of + +PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments +23 +0 +100 +200 +300 +400 +500 +600 +700 +800 +Optimization Iteration +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +Objective Value +×108 +U +0 +150 +300 +450 +600 +750 +900 +1050 +Experimental Designs +−3 +−2 +−1 +0 +1 +2 +3 +Objective Value +×108 +Fig. 7. Subset of the plots generated by running the code in snippet 20. Left: value of the utility (objective) function, i.e., the penalized +OED criterion, over consecutive iterations of the optimization algorithm. Middle: optimal solution, showing optimal sensor locations +in the domain. Right: value of the objective of the optimal solution (red star) returned by algorithm 1, compared with the global +optimum solution (black 𝑥 mark), and all possible solutions marked as blue circles; the x-axis shows the indexes of all possible binary +designs from 1 to 2𝑛s=10 = 1024, and the y-axis shows the corresponding values of the optimization objective. +the linear toy model above in 4.1; see snippet 11. Specifically, the relaxation approach (17) can be used to solve the +present optimal sensor placement problem by replacing the code in snippet 18 with the following code in snippet 21, +which demonstrates the simplicity of PyOED interface. Results of snippet 21 are omitted from the presentation here for +clarity and because the main goal here is to discuss usage of the approaches in PyOED rather than assessing the quality +of the solution approach, which is left for interested users of the package and for future benchmarking research. +1 # Formulate and solve using the relaxation approach +2 from pyoed.oed.relaxed_oed import PointwiseRelaxedOED +3 +oed_problem = PointwiseRelaxedOED(inverse_problem=inverse_problem , problem_is_linear=True , ) +4 +oed_problem.register_optimality_criterion('A-opt') +5 +6 # Add penalty (differentiable function and gradient) +7 +penalty_f_l2 = lambda design: np.power(np.sum(design)-budget , 2) +8 +penalty_f_l2_grad = lambda design: 2 * (np.sum(design)-budget) * np.ones_like(design) +9 +oed_problem.register_penalty_term( +10 +penalty_weight =1, +11 +penalty_function=penalty_f_l2 , +12 +penalty_function_gradient=penalty_f_l2_grad , ) +13 +14 # Solve the OED problem +15 +oed_results = oed_problem.solve_oed_problem(oed_evaluation_method='randomized ', ) +Snippet 21. Create an OED object implementing the relaxation approach (17). +Note, however, that we had to change the penalty function in snippet 21 because the ℓ0 penalty function used in +snippet 18 is non differentiable, while the relaxation approach requires the OED objective function to be differentiable. +For details, see, for example, [10]. +5 +CONCLUDING REMARKS +This work describes PyOED, a highly extensible high-level software package for OED in inverse problems and DA. PyOED +aims to be a comprehensive Python toolkit for model-constrained OED. The package targets scientists and researchers +interested in understanding the details of OED formulations and approaches. It is also meant to enable researchers + +24 +Ahmed Attia and Shady E. Ahmed +to experiment with standard and innovative OED technologies within external test problems (e.g., simulations). The +mathematical formulations of OED, inverse problems, and DA overlap significantly, and thus, we plan to extend PyOED +with a plethora of Bayesian inversion, DA, and OED implementations as well as new scientific simulation models, +observation error models, and observation operators. +While we focused the discussions in this paper on specific OED approaches, the current version PyOED (v1.0) provides +several other implementations and emphasizes implementing the essential infrastructure that enables combininig DA +and OED elements with other parts of the package. The main limitation of the initial version of PyOED is scalability. +Specifically, the concept is developed without parallelization capability. In future versions of PyOED, scalability will be +achieved by adding message passing interface (MPI) support, for example using the mpi4py package, and by supporting +PETSc [17]. Performance will also be enhanced by converting or rewriting suitable parts of the package in Cython. +ACKNOWLEDGMENTS +This material is based upon work supported by the U.S. Department of Energy, Office of Science, under contract number +DE-AC02-06CH11357. This work was supported in part by the U.S. Department of Energy, Office of Science, Office +of Advanced Scientific Computing Research and Office of Nuclear Physics, Scientific Discovery through Advanced +Computing (SciDAC) Program through the FASTMath Institute. The work of SEA was supported by Argonne National +Laboratory during his appointment as a 2021 Wallace Givens Associate. +REFERENCES +[1] Alen Alexanderian. 2020. Optimal Experimental Design for Bayesian Inverse Problems Governed by PDEs: A Review. arXiv preprint arXiv:2005.12998 +(2020). +[2] Alen Alexanderian. 2021. Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: a review. 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Optimal sensor location for parameter estimation of distributed processes. International Journal of Control 73, 13 (2000), +1235–1248. + +26 +Ahmed Attia and Shady E. Ahmed +[49] U. Villa, N. Petra, and O. Ghattas. 2016. hIPPYlib: An extensible software framework for large-scale deterministic and linearized Bayesian inversion. +(2016). http://hippylib.github.io. +[50] Umberto Villa, Noemi Petra, and Omar Ghattas. 2018. hIPPYlib: An extensible software framework for large-scale inverse problems. Journal of Open +Source Software 3, 30 (2018). +[51] Curtis R Vogel. 2002. Computational methods for inverse problems. SIAM. +[52] Bob Wheeler and Maintainer Jerome Braun. 2019. Package ‘AlgDesign’. R Proj. Stat. Comput 1, 0 (2019), 1–25. +The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne +National Laboratory (“Argonne"). Argonne, a U.S. Department of Energy Office of Science labo- +ratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for +itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in +said article to reproduce, prepare derivative works, distribute copies to the public, and perform +publicly and display publicly, by or on behalf of the Government. The Department of Energy +will provide public access to these results of federally sponsored research in accordance with +the DOE Public Access Plan. http://energy.gov/downloads/doe-public-access-plan. + diff --git a/etE_T4oBgHgl3EQf1hz1/content/tmp_files/load_file.txt b/etE_T4oBgHgl3EQf1hz1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..221f080180faa7b1f036099ed99879c3b850b3c9 --- /dev/null +++ b/etE_T4oBgHgl3EQf1hz1/content/tmp_files/load_file.txt @@ -0,0 +1,1053 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf,len=1052 +page_content='PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments AHMED ATTIA∗, Mathematics and Computer Science Division, Argonne National Laboratory, USA SHADY E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' AHMED†, Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Labora- tory, USA This paper describes the first version (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0) of PyOED, a highly extensible scientific package that enables developing and testing model-constrained optimal experimental design (OED) for inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Specifically, PyOED aims to be a comprehensive Python toolkit for model-constrained OED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The package targets scientists and researchers interested in understanding the details of OED formulations and approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' It is also meant to enable researchers to experiment with standard and innovative OED technologies with a wide range of test problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', simulation models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' OED, inverse problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', Bayesian inversion), and data assimilation (DA) are closely related research fields, and their formulations overlap significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, PyOED is continuously being expanded with a plethora of Bayesian inversion, DA, and OED methods as well as new scientific simulation models, observation error models, and observation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' These pieces are added such that they can be permuted to enable testing OED methods in various settings of varying complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The PyOED core is completely written in Python and utilizes the inherent object-oriented capabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' however, the current version of PyOED is meant to be extensible rather than scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Specifically, PyOED is developed to “enable rapid development and benchmarking of OED methods with minimal coding effort and to maximize code reutilization.” PyOED will be continuously expanded with a plethora of Bayesian inversion, DA, and OED methods as well as new scientific simulation models, observation error models, and observation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This paper provides a brief description of the PyOED layout and philosophy and provides a set of exemplary test cases and tutorials to demonstrate how the package can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Additional Key Words and Phrases: Optimal experimental design, OED, inverse problems, data assimilation 1 INTRODUCTION Recently, interest has increased in developing scalable data assimilation (DA) and uncertainty quantification methodolo- gies for solving large-scale inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' An inverse problem refers to the retrieval of a quantity of interest (QoI) associated with or stemming from a physical phenomenon underlying partial noisy experimental or observational data of that physical system [7, 44, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The QoI could be, for example, the model state, initial condition, or other physics quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Inverse problems are prominent in a wide spectrum of applications including power grids and atmospheric numerical weather prediction [26, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In these problems, the prediction of the physical phenomena is often formulated as an initial value problem, while the initial condition of the simulator is corrected by fusing all available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Algorithmic approaches for solving inverse problems seek either a single-point estimate of the target QoI or a full probabilistic description of the knowledge about the QoI given all available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The underlying principle of these methods is that information collected from observational systems is fused into computational models, along with associated uncertainties, to produce an accurate estimate of the ground truth of the physical phenomena of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' ∗The first author led the development of PyOED and this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' †The second author developed a set of simulation models for initial testing of PyOED, and contributed to the introduction and revision of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Authors’ addresses: Ahmed Attia, Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, Illinois, USA, 60349, aattia@anl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='gov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed, Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Laboratory, Richland, Washington, USA, 99352, shady.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='ahmed@pnnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='08336v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='MS] 19 Jan 2023 2 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed In the former approach seeking a single QoI estimate, the solution of an inverse problem is obtained by solving an optimization problem with an objective to minimize the mismatch between observational data and model simulations, possibly regularized by prior knowledge and uncertainty models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The latter approach, commonly known as Bayesian inversion, seeks to characterize the probability distribution of the QoI through the posterior formulated by applying Bayes’ rule, that is, the probability distribution of the QoI conditioned by all available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' DA methods [12–16, 18, 20, 26, 35, 43] aim to solve large- to extreme-scale inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' They work by fusing information obtained from multiple sources, such as the dynamical model, prior knowledge, noisy and incomplete measurements, and error models, in order to better estimate the state and parameters of the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This estimate improves the predictability of the simulation systems developed to make future predictions about the physical phenomena of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The quality of DA systems, and hence the accuracy of their predictions, is heavily influenced by the extent to which the mathematical assumptions reflect reality and depends on the quality of the collected measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Optimal data acquisition is the problem of determining the optimal observational strategy, for example, from a large set of candidate observational schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This problem is widely formulated as an optimal experimental design (OED) problem [22, 39], where the design parameterizes and thus determines the observational configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In an OED problem, a design is defined to characterize a candidate configuration or a control, and the quality of the design is quantified by using a utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The optimal design is then defined as the one that maximizes this utility function or, equivalently, minimizes some OED criterion [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Since the aim of Bayesian inference is to estimate the QoI posterior, Bayesian OED seeks an observational configuration that, when combined with the underlying dynamics, would maximize information gain from the data or minimize the posterior uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, an optimal design is found by solving an optimization problem with the objective to maximize a utility function that quantifies the quality of the design and its influence on the solution of the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' OED for inverse problems has experienced a recent surge in interest by the scientific computing community;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see, for example, [2] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Numerical testing and experiments are critical for developing efficient OED formulations and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This process is elementary for successful scientific research in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Although statisticians have developed a plethora of mathematical formulations and algorithmic approaches for general-purpose OED algorithms, most of the available and publicly accessible OED software tools are limited to idealized formulations and specific applications such as finding optimal collocation points for regression problems or designing clinical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In addition, they are written in the R programming language or MATLAB, thus limiting code reutilization and accessibility by a wider audience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see, for example, [24, 40, 41, 46, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Unfortunately, these tools do not align well with the interests of the increasingly large computational science community for developing new OED formulations and algorithmic approaches for inverse problems, DA, and model-constrained OED [4, 8, 19, 23, 27, 28, 30, 37, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' As a first step in alleviating this limitation, we present and describe PyOED, a highly extensible open-source Python package written mainly to enable computational scientists to formulate and rapidly test OED—as well as DA—formulations and algorithmic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED is unique in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' First, to the best of our knowledge, it is the first open-source package for scientific computing that allows implementing and testing the individual components of DA as well as OED systems in a unified and streamlined environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Second, it is written in Python, which is arguably the most popular and adopted programming language for recent algorithmic developments in the computational science disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' It has a huge user-support community, and the learning curve is relatively smoother than that of other lower-level programming PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 3 languages such as C/C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Third, PyOED is designed in an object-oriented programming (OOP) fashion, which enables practitioners to reconfigure and reuse the individual building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Moreover, it is easy to combine PyOED with other user-defined routines, such as numerical integration of simulation models and optimization routines, which makes PyOED highly extensible and adaptable to a wide range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Fourth, PyOED is not limited to a specific inverse problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus new DA and OED methods can be implemented and even interface with other inversion packages, for example, hIPPYlib [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Fifth, PyOED leverages best practices in software development, including detailed documentation with hands-on examples and robust unit-testing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Section 2 provides the mathematical formalism of inverse problems, DA, and OED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Section 3 describes the structure and the philosophy of the PyOED package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In Section 4 we provide a list of numerical experiments to demonstrate the general workflow and usage of PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Concluding remarks are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 2 MATHEMATICAL BACKGROUND In this section we provide a brief overview of the mathematical background of the PyOED core, which is important for approaching the OED problem for Bayesian inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In this presentation we focus on sensor placement for Bayesian inversion as a modal OED formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' We start by discussing the forward problem in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' then we introduce the inverse problem in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='2 and the OED formalism in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1 The forward problem The forward problem maps the model parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', the initial condition) onto the observation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Consider the forward problem described by y = F (𝜃) + 𝛿 , (1a) where 𝜃 is the model parameter of interest, y ∈ RNobs is the observation, and 𝛿 ∈ RNobs is a noise term that accounts for the inaccuracy of the observational system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The forward operator F is occasionally referred to as the “parameter-to- observable” map and generally represents a composition of a simulation/solution model S and an observation operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The simulation model S describes the evolution of the physical phenomena, for example, space-time advection and diffusion of a contaminant simulated over a predefined model grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The observation operator O projects the simulated state onto the observational grid, for example, by interpolation or restriction to the observation grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, the forward problem (1a) can be rewritten as y = O ◦ S(𝜃) + 𝛿 , (1b) where ◦ is the composition operator, that is, O ◦ S(𝜃) ≡ O (S(𝜃)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' It is impossible to find a simple unique formalism of the simulation model that accurately represents all possible dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In this work and in the implementation of PyOED, we differentiate two types of simulation models: time-independent and time-dependent simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A wide range of time-dependent simulation models for dynamical systems governed by partial differential equations (PDEs) can be described as 𝜕𝒖 𝜕𝑡 = 𝒇 (𝒖(𝒙,𝑡, 𝜇)), (2) 4 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed where 𝒖 represents the prognostic variable(s) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', physics), 𝒙 denotes the spatial coordinates, and 𝜇 defines the physics parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' To numerically solve the simulation model S equations, we utilize spatial discretization and temporal integration routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' If we follow a discretize-then-optimize approach, we can rewrite (2) in the form u𝑛 := u(𝑡𝑛) = S𝑡𝑛−1→𝑡𝑛 (u0, 𝜇), (3) where u𝑛 := u(𝑡𝑛) ≡ u(𝑡𝑛, 𝒙, 𝜇) is the model state at time instance 𝑡𝑛 and S𝑡→𝑡+Δ𝑡 is a one-time step transition mapping that results from the application of a standard spatial discretization method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', finite difference, finite volume, or finite element) and time integration scheme (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', Runge–Kutta routine) with a step size Δ𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, the prediction at time 𝑡𝑛 can be related to the initial condition u(𝑡0) and the model parameters using the recursive application of the mapping S over a time interval [𝑡0,𝑡𝑛] as u𝑛 = S𝑡𝑛−1→𝑡𝑛 ◦ · · · ◦ S𝑡0→𝑡1 (u0, 𝜇) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' (4) In space-time formulations, one can stack the model state in one long vector u := � uT𝑛, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' , uT 0 �T and define the solution operator S as a block operator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', a block matrix) that operates recursively on the the components of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This would enable unifying the formulation—to some extent—of the forward problem into the form (1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Since the observational measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', sensory data) are not necessarily the same as the model state, we define the state-to-observable mapping O𝑛(·) to map the state u𝑛 onto the observation space, for example, by restricting that state onto the observational grid points, as y(𝑡𝑛) = O𝑛(u(𝑡𝑛)) , (5) which then can be used to construct a general observational vector y ∈ RNobs representing spatiotemporal data, for example, by stacking observations at multiple time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Note that while we focused the discussion above on time-dependent problems, the case of time-independent simulations can be thought of as a special case of (4) where S maps, for example, the physics parameter 𝜇 to a model state u and the time index is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In most inverse problem formulations, based on the application of interest, the inversion parameter 𝜃 stated in (1a) stands for the model physics 𝜇, the model initial condition u0, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Note also that we have omitted details including adaptive time stepping where Δ𝑡 is adaptively adjusted to guarantee stability and accuracy of the time integration methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' We intentionally remove such details from the discussion here to simplify the presentation and focus more on OED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In fact, the OED routines in PyOED are designed to solve OED problems where the utility function is regarded as a black box, thus abstracting the OED capabilities from the inverse problem definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For these reasons, we take (1) to be an acceptable simplification that describes a forward problem setup that is general enough for our purposes in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Both the simulation model S and the observation operator O are imperfect and generally include sensory noise and representativeness errors characterizing imperfection of the map between the model space and observation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The fact that model observations F (𝜃) are not perfectly aligned with observational data (y) is modeled—assuming additive noise—by adding the noise term 𝛿 to the simulated observations F (𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In most applications, the observational noise follows a Gaussian distribution 𝛿 ∼ N (0, Γnoise), where Γnoise is the observation error covariance matrix that captures uncertainty stemming from sensory noise and representativeness errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In this case, the data likelihood is P (y|𝜃) ∝ exp � −1 2 ∥F (𝜃) − y∥2 Γ−1 noise � , (6) PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 5 where the matrix-weighted norm in (6) is defined as ∥x∥2 A = xTAx for a vector x and a square symmetric matrix A of conformable sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='2 The inverse problem An inverse problem refers to the retrieval of the model parameter 𝜃 from noisy observation y, conditioned by the model dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This can be achieved by finding a point estimate or by building a complete probabilistic description as discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In the former, an optimization problem is solved to minimize the mismatch between the expected observations (through simulation models) and real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This is typically employed in variational DA methods where the estimate of the true parameter is obtained by minimizing a regularized log-likelihood objective, where regularization is employed to enforce smoothness or background information on the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In this case, a point estimate of the true 𝜃 is obtained by solving arg min 𝜃 J (𝜃) := 1 2 ∥F (𝜃) − y∥2 Γ−1 noise + 1 2 ��𝜃 − 𝜃pr ��2 Γ−1 pr , (7) where 𝜃pr is an initial guess of the unknown true value of 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In general, the second term is added to enforce regularization or prior knowledge on the solution, for example, if the solution is assumed a priori to follow a Gaussian distribution 𝜃 ∼ N �𝜃pr, Γpr �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Uncertainty envelopes around the single-point estimate obtained by solving (7) can be developed, for example, by using Laplacian approximation [47] where the posterior is approximated by a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This approach has been successfully employed in infinite-dimensional Bayesian inversion problems [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A fully Bayesian approach, on the other hand, aims to provide a consistent probabilistic description of the unknown parameter along with the associated uncertainties and is not limited to Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This is achieved by describing the posterior, that is, the probability distribution of the model parameter 𝜃 conditioned by the available simulations and noisy data y, and is obtained by applying a form of Bayes’ theorem P (𝜃|y) ∝ P (y|𝜃) P(𝜃) , (8) where P(𝜃) is the prior, P (y|𝜃) is the data likelihood, and ∝ indicates removal of a normalizing constant in the right-hand side of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For further details on Bayesian inversion see, for example„ [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Given the posterior (8), one can use the maximum a posteriori (MAP) point as an estimate of the true unknown QoI or follow a Monte Carlo approach to sample the posterior, thus building a complete probabilistic picture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see, for example, [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Both the variational and the Bayesian inference approaches provide a plethora of techniques for statistical data analysis in general, and specifically for solving inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The Bayesian perspective provides a formal mathematical ground for estimating the physical QoI, for example, the model parameter 𝜃, along with the associated uncertainties given the available sources of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In many cases, however, this inversion is an intermediate step, and the goal QoI is a function of the model parameter, that is, 𝜌 := P(𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A goal-oriented approach is followed in this case where one aims to inspect the posterior of the QoI conditioned by the available data [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 6 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed The ideal case: linear Gaussian problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' If the forward operator F is linear (or linearized), and assuming Gaussian observational noise and a Gaussian prior N �𝜃pr, Γpr �, then the posterior is Gaussian N � 𝜃y post, Γpost � with Γpost = � F∗Γ−1 noiseF + Γ−1 pr �−1 , 𝜃y post = Γpost � Γ−1 pr𝜃pr + F∗Γ−1 noise y � , (9) where F ≡ F is the forward model and F∗ is the associated adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Despite being simple, this setup (9) is of utmost importance in the Bayesian inversion and OED literature and is elementary for testing implementations of new DA and OED approaches, mainly because the posterior can be formulated exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Moreover, in many large-scale applications, the posterior can be approximated, to an acceptable degree, by a Gaussian distribution obtained by linearizing the nonlinear operator F around the MAP estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The linearized model is also known as the tangent linear model (TLM) obtained by differentiating F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' As mentioned earlier, inversion for the parameter 𝜃 is often an intermediate stage, and the end-goal is to describe the posterior of a general QoI that is not the model parameter 𝜃 but rather a goal quantity 𝜌 that depends on the inversion parameter 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Specifically, goal-oriented inversion seeks the posterior P (𝜌|y) ∝ L(y|𝜌,𝜃)P(𝜌) , where P(𝜌) is a prior on the goal QoI and L(y|𝜃) = L(y|𝜌,𝜃) is the data-likelihood (6), where 𝜌 is determined completely based on 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' We focus the discussion here on the case of linear prediction operators P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' That is, we consider prediction quantities of the form 𝜌 = P𝜃, (10) where P is a linear prediction operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Within the Gaussian linear setting, the prior of the goal QoI 𝜌 is N �𝜌pr, Σpr � with 𝜌pr = P𝜃pr, Σpr = PΓprP∗, (11) where P∗ is the adjoint of the prediction operator P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The posterior distribution of the prediction 𝜌, conditioned by the observations y, is also Gaussian and is given by N �𝜌post, Σpost �, where 𝜌post = P𝜃y post , Σpost = PΓpostP∗ = P � F∗Γ−1 noiseF + Γ−1 pr � P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' (12) Note that the goal-oriented Bayesian inversion falls back to the standard formulation of a Bayesian inverse problem if the prediction operator P is an identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='3 Optimal experimental design Here we outline the basics of an OED problem for Bayesian inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' An excellent review of recent advances on model-constrained OED can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' An OED optimization problem takes the general form 𝜻opt = arg max 𝜻 U(𝜻) , (13) where U is a predefined utility function that quantifies the quality of the design 𝜻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The nature of 𝜻 depends on the application at hand, and the utility function U is chosen to enable defining what an “optimal” design is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The optimization problem (15) is often associated with an auxiliary sparsity-enforcing term −𝛼Φ(𝜻) to prevent dense designs and to reduce the cost associated with deploying observational sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The utility function can be rewritten as U(𝜻) = Ψ(𝜻) − 𝛼Φ(𝜻) , (14) PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 7 where Ψ(·) is an OED optimality objective, referred to hereafter as the “optimality criterion,” which is defined based on the inverse problem at hand and on a chosen criterion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', from the well-known OED alphabetic criteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The function Φ(𝜻) asserts regularization or sparsity on the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, this could be a resource constraint: �𝑛s 𝑖 𝜻𝑖 = ∥𝜻 ∥ ≤ 𝑘 , or �𝑛s 𝑖 𝜻𝑖 = ∥𝜻 ∥ = 𝑘 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='𝑘 ∈ Z+ , for example, an upper bound (or exact budget) on the number of sensors (in OED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' It also could be a sparsifying (possibly nondifferentiable) function, for example, ∥𝜻 ∥0, ∥𝜻 ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Generally speaking, we seek a design that maximizes the utility function U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Other formulations, however, involve minimization of an OED optimality criterion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see, for example, [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Both formulations are adopted in the OED literature and in PyOED and often are even equivalent, as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In inverse problems, a design can be associated with the observational configuration and thus can be used to optimally select an optimal observational policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, a design can be defined to select sensor location or temporal observation frequency that can help provide accurate prediction with minimum uncertainty, or it can be defined to select an observational configuration that guarantees maximum information gain from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' OED for sensor placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In sensor placement we associate a binary design variable 𝜻𝑖 with the 𝑖th candidate sensor location with 1 meaning activating the sensor and deactivating it otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This defines the design as a binary vector 𝜻 ≡ 𝜻b ∈ {0, 1}𝑛s, which collectively define the observational configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In this case, the OED problem (13) takes the form 𝜻opt = arg max 𝜻 ∈{0,1}𝑛s U(𝜻) := Ψ(𝜻) − 𝛼Φ(𝜻) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' (15) In Bayesian OED for sensor placement, the observation covariance Γnoise is replaced with a weighted version WΓ(𝜻), resulting in the weighted data-likelihood L(y|𝜃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 𝜻) ∝ exp � −1 2 ∥F(𝜃) − y∥2 WΓ (𝜻) � , (16a) where the weighted observation error covariance matrix takes the form WΓ(𝜻) := � W(𝜻)ΓnoiseW(𝜻) �† = LT(𝜻) � L(𝜻) � W(𝜻)ΓnoiseW(𝜻) � LT(𝜻) �−1 L(𝜻) , (16b) where † denotes the Moore–Penrose (pseudo) inverse and W(𝜻) := diag (𝜻) is a diagonal matrix with the binary design 𝜻 ∈ {0, 1}𝑛s on its diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' L(𝜻) is a sparse matrix that extracts nonzero rows/columns from the design matrix WΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see [9] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The utility function .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In the case of linear Bayesian inversion, the posterior is Gaussian with the covariance being independent from the actual realizations of the data, as shown by (9) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This fact enables designing observational policies before actually deploying the observational sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Specifically, in linear Bayesian OED, we set the objective to minimize a scalar summary of the posterior uncertainty, that is, the posterior covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This is the underlying principle of the alphabetical criteria [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, an A-optimal design is the one that minimizes the trace of the posterior covariance matrix, and a D-optimal design is the one that minimizes its determinant (or equivalently the log-determinant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Note that in the case of a linear model F, the Fisher information matrix FIM is equal to the inverse of the posterior covariance matrix, that is, FIM = Γ−1 post(𝜻) = F∗WΓ(𝜻)F + Γ−1 pr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, in this case the utility function— discarding the penalty term—is set to U(𝜻) := Tr (FIM(𝜻)) for A-optimal designs and U(𝜻) := log det (FIM(𝜻)) for D-optimal designs, and then the utility function is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 8 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed When the model F is nonlinear, the FIM requires evaluating the TLM at the true parameter, that is, F = 𝜕F |𝜃=𝜃true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, to obtain an optimal design, one can iterate over finding the MAP estimate of 𝜃 and solving an OED problem with Gaussian approximation around that estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Other utility functions employed in nonlinear OED problems or non-Gaussian distributions include the Kullback–Leibler divergence between the posterior and the prior [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Popular solution approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The OED problem (15) can be viewed as a mixed-integer program and can be solved by using branch-and-bound [25, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' However, this type of research has not yet been applied to model-constrained-OED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A common approach to solving (15) is to replace the binary optimization with the following relaxation: 𝜻opt = arg max 𝜻 ∈[0,1]𝑛s U(𝜻) , (17) where the design variables are relaxed to take any values in the interval [0, 1] rather than only the binary values {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This approach, if carried out properly, has the effect of generating a continuous relaxation surface that connects the values of the objective evaluated at the binary designs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see [9] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A gradient-based optimization approach is generally used to solve (17), which requires developing the gradient of both the optimality criterion Ψ and the penalty Φ with respect to the design 𝜻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' As mentioned earlier, the penalty term is generally chosen to promote sparsity on the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A popular penalty function is based on the ℓ0 norm to promote design sparsity and is thus nonsmooth and consequently is nondifferentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This difficulty can be alleviated, for example, by approximating the effect of ℓ0 with a sequence of differentiable functions that converge in effect to the ℓ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see, for example, [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In order to guarantee continuity of the relaxation surface, the weighted precision matrix is defined in the general form WΓ(𝜻) := � W(𝜻) ⊙ Γnoise �† , W𝑖,𝑗 (𝜻) := \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 𝜔𝑖 𝜔𝑗 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 𝑖 ≠ 𝑗 \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 𝜔𝑖 = 0 1 𝜔2 𝑖 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='𝜔𝑖 ≠ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 𝑖 = 𝑗 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 𝑖, 𝑗 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' ,𝑛s , 𝑚,𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' ,𝑛𝑡 , (18) where ⊙ is the Hadamard (Schur) product of matrices and 𝜔𝑖 ∈ [0, 1] is a weight calculated by using 𝜻𝑖, for example, 𝜔𝑖 := 𝜻𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see [9] for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The formulation (18) guarantees that for 𝜻𝑖 ∈ [0, 1]𝑛s it holds that lim𝜻→𝜻 b WΓ(𝜻) = WΓ(𝜻b) for a binary design 𝜻b ∈ {0, 1}𝑛s and thus guarantees continuity of the relaxation surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, the solution of the relaxed OED optimization problem (17) is guaranteed to match the solution of the original binary OED optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The A- and D-optimal design relaxed optimization problems (17) take the following respective forms: 𝜻A−opt = arg max 𝜻 ∈[0,1]𝑛s := Tr � P � F∗� Γnoise ⊙ W(𝜻) �† F + Γ−1 pr �−1 P∗ �−1 − 𝛼Φ(𝜻) , (19a) 𝜻D−opt = arg max 𝜻 ∈[0,1]𝑛s := log det � P � F∗� Γnoise ⊙ W(𝜻) �† F + Γ−1 pr �−1 P∗ �−1 − 𝛼Φ(𝜻) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' (19b) The most important piece of information for solving the relaxation (17) is the gradient of the utility function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see, for example, [9] for a detailed derivation of the gradients of the objective functions in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Gradient formulation, however, PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 9 is mathematically involved and can be extremely computationally demanding because it requires numerous evaluations of the forward operator, the goal operator, and the corresponding adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Moreover, the penalty function Φ(·) is required to be differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A stochastic learning approach to binary OED has been recently presented in [10], to solve the binary optimization problem (15) without the need for relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This approach does not require differentiability of the utility function U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In this approach the optimal design is defined as popt = arg max p∈[0,1]𝑛s E𝜻∼P(𝜻 |p) � U(𝜻) − 𝑏 � , (20) where P (𝜻 |p) is a multivariate Bernoulli distribution with parameter p specifying probabilities of success/activation of each entry of 𝜻, that is, p𝑖 ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Here 𝑏 is a constant “baseline” used to minimize variability of the stochastic estimate of the gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see [10] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Algorithm 1 summarizes the procedure followed to solve (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Algorithm 1 Stochastic optimization for binary OED with the optimal baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Input: Initial distribution parameter p(0), step size schedule 𝜂 (𝑛), sample sizes Nens, 𝑚, baseline batch size 𝑏𝑚 Output: 𝜻opt 1: initialize 𝑛 = 0 2: while Not Converged do 3: Update 𝑛 ← 𝑛 + 1 4: Sample {𝜻 [𝑗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 𝑗 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' , Nens} ∼ P � 𝜻 |p(𝑛)� 5: Calculate 𝑏 = OptimalBaseline(p(𝑛), Nens, 𝑏𝑚) 6: Calculate g(𝑛) = 1 Nens �Nens 𝑗=1 (J (𝜻 [𝑗] − 𝑏)) �𝑛s 𝑖=1 � 𝜻𝑖 [𝑗 ] p𝑖 + 𝜻 [𝑗 ]𝑖−1 1−p𝑖 � e𝑖 7: Update p(𝑛+1) = L � p(𝑛) − 𝜂 (𝑛)𝑔(𝑛)� 8: end while 9: Set popt = p(𝑛) 10: Sample {𝜻 [𝑗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 𝑗 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' ,𝑚} ∼ P �𝜻 |popt�, and calculate J (𝜻 [𝑗]) 11: return 𝜻opt: the design 𝜻 with smallest value of J in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 12: function OptimalBaseline(𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Nens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 𝑏𝑚) 13: Initialize 𝑏 ← 0 14: for 𝑒 ← 1 to 𝑏𝑚 do 15: for 𝑗 ← 1 to Nens do 16: Sample 𝜻 [𝑗] ∼ P (𝜻 |p) 17: Calculate r[𝑗] = �𝑛s 𝑖=1 � 𝜻𝑖 [𝑗 ] p𝑖 + 𝜻 [𝑗 ]𝑖−1 1−p𝑖 � e𝑖 18: end for 19: Calculate d[𝑒] = 1 Nens �Nens 𝑗=1 r[𝑗] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' and g[𝑒] = 1 Nens �Nens 𝑗=1 J (𝜻 [𝑗]) r[𝑗] 20: Update 𝑏 ← 𝑏 + (g[𝑒])T d[𝑒] 21: end for 22: Update 𝑏 ← 𝑏 Nens / � 𝑏𝑚 �𝑛s 𝑖=1 1 p𝑖−p2 𝑖 � 23: return 𝑏 24: end function Note that we do not provide an exclusive set of formulations or solution approaches in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' We provide here only an exemplary set of formulations and algorithms used to inspire the development of PyOED, which itself can be used to test further formulations and algorithmic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 10 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed 3 PYOED: STRUCTURE AND PHILOSOPHY PyOED aims to provide a unified platform for implementing and testing model-constrained OED algorithmic approaches including formalisms (15), (17), (19), and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Solving model-constrained OED and inverse problems requires proper understanding and formulation of the underlying dynamical system, observational configuration, uncertainty models, DA and inversion algorithms, and the OED objective [48] and the selected utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED is a stand-alone, yet extensible, Python package that provides users and researchers in the computational science and engineering disciplines with a testing suite that effectively glues these components in an OOP fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, PyOED provides a variety of time-dependent and time-independent simulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' These include systems governed by linear algebraic equations, ordinary differential equations, and PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED is also equipped with a set of classes implementing various observational operators, probabilistic uncertainty models, and DA and OED methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A high-level overview of the PyOED major components and their coupling for solving DA and OED problems is provided in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1 we briefly describe the main components of PyOED and outline the functionality they provide in correspondence with the diagram 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' High-level overview of the main components of PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1 Code structure Figure 2 shows the main subpackages (ordered alphabetically) shipped with the current version of PyOED (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The rest of this section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1 provides a high-level description of the packages/subpackages of PyOED as displayed in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Following the convention in the DATeS package [15], we use the word “model” to refer to three entities: the simulation model, the observation model (or operator), and the error models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The simulation model provides a prediction about the behavioral pattern of the physical phenomena of concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In the current version of PyOED (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0) we provide various simulation models under the pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='simulation_models, including several versions of the Lorenz system [34] and advection-diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The structure of these prototyp- ical simulation models should provide clear guidelines to practitioners willing to adopt PyOED for their particular applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Assimilation/lnversion OED Bayesian inversion, 3D-Var, 4D-Var, Kalman filtering, etc Optimize Simulation Observation Noisy Data Acquisition Model Operator Forward, adjoint, etc Forward, adjoint, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' applylobserve, Jacobian, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Data Sensors Uncertainty/Error Model Satellites, Doppler Lidar, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' prior, observation noise, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Main subpackages (ordered alphabetically) available in the current version of PyOED (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The observation operator maps the model state onto the observation grid, thus providing a functional mapping between the model state and observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Two of the most prominent observation operators in experimen- tal settings are the identity operator and an interpolator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED provides implementations of several observa- tion operators including these two, with an observational design properly incorporated to enable altering obser- vational configurations at any point in the DA or OED solution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Observation operators are provided in the pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='observation_operators subpackage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The error models quantify the uncertainty associated with the model parameter, model state, and observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' An experimental design can be associated with any of these pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, in sensor placement, an experimental design is associated with the observational grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' thus, modifying the observational design affects the observational error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, in the relaxation approach (19), the design weights scale the entries of the covariance matrix, and the stochastic approach (20) works by removing rows/columns of the observation error covariance matrix corresponding to zero design variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED provides various implementations of error models suitable for modeling priors, as well as observational errors in Bayesian inversion, where a design variable is consistently implemented to enable modifying the experimental design during any step of the DA and/or OED solution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The current version of PyOED (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0) provides various error model implementations through the subpackage pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='error_models, including a Gaussian model and Laplacian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED provides a set of DA tools that include algorithms for “filtering” and “smoothing.” These two terms are widely used in the DA literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The former algorithm solves inverse problems that involve time-independent or time-dependent simulation models, while the latter algorithm is restricted to time-dependent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Filtering involves prediction (of observation) using the parameter-to-observable map, followed by a correction filtering assimilation smoothing examples ml error_models models observation operators pyoed oed simulation models optimization stats tutorials utility12 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed procedure to correct knowledge of the QoI given the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In filtering for time-dependent simulations, the observational data is assimilated sequentially, with one observation time per assimilation window/cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Examples of filtering DA methods include three-dimensional variational DA, and Kalman filtering [6, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Smoothing, on the other hand, is concerned with history matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' these algorithms try to find the QoI that best matches multiple spatiotemporal observations (a trajectory) and is usually defined as an initial value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Examples include space-time Bayesian inversion [45] and four-dimensional variational DA [6], for which vanilla implementations are provided in PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Implementations of filtering DA algorithms are provided through pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='filtering, and smoothing algorithms are provided in pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Numerical optimization routines are elementary for solving OED optimization problems, as well as the variational approaches for solving DA problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A variety of optimization software packages can be used for solving numerical optimization problems including those described in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED enables using external optimization packages, including Python’s Scipy package, to solve DA and OED optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED, however, provides specific implementations of optimization procedures not available in popular optimization packages, such as the stochastic algorithm described by Algorithm 1, various versions of the stochastic average approximation (SAA) algorithm, and robust optimization [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This subpackage is intended to provide implementations of machine learning algorithms useful for DA and OED applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, the stochastic learning approach to OED (20) can be seen as a reinforcement learning (RL) approach to solving the OED problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The module pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='reinforcement_learning under this package provides implementation of RL components, including an agent, a policy, transition probability, actions, and utility functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This package aims to collect statistical procedures used by other parts of the package, such as sampling routines, and implementation of random variables and their probabilistic utility functions including density evaluation and log-probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This version of PyOED (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0) provides an exemplary implementation of a multivariate Bernoulli distribution required by the RL algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Since statistical tools are crucial for various DA and OED algorithms, we chose to keep the subpackage pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='stats rather than moving these implementations to other parts of the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This approach is advantageous because we continuously extend the package with various statistical tools, for example, for randomized approximation methods for Bayesian inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='oed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' OED is the main component of PyOED that provides implementations of various algorithmic approaches for solving OED problems, including relaxation (19) and stochastic learning (20), as well as recent developments including robust OED [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Most implementations in this package take an inverse problem (DA object) as input and use it to access all the underlying components, thus gaining access to the simulation model, error models, and observation operator as well as the experimental design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This approach enables the user to modify an experimental design, solve the DA problem if needed, and solve the underlying OED optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The core OED functionalities in most PyOED routines, however, can be used with black-box utility functions, waiving the need for an inverse problem if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This subpackage aims to collect implementations of general-purpose functionality, such as file I/O and visualization, as well as general mathematical and statistical procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The subpackage includes matrix-free PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 13 implementations of expensive operations such as evaluating the trace and log-determinant of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' It also provides routines to approximate matrix trace using statistical randomization [5, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This subpackage provides various example scripts that users can follow to learn how to effectively use various pieces of the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The modules in this subpackage explain how to load all pieces of the subpackage independently and explain how to properly coordinate these components to design a consistent DA and/or OED experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='tutorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Given the popularity of Jupyter Notebooks in the computational science community, we converted some of the examples in the subpackage pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='examples to Jupyter Notebooks and provided them in this subpackage pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='tutorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' We employ them in the test cases presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' We can add more tutorials on reasonable demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='2 PyOED utilization workflow While the components of DA and OED problems can be used independently of each other, some level of ordering is mandatory for proper utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, an inverse problem (DA) object cannot be instantiated before a simulation model, an observation operator, and error model objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Similarly, an OED problem for Bayesian inversion cannot be solved before creating an inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The general workflow for utilizing PyOED components is displayed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A practical guide that illustrates how to follow this simple workflow is described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Workflow describing initialization order and access level of PyOED components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='3 Extending and contributing to PyOED PyOED is meant to be extensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, we continuously interface other software tools that provide efficient imple- mentations of the components of DA and OED problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, hIPPYlib [50] is a software package for solving high-dimensional inverse problems following an optimize-then-discretize approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' It has been employed to empirically verify several inversion and OED algorithmic developments recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Instead of rebuilding the functionality of hIPPYlib and similar packages, we have interfaced with some of its components to show how easily and efficiently PyOED can extend other successful packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, PyOED interfaces with finite-element (FE) implementations of Models Simulation model Observation operator Error models Inverse problem (DA) Data Design Prior Observation noise IP & OED Inversion (analysis/posterior) OED (optimal design)14 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed advection-diffusion and Poisson models from hIPPYlib as well-as point observation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' However, such extension does not hinder the functionality or limit the extensibility of PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Specifically, interfacing with such external packages is optional and is not provided in the core of PyOED, mainly because the backbone of these packages is not guaranteed not to be quickly outdated or be unmaintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, these extensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', interfacing with hIPPYlib) are made optional during the import process of PyOED subpackages, and dependent functionality is used only when properly installed and available on the current architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='4 Code availability The development version of PyOED is available from https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='com/ahmedattia/pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 4 TEST CASES PyOED comes with a set of prototypical test problems with increasing complexity for both DA and OED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' An ideal case typically used in scientific publications is the linear Gaussian setup, where the simulation model and the observation operator are both linear and the error models (observation noise and the prior) are both Gaussian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In this case, the posterior is also a Gaussian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' the solution of the inverse problem is unique—the posterior mean and mode (MAP) are identical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' and posterior moments (mean and covariance) both have closed forms that can be obtained by applying the Kalman filter theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Such a simplified setup can be used for testing new formulations in both DA and OED, and thus it is provided in PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' We discuss this formulation and in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1 show how it can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='2 we discuss in further detail a standard experiment widely used in OED scientific research and offered by PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1 An ideal setup: linear Gaussian toy problem Consider a time-dependent forward problem defined at time instances 𝑡0 + 𝑖Δ𝑡,𝑖 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' ,𝑛𝑡, for a fixed step size Δ𝑡, as follows: u𝑛 = A u𝑛−1 , y𝑛 = Iu𝑛 + 𝛿 , 𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' , (21) where u𝑛 ∈ RNstate is the discrete model state at time instance 𝑡𝑛, A ∈ RNstate×Nstate is a matrix representing model evolution over time interval [𝑡𝑛−1,𝑡𝑛], and I is the identity observation operator/matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' If we assume 𝛿 ∼ N (0, R) and assume a Gaussian prior u0 ∼ N � upr 0 , Γpr � , then the posterior is Gaussian N � upost 0 , Γpost � with Γpost = � ATR−1A + Γpr−1�−1 , upost 0 = Γpost � Γpr−1upr 0 + 𝑛𝑡 ∑︁ 𝑖=1 ATR−1 y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' (22) Since (22) is a closed form of the posterior, we can use it to test and debug new DA and OED implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This fact is highly utilized in the unit tests developed in PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' To create a proper experiment, we will follow the workflow described by Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In the rest of this section (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1) we describe how to initialize an inverse problem in PyOED with the settings (22), and we provide a simple scheme that can be followed to initialize other experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The code summarized here is provided in the pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='fourDVar_driver module with additional comments, details, and capabilities that can help the user understand the workflow for creating and solving an inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 15 Jupyter Notebook pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='tutorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='toy_linear is also available and can be used to regenerate the numerical results presented in this section (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Creating the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Assuming PyOED is already in the Python path, the first step is to import/load the simulation model (that describes A), the observation operator (here an identity operator), and the error models to create the prior and the observation error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This can be done as described in the code snippet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='simulation_models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='toy_linear import ToyLinearTimeDependent 2 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='observation_operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='identity import Identity 3 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='error_models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='Gaussian import GaussianErrorModel Snippet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Import essential modules to create the simulation model object, the observation operator, the prior, and the observation error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Note that we have imported only the classes we need in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' However, PyOED provides several other implementations of the simulation models, observation operators, and error models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In order to create a simulation model, an object of ToyLinearTimeDepndent is instantiated as described by snippet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This generates an internal two-dimensional array of size 5 × 5 that represents the forward model A, which integrates the model state forward by a timestep 𝑑𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' That internal array can be reproduced by setting the random_seed parameter in the passed configurations dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=" 1 model = ToyLinearTimeDependent(configs ={'nx':5, 'dt':0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="1, 'random_seed ':123}) Snippet 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Instantiate the simulation model object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Since each simulation model has its own configurations, many of which are assigned default values, we follow the strategy of DATeS [15] and use dictionaries to pass model arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED aggregates and validates the passed dictionary against the default values and initiates the model accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, in snippet 2 we specify a random_seed argument that guarantees reproducibility of any randomly generated data inside the model object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This is done by keeping an internal random state inside the model object that is independent from other objects and is initialized to the passed random seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, if no random seed is passed, each time the same model object is instantiated, completely random sequences will be generated if requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Implementations of simulations model must provide an implementation of a class-level method get_default_configs that returns a dictionary with all default values used if not passed upon instantiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In order to ensure that, all PyOED simulation models inherit the class pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='simulation_models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='SimulationModel that guarantees enforcing the implementation of mandatory methods required for the seamless integration of various components in PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The final configurations of a simulation model is a combination of those in the passed configurations dictionary and the default values, with precedence given to the passed configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The simulation model’s configurations (a copy of it, in fact) can be accessed through the attribute configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' We generally choose to return a copy to guarantee that all settings are validated before modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, one cannot modify the time step 𝑑𝑡 without verifying whether the timestepping implementation is tied to that time step or not and updating dependencies accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The observation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Similar to simulation models, an observation operator is created by passing the settings in the configurations dictionary to the observation operator class constructor as shown in snippet 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, the observation operator has access to the model grid and other useful attributes to create and manipulate data structures (such as the 16 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed model state) without having to provide any new implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This is mainly because observations in this case are the same as the corresponding model states (discarding observation noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=" 1 obs_oper = Identity(configs ={'model ':model }) Snippet 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create an identity observation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The prior and the data noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The error models (prior and observation noise) are created in this example as shown in snippet 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Note that the random_seed configurations variable is used to set the random number generator for reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This enables us regenerate a set of experiments and generate proper benchmarks for a fair comparison between various implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' As mentioned earlier, if no random seed is passed, each instance of the error model is assigned a randomly generated state that guarantees that each instance has its own different sequence of random numbers/vectors realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=" 1 prior = GaussianErrorModel(configs ={'size':model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="state_size , 'mean':1, 'variance ':1, 'random_seed ':1}) 2 obs_noise = GaussianErrorModel(configs ={'size':obs_oper." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="shape [0], 'variance ':0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="01 , 'random_seed ':1}) Snippet 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create the prior and the observation error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The inverse problem (DA) object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The next step is to put these models together in action and use them to create an inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' We illustrate the utilization of a DA object to solve the inverse problem following a 4DVar formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The literature provides a plethora of variants of the general 4DVar scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED provides a few implementations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' however, the most basic (vanilla) implementation is used here for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Two approaches are followed in PyOED for instantiating a DA object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The first is to pass all configurations (upon initialization) in the configurations dictionary configs similar to the case of simulation models, error models, and observation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The second approach is to use the proper registration methods associated with the created object after instantiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The latter can be also used to update components of the DA object after initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, one might want to change the settings of the assimilation time window, register new observations or remove the old ones, or modify or even change the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Since the first approach has already been explained with the simulation and error models, we demonstrate the second approach here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Specifically, a 4DVar assimilation object is created as in snippet 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 inverse_problem = VanillaFourDVar () 2 inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='register_model(model) 3 inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='register_observation_operator(obs_oper) 4 inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='register_prior_model(prior) 5 inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='register_observation_error_model(obs_noise) Snippet 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create the inverse problem object with default settings, and then add (register) all the pieces created above, that is, the simulation model, the observation operator, the prior, and the observation error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The next step is to register observational data (along with observation times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A standard strategy in experimentation is to create synthetic data from a ground truth (known as a twin experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This is explained by snippet 6, where we define the assimilation timespan (window) to be the interval [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='3], and the observations are taken at 3 time instances 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The observational data is mimicked by adding random noise (using the observation error model) to the observed ground truth at the corresponding observation time instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 17 1 # Set the assimilation/simulation time window 2 tspan = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='3) 3 inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='register_assimilation_window(tspan) 4 5 # Create truth (true initial state and trajectory) 6 true_IC = model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='create_initial_condition () 7 checkpoints = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='3] 8 _, true_traject = model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='integrate_state(true_IC , tspan=tspan , checkpoints=checkpoints) 9 10 # Create synthetic data (perturbation to truth) and register them 11 for t, state in zip(checkpoints , true_traject): 12 obs = obs_noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='add_noise(obs_oper(state)) 13 inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='register_observation(t=t, observation=obs) Snippet 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create synthetic noisy observations, and register all observation time points and observational data to the inverse problem object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The final step in the DA procedure is to solve the inverse problem and assess the quality of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For this setup, we know the ground truth, and thus one can evaluate the root mean squared error (RMSE), which is a standard error metric in statistics in the DA literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In order to solve the inverse problem, the solve_inverse_problem method of the 4DVar DA object is called (snippet 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This method will raise an instructive error if any of the essential elements, for example, the simulation model, are not registered properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Note that this function is flexible and allows the posterior covariance to be constructed if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' It also allows waiving finding the MAP estimate, which can be advantageous in OED applications due to associated computational savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, one might want to estimate the posterior covariance in the linear Gaussian case without evaluating the MAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='solve_inverse_problem(init_guess=prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='mean , update_posterior=True) Snippet 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Solve the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The optimization initial point is set by default to the prior mean;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' however, it can be modified if a better initial guess is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Here, the posterior covariance is evaluated, and consequently the posterior is updated with both the mean and the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED provides several utility functions to evaluate statistics, such as the RMSE, which can be used to quantify the accuracy of the inverse problem solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Snippet 8 shows how to call the utility function calculate_rmse and use it to evaluate the prior and the analysis (posterior) RMSE, which are then printed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 from pyoed import utility 2 prior_rmse = utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='calculate_rmse(true_IC , inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='mean) 3 posterior_rmse = utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='calculate_rmse(true_IC , inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='mean) 4 print(f"Prior RMSE: {prior_rmse}") 5 print(f"Posterrior RMSE: {posterior_rmse}") Snippet 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Calculate and print the RMSE values associated with the prior mean (initial guess here) and the posterior mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The same procedure can be easily followed to inspect the RMSE results over the whole assimilation timespan as described by Snippet 9 with results plotted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 checkpoints , true_traject = model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='integrate_state(true_IC , tspan=tspan) 2 _, prior_traject = model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='integrate_state(inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='mean , tspan=tspan) 3 _, posterior_traject = model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='integrate_state(inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='mean , tspan=tspan) 4 prior_rmse = [utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='calculate_rmse(xp, xt) for xp , xt in zip(prior_traject , true_traject)] 18 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed 5 posterior_rmse = [utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='calculate_rmse(xp , xt) for xp , xt in zip(posterior_traject , true_traject)] Snippet 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Generate RMSE over the whole assimilation window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='30 Time 0 10 20 30 40 50 60 RMSE Prior Posterior Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' RMSE results of the solution of the inverse problem presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The RMSE results of both the prior and the posterior trajectories plotted here are obtained by running the code in snippet 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' One can also analyze the posterior covariances, for example, by generating and plotting the posterior covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Given the linear Gaussian settings in the present setup, one can validate the generated posterior covariance matrix against the exact formula (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' One way to construct the posterior covariance matrix is to invoke the posterior model as shown in snippet 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 post_cov = inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='covariance_matrix () Snippet 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Construct and retrieve the posterior covariance matrix Note, however, that one should avoid constructing the covariance matrix for high-dimensional error models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Alterna- tively, matrix-free implementations of covariance (and precision) matrix-vector product should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, to multiply the prior covariance by state, one should call prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='covariance_matvec(state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The error models provide many attributes to efficiently access the statistics of the model, such as covariance diagonal, and trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The posterior covariance matrix, constructed by employing the posterior functionality as in snippet 10 and the covariance matrix evaluated by applying (22) along with the mismatch errors are plotted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='05 1 2 3 ×10−12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Entries of the posterior covariance matrix and the associated errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Left: the posterior covariance matrix generated by snippet 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Middle: the closed-form posterior covariance matrix given by (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Right: RMSE obtained by pointwise comparison of the covariance matrices obtained by solving the inverse problem (left) and by using the closed form (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 19 An OED experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The simulation model is instantiated with a model grid of size nx=5, and the observation operator copies the model state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In fact, one can inspect the model array representation A for this toy linear model by calling model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='get_model_array().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Both the model state and the observation vector sizes here are 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Thus, there are actually 5 candidate sensor locations (observation gridpoints), and one can try to find the optimal subset of sensors by using an OED implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Here we briefly illustrate utilizing an OED object to find the A-optimal design for the toy linear example discussed above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see snippet 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' First, the proper OED module (here following [9]) is imported and is used to create the oed_problem instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The A-optimality criterion is registered (which can be changed later by registering a proper OED criterion), and the OED problem is then solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The results of solving the OED problem, for example, the optimal observational design, are then stored in oed_result, which is an instance of (or derived from) the pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='oed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='OEDResults class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This class provides access to various attributes of the OED problem and the solution process (such as the optimization trajectory and brute force solution if requested).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This gives a taste of how simple it is to create and test DA and OED problems in PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Further details on OED implementations in PyOED are discussed in the following section (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='oed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='relaxed_oed import RelaxedOED 2 oed_problem = RelaxedOED(inverse_problem=inverse_problem , problem_is_linear=True) 3 oed_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="register_optimality_criterion('A-opt') 4 oed_results = oed_problem." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='solve_oed_problem () Snippet 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create an OED object, and solve the OED optimization problem for the toy linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='2 A standard model-constrained OED experiment Parameter identification for an advection-diffusion (AD) model is the foundation of an experiment widely used in the model-constrained OED literature for validating theoretical developments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see, for example, [4, 8, 19, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Comparing independent scientific OED developments is admittedly hard, mainly because of the lack of availability of open software packages developed for OED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This is one of the main goals and features of PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Specifically, PyOED will enable OED researchers to compare the performance of new OED algorithmic approaches with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Moreover, it enables comparison with solution by brute force search for small- to moderate-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In this section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='2 we describe in detail the steps required to construct and solve an OED problem in PyOED with an AD simulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This problem has been utilized independently in several OED developments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see, for example, [4, 8–10, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Here, we show how PyOED can be used to solve and benchmark this OED problem, thus providing a starting point for utilizing and developing multiple approaches for solving OED problems in general in PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Following the same approach as in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1, we start by describing the components of the inverse problem and briefly show how they are initialized in PyOED;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' then we initialize and solve the OED problem using the efficient stochastic approach summarized by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Additionally, we discuss the steps that should be modified to utilize other solution formulations and methods such as the relaxation approach (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The code summarized here is provided in the pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='OED_AD_FE module with additional comments, details, and capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A Jupyter Notebook pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='tutorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='OED_AD_FE is also available and can be used to regenerate the numerical results presented in this section (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 20 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed The simulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The governing equation of the contaminant field 𝑢 = 𝑢(x,𝑡) is modeled by the following AD model equations with the associated boundary conditions: 𝑢𝑡 − 𝜅Δ𝑢 + v · ∇𝑢 = 0 in D × [0,𝑇], 𝑢(𝑥, 0) = 𝜃 in D, 𝜅∇𝑢 · n = 0 on 𝜕D × [0,𝑇], (23) where 𝜅 > 0 is the diffusivity, 𝑇 is the simulation final time and v is the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The domain is D := (0, 1) × (0, 1) with two rectangular regions modeling two buildings inside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The velocity field v is known exactly and is obtained by solving a steady Navier–Stokes equation, with the side walls driving the flow, as detailed in [9, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' To create a simulation model object implementing (23), ground truth of the initial condition, and plot the domain (with finite elements discretization) as well as the velocity field, one can use Snippet 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' the output is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 # Create the simulation model (AD with FE discretization) 2 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='simulation_models import fenics_models 3 model_timestep = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0 4 model = fenics_models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='create_AdvectionDiffusion2D_model(dt=model_timestep) 5 6 # Ground truth of the inversion parameter (initial condition here) 7 true_model_state = model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='create_initial_condition () 8 9 # Plot the domain 10 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='plot_domain () 11 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='plot_velocity_field () Snippet 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create an object representing the simulation model (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Left: finite elements discretization of the domain D of the AD problem (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Right: the velocity field v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In this setup, following [8, 36, 49], we choose a Laplacian prior of the parameter 𝜃 is N �𝜃pr, Γpr �, with Γpr being a discretization of A−2, where A is a Laplacian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In PyOED, a Laplacian prior can be created as described in snippet 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='error_models import Laplacian 2 configs = dict(Vh=model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='parameter_dof , 3 mean=model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='parameter_vector(init_val =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='5) , PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 21 4 gamma =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0, 5 delta=16, 6 random_seed =123, ) 7 prior = Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='DolfinBiLaplacianErrorModel(configs) Snippet 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create a Laplacian prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The observation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A common observational configuration is to consider uniformly distributed candidate sensor locations and solve an OED problem to choose the optimal subset of candidate sensor locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' A uniform observation operator can be created and incorporated in this problem as described in snippet 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='observation_operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='fenics_observation_operators import( 2 create_pointwise_observation_operator , 3 ) 4 num_candidate_sensors = 10 5 obs_oper = create_pointwise_observation_operator(model=model , num_obs_points=num_candidate_sensors , ) Snippet 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create a uniform observation operator with 10 candidate locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Assuming Gaussian observational noise model, a Gaussian observation error model is created as described in 15 1 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='error_models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="Gaussian import GaussianErrorModel 2 obs_noise = GaussianErrorModel( 3 configs ={'size':obs_oper." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="shape[0], 'variance ':0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="1, 'random_seed ':2345} , 4 ) Snippet 15." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create a Gaussian noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The inverse problem: 4DVar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' As with the case of the toy linear problem described above in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1, the elements of the inverse problem here can be created as described by snippet 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Similarly, synthetic observations (data) can be created as in snippet 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Note that all steps followed so far in this example are similar to those followed in the case of the toy linear model discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 import numpy as np 2 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='smoothing import fourDVar 3 checkpoints = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='arange(0, model_timestep *(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='5) , model_timestep) 4 DA_configs = dict(assimilation_window =( checkpoints [0], checkpoints [-1]), 5 model=model , 6 prior_model=prior , 7 observation_operator=obs_oper , 8 observation_error_model=obs_noise , ) 9 inverse_problem = fourDVar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='VanillaFourDVar(configs=DA_configs) Snippet 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create the DA object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 # Create and register observations (perturb observation from true model trajectory) 2 obs_times , true_obs = model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='integrate_state(true_model_state , 3 tspan =( checkpoints [0], checkpoints [-1]), 4 checkpoints=checkpoints [1: ], 5 ) 6 # Perturb with noise and register with the inverse problem 7 for t, y in zip(obs_times , true_obs): 22 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed 8 y = obs_oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='apply(y) 9 yobs = obs_noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='add_noise(y) 10 inverse_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='register_observation(t=t, observation=yobs) Snippet 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create synthetic observations, and associate them to the inverse problem object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The OED problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The discussion above is valid for all model-constrained OED approaches in PyOED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In what follows, we describe the steps needed to create an OED object that follows the stochastic approach (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' An OED object is created in PyOED by following the steps in snippet 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Here, we seek to activate only 4 sensors out of the candidate 10, and we use the trace of the Fisher information matrix (FIM) as the utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' To enforce the budget, we use an ℓ0 penalty term as detailed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 # Create OED problem (stochastic formulation) 2 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='oed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='binary_oed import BinaryOED 3 oed_problem = BinaryOED(inverse_problem=inverse_problem , problem_is_linear=True ,) 4 5 # Register the utility function: trace of the FIM (A-optimality) 6 oed_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="register_optimality_criterion('A-opt') 7 8 # Register penalty/regularization term (with desired budge of only 4 active sensor) 9 penalty_f = lambda design: np." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='abs(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='count_nonzero(design) - 4) 10 oed_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='register_penalty_term( 11 penalty_function=penalty_f , 12 penalty_weight =-1e+8, # Negative as the objective (trace of the FIM below) 13 ) Snippet 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create an OED object implementing the stochastic approach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The OED problem can be solved as described by snippet 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 oed_results = oed_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="solve_oed_problem( 2 oed_evaluation_method='randomized ', # randomized approximation of FIM trace 3 learning_rate =1e-10, 4 batch_size =32, 5 bruteforce=True , # To compare the solution to search by enumeration (bruteforce search) 6 ) Snippet 19." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Solve the OED problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The resulting oed_results object can be used to generate several analysis plots as described by the code in snippet 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This generates multiple standard plots, including the performance of the optimization algorithm over consecutive iterations in Figure 7(left), the optimal sensor locations generated by the optimization algorithm in Figure 7(middle), and comparison of the quality of the solution with respect to brute force search shown in Figure 7(right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 # Create standard plots of the OED results 2 oed_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='plot_results(oed_results) Snippet 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create standard plots for assessing the performance of the optimization routine and the quality of the generated design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' To use other OED formulations to solve the same problem, the user only needs to update the code in the snippet 18 with the proper OED implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For example, the relaxation approach (17) can be used as illustrated in the case of PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments 23 0 100 200 300 400 500 600 700 800 Optimization Iteration 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='75 Objective Value ×108 U 0 150 300 450 600 750 900 1050 Experimental Designs −3 −2 −1 0 1 2 3 Objective Value ×108 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Subset of the plots generated by running the code in snippet 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Left: value of the utility (objective) function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', the penalized OED criterion, over consecutive iterations of the optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Middle: optimal solution, showing optimal sensor locations in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Right: value of the objective of the optimal solution (red star) returned by algorithm 1, compared with the global optimum solution (black 𝑥 mark), and all possible solutions marked as blue circles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' the x-axis shows the indexes of all possible binary designs from 1 to 2𝑛s=10 = 1024, and the y-axis shows the corresponding values of the optimization objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' the linear toy model above in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' see snippet 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Specifically, the relaxation approach (17) can be used to solve the present optimal sensor placement problem by replacing the code in snippet 18 with the following code in snippet 21, which demonstrates the simplicity of PyOED interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Results of snippet 21 are omitted from the presentation here for clarity and because the main goal here is to discuss usage of the approaches in PyOED rather than assessing the quality of the solution approach, which is left for interested users of the package and for future benchmarking research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 1 # Formulate and solve using the relaxation approach 2 from pyoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='oed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='relaxed_oed import PointwiseRelaxedOED 3 oed_problem = PointwiseRelaxedOED(inverse_problem=inverse_problem , problem_is_linear=True , ) 4 oed_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="register_optimality_criterion('A-opt') 5 6 # Add penalty (differentiable function and gradient) 7 penalty_f_l2 = lambda design: np." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='power(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='sum(design)-budget , 2) 8 penalty_f_l2_grad = lambda design: 2 * (np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='sum(design)-budget) * np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='ones_like(design) 9 oed_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='register_penalty_term( 10 penalty_weight =1, 11 penalty_function=penalty_f_l2 , 12 penalty_function_gradient=penalty_f_l2_grad , ) 13 14 # Solve the OED problem 15 oed_results = oed_problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content="solve_oed_problem(oed_evaluation_method='randomized ', ) Snippet 21." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Create an OED object implementing the relaxation approach (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Note, however, that we had to change the penalty function in snippet 21 because the ℓ0 penalty function used in snippet 18 is non differentiable, while the relaxation approach requires the OED objective function to be differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' For details, see, for example, [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' 5 CONCLUDING REMARKS This work describes PyOED, a highly extensible high-level software package for OED in inverse problems and DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' PyOED aims to be a comprehensive Python toolkit for model-constrained OED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The package targets scientists and researchers interested in understanding the details of OED formulations and approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' It is also meant to enable researchers 24 Ahmed Attia and Shady E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Ahmed to experiment with standard and innovative OED technologies within external test problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=', simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The mathematical formulations of OED, inverse problems, and DA overlap significantly, and thus, we plan to extend PyOED with a plethora of Bayesian inversion, DA, and OED implementations as well as new scientific simulation models, observation error models, and observation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' While we focused the discussions in this paper on specific OED approaches, the current version PyOED (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='0) provides several other implementations and emphasizes implementing the essential infrastructure that enables combininig DA and OED elements with other parts of the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The main limitation of the initial version of PyOED is scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Specifically, the concept is developed without parallelization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' In future versions of PyOED, scalability will be achieved by adding message passing interface (MPI) support, for example using the mpi4py package, and by supporting PETSc [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Performance will also be enhanced by converting or rewriting suitable parts of the package in Cython.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' ACKNOWLEDGMENTS This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Department of Energy, Office of Science, under contract number DE-AC02-06CH11357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' This work was supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' Department of 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publicly, by or on behalf of the Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content=' http://energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} +page_content='gov/downloads/doe-public-access-plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE_T4oBgHgl3EQf1hz1/content/2301.08336v1.pdf'} diff --git a/ftE2T4oBgHgl3EQfbwdB/content/tmp_files/2301.03888v1.pdf.txt b/ftE2T4oBgHgl3EQfbwdB/content/tmp_files/2301.03888v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..99dfae5e4ecd13e51770254ea0aac509ea65c837 --- /dev/null +++ b/ftE2T4oBgHgl3EQfbwdB/content/tmp_files/2301.03888v1.pdf.txt @@ -0,0 +1,856 @@ +Joint Hybrid Beamforming and User Scheduling +for Multi-Satellite Cooperative Networks +Xuan Zhang∗, Shu Sun∗, Meixia Tao∗, Qin Huang† and Xiaohu Tang‡ +∗Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China +†School of Electronic and Information Engineering, Beihang University, Beijing, China +‡School of Information Science and Technology, Southwest Jiaotong University, Chengdu, Sichuan, China +Emails: ∗{zhangxuan1998, shusun, mxtao}@sjtu.edu.cn, †qhuang.smash@gmail.com, ‡xhutang@swjtu.edu.cn +Abstract—In this paper, we consider a cooperative commu- +nication network where multiple satellites provide services for +ground users (GUs) (at the same time and on the same frequency). +The communication and computational resources on satellites are +usually restricted and the satellite-GU link determination affects +the communication performance significantly when multiple +satellites provide services for multiple GUs in a collaborative +manner. Therefore, considering the limitation of the on-board +radio-frequency chains, we first propose a hybrid beamforming +method consisting of analog beamforming for beam alignment +and digital beamforming for interference mitigation. Then, to +establish appropriate connections between satellites and GUs, we +propose a heuristic user scheduling algorithm which determines +the connections according to the total spectral efficiency (SE) +increment of the multi-satellite cooperative network. Next, a joint +hybrid beamforming and user scheduling scheme is proposed +to dramatically improve the performance of the multi-satellite +cooperative network. Moreover, simulations are conducted to +compare the proposed schemes with representative baselines and +analyze the key factors influencing the performance of the multi- +satellite cooperative network. It is shown that the proposed joint +beamforming and user scheduling approach can provide 47.2% +SE improvement on average as compared with its non-joint +counterpart. +Index Terms—Satellite communication, hybrid beamforming, +user scheduling. +I. INTRODUCTION +Satellites have a distinctive ability of covering wide geo- +graphical areas through a minimum amount of infrastructure +on the ground. Currently, the field of satellite communica- +tions is drawing increasing attention in the global telecom- +munications market. Several network operators start using +satellites in backhaul infrastructures for connectivity and for +5G system integration [1]. Furthermore, satellites are likely +to play an increasingly important role in the 6G era to +provide global seamless coverage and support space-terrestrial +integrated networks. The integrated architectures, applications, +and challenges of satellite-terrestrial networks toward 6G are +presented in [2]. +In [3], the authors introduced several satellite communica- +tion networks which are categorized by different architectures, +including land mobile satellite communication networks, hy- +brid satellite-terrestrial relay networks, and satellite-terrestrial +This work is supported by the National Natural Science Foundation of +China under grant 61941106. +integrated networks. The authors in [4] made a technical com- +parison of three low-Earth-orbit (LEO) satellite constellation +systems: OneWeb’s, SpaceX’s, and Telesat’s. The advantages +and technical challenges of multi-satellite cooperative trans- +mission systems in 5G were discussed in [5]. +In satellite communication systems, due to the limitation +of the on-board radio-frequency (RF) chains caused by the +confined device complexity and transmission power, hybrid +analog and digital beamforming is a promising method to +balance performances and hardware constraints. In [6], a +hybrid beamforming strategy was used for massive multiple- +input multiple-output LEO satellite communications. Later, +the authors in [7] investigated hybrid beamforming method +for reconfigurable intelligent surface-assisted secure integrated +terrestrial-aerial networks. Besides, user scheduling is another +factor that affects the performances of satellite communication +networks. In [8], an adaptive user scheduling method was +proposed to mitigate intra-beam and inter-beam interference +under the single-satellite scenario. Nevertheless, cooperation +among multiple satellites was not taken into account in [6]– +[8]. The authors in [9] proposed a multilevel clustering algo- +rithm and a cross-cluster grouping algorithm to realize user +scheduling, which considered the multi-satellite cooperation, +but the user scheduling was only dependent on the positions +of users. Furthermore, full frequency reuse (FFR) is widely +adopted to improve the spectral efficiency (SE). However, +aggressive frequency reuse will inevitably cause severe inter- +beam interference. As such, multi-satellite collaboration via +cooperative digital beamforming is crucial for inter-beam in- +terference mitigation. The authors in [10] compared the perfor- +mances of several common digital beamforming designs: zero- +forcing (ZF), regularized ZF and MMSE digital beamformer. +In [8] and [11], beamforming design and resource allocation +were considered for the integrated terrestrial-satellite network. +The authors in [12] studied the transmit beamforming design +for spectral coexistence of satellite and terrestrial networks. +However, to our best knowledge, there is little existing work +on the joint design of beamforming and user scheduling for +multi-satellite cooperative networks without the assistance of +terrestrial systems. +In this paper, we investigate beamforming and user schedul- +ing in a multi-satellite cooperative network with FFR. Con- +sidering the limitation of on-board device complexity and +arXiv:2301.03888v1 [cs.IT] 10 Jan 2023 + +transmission power, we propose a hybrid beamforming method +based on the hybrid architecture of satellite antennas. It +consists of analog beamforming based on a discrete Fourier +transform (DFT) codebook for beam alignment and digital +beamforming for interference mitigation. The user scheduling +method is also important when multiple satellites work coop- +eratively. To reasonably arrange multiple satellites to provide +services for ground users (GUs), we propose a low-complexity +heuristic user scheduling algorithm. Considering the intrinsic +connection between beamforming and user scheduling, we +propose a joint hybrid beamforming and user scheduling +scheme for maximizing the network SE. Simulation results +demonstrate that the proposed joint scheme outperforms a +representative non-joint baseline strategy dramatically and can +achieve 47.2% SE increase on average. +II. SYSTEM MODEL +A. System Architecture +As depicted in Fig. 1, we consider a downlink communi- +cation scenario of a multi-satellite communication network +at a certain moment, where multiple satellites cooperatively +provide services for GUs. We assume that there are Nu GUs +requesting services and Ns satellites visible to at least one of +these GUs. We use Vg to denote the set of visible satellites of +GU g and Vs for the set of GUs that satellite s is serving. We +also assume that all the satellites are equipped with regener- +ative payload and belong to an LEO constellation operating +in the Ka-band with FFR adopted. Furthermore, there are +optical inter-satellite links (ISLs) to exchange data among +satellites, and the satellites can perform on-board distributed +computing to share computation load [13]. The earth station +periodically sends topological relationships of the constellation +to the satellites via its line of sight (LoS) and satellites can +share the topological relationships through ISLs. +Consider that each GU is equipped with a very small +aperture terminal (VSAT) which is a single antenna system, +and each satellite is equipped with a uniform planner array +(UPA). Different from purely analog or digital beamforming +architecture, the UPA here adopts a hybrid architecture. The +UPA is composed of Nb = N sub +x +× N sub +y +sub-arrays where +x denotes the axis pointing in the direction of the satellite’s +movement and y denotes the axis pointing in the direction +orthogonal to the satellite’s movement. Each sub-array consists +of N = Nx × Ny antenna elements, and each sub-array is +connected with one RF chain, generating one independent spot +beam. Therefore, one satellite can generate Nb independent +spot beams at most. Besides, we assume that each independent +beam serves one GU. This indicates that one satellite can +provide services for up to Nb GUs simultaneously. +B. Channel Model +We consider the scenario where the satellite-GU link is +under the condition of LoS and clear sky (no rain and cloud +attenuation), and all the GUs are distributed in suburban areas. +The channel is modeled according to the technical reports of +3GPP [14] and ITU-R [15]. The multiple-input single-output +Earth Station +ISL +ISL +Fig. 1. System architecture of the multi-satellite cooperative communication +network in this work. ISL denotes inter-satellite link. +(MISO) channel between satellite s, for s ∈ {1, 2, . . . , Ns} +and GU g for g ∈ {1, 2, . . . , Nu} can be modeled as +hsg = ξsg · hsgs, +(1) +where ξsg and hsgs ∈ CN×1 stand for the radio propagation +loss and the small-scale fading channel between satellite s and +GU g, respectively. Here, ξsg is directly relevant to the large- +scale path loss (PL). The PL can be calculated as follows: +PL[dB] = PLb[dB] + PLg[dB] + PLs[dB], +(2) +where PLb denotes the basic path loss, including the distance- +and frequency-dependent free space path loss and the log- +normal distributed shadow fading, PLg denotes the attenuation +due to atmospheric gasses, and PLs denotes the attenuation due +to tropospheric scintillation. +The small-scale fading channel model follows a Loo distri- +bution, where the received signal is the sum of two compo- +nents: the direct path and the diffuse multipath. We assume +that there are Ncl clusters with Nray propagation paths in each +cluster. The small-scale fading channel hsgs can be modeled +as +hsgs = δ +� +m0 aT(φsg, θsg) + +Ncl +� +l=1 +Nray +� +i=1 +mliaT(φsg,li, θsg,li) +� +, +(3) +where m0 and mli are complex coefficients of the direct path +and the diffuse multipath, respectively. The amplitude of m0 is +subject to the normal distribution, while the amplitude of mli +obeys the Rayleigh distribution. The phases of both m0 and +mli follow a uniform distribution from 0 to 2π. The normal- +ization factor δ is introduced to satisfy E +� +∥hsgs∥2� += 1. The +vector aT(φ, θ) ∈ CN×1 is the normalized antenna steering +vector of the satellite’s sub-array, which can be written as +aT(φ, θ) = +1 +√ +N +� +1, . . . , e−j 2π +λ d(p cos θ cos φ+q cos θ sin φ), +. . . , e−j 2π +λ d((Nx−1) cos θ cos φ+(Ny−1) cos θ sin φ)�T +, +(4) +where λ and d are the carrier wavelength and the antenna +element spacing, respectively. In our study, we assume d = λ +2 + +to guarantee that there is no grating lobe when a beam is +steered towards ± 90◦. Besides, φ and θ denote the azimuth +and elevation angles from the perspective of satellite. +C. Signal Model and Problem Formulation +As mentioned in Section II-A, multiple satellites provide +communication services for their GUs cooperatively with FFR. +Therefore, each GU will suffer interference from all the +satellites that are in this GU’s LoS. When receiving signals, the +GU antenna will aim at the serving satellite and the intended +signal and interference from this satellite will experience the +maximum gain of the GU antenna. While the interference from +other visible satellites will experience a gain decrease due to +the off-boresight angles of these satellite-GU links and the +narrow-beam characteristics of the VSAT considered herein. +Note that the set of visible satellites of each GU is known, +but the specific links between satellites and GUs are unknown +and need to be determined. Based on this fact, we introduce a +discrete variable αsg to indicate whether the link is established, +where αsg = 1 if satellite s is providing service for GU g, +and αsg = 0 otherwise. +The received signal of GU g can be expressed as [8], [11] +yg = +� +s∈Vg +αsghH +sgwsgxg + +� +g′̸= g +� +s∈Vg +αsg′ hH +sgwsg′ xg′ + ng, +(5) +where hsg ∈ CN×1 is the channel vector between satellite s +and GU g, wsg ∈ CN×1 represents the beamforming vector, +and xg is the requested data of GU g, which is assumed to +be independent and satisfy E +� +|xg|2� += 1. The first term in +(5) is the intended data for GU g, the second term is the +interference coming from the communication services for the +other GUs, and the third term is the complex additive white +Gaussian noise following the distribution CN +� +0, σ2 +s +� +, where +σ2 +s denotes the noise power. +Then the received signal-to-interference-plus-noise ratio +(SINR) of GU g can be obtained as +γg = +| � +s∈Vg αsghH +sgwsg|2 +� +g′̸=g | � +s∈Vg αsg′ hH +sgwsg′ |2 + σ2s +. +(6) +According to the Shannon theorem, the SE per channel use +of GU g can be calculated by +Rg = log2 (1 + γg). +(7) +As a result, the total SE of the multi-satellite cooperative +network can be expressed as +R = +Nu +� +g=1 +Rg. +(8) +Our objective is to maximize the total SE of the multi- +satellite cooperative network by means of calculating the +beamforming vector wsg and adjusting the satellite-GU links. +Therefore, we formulate the optimization problem (OP) as +follows: +OP : +max +{wsg,αsg} +Nu +� +g=1 +log2 (1 + γg) +(9) +s.t. +C1 : +� +g +αsg∥wsg∥2 ≤ PT, ∀s ∈ {1, 2, . . . , Ns}, +(10) +C2 : +� +g +αsg ≤ Nb, ∀s ∈ {1, 2, . . . , Ns}, +(11) +C3 : +� +s∈Vg +αsg = 1, ∀g ∈ {1, 2, . . . , Nu}, +(12) +C4 : αsg ∈ {0, 1}, ∀s, g. +(13) +Here, constraint C1 is the power constraint of each satellite, +constraint C2 means that the number of GUs connected to +the same satellite cannot exceed the maximum number of +spot beams that one satellite can generate, namely Nb, and +constraint C3 is the GU connection constraint. Notably, the +OP is not always feasible because of the coexistence of C2 +and C3. When C2 and C3 contradict each other, we give +priority to guaranteeing C2 and try to connect as many GUs as +possible. The OP cannot be solved directly due to the fact that +the objective function and constraints are non-convex. Thus, +we propose to solve it by the following steps: +1) Given hsg in (1), the analog beamforming vector wA +sg is +obtained based on a DFT codebook (with more details +to be shown in Algorithm 1). +2) The vector wA +sg is set as the initial beamforming vector. +3) Based on 2), we propose two schemes to solve OP, +which will be further discussed in the following sections. +III. HYBRID BEAMFORMING +A. Analog Beamforming Based on Codebook +We consider analog beamforming for beam alignment based +on a codebook, which is widely used in terrestrial systems. +Codebook-based beamforming design can reduce the overhead +of channel state information (CSI) feedback. In this work, +we assume that the GU side has perfect CSI and the GU- +satellite CSI feedback link is lossless. As described in Section +II-A, every satellite is equipped with a UPA, thus the two- +dimensional (2D) DFT codebook is applicable and it can +be seen as the synthesis of two 1D DFT codebooks in the +directions of x and y axes, Dx, Dy. The N × N 2D DFT +codebook matrix, denoted as D, can be expressed as +D = Dx ⊗ Dy. +(14) +The core idea of analog beamforming herein is to select +the best K (≤ N) codewords from D and combine them into +a new codeword that satisfies the equal-amplitude constraint +of analog beamforming. Considering the communication over- +head of CSI feedback, K should not be too large and we +assume K = 4 based on the results of trial simulations. The +steps of the proposed analog beamforming method are given +in Algorithm 1. + +Algorithm 1: Codebook-Based Analog Beamforming +Input: GU-side channel vector hsg, ∀s, g, codebook D. +Output: Analog beamforming vector wA +sg. +1 For each codeword D:,k, calculate |hH +sgD:,k|2 and find +the best K codewords maximizing it, c1, . . . , cK; +2 DK = [c1, . . . , cK], solve the equations DKx = hsg +and obtain the least square solution ˆx = (DK)†hsg; +3 The GU sends the codeword combination coefficients +ˆx and codewords’ indices to the satellite; +4 Satellite-side combination: w +′ +sg = DKˆx; +5 for i ∈ [1, N] do +6 +wA +sg(i) = w +′ +sg(i) +1 +√ +N +|w′ +sg(i)| +B. Digital Beamforming +Based on the link information between satellites and GUs, +the channel matrix of satellite s can be written as +Hs = [. . . , hsg, . . . ]H ∈ CN s +u×N, g ∈ Vs, +(15) +where N s +u is the number of GUs that satellite s is serving. +Similarly, the analog beamforming matrix of satellite s can be +written as +FA +s = +� +. . . , wA +sg, . . . +� +∈ CN×N s +u, g ∈ Vs. +(16) +Hence, we can write the generalized channel matrix between +satellite s and the GUs as +�Hs = HsFA +s , +(17) +thus the hybrid beamforming is reduced to a digital beam- +forming problem to mitigate the inter-beam interference of +satellite s. In this work, we adopt the regularized ZF and the +corresponding digital beamforming matrix is +FD +s = √η �HH +s ( �Hs �HH +s + βIN s +u)−1 ∈ CN s +u×N s +u, +(18) +where √η is the power scaling factor to guarantee the satellite +is operating at its maximum power, and β is an adjustable +parameter where βopt = N s +uσ2 +s +PT +in the large system limit [16]. +Thus, combining (16) and (18), the hybrid beamforming +matrix can be expressed as +FHY +s += FA +s · FD +s = [. . . , wsg, . . . ] , ∈ CN×N s +u, g ∈ Vs. +(19) +IV. USER SCHEDULING AND IMPLEMENTATION SCHEMES +A. User Scheduling +As described in Section II-C, the links between multi- +ple satellites and GUs need to be determined. The optimal +exhaustive search is infeasible here since the computational +complexity grows exponentially with the number of GUs. +Thus, we propose a heuristic user scheduling algorithm which +can achieve a good performance with polynomial complexity +as shown in Algorithm 2. Therein, L denotes an Ns ×Nu link +matrix where αsg lies in its s-th row and g-th column, and +Lsg denotes the matrix whose s-th row and g-th column is 1 +Algorithm 2: Heuristic User Scheduling Algorithm +Input: Channel matrix Hs, beamforming matrix Fs, +and the set of visible satellites Vg, +∀s ∈ {1, 2, . . . , Ns}, ∀g ∈ {1, 2, . . . , Nu}. +Output: Link matrix L. +1 Initialize +S = {1, . . . , Ns}, Gn +�� +n=Nu = {1, . . . , Nu}, L = 0; +2 for g ∈ [1, Nu] do +3 +if length(Vg) == 1 then +4 +L(Vg, g) = 1; +5 +Remove g from Gn and n = n − 1; +6 repeat +7 +for each possible link Lsg, s ∈ S, g ∈ Gn do +8 +△Rsg = R(L + Lsg, Hs, Fs) − R(L, Hs, Fs); +9 +[ˆs, ˆg] = arg max +s,g +△Rsg; +10 +if satellite ˆs has spare resource then +11 +L(ˆs, ˆg) = 1; +12 +remove ˆg from Gn and n = n − 1; +13 +else +14 +remove ˆs from S; +15 until n == 0; +and other places are 0. The set of satellites that have spare +resource is denoted as S. Here, the resource constraint is that +the number of GUs that one satellite is serving simultaneously +can not exceed Nb. The set of unserved GUs is denoted by Gn, +where n is the number of GUs in this set. △R indicates the +increment of total SE. The total SE is mainly associated with +the link matrix L, channel matrix Hs and the beamforming +matrix Fs, which is equivalent to (8) and can be abstracted as +R = R(L, Hs, Fs), +(20) +where R indicates a function for calculating the total SE. +B. Separate & Joint Hybrid Beamforming and User Schedul- +ing Schemes +In Section III, we have introduced the hybrid beamforming +method. Within the hybrid beamforming, the analog beam- +forming can be completed independently. As for the digital +beamforming and user scheduling, there are two ways: +1) Separate (SHU): In the SHU scheme, we perform +digital beamforming and user scheduling separately and in- +dependently. Analog beamforming matrix FA +s is taken as the +input of Algorithm 2 and user scheduling is performed based +on FA +s , i.e., Fs = FA +s . Digital beamforming is conducted after +user scheduling according to (18), utilizing the final links to +mitigate the interference and improve the performance. Finally, +we use FHY +s +in (19) to calculate the total SE when SHU is +adopted. +2) Joint (JHU): The JHU scheme is based on alternating +optimization. Different from SHU, user scheduling and digital +beamforming in JHU are designed jointly. The beamforming + +4 +5 +6 +7 +8 +9 +10 +11 +Time (a.m.) +0 +2 +4 +6 +8 +Total Spectral Efficiency (bps/Hz) +Total SEs in 24 Experiments +AU +SHU +JHU +Mean Total SEs of 24 Experiments +1.7726 +1.9535 +2.8764 +AU +SHU +JHU +0 +0.5 +1 +1.5 +2 +2.5 +3 +Spectral Efficiency(bps/Hz) +Fig. 2. Total SEs and mean total SEs of different schemes. +matrix is updated in real time within the user scheduling. +As in SHU, FA +s is taken as the initial input of Algorithm +2. The difference is that each time before calculating the +SE increment △Rsg, the hybrid beamforming matrix FHY +s +is +computed based on the current link matrix. If we abstract (18) +and (19) as +FHY +s += F(L, Hs, FA +s ), +(21) +where F is a function for calculating the hybrid beamforming +matrix, then Step 8 of Algorithm 2 when JHU is adopted can +be replaced by +△Rsg = R(L + Lsg, Hs, F(L + Lsg, Hs, FA +s )) +− R(L, Hs, F(L, Hs, FA +s )). +(22) +V. PERFORMANCE EVALUATION +We consider a 6 × 8 LEO Walker constellation with an +inclination of 40◦ as an example, which has the ability to +achieve full-time coverage of the entire China according to the +simulation result of a popular aerospace simulation software +Systems Tool Kit (STK) [18]. We use STK to simulate +the movement of the constellation and we sample every 20 +minutes within eight hours from 4am to 12pm in Beijing +time, resulting in 24 experiments to test the performances of +the proposed algorithms and schemes. Besides, all GUs are +located in suburban areas of 80 representative cities in China. +TABLE I +SIMULATION PARAMETERS +Parameter +Value +Number of orbital planes +6 +Number of satellites per orbital plane +8 +Orbital plane inclination +40◦ +Orbital height +1200 km (LEO) [17] +Number of sub-arrays +Nsub +x = 8,Nsub +y = 4 +Number of antenna elements per sub-array +Nx = Ny = 8 +Number of GUs +Nu = 80 +Downlink carrier frequency +20 GHz [17] +Bandwidth +400 MHz [17] +Receiver noise temperature +24 dBK [6] +Satellite antenna gain* +21.5 dBi +VSAT maximum antenna gain +40 dBi [17] +Transmission power per satellite +PT = 80 W +* The satellite antenna is a kind of phased array antenna, whose gain +can be calculated according to the number and size of antenna +elements and the carrier wavelength. +The simulation parameters are given in Table I. Note that the +proposed algorithm and main observations still hold for other +types of LEO constellations. +The baseline scheme is analog beamforming and user +scheduling scheme, denoted as AU. The specific steps of AU +scheme are: (1) analog beamforming as described in Section +III-A, (2) user scheduling using Algorithm 2, and (3) power +scaling. The SE performances of AU, SHU and JHU are shown +in Fig. 2, where the left picture illustrates the total SEs in +24 experiments and the right picture shows the corresponding +mean total SEs averaged over the 24 experiments. It can +be seen that AU performs the worst because of the lack of +interference mitigation. By performing digital beamforming +after user scheduling to mitigate interference, the performance +of SHU increases by 10.2% compared with AU on average. +However, SHU cannot make full use of link information since +digital beamforming is implemented independently of user +scheduling. In JHU, the digital beamforming matrix is updated +in real time when calculating the total SE increment and es- +tablishing links, leading to increase of SE by 47.2% compared +with SHU and 62.3% compared with AU on average. +From Fig. 2, we can observe that the network total SE +fluctuates significantly with time. The reason is that the +topological relationships between satellites and GUs change +rapidly with time. Taking GUs in five typical cities as exam- +ples, we calculate their SEs with JHU and show the results in +Fig. 3. These five cities lie in the northern, eastern, middle, +western, and southern part of China, respectively. The SEs +of GUs in these five cities fluctuate differently due to their +different geographical positions and topological relationships. +For the GU in each city, the SE varies with time because +of the change of topological relationship, which is caused by +satellites’ movement. These five GUs have very different SEs +at the same time because of different sets of visible satellites +for these GUs and different elevation angles and distances +between satellites and GUs, leading to different path losses. +The different sets of visible satellites and path losses are both +caused by discrepant geographical positions of these GUs. +In Fig. 3, we can also find that the maximum SEs of GUs +in Kashi and Nansha are higher than those in other cities. Part +of the reason lies in the density of GUs. Sparse GUs refer to + +4 +6 +8 +10 +12 +Time (a.m.) +0 +0.2 +0.4 +0.6 +0.8 +Per-user SE (bps/Hz) +Beijing +(a) Beijing +4 +6 +8 +10 +12 +Time (a.m.) +0 +0.2 +0.4 +0.6 +0.8 +Per-user SE (bps/Hz) +Shanghai +(b) Shanghai +4 +6 +8 +10 +12 +Time (a.m.) +0 +0.2 +0.4 +0.6 +0.8 +Per-user SE (bps/Hz) +Wuhan +(c) Wuhan +4 +6 +8 +10 +12 +Time (a.m.) +0 +0.2 +0.4 +0.6 +0.8 +Per-user SE (bps/Hz) +Kashi +(d) Kashi +4 +6 +8 +10 +12 +Time (a.m.) +0 +0.2 +0.4 +0.6 +0.8 +Per-user SE (bps/Hz) +Nansha +(e) Nansha +Fig. 3. Per-user SEs of GUs in five typical cities. +TABLE II +SE STATISTICS OF DENSE AND SPARSE GUS +Mean Value of SEs (bps/Hz) +Variance of SEs (bps/Hz)2 +Dense GUs +0.028569 +0.035576 +Sparse GUs +0.049798 +0.064577 +the users in an area where inter-GU distances are all greater +than D. The opposite holds for dense GUs. In our study, we +set D to 400km as an example. Among these 80 GUs, there +are 12 sparse GUs and 68 dense GUs. Correspondingly, the +GUs in Kashi and Nansha are sparse GUs and the other three +GUs are dense GUs. In order to obtain more comprehensive +observations, we calculate the SEs of all dense and sparse GUs +in 24 experiments and obtain their respective mean value and +variance in Table II. The mean SE of sparse GUs are nearly +twice as much as that of dense GUs. The potential reasons are: +The sparse GUs are surrounded by fewer GUs and suffer less +interference from others hence having higher SINR; Most of +the sparse GUs are located in the border areas of China, thus +they have greater probability of monopolizing one satellite and +getting larger transmission power. Additionally, the variance +of SE of sparse GUs is much larger than that of dense +GUs, which indicates that the SEs of sparse GUs fluctuate +more substantially than dense GUs due to their geographical +positions. +VI. CONCLUSION +In this paper, we first provide a hybrid beamforming ar- +chitecture of UPAs which is suitable for satellites because of +the limitation of on-board RF chains. According to the hybrid +architecture, we introduce an analog beamforming method +based on the 2D DFT codebook which generates a desired +codeword by linearly combining four selected codewords. +Then digital beamforming in accordance with regularized ZF +is employed to mitigate the inter-beam interference and scale +the transmission power. Moreover, we propose a heuristic user +scheduling algorithm to determine the links between satellites +and GUs. Subsequently, we propound two implementation +schemes: a separate scheme and a joint scheme. 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Qian, “Joint beamforming design +and resource allocation for terrestrial-satellite cooperation system,” IEEE +Transactions on Communications, vol. 68, no. 2, pp. 778–791, 2020. +[12] S. K. Sharma, S. Chatzinotas, and B. Ottersten, “Transmit beamforming +for spectral coexistence of satellite and terrestrial networks,” in 8th In- +ternational Conference on Cognitive Radio Oriented Wireless Networks, +pp. 275–281, 2013. +[13] B. C. Gunter and D. C. Maessen, “Space-based distributed computing +using a networked constellation of small satellites,” Journal of Space- +craft and Rockets, vol. 50, no. 5, pp. 1086–1095, 2013. +[14] 3GPP, “Study on new radio (NR) to support non-terrestrial networks,” +Technical report (TR) 38.811, 3rd Generation Partnership Project +(3GPP), 10 2020. Version 15.4.0. +[15] P. Series, “Propagation data required for the design of earth-space land +mobile telecommunication systems,” Recommendation ITU-R, pp. 681– +10, 2017. +[16] R. Muharar and J. 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Version 16.1.0. + +[18] “System Tool Kit (STK).” https://licensing.agi.com/stk. + diff --git a/ftE2T4oBgHgl3EQfbwdB/content/tmp_files/load_file.txt b/ftE2T4oBgHgl3EQfbwdB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9f0bd65eedc38ecc2a5df3b14f9fa6e9dff12c7 --- /dev/null +++ b/ftE2T4oBgHgl3EQfbwdB/content/tmp_files/load_file.txt @@ -0,0 +1,452 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf,len=451 +page_content='Joint Hybrid Beamforming and User Scheduling for Multi-Satellite Cooperative Networks Xuan Zhang∗, Shu Sun∗, Meixia Tao∗, Qin Huang† and Xiaohu Tang‡ ∗Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China †School of Electronic and Information Engineering, Beihang University, Beijing, China ‡School of Information Science and Technology, Southwest Jiaotong University, Chengdu, Sichuan, China Emails: ∗{zhangxuan1998, shusun, mxtao}@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='cn, †qhuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='smash@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='com, ‡xhutang@swjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='cn Abstract—In this paper, we consider a cooperative commu- nication network where multiple satellites provide services for ground users (GUs) (at the same time and on the same frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The communication and computational resources on satellites are usually restricted and the satellite-GU link determination affects the communication performance significantly when multiple satellites provide services for multiple GUs in a collaborative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Therefore, considering the limitation of the on-board radio-frequency chains, we first propose a hybrid beamforming method consisting of analog beamforming for beam alignment and digital beamforming for interference mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Then, to establish appropriate connections between satellites and GUs, we propose a heuristic user scheduling algorithm which determines the connections according to the total spectral efficiency (SE) increment of the multi-satellite cooperative network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Next, a joint hybrid beamforming and user scheduling scheme is proposed to dramatically improve the performance of the multi-satellite cooperative network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Moreover, simulations are conducted to compare the proposed schemes with representative baselines and analyze the key factors influencing the performance of the multi- satellite cooperative network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' It is shown that the proposed joint beamforming and user scheduling approach can provide 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2% SE improvement on average as compared with its non-joint counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Index Terms—Satellite communication, hybrid beamforming, user scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' INTRODUCTION Satellites have a distinctive ability of covering wide geo- graphical areas through a minimum amount of infrastructure on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Currently, the field of satellite communica- tions is drawing increasing attention in the global telecom- munications market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Several network operators start using satellites in backhaul infrastructures for connectivity and for 5G system integration [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Furthermore, satellites are likely to play an increasingly important role in the 6G era to provide global seamless coverage and support space-terrestrial integrated networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The integrated architectures, applications, and challenges of satellite-terrestrial networks toward 6G are presented in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In [3], the authors introduced several satellite communica- tion networks which are categorized by different architectures, including land mobile satellite communication networks, hy- brid satellite-terrestrial relay networks, and satellite-terrestrial This work is supported by the National Natural Science Foundation of China under grant 61941106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' integrated networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The authors in [4] made a technical com- parison of three low-Earth-orbit (LEO) satellite constellation systems: OneWeb’s, SpaceX’s, and Telesat’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The advantages and technical challenges of multi-satellite cooperative trans- mission systems in 5G were discussed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In satellite communication systems, due to the limitation of the on-board radio-frequency (RF) chains caused by the confined device complexity and transmission power, hybrid analog and digital beamforming is a promising method to balance performances and hardware constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In [6], a hybrid beamforming strategy was used for massive multiple- input multiple-output LEO satellite communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Later, the authors in [7] investigated hybrid beamforming method for reconfigurable intelligent surface-assisted secure integrated terrestrial-aerial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Besides, user scheduling is another factor that affects the performances of satellite communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In [8], an adaptive user scheduling method was proposed to mitigate intra-beam and inter-beam interference under the single-satellite scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Nevertheless, cooperation among multiple satellites was not taken into account in [6]– [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The authors in [9] proposed a multilevel clustering algo- rithm and a cross-cluster grouping algorithm to realize user scheduling, which considered the multi-satellite cooperation, but the user scheduling was only dependent on the positions of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Furthermore, full frequency reuse (FFR) is widely adopted to improve the spectral efficiency (SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' However, aggressive frequency reuse will inevitably cause severe inter- beam interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' As such, multi-satellite collaboration via cooperative digital beamforming is crucial for inter-beam in- terference mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The authors in [10] compared the perfor- mances of several common digital beamforming designs: zero- forcing (ZF), regularized ZF and MMSE digital beamformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In [8] and [11], beamforming design and resource allocation were considered for the integrated terrestrial-satellite network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The authors in [12] studied the transmit beamforming design for spectral coexistence of satellite and terrestrial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' However, to our best knowledge, there is little existing work on the joint design of beamforming and user scheduling for multi-satellite cooperative networks without the assistance of terrestrial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In this paper, we investigate beamforming and user schedul- ing in a multi-satellite cooperative network with FFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Con- sidering the limitation of on-board device complexity and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='03888v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='IT] 10 Jan 2023 transmission power, we propose a hybrid beamforming method based on the hybrid architecture of satellite antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' It consists of analog beamforming based on a discrete Fourier transform (DFT) codebook for beam alignment and digital beamforming for interference mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The user scheduling method is also important when multiple satellites work coop- eratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' To reasonably arrange multiple satellites to provide services for ground users (GUs), we propose a low-complexity heuristic user scheduling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Considering the intrinsic connection between beamforming and user scheduling, we propose a joint hybrid beamforming and user scheduling scheme for maximizing the network SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Simulation results demonstrate that the proposed joint scheme outperforms a representative non-joint baseline strategy dramatically and can achieve 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2% SE increase on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' System Architecture As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 1, we consider a downlink communi- cation scenario of a multi-satellite communication network at a certain moment, where multiple satellites cooperatively provide services for GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' We assume that there are Nu GUs requesting services and Ns satellites visible to at least one of these GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' We use Vg to denote the set of visible satellites of GU g and Vs for the set of GUs that satellite s is serving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' We also assume that all the satellites are equipped with regener- ative payload and belong to an LEO constellation operating in the Ka-band with FFR adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Furthermore, there are optical inter-satellite links (ISLs) to exchange data among satellites, and the satellites can perform on-board distributed computing to share computation load [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The earth station periodically sends topological relationships of the constellation to the satellites via its line of sight (LoS) and satellites can share the topological relationships through ISLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Consider that each GU is equipped with a very small aperture terminal (VSAT) which is a single antenna system, and each satellite is equipped with a uniform planner array (UPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Different from purely analog or digital beamforming architecture, the UPA here adopts a hybrid architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The UPA is composed of Nb = N sub x × N sub y sub-arrays where x denotes the axis pointing in the direction of the satellite’s movement and y denotes the axis pointing in the direction orthogonal to the satellite’s movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Each sub-array consists of N = Nx × Ny antenna elements, and each sub-array is connected with one RF chain, generating one independent spot beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Therefore, one satellite can generate Nb independent spot beams at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Besides, we assume that each independent beam serves one GU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' This indicates that one satellite can provide services for up to Nb GUs simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Channel Model We consider the scenario where the satellite-GU link is under the condition of LoS and clear sky (no rain and cloud attenuation), and all the GUs are distributed in suburban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The channel is modeled according to the technical reports of 3GPP [14] and ITU-R [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The multiple-input single-output Earth Station ISL ISL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' System architecture of the multi-satellite cooperative communication network in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' ISL denotes inter-satellite link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' (MISO) channel between satellite s, for s ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , Ns} and GU g for g ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , Nu} can be modeled as hsg = ξsg · hsgs, (1) where ξsg and hsgs ∈ CN×1 stand for the radio propagation loss and the small-scale fading channel between satellite s and GU g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Here, ξsg is directly relevant to the large- scale path loss (PL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The PL can be calculated as follows: PL[dB] = PLb[dB] + PLg[dB] + PLs[dB], (2) where PLb denotes the basic path loss, including the distance- and frequency-dependent free space path loss and the log- normal distributed shadow fading, PLg denotes the attenuation due to atmospheric gasses, and PLs denotes the attenuation due to tropospheric scintillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The small-scale fading channel model follows a Loo distri- bution, where the received signal is the sum of two compo- nents: the direct path and the diffuse multipath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' We assume that there are Ncl clusters with Nray propagation paths in each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The small-scale fading channel hsgs can be modeled as hsgs = δ � m0 aT(φsg, θsg) + Ncl � l=1 Nray � i=1 mliaT(φsg,li, θsg,li) � , (3) where m0 and mli are complex coefficients of the direct path and the diffuse multipath, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The amplitude of m0 is subject to the normal distribution, while the amplitude of mli obeys the Rayleigh distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The phases of both m0 and mli follow a uniform distribution from 0 to 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The normal- ization factor δ is introduced to satisfy E � ∥hsgs∥2� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The vector aT(φ, θ) ∈ CN×1 is the normalized antenna steering vector of the satellite’s sub-array, which can be written as aT(φ, θ) = 1 √ N � 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , e−j 2π λ d(p cos θ cos φ+q cos θ sin φ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , e−j 2π λ d((Nx−1) cos θ cos φ+(Ny−1) cos θ sin φ)�T , (4) where λ and d are the carrier wavelength and the antenna element spacing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In our study, we assume d = λ 2 to guarantee that there is no grating lobe when a beam is steered towards ± 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Besides, φ and θ denote the azimuth and elevation angles from the perspective of satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Signal Model and Problem Formulation As mentioned in Section II-A, multiple satellites provide communication services for their GUs cooperatively with FFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Therefore, each GU will suffer interference from all the satellites that are in this GU’s LoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' When receiving signals, the GU antenna will aim at the serving satellite and the intended signal and interference from this satellite will experience the maximum gain of the GU antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' While the interference from other visible satellites will experience a gain decrease due to the off-boresight angles of these satellite-GU links and the narrow-beam characteristics of the VSAT considered herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Note that the set of visible satellites of each GU is known, but the specific links between satellites and GUs are unknown and need to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Based on this fact, we introduce a discrete variable αsg to indicate whether the link is established, where αsg = 1 if satellite s is providing service for GU g, and αsg = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The received signal of GU g can be expressed as [8], [11] yg = � s∈Vg αsghH sgwsgxg + � g′̸= g � s∈Vg αsg′ hH sgwsg′ xg′ + ng, (5) where hsg ∈ CN×1 is the channel vector between satellite s and GU g, wsg ∈ CN×1 represents the beamforming vector, and xg is the requested data of GU g, which is assumed to be independent and satisfy E � |xg|2� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The first term in (5) is the intended data for GU g, the second term is the interference coming from the communication services for the other GUs, and the third term is the complex additive white Gaussian noise following the distribution CN � 0, σ2 s � , where σ2 s denotes the noise power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Then the received signal-to-interference-plus-noise ratio (SINR) of GU g can be obtained as γg = | � s∈Vg αsghH sgwsg|2 � g′̸=g | � s∈Vg αsg′ hH sgwsg′ |2 + σ2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' (6) According to the Shannon theorem, the SE per channel use of GU g can be calculated by Rg = log2 (1 + γg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' (7) As a result, the total SE of the multi-satellite cooperative network can be expressed as R = Nu � g=1 Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' (8) Our objective is to maximize the total SE of the multi- satellite cooperative network by means of calculating the beamforming vector wsg and adjusting the satellite-GU links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Therefore, we formulate the optimization problem (OP) as follows: OP : max {wsg,αsg} Nu � g=1 log2 (1 + γg) (9) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' C1 : � g αsg∥wsg∥2 ≤ PT, ∀s ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , Ns}, (10) C2 : � g αsg ≤ Nb, ∀s ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , Ns}, (11) C3 : � s∈Vg αsg = 1, ∀g ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , Nu}, (12) C4 : αsg ∈ {0, 1}, ∀s, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' (13) Here, constraint C1 is the power constraint of each satellite, constraint C2 means that the number of GUs connected to the same satellite cannot exceed the maximum number of spot beams that one satellite can generate, namely Nb, and constraint C3 is the GU connection constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Notably, the OP is not always feasible because of the coexistence of C2 and C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' When C2 and C3 contradict each other, we give priority to guaranteeing C2 and try to connect as many GUs as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The OP cannot be solved directly due to the fact that the objective function and constraints are non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Thus, we propose to solve it by the following steps: 1) Given hsg in (1), the analog beamforming vector wA sg is obtained based on a DFT codebook (with more details to be shown in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 2) The vector wA sg is set as the initial beamforming vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 3) Based on 2), we propose two schemes to solve OP, which will be further discussed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' HYBRID BEAMFORMING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Analog Beamforming Based on Codebook We consider analog beamforming for beam alignment based on a codebook, which is widely used in terrestrial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Codebook-based beamforming design can reduce the overhead of channel state information (CSI) feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In this work, we assume that the GU side has perfect CSI and the GU- satellite CSI feedback link is lossless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' As described in Section II-A, every satellite is equipped with a UPA, thus the two- dimensional (2D) DFT codebook is applicable and it can be seen as the synthesis of two 1D DFT codebooks in the directions of x and y axes, Dx, Dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The N × N 2D DFT codebook matrix, denoted as D, can be expressed as D = Dx ⊗ Dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' (14) The core idea of analog beamforming herein is to select the best K (≤ N) codewords from D and combine them into a new codeword that satisfies the equal-amplitude constraint of analog beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Considering the communication over- head of CSI feedback, K should not be too large and we assume K = 4 based on the results of trial simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The steps of the proposed analog beamforming method are given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Algorithm 1: Codebook-Based Analog Beamforming Input: GU-side channel vector hsg, ∀s, g, codebook D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Output: Analog beamforming vector wA sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 1 For each codeword D:,k, calculate |hH sgD:,k|2 and find the best K codewords maximizing it, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , cK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 2 DK = [c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , cK], solve the equations DKx = hsg and obtain the least square solution ˆx = (DK)†hsg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 3 The GU sends the codeword combination coefficients ˆx and codewords’ indices to the satellite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 4 Satellite-side combination: w ′ sg = DKˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 5 for i ∈ [1, N] do 6 wA sg(i) = w ′ sg(i) 1 √ N |w′ sg(i)| B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Digital Beamforming Based on the link information between satellites and GUs, the channel matrix of satellite s can be written as Hs = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , hsg, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' ]H ∈ CN s u×N, g ∈ Vs, (15) where N s u is the number of GUs that satellite s is serving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Similarly, the analog beamforming matrix of satellite s can be written as FA s = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , wA sg, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' � ∈ CN×N s u, g ∈ Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' (16) Hence, we can write the generalized channel matrix between satellite s and the GUs as �Hs = HsFA s , (17) thus the hybrid beamforming is reduced to a digital beam- forming problem to mitigate the inter-beam interference of satellite s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In this work, we adopt the regularized ZF and the corresponding digital beamforming matrix is FD s = √η �HH s ( �Hs �HH s + βIN s u)−1 ∈ CN s u×N s u, (18) where √η is the power scaling factor to guarantee the satellite is operating at its maximum power, and β is an adjustable parameter where βopt = N s uσ2 s PT in the large system limit [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Thus, combining (16) and (18), the hybrid beamforming matrix can be expressed as FHY s = FA s · FD s = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , wsg, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' ] , ∈ CN×N s u, g ∈ Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' (19) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' USER SCHEDULING AND IMPLEMENTATION SCHEMES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' User Scheduling As described in Section II-C, the links between multi- ple satellites and GUs need to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The optimal exhaustive search is infeasible here since the computational complexity grows exponentially with the number of GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Thus, we propose a heuristic user scheduling algorithm which can achieve a good performance with polynomial complexity as shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Therein, L denotes an Ns ×Nu link matrix where αsg lies in its s-th row and g-th column, and Lsg denotes the matrix whose s-th row and g-th column is 1 Algorithm 2: Heuristic User Scheduling Algorithm Input: Channel matrix Hs, beamforming matrix Fs, and the set of visible satellites Vg, ∀s ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , Ns}, ∀g ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , Nu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Output: Link matrix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 1 Initialize S = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , Ns}, Gn �� n=Nu = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' , Nu}, L = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 2 for g ∈ [1, Nu] do 3 if length(Vg) == 1 then 4 L(Vg, g) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 5 Remove g from Gn and n = n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 6 repeat 7 for each possible link Lsg, s ∈ S, g ∈ Gn do 8 △Rsg = R(L + Lsg, Hs, Fs) − R(L, Hs, Fs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 9 [ˆs, ˆg] = arg max s,g △Rsg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 10 if satellite ˆs has spare resource then 11 L(ˆs, ˆg) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 12 remove ˆg from Gn and n = n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 13 else 14 remove ˆs from S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 15 until n == 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' and other places are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The set of satellites that have spare resource is denoted as S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Here, the resource constraint is that the number of GUs that one satellite is serving simultaneously can not exceed Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The set of unserved GUs is denoted by Gn, where n is the number of GUs in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' △R indicates the increment of total SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The total SE is mainly associated with the link matrix L, channel matrix Hs and the beamforming matrix Fs, which is equivalent to (8) and can be abstracted as R = R(L, Hs, Fs), (20) where R indicates a function for calculating the total SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Separate & Joint Hybrid Beamforming and User Schedul- ing Schemes In Section III, we have introduced the hybrid beamforming method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Within the hybrid beamforming, the analog beam- forming can be completed independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' As for the digital beamforming and user scheduling, there are two ways: 1) Separate (SHU): In the SHU scheme, we perform digital beamforming and user scheduling separately and in- dependently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Analog beamforming matrix FA s is taken as the input of Algorithm 2 and user scheduling is performed based on FA s , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=', Fs = FA s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Digital beamforming is conducted after user scheduling according to (18), utilizing the final links to mitigate the interference and improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Finally, we use FHY s in (19) to calculate the total SE when SHU is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 2) Joint (JHU): The JHU scheme is based on alternating optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Different from SHU, user scheduling and digital beamforming in JHU are designed jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The beamforming 4 5 6 7 8 9 10 11 Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=') 0 2 4 6 8 Total Spectral Efficiency (bps/Hz) Total SEs in 24 Experiments AU SHU JHU Mean Total SEs of 24 Experiments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='7726 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='9535 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='8764 AU SHU JHU 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='5 3 Spectral Efficiency(bps/Hz) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Total SEs and mean total SEs of different schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' matrix is updated in real time within the user scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' As in SHU, FA s is taken as the initial input of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The difference is that each time before calculating the SE increment △Rsg, the hybrid beamforming matrix FHY s is computed based on the current link matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' If we abstract (18) and (19) as FHY s = F(L, Hs, FA s ), (21) where F is a function for calculating the hybrid beamforming matrix, then Step 8 of Algorithm 2 when JHU is adopted can be replaced by △Rsg = R(L + Lsg, Hs, F(L + Lsg, Hs, FA s )) − R(L, Hs, F(L, Hs, FA s )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' (22) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' PERFORMANCE EVALUATION We consider a 6 × 8 LEO Walker constellation with an inclination of 40◦ as an example, which has the ability to achieve full-time coverage of the entire China according to the simulation result of a popular aerospace simulation software Systems Tool Kit (STK) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' We use STK to simulate the movement of the constellation and we sample every 20 minutes within eight hours from 4am to 12pm in Beijing time, resulting in 24 experiments to test the performances of the proposed algorithms and schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Besides, all GUs are located in suburban areas of 80 representative cities in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' TABLE I SIMULATION PARAMETERS Parameter Value Number of orbital planes 6 Number of satellites per orbital plane 8 Orbital plane inclination 40◦ Orbital height 1200 km (LEO) [17] Number of sub-arrays Nsub x = 8,Nsub y = 4 Number of antenna elements per sub-array Nx = Ny = 8 Number of GUs Nu = 80 Downlink carrier frequency 20 GHz [17] Bandwidth 400 MHz [17] Receiver noise temperature 24 dBK [6] Satellite antenna gain* 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='5 dBi VSAT maximum antenna gain 40 dBi [17] Transmission power per satellite PT = 80 W The satellite antenna is a kind of phased array antenna, whose gain can be calculated according to the number and size of antenna elements and the carrier wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The simulation parameters are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Note that the proposed algorithm and main observations still hold for other types of LEO constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The baseline scheme is analog beamforming and user scheduling scheme, denoted as AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The specific steps of AU scheme are: (1) analog beamforming as described in Section III-A, (2) user scheduling using Algorithm 2, and (3) power scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The SE performances of AU, SHU and JHU are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 2, where the left picture illustrates the total SEs in 24 experiments and the right picture shows the corresponding mean total SEs averaged over the 24 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' It can be seen that AU performs the worst because of the lack of interference mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' By performing digital beamforming after user scheduling to mitigate interference, the performance of SHU increases by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2% compared with AU on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' However, SHU cannot make full use of link information since digital beamforming is implemented independently of user scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In JHU, the digital beamforming matrix is updated in real time when calculating the total SE increment and es- tablishing links, leading to increase of SE by 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2% compared with SHU and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='3% compared with AU on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 2, we can observe that the network total SE fluctuates significantly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The reason is that the topological relationships between satellites and GUs change rapidly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Taking GUs in five typical cities as exam- ples, we calculate their SEs with JHU and show the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' These five cities lie in the northern, eastern, middle, western, and southern part of China, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The SEs of GUs in these five cities fluctuate differently due to their different geographical positions and topological relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' For the GU in each city, the SE varies with time because of the change of topological relationship, which is caused by satellites’ movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' These five GUs have very different SEs at the same time because of different sets of visible satellites for these GUs and different elevation angles and distances between satellites and GUs, leading to different path losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The different sets of visible satellites and path losses are both caused by discrepant geographical positions of these GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 3, we can also find that the maximum SEs of GUs in Kashi and Nansha are higher than those in other cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Part of the reason lies in the density of GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Sparse GUs refer to 4 6 8 10 12 Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=') 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='8 Per-user SE (bps/Hz) Beijing (a) Beijing 4 6 8 10 12 Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=') 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='8 Per-user SE (bps/Hz) Shanghai (b) Shanghai 4 6 8 10 12 Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=') 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='8 Per-user SE (bps/Hz) Wuhan (c) Wuhan 4 6 8 10 12 Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=') 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='8 Per-user SE (bps/Hz) Kashi (d) Kashi 4 6 8 10 12 Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=') 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='8 Per-user SE (bps/Hz) Nansha (e) Nansha Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Per-user SEs of GUs in five typical cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' TABLE II SE STATISTICS OF DENSE AND SPARSE GUS Mean Value of SEs (bps/Hz) Variance of SEs (bps/Hz)2 Dense GUs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='028569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='035576 Sparse GUs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='049798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='064577 the users in an area where inter-GU distances are all greater than D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The opposite holds for dense GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In our study, we set D to 400km as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Among these 80 GUs, there are 12 sparse GUs and 68 dense GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Correspondingly, the GUs in Kashi and Nansha are sparse GUs and the other three GUs are dense GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' In order to obtain more comprehensive observations, we calculate the SEs of all dense and sparse GUs in 24 experiments and obtain their respective mean value and variance in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The mean SE of sparse GUs are nearly twice as much as that of dense GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' The potential reasons are: The sparse GUs are surrounded by fewer GUs and suffer less interference from others hence having higher SINR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Most of the sparse GUs are located in the border areas of China, thus they have greater probability of monopolizing one satellite and getting larger transmission power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Additionally, the variance of SE of sparse GUs is much larger than that of dense GUs, which indicates that the SEs of sparse GUs fluctuate more substantially than dense GUs due to their geographical positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' CONCLUSION In this paper, we first provide a hybrid beamforming ar- chitecture of UPAs which is suitable for satellites because of the limitation of on-board RF chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' According to the hybrid architecture, we introduce an analog beamforming method based on the 2D DFT codebook which generates a desired codeword by linearly combining four selected codewords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Then digital beamforming in accordance with regularized ZF is employed to mitigate the inter-beam interference and scale the transmission power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Moreover, we propose a heuristic user scheduling algorithm to determine the links between satellites and GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Subsequently, we propound two implementation schemes: a separate scheme and a joint scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Simulation results show that the JHU scheme outperforms SHU because of the joint design of beamforming and user scheduling, and can achieve an average gain of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content='2% compared with SHU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Furthermore, based on the simulation results, we analyze the key factors influencing GUs’ SEs, including geographical positions, topological relationships, and the density of GUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Future work may consider the impact of the satellite constel- lation design on the network’s communication performance and different beamforming architectures or algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' REFERENCES [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfbwdB/content/2301.03888v1.pdf'} +page_content=' Giambene, S.' metadata={'source': 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diff --git a/gNE4T4oBgHgl3EQfqw2j/vector_store/index.faiss b/gNE4T4oBgHgl3EQfqw2j/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..32787496d1cc0e68f80ab2e1401450b4d9737023 --- /dev/null +++ b/gNE4T4oBgHgl3EQfqw2j/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d13015b7a726bab4f4d33f8d183e37b90c82865ca19d4b8ff18f74dec13186c5 +size 10944557 diff --git a/gdAzT4oBgHgl3EQf4P5h/content/tmp_files/2301.01841v1.pdf.txt b/gdAzT4oBgHgl3EQf4P5h/content/tmp_files/2301.01841v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..96532d2eefe4da7356388335e44b01670fcde562 --- /dev/null +++ b/gdAzT4oBgHgl3EQf4P5h/content/tmp_files/2301.01841v1.pdf.txt @@ -0,0 +1,2445 @@ +Highlights + +1. An automatic pipeline leverages ALS and aerial camera to quantify decaying +trees + +2. Five decay stages of individual coniferous trees are identified for the first time + +3. Image texture is of the highest relevance to the success of categorizing tree decay + +4. Multispectral colorized point clouds underperform side-view projected imagery. + +5. ALS data with laser intensity failed to quantify the tree decay stages + +6. Landscape-wide assessment of dead wood amount and quality can be achieved + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +*Corresponding author +Automatic Classification of Single Tree Decay Stages from +Combined ALS Data and Aerial Imagery using Machine +Learning +Tsz-Chung Wong1, Abubakar Sani-Mohammed1, Wei Yao1,2*, Marco Heurich3,4,5 +1 Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic +University, Hung Hom, Kowloon, Hong Kong +2 The Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen, China +3 Dept. for National Park Monitoring and Animal Management, Bavarian Forest National Park, +94481 Grafenau, Germany +4 Chair of Wildlife Ecology and Management, Faculty of Environment and Natural +Resources, University of Freiburg, Freiburg, Germany +5 Department of Forestry and Wildlife Management, Campus Evenstad, Inland Norway +University of Applied Sciences, Koppang, Norway +- tszchung.wong@connect.polyu.hk, abubakar.sanimohammed@connect.polyu.hk, +wei.hn.yao@polyu.edu.hk, marco.heurich@npv-bw.bayern.de +Keywords: Tree decay stages; Airborne laser scanning; Color infrared imagery, +Convolutional Neural Network; Machine Learning; Dead wood; Forest disturbances +Abstract: Understanding forest health is of great importance for the conservation of the +integrity of forest ecosystems. The monitoring of forest health is, therefore, indispensable for +the long-term conservation of forests and their sustainable management. In this regard, +evaluating the amount and quality of dead wood is of utmost interest as they are favorable +indicators of biodiversity. Apparently, remote sensing-based machine learning techniques +have proven to be more efficient and sustainable with unprecedented accuracy in forest +inventory. However, the application of these techniques is still in its infancy with respect to + + + +dead wood mapping. This study investigates for the first time the automatic classification of +individual coniferous trees into five decay stages (live, declining, dead, loose bark, and clean) +from combined airborne laser scanning (ALS) point clouds and color infrared (CIR) images +using three different Machine Learning methods - 3D point cloud-based deep learning +(PointNet), Convolutional Neural Network (CNN), and Random Forest (RF). First, CIR +colorized point clouds are created by fusing the ALS point clouds and the color infrared +images. Then, with the CIR colorized point cloud, individual tree segmentation is conducted +using a semi-automatic approach to extract single tree objects, which are further projected +onto four orthogonal planes displaying the side views of the trees in 2D. Finally, the +classification is conducted on the two datasets (3D multispectral point clouds and 2D +projected images) based on the three Machine Learning algorithms. All models achieved +promising results, reaching overall accuracy (OA) of up to 90.9%, 90.6%, and 80.6% for +CNN, RF, and PointNet, respectively. The experimental results reveal that the image-based +approach notably outperformed the 3D point cloud-based one, while spectral image texture is +of the highest relevance to the success of categorizing tree decay. Our models could therefore +be used for automatic determination of single tree decay stages and landscape-wide +assessment of dead wood amount and quality using modern airborne remote sensing +techniques with machine/deep learning. The proposed method can contribute as an important +and rigorous tool for monitoring biodiversity in forest ecosystems. + + + + + +1. Introduction +Forests are a precious natural resources that deliver important ecosystem services to the +benefit of humankind(Brockerhoff et al., 2017). Therefore maintaining and monitoring forest +health is of great importance for the conservation of the integrity of forest ecosystems. Among +others forests play a critical role in maintaining biodiversity and in the carbon cycle +(Paniagua-Ramirez et al., 2021; Houghton 2005; Polewski et al. 2021) and can contribute to +reducing the greenhouse effect and consquently alleviating global warming if sustainably +managed. However, various diseases and disturbances could cause tree decay, including +microorganisms, insect infestation, droughts, wildfires, and extreme weather conditions +(Blanchette, 1998; Trumbore et al., 2015; Seibold et al., 2021). For instance, bark beetle +attacks have caused large-scale disturbances in recent years (Latifi, 2014; Huang et al., 2020; +Hlásny et al., 2021; Sani-Mohammed, et al., 2022). Such areas can harbour large amounts of +dead wood, making assessing the carbon stored a critical task. Moreover, dead wood is an +important resource for biodiversity (Jonsson et al., 2005; Lassauce et al., 2011; Stokland et al., +2012; Seibold et al., 2015). However, to harbor a high biodiversity, the quantity and quality of +the dead wood is crucial (Similä et al., 2003; Lassauce et al., 2011; Seibold et al., 2016). +Therefore, not just the estimation of the amount of dead wood but also the assessment of the +dead wood quality is a decisive precondition for sustainable forest management and +conservation of biodiversity. Thus, early detection of declining trees can help identify the +origin of bark beetle infestation for quick preventive measures before it spreads widely in the +forest. This would significantly improve forest management and protect biodiversity +(Abdullah et al., 2019). For standing dead wood, Thomas et al. (1979) developed a +framework for classifying different decay stages, which proved to be crucial for the habitat +modelling of different groups of species, such as birds, bats and beetles (Similä et al., 2003; +Cousins et al., 2015; Zielewska-Büttner et al., 2018; Kortmann etal., 2018). However, the + + + +assessment of the different decay stages still depends on a visual evaluation by field crews. + Conventional forest management methods such as field surveying are very time and +cost-consuming due to the large area of forests. With the advancement of remote sensing +technologies, the health of forests can be sensed without on-site measurements in a cost- +efficient way with high quality (Lausch et al., 2016). Among various remote sensing +techniques, airborne laser scanning (ALS) has been widely used in forestry applications as it +can accurately map the 3-dimensional structures of the forests. Based on this data cruical +information can be extracted from the point cloud data from the tree to the landscape level +(Maltamo et al., 2014; Latifi et al., 2015; Moudrý et al., 2022). +In the past, various studies have been conducted to classify and detect standing dead +trees using lidar point clouds. Yao et al. (2012a) identified standing dead trees in a temperate +forest park using full-waveform lidar data by training a binary support vector machine (SVM) +classifier, achieving overall accuracy (OA) of 73%. Polewski et al. (2015) and Polewski et al. +(2016) proposed a method of integrating ALS data and aerial infrared imagery to detect +standing dead trees by using an active learning approach, reaching an OA of 89%. Wing et al. +(2015) analyzed the spatial distribution of snags and provided a better understanding of +wildlife snag use dynamics using neighbourhood attribute filtered ALS data followed by +individual tree detection. The overall detection rate for snags with DBH >=25 cm was 56% +with low commission error rates, while detection rates ranged from 43 to 100%, increasing +with the size of the snag. A multitude of research hase been conducted for classifying tree +species together with standing dead trees using lidar data and multispectral images (Yao, et al, +2012b; Polewski et al.,2020). Kamińska et al. (2018) conducted a species-related individual +dead tree detection by combining multi-temporal ALS point cloud and color infrared imagery +using a Random Forest (RF) classifier, resulting in OA of 94.3%. Krzystek et al. (2020) + + + +studied large-scale mapping of individual conifers, broadleaf trees, and standing dead trees +using lidar data and multispectral imagery. They reached an overall accuracy of better than +90% after training RF and logistic regression models. Marchi et al. (2018) reviewed studies +based on the application of airborne and terrestrial laser scanning for the identification of +large deadwood components (e.g., snags, logs, stumps). They concluded that efforts should be +made to transfer the available single-tree methodologies to an operational level but did not +consider further characterizing tree conditions. To the best of our knowledge no research has +been published that quantifies different decay classes of standing dead wood using remote +sensing techniques until know.Until recently studies have extracted handcrafted features from +point clouds and multispectral images for classification tasks. Deep leaning methods are +currently becoming more and more used as classification tools for similar tasks in recent years. +For example, Hamraz et al., (2019) constructed a deep convolutional neural network (CNN), +for binary classification of coniferous and deciduous trees using 2D representations of +individual trees and achieved an OA of around 90%. Briechle et al., (2020) also studied the +possibility of using the 3D deep neural network PointNet++ for classifying tree species and +standing dead trees. Their results showed that the 3D PointNet++ approach (OA = 90.2%) +outperformed the RF classifier and handcrafted features approach (OA=85.3%). Furthermore, +Seidel et al. (2021) compared the performance of image-based CNN approach and point- +cloud-based PointNet approach for tree species classification tasks. Their study revealed that +the CNN approach achieved higher accuracy and computational efficiency than the PointNet +approach. However, these studies have only focused on utilizing LiDAR data and +multispectral images for classifying tree species or standing dead trees. Here we propose that +these kinds of data could be applied to broader applications, such as categorising individual +trees into several decay stages rather than just living and dead trees. +However, to the best of our knowledge, the possibility of using ALS data and + + + +multispectral aerial images for classifying tree decay stages is yet to be studied, especially +using Machine Learning. Therefore, the objectives of this study are; (1) to define five tree +decay stages that could be detected by ALS and multispectral images, (2) to generate and +curate two datasets (3D multispectral colorized point clouds and 2D side-view projected +images) of individual tree decay stages as a basis for their classification (training), by +combining ALS data and CIR aerial imagery, and (3) to assess the performance of three +Machine/Deep Learning algorithms (PointNet, CNN, and RF) in classifying the five decay +stages of individual trees. +The remainder of this paper is structured as follows; Section two describes the study area +and materials used for the experiments, including the datasets acquisition and preparation. In +section three, our three classification approaches using Machine/deep learning are explained +in detail. The results are presented and discussed in section four and five. Lastly, the +conclusions are stated in section six. + + + + + +2. Materials +2.1 Study area +The experiments were carried out in the Bavarian Forest National Park (BFNP) (49° 3’ +19” N, 13° 12’ 9” E), located in Germany and along the border with the Czech Republic +(Figure 1). Established as the first National Park in Germany, the park covers an area of +approximately 24,000 ha with altitudes ranging from 650 to 1453 m above sea level(Heurich +et al., 2010). The park is dominated by Norway Spruce (Picea abies), European Beech (Fagus +sylvatica), European Silver fir (Abies alba), and other Spruce trees(van der Knaap et al., +2020). Despite experiencing severe thunderstorms in the 1980s and a bark beetle attack during +the 1990s (Lausch et al., 2013), the park was protected from human intervention due to the +policy of the authority, resulting in a total tree mortality rate of 22% (Müller & Job, 2009). + +Figure 1. Map of the BFNP and the three transects of ALS point clouds (marked in blue) + + + +4580000 +4585000 +4590000 +4595000 +4600000 +4605000 +4610000 +4615000 +4620000 +Bayerisch +N +cise +5440000- +Bavar +Srnt +5440000 +Bodenmais +5435000 +5435000 +Lindberg +Filipova Hut +EGEN +Kvilda +Langdorf +Zwiesel +Sumava +National +5430000- +Borova +5430000 +Frauenau +Lada +Regen +5425000 +5425000 +Rinchnach +GERMANY +Leipzig +Finsterau +Dresden +Waldhauser +Chemnitz +ofsmais +furt +5420000 +hain +Prague +dorf +Vald +-5420000 +im +CZECH +Nuremberg +Mauth +pttgart +Munich +Vi +5415000 +Mat +5415000 +AUSTRIA +Grafenau +Hohenau +0 +100 +200 +400Kilometers +Lalling +Schonberg +4580000 +4585000 +4590000 +4595000 +4600000 +4605000 +4610000 +4615000 +4620000 + +2.2 Data acquisition + +ALS point clouds and color infrared images were acquired in 2016 for the BFNP +administration within the framework of the data pool initiative for the Bohemian Forest +Ecosystem(Latifi et al., 2021). Detailed information on the data is described in sections 2.2.1 +and 2.2.2 respectively. +2.2.1 ALS data + +Three transects of ALS point clouds (leaf-on) were acquired on 18th August 2016 with an +LMS-Q680i - 400 kHz scanner in a flight operated by the Milan Geoservice GmbH, at an +altitude of 300 m above sea level. This exercise covered a total area of 13.5 square km with an +estimated point density of 70 points per square meter. The raw data was processed and +corrected to LAZ 1.2 format (with the X, Y, Z and Intensity), while the WGS84/UTM +coordinates were calculated with a transformation fitted according to the Gauss-Krüger +coordinate system. Figure 2 shows the top view of the three transects of ALS point clouds +visualized by intensity values. + + +Figure. 2 The transect of ALS data displayed by the laser intensity; 1st (left), 2nd and 3rd (right) +2.2.2 Color infrared images + +Three-band CIR aerial images (G, R, NIR) covering the entire BFNP park were acquired + + + +by a Leica DMC 122 camera from a Cessna 207 Aircraft. The flight was conducted on 23rd +June 2016 by ILV Remote Sensing GmbH at an average altitude of 2918 m above sea level. +The longitudinal overlap of the flight was 75%, while the cross-coverage was 60%. The +camera was calibrated in 2014, with a focal length of 120 mm, and was used to capture the +images with a ground resolution of 20 cm. Similar to the point cloud data, the images were +geo-referenced according to the Gauss-Krüger coordinate system. The images were +radiometrically corrected and orthorectified. Figure 3 shows an example of a cropped CIR +image. Unhealthy trees are identified by the green band of this false-color image because +defoliated trees less reflect near-infrared radiation. In contrast, healthy trees appear in red as +they reflect a high proportion of near-infrared radiation (Knipling, 1970). + +Figure 3. CIR aerial image captured in BFNP +2.3 Definition of tree decay stages +To achieve our objective of classifying individual trees into their decay levels, it is +important to first define them. Thomas et al. (1979) defined nine different decay stages, +starting from the “live” to the “stump” stage. However, in this study, we are considering only +the first five stages of decay (live, declining, dead, loose bark, and clean) proposed by +Thomas et al. (1979) for experiments. For easier differentiation, we renamed these five stages +starting from Decay Level 1 to Decay Level 5, respectively. Figure 4 illustrates individually + + + +segmented trees at the five stages of decay/decomposition visualized with aerial multispectral +point clouds. +Live +Dead +Decay stages +1. Live +2. Declining +3. Dead +4. Loose bark +5. Clean + + + + + +Description +No decay; tree has +valuable habitat +characteristics such +as large, clustered, +or gnarled branches, +or horizontal, +thickly moss- +covered branches. +Internal decay +or growth +deformities +(including insect +damage, broken +tops); dying +tree. +Needles or +twigs may be +present; roots +sound. +(Almost) No +needles/ twigs; +50% of +branches lost; +loose bark; top +usually +broken; roots +stable +Most branches/ +bark absent; +some internal +decay; roots of +larger trees +stable. +Visualization by corresponding multispectral ALS point clouds + + + + + +Figure 4. System for decay stages for wildlife (coniferous) standing dead tree (Thomas et al., +1979) and corresponding decay of individual spruce trees from multispectral point clouds. + +！25 +20 +15 +10 +5 +5 +5 +040 +35 +30 +25 +20 +15 +10 +5 +5 +5 +0 +020 +15 +10- +5 +2 +0 +2 +020 +15 +10 +5 +2 +2 +0 +015 +10 +5 +0.25 +Φ25- + +2.4 Reference data curation for tree decay analysis + +Reference data were collected by combing manual interpretation of multispectral point +clouds of individual trees with field data according to the defined stages of wildlife tree decay +described in Section 2.3. This important exercise was conducted with expert advice and the +additional support of the CIR imagery. Several reference plots in the BFNP were considered +in such a way that there was a balance in the diversity of decay levels. It is noted that such a +manual point cloud labelling method was used due to a lack of complete and precise field data. +In total, 1,030 samples were interpreted and labeled. Specifically, the number of labeled +samples for the classes was 233, 167, 236, 239, and 155, for decay stages 1, decay stages 2 +decay stages 3, decay level 4, and decay level 5, respectively. + + + + + +3. Methodology +3.1 Methodological Overview + +After data acquisition, the data was pre-processed and prepared for input for the +classification algorithms. We merged the ALS point clouds with the CIR images to extract +colors (CIR) for the point clouds. Then we conducted a semi-automatic individual tree +segmentation method to segment individual trees. These segmented trees (point clouds) +formed the dataset for a 3D deep learning model based on the PointNet to perform a point- +cloud-based classification. Also, we transformed each individual tree (point cloud) into four +2D projected images via multi-view orthographic projection to form an image dataset. These +2D projections were the dataset used for performing image-based classification using the +CNN and the RF approach. These two sets of data were then trained in the three +Machine/deep learning algorithms. Finally, the trained models were tested, and the results +were evaluated to assess their accuracies. Figure 5 illustrates the general workflow of the +methodology. + +Figure 5. General workflow of the methodology + + + +3.2 Data preparation +3.2.1 Multispectral point cloud curation +We generated colors for the point clouds of the selected plots of 30 m radius by +combining the ALS point cloud (X, Y, Z, I) with georeferenced CIR images (NIR, R, G). Thus, +the ALS point cloud was colorized by the georeferenced CIR images in such a way that points +pick the exact corresponding colors from the spectral bands of the images. For instance, the +NIR, R, G values of the CIR imagery at (xi, yi) would be applied to the ALS point cloud at +(Xi, Yi). This combination generated three more values to the existing four resulting in seven +values (X, Y, Z, I, NIR, R, G). However, only six values were considered for the classification, +excluding the intensity (I) (as seen in Figure 6). This processing was executed using +LiDAR360 software v5.2 (GreenValley International) and QT Modeler v8.1.1425. + +Figure 6. A sample plot of combined ALS data and CIR image resulting in multispectral +colorized point clouds +3.3 Individual tree segmentation +A semi-automatic individual tree segmentation approach was conducted to select + + + +appropriate samples for training. First, we conducted an automatic individual tree +segmentation following the point cloud segmentation algorithm proposed by Li et al. (2012). +The ground points were classified in this automatic stage using an improved progressive TIN +densification filtering algorithm (Zhao et al., 2016). We then normalized the point clouds with +reference to the ground points to remove the effects of topographic relief. One critical +parameter of this algorithm is the 2D Euclidean distance threshold between two adjacent trees +because inappropriate thresholds could cause under-segmentation or over-segmentation. As +the target trees in this research were coniferous, the threshold value was set as 0.5 m due to +their narrow tree crowns, which led to a shorter distance between two neighboring trees. +Nevertheless, the segmented trees from the automatic approach still needed some final +evaluationto not affect the classification performance in subsequent stages. +Thus, the automatically segmented trees were manually edited and refined in an expert +interactive mode by deleting extra point clouds that do not belong to a single tree. For +instance, the unwanted point clouds of bushes and young trees around individual trees were +manually cleaned (see Figure 7). However, trees with incomplete point clouds were not +selected. With this approach, a total of 1030 individual tree samples were extracted and +constituted the data sed(for the 3D PointNet classification). + + +Figure 7. A sample of automatic individual tree segmentation (left) and semi-automatic + + + +individual tree segmentation (right) +3.4 Image projection + +In addition to the 3D point clouds of individual trees, another image dataset was +generated from the 3D point clouds for the image classification approach. Each individual tree +(point cloud) was transformed into four 2D projected images via multi-view orthographic +(parallel) projection; the four mutually perpendicular side-views that are produced by four +mutually perpendicular planes of projection (Front view 0°, Left-side view 90°, Rear view +180°, and Right-side view 270°) as in Figure 8. This projection from different viewing angles +could minimize information loss caused by 3D to 2D representation (Seidel et al., 2021). +Initially, the image size of each projected image was 644 x 658 pixels. To achieve a balance +between classification efficiency and accuracy, the image sizes were downscaled to 129 x 132 +pixels before the training and validation stages. Table 1 summarizes the distribution of +samples in both the 3D point cloud dataset and the 2D image dataset. + +(a) + +(b) + +(c) + +(d) +Figure 8. Images of the same live tree projected from (a) front view, (b) left-side view, (c) +rear view, and (d) right-side view + + + +Table 1. Number of tree samples in point cloud dataset and image dataset +Class label +Point cloud dataset +Image dataset +Level 1 (live) +233 +932 +Level 2 (declining) +167 +668 +Level 3 (dead) +236 +944 +Level 4 (loose bark) +239 +956 +Level 5 (clean) +155 +620 +Total +1030 samples +4120 images + +3.5 Model architecture +3.5.1 PointNet + +Proposed by Qi et al. (2017), PointNet is a deep learning architecture that simply takes +point clouds as input to represent 3D geometry instead of typical formats such as images and +voxels for classification tasks. Each set of point cloud input {Pi | i = 1, ..., n} is represented by +its x, y, and z coordinates, while additional feature channels such as color and intensity could +be added to the vector. Figure 9 illustrates the model architecture of PointNet. The size of +input points (n x 3) changes depending on the number of feature channels. For multispectral +point clouds that contain r, g, and b (NIR, R, and G) information, the input size would be n x 6. +Two shared multi-layer perceptron (MLP) map each point cloud to larger dimensions before +max pooling. A three-layer fully connected classification model consisting of a shared MLP +and a softmax activation map the global feature vector to k classification scores. + + + + +Figure 9. PointNet architecture +Data preparation +PointNet requires a fixed number of points per feature. So, we performed a point +sampling procedure. Due to the substantial geometrical and size difference of trees within the +five decay classes, the number of points per sample varies greatly. The minimum point count, +maximum point count, and mean point count of each class are listed in Table 2. Because of +the high variability in the number of points per sample, down-sampling and up-sampling of +points must be conducted based on an optimal value. This value must represent the individual +trees' shape after down-sampling while not increasing the computational cost enormously +during up-sampling. Therefore, we used 2048 as the fixed number of points in this research. + +Moreover, we applied data normalization as the values of each feature tree has vastly +different ranges. Therefore, the x, y, and z coordinates of the point cloud data were +normalized to range between 0 and 1. Also, the r, g, and b (NIR, R, and G) values of the point +cloud data initially range from 0 to 255. These values were also normalized to range between +0 to 1 by dividing all values by 255, to lower the computational cost. + + + +n x (no. of feature channels) +Input +n x (no. of feature channels) +Feature +mlp. +mlp (64,64) +Max +transform +transform +(64,128,1024) +pool +mlp. +(512,256,k) +Input points +n x 64 +4 +1024 +shared +nx +shared +nx 1024 +Global feature +Output scores +3x3 +64x64 +T-Net +T-Net +transform +transform +Matrix +Matrix +multiply +multiply + +Table 2. Minimum point count, maximum point count, and mean point count of each class +Class +minPointCount +maxPointCount +meanPointCount +Level 1 +845 +10295 +3346 +Level 2 +1586 +14405 +4785 +Level 3 +644 +5963 +2139 +Level 4 +121 +2799 +637 +Level 5 +16 +1418 +161 + +3.5.2 CNN +Unlike PointNet, the CNN takes 2-dimensional images as input. In general, the CNN +consists of an input layer, multiple hidden layers, and an output layer (Figure 10). The input +layer takes images of the same image size, generating a shape of (number of inputs) × height × +width × (number of channels). The hidden layers include several convolution layers, pooling +layers, a flatten layer and fully connected or dense layers. The convolution layers extract +features from the images with the aid of kernels, abstracting the images to a feature map with +the shape of (number of inputs) × (height of feature map) × (height of feature map) × (number +of channels). The pooling layer follows a convolution layer to reduce the dimensions of the +feature maps, thus reducing the computational cost. The most widely used pooling method for +image classification is max pooling. Max pooling was applied using Equation (1): +𝑓𝑚(𝑣) = 𝑚𝑎𝑥𝑖𝑣𝑖 + + + + + + + +(1) +where the vector 𝑣 is a single 𝑃-dimensional column in a 𝑃 × 𝑘 matrix reduced by a pooling +operation 𝑓 (Boureau, 2010). The feature maps extracted in the final convolution and pooling +layer are flattened into a vector using the flatten layer before connecting to the fully connected +layers to perform classification. We built the CNN from scratch for our experiments. Further + + + +information on our hyperparameters for the model are explained in Section 4.2. + +Figure 10. Architecture of the CNN model built for this study +Dataset preparation + +Initially, the size per image was 644 x 658 pixels. To reduce the computational cost, all +images were downscaled by 0.2, generating images with sizes of 129 x 132 pixels. +Furthermore, all pixel values of the three channels were normalized between 0 and 1 for a +lower computational cost. These images were used as input for the CNN as DL requires no +manual feature extraction. +3.5.3 Random Forest + +Random Forest is a supervised ensemble learning algorithm proposed by Breiman (2001). +Ensemble learning aims at improving model performance by running a base learning +algorithm several times. Combining the hypotheses generated by the algorithm at each time, a +voted hypothesis could be obtained, which usually leads to a better prediction (Dietterich, +2002). Random Forest is an improved version of the decision trees algorithm by reducing the +variance of the model. It generates multiple decision trees randomly, then combines and + +Fully +connected +layers +Output +Conv layer 1 +Conv layer 2 +Conv layer 3 +Conv layer 4 +Input +125x128x32 +60x62x64 +28x29x128 +12x12x256 +129x132x3 +Max- +Max- +Max- +Max- +pooling +pooling +pooling +pooling +2x2 +2x2 +2x2 +2x2 + +averages the outputs of the decision trees to obtain a final prediction. Figure 11 illustrates the +model architecture of a random forest classifier. + +Figure 11. Model architecture of a random forest classifier +Dataset preparation +The processing on the image dataset for Random Forest classification was the same as +that for CNN image classification, which is mentioned in section 3.5.2. +Feature extraction + +Unlike DL, RF being a traditional machine learning algorithm require handcrafted +feature extraction. Thus, we extracted handcrafted features from the processed image dataset, +and performed a feature importance analysis. The initial handcrafted features include Hu +moments (Hu, 1962), Haralick texture (Haralick et al., 1973), Histogram of Hue Saturation +Value (HSV), Histogram of Oriented Gradients (HOG) (Dalal & Triggs, 2005), and Harris +Corner Detection (Harris & Stephens, 1988). This feature importance analysis was computed +using the RF Regressor. Figure 12 shows the ranking of the feature importance. + + +Training +Training +Training +Data 1 +Data 2 +Data n +Training Set +Decision +Decision +Decision +Tree l +Tree 2 +Tree n +Test Set +Voting +Prediction + + +Figure 12. Feature importance ranking for Random Forest classification + +Based on the outcome of the feature importance analysis, we selected the top three +feature sets (Haralick texture, HSV histogram, and Hu moments) as the final handcrafted +features and extracted them from each processed image. These features were combined into a +global feature vector as the input for the RF Classifier. Figure 13 depicts a 2D feature map +(including 1st and 2nd principal component) generated by Principal Component Analysis (PCA) +with the five classes of tree decay. + +Figure 13. A 2-D feature space generated after feature extraction and PCA +3.6 Model evaluation + +The performance of our models’ classification was evaluated based on the overall + +0.8 +0.7 - +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 - +0.0 +Haralick +HSV +Humoments +Harris +HOG +FeatureDescriptor2.5 +Lv1 +2.0 +Lv2 +Lv3 +1.5 +Lv4 +Lv5 +1.0 +PC2 +0.5 +0.0 +-0.5 +-1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +PC1 + +accuracy (OA), F1- scores, and Cohen’s kappa coefficient (κ). The OA is expressed as the +sum of correctly classified features relative to the total number of features of the confusion +table as represented in Equation (2): +𝑂𝐴 = +𝑇𝑃+𝑇𝑁 +𝑇𝑃+𝑇𝑁+𝐹𝑃+𝐹𝑁 + + + + + + +(2) +where TP, TN, FP, and FN represent true positive, true negative, false positive, and false +negative respectively. F1-score is the weighted average of recall and precision which can be +computed as expressed in Equation (3): +𝐹1 = 2 ∙ +𝑟𝑒𝑐𝑎𝑙𝑙 ∙𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 +𝑟𝑒𝑐𝑎𝑙𝑙+𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 + + + + + +(3) +where 𝑅𝑒𝑐𝑎𝑙𝑙 = +𝑇𝑃 +𝑇𝑃+𝐹𝑁 and 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = +𝑇𝑃 +𝑇𝑃+𝐹𝑃. Cohen’s kappa coefficient (𝜅) measures the +agreement between two classifiers by considering the probability of the observed agreement +and the probability of agreement by chance (Vieira, 2010). This is expressed as in Equation +(4): +κ = +𝑝𝑜−𝑝𝑐 +1−𝑝𝑐 + + + + + + + +(4) +where the Probability of Observed Agreement is calculated by 𝑝𝑜 = +∑ 𝑥𝑖𝑖 +𝑁 and the Probability +of Agreement by Chance is derived from the formula 𝑝𝑐 = +∑ 𝑥𝑖+𝑥+𝑖 +𝑁2 +. Then, a comparative +analysis between the three classifiers was performed to propose the best performing model. + + + + + +4. Experiments and results +4.1 Experimental design + +The PointNet process point clouds with user-defined feature channels. Thus, we designed +three scenarios for our experiments. In the first scenario we applied the X, Y, and Z values of +the raw point clouds only as input. In the second scenario we used the X, Y, Z, NIR, R, G +values of the multispectral point clouds as input. For the third scenario, we used the X, Y, Z, +and I (intensity) values of the raw ALS point clouds as input. We applied these dynamics to +perform a sensitivity analysis relating to the color effect as well as the intensity in our +evaluations. We applied a 5-fold cross-validation technique for a robust classification training +process. We randomly divided the dataset into five different datasets; so that the training is +conducted five times. Thus, for each training, four parts are used for training and validation +while the remaining part is used for evaluation. This means each one of the five folds has +been used once for testing. + +Data augmentation was applied to modify the training dataset slightly to avoid model +over-fitting. The techniques used for data augmentation include random rotation of the point +cloud, random removal of points, and random point jittering with Gaussian noise. +The 5-fold cross-validation was also used for training and validating the CNN and RF +classification schemes following the same procedure as described for the PointNet. +4.2 Hyperparameter settings +i) PointNet + + Table 3 shows the hyperparameter- settings considered when training the PointNet deep +learning model. The hyperparameters were adjusted by a manual search approach to achieve + + + +improved classification performance. +Table 3. PointNet hyperparameters and corresponding values +Hyperparameter +Value +Description +numPoints +2048 +Number of points per sample +inputChannelSize +6 +Number of input channels per sample (X, +Y, Z, NIR, R, G) +numEpochs +20 +Number of epochs +learnRate +0.001 +Initial learning rate +miniBatchSize +32 +Number of samples per batch +l2Regularization +0.0001 +L2 regularization factor +learnRateDropPeriod +15 +Number of epochs before reducing the +learning rate +learnRateDropFactor +0.6 +Factor of the learning rate drop +optimizer +Adam +Algorithm of the optimizer +gradientDecayFactor +0.9 +A parameter of Adam optimizer +squaredGradientDecayFactor +0.999 +A parameter of Adam optimizer + + ii) CNN + +In summary, our network consists of four convolutional layers, four max-pooling layers, +one flattened layer, and two dense layers. We built this CNN model architecture from scratch +as illustrated in Table 4. The input shape of each CIR image was (129, 132, 3). The first +convolutional layer consists of 32 filters with a kernel size of 5 x 5. This kernel size was used +due to the relatively large input of the processed image dataset. The second convolutional +layer consists of 64 filters with a kernel size of 3 x 3, followed by the third convolutional + + + +layer, which comprises 128 filters with the same kernel size. Lastly, the fourth convolutional +layer was made up of 256 3 x 3 filters. Max pooling layers with 2 x 2 filters were added +between every two convolutional layers. Throughout the convolutional base of the network, +the Rectified Linear Unit (ReLU) was used as the activation function. To complete the +convolutional architecture, a flatten layer was used to flatten the three-dimensional feature to +one-dimensional feature for easy pass to the dense layers. The two dense layers with the shape +of 256 and 5, respectively. The ReLU was used for the first dense layer, while the softmax +activation function was used for the second dense layer (output layer). The softmax activation +function helps transform the input vector into a probability vector for performing +classification. The output shape (none, 5) in the output layer represents the five classes of tree +decay levels. +Table 4. Description of the CNN model architecture +Layer (type) +Output Shape +Param # +conv2d (Conv2D) +(None, 125, 128, 32) +2432 +max_pooling2d (MaxPooling2D) +(None, 62, 64, 32) +0 +conv2d_1 (Conv2D) +(None, 60, 62, 64) +18496 +max_pooling2d_1 (MaxPooling2D) +(None, 30, 31, 64) +0 +conv2d_2 (Conv2D) +(None, 28, 29, 128) +73856 +max_pooling2d_2 (MaxPooling2D) +(None, 14, 14, 128) +0 +conv2d_3 (Conv2D) +(None, 12, 12, 256) +295168 +max_pooling2d_3 (MaxPooling2D) +(None, 6, 6, 256) +0 +flatten (Flatten) +(None, 9216) +0 +dense (Dense) +(None, 256) +2359552 +dense_1 (Dense) +(None, 5) +1285 +Total params: 2,750,789 + + + +Trainable params: 2,750,789 +Non-trainable params: 0 + +We used the sparse categorical cross-entropy loss function because of the integers used +to represent the labels during our dataset processing. The number of training epochs per +iteration was set to 20 while the loss was optimized with the Adam optimizer. + iii) Random Forest + +For the RF classifier, we applied automatic grid search for tuning the hyperparameters. +We selected a combination with the highest cross-validation score as the final +hyperparameters illustrated in Table 5. Other unmentioned hyperparameters were kept as +default values. +Table 5. Random Forest hyperparameters and corresponding values +Hyperparameter +Value +Description +n_estimators +800 +Number of estimators +max_depth +64 +Maximum depth of the decision tree +random_state +42 +Parameter to control the random number generator +class_weight +balanced +Parameter to adjust classification weights + +4.3 Results +4.3.1 PointNet +Tables 6, 7, and 8 summarize the OAs, Kappa scores, and F1-scores of our three +experiments using PointNet. Overall, scenario 2 (X, Y, Z, NIR, R, G) achieved the best results +with an average OA and Kappa score reaching 80.6% and 0.754 respectively, throughout the +5-fold cross-validation, indicating substantial agreement. In terms of the classes of tree decay +levels, Level 1 outperformed the others with the highest average F1-score reaching 0.989, + + + +while Level 4 achieved the lowest average F1-score reaching 0.702. Scenario 3 performed +poorly and produced the worst results with an overall Kappa score of 0.047, implying there +was almost no agreement. Figures 14, 15, and 16 show the confusion matrices of each +iteration for scenarios 1, 2, and 3, respectively. +Table 6. The OA, Kappa score, and F1-score of PointNet classification (scenario 1) + +OA +Kappa +score (κ) +F1-score +Level 1 +Level 2 +Level 3 +Level 4 +Level 5 +1st iteration +0.602 +0.500 +0.472 +0.629 +0.636 +0.578 +0.704 +2nd iteration +0.651 +0.562 +0.590 +0.619 +0.637 +0.694 +0.759 +3rd iteration +0.583 +0.472 +0.447 +0.526 +0.554 +0.679 +0.762 +4th iteration +0.602 +0.500 +0.563 +0.610 +0.578 +0.609 +0.645 +5th iteration +0.587 +0.480 +0.379 +0.568 +0.648 +0.613 +0.681 +Average +0.605 +0.503 +0.490 +0.590 +0.611 +0.635 +0.710 +(The best results of each parameter are marked in bold text.) +Table 7. The OA, Kappa score, and F1-score of PointNet classification (scenario 2) + +OA +Kappa +score (κ) +F1-score +Level 1 +Level 2 +Level 3 +Level 4 +Level 5 +1st iteration +0.767 +0.704 +0.989 +0.667 +0.702 +0.680 +0.793 +2nd iteration +0.816 +0.768 +1.000 +0.792 +0.742 +0.685 +0.828 +3rd iteration +0.869 +0.833 +1.000 +0.906 +0.792 +0.774 +0.870 +4th iteration +0.786 +0.732 +0.976 +0.862 +0.680 +0.646 +0.849 +5th iteration +0.791 +0.734 +0.978 +0.764 +0.700 +0.727 +0.821 +Average +0.806 +0.754 +0.989 +0.798 +0.723 +0.702 +0.832 +(The best results of each parameter are marked in bold text.) + + + +Table 8. The OA, Kappa score, and F1-score of PointNet classification (scenario 3) + +OA +Kappa +score (κ) +F1-score +Level 1 +Level 2 +Level 3 +Level 4 +Level 5 +1st iteration +0.233 +0.073 +0.500 +0.278 +0.000 +0.000 +0.000 +2nd iteration +0.238 +0.007 +0.379 +0.000 +0.000 +0.000 +0.059 +3rd iteration +0.306 +0.172 +0.494 +0.000 +0.000 +0.447 +0.280 +4th iteration +0.194 +-0.012 +0.320 +0.000 +0.000 +0.000 +0.049 +5th iteration +0.209 +-0.006 +0.345 +0.000 +0.000 +0.000 +0.000 +Average +0.236 +0.047 +0.408 +0.056 +0.000 +0.089 +0.078 +(The best results of each parameter are marked in bold text.) + +(a) 1st iteration + +(b) 2nd iteration + +(c) 3rd iteration + +(d) 4th iteration + +(e) 5th iteration +Figure 14 Confusion matrices of PointNet classification (scenario 1) + +Lv1 +17 +20 +7 +1 +Lv2 +5 +28 +1 +Class +Lv3 +5 +6 +34 +6 +Lv4 +1 +11 +26 +Lv5 +13 +19 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +23 +21 +4 +Lv2 +4 +35 +2 +Class +Lv3 +2 +16 +29 +2 +Lv4 +1 +6 +25 +3 +Lv5 +1 +10 +22 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +17 +22 +14 +1 +Lv2 +20 +11 +Class +Lv3 +2 +31 +True +16 +Lv4 +7 +36 +4 +Lv5 +6 +16 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +20 +19 +3 +Lv2 +3 +25 +1 +Class +Lv3 +2 +8 +24 +5 +True +Lv4 +4 +1 +14 +35 +5 +Lv5 +1 +16 +20 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +11 +29 +4 +Lv2 +2 +25 +2 +Class +Lv3 +1 +4 +35 +True +5 +Lv4 +1 +22 +34 +Lv5 +15 +16 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted Class + + +(a) 1st iteration + +(b) 2nd iteration + +(c) 3rd iteration + +(d) 4th iteration + +(e) 5th iteration +Figure 15 Confusion matrices of PointNet classification (scenario 2) + + +(a) 1st iteration + +(b) 2nd iteration + +(c) 3rd iteration + + +Lv1 +44 +1 +Lv2 +18 +15 +1 +Class +Lv3 +2 +40 +12 +Lv4 +5 +33 +3 +Lv5 +9 +23 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +48 +Lv2 +38 +2 +1 +class +Lv3 +13 +33 +3 +True +Lv4 +4 +5 +25 +1 +Lv5 +9 +24 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +54 +Lv2 +29 +5 +Class +Lv3 +1 +40 +8 +Lv4 +7 +36 +4 +Lv5 +2 +20 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +41 +1 +Lv2 +25 +4 +lass +Lv3 +4 +33 +True +2 +Lv4 +21 +32 +Lv5 +1 +5 +31 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +44 +Lv2 +21 +8 +Class +Lv3 +4 +35 +True +6 +Lv4 +1 +1 +12 +40 +3 +Lv5 +1 +7 +23 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +19 +26 +Lv2 +5 +29 +Class +Lv3 +4 +50 +True +Lv4 +41 +Lv5 +3 +29 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +48 +Lv2 +41 +Class +Lv3 +49 +Lv4 +35 +Lv5 +32 +1 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +20 +4 +30 +Lv2 +2 +7 +25 +class +Lv3 +3 +22 +24 +Lv4 +2 +23 +22 +Lv5 +2 +20 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted Class + + +(d) 4th iteration + +(e) 5th iteration +Figure 16 Confusion matrices of PointNet classification (scenario 3) +4.3.2 CNN +Table 8 summarizes the OAs, kappa scores, and F1-scores obtained by the CNN +approach. In general, this approach attained an average OA reaching 90.9% with an average +kappa score of 0.886 (indicating a nearly perfect agreement) throughout our experiments. +Among the five classes of tree decay levels, Decay Level 1 was classified of best, reaching an +average F1-score of 0.999, while Level 3 performed worst, reaching an average F1-score of +0.858. The confusion matrices of the five iterations are shown in Figure 17. +Table 8. The OA, Kappa score, and F1-score of the CNN classification + +OA +κ +F1-score +Level 1 +Level 2 +Level 3 +Level 4 +Level 5 +1st iteration +0.926 +0.906 +1.000 +0.941 +0.889 +0.876 +0.919 +2nd iteration +0.899 +0.874 +1.000 +0.899 +0.843 +0.844 +0.915 +3rd iteration +0.916 +0.895 +1.000 +0.939 +0.870 +0.867 +0.925 +4th iteration +0.913 +0.889 +0.998 +0.888 +0.856 +0.887 +0.921 +5th iteration +0.893 +0.865 +0.997 +0.912 +0.833 +0.849 +0.895 +Average +0.909 +0.886 +0.999 +0.916 +0.858 +0.865 +0.915 +(The best results of each parameter are marked in bold text.) + +Lv1 +39 +3 +Lv2 +29 +C +Lv3 +39 +True +Lv4 +59 +Lv5 +36 +1 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +43 +1 +Lv2 +29 +lass +Lv3 +45 +Lv4 +57 +Lv5 +31 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted Class + + +(a) 1st iteration + +(b) 2nd iteration + +(c) 3rd iteration + +(d) 4th iteration + +(e) 5th iteration +Figure 17. Confusion matrices of image-based CNN classification +4.3.3 Random Forest + The RF approach achieved an average OA reaching 90.6% and an average kappa score +of 0.881, indicating a nearly perfect agreement. Decay Level 1 attained the highest average +F1-score reaching the perfection of 1, while Level 3 attained the lowest with (F1-score = +0.859). It should be highlighted that all Decay Level 1 samples in all iterations were perfectly +predicted. Figure 18 shows the confusion matrices using the RF approach. +Table 9. The OA, Kappa score, and F1-score of RF classification + +OA +κ +F-1 score +Level 1 +Level 2 +Level 3 +Level 4 +Level 5 +1st iteration +0.920 +0.896 +1.000 +0.933 +0.887 +0.860 +0.910 + +206 +0 +0 +0 +0 +Lv2 +0 +120 +11 +0 +0 +Class +Lv3 +0 +4 +177 +18 +0 +True +0 +0 +11 +163 +0 +5- +0 +0 +0 +17 +97 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted Clas5Lv1 +182 +0 +0 +0 +0 +Lv2 +0 +125 +18 +0 +0 +True Class +LV3 +0 +10 +161 +25 +0 +0 +0 +7 +149 +15 +5 +0 +0 +0 +8 +124 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +171 +0 +0 +0 +0 +Lv2 +0 +131 +10 +0 +0 +Class +Lv3 +0 +7 +167 +18 +0 +0 +0 +15 +169 +12 +5- +0 +0 +0 +7 +117 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +201 +0 +0 +0 +0 +Lv2 +1 +95 +20 +0 +0 +Lv3 +0 +3 +152 +26 +0 +0 +0 +2 +188 +17 +0 +0 +0 +3 +116 +1 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted Clas5Lv1 +171 +0 +1 +0 +0 +Lv2 +0 +119 +18 +0 +0 +Class +Lv3 +0 +5 +155 +16 +0 +True +0 +0 +22 +180 +6 +5 +0 +0 +0 +20 +111 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted Class + +2nd iteration +0.898 +0.872 +1.000 +0.863 +0.849 +0.865 +0.914 +3rd iteration +0.905 +0.881 +1.000 +0.897 +0.856 +0.872 +0.918 +4th iteration +0.919 +0.897 +1.000 +0.887 +0.887 +0.897 +0.900 +5th iteration +0.888 +0.859 +1.000 +0.881 +0.818 +0.854 +0.901 +Average +0.906 +0.881 +1.000 +0.892 +0.859 +0.870 +0.909 +(The best results of each parameter are marked in bold text.) + + +(a) 1st iteration + +(b) 2nd iteration + +(c) 3rd iteration + +(d) 4th iteration + +(e) 5th iteration +Figure 18. Confusion matrices of image-based Random Forest classification + + + +Lv1 +206 +0 +0 +0 +0 +Lv2 +0 +119 +12 +0 +0 +Class +Lv3 +0 +5 +181 +13 +0 +True +0 +0 +16 +151 +7 +5- +0 +0 +0 +13 +101 +Lv1 +Lv2 +LV3 +Lv4 +Lv5 +Predicted Class182 +0 +0 +0 +0 +0 +113 +29 +1 +0 +lass +Lv3 +0 +6 +171 +19 +0 +-切 +0 +0 +7 +157 +7 +5- +0 +0 +0 +15 +117 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +171 +0 +0 +0 +0 +Lv2 +0 +118 +23 +0 +0 +Class +Lv3 +0 +4 +172 +16 +0 +anl +0 +0 +15 +173 +8 +5- +0 +0 +0 +12 +112 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +201 +0 +0 +0 +0 +Lv2 +0 +98 +18 +0 +0 +Class +Lv3 +0 +7 +168 +6 +0 +True +- +0 +0 +12 +182 +13 +0 +0 +0 +11 +108 +Lv1 +- +Lv2 +Lv3 +Lv4 +Lv5 +Predicted ClassLv1 +172 +0 +0 +0 +0 +Lv2 +0 +115 +22 +0 +0 +Class +Lv3 +0 +9 +148 +19 +0 +True +- +0 +0 +16 +179 +13 +0 +0 +0 +13 +118 +Lv1 +Lv2 +Lv3 +Lv4 +Lv5 +Predicted Class + +5. Discussion +Our analysis reveals that our method obtained a very good accuracy for characterising +different decay stages of Norway spruce trees. Overall, the CNN-based approach +outperformed (OA = 0.909, κ = 0.886) in our experiments, followed by the RF approach (OA += 0.906, κ = 0.881). Compared to these two, the PointNet model using data input in scenario 2 +had lower performance (OA = 0.806, κ = 0.754). +Based on the results generated by the three scenarios of PointNet classification, it was +found that using multispectral point clouds (X, Y, Z, NIR, R, G) as input led to better results +than simply using the X, Y, and Z values. This proved that the fusion of ALS point clouds and +CIR images was crucial in classifying tree decay stages. By comparing Figure 14 and Figure +15, it is evident that the classification model in scenario 1 was confused with the +differentiation of decay levels 1 and 2 while the model in scenario 2 produced predictions far +more accurately. This showed the significance of color differences in CIR images which +helped distinguish healthy trees and unhealthy trees even though they appeared similar in +terms of their geometry and size. For scenario 3 which takes X, Y, Z, and intensity +information as input, the performance was even worse than simply using coordinates values. +This implied that the intensity information of ALS point clouds did not benefit the +classification of tree decay levels and further confused the model to perform predictions. + +Moreover, the results show that using a 2D image classification approach works better +than a 3D point cloud classification approach in classifying tree decay levels. A possible +explanation is the difference in datasets used by the two kinds of approaches in this study. +While preparing the 2D image dataset, all individual trees were projected from four viewing +angles. As a result, the number of images was four times larger than the number of individual +samples. On the other hand, the number of point cloud samples is the same as the number of + + + +individual tree samples. Consequently, the comparatively smaller number of training and test +samples could result in lower classification accuracy by the PointNet model. Besides, point +cloud sampling was carried out on all point cloud samples due to the requirements of +PointNet. Although an optimal value was chosen for the number of points per sample, +information loss could not be prevented during down-sampling. This varies from the 2D +image classification approaches, which project all point clouds onto an image without point +sampling. + +Comparing the two image-based approaches, both achieved a high classification +accuracy on the image dataset. The slightly higher overall accuracy attained by the CNN +approach reflects the advantages of using deep learning for image classification over +traditional machine learning. Apart from the difference in model architectures, the CNN +model extracted features by itself rather than the manual extraction of handcrafted features, +such that the most distinctive features between different classes could be extracted +automatically. We only extracted three handcrafted features from the images for RF +classification, so the accuracy might still be improved if more representative features were +extracted. Overall, it is found that the image-based approaches have performed well in +distinguishing the five decay stages of individual spruce trees. + + + + + +6. Conclusion +Monitoring forest health is very critical for forest ecosystem management and +environmental protection. Especially dead wood is an important indicator for assessing forest +health and conversartion as forest biodiversity is closely linked to the amount and quality of +dead wood. Tree decay detection and classification is also very important for safety +assessments of forests (Mattheck & Bethge, 1993) and analysis of nest webs of cavity-nesting +species (Altamirano et al., 2017). In recent years, applications of remote sensing-based +machine learning techniques in forest health assessment are rapidly increasing but they are +still in their infancy. In this study, we developed a classification scheme of different tree decay +stages from ALS point clouds and CIR images. Our findings demonstrate the robustness of +machine/deep learning algorithms for determining different decay stages in Norway spruce +from airborne remote sensing platforms for the first time. We expect that our approach can +also be easily transferred to other coniferous tree species as these class of trees shows similar +properties and decay processes. The performance of the CNN being the best further shows the +prowess of CNN in image-based DL. Our investigation also reveals that the combination of +ALS point clouds and CIR imagery significantly improved accuracies. We suggest that future +research should focus on testing other 3D point-cloud-based approaches to improve the +classification accuracy. Applying our CNN-based framework will open new opportunities for +monitoring dead wood quality on a landscape scale, which is a crucial indicator of forest +biodiversity. This will enable a better assessment of forest health and consequently allow +more effective and sustainable forest ecosystem management and conservation. Moreover, the +method can be also applied to other tasks such as the assessment of trees in regard of public +safety close to roads and houses. + + + + +Acknowledgement +This work was supported by the National Natural Science Foundation of China (Project No. +42171361), the Research Grants Council of the Hong Kong Special Administrative Region, +China, under Project PolyU 25211819. 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Remote Sensing, 10(12), 1972. + diff --git a/gdAzT4oBgHgl3EQf4P5h/content/tmp_files/load_file.txt b/gdAzT4oBgHgl3EQf4P5h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..875a507d6e4c0bfdcfb00c0e76d868a699943c00 --- /dev/null +++ b/gdAzT4oBgHgl3EQf4P5h/content/tmp_files/load_file.txt @@ -0,0 +1,2055 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf,len=2054 +page_content='Highlights 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' An automatic pipeline leverages ALS and aerial camera to quantify decaying trees 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Five decay stages of individual coniferous trees are identified for the first time 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Image texture is of the highest relevance to the success of categorizing tree decay 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Multispectral colorized point clouds underperform side-view projected imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' ALS data with laser intensity failed to quantify the tree decay stages 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Landscape-wide assessment of dead wood amount and quality can be achieved Corresponding author Automatic Classification of Single Tree Decay Stages from Combined ALS Data and Aerial Imagery using Machine Learning Tsz Chung Wong1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Abubakar Sani Mohammed1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Wei Yao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Marco Heurich3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 1 Department of Land Surveying and Geo Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The Hong Kong Polytechnic University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Hung Hom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Kowloon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Hong Kong 2 The Hong Kong Polytechnic University Shenzhen Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' China 3 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' for National Park Monitoring and Animal Management, Bavarian Forest National Park, 94481 Grafenau, Germany 4 Chair of Wildlife Ecology and Management, Faculty of Environment and Natural Resources, University of Freiburg, Freiburg, Germany 5 Department of Forestry and Wildlife Management, Campus Evenstad, Inland Norway University of Applied Sciences, Koppang, Norway tszchung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='wong@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='hk, abubakar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='sanimohammed@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='hk, wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='yao@polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='hk, marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='heurich@npv bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='bayern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='de Keywords: Tree decay stages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Airborne laser scanning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Color infrared imagery, Convolutional Neural Network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Machine Learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Dead wood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Forest disturbances Abstract: Understanding forest health is of great importance for the conservation of the integrity of forest ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The monitoring of forest health is, therefore, indispensable for the long term conservation of forests and their sustainable management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In this regard, evaluating the amount and quality of dead wood is of utmost interest as they are favorable indicators of biodiversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Apparently, remote sensing based machine learning techniques have proven to be more efficient and sustainable with unprecedented accuracy in forest inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' However, the application of these techniques is still in its infancy with respect to dead wood mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This study investigates for the first time the automatic classification of individual coniferous trees into five decay stages (live, declining, dead, loose bark, and clean) from combined airborne laser scanning (ALS) point clouds and color infrared (CIR) images using three different Machine Learning methods - 3D point cloud-based deep learning (PointNet), Convolutional Neural Network (CNN), and Random Forest (RF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' First, CIR colorized point clouds are created by fusing the ALS point clouds and the color infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Then, with the CIR colorized point cloud, individual tree segmentation is conducted using a semi-automatic approach to extract single tree objects, which are further projected onto four orthogonal planes displaying the side views of the trees in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Finally, the classification is conducted on the two datasets (3D multispectral point clouds and 2D projected images) based on the three Machine Learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' All models achieved promising results, reaching overall accuracy (OA) of up to 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='9%, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='6%, and 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='6% for CNN, RF, and PointNet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The experimental results reveal that the image-based approach notably outperformed the 3D point cloud-based one, while spectral image texture is of the highest relevance to the success of categorizing tree decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Our models could therefore be used for automatic determination of single tree decay stages and landscape-wide assessment of dead wood amount and quality using modern airborne remote sensing techniques with machine/deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The proposed method can contribute as an important and rigorous tool for monitoring biodiversity in forest ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Introduction Forests are a precious natural resources that deliver important ecosystem services to the benefit of humankind(Brockerhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Therefore maintaining and monitoring forest health is of great importance for the conservation of the integrity of forest ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Among others forests play a critical role in maintaining biodiversity and in the carbon cycle (Paniagua-Ramirez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Houghton 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Polewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 2021) and can contribute to reducing the greenhouse effect and consquently alleviating global warming if sustainably managed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' However, various diseases and disturbances could cause tree decay, including microorganisms, insect infestation, droughts, wildfires, and extreme weather conditions (Blanchette, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Trumbore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Seibold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' For instance, bark beetle attacks have caused large-scale disturbances in recent years (Latifi, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Hlásny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Sani-Mohammed, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Such areas can harbour large amounts of dead wood, making assessing the carbon stored a critical task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Moreover, dead wood is an important resource for biodiversity (Jonsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Lassauce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Stokland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Seibold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' However, to harbor a high biodiversity, the quantity and quality of the dead wood is crucial (Similä et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Lassauce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Seibold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Therefore, not just the estimation of the amount of dead wood but also the assessment of the dead wood quality is a decisive precondition for sustainable forest management and conservation of biodiversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Thus, early detection of declining trees can help identify the origin of bark beetle infestation for quick preventive measures before it spreads widely in the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This would significantly improve forest management and protect biodiversity (Abdullah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' For standing dead wood, Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (1979) developed a framework for classifying different decay stages, which proved to be crucial for the habitat modelling of different groups of species, such as birds, bats and beetles (Similä et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Cousins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Zielewska-Büttner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Kortmann etal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' However, the assessment of the different decay stages still depends on a visual evaluation by field crews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Conventional forest management methods such as field surveying are very time and cost-consuming due to the large area of forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' With the advancement of remote sensing technologies, the health of forests can be sensed without on-site measurements in a cost- efficient way with high quality (Lausch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Among various remote sensing techniques, airborne laser scanning (ALS) has been widely used in forestry applications as it can accurately map the 3-dimensional structures of the forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Based on this data cruical information can be extracted from the point cloud data from the tree to the landscape level (Maltamo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Latifi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Moudrý et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In the past, various studies have been conducted to classify and detect standing dead trees using lidar point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2012a) identified standing dead trees in a temperate forest park using full-waveform lidar data by training a binary support vector machine (SVM) classifier, achieving overall accuracy (OA) of 73%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Polewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2015) and Polewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2016) proposed a method of integrating ALS data and aerial infrared imagery to detect standing dead trees by using an active learning approach, reaching an OA of 89%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Wing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2015) analyzed the spatial distribution of snags and provided a better understanding of wildlife snag use dynamics using neighbourhood attribute filtered ALS data followed by individual tree detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The overall detection rate for snags with DBH >=25 cm was 56% with low commission error rates, while detection rates ranged from 43 to 100%, increasing with the size of the snag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' A multitude of research hase been conducted for classifying tree species together with standing dead trees using lidar data and multispectral images (Yao, et al, 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Polewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=',2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Kamińska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2018) conducted a species-related individual dead tree detection by combining multi-temporal ALS point cloud and color infrared imagery using a Random Forest (RF) classifier, resulting in OA of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Krzystek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2020) studied large-scale mapping of individual conifers, broadleaf trees, and standing dead trees using lidar data and multispectral imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' They reached an overall accuracy of better than 90% after training RF and logistic regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Marchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2018) reviewed studies based on the application of airborne and terrestrial laser scanning for the identification of large deadwood components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', snags, logs, stumps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' They concluded that efforts should be made to transfer the available single-tree methodologies to an operational level but did not consider further characterizing tree conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' To the best of our knowledge no research has been published that quantifies different decay classes of standing dead wood using remote sensing techniques until know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='Until recently studies have extracted handcrafted features from point clouds and multispectral images for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Deep leaning methods are currently becoming more and more used as classification tools for similar tasks in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' For example, Hamraz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', (2019) constructed a deep convolutional neural network (CNN), for binary classification of coniferous and deciduous trees using 2D representations of individual trees and achieved an OA of around 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Briechle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', (2020) also studied the possibility of using the 3D deep neural network PointNet++ for classifying tree species and standing dead trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Their results showed that the 3D PointNet++ approach (OA = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2%) outperformed the RF classifier and handcrafted features approach (OA=85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Furthermore, Seidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2021) compared the performance of image-based CNN approach and point- cloud-based PointNet approach for tree species classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Their study revealed that the CNN approach achieved higher accuracy and computational efficiency than the PointNet approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' However, these studies have only focused on utilizing LiDAR data and multispectral images for classifying tree species or standing dead trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Here we propose that these kinds of data could be applied to broader applications, such as categorising individual trees into several decay stages rather than just living and dead trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' However, to the best of our knowledge, the possibility of using ALS data and multispectral aerial images for classifying tree decay stages is yet to be studied, especially using Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Therefore, the objectives of this study are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (1) to define five tree decay stages that could be detected by ALS and multispectral images, (2) to generate and curate two datasets (3D multispectral colorized point clouds and 2D side-view projected images) of individual tree decay stages as a basis for their classification (training), by combining ALS data and CIR aerial imagery, and (3) to assess the performance of three Machine/Deep Learning algorithms (PointNet, CNN, and RF) in classifying the five decay stages of individual trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The remainder of this paper is structured as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Section two describes the study area and materials used for the experiments, including the datasets acquisition and preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In section three, our three classification approaches using Machine/deep learning are explained in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The results are presented and discussed in section four and five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Lastly, the conclusions are stated in section six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Materials 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1 Study area The experiments were carried out in the Bavarian Forest National Park (BFNP) (49° 3’ 19” N, 13° 12’ 9” E), located in Germany and along the border with the Czech Republic (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Established as the first National Park in Germany, the park covers an area of approximately 24,000 ha with altitudes ranging from 650 to 1453 m above sea level(Heurich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The park is dominated by Norway Spruce (Picea abies), European Beech (Fagus sylvatica), European Silver fir (Abies alba), and other Spruce trees(van der Knaap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Despite experiencing severe thunderstorms in the 1980s and a bark beetle attack during the 1990s (Lausch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2013), the park was protected from human intervention due to the policy of the authority, resulting in a total tree mortality rate of 22% (Müller & Job, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Map of the BFNP and the three transects of ALS point clouds (marked in blue) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='4580000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='4585000 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='4610000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='4615000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='4620000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 Data acquisition ALS point clouds and color infrared images were acquired in 2016 for the BFNP administration within the framework of the data pool initiative for the Bohemian Forest Ecosystem(Latifi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Detailed information on the data is described in sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1 ALS data Three transects of ALS point clouds (leaf-on) were acquired on 18th August 2016 with an LMS-Q680i - 400 kHz scanner in a flight operated by the Milan Geoservice GmbH, at an altitude of 300 m above sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This exercise covered a total area of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 square km with an estimated point density of 70 points per square meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The raw data was processed and corrected to LAZ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 format (with the X, Y, Z and Intensity), while the WGS84/UTM coordinates were calculated with a transformation fitted according to the Gauss-Krüger coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 2 shows the top view of the three transects of ALS point clouds visualized by intensity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 2 The transect of ALS data displayed by the laser intensity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 1st (left), 2nd and 3rd (right) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 Color infrared images Three-band CIR aerial images (G, R, NIR) covering the entire BFNP park were acquired by a Leica DMC 122 camera from a Cessna 207 Aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The flight was conducted on 23rd June 2016 by ILV Remote Sensing GmbH at an average altitude of 2918 m above sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The longitudinal overlap of the flight was 75%, while the cross-coverage was 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The camera was calibrated in 2014, with a focal length of 120 mm, and was used to capture the images with a ground resolution of 20 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Similar to the point cloud data, the images were geo-referenced according to the Gauss-Krüger coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The images were radiometrically corrected and orthorectified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 3 shows an example of a cropped CIR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Unhealthy trees are identified by the green band of this false-color image because defoliated trees less reflect near-infrared radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In contrast, healthy trees appear in red as they reflect a high proportion of near-infrared radiation (Knipling, 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' CIR aerial image captured in BFNP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3 Definition of tree decay stages To achieve our objective of classifying individual trees into their decay levels, it is important to first define them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (1979) defined nine different decay stages, starting from the “live” to the “stump” stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' However, in this study, we are considering only the first five stages of decay (live, declining, dead, loose bark, and clean) proposed by Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (1979) for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' For easier differentiation, we renamed these five stages starting from Decay Level 1 to Decay Level 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 4 illustrates individually segmented trees at the five stages of decay/decomposition visualized with aerial multispectral point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Live Dead Decay stages 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Live 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Declining 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Dead 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Loose bark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Clean Description No decay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' tree has valuable habitat characteristics such as large, clustered, or gnarled branches, or horizontal, thickly moss- covered branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Internal decay or growth deformities (including insect damage, broken tops);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' dying tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Needles or twigs may be present;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' roots sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (Almost) No needles/ twigs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 50% of branches lost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' loose bark;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' top usually broken;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' roots stable Most branches/ bark absent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' some internal decay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' roots of larger trees stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Visualization by corresponding multispectral ALS point clouds Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' System for decay stages for wildlife (coniferous) standing dead tree (Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 1979) and corresponding decay of individual spruce trees from multispectral point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='25 20 15 10 5 5 5 040 35 30 25 20 15 10 5 5 5 0 020 15 10 5 2 0 2 020 15 10 5 2 2 0 015 10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='25 Φ25- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='4 Reference data curation for tree decay analysis Reference data were collected by combing manual interpretation of multispectral point clouds of individual trees with field data according to the defined stages of wildlife tree decay described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This important exercise was conducted with expert advice and the additional support of the CIR imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Several reference plots in the BFNP were considered in such a way that there was a balance in the diversity of decay levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' It is noted that such a manual point cloud labelling method was used due to a lack of complete and precise field data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In total, 1,030 samples were interpreted and labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Specifically, the number of labeled samples for the classes was 233, 167, 236, 239, and 155, for decay stages 1, decay stages 2 decay stages 3, decay level 4, and decay level 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1 Methodological Overview After data acquisition, the data was pre-processed and prepared for input for the classification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We merged the ALS point clouds with the CIR images to extract colors (CIR) for the point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Then we conducted a semi-automatic individual tree segmentation method to segment individual trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' These segmented trees (point clouds) formed the dataset for a 3D deep learning model based on the PointNet to perform a point- cloud-based classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Also, we transformed each individual tree (point cloud) into four 2D projected images via multi-view orthographic projection to form an image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' These 2D projections were the dataset used for performing image-based classification using the CNN and the RF approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' These two sets of data were then trained in the three Machine/deep learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Finally, the trained models were tested, and the results were evaluated to assess their accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 5 illustrates the general workflow of the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' General workflow of the methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 Data preparation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1 Multispectral point cloud curation We generated colors for the point clouds of the selected plots of 30 m radius by combining the ALS point cloud (X, Y, Z, I) with georeferenced CIR images (NIR, R, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Thus, the ALS point cloud was colorized by the georeferenced CIR images in such a way that points pick the exact corresponding colors from the spectral bands of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' For instance, the NIR, R, G values of the CIR imagery at (xi, yi) would be applied to the ALS point cloud at (Xi, Yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This combination generated three more values to the existing four resulting in seven values (X, Y, Z, I, NIR, R, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' However, only six values were considered for the classification, excluding the intensity (I) (as seen in Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This processing was executed using LiDAR360 software v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 (GreenValley International) and QT Modeler v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' A sample plot of combined ALS data and CIR image resulting in multispectral colorized point clouds 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3 Individual tree segmentation A semi-automatic individual tree segmentation approach was conducted to select appropriate samples for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' First, we conducted an automatic individual tree segmentation following the point cloud segmentation algorithm proposed by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The ground points were classified in this automatic stage using an improved progressive TIN densification filtering algorithm (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We then normalized the point clouds with reference to the ground points to remove the effects of topographic relief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' One critical parameter of this algorithm is the 2D Euclidean distance threshold between two adjacent trees because inappropriate thresholds could cause under-segmentation or over-segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' As the target trees in this research were coniferous, the threshold value was set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 m due to their narrow tree crowns, which led to a shorter distance between two neighboring trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Nevertheless, the segmented trees from the automatic approach still needed some final evaluationto not affect the classification performance in subsequent stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Thus, the automatically segmented trees were manually edited and refined in an expert interactive mode by deleting extra point clouds that do not belong to a single tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' For instance, the unwanted point clouds of bushes and young trees around individual trees were manually cleaned (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' However, trees with incomplete point clouds were not selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' With this approach, a total of 1030 individual tree samples were extracted and constituted the data sed(for the 3D PointNet classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' A sample of automatic individual tree segmentation (left) and semi-automatic individual tree segmentation (right) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='4 Image projection In addition to the 3D point clouds of individual trees, another image dataset was generated from the 3D point clouds for the image classification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Each individual tree (point cloud) was transformed into four 2D projected images via multi-view orthographic (parallel) projection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' the four mutually perpendicular side-views that are produced by four mutually perpendicular planes of projection (Front view 0°, Left-side view 90°, Rear view 180°, and Right-side view 270°) as in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This projection from different viewing angles could minimize information loss caused by 3D to 2D representation (Seidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Initially, the image size of each projected image was 644 x 658 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' To achieve a balance between classification efficiency and accuracy, the image sizes were downscaled to 129 x 132 pixels before the training and validation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Table 1 summarizes the distribution of samples in both the 3D point cloud dataset and the 2D image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Images of the same live tree projected from (a) front view, (b) left-side view, (c) rear view, and (d) right-side view Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Number of tree samples in point cloud dataset and image dataset Class label Point cloud dataset Image dataset Level 1 (live) 233 932 Level 2 (declining) 167 668 Level 3 (dead) 236 944 Level 4 (loose bark) 239 956 Level 5 (clean) 155 620 Total 1030 samples 4120 images 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 Model architecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1 PointNet Proposed by Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (2017), PointNet is a deep learning architecture that simply takes point clouds as input to represent 3D geometry instead of typical formats such as images and voxels for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Each set of point cloud input {Pi | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', n} is represented by its x, y, and z coordinates, while additional feature channels such as color and intensity could be added to the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 9 illustrates the model architecture of PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The size of input points (n x 3) changes depending on the number of feature channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' For multispectral point clouds that contain r, g, and b (NIR, R, and G) information, the input size would be n x 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Two shared multi-layer perceptron (MLP) map each point cloud to larger dimensions before max pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' A three-layer fully connected classification model consisting of a shared MLP and a softmax activation map the global feature vector to k classification scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' PointNet architecture Data preparation PointNet requires a fixed number of points per feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' So, we performed a point sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Due to the substantial geometrical and size difference of trees within the five decay classes, the number of points per sample varies greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The minimum point count, maximum point count, and mean point count of each class are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Because of the high variability in the number of points per sample, down-sampling and up-sampling of points must be conducted based on an optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=" This value must represent the individual trees' shape after down-sampling while not increasing the computational cost enormously during up-sampling." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Therefore, we used 2048 as the fixed number of points in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Moreover, we applied data normalization as the values of each feature tree has vastly different ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Therefore, the x, y, and z coordinates of the point cloud data were normalized to range between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Also, the r, g, and b (NIR, R, and G) values of the point cloud data initially range from 0 to 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' These values were also normalized to range between 0 to 1 by dividing all values by 255, to lower the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' n x (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' of feature channels) Input n x (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' of feature channels) Feature mlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' mlp (64,64) Max transform transform (64,128,1024) pool mlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' (512,256,k) Input points n x 64 4 1024 shared nx shared nx 1024 Global feature Output scores 3x3 64x64 T-Net T-Net transform transform Matrix Matrix multiply multiply Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Minimum point count, maximum point count, and mean point count of each class Class minPointCount maxPointCount meanPointCount Level 1 845 10295 3346 Level 2 1586 14405 4785 Level 3 644 5963 2139 Level 4 121 2799 637 Level 5 16 1418 161 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 CNN Unlike PointNet, the CNN takes 2-dimensional images as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In general, the CNN consists of an input layer, multiple hidden layers, and an output layer (Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The input layer takes images of the same image size, generating a shape of (number of inputs) × height × width × (number of channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The hidden layers include several convolution layers, pooling layers, a flatten layer and fully connected or dense layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The convolution layers extract features from the images with the aid of kernels, abstracting the images to a feature map with the shape of (number of inputs) × (height of feature map) × (height of feature map) × (number of channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The pooling layer follows a convolution layer to reduce the dimensions of the feature maps, thus reducing the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The most widely used pooling method for image classification is max pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Max pooling was applied using Equation (1): 𝑓𝑚(𝑣) = 𝑚𝑎𝑥𝑖𝑣𝑖 (1) where the vector 𝑣 is a single 𝑃-dimensional column in a 𝑃 × 𝑘 matrix reduced by a pooling operation 𝑓 (Boureau, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The feature maps extracted in the final convolution and pooling layer are flattened into a vector using the flatten layer before connecting to the fully connected layers to perform classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We built the CNN from scratch for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Further information on our hyperparameters for the model are explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Architecture of the CNN model built for this study Dataset preparation Initially, the size per image was 644 x 658 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' To reduce the computational cost, all images were downscaled by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2, generating images with sizes of 129 x 132 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Furthermore, all pixel values of the three channels were normalized between 0 and 1 for a lower computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' These images were used as input for the CNN as DL requires no manual feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3 Random Forest Random Forest is a supervised ensemble learning algorithm proposed by Breiman (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Ensemble learning aims at improving model performance by running a base learning algorithm several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Combining the hypotheses generated by the algorithm at each time, a voted hypothesis could be obtained, which usually leads to a better prediction (Dietterich, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Random Forest is an improved version of the decision trees algorithm by reducing the variance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' It generates multiple decision trees randomly, then combines and Fully connected layers Output Conv layer 1 Conv layer 2 Conv layer 3 Conv layer 4 Input 125x128x32 60x62x64 28x29x128 12x12x256 129x132x3 Max Max Max Max pooling pooling pooling pooling 2x2 2x2 2x2 2x2 averages the outputs of the decision trees to obtain a final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 11 illustrates the model architecture of a random forest classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Model architecture of a random forest classifier Dataset preparation The processing on the image dataset for Random Forest classification was the same as that for CNN image classification, which is mentioned in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Feature extraction Unlike DL, RF being a traditional machine learning algorithm require handcrafted feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Thus, we extracted handcrafted features from the processed image dataset, and performed a feature importance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The initial handcrafted features include Hu moments (Hu, 1962), Haralick texture (Haralick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 1973), Histogram of Hue Saturation Value (HSV), Histogram of Oriented Gradients (HOG) (Dalal & Triggs, 2005), and Harris Corner Detection (Harris & Stephens, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This feature importance analysis was computed using the RF Regressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 12 shows the ranking of the feature importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Training Training Training Data 1 Data 2 Data n Training Set Decision Decision Decision Tree l Tree 2 Tree n Test Set Voting Prediction Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Feature importance ranking for Random Forest classification Based on the outcome of the feature importance analysis, we selected the top three feature sets (Haralick texture, HSV histogram, and Hu moments) as the final handcrafted features and extracted them from each processed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' These features were combined into a global feature vector as the input for the RF Classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 13 depicts a 2D feature map (including 1st and 2nd principal component) generated by Principal Component Analysis (PCA) with the five classes of tree decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' A 2-D feature space generated after feature extraction and PCA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='6 Model evaluation The performance of our models’ classification was evaluated based on the overall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 Haralick HSV Humoments Harris HOG FeatureDescriptor2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 Lv1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 Lv2 Lv3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 Lv4 Lv5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 PC2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0 PC1 accuracy (OA), F1- scores, and Cohen’s kappa coefficient (κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The OA is expressed as the sum of correctly classified features relative to the total number of features of the confusion table as represented in Equation (2): 𝑂𝐴 = 𝑇𝑃+𝑇𝑁 𝑇𝑃+𝑇𝑁+𝐹𝑃+𝐹𝑁 (2) where TP, TN, FP, and FN represent true positive, true negative, false positive, and false negative respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' F1-score is the weighted average of recall and precision which can be computed as expressed in Equation (3): 𝐹1 = 2 ∙ 𝑟𝑒𝑐𝑎𝑙𝑙 ∙𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑟𝑒𝑐𝑎𝑙𝑙+𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 (3) where 𝑅𝑒𝑐𝑎𝑙𝑙 = 𝑇𝑃 𝑇𝑃+𝐹𝑁 and 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = 𝑇𝑃 𝑇𝑃+𝐹𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Cohen’s kappa coefficient (𝜅) measures the agreement between two classifiers by considering the probability of the observed agreement and the probability of agreement by chance (Vieira, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This is expressed as in Equation (4): κ = 𝑝𝑜−𝑝𝑐 1−𝑝𝑐 (4) where the Probability of Observed Agreement is calculated by 𝑝𝑜 = ∑ 𝑥𝑖𝑖 𝑁 and the Probability of Agreement by Chance is derived from the formula 𝑝𝑐 = ∑ 𝑥𝑖+𝑥+𝑖 𝑁2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Then, a comparative analysis between the three classifiers was performed to propose the best performing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Experiments and results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1 Experimental design The PointNet process point clouds with user-defined feature channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Thus, we designed three scenarios for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In the first scenario we applied the X, Y, and Z values of the raw point clouds only as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In the second scenario we used the X, Y, Z, NIR, R, G values of the multispectral point clouds as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' For the third scenario, we used the X, Y, Z, and I (intensity) values of the raw ALS point clouds as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We applied these dynamics to perform a sensitivity analysis relating to the color effect as well as the intensity in our evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We applied a 5-fold cross-validation technique for a robust classification training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We randomly divided the dataset into five different datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' so that the training is conducted five times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Thus, for each training, four parts are used for training and validation while the remaining part is used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This means each one of the five folds has been used once for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Data augmentation was applied to modify the training dataset slightly to avoid model over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The techniques used for data augmentation include random rotation of the point cloud, random removal of points, and random point jittering with Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The 5-fold cross-validation was also used for training and validating the CNN and RF classification schemes following the same procedure as described for the PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 Hyperparameter settings i) PointNet Table 3 shows the hyperparameter- settings considered when training the PointNet deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The hyperparameters were adjusted by a manual search approach to achieve improved classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' PointNet hyperparameters and corresponding values Hyperparameter Value Description numPoints 2048 Number of points per sample inputChannelSize 6 Number of input channels per sample (X, Y, Z, NIR, R, G) numEpochs 20 Number of epochs learnRate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='001 Initial learning rate miniBatchSize 32 Number of samples per batch l2Regularization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='0001 L2 regularization factor learnRateDropPeriod 15 Number of epochs before reducing the learning rate learnRateDropFactor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='6 Factor of the learning rate drop optimizer Adam Algorithm of the optimizer gradientDecayFactor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='9 A parameter of Adam optimizer squaredGradientDecayFactor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='999 A parameter of Adam optimizer ii) CNN In summary, our network consists of four convolutional layers, four max-pooling layers, one flattened layer, and two dense layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We built this CNN model architecture from scratch as illustrated in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The input shape of each CIR image was (129, 132, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The first convolutional layer consists of 32 filters with a kernel size of 5 x 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This kernel size was used due to the relatively large input of the processed image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The second convolutional layer consists of 64 filters with a kernel size of 3 x 3, followed by the third convolutional layer, which comprises 128 filters with the same kernel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Lastly, the fourth convolutional layer was made up of 256 3 x 3 filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Max pooling layers with 2 x 2 filters were added between every two convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Throughout the convolutional base of the network, the Rectified Linear Unit (ReLU) was used as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' To complete the convolutional architecture, a flatten layer was used to flatten the three-dimensional feature to one-dimensional feature for easy pass to the dense layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The two dense layers with the shape of 256 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The ReLU was used for the first dense layer, while the softmax activation function was used for the second dense layer (output layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The softmax activation function helps transform the input vector into a probability vector for performing classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The output shape (none, 5) in the output layer represents the five classes of tree decay levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Description of the CNN model architecture Layer (type) Output Shape Param # conv2d (Conv2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 32) 2432 max_pooling2d (MaxPooling2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 62,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 32) 0 conv2d_1 (Conv2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 60,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 62,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 64) 18496 max_pooling2d_1 (MaxPooling2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 64) 0 conv2d_2 (Conv2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 128) 73856 max_pooling2d_2 (MaxPooling2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 128) 0 conv2d_3 (Conv2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 256) 295168 max_pooling2d_3 (MaxPooling2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 256) 0 flatten (Flatten) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 9216) 0 dense (Dense) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 256) 2359552 dense_1 (Dense) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 5) 1285 Total params: 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='750,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='789 Trainable params: 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='750,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='789 Non trainable params: 0 We used the sparse categorical cross-entropy loss function because of the integers used to represent the labels during our dataset processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The number of training epochs per iteration was set to 20 while the loss was optimized with the Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' iii) Random Forest For the RF classifier, we applied automatic grid search for tuning the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We selected a combination with the highest cross-validation score as the final hyperparameters illustrated in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Other unmentioned hyperparameters were kept as default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Random Forest hyperparameters and corresponding values Hyperparameter Value Description n_estimators 800 Number of estimators max_depth 64 Maximum depth of the decision tree random_state 42 Parameter to control the random number generator class_weight balanced Parameter to adjust classification weights 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3 Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='1 PointNet Tables 6, 7, and 8 summarize the OAs, Kappa scores, and F1-scores of our three experiments using PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Overall, scenario 2 (X, Y, Z, NIR, R, G) achieved the best results with an average OA and Kappa score reaching 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='6% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='754 respectively, throughout the 5-fold cross-validation, indicating substantial agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In terms of the classes of tree decay levels, Level 1 outperformed the others with the highest average F1-score reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='989, while Level 4 achieved the lowest average F1-score reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Scenario 3 performed poorly and produced the worst results with an overall Kappa score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='047, implying there was almost no agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figures 14, 15, and 16 show the confusion matrices of each iteration for scenarios 1, 2, and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The OA, Kappa score, and F1-score of PointNet classification (scenario 1) OA Kappa score (κ) F1-score Level 1 Level 2 Level 3 Level 4 Level 5 1st iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='629 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='704 2nd iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='590 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='759 3rd iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='583 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='447 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='762 4th iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='610 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='645 5th iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='480 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='681 Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='605 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='590 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='611 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='710 (The best results of each parameter are marked in bold text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=') Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The OA, Kappa score, and F1-score of PointNet classification (scenario 2) OA Kappa score (κ) F1-score Level 1 Level 2 Level 3 Level 4 Level 5 1st iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='793 2nd iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='768 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='828 3rd iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='833 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='870 4th iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='849 5th iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='821 Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='832 (The best results of each parameter are marked in bold text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=') Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The OA, Kappa score, and F1-score of PointNet classification (scenario 3) OA Kappa score (κ) F1-score Level 1 Level 2 Level 3 Level 4 Level 5 1st iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 2nd iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='059 3rd iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='(d) 4th iteration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='(e) 5th iteration Figure 16 Confusion matrices of PointNet classification (scenario 3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='2 CNN Table 8 summarizes the OAs, kappa scores, and F1-scores obtained by the CNN approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In general, this approach attained an average OA reaching 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='9% with an average kappa score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='886 (indicating a nearly perfect agreement) throughout our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Among the five classes of tree decay levels, Decay Level 1 was classified of best, reaching an average F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='999, while Level 3 performed worst, reaching an average F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The confusion matrices of the five iterations are shown in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The OA, Kappa score, and F1-score of the CNN classification OA κ F1-score Level 1 Level 2 Level 3 Level 4 Level 5 1st iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='906 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='919 2nd iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='874 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='915 3rd iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='895 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='925 4th iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='921 5th iteration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='895 Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} 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iteration (e) 5th iteration Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Confusion matrices of image-based CNN classification 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='3 Random Forest The RF approach achieved an average OA reaching 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='6% and an average kappa score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='881, indicating a nearly perfect agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Decay Level 1 attained the highest average F1-score reaching the perfection of 1, while Level 3 attained the lowest with (F1-score = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='859).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' It should be highlighted that all Decay Level 1 samples in all iterations were perfectly predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Figure 18 shows the confusion matrices using the RF approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Table 9.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='Predicted Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Discussion Our analysis reveals that our method obtained a very good accuracy for characterising different decay stages of Norway spruce trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Overall, the CNN-based approach outperformed (OA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='909, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='886) in our experiments, followed by the RF approach (OA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='906, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='881).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Compared to these two, the PointNet model using data input in scenario 2 had lower performance (OA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='806, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='754).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Based on the results generated by the three scenarios of PointNet classification, it was found that using multispectral point clouds (X, Y, Z, NIR, R, G) as input led to better results than simply using the X, Y, and Z values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This proved that the fusion of ALS point clouds and CIR images was crucial in classifying tree decay stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' By comparing Figure 14 and Figure 15, it is evident that the classification model in scenario 1 was confused with the differentiation of decay levels 1 and 2 while the model in scenario 2 produced predictions far more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This showed the significance of color differences in CIR images which helped distinguish healthy trees and unhealthy trees even though they appeared similar in terms of their geometry and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' For scenario 3 which takes X, Y, Z, and intensity information as input, the performance was even worse than simply using coordinates values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This implied that the intensity information of ALS point clouds did not benefit the classification of tree decay levels and further confused the model to perform predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Moreover, the results show that using a 2D image classification approach works better than a 3D point cloud classification approach in classifying tree decay levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' A possible explanation is the difference in datasets used by the two kinds of approaches in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' While preparing the 2D image dataset, all individual trees were projected from four viewing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' As a result, the number of images was four times larger than the number of individual samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' On the other hand, the number of point cloud samples is the same as the number of individual tree samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Consequently, the comparatively smaller number of training and test samples could result in lower classification accuracy by the PointNet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Besides, point cloud sampling was carried out on all point cloud samples due to the requirements of PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Although an optimal value was chosen for the number of points per sample, information loss could not be prevented during down-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This varies from the 2D image classification approaches, which project all point clouds onto an image without point sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Comparing the two image-based approaches, both achieved a high classification accuracy on the image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The slightly higher overall accuracy attained by the CNN approach reflects the advantages of using deep learning for image classification over traditional machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Apart from the difference in model architectures, the CNN model extracted features by itself rather than the manual extraction of handcrafted features, such that the most distinctive features between different classes could be extracted automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We only extracted three handcrafted features from the images for RF classification, so the accuracy might still be improved if more representative features were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Overall, it is found that the image-based approaches have performed well in distinguishing the five decay stages of individual spruce trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Conclusion Monitoring forest health is very critical for forest ecosystem management and environmental protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Especially dead wood is an important indicator for assessing forest health and conversartion as forest biodiversity is closely linked to the amount and quality of dead wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Tree decay detection and classification is also very important for safety assessments of forests (Mattheck & Bethge, 1993) and analysis of nest webs of cavity-nesting species (Altamirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In recent years, applications of remote sensing-based machine learning techniques in forest health assessment are rapidly increasing but they are still in their infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' In this study, we developed a classification scheme of different tree decay stages from ALS point clouds and CIR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Our findings demonstrate the robustness of machine/deep learning algorithms for determining different decay stages in Norway spruce from airborne remote sensing platforms for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We expect that our approach can also be easily transferred to other coniferous tree species as these class of trees shows similar properties and decay processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' The performance of the CNN being the best further shows the prowess of CNN in image-based DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Our investigation also reveals that the combination of ALS point clouds and CIR imagery significantly improved accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' We suggest that future research should focus on testing other 3D point-cloud-based approaches to improve the classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Applying our CNN-based framework will open new opportunities for monitoring dead wood quality on a landscape scale, which is a crucial indicator of forest biodiversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This will enable a better assessment of forest health and consequently allow more effective and sustainable forest ecosystem management and conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Moreover, the method can be also applied to other tasks such as the assessment of trees in regard of public safety close to roads and houses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Acknowledgement This work was supported by the National Natural Science Foundation of China (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 42171361), the Research Grants Council of the Hong Kong Special Administrative Region, China, under Project PolyU 25211819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' This work was partially supported by The Hong Kong Polytechnic University under Project 1-ZVN6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' References Abdullah, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', Skidmore, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', Darvishzadeh, R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', Krzystek, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' and Heurich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2012a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Identifying standing dead trees in forest areas based on 3D single tree detection from full waveform lidar data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' ISPRS Annals of the Protogrammetry, Remote Sensing and Spatial Information Sciences, 1(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Yao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', Krzystek, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', & Heurich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 2012b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Tree species classification and estimation of stem volume and DBH based on single tree extraction by exploiting airborne full-waveform LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Remote Sensing of Environment, 123, 368-380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', Guo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' and Xue, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Improved progressive TIN densification filtering algorithm for airborne LiDAR data in forested areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' ISPRS Journal of Photogrammetry and Remote Sensing, 117, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content='79-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Zielewska-Büttner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', Heurich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', Müller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=', & Braunisch, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Remotely sensed single tree data enable the determination of habitat thresholds for the three-toed woodpecker (Picoides tridactylus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} +page_content=' Remote Sensing, 10(12), 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQf4P5h/content/2301.01841v1.pdf'} diff --git a/gdE4T4oBgHgl3EQfRwwk/content/tmp_files/2301.04992v1.pdf.txt b/gdE4T4oBgHgl3EQfRwwk/content/tmp_files/2301.04992v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c13dc3f4d5edbcd3fe6d9f88711622a8189f770 --- /dev/null +++ b/gdE4T4oBgHgl3EQfRwwk/content/tmp_files/2301.04992v1.pdf.txt @@ -0,0 +1,2025 @@ +The Bending of C3: Experimentally Probing the l-type Doubling and Resonance +Marie-Aline Martin-Drumela,1, Qiang Zhangb,∗∗, Kirstin D. Doneyc, Olivier Piralia,d, Michel Vervloeta, Dennis Tokaryke, Colin +Westernf, Harold Linnartzc, Yang Chenb, Dongfeng Zhaob,∗ +aUniversité Paris-Saclay, CNRS, Institut des Sciences Moléculaires d’Orsay, Orsay, F-91405, France +bCAS Center for Excellence in Quantum Information and Quantum Physics, and Hefei National Laboratory for Physical Sciences at the Microscale, University of +Science and Technology of China, Hefei, Anhui, 230026, P. R. China +cLaboratory for Astrophysics, Leiden Observatory, Leiden University, PO Box 9513, RA Leiden, NL2300, the Netherlands +dSOLEIL Synchrotron, AILES beamline, l’Orme des Merisiers, Saint-Aubin, Gif-sur-Yvette, F-91190, France +eDepartment of Physics, University of New Brunswick, Fredericton, NB, Canada +fSchool of Chemistry, University of Bristol, Bristol, United Kingdom +Abstract +C3, a pure carbon chain molecule that has been identified in different astronomical environments, is considered a good probe of kinetic +temperatures through observation of transitions involving its low-lying bending mode (ν2) in its ground electronic state. The present +laboratory work aims to investigate this bending mode with multiple quanta of excitation by combining recordings of high resolution +optical and infrared spectra of C3 produced in discharge experiments. The optical spectra of rovibronic ( ˜A 1Πu − ˜X 1Σ+ +g) transitions +have been recorded by laser induced fluorescence spectroscopy using a single longitude mode optical parametric oscillator as narrow +bandwidth laser source at the University of Science and Technology of China. 36 bands originating from ˜X(0v20), v2 = 0 − 5, +are assigned. The mid-infrared spectrum of the rovibrational ν3 band has been recorded by Fourier-transform infrared spectroscopy +using a globar source on the AILES beamline of the SOLEIL synchrotron facility. The spectrum reveals hot bands involving up to +5 quanta of excitation in ν2. From combining analyses of all the presently recorded spectra and literature data, accurate rotational +parameters and absolute energy levels of C3, in particular for states involving the bending mode, are determined. A single PGOPHER +file containing all available data involving the ˜X and ˜A states (literature and present study) is used to fit all the data. The spectroscopic +information derived from this work enables new interstellar searches for C3, not only in the infrared and optical regions investigated +here but also notably in the ν2 band region (around 63 cm−1) where vibrational satellites can now be accurately predicted. This makes +C3 a universal diagnostic tool to study very different astronomical environments, from dark and dense to translucent clouds. +1. Introduction +The bare carbon chain molecule propadienediylidene, C3, has +for long been the object of considerable interest to astronomers +and spectroscopists alike. It is one of the few species for which +astronomical observations have preceded laboratory identifica- +tion. Puzzling, unidentified lines around 4051 Å were first de- +tected in cometary optical emission in 1881 by Huggins [1] +then observed toward many cometary bodies [e.g., 2–4]. These +lines were successfully reproduced in the lab by Raffety [5] and +Herzberg [6] but their molecular carrier remained elusive until the +work of Douglas [7] who conclusively identified C3 through an +isotope-substitution experiment. First actual rovibronic assign- +ments of transitions in this band to transitions of the ˜A 1Πu− ˜X 1Σ+ +g +system were delayed until the work of Gausset et al. [8, 9]. Be- +cause of its lack of a permanent dipole moment, C3 does not +possess a pure rotational spectrum and is thus not accessible by +radio astronomy. Instead, its electronic and vibrational spectra +in the optical and infrared regions, respectively, provide useful +spectroscopic approaches for detecting and tracing this simple +molecule in the interstellar medium (ISM). Optical detections of +∗dzhao@ustc.edu.cn +∗∗qiangzhang@ustc.edu.cn +1marie-aline.martin@universite-paris-saclay.fr +the ˜A 1Πu − ˜X 1Σ+ +g band of C3 have established the species as an +ubiquitous component of diffuse interstellar matter [e.g., 10–15] +making it an ideal probe of the physical and chemical conditions +in such environments. In the work of Schmidt et al. [15], the most +extensive rovibronic C3 series detected in space to date, seven ˜A– +˜X rovibronic bands in addition to the origin band were observed +in absorption towards HD 169454, a reddened star (because of +molecular extinction along a line of sight crossing one or more +molecular clouds), in a high-quality absorption spectrum. Rovi- +bronic series have also been reported in cometary spectra [e.g., +16]. In the past decades, C3 has also been detected in circum- +stellar shells and in the dense ISM via detection of transitions in +its asymmetric stretching (ν3 ∼ 2040 cm−1) [e.g., 17] and bend- +ing (ν2 ∼ 63 cm−1) [e.g., 18–20] vibrational bands. It is partic- +ularly interesting to note that the observation of several rovibra- +tional transitions involving the low-lying ν2 fundamental mode +have been reported in the carbon-rich star IRC+10216 [18], in +the molecular cloud Sagittarius B2 [18, 19], and in the star form- +ing cores W31C and W49N [20]. The unusually low frequency +of the bending mode makes vibrationally excited C3 a powerful +probe of the kinetic temperature in various interstellar environ- +ments [18]. +Besides the case of the 4051 Å comet band system, astronom- +ical observations have also prompted other laboratory studies on +C3. Following the early work by Douglas [7] and Gausset et al. +arXiv:2301.04992v1 [astro-ph.GA] 12 Jan 2023 + +[9], dedicated experiments have been conducted, aiming for a +comprehensive characterization of the ˜A 1Πu − ˜X 1Σ+ +g electronic +band system of C3 at low- [see, e.g., 21–27] and high- [e.g., +15, 20, 28–37] resolution. The work of McCall et al. [38] led +to a reassignment of the R(0) transition of the ˜A 1Πu − ˜X 1Σ+ +g +(000)–(000) band, in agreement with astronomical data [13]. Op- +tical measurements have enabled the accurate description of the +vibrational pattern in the electronic ground state (GS), notably the +determination of the infrared inactive ν1 symmetric stretch band +center (∼ 1224 cm−1) [34]. +In almost simultaneous studies, the ν3 fundamental vibra- +tional band was observed in the laboratory [39] and in space +[17]. +The ν2 bending mode was investigated by far-infrared +spectroscopy [40] subsequently enabling its interstellar detection +[18, 41] which then prompted further high resolution laboratory +investigations of the band [42, 43]. The ν1 + ν3 combination +band was also granted some interest: first measured in the lab- +oratory by Krieg et al. [44], it was subsequently re-investigated +by Schröder et al. [45] who extended the known stretching vi- +brational manifold, with levels involving up to seven quanta of +excitation in ν1 and three quanta in ν3, thus probing the poten- +tial energy surface (PES) of this “floppy” molecule to high en- +ergies along the stretching coordinate and providing precise sets +of molecular constants. In contrast, the ν2 bending vibrational +manifold remains unstudied at this level of thoroughness with +only a limited number of dedicated investigations. +Relatively +low-resolution stimulated-emission pumping (SEP) studies [e.g., +24, 25, 46] have resulted in the observation of levels involving +v2 = 0 − 34 in the electronic GS while the rovibronic investi- +gations of the ˜A 1Πu − ˜X 1Σ+ +g band system by Gausset et al. [9] +revealed bands involving up to v2 = 4. More extensive high- +resolution data from direct absorption infrared studies, by prob- +ing hot bands in either ν3 [47] or ν1 + ν3 [44], have enabled the +detection of levels with up to (only) two quanta of excitation in +ν2. High resolution spectroscopic information for transitions in- +volving higher quanta of excitation in ν2 is highly desirable to +characterize the PES along the bending coordinate. Such data +will also serve the astronomical community as the observation of +vibrationally excited C3 will allow to probe kinetic temperature +of various interstellar environments. +Even though an exhaustive review of the extensive literature +on C3 is beyond the scope of this paper, it is worth mentioning +that because of its relevance not only in astrophysics but also in +molecular physics, many other studies have been dedicated to this +species. Many theoretical investigations were conducted on its +potential energy surface [e.g. 48–50] aiming specifically at estab- +lishing its equilibrium structure (and resolving the long-standing +ambiguity linear vs. bent; the former being the census to date) +and providing reliable energy term values and vibrational con- +stants for excited vibrational levels in the electronic GS. Exper- +imentally, other electronic band systems have been investigated +[e.g., 51–54] while C3 isotopologues have been the subject of op- +tical [55, 56] and infrared [57] studies. The vacuum ultraviolet +photoionization spectrum of C3 was also reported [58]. Weltner +and Van Zee [59] and Van Orden and Saykally [60] have pub- +lished detailed reviews on the experimental works on C3 as of +the late 1990’s. C3 is also included in many astronomical models +[e.g., 61, 62] and dedicated experiments have been conducted to +investigate its reactivity with neutral species in space [see 63]. +In this paper, we report a joint optical ( ˜A 1Πu− ˜X 1Σ+ +g band) and +infrared (ν3 band) investigation of C3 in which we detected and +assigned bands involving levels with up to 5 quanta of excitation +in ν2. Many of these bands are reported here for the first time. The +combined analysis of these two data sets enables the accurate de- +termination of the spectroscopic constants of the species as well +as absolute rovibrational level energies for the v2 = 0 − 5 levels +in the electronic GS. From this, accurate far-infrared transitions +are derived for ro-vibrational transitions involving the ν2 mode +and its hot band sequences guiding reliable interstellar searches. +The paper is organized as follows: in Section 2, we describe +the two experimental approaches used in this work; in Section +3, the experimental results and spectral analysis are presented; +in Section 4, a prediction of the line position of the ν2 rovibra- +tional hot bands in the far-infrared is reported; and in Section 5, +the astronomical implications of the present data are discussed. +The recorded line positions, experimental spectra, and fit files are +available in the extensive supplementary material. +2. Experimental methods +2.1. Optical Spectroscopy +The optical spectroscopic study of the ˜A 1Πu− ˜X 1Σ+ +g transition +of C3 has been performed between 380 and 410 nm at the Univer- +sity of Science and Technology of China by using a laser induced +fluorescence (LIF) setup that has been described in Zhang et al. +[64, 65, 66]. In this experiment, C3 molecules are produced in a +pulsed DC discharge nozzle using two flat-top stainless steel elec- +trodes [67]. A gas mixture of 0.3 % acetylene (C2H2) diluted in +argon is introduced into the nozzle by a General Valve (Series 9, +0.5 mm orifice). High voltage pulses (≃ −2000 V, 20 µs, 10 Hz) +are applied to one of the electrodes while the other is grounded +in order to produce an intense pulsed plasma. The plasma con- +taining C3 molecules is then expanded and adiabatically cooled +by collisions with the buffer gas. +About 10 mm downstream, +the molecular beam is crossed perpendicularly by a laser beam, +which suppresses the Doppler broadening in the recorded spectra. +Fluorescent emission from laser excited C3 molecules is collected +by a lens system perpendicular to the laser beam, guided into a +grating spectrometer (Zolix, 0.5 m) and then detected by a photo- +multiplier tube (Hamamatsu, R928). +A home-built single-longitude-mode optical parametric oscil- +lator (SLM-OPO) is employed as the laser source [68]. In the +present study, the signal output of the OPO is frequency-doubled +in a KDP (KH2PO4) crystal to obtain tunable radiation between +350 and 450 nm. +A small portion (∼ 5 %) of the OPO sig- +nal output is injected into a wavelength meter (High Finesse, +WS7-60) for calibration. +The wavelength meter is calibrated +with a stabilized He-Ne laser, providing a frequency accuracy of +∼ 0.002 cm−1 for the laser source. +Special care had to be taken in the measurement of the rela- +tively weak hot vibronic bands. This is because in the supersonic +jet the population of an excited vibrational level is much lower +than that in the v = 0 level and, in most cases, the hot bands are +overlapped with stronger fundamental vibronic bands. To over- +come these difficulties, the spectrometer is used as a narrow band +2 + +pass filter which helps to distinguish fluorescence of the hot bands +from other overlapping bands. As the fluorescence from different +upper states often results in different dispersed spectra, and since +the bandpass of the monochromator is extremely narrow (∼ 0.5 +nm), it becomes possible to only detect the fluorescence emission +from upper states involving specific GS hot bands. This was real- +ized by selecting a dispersion wavelength of the monochromator +corresponding to the fluorescence from the upper electronic state +(itself optically-pumped from vibrationaly excited states of the ˜X +state) to the lowest allowed vibrational level of the GS. +A laser excitation spectrum is recorded by measuring the inten- +sity of the fluorescence as a function of the continuously tuned +laser wavelength. This technique yields a very good signal-to- +noise (S/N) ratio and a spectral resolution of ∼ 0.02 cm−1 (cor- +responding to a resolving power of ∼ 1,200,000) for the strong +bands. Based on the line width of the rovibronic transitions, the +accuracy of the extracted line positions from our spectra can con- +fidently be assumed to be of about 0.002 cm−1 for the bands aris- +ing from v′′ +2 = 0, 1 in the ground electronic state [with the ex- +ception of the ˜A(000)– ˜X(000) band for which a 0.005 cm−1 un- +certainty is used] and 0.005 cm−1 for the other hot bands (with +v′′ +2 ≥ 2). Absolute frequency accuracies may be subject to small +shifts because of combining spectra from different wavelength +regimes that are recorded in different experimental setups, but +comparison with previous studies [e.g., 31, 40, 42] allows for re- +liable calibration (see later in section 2.3). +2.2. Infrared Spectroscopy +The absorption spectrum of C3 has been recorded in the ν3 +asymmetric stretch region (around 2000 cm−1/ 5 µm) using an +experimental set-up available on the AILES beamline of syn- +chrotron SOLEIL previously used to investigate the far-infrared +spectra of various reactive species [see 69–71]. The schematic +representation of the discharge set-up together with its implemen- +tation on the AILES beamline is given in Figure 1 of Martin- +Drumel et al. [69]. We briefly describe in this paper the main +characteristics of the set-up and the discharge conditions we used +to optimize the signal of C3. The discharge cell consists of a +1.1 m long Pyrex tube (13 cm inner diameter) equipped with mul- +tipass optics (White-type) allowing for 24 m of absorption path- +length. The cell is connected under vacuum to a Bruker Fourier- +transform (FT) infrared spectrometer and is separated from it by +two wedged CaF2 windows. A total of seven cell inlets are used: +two are connected to the two water-cooled electrodes (separated +by 70 cm), four allow for gas injection (buffer gas or sample) at +different locations in the cell, and one located at the center of the +cell is connected to a vacuum system (250 m3/h Roots blower). +To probe the positive column of the plasma, one of the electrodes +is connected to the high voltage while the second is grounded. In +the present work, C3 is synthesized using a discharge of He (in- +jected both through the electrodes and through the two gas inlets +closest to the multipass optics) seeded with a small amount of +CH4 (injected using the inlets located closer to the center of the +cell). +Under our experimental conditions, we find that the abundance +of C3 is very sensitive to both the discharge current and the pres- +sure of CH4 precursor. To optimize the production of C3, we use +a rapid scan optical fiber spectrometer (Ocean Optics) to monitor +the 4051 Å emission band. The CH4 pressure is slowly increased +until reaching the maximum of the visible emission while simul- +taneously adjusting the discharge current. In the optimum con- +ditions, about 1 A of current (at 1 kV DC) drives the discharge +through a ballast resistor of 100 Ω. A total pressure of about +1 mbar of a mixture of CH4 seeded in He is injected in the cell +and a continuous flow of gas is maintained. To record the ab- +sorption spectra in the 5 µm region, we use the globar internal +source of the Bruker FTIR installed on the AILES beamline of the +SOLEIL synchrotron, a CaF2 beamsplitter, and an Indium Anti- +monide (InSb) detector. These experimental conditions, together +with the use of a bandpass optical filter, limit the bandwidth of our +acquisition to the 1850–2100 cm−1 range. The unapodized spec- +tral resolution is set to 0.004 cm−1 and 138 scans are co-added. +Despite the effort to optimize the synthesis of C3 in our cell and +reduce the noise floor, the most intense lines of our spectra corre- +spond to only about 5 % of signal absorption resulting in a SNR +of 10 at best. +Aside from C3, intense absorption lines of CO (possibly pro- +duced by reaction with residual H2O) belonging to the ∆v = 1 se- +quences (with v′′ = 0−14, the 4 – 3 band being the strongest one) +of the species in its electronic GS are observed over the full spec- +tral range covered in this study (Figure S1 in the supplementary +material). Even though these series of intense lines hinder identi- +fication of weaker series of C3 lines, they enable spectral calibra- +tion by comparison with the accurate wavenumbers values from +Refs. [72–74]. After calibration, the frequency accuracy is esti- +mated for each C3 line based on its full-width at half-maximum +and signal-to-noise ratio [75] and ranges from 0.0005 cm−1 to +0.006 cm−1. The rotational contour of the CO bands also allows +for estimation of its rotational temperature (see Figure S2 in the +supplementary material) from comparison with simulations us- +ing the PGOPHER software [76]. Under our experimental con- +ditions, a rotational temperature of 500 K is found in each CO +band. Since the observation of CO hot bands is limited by the +spectral coverage rather than the levels population, no vibrational +temperature value can be asserted. Besides CO lines, a few weak +transitions of the fundamental bending mode ν2 of H2O are also +detected [77] and several lines are assigned to the R branch of the +fundamental ν3 band of C2H in the 1860–1895 cm−1 range [78] +(Figure S1). +2.3. Methodology +As mentioned previously, the ν2 vibrational band lies in the far- +infrared region and is significantly weaker than the ν3 band [48]; +for these reasons, it remains challenging to measure the hot band +sequences involving v2 which would provide vibrational energies +in the v2 progression. Alternatively, such information can be de- +rived from a combined rovibronic and rovibrational analysis of +bands involving various quanta of ν2 excitation in the electronic +GS. This approach is used in the present study exploiting C3 spec- +tra in the optical and mid-infrared region. +Initial analyses of optical and infrared data have been +performed independently, mostly exploiting combination dif- +ferences. +Subsequently, all literature and newly measured +rotationally-resolved transitions have been imported into the +PGOPHER software [76] which has been used successfully to +analyze specific bands of C3 previously [37, 45, 51, 56]. Some +3 + +features of this software have proven particularly powerful in the +present study. First, all assignments can be treated as separate +input files (for instance, one file per band) hence easing the treat- +ment of large datasets. Then, the software allows for a relatively +straightforward simultaneous treatment of rovibrational and rovi- +bronic data. Last but not least, graphical representations of resid- +uals, simulated bands, and term values plots is an invaluable tool +to refine assignments and detect perturbations. We thus perform +a combined fit using PGOPHER of the presently recorded data +together with all available rovibronic ( ˜A 1Πu − ˜X 1Σ+ +g only) and +rovibrational (all known bands in the ˜X state) literature data al- +lowing for the most complete spectroscopic description of the C3 +molecule to date. +2.4. Spectroscopy of C3 +As already mentioned in the introduction, detailed descriptions +of the spectroscopy of C3 can be found in Gausset et al. [9] and +Rousselot et al. [62]; here we recall some aspects pertinent to +this work. In the ˜X 1Σ+ +g electronic GS, as a result of the pres- +ence of a doubly-degenerate bending vibration, the species ex- +hibits l-type doubling. Levels with v2 > 0 are split into multiple +l-states, with l the vibrational angular momentum, such that l = +v, v − 2, ..., 0 or 1. Because the C3 bending frequency is unusu- +ally small (ω2 ∼ 63 cm−1), the l-type doubling is unusually large. +In addition, the ˜A 1Πu electronic state is subject to Renner-Teller +coupling, splitting the v2 bending states into K = |l ± Λ| compo- +nents, where K is the total vibronic angular momentum and Λ +the projection of the electronic orbital angular momentum onto +the molecular axis of the molecule. Again, the very small bend- +ing frequency yields significant effects on C3 spectroscopy, here +resulting in a large Renner-Teller interaction. A typical represen- +tation of splittings for the bending mode of linear molecules expe- +riencing Renner-Teller interaction is given in Figure 9 of Gausset +et al. [9]. Because C3 is linear, thus centrosymmetric, and con- +tains identical nuclei of zero spin, all the antisymmetric levels +have zero weight by spin statistics. In other words, half of the +rotational levels are missing. For example, for Σ+ +g states, only +even-J (e) levels exist while for a Σ+ +u state, only odd-J ( f) lev- +els exist. For other states, all J levels exist but with alternate e/ f +labels. +In the following, each vibronic level is labeled M(v1v2v3) Ns +where M is the electronic state ( ˜X or ˜A), vi (i = 1 − 3) refers to +the quanta of excitation in each vibrational level, N is the Greek +letter associated to the K quantum number, and s the symmetry of +the state (g or u). For instance, the ˜X(020) ∆g state corresponds +to the v2 = 2, K = l = 2 vibrational level in the ˜X 1Σ+ +g electronic +state; the ˜A(020) Φu state corresponds to the v2 = 2, K = 3 +(hence l = 2) vibrational level in the ˜A 1Πu electronic state. When +discussing rovibrational transitions in the ˜X 1Σ+ +g state, the ˜X is +often omitted. +3. Results and discussion +Our rovibronic and rovibrational sets of data are extremely +complementary as illustrated in Figure 1, allowing to retrieve +both accurate energies and spectroscopic constants for levels in- +volving v2 = 0 − 5 in the ˜X 1Σ+ +g state. With regard to the ˜X(0v20) +vibrational levels with v2 = 0−5, the single fundamental piece of +information not accessible using our combined data set alone is +the energy of the ˜X(010) Πu state (see Figure 1); a value that has +fortunately been determined accurately by Schmuttenmaer et al. +[40]. +3.1. Optical data +In the 24300–26400 cm−1 region, 36 ˜A 1Πu − ˜X 1Σ+ +g vibronic +bands have been recorded by LIF spectroscopy as summarized +in Table 1 and visible on Figure 2. Of these, 17 are reported +for the first time. The strongest bands in the optical spectra be- +long to transitions arising from the ˜X(000) state (Figure 2). The +PGOPHER software allows to estimate the rotational tempera- +ture of each band; it ranges for 20 K to 150 K (see Table S1 in +the supplementary material). Typical linewidths reproducing +the experimental spectra range from 0.02 cm−1 to 0.04 cm−1. The +present measurements are in good agreement with band values +previously reported in the literature and often provide more ac- +curate frequencies (see Table S2 in the supplementary mate- +rial). Thanks to the relatively low rotational temperature com- +bined with the high resolution resulting in well-resolved features, +assignments of the new bands are relatively straightforward. Sev- +eral bands appear heavily perturbed, in particular those involving +the ˜A(020) Π(−) +u +and ˜A(040) Π(+) +u +states. +In the ˜A(000)– ˜X(020) band region, we assign for the first +time transitions involving the uΣu and uPu upper state perturbers, +previously only observed in the ˜A(000)– ˜X(000) band region +[34, 37, 38]. In total, 22 transitions are assigned in the bands +involving ˜X(020): 11 in the uΣu–Σ+ +g band (P-, Q-, R-branch; al- +though the single Q-branch assignment remains tentative), 5 in +the uPu–Σ+ +g band (P- and R-branch), and 6 transitions in the uΣu– +∆g band (P- and R-branch). No transitions are observed for the +uPu–∆g branch. Figure 3 presents an overview of this band sys- +tem with the sub-bands highlighted in various colors. These tran- +sitions are predicted by PGOPHER without setting up a transition +moment for the corresponding bands (Table S1 in the supple- +mentary material). One can notice that the simulated intensi- +ties of the perturber transitions do not perfectly reproduce the ex- +perimental spectrum on Figure 3 (in particular, the uΣu–∆g band +intensity is overestimated); we assume that some intensity per- +turbations are not properly taken into account. The agreement in +line position, however, is quite satisfactory. The present assign- +ments thus confirm the assignment of the perturber lines observed +in the ˜A(000)– ˜X(000) region. It is also worth noticing that, on +Figure 3, the R-branches of the Πu–Σ+ +g and Πu–∆g bands (above +24545 cm−1) appear stronger in the experimental spectrum than +on the simulation, and stronger than the Q-branches (the strongest +features around 24540 cm−1). Since no rotational temperature al- +lows proper reproduction of such intensity ratios, this intensity +difference is either the result of an experimental adjustment dur- +ing the scan or of some discrepancy in the model that does not +properly apply to this band system. For the Πu–∆g band, half of +the ∆g levels, with even J′′ values, was previously observed by +Gausset et al. [9]. This peculiarity has led the authors to a better +understanding of the l-uncoupling in the ground electronic state +of C3. In the present study, we are able to assign transitions in- +volving both even and odd J′′ values. The transitions involving +odd J′′ values appear about 5 times weaker on the experimental +4 + +Figure 1: Schematic vibrational energy level diagram of C3 together with optical (in dashed lines) and infrared (in plain lines) bands observed in this work. Energy +levels are represented with increasing quanta of excitation in v2 from left to right. Energies are from the combined fit performed in this work. Only the levels involved +in bands observed in the present study are plotted, with the exception of the ˜A(0v20) levels, v2 = 0 − 5, for which all levels included in the combined fit are shown. +Transitions in the same color arise from the same ˜X(0v20) lower state. +Figure 2: Overview of the electronic spectra of C3 recorded in this work. Top traces: Experimental spectra. Bottom traces: PGOPHER simulations (at thermal +equilibrium and rotational temperatures reflecting each experimental band, from 20 to 150 K; see Table S1 in the supplementary material). Intensities are in arbitrary +units and the ratio between simulated bands is adjusted to reflect the experimental traces. Each simulated band is color coded according to the hot band of ν2 involved +in the ˜X 1Σ+ +g state (the color sequence ranges from purple to red using the same color coding as in Figure 1). At this scale, bands involving different l values in the +˜X state, e.g., ˜A(000)– ˜X(020) Πu–Σ+ +g and Πu–∆g, are overlapping. Zooms onto the ˜A(000)– ˜X(020) (highlighted by a star symbol on the figure) and ˜A(010)– ˜X(030) +Σ− +g –Πu and Σ− +g –Φu (highlighted by a triangle) band systems are presented in Figure 3 and 4, respectively. +spectrum than predicted using PGOPHER, and are thus signifi- +cantly weaker than those involving even J′′ values, which may +explain why they remained unobserved thus far. +Three ∆K = −3 bands are observed in this study, namely +˜A(010)– ˜X(030) Σ− +g–Φu (P-, Q-, and R-branches, Figure 4), +˜A(000)– ˜X(040) Πu–Γg (P- and Q-branches, tentative assignments +in the R-branch, see Figure S3 in the supplementary material), +and ˜A(020)– ˜X(040) Π(−) +u –Γg (P- and Q-branches, Figure S4 in the +supplementary material). To our knowledge, it is the first time +that these transitions are observed for the ˜A 1Πu − ˜X 1Σ+ +g band. +5 + +Table 1: ˜A 1Πu − ˜X 1Σ+ +g rovibronic bands of the C3 molecule observed in the +present work. J′′ +max values are reported for this work and the literature, together +with the references of the previous works. When no literature value is reported, +the band is observed for the first time in this work. Horizontal lines group bands +arising from lower state levels with the same number of quanta of excitation in +ν2. See Table S2 in the supplementary material for detailed information on the +literature data and the present dataset. +Vibronic assignment +Band Origina +This work +Literature +/ cm−1 +J′′ +max +J′′ +max +Refs. +˜A(000)– ˜X(000) +Πu–Σ+ +g +24676 +24 +64 +[9, 15, 33, 34, 37, 38] +uΣu–Σ+ +g +b +24679 +14 +16 +[34, 37, 38] +uPu–Σ+ +g +b +24676 +6 +8 +[34, 37, 38] +˜A(020)– ˜X(000) +Π(−) +u –Σ+ +g +25039 +14 +30 +[15, 31] +Π(+) +u –Σ+ +g +25529 +18 +50 +[9, 15] +˜A(040)– ˜X(000) +Π(−) +u –Σ+ +g +25441 +14 +50 +[9, 15] +Π(+) +u –Σ+ +g +26296 +16 +4 +[15] +˜A(002)– ˜X(000) +Πu–Σ+ +g +26347 +14 +10 +[15] +˜A(100)– ˜X(000) +Πu–Σ+ +g +25761 +18 +28 +[9, 15] +˜A(120)– ˜X(000) +Πu–Σ+ +g +26128 +20 +˜A(010)– ˜X(010) +Σ− +g –Πu +24749 +17 +47 +[9] +∆g–Πu +24872 +18 +51 +[9] +Σ+ +g –Πu +25093 +17 +39 +[9] +˜A(000)– ˜X(020) +Πu–Σ+ +g +24543 +26 +50 +[9] +Πu–∆g +24544 +30 +48 +[9] +uΣu–Σ+ +g +b +25546 +14 +uΣu–∆g b +25458 +14 +uPu–Σ+ +g +b +25542 +6 +˜A(020)– ˜X(020) +Π(−) +u –Σ+ +g +25906 +26 +28 +[31] +Π(−) +u –∆g +25906 +25 +30 +[31] +˜A(100)– ˜X(020) +Πu–Σ+ +g +25629 +26 +Πu–∆g +25629 +23 +˜A(010)– ˜X(030) +Σ− +g –Πu +24605 +34 +35 +[9] +Σ− +g –Φu +24607 +30 +∆g–Πu +24728 +28 +30 +[9] +∆g–Φu +24729 +25 +˜A(000)– ˜X(040) +Πu–Σ+ +g +24389 +28 +38 +[9] +Πu–∆g +24389 +28 +Πu–Γg +24391 +30 +˜A(020)– ˜X(040) +Π(−) +u –Σ+ +g +24752 +20 +Π(−) +u –∆g +24752 +21 +Π(−) +u –Γg +24754 +21 +˜A(100)– ˜X(040) +Πu–Σ+ +g +25475 +24 +Πu–∆g +25475 +24 +˜A(010)– ˜X(050) +∆g–Πu +24565 +26 +∆g–Φu +24656 +22 +a Accurate values can be retrieved from the energies reported in Table S4. +b uΣu and uPu are perturber states of the ˜A(000) state, unidentified so far (see text). +These bands are spectrally intertwined with the corresponding +∆K = ±1 bands arising from the same upper state and their in- +tensity is properly predicted by PGOPHER by perturbation (i.e., +even though no transition moment is used to predict them, see Ta- +ble S1 in the supplementary material). These nominally forbid- +den transitions appear quite strong both on the simulation and the +experimental trace, as visible on Figure 4. We note, however, that +again the R-branches of both bands (located above 24605 cm−1) +are stronger on the experimental spectrum than predicted by the +simulation. +Two bands of the ˜A(010)– ˜X(050) band system, namely the +∆g–Πu and ∆g–Φu bands, are observed in this work (see Figure +S5 in the supplementary material). Again, to the best of our +knowledge, these bands are the first bands of the ˜A 1Πu − ˜X 1Σ+ +g +transition probing the ˜X(050) state at high resolution. A single +band probing higher quanta of excitation in ν2 was previously +reported in the literature, the ˜A(000)– ˜X(060) Πu–Σ+ +g band for +0.0 +0.5 +1.0 +24520 +24530 +24540 +24550 + +0.0 +0.5 +1.0 + +0.0 +0.1 +0.2 +0.3 +24526 +24528 +24530 + +0.0 +0.1 +0.2 +0.3 +Intensity / arb. u. +Wavenumber / cm +1 +Experimental spectrum +A(000) +X(020) simulation +u ++ +g +u +g(e) +u +g(f) +u +u ++ +g +u +u +g +uPu ++ +g +Figure 3: The ˜A(000)– ˜X(020) band system observed around 24540 cm−1 (top +traces) and comparison with a PGOPHER simulation at 70 K (lower traces). (Left +panel) Overview of the band; (right panel) zoom onto a portion of the spectral +range. In the simulation, transitions within each sub-band are plotted in various +colors. For the Πu–∆g band, transitions involving even and odd J′′ values (e and +f levels in the lower state) are plotted in two shades of green with the intensity +of the simulated f symmetry divided by a factor 5 compared to the PGOPHER +simulation. For the transitions involving the perturber states in the ˜A 1Πu state, no +scaling factor was used for the intensities as we could not define one that would +be valid over the full spectral range. +0.0 +0.5 +1.0 +24570 +24580 +24590 +24600 +24610 + +0.0 +0.5 +1.0 + +0.0 +0.2 +0.4 +0.6 +24585 +24590 +24595 + +0.0 +0.2 +0.4 +0.6 +Intensity / arb. u. +Wavenumber / cm +1 +Experimental spectrum +A(010) +X(030) simulation +g +u +g +u +Figure 4: The ˜A(010)– ˜X(030) Σ− +g –Πu and Σ− +g –Φu band system observed around +24600 cm−1 (top traces) and comparison with a PGOPHER simulation at 150 K +(lower traces). (Left panel) Overview of the band; (right panel) zoom onto a +portion of the spectral range. In the simulation, transitions from the ∆K = −1 +band are in gray and those from the ∆K = −3 band are highlighted in orange. +which Q-branches assignments were proposed by Merer [28]. In +the present work, the spectral range where this band lies (around +24217 cm−1) has not been investigated and no other band involv- +ing the ˜X(060) level is observed, preventing assignments confir- +mation by combination differences. Additional figures showing +overviews of all ˜A 1Πu − ˜X 1Σ+ +g bands observed in this study are +reported in the supplementary material (Figure S6–S17). +3.2. Infrared data +The full FTIR absorption spectrum of C3 recorded in this work +is shown in Figure 5. The intense absorption features from CO, +H2O, and C2H have been removed from the experimental trace for +clarity. The simulated spectrum given in the lower part of the fig- +ure is obtained using the PGOPHER software [76] using the band +center and rotational constants obtained from our fit (see section +3.3), a Gaussian lineshape with a 0.005 cm−1 width, and assuming +a 700 K rotational temperature (which was found to better repro- +duce the fundamental ν2 band; we note that this temperature is +6 + +0.97 +0.98 +0.99 +1.00 +Transmittance +1900 +1925 +1950 +1975 +2000 +2025 +2050 +2075 +Wavenumber / cm +1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Intensity / arb. u. +C3 +3 simulation +(001) +(000) +(011) +(010) +(021) +(020) +(031) +(030) +(041) +(040) +(051) +(050) +Experimental spectrum +Figure 5: Spectrum of C3 in the 1850–2100 cm−1 spectral range. Top trace: Absorption spectrum presented in transmittance. The intense absorption lines of CO as +well as the lines of H2O and C2H have been removed from the experimental trace for clarity. Bottom trace: A 700 K PGOPHER simulation (at thermal equilibrium) +of the ν3 band of C3 and its hot bands involving up to five quanta of excitation in ν2. The hot band sequence is plotted in a color sequence ranging from purple to dark +red (the color coding is the same as in Figure 1). The simulation is normalized to the strongest transition of the fundamental ν3 band. +Table 2: Rovibrational bands of the C3 molecule observed in the present work. +J′′ +max values are reported for this work and the literature. When no literature value +is reported, the band is observed for the first time in this work. Horizontal lines +group bands arising from lower state levels with the same number of quanta of ex- +citation in ν2. See Table S3 in the supplementary material for more information +on the literature data and the present dataset. +Vibronic assignment +Band Origina +This work +Literature +/ cm−1 +J′′ +max +J′′ +max +Refs. +˜X(001)– ˜X(000) +Σ+ +u –Σ+ +g +2040 +60 +52 +[39, 47] +˜X(011)– ˜X(010) +Πg–Πu +2015 +60 +41 +[47] +˜X(021)– ˜X(020) +Σ+ +u –Σ+ +g +2001 +54 +32 +[47] +∆u–∆g +1994 +55 +32 +[47] +Σ+ +u –∆g +2003 +20b +∆u–Σ+ +g +1992 +18b +˜X(031)– ˜X(030) +Πg–Πu +1985 +51 +Φg–Φu +1976 +55 +Πg–Φu +1988 +23 +Φg–Πu +1973 +31 +˜X(041)– ˜X(040) +Σ+ +u –Σ+ +g +1973 +48 +∆u–∆g +1970 +49 +Γu–Γg +1960 +49 +Γu–∆g +1956 +30 +∆u–Γg +1974 +28 +˜X(051)– ˜X(050) +Πg–Πu +1961 +48 +Φg–Φu +1956 +34 +Hg–Hu +1945 +44 +a Accurate values can be retrieved from the energies reported in Table 3. +b Tentative assignment. +higher than that found for CO). The hot band sequence involving +increasing values of v′′ +2 extends toward lower frequencies as vi- +sually indicated by the colored sequence. Figure 5 also illustrates +the strong overlap of all the bands observed in this work which +causes most of the difficulties in the assignment process. For ex- +ample, the lines arising from the (051)–(050) bands span over +most of the spectral region. Since they are weak and hindered by +many other more intense lines, their assignment was challenging. +In total, 18 rovibrational bands of C3 have been observed in this +study, 14 of them for the first time (see Table 2). A detailed ac- +count on this work and available literature of rovibrational data +is available in Table S3 in the supplementary material. In this +work, the assignment of the infrared data was performed sepa- +rately from the optical measurements presented in the previous +sections and strongly relied on the literature data (including opti- +cal studies). +The analysis of the (001)–(000) Σ+ +u–Σ+ +g, (011)–(010) Πg–Πu, +and (021)–(020) Σ+ +u–Σ+ +g and ∆u–∆g bands is rather straightfor- +ward thanks to the diode laser experiments performed by Mat- +sumura et al. [39] and Kawaguchi et al. [47]. For all these bands, +our observations are perfectly consistent with the literature mea- +surements and allow for an extension of the dataset toward high- +J values (up to 55–60, see Table 2). The frequency accuracy is +slightly improved for the (001)–(000) and (011)–(010) bands (by +up to a factor 2, with values as low as 0.0005 cm−1) and similarly +for the (021)–(020) bands (see Table S3 in the supplementary +material). Our extended dataset and the use of ∆up +2 (J) (see the +supplementary material for a detailed explanation) allowed to +identify local perturbations in the upper ˜X(021) ∆u levels man- +ifold. Figure 6 shows the diagrams of the first derivative of the +second differences, ∆up +2 (J) values, calculated in the upper states. +The highest shift occurs for the ˜X(021) ∆u e manifold at J′ = 35 +(the shift corresponds to about 10 times the C3 linewidth), and +similar but smaller shifts are observed for the ˜X(021) f mani- +fold. No notable level shifts are observed in (021) Σ+ +g state. The +observed perturbations in the ∆u state are likely caused by a Cori- +olis interaction with the ˜X(190) Π manifold, which mixes energy +levels with ∆J = 0, e ↔ e or f ↔ f, and ∆l = odd. While the +˜X(190) Π state has not yet been detected in SEP measurements, +observations of ˜X(180) Σ at about 1993 cm−1 puts the ˜X(190) Π +state in the right energy range [25]. +We report for the first time numerous infrared transitions in- +volving the (031)–(030), (041)–(040), and (051)–(050) hot bands. +7 + +0 +500 +1000 +1500 +2000 +2500 +3000 +J(J + 1) +3.35 +3.40 +3.45 +3.50 +3.55 +3.60 +3.65 +3.70 +2(J + 1) +2(J +1) / cm +1 +X (021) e +X (021) f +Figure 6: Perturbation analysis of the ˜X(021) ∆u e and f manifolds. +The assignment procedure relied mainly on combination differ- +ences using the work of Gausset et al. [9] (see the supplemen- +tary material for a detailed explanation on the procedure). As no +combination differences are available for the (030) Φu state, we +used the results of successive fits that included l-type resonance +with the Π state to secure the assignments. Further confirma- +tion is obtained by GS combination differences using the present +optical data. SEP data from Rohlfing and Goldsmith [46] and +Northrup and Sears [24], despite their limited resolution, have +provided crucial information on states for which little was known +so far (vibrational energies and estimated B values). In several +cases, for example the (051)–(050) band manifold, this provided +sufficient information to initiate an analysis. +Rohlfing and Goldsmith [46] reported the vibrational energy +of the ˜X(051) Πg state [2330.9(5) cm−1] as well as an estima- +tion of the rotational constants for the lowest J values [B = +0.4718(13) cm−1]. Using this information, we started our anal- +ysis for the e and f states. Once Πg–Πu transitions were found +(up to J′′ = 40 for both e and f transitions), we used the l-type +resonance to secure the assignments of Φg–Φu and Hg–Hu bands. +The intensities of the P branches involving high J values are very +weak (red-end of our spectrum) and, as for the ∆g states of (021), +some Coriolis-type resonances seem to complicate the assign- +ments. Figures S18–S23 in the supplementary material show +the different infrared band manifolds observed in this study. +The rotational assignments proposed in this study are con- +firmed by the observation of several Q-branches, i.e., no shift by +one or more quanta of J is possible in the proposed assignments. +Such a Q-branch is shown on Figure 7 for the (051)–(050) Hg– +Hu band, another example is provided for the (041)–(040) Γu– +Γg band in Figure S24 in the supplementary material. Overall, +Q-branch transitions were observed, for both e and f levels, for +the (011)–(010) Πg–Πu [Q(1),Q(3)], (021)–(020) ∆u–∆g [Q(2), +tentative], (031)–(030) Φg–Φu [Q(3)–Q(6)], (041)–(040) Γu–Γg +[Q(4)–Q(7)], and (051)–(050) Φg–Φu [Q(3)–Q(5), Q(6) tentative] +and Hg–Hu [Q(5)–Q(11)] bands. +Once all the data were included in PGOPHER, we have also +been able to assign some transitions involving ∆l = ±2 bands. +An interesting feature of PGOPHER is that it predicts these tran- +sitions without inputting a corresponding transition moment for +these nominally “forbidden” transitions (similarly to what was +Figure 7: Zoom onto the Q-branch of the ˜X(051)– ˜X(050) Hg–Hu band. Top trace: +experimental spectrum, in transmittance, after removal of CO, H2O, and C2H +lines. Bottom trace: PGOPHER simulations at 700 K (final set of parameters). +described previously for the electronic spectra). For the (021)– +(020) bands, the ∆l = ±2 bands are predicted with relatively low +intensities; and only features with relatively poor SNR are ob- +served on the experimental spectrum. Only tentative assignments +are made (7 for the ∆u–Σ+ +g and 4 for the Σ+ +u–∆g band) and these +are not included in the fit. For hot bands involving higher quanta +of excitation in ν2, however, these predicted ∆l = ±2 transitions +have significant intensity, in particular for high l values. Indeed, +several features with reasonable SNR are observed on the spec- +trum as visible on Figure 8 for the (041)–(040) Γu–∆g band. Ad- +ditional examples are provided in the supplementary material +for the (031)–(030) Πg–Φu and Φg–Πu bands (Figure S25) and +the (041)–(040) ∆u–Γg (Figure S26). These “cross-ladder” tran- +sitions (if we refer to levels of a given l value for a specific state +as a ladder) provide constraints on the energy difference between +the various l levels of the vibrational levels for which they are +observed. +Figure 8: Zoom onto three consecutive R-branch transitions of the (041)–(040) +Γu–∆g band. Top trace: experimental spectrum, in transmittance, after removal +of CO, H2O, and C2H lines. Bottom trace: PGOPHER simulations at 700 K +(final set of parameters, normalized to the strongest transition of the fundamental +ν3 band) where the Γu–∆g transitions are highlighted in green. The rest of the +simulated transitions of the (041)–(040) bands visible in this range are plotted in +shades of red while the other C3 bands are simulated in gray. +In the high frequency part of the spectrum, most of the tran- +8 + +sitions can be assigned to the C3 molecule (see for instance fig- +ure 8 and figure S27 in the supplementary material). Toward +the lower end of the spectrum, however, many transitions remain +unassigned. These transitions probably arise from the R-branches +of the (061)–(060) bands but assigning this spectrum has not been +possible despite our best efforts, mainly because the signal-to- +noise ratio is rather poor in that region and spectroscopic assign- +ments are challenging on the basis on a single branch. +3.3. Combined fit +3.3.1. Dataset +The unique feature of the present work is that it becomes pos- +sible to combine the optical and infrared data sets. Available data +from the literature and the present work for the ˜A 1Πu − ˜X 1Σ+ +g +rovibronic and all ˜X 1Σ+ +g − ˜X 1Σ+ +g rovibrational transitions are +listed in Tables S2 and S3 in the supplementary material. These +detailed tables contain the Jmax values, frequency uncertainties +used in the combined fit, and the frequency offset eventually ap- +plied to the data. All available ˜A 1Πu − ˜X 1Σ+ +g data and rovi- +brational transitions in the electronic GS (i.e., data with list of +assignments provided in the literature) are included in the present +fit with one exception, namely the work of Balfour et al. [27] who +reported the spectroscopic assignments for five ˜A 1Πu − ˜X 1Σ+ +g +transitions. +One of the rovibronic band observed by Balfour +et al. [27], the ˜A(020)– ˜X(000) Πu–Σ+ +g band, was re-investigated +by Tokaryk and Chomiak [31] who proposed a different spec- +troscopic assignment. In the present study, our assignments are +in line with those of Tokaryk and Chomiak [31]. Additionally, +we propose in this work an alternative spectroscopic assignment +for the ˜A(002)– ˜X(000) Πu–Σ+ +g band compared to Balfour et al. +[27]. Finally, when performing the combined fit, we noticed that +the spectroscopic assignments proposed by Balfour et al. [27] in +the ˜A(200)– ˜X(000) Πu–Σ+ +g band are incompatible with those of +the ˜A(200)– ˜X(200) Πu–Σ+ +g from Merer [28]. Since assignments +from Merer [28] have proven consistent with literature data for +the other bands they reported, we decided to only include the +Merer values. There is no literature data able to confirm or in- +firm the spectroscopic assignments of the remaining two bands +observed by Balfour et al. [27]; at this stage we have chosen not +to include these data in our fit. +As previously noted by Saha and Western [51], a serious dif- +ficulty arises in high resolution combination fits when rovibronic +data are included from different light sources, as (small) spectral +offsets are intrinsic to this approach. This can be overcome by +shifting the frequencies of one dataset by this offset value. It is +often challenging, however, to determine which dataset presents +an offset from the absolute transitions rest frequencies. The im- +pact of this issue is relatively limited because only the absolute +energy of the upper state is affected while the accuracy of the +rotational constants and overall fit is not. In the present study, +small offsets (typically of the order of several hundredths of a +cm−1) are applied to several rovibronic datasets (see Table S2 for +a detailed list of concerned data and offset values). We use the +frequency offsets established by other authors from the literature +when available (for instance, an offset of +0.04 cm−1 was deter- +mined for the data of Gausset et al. [9] by Tanabashi et al. [33]) +for consistency reasons. When not available in the literature, we +determine these offsets ourselves. Overall, offsets values range +from 0.02 to 0.12 cm−1, hence the absolute energies in the ˜A 1Πu +state may be affected by these amounts. +As much as possible, data are included in the fit at their experi- +mental accuracy (see Table S2 for typical values for each dataset). +When no frequency accuracy was provided, we assume a value +based on the dispersion of the frequencies from our best model. +In some instances, the literature data are provided with an upper +limit for the frequency error but the residuals from the fit show +that this value is over-estimated. For example, the uncertainty +on line frequency is assumed to be better than 0.01 cm−1 for the +˜A(000)– ˜X(000) transitions reported by Zhang et al. [34] while the +residuals from our fit show that the line accuracy is probably more +of the order of 0.005 cm−1; that value is thus used in the present +fit. It has proven challenging to treat such a large dataset for one +molecule subject to clear perturbations with transitions signifi- +cantly deviating from the fit. Transitions severely perturbed are +excluded from the fit while transitions slightly diverging are kept +in the fit but with an increased frequency error (hence a lower +weight), typically by a factor 10, in order to maintain a global +5σ deviation for the combined fit. Overall, the dataset contains +4425 observations (3957 rovibronic and 1468 rovibrational tran- +sitions) including 2046 (1106 rovibronic and 940 rovibrational +transitions) from this work. The full dataset is available as elec- +tronic files as part of the supplementary material (these files +also contain the transitions not included in the present fit). +3.3.2. Hamiltonian +The PGOPHER Hamiltonian used for the combined fit is sim- +ilar to the previous studies using PGOPHER on C3. One differ- +ence with the work of Haddad et al. [37] is that we expressed the +Hamiltonian in terms of the rotational angular momentum of the +nuclear framework, ˆR, in order to obtain energy levels and rota- +tional constants comparable with most of the literature data. It +is worth noting that the specific Hamiltonian used by the PGO- +PHER software differs from the conventional Hamiltonian for a +linear molecule developed for instance by Yamada et al. [79]. +The off-diagonal constants accounting for l-type doubling are ex- +pressed as perturbation terms between two states, which results +in several q values for a given vibrational level instead of a more +physically-relevant single one (see Table S4). For example, for +the ˜X(050) vibrational level there are three different q values: one +for the Π state, and two defined as ⟨Πu| q |Φu⟩ and ⟨Φu| q |Hu⟩. The +resulting PGOPHER files together with details on their construc- +tion are provided in the Supplementary Material. +The perturbation analysis for the ˜A(000) state is carried out +using the same effective Hamiltonian as reported in Haddad et al. +[37], with the two perturbing Σ and P = 1 states identified in +the literature treated as three perturbing states with the e and f +levels of the P = 1 state treated separately. The resulting states +are labeled uΣ, uPe, and uPf . As in Haddad et al. [37], the spin- +spin interaction constant λ was fixed to 0.1 cm−1 in the uΣ state. +Table S5 in the supplementary material presents the resulting +constants in these perturbing states. +3.3.3. Fit results +A total of 340 parameters have been adjusted to reproduce +more than 4400 experimental data with a rms of 0.041 cm−1 and +9 + +a reduced standard deviation of 1.3. Figure 9 displays the resid- +uals of the fit with a color coding based on the ˜X(v1v2v3) level +involved in the transition (the same as in Figure 1). Overall, the +fit is quite satisfactory. One can notice that the infrared (021)– +(020) transitions (in green, around observation #400) are among +those the least well reproduced by the fit as a result of severe per- +turbations in the ˜X(021) level. +Figure 9: Residuals of the combined fit. Transitions involving the ˜X(0v20) vi- +brational level, with up to five quanta of excitation in ν2, are plotted in a color +sequence ranging from purple to dark red (same as in Figure 1); data in black +arise from other ˜X vibrational levels. The vertical dashed line separates the rovi- +brational data (low observation numbers) from the ˜A 1Πu − ˜X 1Σ+ +g electronic data. +“Weighted Obs-Calc” corresponds to the Obs-Calc value divided by the frequency +error of the transition. +The full list of parameters is reported in Tables S4 and S5 in +the supplementary material, and a subset of parameters per- +taining to the ˜X(0v20) levels is reported in Table 3 where they +are compared to available literature data. Only five parameters +could not be determined in the present fit, the energies of five lev- +els [ ˜X(110), ˜X(300), ˜X(400), ˜X(500), ˜X(600)] involved each in a +single rovibrational transition for which no cross-correlating data +exist. +4. Prediction of ν2 hot bands +The present modeling of C3 in its electronic ground and ˜A +states can be used to predict further transitions not directly ob- +servable. This is particularly interesting for the hot bands of the +ν2 fundamental that remain elusive in the laboratory to date. Ta- +ble 4 contains lists of predicted transitions for the ˜X(020)– ˜X(010) +and ˜X(030)– ˜X(020) band systems. These transitions have been +calculated using PGOPHER and the final set of parameters re- +ported in Table 3. One limitation of PGOPHER is that the list +of energies does not carry frequency error information, hence we +cannot convey these errors to the transitions frequencies. How- +ever, based of the frequency errors of the transitions used to de- +termine these energies, and the overall good quality of the fit, we +estimate the accuracy of these transition frequencies to be of the +order of 0.005 cm−1 (150 MHz). +5. Astronomical implications +Due to the extremely low bending frequency of C3, even in +an environment at moderate temperature its low excited bending +vibrational levels ˜X(010), at ∼ 63 cm−1 (91 K), and ˜X(020), at +∼ 132 cm−1 (191 K), can be thermally populated significantly. In- +deed, at 100 K, 29 % and 13 % of the GS population lies in each +of these levels; at 50 K, these numbers drop down to 14 % and +2 %, which remains significant for an abundant molecule. Hence, +ν2 hot bands may be detectable in various environments of the +interstellar medium where C3 is abundant. Unambiguous detec- +tions, i.e., beyond the line confusion limit, become possible when +astronomical data can be compared to accurate submillimeter lab- +oratory data (with sub-MHz resolution) that will strongly bene- +fit from the predictions made here. Given the accurate parame- +ters derived here, even without such laboratory data it should be +possible to identify these ν2 excited transitions in astronomical +data. The 0.005 cm−1 (150 MHz) accuracy of our predictions +corresponds to a velocity uncertainty of ∆V = 23.6 km·s−1. Since +the linewidth of observed spectra of C3 ν2 fundamental band in +star forming region ranges from 5 to 12 km·s−1 [19, 20, 80, 81], +we conclude that our data can be used to search for C3 ν2 hot +bands. Measurement of C3 in excited states and determination of +its abundance and excitation temperature may give new insights +into the chemistry of its formation and will add further informa- +tion to derive the origin of small and possibly also longer carbon +chains in the ISM. +6. Conclusion +This work presents the most complete spectroscopic study of +C3 to date. Through the combination of infrared and optical tran- +sitions precise vibrational energies and rotational constants for +the low-lying bending modes of C3 are determined up to v2 = 5, +significantly extending our knowledge of the rovibrational mani- +fold of the electronic GS. The measured and predicted transition +frequencies presented here can be used in astronomical obser- +vations in both the optical, infrared, and far-infrared, increasing +the effectiveness of C3 as a probe of the physical and chemical +environment of the target. PGOPHER files allowing the full sim- +ulation of all known bands of ˜A 1Πu − ˜X 1Σ+ +g system are given in +the supplementary material so that further improvements of the +laboratory spectroscopy of this molecule can be directly incorpo- +rated in the model. +7. Acknowledgments +This manuscript comprises of two datasets. The project started +with LIF measurements at USTC (recorded in 2017) and upon +analysis it became clear that combining these with non-published +infrared measurements recorded at SOLEIL in 2010 offered a +unique opportunity for a very complete spectroscopic study of +the C3 radical. PGOPHER is ideal to merge the two large data +sets. Given all involved spectroscopic challenges, we asked Colin +Western for help, and as usual were helped on the spot and far +beyond, resulting in his co-authorship. Along the way of finish- +ing this manuscript, Colin sadly passed away. We dedicate this +manuscript to his memory. +We acknowledge financial support from the National Natural Sci- +ence Foundation of China (22173089 and 21827804), the Nether- +lands Organization for Scientific Research (NWO) through a +10 + +Table 3: Spectroscopic constants (in cm−1) of C3 in the ˜X(0v20) state, with v2 = 0 − 5. Values derived in this study are compared to literature data, when available. +Numbers in parentheses are 1σ deviations of the fit, in units of the last digit of the parameter (for literature data, the information is sometimes not available). +Level +E +B +D × 106 +H × 1010 +L × 1014 +−q/2 × 103 +−qD/2 × 107 +−qH/2 × 1011 +−qL/2 × 1014 +−qM/2 × 1018 +˜X(000) +Σ+ +g +This work +0 +0.430 587 17(17) +1.5337(25) +1.905(17) +−1.599(32) +Ref. [47] +0 +0.430 579(17) +1.485(22) +1.385(77) +˜X(010) +Πu +This work +63.416 591 12(48) +0.442 415 78(25) +2.3525(28) +2.675(23) +−2.060(53) +−2.850 533(79) +5.048(19) +−12.14(35) +2.52(21) +−2.66(37) +Ref. [47]a +E010 +0.442 381(18) +2.328(37) +2.62(11) +˜X(020) +Σ+ +g +This work +132.800 29(78) +0.451 584 0(62) +2.419(10) +1.571(31) +Ref. [47] +132.7993(19) +0.451 632(41) +2.57 +∆g +This work +133.035 64(82) +0.453 099 5(67) +2.929(14) +3.185(88) +−2.52(17) +Ref. [47]a +133.065(29) +0.453 088(31) +2.775(50) +5.11(49) +−3.788(23) +4.3(15) +� +Σ+ +g +���∆g +� +This work +−3.8250(16) +5.074(28) +−5.143(89) +˜X(030) +Πg +This work +207.4240(11) +0.460 472 9(66) +2.6930(94) +1.799(45) +−0.465(76) +−5.1990(35) +5.928(60) +−5.20(20) +Ref. [9] +E010 + 143.88 +0.4600 +2.2 +−5.6 +Φg +This work +208.3943(15) +0.462 887 4(98) +3.248(18) +2.79(11) +−1.45(23) +⟨Πu|Φu⟩ +This work +4.5111(22) +−5.249(43) +4.79(14) +˜X(040) +Σ+ +g +This work +286.5641(13) +0.467 958(12) +2.762(21) +1.585(80) +Ref. [9] +286.52 +0.4675 +3.248(18) +∆g +This work +287.2198(11) +0.468 951 8(86) +2.884(13) +1.569(51) +−1.17(14) +Γg +This work +289.1579(22) +0.472 027(11) +3.527(15) +2.484(52) +� +Σ+ +g +���∆g +� +This work +6.1731(30) +−6.804(77) +9.52(56) +−1.17(14) +� +∆g +���Γg +� +This work +−5.0522(30) +5.130(65) +−3.88(25) +˜X(050) +Πg +This work +370.4479(18) +0.475 305(15) +3.064(26) +3.28(10) +−7.3000(90) +4.75(25) +Φg +This work +372.0318(17) +0.477 088(16) +2.648(28) +Hg +This work +375.121(12) +0.480 649(30) +4.448(64) +4.47(22) +⟨Πu|Φu⟩ +This work +6.9550(84) +−8.07(24) +⟨Φu|Hu⟩ +This work +−5.228(27) +1.56(25) +a Average values of the e and f components reported by the authors. +Table 4: Low-J far-infrared transitions of the ˜X(020)– ˜X(010) and ˜X(030)– ˜X(020) hot bands of ν2 predicted using our model (in cm−1). The e/f label of the lower +state is reported. +J′′ +P +Q +R +P +Q +R +P +Q +R +(020)–(010) Σ − Π +(020)–(010) ∆ − Π +1 +68.947(e) +71.657(e) +70.088(e) +2 +69.864(f) +68.296(f) +71.015( f) +3 +67.261(e) +73.589(e) +65.693(e) +68.412(e) +72.029(e) +4 +69.959(f) +64.782( f) +68.399( f) +72.936( f) +5 +65.678(e) +75.634(e) +64.118(e) +68.655(e) +74.058(e) +6 +70.123(f) +63.144( f) +68.547( f) +74.918( f) +(030)–(020) Π − Σ +(030)–(020) Π − ∆ +(030)–(020) Φ − ∆ +0 +75.074(e) +1 +2 +72.364(e) +74.247(e) +76.917(e) +73.933(e) +75.816(e) +78.486(e) +75.841(e) +3 +73.097( f) +75.767( f) +79.617( f) +73.122( f) +76.824( f) +4 +70.589(e) +74.440(e) +78.788(e) +72.149(e) +76.000(e) +80.348(e) +69.505(e) +73.206(e) +77.830(e) +5 +71.463(f) +75.811( f) +81.712( f) +68.670( f) +73.294( f) +78.842( f) +6 +68.831(e) +74.732(e) +80.682(e) +70.408(e) +76.309(e) +82.258(e) +67.890(e) +73.438(e) +79.902(e) +VICI grant, the Netherlands Research School for Astronomy +(NOVA), and the Programme National “Physique et Chimie du +Milieu Interstellaire” (PCMI) of CNRS/INSU with INC/INP co- +funded by CEA and CNES. DT acknowledges funding from the +Natural Sciences and Engineering Research Council of Canada in +the form of a Discovery grant. Part of this work was performed +at the SOLEIL facility under the proposal 20100296. +8. Authors contributions +Marie-Aline Martin-Drumel: Investigation, Formal analy- +sis, Writing – original draft; Writing – review & editing; Qiang +Zhang: Investigation, Formal analysis, Methodology, Writing – +original draft, Writing — review & editing; Kirstin Doney: In- +vestigation, Formal analysis, Writing – original draft; Writing +– review & editing; Olivier Pirali: Conceptualization, Investi- +gation, Formal analysis, Writing – original draft; Writing – re- +view & editing; Michel Vervloet: Investigation, Formal analysis, +Writing – review & editing; Dennis Tokaryk: Conceptualiza- +tion, Investigation, Writing – review & editing; Colin Western: +Software; Harold Linnartz: Resources, Supervision, Writing – +review & editing; Yang Chen: Conceptualization, Investigation; +Supervision; Writing -– review & editing Dongfeng Zhao: Con- +ceptualization, Investigation, Supervision, Writing — review & +editing. +References +[1] W. 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Stutzki, First detection of the carbon +chain molecules 13ccc and c13cc towards sgrb2(m), Astron. +Astrophys. 633 (2020) A120. doi:10.1051/0004-6361/ +201936538. +15 + diff --git a/gdE4T4oBgHgl3EQfRwwk/content/tmp_files/load_file.txt b/gdE4T4oBgHgl3EQfRwwk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d6eb87cda6d4af1c18db1291410f01b1b9ed9ca3 --- /dev/null +++ b/gdE4T4oBgHgl3EQfRwwk/content/tmp_files/load_file.txt @@ -0,0 +1,2045 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf,len=2044 +page_content='The Bending of C3: Experimentally Probing the l-type Doubling and Resonance Marie-Aline Martin-Drumela,1, Qiang Zhangb,∗∗, Kirstin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Doneyc, Olivier Piralia,d, Michel Vervloeta, Dennis Tokaryke, Colin Westernf, Harold Linnartzc, Yang Chenb, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' PO Box 9513,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' RA Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' NL2300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' the Netherlands dSOLEIL Synchrotron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' AILES beamline,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' l’Orme des Merisiers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Saint-Aubin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' F-91190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' France eDepartment of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' University of New Brunswick,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Fredericton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Canada fSchool of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' University of Bristol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Bristol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' United Kingdom Abstract C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' a pure carbon chain molecule that has been identified in different astronomical environments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' is considered a good probe of kinetic temperatures through observation of transitions involving its low-lying bending mode (ν2) in its ground electronic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The present laboratory work aims to investigate this bending mode with multiple quanta of excitation by combining recordings of high resolution optical and infrared spectra of C3 produced in discharge experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The optical spectra of rovibronic ( ˜A 1Πu − ˜X 1Σ+ g) transitions have been recorded by laser induced fluorescence spectroscopy using a single longitude mode optical parametric oscillator as narrow bandwidth laser source at the University of Science and Technology of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 36 bands originating from ˜X(0v20), v2 = 0 − 5, are assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The mid-infrared spectrum of the rovibrational ν3 band has been recorded by Fourier-transform infrared spectroscopy using a globar source on the AILES beamline of the SOLEIL synchrotron facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The spectrum reveals hot bands involving up to 5 quanta of excitation in ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' From combining analyses of all the presently recorded spectra and literature data, accurate rotational parameters and absolute energy levels of C3, in particular for states involving the bending mode, are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A single PGOPHER file containing all available data involving the ˜X and ˜A states (literature and present study) is used to fit all the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The spectroscopic information derived from this work enables new interstellar searches for C3, not only in the infrared and optical regions investigated here but also notably in the ν2 band region (around 63 cm−1) where vibrational satellites can now be accurately predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' This makes C3 a universal diagnostic tool to study very different astronomical environments, from dark and dense to translucent clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Introduction The bare carbon chain molecule propadienediylidene, C3, has for long been the object of considerable interest to astronomers and spectroscopists alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' It is one of the few species for which astronomical observations have preceded laboratory identifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Puzzling, unidentified lines around 4051 Å were first de- tected in cometary optical emission in 1881 by Huggins [1] then observed toward many cometary bodies [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' These lines were successfully reproduced in the lab by Raffety [5] and Herzberg [6] but their molecular carrier remained elusive until the work of Douglas [7] who conclusively identified C3 through an isotope-substitution experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' First actual rovibronic assign- ments of transitions in this band to transitions of the ˜A 1Πu− ˜X 1Σ+ g system were delayed until the work of Gausset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Be- cause of its lack of a permanent dipole moment, C3 does not possess a pure rotational spectrum and is thus not accessible by radio astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Instead, its electronic and vibrational spectra in the optical and infrared regions, respectively, provide useful spectroscopic approaches for detecting and tracing this simple molecule in the interstellar medium (ISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Optical detections of ∗dzhao@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='cn ∗∗qiangzhang@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='cn 1marie-aline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='martin@universite-paris-saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='fr the ˜A 1Πu − ˜X 1Σ+ g band of C3 have established the species as an ubiquitous component of diffuse interstellar matter [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 10–15] making it an ideal probe of the physical and chemical conditions in such environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the work of Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [15], the most extensive rovibronic C3 series detected in space to date, seven ˜A– ˜X rovibronic bands in addition to the origin band were observed in absorption towards HD 169454, a reddened star (because of molecular extinction along a line of sight crossing one or more molecular clouds), in a high-quality absorption spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Rovi- bronic series have also been reported in cometary spectra [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the past decades, C3 has also been detected in circum- stellar shells and in the dense ISM via detection of transitions in its asymmetric stretching (ν3 ∼ 2040 cm−1) [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 17] and bend- ing (ν2 ∼ 63 cm−1) [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 18–20] vibrational bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' It is partic- ularly interesting to note that the observation of several rovibra- tional transitions involving the low-lying ν2 fundamental mode have been reported in the carbon-rich star IRC+10216 [18], in the molecular cloud Sagittarius B2 [18, 19], and in the star form- ing cores W31C and W49N [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The unusually low frequency of the bending mode makes vibrationally excited C3 a powerful probe of the kinetic temperature in various interstellar environ- ments [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Besides the case of the 4051 Å comet band system, astronom- ical observations have also prompted other laboratory studies on C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Following the early work by Douglas [7] and Gausset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='04992v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='GA] 12 Jan 2023 [9], dedicated experiments have been conducted, aiming for a comprehensive characterization of the ˜A 1Πu − ˜X 1Σ+ g electronic band system of C3 at low- [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 21–27] and high- [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 15, 20, 28–37] resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The work of McCall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [38] led to a reassignment of the R(0) transition of the ˜A 1Πu − ˜X 1Σ+ g (000)–(000) band, in agreement with astronomical data [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Op- tical measurements have enabled the accurate description of the vibrational pattern in the electronic ground state (GS), notably the determination of the infrared inactive ν1 symmetric stretch band center (∼ 1224 cm−1) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In almost simultaneous studies, the ν3 fundamental vibra- tional band was observed in the laboratory [39] and in space [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The ν2 bending mode was investigated by far-infrared spectroscopy [40] subsequently enabling its interstellar detection [18, 41] which then prompted further high resolution laboratory investigations of the band [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The ν1 + ν3 combination band was also granted some interest: first measured in the lab- oratory by Krieg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [44], it was subsequently re-investigated by Schröder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [45] who extended the known stretching vi- brational manifold, with levels involving up to seven quanta of excitation in ν1 and three quanta in ν3, thus probing the poten- tial energy surface (PES) of this “floppy” molecule to high en- ergies along the stretching coordinate and providing precise sets of molecular constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In contrast, the ν2 bending vibrational manifold remains unstudied at this level of thoroughness with only a limited number of dedicated investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Relatively low-resolution stimulated-emission pumping (SEP) studies [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 24, 25, 46] have resulted in the observation of levels involving v2 = 0 − 34 in the electronic GS while the rovibronic investi- gations of the ˜A 1Πu − ˜X 1Σ+ g band system by Gausset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [9] revealed bands involving up to v2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' More extensive high- resolution data from direct absorption infrared studies, by prob- ing hot bands in either ν3 [47] or ν1 + ν3 [44], have enabled the detection of levels with up to (only) two quanta of excitation in ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' High resolution spectroscopic information for transitions in- volving higher quanta of excitation in ν2 is highly desirable to characterize the PES along the bending coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Such data will also serve the astronomical community as the observation of vibrationally excited C3 will allow to probe kinetic temperature of various interstellar environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Even though an exhaustive review of the extensive literature on C3 is beyond the scope of this paper, it is worth mentioning that because of its relevance not only in astrophysics but also in molecular physics, many other studies have been dedicated to this species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Many theoretical investigations were conducted on its potential energy surface [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 48–50] aiming specifically at estab- lishing its equilibrium structure (and resolving the long-standing ambiguity linear vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' bent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' the former being the census to date) and providing reliable energy term values and vibrational con- stants for excited vibrational levels in the electronic GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Exper- imentally, other electronic band systems have been investigated [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 51–54] while C3 isotopologues have been the subject of op- tical [55, 56] and infrared [57] studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The vacuum ultraviolet photoionization spectrum of C3 was also reported [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Weltner and Van Zee [59] and Van Orden and Saykally [60] have pub- lished detailed reviews on the experimental works on C3 as of the late 1990’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' C3 is also included in many astronomical models [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 61, 62] and dedicated experiments have been conducted to investigate its reactivity with neutral species in space [see 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In this paper, we report a joint optical ( ˜A 1Πu− ˜X 1Σ+ g band) and infrared (ν3 band) investigation of C3 in which we detected and assigned bands involving levels with up to 5 quanta of excitation in ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Many of these bands are reported here for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The combined analysis of these two data sets enables the accurate de- termination of the spectroscopic constants of the species as well as absolute rovibrational level energies for the v2 = 0 − 5 levels in the electronic GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' From this, accurate far-infrared transitions are derived for ro-vibrational transitions involving the ν2 mode and its hot band sequences guiding reliable interstellar searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The paper is organized as follows: in Section 2, we describe the two experimental approaches used in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' in Section 3, the experimental results and spectral analysis are presented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' in Section 4, a prediction of the line position of the ν2 rovibra- tional hot bands in the far-infrared is reported;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' and in Section 5, the astronomical implications of the present data are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The recorded line positions, experimental spectra, and fit files are available in the extensive supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Experimental methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Optical Spectroscopy The optical spectroscopic study of the ˜A 1Πu− ˜X 1Σ+ g transition of C3 has been performed between 380 and 410 nm at the Univer- sity of Science and Technology of China by using a laser induced fluorescence (LIF) setup that has been described in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [64, 65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In this experiment, C3 molecules are produced in a pulsed DC discharge nozzle using two flat-top stainless steel elec- trodes [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A gas mixture of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3 % acetylene (C2H2) diluted in argon is introduced into the nozzle by a General Valve (Series 9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='5 mm orifice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' High voltage pulses (≃ −2000 V, 20 µs, 10 Hz) are applied to one of the electrodes while the other is grounded in order to produce an intense pulsed plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The plasma con- taining C3 molecules is then expanded and adiabatically cooled by collisions with the buffer gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' About 10 mm downstream, the molecular beam is crossed perpendicularly by a laser beam, which suppresses the Doppler broadening in the recorded spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Fluorescent emission from laser excited C3 molecules is collected by a lens system perpendicular to the laser beam, guided into a grating spectrometer (Zolix, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='5 m) and then detected by a photo- multiplier tube (Hamamatsu, R928).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A home-built single-longitude-mode optical parametric oscil- lator (SLM-OPO) is employed as the laser source [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the present study, the signal output of the OPO is frequency-doubled in a KDP (KH2PO4) crystal to obtain tunable radiation between 350 and 450 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A small portion (∼ 5 %) of the OPO sig- nal output is injected into a wavelength meter (High Finesse, WS7-60) for calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The wavelength meter is calibrated with a stabilized He-Ne laser, providing a frequency accuracy of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='002 cm−1 for the laser source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Special care had to be taken in the measurement of the rela- tively weak hot vibronic bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' This is because in the supersonic jet the population of an excited vibrational level is much lower than that in the v = 0 level and, in most cases, the hot bands are overlapped with stronger fundamental vibronic bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' To over- come these difficulties, the spectrometer is used as a narrow band 2 pass filter which helps to distinguish fluorescence of the hot bands from other overlapping bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' As the fluorescence from different upper states often results in different dispersed spectra, and since the bandpass of the monochromator is extremely narrow (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='5 nm), it becomes possible to only detect the fluorescence emission from upper states involving specific GS hot bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' This was real- ized by selecting a dispersion wavelength of the monochromator corresponding to the fluorescence from the upper electronic state (itself optically-pumped from vibrationaly excited states of the ˜X state) to the lowest allowed vibrational level of the GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A laser excitation spectrum is recorded by measuring the inten- sity of the fluorescence as a function of the continuously tuned laser wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' This technique yields a very good signal-to- noise (S/N) ratio and a spectral resolution of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='02 cm−1 (cor- responding to a resolving power of ∼ 1,200,000) for the strong bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Based on the line width of the rovibronic transitions, the accuracy of the extracted line positions from our spectra can con- fidently be assumed to be of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='002 cm−1 for the bands aris- ing from v′′ 2 = 0, 1 in the ground electronic state [with the ex- ception of the ˜A(000)– ˜X(000) band for which a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='005 cm−1 un- certainty is used] and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='005 cm−1 for the other hot bands (with v′′ 2 ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Absolute frequency accuracies may be subject to small shifts because of combining spectra from different wavelength regimes that are recorded in different experimental setups, but comparison with previous studies [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 31, 40, 42] allows for re- liable calibration (see later in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Infrared Spectroscopy The absorption spectrum of C3 has been recorded in the ν3 asymmetric stretch region (around 2000 cm−1/ 5 µm) using an experimental set-up available on the AILES beamline of syn- chrotron SOLEIL previously used to investigate the far-infrared spectra of various reactive species [see 69–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The schematic representation of the discharge set-up together with its implemen- tation on the AILES beamline is given in Figure 1 of Martin- Drumel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' We briefly describe in this paper the main characteristics of the set-up and the discharge conditions we used to optimize the signal of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The discharge cell consists of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1 m long Pyrex tube (13 cm inner diameter) equipped with mul- tipass optics (White-type) allowing for 24 m of absorption path- length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The cell is connected under vacuum to a Bruker Fourier- transform (FT) infrared spectrometer and is separated from it by two wedged CaF2 windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A total of seven cell inlets are used: two are connected to the two water-cooled electrodes (separated by 70 cm), four allow for gas injection (buffer gas or sample) at different locations in the cell, and one located at the center of the cell is connected to a vacuum system (250 m3/h Roots blower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' To probe the positive column of the plasma, one of the electrodes is connected to the high voltage while the second is grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the present work, C3 is synthesized using a discharge of He (in- jected both through the electrodes and through the two gas inlets closest to the multipass optics) seeded with a small amount of CH4 (injected using the inlets located closer to the center of the cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Under our experimental conditions, we find that the abundance of C3 is very sensitive to both the discharge current and the pres- sure of CH4 precursor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' To optimize the production of C3, we use a rapid scan optical fiber spectrometer (Ocean Optics) to monitor the 4051 Å emission band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The CH4 pressure is slowly increased until reaching the maximum of the visible emission while simul- taneously adjusting the discharge current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the optimum con- ditions, about 1 A of current (at 1 kV DC) drives the discharge through a ballast resistor of 100 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A total pressure of about 1 mbar of a mixture of CH4 seeded in He is injected in the cell and a continuous flow of gas is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' To record the ab- sorption spectra in the 5 µm region, we use the globar internal source of the Bruker FTIR installed on the AILES beamline of the SOLEIL synchrotron, a CaF2 beamsplitter, and an Indium Anti- monide (InSb) detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' These experimental conditions, together with the use of a bandpass optical filter, limit the bandwidth of our acquisition to the 1850–2100 cm−1 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The unapodized spec- tral resolution is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='004 cm−1 and 138 scans are co-added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Despite the effort to optimize the synthesis of C3 in our cell and reduce the noise floor, the most intense lines of our spectra corre- spond to only about 5 % of signal absorption resulting in a SNR of 10 at best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Aside from C3, intense absorption lines of CO (possibly pro- duced by reaction with residual H2O) belonging to the ∆v = 1 se- quences (with v′′ = 0−14, the 4 – 3 band being the strongest one) of the species in its electronic GS are observed over the full spec- tral range covered in this study (Figure S1 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Even though these series of intense lines hinder identi- fication of weaker series of C3 lines, they enable spectral calibra- tion by comparison with the accurate wavenumbers values from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [72–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' After calibration, the frequency accuracy is esti- mated for each C3 line based on its full-width at half-maximum and signal-to-noise ratio [75] and ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0005 cm−1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='006 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The rotational contour of the CO bands also allows for estimation of its rotational temperature (see Figure S2 in the supplementary material) from comparison with simulations us- ing the PGOPHER software [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Under our experimental con- ditions, a rotational temperature of 500 K is found in each CO band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Since the observation of CO hot bands is limited by the spectral coverage rather than the levels population, no vibrational temperature value can be asserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Besides CO lines, a few weak transitions of the fundamental bending mode ν2 of H2O are also detected [77] and several lines are assigned to the R branch of the fundamental ν3 band of C2H in the 1860–1895 cm−1 range [78] (Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Methodology As mentioned previously, the ν2 vibrational band lies in the far- infrared region and is significantly weaker than the ν3 band [48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' for these reasons, it remains challenging to measure the hot band sequences involving v2 which would provide vibrational energies in the v2 progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Alternatively, such information can be de- rived from a combined rovibronic and rovibrational analysis of bands involving various quanta of ν2 excitation in the electronic GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' This approach is used in the present study exploiting C3 spec- tra in the optical and mid-infrared region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Initial analyses of optical and infrared data have been performed independently, mostly exploiting combination dif- ferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Subsequently, all literature and newly measured rotationally-resolved transitions have been imported into the PGOPHER software [76] which has been used successfully to analyze specific bands of C3 previously [37, 45, 51, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Some 3 features of this software have proven particularly powerful in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' First, all assignments can be treated as separate input files (for instance, one file per band) hence easing the treat- ment of large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Then, the software allows for a relatively straightforward simultaneous treatment of rovibrational and rovi- bronic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Last but not least, graphical representations of resid- uals, simulated bands, and term values plots is an invaluable tool to refine assignments and detect perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' We thus perform a combined fit using PGOPHER of the presently recorded data together with all available rovibronic ( ˜A 1Πu − ˜X 1Σ+ g only) and rovibrational (all known bands in the ˜X state) literature data al- lowing for the most complete spectroscopic description of the C3 molecule to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Spectroscopy of C3 As already mentioned in the introduction, detailed descriptions of the spectroscopy of C3 can be found in Gausset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [9] and Rousselot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [62];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' here we recall some aspects pertinent to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the ˜X 1Σ+ g electronic GS, as a result of the pres- ence of a doubly-degenerate bending vibration, the species ex- hibits l-type doubling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Levels with v2 > 0 are split into multiple l-states, with l the vibrational angular momentum, such that l = v, v − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Because the C3 bending frequency is unusu- ally small (ω2 ∼ 63 cm−1), the l-type doubling is unusually large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In addition, the ˜A 1Πu electronic state is subject to Renner-Teller coupling, splitting the v2 bending states into K = |l ± Λ| compo- nents, where K is the total vibronic angular momentum and Λ the projection of the electronic orbital angular momentum onto the molecular axis of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Again, the very small bend- ing frequency yields significant effects on C3 spectroscopy, here resulting in a large Renner-Teller interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A typical represen- tation of splittings for the bending mode of linear molecules expe- riencing Renner-Teller interaction is given in Figure 9 of Gausset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Because C3 is linear, thus centrosymmetric, and con- tains identical nuclei of zero spin, all the antisymmetric levels have zero weight by spin statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In other words, half of the rotational levels are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For example, for Σ+ g states, only even-J (e) levels exist while for a Σ+ u state, only odd-J ( f) lev- els exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For other states, all J levels exist but with alternate e/ f labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the following, each vibronic level is labeled M(v1v2v3) Ns where M is the electronic state ( ˜X or ˜A), vi (i = 1 − 3) refers to the quanta of excitation in each vibrational level, N is the Greek letter associated to the K quantum number, and s the symmetry of the state (g or u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For instance, the ˜X(020) ∆g state corresponds to the v2 = 2, K = l = 2 vibrational level in the ˜X 1Σ+ g electronic state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' the ˜A(020) Φu state corresponds to the v2 = 2, K = 3 (hence l = 2) vibrational level in the ˜A 1Πu electronic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' When discussing rovibrational transitions in the ˜X 1Σ+ g state, the ˜X is often omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Results and discussion Our rovibronic and rovibrational sets of data are extremely complementary as illustrated in Figure 1, allowing to retrieve both accurate energies and spectroscopic constants for levels in- volving v2 = 0 − 5 in the ˜X 1Σ+ g state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' With regard to the ˜X(0v20) vibrational levels with v2 = 0−5, the single fundamental piece of information not accessible using our combined data set alone is the energy of the ˜X(010) Πu state (see Figure 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' a value that has fortunately been determined accurately by Schmuttenmaer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Optical data In the 24300–26400 cm−1 region, 36 ˜A 1Πu − ˜X 1Σ+ g vibronic bands have been recorded by LIF spectroscopy as summarized in Table 1 and visible on Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Of these, 17 are reported for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The strongest bands in the optical spectra be- long to transitions arising from the ˜X(000) state (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The PGOPHER software allows to estimate the rotational tempera- ture of each band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' it ranges for 20 K to 150 K (see Table S1 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Typical linewidths reproducing the experimental spectra range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='02 cm−1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='04 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The present measurements are in good agreement with band values previously reported in the literature and often provide more ac- curate frequencies (see Table S2 in the supplementary mate- rial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Thanks to the relatively low rotational temperature com- bined with the high resolution resulting in well-resolved features, assignments of the new bands are relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Sev- eral bands appear heavily perturbed, in particular those involving the ˜A(020) Π(−) u and ˜A(040) Π(+) u states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the ˜A(000)– ˜X(020) band region, we assign for the first time transitions involving the uΣu and uPu upper state perturbers, previously only observed in the ˜A(000)– ˜X(000) band region [34, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In total, 22 transitions are assigned in the bands involving ˜X(020): 11 in the uΣu–Σ+ g band (P-, Q-, R-branch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' al- though the single Q-branch assignment remains tentative), 5 in the uPu–Σ+ g band (P- and R-branch), and 6 transitions in the uΣu– ∆g band (P- and R-branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' No transitions are observed for the uPu–∆g branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Figure 3 presents an overview of this band sys- tem with the sub-bands highlighted in various colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' These tran- sitions are predicted by PGOPHER without setting up a transition moment for the corresponding bands (Table S1 in the supple- mentary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' One can notice that the simulated intensi- ties of the perturber transitions do not perfectly reproduce the ex- perimental spectrum on Figure 3 (in particular, the uΣu–∆g band intensity is overestimated);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' we assume that some intensity per- turbations are not properly taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The agreement in line position, however, is quite satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The present assign- ments thus confirm the assignment of the perturber lines observed in the ˜A(000)– ˜X(000) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' It is also worth noticing that, on Figure 3, the R-branches of the Πu–Σ+ g and Πu–∆g bands (above 24545 cm−1) appear stronger in the experimental spectrum than on the simulation, and stronger than the Q-branches (the strongest features around 24540 cm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Since no rotational temperature al- lows proper reproduction of such intensity ratios, this intensity difference is either the result of an experimental adjustment dur- ing the scan or of some discrepancy in the model that does not properly apply to this band system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For the Πu–∆g band, half of the ∆g levels, with even J′′ values, was previously observed by Gausset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' This peculiarity has led the authors to a better understanding of the l-uncoupling in the ground electronic state of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the present study, we are able to assign transitions in- volving both even and odd J′′ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The transitions involving odd J′′ values appear about 5 times weaker on the experimental 4 Figure 1: Schematic vibrational energy level diagram of C3 together with optical (in dashed lines) and infrared (in plain lines) bands observed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Energy levels are represented with increasing quanta of excitation in v2 from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Energies are from the combined fit performed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Only the levels involved in bands observed in the present study are plotted, with the exception of the ˜A(0v20) levels, v2 = 0 − 5, for which all levels included in the combined fit are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Transitions in the same color arise from the same ˜X(0v20) lower state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Figure 2: Overview of the electronic spectra of C3 recorded in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Top traces: Experimental spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Bottom traces: PGOPHER simulations (at thermal equilibrium and rotational temperatures reflecting each experimental band, from 20 to 150 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' see Table S1 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Intensities are in arbitrary units and the ratio between simulated bands is adjusted to reflect the experimental traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Each simulated band is color coded according to the hot band of ν2 involved in the ˜X 1Σ+ g state (the color sequence ranges from purple to red using the same color coding as in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' At this scale, bands involving different l values in the ˜X state, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', ˜A(000)– ˜X(020) Πu–Σ+ g and Πu–∆g, are overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Zooms onto the ˜A(000)– ˜X(020) (highlighted by a star symbol on the figure) and ˜A(010)– ˜X(030) Σ− g –Πu and Σ− g –Φu (highlighted by a triangle) band systems are presented in Figure 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' spectrum than predicted using PGOPHER, and are thus signifi- cantly weaker than those involving even J′′ values, which may explain why they remained unobserved thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Three ∆K = −3 bands are observed in this study, namely ˜A(010)– ˜X(030) Σ− g–Φu (P-, Q-, and R-branches, Figure 4), ˜A(000)– ˜X(040) Πu–Γg (P- and Q-branches, tentative assignments in the R-branch, see Figure S3 in the supplementary material), and ˜A(020)– ˜X(040) Π(−) u –Γg (P- and Q-branches, Figure S4 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' To our knowledge, it is the first time that these transitions are observed for the ˜A 1Πu − ˜X 1Σ+ g band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 5 Table 1: ˜A 1Πu − ˜X 1Σ+ g rovibronic bands of the C3 molecule observed in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' J′′ max values are reported for this work and the literature, together with the references of the previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' When no literature value is reported, the band is observed for the first time in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Horizontal lines group bands arising from lower state levels with the same number of quanta of excitation in ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' See Table S2 in the supplementary material for detailed information on the literature data and the present dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Vibronic assignment Band Origina This work Literature / cm−1 J′′ max J′′ max Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' ˜A(000)– ˜X(000) Πu–Σ+ g 24676 24 64 [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 38] uΣu–Σ+ g b 24679 14 16 [34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 38] uPu–Σ+ g b 24676 6 8 [34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 38] ˜A(020)– ˜X(000) Π(−) u –Σ+ g 25039 14 30 [15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 31] Π(+) u –Σ+ g 25529 18 50 [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 15] ˜A(040)– ˜X(000) Π(−) u –Σ+ g 25441 14 50 [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 15] Π(+) u –Σ+ g 26296 16 4 [15] ˜A(002)– ˜X(000) Πu–Σ+ g 26347 14 10 [15] ˜A(100)– ˜X(000) Πu–Σ+ g 25761 18 28 [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 15] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(120)– ˜X(000) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='26128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(010)– ˜X(010) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Σ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g –Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24749 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='∆g–Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24872 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g –Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25093 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(000)– ˜X(020) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24543 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24544 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='uΣu–Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25546 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='uΣu–∆g b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25458 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='uPu–Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(020)– ˜X(020) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Π(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='u –Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25906 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[31] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Π(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='u –∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25906 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[31] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(100)– ˜X(020) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25629 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25629 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(010)– ˜X(030) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Σ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g –Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24605 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Σ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g –Φu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24607 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='∆g–Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24728 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='∆g–Φu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24729 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(000)– ˜X(040) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24389 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24389 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–Γg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24391 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(020)– ˜X(040) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Π(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='u –Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24752 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Π(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='u –∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24752 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Π(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='u –Γg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24754 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(100)– ˜X(040) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25475 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πu–∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='25475 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜A(010)– ˜X(050) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='∆g–Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24565 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='∆g–Φu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='24656 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='a Accurate values can be retrieved from the energies reported in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' b uΣu and uPu are perturber states of the ˜A(000) state, unidentified so far (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' These bands are spectrally intertwined with the corresponding ∆K = ±1 bands arising from the same upper state and their in- tensity is properly predicted by PGOPHER by perturbation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', even though no transition moment is used to predict them, see Ta- ble S1 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' These nominally forbid- den transitions appear quite strong both on the simulation and the experimental trace, as visible on Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' We note, however, that again the R-branches of both bands (located above 24605 cm−1) are stronger on the experimental spectrum than predicted by the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Two bands of the ˜A(010)– ˜X(050) band system, namely the ∆g–Πu and ∆g–Φu bands, are observed in this work (see Figure S5 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Again, to the best of our knowledge, these bands are the first bands of the ˜A 1Πu − ˜X 1Σ+ g transition probing the ˜X(050) state at high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A single band probing higher quanta of excitation in ν2 was previously reported in the literature, the ˜A(000)– ˜X(060) Πu–Σ+ g band for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 24520 24530 24540 24550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3 24526 24528 24530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3 Intensity / arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Wavenumber / cm 1 Experimental spectrum A(000) X(020) simulation u + g u g(e) u g(f) u u + g u u g uPu + g Figure 3: The ˜A(000)– ˜X(020) band system observed around 24540 cm−1 (top traces) and comparison with a PGOPHER simulation at 70 K (lower traces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' (Left panel) Overview of the band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' (right panel) zoom onto a portion of the spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the simulation, transitions within each sub-band are plotted in various colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For the Πu–∆g band, transitions involving even and odd J′′ values (e and f levels in the lower state) are plotted in two shades of green with the intensity of the simulated f symmetry divided by a factor 5 compared to the PGOPHER simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For the transitions involving the perturber states in the ˜A 1Πu state, no scaling factor was used for the intensities as we could not define one that would be valid over the full spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 24570 24580 24590 24600 24610 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='6 24585 24590 24595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='6 Intensity / arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Wavenumber / cm 1 Experimental spectrum A(010) X(030) simulation g u g u Figure 4: The ˜A(010)– ˜X(030) Σ− g –Πu and Σ− g –Φu band system observed around 24600 cm−1 (top traces) and comparison with a PGOPHER simulation at 150 K (lower traces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' (Left panel) Overview of the band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' (right panel) zoom onto a portion of the spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the simulation, transitions from the ∆K = −1 band are in gray and those from the ∆K = −3 band are highlighted in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' which Q-branches assignments were proposed by Merer [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the present work, the spectral range where this band lies (around 24217 cm−1) has not been investigated and no other band involv- ing the ˜X(060) level is observed, preventing assignments confir- mation by combination differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Additional figures showing overviews of all ˜A 1Πu − ˜X 1Σ+ g bands observed in this study are reported in the supplementary material (Figure S6–S17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Infrared data The full FTIR absorption spectrum of C3 recorded in this work is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The intense absorption features from CO, H2O, and C2H have been removed from the experimental trace for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The simulated spectrum given in the lower part of the fig- ure is obtained using the PGOPHER software [76] using the band center and rotational constants obtained from our fit (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3), a Gaussian lineshape with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='005 cm−1 width, and assuming a 700 K rotational temperature (which was found to better repro- duce the fundamental ν2 band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' we note that this temperature is 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='00 Transmittance 1900 1925 1950 1975 2000 2025 2050 2075 Wavenumber / cm 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0 Intensity / arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' C3 3 simulation (001) (000) (011) (010) (021) (020) (031) (030) (041) (040) (051) (050) Experimental spectrum Figure 5: Spectrum of C3 in the 1850–2100 cm−1 spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Top trace: Absorption spectrum presented in transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The intense absorption lines of CO as well as the lines of H2O and C2H have been removed from the experimental trace for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Bottom trace: A 700 K PGOPHER simulation (at thermal equilibrium) of the ν3 band of C3 and its hot bands involving up to five quanta of excitation in ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The hot band sequence is plotted in a color sequence ranging from purple to dark red (the color coding is the same as in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The simulation is normalized to the strongest transition of the fundamental ν3 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Table 2: Rovibrational bands of the C3 molecule observed in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' J′′ max values are reported for this work and the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' When no literature value is reported, the band is observed for the first time in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Horizontal lines group bands arising from lower state levels with the same number of quanta of ex- citation in ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' See Table S3 in the supplementary material for more information on the literature data and the present dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Vibronic assignment Band Origina This work Literature / cm−1 J′′ max J′′ max Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' ˜X(001)– ˜X(000) Σ+ u –Σ+ g 2040 60 52 [39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 47] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜X(011)– ˜X(010) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πg–Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[47] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜X(021)– ˜X(020) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='u –Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[47] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='∆u–∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1994 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='[47] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='u –∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2003 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='20b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='∆u–Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1992 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='18b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜X(031)– ˜X(030) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πg–Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1985 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Φg–Φu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1976 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πg–Φu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1988 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Φg–Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1973 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜X(041)– ˜X(040) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='u –Σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1973 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='∆u–∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1970 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Γu–Γg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1960 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Γu–∆g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1956 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='∆u–Γg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1974 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='˜X(051)– ˜X(050) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Πg–Πu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1961 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Φg–Φu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1956 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='Hg–Hu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1945 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='a Accurate values can be retrieved from the energies reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' b Tentative assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' higher than that found for CO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The hot band sequence involving increasing values of v′′ 2 extends toward lower frequencies as vi- sually indicated by the colored sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Figure 5 also illustrates the strong overlap of all the bands observed in this work which causes most of the difficulties in the assignment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For ex- ample, the lines arising from the (051)–(050) bands span over most of the spectral region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Since they are weak and hindered by many other more intense lines, their assignment was challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In total, 18 rovibrational bands of C3 have been observed in this study, 14 of them for the first time (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' A detailed ac- count on this work and available literature of rovibrational data is available in Table S3 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In this work, the assignment of the infrared data was performed sepa- rately from the optical measurements presented in the previous sections and strongly relied on the literature data (including opti- cal studies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The analysis of the (001)–(000) Σ+ u–Σ+ g, (011)–(010) Πg–Πu, and (021)–(020) Σ+ u–Σ+ g and ∆u–∆g bands is rather straightfor- ward thanks to the diode laser experiments performed by Mat- sumura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [39] and Kawaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For all these bands, our observations are perfectly consistent with the literature mea- surements and allow for an extension of the dataset toward high- J values (up to 55–60, see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The frequency accuracy is slightly improved for the (001)–(000) and (011)–(010) bands (by up to a factor 2, with values as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0005 cm−1) and similarly for the (021)–(020) bands (see Table S3 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Our extended dataset and the use of ∆up 2 (J) (see the supplementary material for a detailed explanation) allowed to identify local perturbations in the upper ˜X(021) ∆u levels man- ifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Figure 6 shows the diagrams of the first derivative of the second differences, ∆up 2 (J) values, calculated in the upper states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The highest shift occurs for the ˜X(021) ∆u e manifold at J′ = 35 (the shift corresponds to about 10 times the C3 linewidth), and similar but smaller shifts are observed for the ˜X(021) f mani- fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' No notable level shifts are observed in (021) Σ+ g state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The observed perturbations in the ∆u state are likely caused by a Cori- olis interaction with the ˜X(190) Π manifold, which mixes energy levels with ∆J = 0, e ↔ e or f ↔ f, and ∆l = odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' While the ˜X(190) Π state has not yet been detected in SEP measurements, observations of ˜X(180) Σ at about 1993 cm−1 puts the ˜X(190) Π state in the right energy range [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' We report for the first time numerous infrared transitions in- volving the (031)–(030), (041)–(040), and (051)–(050) hot bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 7 0 500 1000 1500 2000 2500 3000 J(J + 1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='70 2(J + 1) 2(J 1) / cm 1 X (021) e X (021) f Figure 6: Perturbation analysis of the ˜X(021) ∆u e and f manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The assignment procedure relied mainly on combination differ- ences using the work of Gausset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [9] (see the supplemen- tary material for a detailed explanation on the procedure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' As no combination differences are available for the (030) Φu state, we used the results of successive fits that included l-type resonance with the Π state to secure the assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Further confirma- tion is obtained by GS combination differences using the present optical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' SEP data from Rohlfing and Goldsmith [46] and Northrup and Sears [24], despite their limited resolution, have provided crucial information on states for which little was known so far (vibrational energies and estimated B values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In several cases, for example the (051)–(050) band manifold, this provided sufficient information to initiate an analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Rohlfing and Goldsmith [46] reported the vibrational energy of the ˜X(051) Πg state [2330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='9(5) cm−1] as well as an estima- tion of the rotational constants for the lowest J values [B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='4718(13) cm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Using this information, we started our anal- ysis for the e and f states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Once Πg–Πu transitions were found (up to J′′ = 40 for both e and f transitions), we used the l-type resonance to secure the assignments of Φg–Φu and Hg–Hu bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The intensities of the P branches involving high J values are very weak (red-end of our spectrum) and, as for the ∆g states of (021), some Coriolis-type resonances seem to complicate the assign- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Figures S18–S23 in the supplementary material show the different infrared band manifolds observed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The rotational assignments proposed in this study are con- firmed by the observation of several Q-branches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', no shift by one or more quanta of J is possible in the proposed assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Such a Q-branch is shown on Figure 7 for the (051)–(050) Hg– Hu band, another example is provided for the (041)–(040) Γu– Γg band in Figure S24 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Overall, Q-branch transitions were observed, for both e and f levels, for the (011)–(010) Πg–Πu [Q(1),Q(3)], (021)–(020) ∆u–∆g [Q(2), tentative], (031)–(030) Φg–Φu [Q(3)–Q(6)], (041)–(040) Γu–Γg [Q(4)–Q(7)], and (051)–(050) Φg–Φu [Q(3)–Q(5), Q(6) tentative] and Hg–Hu [Q(5)–Q(11)] bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Once all the data were included in PGOPHER, we have also been able to assign some transitions involving ∆l = ±2 bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' An interesting feature of PGOPHER is that it predicts these tran- sitions without inputting a corresponding transition moment for these nominally “forbidden” transitions (similarly to what was Figure 7: Zoom onto the Q-branch of the ˜X(051)– ˜X(050) Hg–Hu band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Top trace: experimental spectrum, in transmittance, after removal of CO, H2O, and C2H lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Bottom trace: PGOPHER simulations at 700 K (final set of parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' described previously for the electronic spectra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For the (021)– (020) bands, the ∆l = ±2 bands are predicted with relatively low intensities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' and only features with relatively poor SNR are ob- served on the experimental spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Only tentative assignments are made (7 for the ∆u–Σ+ g and 4 for the Σ+ u–∆g band) and these are not included in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For hot bands involving higher quanta of excitation in ν2, however, these predicted ∆l = ±2 transitions have significant intensity, in particular for high l values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Indeed, several features with reasonable SNR are observed on the spec- trum as visible on Figure 8 for the (041)–(040) Γu–∆g band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Ad- ditional examples are provided in the supplementary material for the (031)–(030) Πg–Φu and Φg–Πu bands (Figure S25) and the (041)–(040) ∆u–Γg (Figure S26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' These “cross-ladder” tran- sitions (if we refer to levels of a given l value for a specific state as a ladder) provide constraints on the energy difference between the various l levels of the vibrational levels for which they are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Figure 8: Zoom onto three consecutive R-branch transitions of the (041)–(040) Γu–∆g band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Top trace: experimental spectrum, in transmittance, after removal of CO, H2O, and C2H lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Bottom trace: PGOPHER simulations at 700 K (final set of parameters, normalized to the strongest transition of the fundamental ν3 band) where the Γu–∆g transitions are highlighted in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The rest of the simulated transitions of the (041)–(040) bands visible in this range are plotted in shades of red while the other C3 bands are simulated in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the high frequency part of the spectrum, most of the tran- 8 sitions can be assigned to the C3 molecule (see for instance fig- ure 8 and figure S27 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Toward the lower end of the spectrum, however, many transitions remain unassigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' These transitions probably arise from the R-branches of the (061)–(060) bands but assigning this spectrum has not been possible despite our best efforts, mainly because the signal-to- noise ratio is rather poor in that region and spectroscopic assign- ments are challenging on the basis on a single branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Combined fit 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Dataset The unique feature of the present work is that it becomes pos- sible to combine the optical and infrared data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Available data from the literature and the present work for the ˜A 1Πu − ˜X 1Σ+ g rovibronic and all ˜X 1Σ+ g − ˜X 1Σ+ g rovibrational transitions are listed in Tables S2 and S3 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' These detailed tables contain the Jmax values, frequency uncertainties used in the combined fit, and the frequency offset eventually ap- plied to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' All available ˜A 1Πu − ˜X 1Σ+ g data and rovi- brational transitions in the electronic GS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', data with list of assignments provided in the literature) are included in the present fit with one exception, namely the work of Balfour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [27] who reported the spectroscopic assignments for five ˜A 1Πu − ˜X 1Σ+ g transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' One of the rovibronic band observed by Balfour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [27], the ˜A(020)– ˜X(000) Πu–Σ+ g band, was re-investigated by Tokaryk and Chomiak [31] who proposed a different spec- troscopic assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the present study, our assignments are in line with those of Tokaryk and Chomiak [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Additionally, we propose in this work an alternative spectroscopic assignment for the ˜A(002)– ˜X(000) Πu–Σ+ g band compared to Balfour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Finally, when performing the combined fit, we noticed that the spectroscopic assignments proposed by Balfour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [27] in the ˜A(200)– ˜X(000) Πu–Σ+ g band are incompatible with those of the ˜A(200)– ˜X(200) Πu–Σ+ g from Merer [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Since assignments from Merer [28] have proven consistent with literature data for the other bands they reported, we decided to only include the Merer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' There is no literature data able to confirm or in- firm the spectroscopic assignments of the remaining two bands observed by Balfour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' at this stage we have chosen not to include these data in our fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' As previously noted by Saha and Western [51], a serious dif- ficulty arises in high resolution combination fits when rovibronic data are included from different light sources, as (small) spectral offsets are intrinsic to this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' This can be overcome by shifting the frequencies of one dataset by this offset value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' It is often challenging, however, to determine which dataset presents an offset from the absolute transitions rest frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The im- pact of this issue is relatively limited because only the absolute energy of the upper state is affected while the accuracy of the rotational constants and overall fit is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In the present study, small offsets (typically of the order of several hundredths of a cm−1) are applied to several rovibronic datasets (see Table S2 for a detailed list of concerned data and offset values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' We use the frequency offsets established by other authors from the literature when available (for instance, an offset of +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='04 cm−1 was deter- mined for the data of Gausset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [9] by Tanabashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [33]) for consistency reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' When not available in the literature, we determine these offsets ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Overall, offsets values range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='02 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='12 cm−1, hence the absolute energies in the ˜A 1Πu state may be affected by these amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' As much as possible, data are included in the fit at their experi- mental accuracy (see Table S2 for typical values for each dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' When no frequency accuracy was provided, we assume a value based on the dispersion of the frequencies from our best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In some instances, the literature data are provided with an upper limit for the frequency error but the residuals from the fit show that this value is over-estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For example, the uncertainty on line frequency is assumed to be better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='01 cm−1 for the ˜A(000)– ˜X(000) transitions reported by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [34] while the residuals from our fit show that the line accuracy is probably more of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='005 cm−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' that value is thus used in the present fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' It has proven challenging to treat such a large dataset for one molecule subject to clear perturbations with transitions signifi- cantly deviating from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Transitions severely perturbed are excluded from the fit while transitions slightly diverging are kept in the fit but with an increased frequency error (hence a lower weight), typically by a factor 10, in order to maintain a global 5σ deviation for the combined fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Overall, the dataset contains 4425 observations (3957 rovibronic and 1468 rovibrational tran- sitions) including 2046 (1106 rovibronic and 940 rovibrational transitions) from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The full dataset is available as elec- tronic files as part of the supplementary material (these files also contain the transitions not included in the present fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Hamiltonian The PGOPHER Hamiltonian used for the combined fit is sim- ilar to the previous studies using PGOPHER on C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' One differ- ence with the work of Haddad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [37] is that we expressed the Hamiltonian in terms of the rotational angular momentum of the nuclear framework, ˆR, in order to obtain energy levels and rota- tional constants comparable with most of the literature data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' It is worth noting that the specific Hamiltonian used by the PGO- PHER software differs from the conventional Hamiltonian for a linear molecule developed for instance by Yamada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The off-diagonal constants accounting for l-type doubling are ex- pressed as perturbation terms between two states, which results in several q values for a given vibrational level instead of a more physically-relevant single one (see Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' For example, for the ˜X(050) vibrational level there are three different q values: one for the Π state, and two defined as ⟨Πu| q |Φu⟩ and ⟨Φu| q |Hu⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The resulting PGOPHER files together with details on their construc- tion are provided in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The perturbation analysis for the ˜A(000) state is carried out using the same effective Hamiltonian as reported in Haddad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [37], with the two perturbing Σ and P = 1 states identified in the literature treated as three perturbing states with the e and f levels of the P = 1 state treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The resulting states are labeled uΣ, uPe, and uPf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' As in Haddad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [37], the spin- spin interaction constant λ was fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1 cm−1 in the uΣ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Table S5 in the supplementary material presents the resulting constants in these perturbing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Fit results A total of 340 parameters have been adjusted to reproduce more than 4400 experimental data with a rms of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='041 cm−1 and 9 a reduced standard deviation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Figure 9 displays the resid- uals of the fit with a color coding based on the ˜X(v1v2v3) level involved in the transition (the same as in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Overall, the fit is quite satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' One can notice that the infrared (021)– (020) transitions (in green, around observation #400) are among those the least well reproduced by the fit as a result of severe per- turbations in the ˜X(021) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Figure 9: Residuals of the combined fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Transitions involving the ˜X(0v20) vi- brational level, with up to five quanta of excitation in ν2, are plotted in a color sequence ranging from purple to dark red (same as in Figure 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' data in black arise from other ˜X vibrational levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The vertical dashed line separates the rovi- brational data (low observation numbers) from the ˜A 1Πu − ˜X 1Σ+ g electronic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' “Weighted Obs-Calc” corresponds to the Obs-Calc value divided by the frequency error of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The full list of parameters is reported in Tables S4 and S5 in the supplementary material, and a subset of parameters per- taining to the ˜X(0v20) levels is reported in Table 3 where they are compared to available literature data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Only five parameters could not be determined in the present fit, the energies of five lev- els [ ˜X(110), ˜X(300), ˜X(400), ˜X(500), ˜X(600)] involved each in a single rovibrational transition for which no cross-correlating data exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Prediction of ν2 hot bands The present modeling of C3 in its electronic ground and ˜A states can be used to predict further transitions not directly ob- servable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' This is particularly interesting for the hot bands of the ν2 fundamental that remain elusive in the laboratory to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Ta- ble 4 contains lists of predicted transitions for the ˜X(020)– ˜X(010) and ˜X(030)– ˜X(020) band systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' These transitions have been calculated using PGOPHER and the final set of parameters re- ported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' One limitation of PGOPHER is that the list of energies does not carry frequency error information, hence we cannot convey these errors to the transitions frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' How- ever, based of the frequency errors of the transitions used to de- termine these energies, and the overall good quality of the fit, we estimate the accuracy of these transition frequencies to be of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='005 cm−1 (150 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Astronomical implications Due to the extremely low bending frequency of C3, even in an environment at moderate temperature its low excited bending vibrational levels ˜X(010), at ∼ 63 cm−1 (91 K), and ˜X(020), at ∼ 132 cm−1 (191 K), can be thermally populated significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' In- deed, at 100 K, 29 % and 13 % of the GS population lies in each of these levels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' at 50 K, these numbers drop down to 14 % and 2 %, which remains significant for an abundant molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Hence, ν2 hot bands may be detectable in various environments of the interstellar medium where C3 is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Unambiguous detec- tions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=', beyond the line confusion limit, become possible when astronomical data can be compared to accurate submillimeter lab- oratory data (with sub-MHz resolution) that will strongly bene- fit from the predictions made here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Given the accurate parame- ters derived here, even without such laboratory data it should be possible to identify these ν2 excited transitions in astronomical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='005 cm−1 (150 MHz) accuracy of our predictions corresponds to a velocity uncertainty of ∆V = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='6 km·s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Since the linewidth of observed spectra of C3 ν2 fundamental band in star forming region ranges from 5 to 12 km·s−1 [19, 20, 80, 81], we conclude that our data can be used to search for C3 ν2 hot bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Measurement of C3 in excited states and determination of its abundance and excitation temperature may give new insights into the chemistry of its formation and will add further informa- tion to derive the origin of small and possibly also longer carbon chains in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Conclusion This work presents the most complete spectroscopic study of C3 to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Through the combination of infrared and optical tran- sitions precise vibrational energies and rotational constants for the low-lying bending modes of C3 are determined up to v2 = 5, significantly extending our knowledge of the rovibrational mani- fold of the electronic GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The measured and predicted transition frequencies presented here can be used in astronomical obser- vations in both the optical, infrared, and far-infrared, increasing the effectiveness of C3 as a probe of the physical and chemical environment of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' PGOPHER files allowing the full sim- ulation of all known bands of ˜A 1Πu − ˜X 1Σ+ g system are given in the supplementary material so that further improvements of the laboratory spectroscopy of this molecule can be directly incorpo- rated in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Acknowledgments This manuscript comprises of two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The project started with LIF measurements at USTC (recorded in 2017) and upon analysis it became clear that combining these with non-published infrared measurements recorded at SOLEIL in 2010 offered a unique opportunity for a very complete spectroscopic study of the C3 radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' PGOPHER is ideal to merge the two large data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Given all involved spectroscopic challenges, we asked Colin Western for help, and as usual were helped on the spot and far beyond, resulting in his co-authorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Along the way of finish- ing this manuscript, Colin sadly passed away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' We dedicate this manuscript to his memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' We acknowledge financial support from the National Natural Sci- ence Foundation of China (22173089 and 21827804), the Nether- lands Organization for Scientific Research (NWO) through a 10 Table 3: Spectroscopic constants (in cm−1) of C3 in the ˜X(0v20) state, with v2 = 0 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Values derived in this study are compared to literature data, when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Numbers in parentheses are 1σ deviations of the fit, in units of the last digit of the parameter (for literature data, the information is sometimes not available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Level E B D × 106 H × 1010 L × 1014 −q/2 × 103 −qD/2 × 107 −qH/2 × 1011 −qL/2 × 1014 −qM/2 × 1018 ˜X(000) Σ+ g This work 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='430 587 17(17) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='5337(25) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='905(17) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='599(32) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [47] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='430 579(17) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='485(22) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='385(77) ˜X(010) Πu This work 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='416 591 12(48) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='442 415 78(25) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3525(28) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='675(23) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='060(53) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='850 533(79) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='048(19) −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='14(35) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='52(21) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='66(37) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [47]a E010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='442 381(18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='328(37) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='62(11) ˜X(020) Σ+ g This work 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='800 29(78) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='451 584 0(62) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='419(10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='571(31) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [47] 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='7993(19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='451 632(41) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='57 ∆g This work 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='035 64(82) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='453 099 5(67) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='929(14) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='185(88) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='52(17) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [47]a 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='065(29) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='453 088(31) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='775(50) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='11(49) −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='788(23) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3(15) � Σ+ g ���∆g � This work −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='8250(16) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='074(28) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='143(89) ˜X(030) Πg This work 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='4240(11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='460 472 9(66) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='6930(94) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='799(45) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='465(76) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1990(35) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='928(60) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='20(20) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' [9] E010 + 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='4600 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='6 Φg This work 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3943(15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='462 887 4(98) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='248(18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='79(11) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='45(23) ⟨Πu|Φu⟩ This work 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='5111(22) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='249(43) 4.' metadata={'source': 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+page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='4675 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='248(18) ∆g This work 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='2198(11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='468 951 8(86) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='884(13) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='569(51) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='17(14) Γg This work 289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1579(22) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='472 027(11) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='527(15) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='484(52) � Σ+ g ���∆g � This work 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='1731(30) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='804(77) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='52(56) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='17(14) � ∆g ���Γg � This work −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0522(30) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='130(65) −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='88(25) ˜X(050) Πg This work 370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='4479(18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='475 305(15) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='064(26) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='28(10) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='3000(90) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='75(25) Φg This work 372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='0318(17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='477 088(16) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='648(28) Hg This work 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='121(12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='480 649(30) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='448(64) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='47(22) ⟨Πu|Φu⟩ This work 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='9550(84) −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='07(24) ⟨Φu|Hu⟩ This work −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='228(27) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='56(25) a Average values of the e and f components reported by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Table 4: Low-J far-infrared transitions of the ˜X(020)– ˜X(010) and ˜X(030)– ˜X(020) hot bands of ν2 predicted using our model (in cm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' The e/f label of the lower state is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' J′′ P Q R P Q R P Q R (020)–(010) Σ − Π (020)–(010) ∆ − Π 1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='947(e) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='657(e) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='088(e) 2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='864(f) 68.' metadata={'source': 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+page_content='118(e) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='655(e) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='058(e) 6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='123(f) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='144( f) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='547( f) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='918( f) (030)–(020) Π − Σ (030)–(020) Π − ∆ (030)–(020) Φ − ∆ 0 75.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='842( f) 6 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='831(e) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='732(e) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='682(e) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='408(e) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='309(e) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='258(e) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='890(e) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='438(e) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content='902(e) VICI grant, the Netherlands Research School for Astronomy (NOVA), and the Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of CNRS/INSU with INC/INP co- funded by CEA and CNES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' DT acknowledges funding from the Natural Sciences and Engineering Research Council of Canada in the form of a Discovery grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Part of this work was performed at the SOLEIL facility under the proposal 20100296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Authors contributions Marie-Aline Martin-Drumel: Investigation, Formal analy- sis, Writing – original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Writing – review & editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Qiang Zhang: Investigation, Formal analysis, Methodology, Writing – original draft, Writing — review & editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Kirstin Doney: In- vestigation, Formal analysis, Writing – original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Writing – review & editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Olivier Pirali: Conceptualization, Investi- gation, Formal analysis, Writing – original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Writing – re- view & editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Michel Vervloet: Investigation, Formal analysis, Writing – review & editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Dennis Tokaryk: Conceptualiza- tion, Investigation, Writing – review & editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Colin Western: Software;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Harold Linnartz: Resources, Supervision, Writing – review & editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Yang Chen: Conceptualization, Investigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Supervision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Writing -– review & editing Dongfeng Zhao: Con- ceptualization, Investigation, Supervision, Writing — review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Huggins, Preliminary note on the photographic spectrum of comet b 1881, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} +page_content=' London 33 (1882) 1–3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfRwwk/content/2301.04992v1.pdf'} diff --git a/gtE1T4oBgHgl3EQfMQPY/content/tmp_files/2301.02988v1.pdf.txt b/gtE1T4oBgHgl3EQfMQPY/content/tmp_files/2301.02988v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1c9013f724a03c3a60cdaf99f1c98924f207912 --- /dev/null +++ b/gtE1T4oBgHgl3EQfMQPY/content/tmp_files/2301.02988v1.pdf.txt @@ -0,0 +1,1732 @@ +Sample-size-reduction of quantum states for the noisy linear problem +Kabgyun Jeong∗ +Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea and +School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea +(Dated: January 10, 2023) +Quantum supremacy poses that a realistic quantum computer can perform a calculation that +classical computers cannot in any reasonable amount of time. It has become a topic of significant +research interest since the birth of the field, and it is intrinsically based on the efficient construction +of quantum algorithms. It has been shown that there exists an expeditious way to solve the noisy +linear (or learning with errors) problems in quantum machine learning theory via a well-posed +quantum sampling over pure quantum states. In this paper, we propose an advanced method to +reduce the sample size in the noisy linear structure, through a technique of randomizing quantum +states, namely, ε-random technique. Particularly, we show that it is possible to reduce a quantum +sample size in a quantum random access memory (QRAM) to the linearithmic order, in terms of +the dimensions of the input-data. Thus, we achieve a shorter run-time for the noisy linear problem. +I. +INTRODUCTION +The Shor’s factoring algorithm [1], first proposed in +1994, states that there is an efficient way to find a prime +factor x for a given integer N = a·x via quantum Fourier +transform (QFT). It changes the computing paradigm +from the classical to the quantum regime. Despite be- +ing a pivotal method across computational science, there +exists a difficulty when a noise is involved. +In 2005, +Regev introduced a powerful conjecture [2] for defense +against quantum attacks, known as the learning with er- +rors (LWE) structure, by adding a small-sized Gaussian +noise δ to the factoring problem. The non-trivial problem +is expressed as N ′ = a·x+δ, and we call this task as the +noisy linear problem (NLP) in a broad sense. This paved +the way for post-quantum cryptography to emerge as a +modern method of secure communications. +While the +most quantum algorithms including Shor’s one mainly +focus on the data-processing with a well-posed quantum +sample underlying a superposition principle, however it +is also possible to take into account updating the initial +quantum data (in a feature space), before the quantum +algorithm run, for algorithmic efficiency. The quantum +data in a higher dimension are essentially different from +the shape of classical big-data, i.e., geometrically, any +quantum states form a unit hypersphere. Here, we try +to reveal the usefulness of initial quantum-data structure +in quantum learning theory, and apply it to the specific +noisy linear problem. Actually, famous Shannon’s noisy +channel coding theorem predicts a communication is al- +ways robust against a small environmental-noise [3], and +it is naturally believed even in the quantum regime. +The noisy linear problem is fundamental in the the- +ory of computer science and machine learning. Partic- +ularly, modern cryptography is based on the difficulty +of the noisy linear problem, and it was widely believed +that it is the strongest candidate for post-quantum cryp- +tography. However, Grilo, Kerenidis, and Zijlstra have +∗ kgjeong6@snu.ac.kr +recently proposed an efficient quantum algorithm, which +is purely based on a quantum oracle for (partially) solv- +ing the noisy linear (especially, LWE) scheme through +an assumption of the well-posed construction of quantum +sample [4] (including a binary strategy [5]), and we call +it shortly ‘GKZ algorithm’. +Specifically, if we assume +that the quantum sample can be prepared in a quan- +tum superposed state relating to the problem, a kind +of quantum Fourier transforms (QFTs) [1], called the +Bernstein-Vazirani (BV) algorithm [6] can solve the noisy +linear problem within polynomial resources and running +times. However, this kind of algorithms has a precondi- +tion such that there exists an efficient quantum memory +(i.e., QRAM). As pointed before by Aaronson [7], the +QRAM issue is not a simple problem so far, but it seems +to be a technical one. +In addition to post-quantum cryptography, quantum- +key-distribution (QKD) protocols [8–10] form another +basis for secure quantum communications. As a special +case of quantum cryptographic primitives, a random uni- +tary channel (RUC), also known as a randomizing map or +a private quantum channel [11], including its discrete [12] +and Gaussian variants [13–15] in quantum cryptographic +schemes, has many crucial implications. It has perfect +information-theoretic security, and can be constructed in +optimal ways [16, 17]. Furthermore, RUC can be directly +utilized for an existence-proof for the superadditivity of +the classical capacity [18, 19] through a pair of RUCs. +Also, it can be utilized for superdense coding scheme [20] +and quantum data-hiding task [21]. +Actually, RUC is +an inverse process of QKD protocols in that two legiti- +mate users with secret keys can always generate the maxi- +mally mixed state, and it can be efficiently constructed by +quantum randomization techniques. However, we cannot +find any studies on quantum algorithms for its speed-up +utilizing the quantum randomization techniques. In this +paper, we first suggest that quantum randomizing meth- +ods (inspired by the efficient RUC-construction) can be +directly applied to reduce the initial quantum sample size +used in quantum learning theory (i.e., NLP), since the ef- +ficiency of the quantum randomizing technique over any +arXiv:2301.02988v1 [quant-ph] 8 Jan 2023 + +2 +quantum states has been analyzed previously [21]. +The quantum sample proposed by Grilo et al. in the +noisy linear problem [4] is formally given by +|ψ⟩DA = +1 +� +qd +� +a∈Fdq +|a⟩D|a · x + δa mod q⟩A ∈ P(Cd+1), +(1) +where x ∈ Fd +q is fixed, a ∈ Fd +q is chosen uniformly at ran- +dom, and the noise term δa ∈ Fq is an independent and +identically distributed (IID) random variable with a cer- +tain error probability distribution [22]. Here, Fq denotes +a finite field of order q. Specifically, a and x are decom- +posed into a1a2 · · · ad and x1x2 · · · xd, respectively, and +aj, xj ∈ Fq for any j. Our main goal is to recover x effi- +ciently in this quantum setting. It can be seen that P(Cd) +denotes the set of all pure quantum states—a set of unit +vectors on the Hilbert space Cd, and D and A denote +the data and ancilla systems, respectively. This process +can be prepared by a quantum oracle function Rψ un- +der a help of QRAM, mapping a set of input pure states +{|aj⟩D}d +j=1 and the ancilla |µ⟩A to the output |ψ⟩DA for +every µ ∈ Fq. +For a given quantum sample |ψ⟩DA as +in the superposition of pure quantum states, applying +QFT (i.e., QFTq|a⟩ = +1 +√q +�q−1 +b=0 ωab|b⟩ with ω = e +2πi +q +the +root of unity) on the sample state, i.e., QFT⊗(d+1) +q +|ψ⟩DA, +returns the appropriate output value x with a high prob- +ability. +This is known as the Bernstein-Vazirani (BV) +algorithm. +In fact, we wish to find x efficiently in this quantum +learning scenario. In this situation, we may apply the +quantum randomizing techniques on {|aj⟩D}d +j=1 ⊂ P(Cd) +(via the ε-net construction) over the data systems. Thus, +as a new data-set, we can obtain a smaller set of net +points {| ˜aj⟩D}m∗ +j=1 (m∗ < d), where m∗ := +� +O(d log d) +and m is sufficiently large. We notice that +� +O(d log d) := +O(√d log d). This means that we can possibly solve the +noisy linear problem more efficiently compared to previ- +ous quantum approach [4] in the framework of its sample +size and running-time. +Here, we provide that in principle we can reduce the +quantum sample-size given by d to m∗ via quantum +randomization techniques through a mathematical tool +known as the ε-net construction [21] and L´evy’s inequal- +ity [23]. These methods discretize a set of pure quantum +states to a finite less-sized set on a unit hypersphere, and +estimate large deviations for random variables, respec- +tively. (See the technical lemmas in the Appendix A +for details.) +This paper is organized as follows. +In Sec. II, we +briefly introduce the original GKZ algorithm for solving +the noisy linear problem in which they make use of well- +posed quantum superposed samples. In Sec. III, we pro- +pose our main result for reducing the quantum sample- +size via quantum randomization techniques. This implies +that we can solve NLP more efficiently than GKZ algo- +rithm in the linearithmic order. We also suggest a new +type of model for QRAM, namely, approximate QRAM +and analyze its performance in Sec. III A. Finally, discus- +sions and remarks are offered in Sec. IV, and some open +questions are raised for future works. +II. +ORIGINAL GKZ ALGORITHM +Before the main result, we shortly review the GKZ +algorithm for solving the noisy linear problem. We as- +sume that the errors satisfy δa = a · ⃗η, for all ⃗η = +(η0, . . . , ηd−1)T ∈ Fd +q. The error vector ⃗η is chosen from a +certain distribution χ over Fq, which is defined by a dis- +crete Gaussian with a small standard deviation at zero. +For a given quantum sample in Eq. (1), the QFTs return +the following outcome: +� +QFT⊗d +q +⊗ QFTq +� +|ψ⟩DA = +1 +qd√q +� +a∈Fdq +� +b∈Fdq +� +c∈Fq +ωa·(b+cx′)|b⟩D ⊗ |c mod q⟩A, +(2) +where x′ = (x + ⃗η). By exploiting the delta function, +∆bj,−cxj := +1 +q +� +aj∈Fq ωaj(bj+cxj) for each register j ∈ +{0, . . . , d − 1}, we can obtain +1 +√q +� +b∈Fdq +� +c∈Fq +| − cx′⟩D ⊗ |c mod q⟩A. +(3) +However, x′ is not equal to the true value x, when ⃗η ̸= 0 +(without the noise term, i.e., ⃗η = 0, we can directly re- +cover the hidden x by simply measuring the ancilla reg- +ister A). Thus, we need to check if x′ = x is satisfied by +substituting b = −cx, formally called the ‘test’ process, +and by calculating the success probability as +Pr [x′ = x] = +1 +q2d+1 +������ +� +c∈Fq +� +a∈Fd +q +ωcδa| − cx⟩D ⊗ |c⟩A +������ +2 +≥ α +t cos2(2πα), +(4) +where α ∈ [0, 1/4) and c ≤ αq +t . Furthermore, we notice +that the condition |a′ · x + δa′ − a′ · x′| ≤ t needs to be +satisfied for every randomly chosen test sample a′ ∈ Fd +q. + +3 +0101101001 +10011001101 +…… +011100110011 +1010100110011 +Data +0101101001 +11011001110 +…… +010100110010 +1011100110101 +Random +Sampling +P(Cd) +AC3nicjVHLSsNAFD2Nr1pfVfiJliEuimJCr +osduOygn1ArTqZjhrMi2QilFLcuRO3/oBb/RzxD/QvDNGUIvohCRnzr3nzNx7nchzE2lZLzljbHxicio/XZiZnZtfKC4uNZMwjblo8NAL47bDEuG5gWhIV3qiHcWC+Y4nWs5lTcVbVyJO3DA4lP1IdH12HrhnLmeSqJPiypHP5AVn3qA+LG +vsOIPa8Li3cVIsWRVL3MU2BkoIVv1sPiMI/QgiOFD4EAkrAHhoSeDmxYiIjrYkBcTMjVcYEhCqRNKUtQBiP2kr7ntOtkbEB75ZloNadTPHpjUpYJ01IeTFhdZqp46l2Vuxv3gPtqe7Wp7+TefnESlwQ+5fuM/O/OlWLxBl2dQ0u1RpRl +XHM5dUd0Xd3PxSlSHiDiFexSPCXOt/OyzqTWJrl31lun4q85UrNrzLDfFm7olDdj+Oc5R0Nys2FsV+2C7VN3LRp3HKtZQpnuoIp91NEg72s84BFPxqlxY9wadx+pRi7TLOPbMu7fAaqwmb8= +" +ACznicjVHLTsJAFD3W9xt16aRmLhqpgjK0ujGJSYi +JEBMWwacUNpmOiUhLj1B9zqZxn/QP/CO2NJdEF0mrZ3zj3nzNx7/SQUqWLsfcFa +XFpeWV1b39jc2t7ZLezt36VxJgNeD+Iwlk3fS3koIl5XQoW8mUjuDf2QN/zBlc43 +RlymIo5u1TjhnaHXj0RPBJ4iqNUeZInqQj6L5QZA47q5Tdqs2cCnOrboWCUsVl +rGS7DjOriHzV4sIb2ugiRoAMQ3BEUBSH8JDS04ILhoSwDiaESYqEyXNMsUHajFic +GB6hA/r2adfK0Yj2jM16oBOCemVpLRxTJqYeJifZpt8plx1ug874nx1Hcb09/P +vYaEKjwQ+pduxvyvTtei0EPV1CopsQgurogd8lMV/TN7R9VKXJICNxl/KS4sAo +Z32jSY1teveib/YZga1fsg52b41LekAc+maM8P7kqOe+qUbsrFi8t81Gs4xBFO +aJ7nuMA1aqibj/jBa9WzRpZU+vxm2ot5JoD/FrW0xdXJRC -net +Q F T +i +˜ +ACz3icjVHLSsNAFD2Nr1pfVZdugkVwFRIV7LoxmUL9gGN +SJO69A0CZOJUkrFrT/gVv9K/AP9C+MKShSdEJm7px7zpm5c/0k5Km07beCsbC4tLxSXC2trW9sbpW3d1pnImANYM4jEXH91IW8og1JZch6ySCeSM/ZG1/eK7y7VsmUh5Hl3KcsKuRN4h4nweJMh1JQ97bOImKZ9elyu2VbXVMH8HjqVXu4J81OPyK1z0ECNAhEY +IkiKQ3hI6evCgY2EsCtMCBMUcZ1nmKJE2oxYjBgeoUOaB7Tr5mhEe+WZanVAp4T0C1KaOCBNTDxBsTrN1PlMOyt0nvdEe6q7jWn1c68RoRI3hP6lmzH/q1O1SPR1TVwqinRiKouyF0y/Srq5ua3qiQ5JISpuEd5QXGglbN3NrUm1bWrt/V0/l0zFar2Qc7N8KFuSQ2 +edGcH7SOLOfYchonldpZ3uoi9rCPQ+rnKWq4QB1N8k7whGe8GA3jzrg3Hr6oRiHX7OLHMB4/AUN+lJw= + +ACx3icjVHLSsNAFD2Nr1pfVZdugkVwFRIV7LoRncV7APa +Isl02g7Ni2RSLMWFP+BW/0z8A/0L74wpKFJ0QmbunHvOmblzvdgXqbTt4KxtLyulZcL21sbm3vlHf3mUJYw3WORHSdtzU+6LkDekD5vxwl3A8/nLW98qfKtCU9SEYW3chrzXuAOQzEQzJUK6sapuCtXbKtq2H+DhxLr3YF+ahH5Vd0UcEhgwBOEJIin24SOnr +wIGNmLAeZoQlFAmd53hAibQZsTgxXELHNA9p18nRkPbKM9VqRqf49CekNHFEmoh4CcXqNFPnM+2s0EXeM+2p7jal1cu9AkIlRoT+pZsz/6tTtUgMUNU1CKop1oiqjuUumX4VdXPzW1WSHGLCVNynfEIx08r5O5tak+ra1du6Ov+umQpVe5ZzM3yoW1KD510FwfNE8s5 +tZybs0rtIm91EQc4xDH18xw1XKGOBnmP8IRnvBjXRmRMjPsvqlHINfv4MYzHT+mKkOY= +ii +iii +iv +v +d +AHicjVHLSsNAFD2Nr1pf +VZdugkVwVWaK2HZXFMRlC/YBtUiSTmtomoTJRChFf8Ctfpv4B/oX3hlT0EXRCUnOnHvPmbn3unHgJ4qx95y1srq2vpHfLGxt7+zuFfcPOkmUSk+0vSiIZM91EhH4oWgrXwWiF0vhTN1AdN3JpY53H4R +M/Ci8UbNYDKbOPRHvucolrDu2KJlRljnHNbA149ZwTq9VqF12yuQ7RKyFYzKr7hFkNE8JBiCoEQinABwk9fXAwxMQNMCdOEvJNXOARBdKmlCUowyF2Qt8x7foZG9JeyZG7dEpAb2SlDZOSBNRni +SsT7NPDXOml3mPTe+m4z+ruZ15RYhXti/9ItMv+r07UojFAzNfhU2wYXZ2XuaSmK/rm9o+qFDnExGk8pLgk7Bnlos+20Smdt1bx8Q/TKZm9d7LclN86lvSgBdTtJeDTqXMz8q8dVZqXGSjzuMIx +zileVbRwDWaBvZ7zg1bqyAiux0u9UK5dpDvFrWU9fuEqPow= +p +O(d log d) +AHicjVHLSsNAFD2N73erSzfBIugmTDXYZld0404Fawt +WJEnHGkyTOJkopdSduPUH3OoniX+gf+GdMQVdiE5Icufc87MvdLwiCVjL0VjLHxicmp6ZnZufmFxaViafkjTPh84Yfh7FoeW7KwyDiDRnIkLcSwd2eF/Kmd7Wn8s0bLtIgjo5lP+FnPbcbBReB70qCzouldnot5OBgo9MO467Z2RyeF8vMcmo1h9kms2zG7OoWB +duO4+ws2IxvcrI12FcfEUbHcTwkaEHjgiS4hAuUnpOUQFDQtgZBoQJigKd5xhilrQZsTgxXEKv6Nul3WmORrRXnqlW+3RKSK8gpYl10sTExSr0ydz7SzQn/zHmhPdbc+/b3cq0eoxCWhf+lGzP/qVC0SF6jpGgKqKdGIqs7PXTLdFXVz81tVkhwSwlTcobyg2N +fKUZ9NrUl17aq3rs6/a6ZC1d7PuRk+1C1pwKMpmr8HJ1tWxbYqR3a5vpuPehqrWMGzbOKOvZxiAZ53+IJz3gxWsadcW8fFGNQq5ZwY9lPH4CKIeWeQ= +⌦⇤ +AHicjVHLSsNAFD2N7/q +unQzWARxUSbic1d040oUrA/aIsk46mCSCclEkNKtP+BWv0v8A/0L74wp6KLohCR3zj3nzNx7wzRSueH8veKNjI6NT0xOVadnZufmawuLZ7kuMiFbQkc6uwiDXEYqkS2jTCQv0kwGcRjJ8/D+wObPH2S +WK52cmsdUduPgNlE3SgSGoMuONiqWOVu/qtV5Y2OL7+1yxhvcLQq2uL+37TO/ROo17GuvaGDa2gIFIghkcBQHCFATk8bPjhSwroEZRpFxeo8qaQtiSWIEhN7T95Z27RJNaG89c6cWdEpEb0ZKhlX +SaOJlFNvTmMsXztmiw7x7ztPe7ZH+YekVE2pwR+hfugHzvzpbi8ENdl0NimpKHWKrE6VL4bpib85+VGXISXMxteUzygWTjnoM3Oa3NVuexu4/IdjWtTuRckt8GlvSQMeTJEND842Gv5mwz/ZrDf3y1 +FPYhkrWKN57qCJQxyjRd4xnvGCV+/IM17P639TvUqpWcKv5T19AZl5kr4= +AC3nicjVHLSsNAFD2Nr1pfVfiJliEuimJ +iLoUu3ElFawWrI/JdGxD8yKZCFKO3fi1h9wq58j/oH+hXfGFNQiOiHJmXPvOTP3Xify3ERa1mvOGBkdG5/ITxampmdm54rzC0dJmMZc1HnohXHDYnw3EDUpSs90YhiwXzHE8dOt6rix1ciTtwOJTXkTj1WTtwL13OJFHnxaWmz2 +SHM6+3y9r7Di9av+stXZeLFkVSy9zGNgZKCFbtbD4giZaCMGRwodAEnYA0NCzwlsWIiIO0WPuJiQq+MCfRIm1KWoAxGbJe+bdqdZGxAe+WZaDWnUzx6Y1KaWCVNSHkxYXWaqeOpdlbsb9497anudk1/J/PyiZXoEPuXbpD5X52qR +eIS27oGl2qKNKOq45lLqruibm5+qUqSQ0Scwi2Kx4S5Vg76bGpNomtXvWU6/qYzFav2PMtN8a5uSQO2f45zGBytV+zNin2wUdrZzUadxzJWUKZ5bmEHe6ihTt43eMQTno0L49a4M+4/U41cplnEt2U8fACmzpnA +N(Cd) +FIG. 1. A model for the quantum state-preparation in an approximate quantum random access memory (i.e., +ε-QRAM): The classical datasets are prepared in sampled pure quantum states ψ ∈ P(Cd) through a random sampling at +step ‘i’. By using quantum ε-random technique, we can obtain net points ˜ψ’s on the unit-sphere at step ‘iii’. Note that the +cardinality of the set of ε-net states (i.e., m∗ = +� +O(d log d)) is less than that of the previously sampled pure states (i.e., +d). While the Grilo et. al.’s quantum algorithm takes a step from i→ii, our algorithm makes use of i→iii→iv before the BV +algorithm over QFTs. Alternatively, we expect that the step v→iv is also possible in the QRAM process with a quantum +oracle. +Under the condition over M trials, we complete the test +accepting x′ = x, that is, the fail probability is given +by Pr [x′ ̸= x] ≤ +� +2t+1 +q +�M +(see details the Lemma 1 in +Ref. [4]). This statement will be also updated for our +approximate scheme. +In Refs. [4, 7], as a hard problem it was seriously +pointed out that a largely superposed state must be pre- +pared in the sampling process—This issue is generally +called a QRAM problem [24, 25] (see also Ref. [26]). +Here, our approach for the noisy linear solving quan- +tum algorithm aims to construct an approximate QRAM +(or ε-QRAM) through an ε-net analysis as in Fig. 1, +when the initial quantum samples are all in pure quan- +tum states. We discuss ε-QRAM thoroughly later in this +paper. +III. +MAIN RESULT +We present the details of the quantum randomizing +methods and their performance to solve the noisy linear +problem. As mentioned before, we should remember that +all communications are not so susceptible to a noise by +the Shannon’s noisy channel coding theorem. +First of all, let us recapitulate the mathematical no- +tations. +Let B(Cd) be the space of (bounded) linear +operators on the d-dimensional Hilbert space Cd, and +U(d) ⊂ B(Cd) be the unitary group on the Hilbert +space. +Note that 1d is simply the d × d identity ma- +trix on the space. For simplicity, we denote a pure state +ψ := |ψ⟩⟨ψ| ∈ P(Cd). +In addition, we assume that a +quantum channel Λ is a completely positive and trace- +preserving (CPT) map. While we make use of m = d +for an input quantum sample-size, we denote m = d2 as +the number of unitary operations in the case of quantum +channel. Now, let us define an ε-randomizing map with +respect to the Schatten p-norm in the trace class [27], +which is an extended version of the statement proposed +by Hayden et al. [21], and it is defined as follows. +For every quantum state ϱ ∈ B(Cd), we call a CPT +map Λ : B(Cd) → B(Cd) an ε-randomizing map with +respect to the Schatten p-norm, if +����Λ(ϱ) − 1d +d +���� +p +≤ +ε +p√ +dp−1 , +(5) +where 1d/d denotes the d-dimensional maximally mixed +state +(MMS), +and +∥M∥p +:= +��d +j=1 |sj|p�1/p += +(Tr(M †M)p/2)1/p is the Schatten p-norm for any matrix +M ∈ B(Cd) (1 ≤ p ≤ ∞) with singular values {sj}d +j=1. +In general, CPT map Λ acting on the quantum state +ϱ can be naturally constructed by +Λ(ϱ) = 1 +m +m +� +j=1 +UjϱU † +j , +(6) +where the unitary operator Uj’s are chosen uniformly at +random from the unitarily invariant measure (or Haar +measure) on the unitary group U(d), and m is the number +of unitary operators depending on the input dimension +d. It is known that m = d2 is optimal to create a perfect +d-dimensional maximally mixed state. However, we pro- +pose that m = O(d log d) is also sufficient to create MMS +in the limit of d → ∞ [21, 27, 28]. This reduction will be +exploited for constructing an efficient quantum samples +used in the noisy linear problem, and thus we call it as +‘ε-random technique’ for quantum algorithms. Here, we +notice that it is important to choose m sufficiently large. +It is also worth noting that if dµ is the unitarily invari- +ant measure chosen uniformly at random on the unitary + +4 +group U(d), then, for every quantum state ϱ ∈ B(Cd), it +satisfies +� +U(d) +UϱU †dµ = 1d +d , +∀U ∈ U(d). +(7) +For example, when d = 2, the above unitaries in the +identity of Eq. (6) take the form of a set of Pauli ma- +trices including 12. The fundamental feature of the uni- +tarily invariant measure in Eq. (7) is that it assures of +the uniform randomness for given any type of quantum +states. If the states are given in the form of a quantum +sample, in principle the measure can be used to check +that the sample is uniform or not. Because most algo- +rithms essentially require a random sample as its input +in the regime of the classical as well as quantum. This +observation offers the possibility for us to construct the +efficient private quantum channel Λ from m = d2 onto +m = O(d log d) unitary operations under a condition of +m ≫ 1. (See Appendix B for the detailed proof of The- +orem 1 below.) Consequently, it is sufficient to create +RUC (or ε-randomizing map) in Eq. (6) with a randomly +chosen unitary operations of O(d log d). Our strategy is +based on this reduction technique, and applies it to the +regime of the quantum sampling problem, and the full +statement of ε-randomizing map is given as follows: +Theorem 1. Let ε > 0 and the dimension d be suf- +ficiently large. There exists a set of unitary operators +{Uj}m +j=1 ⊂ U(d) with cardinality +m ≥ κ +ε2 d log +�10d(p−1)/p +ε +� +(8) +such that Λ(ϕ) += +1 +m +�m +j=1 UjϕU † +j +on B(Cd) is ε- +randomizing map with respect to the Schatten p-norm +(for any p ≥ 1), and κ is a universal constant. +For convenience, +we exploit the notation +˜m +:= +O(d log d)(= m∗2) instead of m as an upper bound. From +Theorem 1 in the field of quantum channel theory, we +try to highlight on the quantum sample-reduction prob- +lem over quantum algorithms, especially, relating to the +solvability of the noisy linear structure in quantum ma- +chine learnings. Here, we further improve the GKZ al- +gorithm in the order of linearithmic over a superposed +quantum sample. Here, we notice that we will take an +upper bound (i.e., O(d log d)) rather than the exact lower +bound in Eq. (8). Before presenting the main result, let +us observe how quantum randomizing technique works. +The intuition is basically conducted by following two lem- +mas. The first one is the ‘ε-net’ construction, and the +second one is ‘L´evy’s inequality’ on the set of pure quan- +tum states. We call, such a combination of two lemmas, +as ε-random technique as mentioned before. +Lemma 2 (ε-net [21]). Let ε > 0, and the dimension +d be sufficiently large. For any pure quantum state |ψ⟩ ∈ +P(Cd), we can choose a net-point | ˜ψ⟩ ∈ N(Cd) ⊂ P(Cd) +satisfying ∥ψ − ˜ψ∥1 ≤ ε, where ∥ · ∥1 is the trace norm. +Then, there exists a set N(Cd) of pure states such that +|N(Cd)| ≤ +�5 +ε +�2d +, +(9) +where | · | denotes the cardinality of the net N. Also, +recall that ψ := |ψ⟩⟨ψ|. +Lemma 3 (L´evy’s inequality [23]). Let F be a function +F : ∂Sd → R defined on the d-dimensional unit ball Sd +and its boundary ∂Sd. Suppose that a point ψ ∈ ∂Sd is +chosen uniformly at random. Then, for every ε′ > 0, +Pr[|F(ψ) − E(F)| ≥ ε′] ≤ C1 exp +� +−C2dε′2 +γ2 +� +, +(10) +where γ := sup |∇F| is the Lipschitz constant of F, and +C1 and C2 are universal constants. +Now, let us define an abstract quantum channel ˜Λ, +which convert a set of pure quantum states {ψ}m +j=1 to +another less-sized set of pure quantum states { ˜ψ}m∗ +j=1 on +the net space by exploiting Lemma 2 and Lemma 3. +Here, we exploit m as in the notion of quantum sample- +size. +This virtual process ˜Λ can be performed by ε- +QRAM (see Subsec. III A). Now, we fixed m = d and +m∗ = +� +O(d log d) for accordance between RUC (Λ) and +ε-QRAM (˜Λ): it changes the representation from density +operator to state one. We notice that, in the framework +of the quantum sample preparation of the GKZ algo- +rithm, when the initial sample may start with d → ∞, +then the success probability of the algorithm can be more +increased with high probability. The Fig. 1 conceptually +shows a quantum state-preparation for the NLP-solving +sample in the ε-QRAM subroutine. +By using those ingredients with ε-QRAM, we are ready +to suggest our main result as follows, and the proof is +essentially equivalent to the GKZ algorithm [4], where +Eq. (1) on the original quantum-sample is substituted +as a netized sample. For convenience, we also make use +of the terms such as ‘Test Candidate (TC)’ and ‘NLP +Algorithm (NLP-A)’ defined in Ref. [4] for the proof. +Proposition 4 (Main result). +Let d be sufficiently +large (thus, m∗ is) and q be a prime number. Assuming +we can efficiently prepare a superposed quantum sample +through the ε-QRAM over N(Cd) ⊂ P(Cd) in the form +of +| ˜ψ⟩DA = +1 +� +qm∗ +� +˜a∈Fm∗ +q +|˜a⟩D|˜a · ˜x + δ˜a mod q⟩A, +(11) +where δ˜a is a random variable chosen noise distribution +with maximum noise magnitude t = poly(m∗). +Then, +there exists a quantum algorithm that outputs ˜x (such +that |˜x| < |x|) with probability +1 +20tqm∗−1 . + +5 +Proof. Let L and M be parameters in order to prove +the proposition. By using Lemma 8 (which is the mod- +ification of Lemma 1 in Ref. [4]) in Appendix C, i.e., +formally, the TC accepts with probability +TC (x′, M) = +� +� +� +1, +if x′ = ˜x; +≤ ( 2t+1 +q +)M, +if x′ ̸= ˜x. +(12) +By using the union bound, the probability that at least +one of independent L-call to TC (x′, log η−1) accepts +some x′ ̸= ˜x is at most (3t/q)M L. Also from Lemma +9 (which is the modification of Theorem 1 in Ref. [4]), +the probability that x is not the output of independent +L-call to BV algorithm is at most +� +1 − +ℓ +20tqm∗ +�L +. +By exploiting the union bound again, we address that +NLP-A (L, M) does not output ˜x with probability at +most +� +1 − +ℓ +20tqm∗ +�L ++ +�3t +q +�M +L, +(13) +where we can choose ℓ = qm∗, L = 20t log η−1, and M = +1 for completion of the proof. ■ +This result implies that, for every η > 0, the BV al- +gorithm achieves sample complexity O(m∗ log η−1) and +running-time in poly(m∗, log η−1) with probability 1 − η +as in the Theorem 2 in Ref. [4]. We notice that the set of +classical information ˜x (representing on the net space) is +ε-close to the original information of x from the quantum +ε-random technique above. +We can summarize our total procedure intuitively in +terms of ε-QRAM, BV algorithm, and quantum mea- +surement as follows: +|a⟩D −→ |ψ⟩DA +ε-QRAM +−−−−−−−−→ +Lemma 2,3 | ˜ψ⟩DA +BV +−−→ QFT⊗⌈m∗⌉+1 +q +| ˜ψ⟩DA +Meas +−−−→ ˜x ∼ x. +(14) +The last step denotes a quantum measurement performed +on the computational basis, and ˜x is ε-close to the origi- +nal secret x. (See Fig. 2 for details.) +A. +Approximate QRAM issue +In computer architecture, the random access memory +(RAM) plays a crucial role storing and calling out data +in computation process. QRAMs [24, 25] are quantum +analogues of classical RAMs, and they perform similar +actions of reading-out and returning datasets in the form +of quantum states. +However, this case is subtle. +The +quantum data can be superposed even in the form of +entangled state [7] because of the following reason: The +QRAM performs that +QRAM : +� +j +αj|j⟩|0⟩ �→ +� +j +αj|j⟩|dj⟩, +(15) +where dj is a quantum datapoint corresponding to the +memory address j, and |0⟩ an ancillary state. Hence, the +data set loads in quantum superposition. Assuming that +a (quantum) big-data set, the problem could be intrigu- +ing for quantum learning theory [29–31]. Unfortunately, +this problem also occurs at the proposed approximate +QRAMs, but our claim is that it is theoretically improv- +able, when the given sets are pure quantum states. Ge- +ometrically, all pure states lie at the boundary on the +Bloch sphere. In this case, we can concentrate a quan- +tum information into a net-point on the unit sphere as +discussed above, and the role of ‘ε-QRAM’ through the +ε-net construction and the Levy’s inequality is described +in Fig. 2. +The reduction of information resources, starting from +the origin of Shannon’s data compression theory [3], is +a core ingredient in information sciences as well as in +quantum computing and quantum simulation [32]. While +classical information datasets have a diverging hypercu- +bic structure in general, they are intractable. Quantum +datasets are geometrically simple in that they have the +shape of a unit-hypersphere (see Appendix A for the +details of discretization method via ε-net on a higher di- +mensional unit sphere, i.e., on d-dimension pure quantum +states). +Specifically, the ε-QRAM acts in operational ways to +discretize all pure quantum states over the net space in +this noisy linear problem, i.e., it outputs a random sam- +ple |˜a⟩ (instead of the total sample-size of |a⟩): +� +j +αj|j⟩|a⟩D ⊗ |µ⟩A �→ +� +j +αj|j⟩|˜a⟩D ⊗ |µ⟩A, +(16) +which in principle can be used as a quantum sam- +ple to resolve the NLP. From this, the quantum ora- +cle gives birth to a quantum sample for solving NLP in +terms of |˜a⟩D ⊗ |˜a · ˜x + δ˜a mod q⟩A. +As a compara- +ble method, we can also devise and modify the bucket- +brigade QRAM [24, 33, 34] in this framework of ε-QRAM +structure. +IV. +DISCUSSION +The noisy linear problem is constituted through a clas- +sical algorithm, fundamentally adding a small Gaussian +noise to the meaningful original dataset. +It is gener- + +6 + + QUANTUM + ORACLE +10011 +10010 +11010 +-QRAM +" +ACznicjVHLTsJAFD3UF+ILdemkZi4agoShR3RjUtM5JEAMW0ZcEJpm+mUhBDi1h +9wq59l/AP9C+MxWiM0du0vXPuPWfuw418HkvbfskYS8srq2vZ9dzG5tb2Tn53rxmHifBYwv9ULRdJ +2Y+D1hDcumzdiSYM3Z91nJHFyremjAR8zC4ltOI9cbOMOAD7jmSoE534gWxdwPg5t8oWjZ2kzbKpUr +5XKVnBRZhApIrR7mn9FHyE8JBiDIYAk34eDmJ4OirAREdbDjDBHtdxhjlyxE0oi1GQ+iIvkM6dV +I0oLPSjDXbo1t8egUxTRwRJ6Q8Qb6zdTxRCsr9DftmdZUtU3p76ZaY0Ilbgn9i7fI/C9P9SIxQEX3w +KmnSCOqOy9VSfRUVOXml64kKUSEKb9PcUG+p5mLOZuaE+ve1WwdHX/VmQpVZy/NTfCmqQF21ZV2en 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A schematic diagram of procedure in the NLP-solving quantum algorithm and the abstract notion of +ε-QRAM. In the state preparation step, ε-QRAM (˜Λ) transforms pure quantum data into a netized sample through specific +blocking operations (−||||||| ), i.e., the output quantum data are given in the form |ψ⟩DA = |˜a⟩D ⊗ |µ⟩A. Here, we assume that +a quantum oracle Rψ via ε-QRAM operates a specific hidden transformation on the input quantum data, and then prepares +a quantum sample | ˜ψ⟩DA as in Eq. (11), to perform the desired noisy linear task. +However, our quantum sample-size is +linearithmically reduced compared to the original quantum sample |ψ⟩DA in Eq. (1). According to the BV kernel method (i.e., +QFT⊗(⌈m∗⌉+1) +q +where m∗ = +� +O(d log d)) and following proper quantum measurements, we can efficiently recover the secret +information x hidden in the noisy linear encodings. We notice that ˜x is ε-close to x for sufficiently large d. +Quantum algorithms +Best classical algorithms +Integer factoring +O(log d) [1] +2o(d) +Database search +O( +√ +d) [39] +O(d) +HHL +O(log d) [40] +O(d) +SVM +O(log d) [41, 42] +O(poly(d)) +Noisy linear problem +O(d)+⋆QRAM [4, 5, 43, 44], This work +2O(d) [45]+RAM +TDA +O(log5 d) [46] +O(d2) +Recommendation +O(poly log d) [25] +♯O(poly log d) [47, 48] +TABLE I. Comparison of sample-complexity for representative quantum algorithms versus best known classical +algorithms. Here, d is the input dimension in Fd +q for the given computational problem. The famous Shor algorithm [1] and +Grover search algorithm [39] can achieve quantum speedups in exponential and quadratic order, respectively. Furthermore, +the Harrow-Hassidim-Lloyd (HHL) [40] sparse matrix inversion, the quantum support-vector-machine (SVM) [41, 42], and +the quantum topological data analysis (TDA) [46] algorithm in quantum machine learning theory have remarkable quantum +advantages. +Beside, for the recommendation system [25], a classical algorithm can achieve quantum efficiency and it was +known as quantum-inspired algorithm [47, 48] (♯). +For the noisy linear problem (NLP), the GKZ algorithm [4] solve the +problem in polynomial time, in which there is an assumption that an well-superposed quantum sample (i.e., Eq. (1)) can +be efficiently prepared by the quantum random access memory (QRAM denoted by (⋆)). We notice that the analysis of the +sample-complexity for the noisy linear model in classical regime is described in Ref. [45]. In this work, we improve the QRAM- +complexity for preparing a quantum sample for NLP by exploiting the quantum ε-random technique, and the subroutine is +called as ‘approximate QRAM (or ε-QRAM)’. Consequently, this work improves the NLP-solving algorithm in the linearithmic +order than the GKZ algorithm [4] including Ref. [5, 43, 44]. +ally believed to be intractable in the presence of super- +powered quantum computers. However, contrary to pop- +ular belief, quantum computing or quantum communica- +tion are very powerful, and recent studies showed that it +is not true under the assumption of a well-posed quan- +tum sample under a quantum random access memory, +by applying quantum Fourier transforms (known as the +Bernstein-Vazirani algorithm), as in the case of the Shor’s +factoring algorithm. Moreover, the difficulty of the noisy +linear algorithmic problem is alleviated using the frame- +work of quantum learning theory. In this study, we at- +tempted to improve its efficiency by exploiting the quan- +tum ε-random technique over all pure quantum states. +This argument is always possible, if we can freely access + +7 +a dataset of pure quantum states, because such quantum +states have the simple geometric structure of a unit hy- +persphere. However, there exists an uncomfortable prob- +lem to overcome for its efficiency argument, and it is com- +monly known as QRAM issue. Here, we tried to provide +a new type of method represented by ε-QRAM. +More precisely, we proposed an efficient way of a lin- +earithmic reduction for a quantum sample size from d to +m∗ = +� +O(d log d) in the input dimension of the data-set +through the quantum ε-random technique, especially, for +solving the noisy linear problem. The quantum random- +izing techniques over quantum algorithms, inspired by +constructing the random unitary channel, rely on purely +mathematical tools known as the ε-net and L´evy’s in- +equality. The key point is that we can always reduce the +consumption of the quantum sample before the expensive +QFT runs; thus, the number of QFTs can be also reduced +in linearithmic order. The impact of this construction in +quantum machine learnings is potentially large, for exam- +ples, a reduction of quantum Fourier transforms in the +circuit-realization as well as in quantum topological data +analysis and quantum-inspired algorithms. (See Table +I.) A caveat of this is that the non-practical Haar mea- +sure problem occurs. However, a specific method such as +unitary design [35, 36] and compressive measurement [37] +settles the theoretical caveats to be a tractable for select- +ing a proper set of random unitaries. Furthermore, the +cardinality of the unitary set can be further tightened to +be m = O(d) with a sharp mathematical argument [38], +as long as d → ∞ in random matrix theory. +While quantum advantage can be achieved in quantum +algorithms including the noisy linear problem under the +assumption of a well-posed quantum sample, it requires +access to a largely-superposed quantum state. However, +a quantum-inspired (or dequantizing) algorithm recently +proposed in Refs. [47, 48], as well as in hybrid algo- +rithms [49] could be good candidates to avoid the draw- +backs of QRAMs. We also believe that it could be possi- +ble to improve via the subtle net analysis (e.g., a mathe- +matical extension of the L´evy’s or McDiarmid’s inequal- +ities) on quantum samples, and a divide-and-conquer +strategy also can be devised in the quantum learning +theory for realizing a noisy intermediate-scale quantum +computing. (See also Refs. in Table I.) +ACKNOWLEDGMENTS +This +work +was +supported +by +the +National +Re- +search Foundation of Korea (NRF) through a grant +funded by the Ministry of Science and ICT (NRF- +2020M3E4A1077861, NRF-2022M3H3A1098237) and the +Ministry of Education (NRF-2021R1I1A1A01042199). +DATA AVAILABILITY +The data that support the findings of this study are +available within the article. +APPENDIX +Here, we derive the theorem, which inspire the efficient +solvability on the noisy linear problem: Let ϕ := |ϕ⟩⟨ϕ| ∈ +P(Cd) be a pure quantum state and dµ be the unitarily +invariant measure on the unitary group U(d). Then, we +obtain the ε-randomizing map in the Schatten p-norm. +Theorem 1. Let ε > 0 and the dimension d be suf- +ficiently large. There exists a set of unitary operators +{Uj}m +j=1 ⊂ U(d) with cardinality +m ≥ κ +ε2 d log +�10d(p−1)/p +ε +� +(17) +such that Λ(ϕ) = +1 +m +�m +j=1 UjϕU † +j on B(Cd) satisfies the +ε-randomizing map with respect to the Schatten p-norm +(for any p ≥ 1), and κ is a universal constant. Here, we +denote the lower bound of m as ˜m := O(d log d). +A. +Technical lemmas +We need several technical lemmas for the proof of the +result (Theorem 1). +Lemma 5. For any r, p such that r > p ≥ 1, a quan- +tum state ϱ ∈ B(Cd) satisfies +����ϱ − 1d +d +���� +r +p +≤ d +r−p +p ∥ϱ∥r +r − d(r−p)/p +dp +, +(18) +where ∥M∥p := +��d +j=1 |sj|p�1/p += (Tr(A†A)p/2)1/p is +the Schatten p-norm in the trace class for a matrix +M ∈ B(Cd) (1 ≤ p ≤ ∞) with singular values {sj}d +j=1 of +M [50]. +Proof. The inequality in Eq. (18) is straightforward +from the fact that ϱ is a density matrix and from the +H¨older’s inequality. ■ +Lemma 6. Let ϕ ∈ P(Cd) be a fixed pure state. If we +define a random variable Y[ϕ] = +��Λ(ϕ) − 1d +d +�� +p, then the +following inequality holds (for all r > p ≥ 1) +E{Uj}Y[ϕ] ≤ +� +p√ +d +mp + +r +mp−1 · +p√ +d +�1/r +, +(19) +where the expectation E := E{Uj} is taken over a set of +unitary matrices {Ui} chosen at random. + +8 +Proof. (i) p = 1 and r = 2 case: We recall that Λ(ϕ) = +1 +m +�m +j=1 UjϕU † +j and Y[ϕ] = +��Λ(ϕ) − 1d +d +�� +1. Then, we have +E∥Λ(ϕ)∥2 +2 = 1 +m + 1 +m2 +m +� +j̸=k +ETr(UjϕU † +j UkϕU † +k) +≤ 1 +m + Tr +�� +U(d) +UjϕU † +j dµ · +� +U(d) +UkϕU † +kdµ +� +(20) += 1 +m + Tr1d +d2 = 1 +m + 1 +d, +where Eq. (20) comes from the definition of the unitarily +invariant measure (i.e., Eq. (7)) with IID unitary sets +in the index j, k. +By exploiting the Cauchy-Schwartz +inequality, we obtain Y 2 +[ϕ] ≤ d∥Λ(ϕ)∥2 +2 − 1, and we can +obtain EY 2 +[ϕ] ≤ dE∥Λ(ϕ)∥2 +2−1 [28]. Thus, for a sufficiently +large d, +EY[ϕ] ≤ +� +EY 2 +[ϕ] ≤ +� +dE ∥Λ(ϕ)∥2 +2 − 1 = +� +d +m. +(21) +(ii) p = 2 and r = 3 case: Now, suppose that Y[ϕ] = +��Λ(ϕ) − 1d +d +�� +2. As in case (i), we can straightforwardly +obtain the inequality: E∥Λ(ϕ)∥3 +3 ≤ +1 +m2 + +3 +md + 1 +d2 . Thus, +by using the H¨older’s inequality on the Schatten p-norm, +E +����Λ(ϕ) − 1d +d +���� +3 +2 +≤ +√ +dE ∥Λ(ϕ)∥3 +3 − d−3/2 ≤ +√ +d +m2 + +3 +m +√ +d +. +(22) +For any r > p ≥ 1 and any matrix M, ∥M∥∞ ≤ ∥M∥r ≤ +∥M∥p ≤ ∥M∥1 holds, and thus, completes the proof. ■ +Furthermore, we need a key lemma known as the Mc- +Diarmid’s, which is a variant of L´evy’ theorem (Lemma +3), defined as follow: +Lemma 7 (McDiarmid’s inequality [51]). Let {Xj}m +j=1 +be m independent random variables with Xj’s chosen +uniformly at random from a set S. Suppose that the mea- +surable function F : Sm → R satisfies |F(x)−F(ˆx)| ≤ cj, +known as the bounded difference, where the vectors x +and ˆx differ only in the j-th position. +If we define +Y = F(X1, . . . , Xm) as a corresponding random variable, +then for any t ≥ 0, we have +Pr[|Y − E(Y )| ≥ t] ≤ 2 exp +� +− +2t2 +�m +j=1 c2 +j +� +, +(23) +where Pr denotes the probability. +Here, we consider the bounded difference in Eq. (23) +in Lemma 7. Let the ε-randomizing map Λ be realized +by a unitary sequence (Uj)m +j=1, and another map ˆΛ be +constructed via (U1, . . . , Uj−1, ˆUj, Uj+1, . . . , Um). Thus, +we have the difference for the function F. That is, +����� +����Λ(ϕ) − 1d +d +���� +p +− +����ˆΛ(ϕ) − 1d +d +���� +p +����� ≤ +���Λ(ϕ) − ˆΛ(ϕ) +��� +p += 1 +m +���UjϕU † +j − ˆUjϕ ˆU † +j +��� +p +≤ 21/p +m . +Let us define a random variable Y[ϕ] := +��Λ(ϕ) − 1d +d +�� +p. +Thus, the McDiarmid’s inequality on the positive part +(i.e., Y[ϕ] − EY[ϕ] > 0) is given by +Pr +� +�Y[ϕ] ≥ t + +� +p√ +d +mp + +r +mp−1 · +p√ +d +�1/r� +� ≤ exp +� +− +mt2 +2(2−p)/p +� +, +(24) +A similar result can be obtained for the negative part. +We can now prove the main proposition. +B. +Proof of Theorem 1 +Now, we completes the proof of Thoerem 1. +Proof. +Let a set {Uj}m +j=1 be IID U(d)-valued ran- +dom variables, distributed according to the unitarily +invariant measure. +We derive that a map containing +Λ(ϕ) = 1 +m +�m +j=1 UjϕU † +j is ε-randomizing with high prob- +ability, that is, for any pure quantum state ϕ ∈ B(Cd), +we find +Pr∀ϕ +� +� +������ +1 +m +m +� +j=1 +UjϕU † +j − 1d +d +������ +p +≥ +ε +p√ +dp−1 +� +� < 1. +(25) +If we fix the net N(Cd) in Lemma 2 in the main text +and define ˜ϕ to be a net point on the (d − 1)-sphere +corresponding to ϕ, then, by the unitary invariance, +∥Λ(ϕ) − Λ( ˜ϕ)∥1 = ∥ϕ − ˜ϕ∥1 ≤ +ε +2 +p√ +dp−1 . +(26) +In addition, from Lemma 2, we obtain a net with the +cardinality |N| ≤ +� +10d(p−1)/p +ε +�2d +. This implies that +Pr∀ϕ +�����Λ(ϕ) − 1d +d +���� +p +≥ +ε +d(p−1)/p +� +≤ Pr∀ϕ, ˜ϕ +� +∥Λ(ϕ) − Λ( ˜ϕ)∥p + +����Λ( ˜ϕ) − 1d +d +���� +p +≥ +ε +d(p−1)/p +� +(27) +≤ Pr∀ ˜ϕ +�����Λ( ˜ϕ) − 1d +d +���� +p +≥ +ε +2d(p−1)/p +� +, +(28) + +9 +because we use ∥Λ(ϕ) − Λ( ˜ϕ)∥p ≤ ∥Λ(ϕ) − Λ( ˜ϕ)∥1 = +∥ϕ − ˜ϕ∥1 ≤ +ε +2 p√ +dp−1 , and the first inequality makes use +of the triangle inequality. These inequalities imply that +we can change infinitely many pure quantum states to a +finite set of pure states (i.e., net points) efficiently. +Finally, by using the union bound, the ε-net (Lemma +2), and the McDiarmid’s inequality (Lemma 7) in the +main text, we have +Pr∀ϕ +�����Λ(ϕ) − 1d +d +���� +p +≥ +ε +d(p−1)/p +� +≤ Pr∀ ˜ϕ +�����Λ( ˜ϕ) − 1d +d +���� +p +≥ +ε +2d(p−1)/p +� +≤ |N| · Pr ˜ϕ′ +�����Λ( ˜ϕ) − 1d +d +���� +p +≥ +ε +2d(p−1)/p +� +(29) +≤ 2 +�10d(p−1)/p +ε +�2d +× exp +� +�− +m +22−2/p +� +ε +2d(p−1)/p − +�d1/d +mp + +r +mp−1d1/p +�1/r�2� +� . +Thus, there is a ε-randomizing map with respect to the +Schatten p-norm, if the probability is upper bounded +by 1, and m ≥ κ·d +ε2 log +� +10d(p−1)/p +ε +� +. This completes the +proof. ■ +It is worth noting that for the estimation of the car- +dinality m, we make use of the condition d < m < d2, +r > p ≥ 1, and the following probability bound: +�10d(p−1)/p +ε +�2d +exp +� +− +m +22−2/p +� +ε +2d(p−1)/p − 2d1/rd +mp/r +�2� +< 1 +2. +For +sufficiently +large +d, +the +bound +gives +rise +to +� +d1/p +mp + +r +mp−1d1/p +�1/r +≤ +� +2d1/p +mp +�1/r +. +Here, +if we +fix +the +dimension +d +and +choose +m +such +that +� +ε +2d(p−1)/p − 2d1/rd +mp/r +�2 += o(ε2), then, up to a constant c, +we have +2d log +�10d(p−1)/p +ε +� +< cmε2 +22−2/p . +Therefore, we can conclude that m ≥ +κ·d +ε2 log 10d(p−1)/p +ε +for a constant κ. +C. +Further components for the reduction of GKZ +algorithm +Here, we briefly introduce two technical lemmas, for +the proof of Proposition 4, which are the natural +modifications of original GKZ algorithm under ε-random +technique we suggest. +Lemma 8 (Modification of Lemma 1 [4]). Let |x⟩ ∈ +P(Cd) and |˜x⟩ ∈ N(Cd), respectively. Then we have +TC (x′, M) = +� +� +� +1, +if x′(= x) = ˜x; +≤ ( 2t+1 +q +)M, +if x′(̸= x) ̸= ˜x. +Proof. +Suppose that there exists a proper quan- +tum state tomography process to read the value x from +the quantum state |x⟩. +By exploiting Lemma 2 and +Lemma 3 in the main context, we can observe that x is +close to ˜x with high probability. ■ +Lemma 9 (Modification of Theorem 1 [4]). Let V ⊆ +Fd +q be a random subset satisfying |V | = qm∗ with fixed +m∗ = O(√d log d) for m ≫ 1. Let +| ˜ψ⟩ = +1 +� +|V | +� +˜a∈Fm∗ +q +|˜a⟩|˜a · ˜x + δ˜a mod q⟩, +where δ˜a is a random variable with absolute value at +most t = poly(m∗). Then, the BV algorithm returns ˜x +with probability +1 +20tqm∗−1 . +Proof. By using Lemma 2 and Lemma 3 again, it +is straightforward, i.e., a and x is close to ˜a and ˜x with +high probability, respectively. ■ +[1] P. W. 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Comb. 141, 148 (1989). + diff --git a/gtE1T4oBgHgl3EQfMQPY/content/tmp_files/load_file.txt b/gtE1T4oBgHgl3EQfMQPY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9263aa8dc5616d57c07032c62ea7da19dc1db8d1 --- /dev/null +++ b/gtE1T4oBgHgl3EQfMQPY/content/tmp_files/load_file.txt @@ -0,0 +1,943 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf,len=942 +page_content='Sample-size-reduction of quantum states for the noisy linear problem Kabgyun Jeong∗ Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea and School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea (Dated: January 10, 2023) Quantum supremacy poses that a realistic quantum computer can perform a calculation that classical computers cannot in any reasonable amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' It has become a topic of significant research interest since the birth of the field, and it is intrinsically based on the efficient construction of quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' It has been shown that there exists an expeditious way to solve the noisy linear (or learning with errors) problems in quantum machine learning theory via a well-posed quantum sampling over pure quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In this paper, we propose an advanced method to reduce the sample size in the noisy linear structure, through a technique of randomizing quantum states, namely, ε-random technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Particularly, we show that it is possible to reduce a quantum sample size in a quantum random access memory (QRAM) to the linearithmic order, in terms of the dimensions of the input-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Thus, we achieve a shorter run-time for the noisy linear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' INTRODUCTION The Shor’s factoring algorithm [1], first proposed in 1994, states that there is an efficient way to find a prime factor x for a given integer N = a·x via quantum Fourier transform (QFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' It changes the computing paradigm from the classical to the quantum regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Despite be- ing a pivotal method across computational science, there exists a difficulty when a noise is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In 2005, Regev introduced a powerful conjecture [2] for defense against quantum attacks, known as the learning with er- rors (LWE) structure, by adding a small-sized Gaussian noise δ to the factoring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The non-trivial problem is expressed as N ′ = a·x+δ, and we call this task as the noisy linear problem (NLP) in a broad sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' This paved the way for post-quantum cryptography to emerge as a modern method of secure communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' While the most quantum algorithms including Shor’s one mainly focus on the data-processing with a well-posed quantum sample underlying a superposition principle, however it is also possible to take into account updating the initial quantum data (in a feature space), before the quantum algorithm run, for algorithmic efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The quantum data in a higher dimension are essentially different from the shape of classical big-data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', geometrically, any quantum states form a unit hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Here, we try to reveal the usefulness of initial quantum-data structure in quantum learning theory, and apply it to the specific noisy linear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Actually, famous Shannon’s noisy channel coding theorem predicts a communication is al- ways robust against a small environmental-noise [3], and it is naturally believed even in the quantum regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The noisy linear problem is fundamental in the the- ory of computer science and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Partic- ularly, modern cryptography is based on the difficulty of the noisy linear problem, and it was widely believed that it is the strongest candidate for post-quantum cryp- tography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' However, Grilo, Kerenidis, and Zijlstra have ∗ kgjeong6@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='kr recently proposed an efficient quantum algorithm, which is purely based on a quantum oracle for (partially) solv- ing the noisy linear (especially, LWE) scheme through an assumption of the well-posed construction of quantum sample [4] (including a binary strategy [5]), and we call it shortly ‘GKZ algorithm’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Specifically, if we assume that the quantum sample can be prepared in a quan- tum superposed state relating to the problem, a kind of quantum Fourier transforms (QFTs) [1], called the Bernstein-Vazirani (BV) algorithm [6] can solve the noisy linear problem within polynomial resources and running times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' However, this kind of algorithms has a precondi- tion such that there exists an efficient quantum memory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', QRAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' As pointed before by Aaronson [7], the QRAM issue is not a simple problem so far, but it seems to be a technical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In addition to post-quantum cryptography, quantum- key-distribution (QKD) protocols [8–10] form another basis for secure quantum communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' As a special case of quantum cryptographic primitives, a random uni- tary channel (RUC), also known as a randomizing map or a private quantum channel [11], including its discrete [12] and Gaussian variants [13–15] in quantum cryptographic schemes, has many crucial implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' It has perfect information-theoretic security, and can be constructed in optimal ways [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Furthermore, RUC can be directly utilized for an existence-proof for the superadditivity of the classical capacity [18, 19] through a pair of RUCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Also, it can be utilized for superdense coding scheme [20] and quantum data-hiding task [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Actually, RUC is an inverse process of QKD protocols in that two legiti- mate users with secret keys can always generate the maxi- mally mixed state, and it can be efficiently constructed by quantum randomization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' However, we cannot find any studies on quantum algorithms for its speed-up utilizing the quantum randomization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In this paper, we first suggest that quantum randomizing meth- ods (inspired by the efficient RUC-construction) can be directly applied to reduce the initial quantum sample size used in quantum learning theory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', NLP), since the ef- ficiency of the quantum randomizing technique over any arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='02988v1 [quant-ph] 8 Jan 2023 2 quantum states has been analyzed previously [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The quantum sample proposed by Grilo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' in the noisy linear problem [4] is formally given by |ψ⟩DA = 1 � qd � a∈Fdq |a⟩D|a · x + δa mod q⟩A ∈ P(Cd+1), (1) where x ∈ Fd q is fixed, a ∈ Fd q is chosen uniformly at ran- dom, and the noise term δa ∈ Fq is an independent and identically distributed (IID) random variable with a cer- tain error probability distribution [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Here, Fq denotes a finite field of order q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Specifically, a and x are decom- posed into a1a2 · · · ad and x1x2 · · · xd, respectively, and aj, xj ∈ Fq for any j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Our main goal is to recover x effi- ciently in this quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' It can be seen that P(Cd) denotes the set of all pure quantum states—a set of unit vectors on the Hilbert space Cd, and D and A denote the data and ancilla systems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' This process can be prepared by a quantum oracle function Rψ un- der a help of QRAM, mapping a set of input pure states {|aj⟩D}d j=1 and the ancilla |µ⟩A to the output |ψ⟩DA for every µ ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' For a given quantum sample |ψ⟩DA as in the superposition of pure quantum states, applying QFT (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', QFTq|a⟩ = 1 √q �q−1 b=0 ωab|b⟩ with ω = e 2πi q the root of unity) on the sample state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', QFT⊗(d+1) q |ψ⟩DA, returns the appropriate output value x with a high prob- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' This is known as the Bernstein-Vazirani (BV) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In fact, we wish to find x efficiently in this quantum learning scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In this situation, we may apply the quantum randomizing techniques on {|aj⟩D}d j=1 ⊂ P(Cd) (via the ε-net construction) over the data systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Thus, as a new data-set, we can obtain a smaller set of net points {| ˜aj⟩D}m∗ j=1 (m∗ < d), where m∗ := � O(d log d) and m is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' We notice that � O(d log d) := O(√d log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' This means that we can possibly solve the noisy linear problem more efficiently compared to previ- ous quantum approach [4] in the framework of its sample size and running-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Here, we provide that in principle we can reduce the quantum sample-size given by d to m∗ via quantum randomization techniques through a mathematical tool known as the ε-net construction [21] and L´evy’s inequal- ity [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' These methods discretize a set of pure quantum states to a finite less-sized set on a unit hypersphere, and estimate large deviations for random variables, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (See the technical lemmas in the Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=') This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' II, we briefly introduce the original GKZ algorithm for solving the noisy linear problem in which they make use of well- posed quantum superposed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' III, we pro- pose our main result for reducing the quantum sample- size via quantum randomization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' This implies that we can solve NLP more efficiently than GKZ algo- rithm in the linearithmic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' We also suggest a new type of model for QRAM, namely, approximate QRAM and analyze its performance in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Finally, discus- sions and remarks are offered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' IV, and some open questions are raised for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' ORIGINAL GKZ ALGORITHM Before the main result, we shortly review the GKZ algorithm for solving the noisy linear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' We as- sume that the errors satisfy δa = a · ⃗η, for all ⃗η = (η0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' , ηd−1)T ∈ Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The error vector ⃗η is chosen from a certain distribution χ over Fq, which is defined by a dis- crete Gaussian with a small standard deviation at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' For a given quantum sample in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (1), the QFTs return the following outcome: � QFT⊗d q ⊗ QFTq � |ψ⟩DA = 1 qd√q � a∈Fdq � b∈Fdq � c∈Fq ωa·(b+cx′)|b⟩D ⊗ |c mod q⟩A, (2) where x′ = (x + ⃗η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' By exploiting the delta function, ∆bj,−cxj := 1 q � aj∈Fq ωaj(bj+cxj) for each register j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' , d − 1}, we can obtain 1 √q � b∈Fdq � c∈Fq | − cx′⟩D ⊗ |c mod q⟩A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (3) However, x′ is not equal to the true value x, when ⃗η ̸= 0 (without the noise term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', ⃗η = 0, we can directly re- cover the hidden x by simply measuring the ancilla reg- ister A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Thus, we need to check if x′ = x is satisfied by substituting b = −cx, formally called the ‘test’ process, and by calculating the success probability as Pr [x′ = x] = 1 q2d+1 ������ � c∈Fq � a∈Fd q ωcδa| − cx⟩D ⊗ |c⟩A ������ 2 ≥ α t cos2(2πα), (4) where α ∈ [0, 1/4) and c ≤ αq t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Furthermore, we notice that the condition |a′ · x + δa′ − a′ · x′| ≤ t needs to be satisfied for every randomly chosen test sample a′ ∈ Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' 3 0101101001 10011001101 ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' 011100110011 1010100110011 Data 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='SHM6+3y9r7Di9av+stXZeLFkVSy9zGNgZKCFbtbD4giZaCMGRwodAEnYA0NCzwlsWIiIO0WPuJiQq+MCfRIm1KWoAxGbJe+bdqdZGxAe+WZaDWnUzx6Y1KaWCVNSHkxYXWaqeOpdlbsb9497anudk1/J/PyiZXoEPuXbpD5X52qR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='eIS27oGl2qKNKOq45lLqruibm5+qUqSQ0Scwi2Kx4S5Vg76bGpNomtXvWU6/qYzFav2PMtN8a5uSQO2f45zGBytV+zNin2wUdrZzUadxzJWUKZ5bmEHe6ihTt43eMQTno0L49a4M+4/U41cplnEt2U8fACmzpnA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='N(Cd) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' A model for the quantum state-preparation in an approximate quantum random access memory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', ε-QRAM): The classical datasets are prepared in sampled pure quantum states ψ ∈ P(Cd) through a random sampling at step ‘i’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' By using quantum ε-random technique, we can obtain net points ˜ψ’s on the unit-sphere at step ‘iii’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Note that the cardinality of the set of ε-net states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', m∗ = � O(d log d)) is less than that of the previously sampled pure states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' While the Grilo et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' al.’s quantum algorithm takes a step from i→ii, our algorithm makes use of i→iii→iv before the BV algorithm over QFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Alternatively, we expect that the step v→iv is also possible in the QRAM process with a quantum oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Under the condition over M trials, we complete the test accepting x′ = x, that is, the fail probability is given by Pr [x′ ̸= x] ≤ � 2t+1 q �M (see details the Lemma 1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' This statement will be also updated for our approximate scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' [4, 7], as a hard problem it was seriously pointed out that a largely superposed state must be pre- pared in the sampling process—This issue is generally called a QRAM problem [24, 25] (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Here, our approach for the noisy linear solving quan- tum algorithm aims to construct an approximate QRAM (or ε-QRAM) through an ε-net analysis as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' 1, when the initial quantum samples are all in pure quan- tum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' We discuss ε-QRAM thoroughly later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' MAIN RESULT We present the details of the quantum randomizing methods and their performance to solve the noisy linear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' As mentioned before, we should remember that all communications are not so susceptible to a noise by the Shannon’s noisy channel coding theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' First of all, let us recapitulate the mathematical no- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Let B(Cd) be the space of (bounded) linear operators on the d-dimensional Hilbert space Cd, and U(d) ⊂ B(Cd) be the unitary group on the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Note that 1d is simply the d × d identity ma- trix on the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' For simplicity, we denote a pure state ψ := |ψ⟩⟨ψ| ∈ P(Cd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In addition, we assume that a quantum channel Λ is a completely positive and trace- preserving (CPT) map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' While we make use of m = d for an input quantum sample-size, we denote m = d2 as the number of unitary operations in the case of quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Now, let us define an ε-randomizing map with respect to the Schatten p-norm in the trace class [27], which is an extended version of the statement proposed by Hayden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' [21], and it is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' For every quantum state ϱ ∈ B(Cd), we call a CPT map Λ : B(Cd) → B(Cd) an ε-randomizing map with respect to the Schatten p-norm, if ����Λ(ϱ) − 1d d ���� p ≤ ε p√ dp−1 , (5) where 1d/d denotes the d-dimensional maximally mixed state (MMS), and ∥M∥p := ��d j=1 |sj|p�1/p = (Tr(M †M)p/2)1/p is the Schatten p-norm for any matrix M ∈ B(Cd) (1 ≤ p ≤ ∞) with singular values {sj}d j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In general, CPT map Λ acting on the quantum state ϱ can be naturally constructed by Λ(ϱ) = 1 m m � j=1 UjϱU † j , (6) where the unitary operator Uj’s are chosen uniformly at random from the unitarily invariant measure (or Haar measure) on the unitary group U(d), and m is the number of unitary operators depending on the input dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' It is known that m = d2 is optimal to create a perfect d-dimensional maximally mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' However, we pro- pose that m = O(d log d) is also sufficient to create MMS in the limit of d → ∞ [21, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' This reduction will be exploited for constructing an efficient quantum samples used in the noisy linear problem, and thus we call it as ‘ε-random technique’ for quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Here, we notice that it is important to choose m sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' It is also worth noting that if dµ is the unitarily invari- ant measure chosen uniformly at random on the unitary 4 group U(d), then, for every quantum state ϱ ∈ B(Cd), it satisfies � U(d) UϱU †dµ = 1d d , ∀U ∈ U(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (7) For example, when d = 2, the above unitaries in the identity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (6) take the form of a set of Pauli ma- trices including 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The fundamental feature of the uni- tarily invariant measure in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (7) is that it assures of the uniform randomness for given any type of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' If the states are given in the form of a quantum sample, in principle the measure can be used to check that the sample is uniform or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Because most algo- rithms essentially require a random sample as its input in the regime of the classical as well as quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' This observation offers the possibility for us to construct the efficient private quantum channel Λ from m = d2 onto m = O(d log d) unitary operations under a condition of m ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (See Appendix B for the detailed proof of The- orem 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=') Consequently, it is sufficient to create RUC (or ε-randomizing map) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (6) with a randomly chosen unitary operations of O(d log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Our strategy is based on this reduction technique, and applies it to the regime of the quantum sampling problem, and the full statement of ε-randomizing map is given as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Let ε > 0 and the dimension d be suf- ficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' There exists a set of unitary operators {Uj}m j=1 ⊂ U(d) with cardinality m ≥ κ ε2 d log �10d(p−1)/p ε � (8) such that Λ(ϕ) = 1 m �m j=1 UjϕU † j on B(Cd) is ε- randomizing map with respect to the Schatten p-norm (for any p ≥ 1), and κ is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' For convenience, we exploit the notation ˜m := O(d log d)(= m∗2) instead of m as an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' From Theorem 1 in the field of quantum channel theory, we try to highlight on the quantum sample-reduction prob- lem over quantum algorithms, especially, relating to the solvability of the noisy linear structure in quantum ma- chine learnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Here, we further improve the GKZ al- gorithm in the order of linearithmic over a superposed quantum sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Here, we notice that we will take an upper bound (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', O(d log d)) rather than the exact lower bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Before presenting the main result, let us observe how quantum randomizing technique works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The intuition is basically conducted by following two lem- mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The first one is the ‘ε-net’ construction, and the second one is ‘L´evy’s inequality’ on the set of pure quan- tum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' We call, such a combination of two lemmas, as ε-random technique as mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Lemma 2 (ε-net [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Let ε > 0, and the dimension d be sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' For any pure quantum state |ψ⟩ ∈ P(Cd), we can choose a net-point | ˜ψ⟩ ∈ N(Cd) ⊂ P(Cd) satisfying ∥ψ − ˜ψ∥1 ≤ ε, where ∥ · ∥1 is the trace norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Then, there exists a set N(Cd) of pure states such that |N(Cd)| ≤ �5 ε �2d , (9) where | · | denotes the cardinality of the net N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Also, recall that ψ := |ψ⟩⟨ψ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Lemma 3 (L´evy’s inequality [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Let F be a function F : ∂Sd → R defined on the d-dimensional unit ball Sd and its boundary ∂Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Suppose that a point ψ ∈ ∂Sd is chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Then, for every ε′ > 0, Pr[|F(ψ) − E(F)| ≥ ε′] ≤ C1 exp � −C2dε′2 γ2 � , (10) where γ := sup |∇F| is the Lipschitz constant of F, and C1 and C2 are universal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Now, let us define an abstract quantum channel ˜Λ, which convert a set of pure quantum states {ψ}m j=1 to another less-sized set of pure quantum states { ˜ψ}m∗ j=1 on the net space by exploiting Lemma 2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Here, we exploit m as in the notion of quantum sample- size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' This virtual process ˜Λ can be performed by ε- QRAM (see Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' III A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Now, we fixed m = d and m∗ = � O(d log d) for accordance between RUC (Λ) and ε-QRAM (˜Λ): it changes the representation from density operator to state one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' We notice that, in the framework of the quantum sample preparation of the GKZ algo- rithm, when the initial sample may start with d → ∞, then the success probability of the algorithm can be more increased with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' 1 conceptually shows a quantum state-preparation for the NLP-solving sample in the ε-QRAM subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' By using those ingredients with ε-QRAM, we are ready to suggest our main result as follows, and the proof is essentially equivalent to the GKZ algorithm [4], where Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (1) on the original quantum-sample is substituted as a netized sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' For convenience, we also make use of the terms such as ‘Test Candidate (TC)’ and ‘NLP Algorithm (NLP-A)’ defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' [4] for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Proposition 4 (Main result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Let d be sufficiently large (thus, m∗ is) and q be a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Assuming we can efficiently prepare a superposed quantum sample through the ε-QRAM over N(Cd) ⊂ P(Cd) in the form of | ˜ψ⟩DA = 1 � qm∗ � ˜a∈Fm∗ q |˜a⟩D|˜a · ˜x + δ˜a mod q⟩A, (11) where δ˜a is a random variable chosen noise distribution with maximum noise magnitude t = poly(m∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Then, there exists a quantum algorithm that outputs ˜x (such that |˜x| < |x|) with probability 1 20tqm∗−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Let L and M be parameters in order to prove the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' By using Lemma 8 (which is the mod- ification of Lemma 1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' [4]) in Appendix C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', formally, the TC accepts with probability TC (x′, M) = � � � 1, if x′ = ˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' ≤ ( 2t+1 q )M, if x′ ̸= ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (12) By using the union bound, the probability that at least one of independent L-call to TC (x′, log η−1) accepts some x′ ̸= ˜x is at most (3t/q)M L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Also from Lemma 9 (which is the modification of Theorem 1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' [4]), the probability that x is not the output of independent L-call to BV algorithm is at most � 1 − ℓ 20tqm∗ �L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' By exploiting the union bound again, we address that NLP-A (L, M) does not output ˜x with probability at most � 1 − ℓ 20tqm∗ �L + �3t q �M L, (13) where we can choose ℓ = qm∗, L = 20t log η−1, and M = 1 for completion of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' ■ This result implies that, for every η > 0, the BV al- gorithm achieves sample complexity O(m∗ log η−1) and running-time in poly(m∗, log η−1) with probability 1 − η as in the Theorem 2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' We notice that the set of classical information ˜x (representing on the net space) is ε-close to the original information of x from the quantum ε-random technique above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' We can summarize our total procedure intuitively in terms of ε-QRAM, BV algorithm, and quantum mea- surement as follows: |a⟩D −→ |ψ⟩DA ε-QRAM −−−−−−−−→ Lemma 2,3 | ˜ψ⟩DA BV −−→ QFT⊗⌈m∗⌉+1 q | ˜ψ⟩DA Meas −−−→ ˜x ∼ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (14) The last step denotes a quantum measurement performed on the computational basis, and ˜x is ε-close to the origi- nal secret x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' 2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=') A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Approximate QRAM issue In computer architecture, the random access memory (RAM) plays a crucial role storing and calling out data in computation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' QRAMs [24, 25] are quantum analogues of classical RAMs, and they perform similar actions of reading-out and returning datasets in the form of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' However, this case is subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The quantum data can be superposed even in the form of entangled state [7] because of the following reason: The QRAM performs that QRAM : � j αj|j⟩|0⟩ �→ � j αj|j⟩|dj⟩, (15) where dj is a quantum datapoint corresponding to the memory address j, and |0⟩ an ancillary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Hence, the data set loads in quantum superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Assuming that a (quantum) big-data set, the problem could be intrigu- ing for quantum learning theory [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Unfortunately, this problem also occurs at the proposed approximate QRAMs, but our claim is that it is theoretically improv- able, when the given sets are pure quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Ge- ometrically, all pure states lie at the boundary on the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' In this case, we can concentrate a quan- tum information into a net-point on the unit sphere as discussed above, and the role of ‘ε-QRAM’ through the ε-net construction and the Levy’s inequality is described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' The reduction of information resources, starting from the origin of Shannon’s data compression theory [3], is a core ingredient in information sciences as well as in quantum computing and quantum simulation [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' While classical information datasets have a diverging hypercu- bic structure in general, they are intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Quantum datasets are geometrically simple in that they have the shape of a unit-hypersphere (see Appendix A for the details of discretization method via ε-net on a higher di- mensional unit sphere, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', on d-dimension pure quantum states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' Specifically, the ε-QRAM acts in operational ways to discretize all pure quantum states over the net space in this noisy linear problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=', it outputs a random sam- ple |˜a⟩ (instead of the total sample-size of |a⟩): � j αj|j⟩|a⟩D ⊗ |µ⟩A �→ � j αj|j⟩|˜a⟩D ⊗ |µ⟩A, (16) which in principle can be used as a quantum sam- ple to resolve the NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' From this, the quantum ora- cle gives birth to a quantum sample for solving NLP in terms of |˜a⟩D ⊗ |˜a · ˜x + δ˜a mod q⟩A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' As a compara- ble method, we can also devise and modify the bucket- brigade QRAM [24, 33, 34] in this framework of ε-QRAM structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' DISCUSSION The noisy linear problem is constituted through a clas- sical algorithm, fundamentally adding a small Gaussian noise to the meaningful original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='It is gener- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='QUANTUM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='ORACLE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='10011 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='10010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='11010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='QRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='ACznicjVHLTsJAFD3UF+ILdemkZi4agoShR3RjUtM5JEAMW0ZcEJpm+mUhBDi1h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='9wq59l/AP9C+MxWiM0du0vXPuPWfuw418HkvbfskYS8srq2vZ9dzG5tb2Tn53rxmHifBYwv9ULRdJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='2Y+D1hDcumzdiSYM3Z91nJHFyremjAR8zC4ltOI9cbOMOAD7jmSoE534gWxdwPg5t8oWjZ2kzbKpUr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='5XKVnBRZhApIrR7mn9FHyE8JBiDIYAk34eDmJ4OirAREdbDjDBHtdxhjlyxE0oi1GQ+iIvkM6dV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='I0oLPSjDXbo1t8egUxTRwRJ6Q8Qb6zdTxRCsr9DftmdZUtU3p76ZaY0Ilbgn9i7fI/C9P9SIxQEX3w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='KmnSCOqOy9VSfRUVOXml64kKUSEKb9PcUG+p5mLOZuaE+ve1WwdHX/VmQpVZy/NTfCmqQF21ZV2en ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='nOn86zZJVPLFOrsqF2nm6iwOcIhj2ucZarhEHQ098Qc84smoGxNjbtx9pBqZlLOPb2bcvwM7oJRf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQfMQPY/content/2301.02988v1.pdf'} +page_content='. +This research was supported by Army Cooperative Agreement +W911NF2120076. +et al., 2016), and deep feature extraction (Krizhevsky et al., +2017), among other innovations. However, a crucial under- +lying aspect of these developments is whether training data +is sufficiently informative. To put this in quantitative terms, +most ML training mechanisms hinge upon training samples +being independent and identically distributed (i.i.d.), which +is often violated in real-world problems, such as natural +language (Liu et al., 2021), financial markets (Heaton et al., +2016), and robotics (Gu et al., 2016), where data exhibits +temporal dependence. Reinforcement learning (RL) algo- +rithms, in particular, are limited by this constraint, as the +data is inherently Markovian, owing to the fact that RL +problem is most commonly represented mathematically as +a Markov Decision Process (MDP) (Sutton, 1988). For this +reason, as well as the numerous applications of RL in recent +years (Li, 2019), we focus on algorithms for RL methods +when data exhibits Markovian dependence. +Under the Markovian sampling setting, many convergence +analyses of iterative methods for RL exist (Qiu et al., 2021b; +Xu et al., 2020b) and typically consider a critical assump- +tion about the rate at which the MDP’s transition dynamics +converge to stationary distribution for a fixed policy. To +establish the analysis, restrictions are placed on mixing time +(τmix): (1) prior oracle knowledge of mixing time is em- +ployed to determine an optimal step-size selection, as in +(Duchi et al., 2012; Nagaraj et al., 2020); or (2) mixing time +decays exponentially fast, such that the data is asymptoti- +cally i.i.d. (Qiu et al., 2021a). In this work, we are interested +in developing RL algorithms with performance certificates +without the aforementioned conditions. +For instance, consider an RL problem where the agent must +navigate through a continuous state space, such as a robot +reaching a target location or a self-driving car traversing +a complex road network. In these cases, the transition dy- +namics can be highly non-linear with sparse rewards, and +the agent may have to explore many states before locating +any rewards. In addition, if the environment’s dynamics are +highly random or there are many obstacles and the agent can +get stuck in certain states for a long time, the total variation +distance to the steady state decreases slowly, i.e., the mixing +rate is slow and hence have a large mixing time. These +issues often manifest in stationary MDPs that are simply +arXiv:2301.12083v1 [cs.LG] 28 Jan 2023 + +Multi-Level Monte Carlo Actor-Critic (MAC) +2 +Table 1. This table compares the total sample complexity of actor-critic (AC) algorithms available in the literature. To our knowledge, this +is the first AC algorithm with an explicit optimal dependence on the underlying mixing time defined as τmix := maxt∈[T ] τ θt +mix where θ +is the policy parameter (see Sec. 4) for detailed discussion). We also remark that our proposed approach is oblivious to mixing time. +References +Sampling +Total complexity +Reward +Fast mixing +Actor +Critic +(Wang et al., 2019) +i.i.d. +i.i.d. +O(ϵ−4) +Discounted +Required +(Kumar et al., 2019) +i.i.d. +i.i.d. +O(ϵ−4) +Discounted +Required +(Qiu et al., 2021a) +i.i.d. +Markovian +˜O(ϵ−3) +Average +Required +(Xu et al., 2020b) +Markovian +Markovian +˜O(ϵ−2) +Average +Required +(Wu et al., 2020) +Markovian +Markovian +˜O(ϵ−2.5) +Average +Required +(Chen & Zhao, 2022) +Markovian +Markovian +˜O(ϵ−2) +Average +Required +This work +Markovian +Markovian +˜O(τ 2 +mix · ϵ−2) +Average +Not required +weakly connected by a few distinct regions, which could +be defined, e.g., by seasonality in data or distinct learning +“tasks” comprised of similar states and sub-goals as detailed +in Riemer et al. (2021). In summary, many RL environments +exhibit a slower than exponential mixing rate due to high +dimensionality, intrinsic volatility, sparse rewards, or that +they contain distinct sub-tasks. +We seek RL methodologies attuned to environments that mix +slowly, especially in the context of actor-critic (AC), due to +the fact that it underlies much of modern deep RL (Konda +& Tsitsiklis, 1999). As previously noted, existing results (cf. +Table 1) hinge upon either i.i.d. (Kumar et al., 2019) or ex- +ponentially fast mixing (Qiu et al., 2021a;b). We, therefore, +aim to come up with a variant of actor-critic that does not +possess these limitations. To do so, inspired by Dorfman +& Levy (2022), we develop a multi-level Monte Carlo gra- +dient estimator and adaptive learning rate for the average +reward, actor, and critic, called Multi-level Monte Carlo +Actor-Critic (MAC). We compare the sample complexity of +different methods in Table 1. Our main contributions are: +• We develop a variant of multi-level Monte Carlo for +the average reward, policy gradient, and temporal dif- +ference estimates, which together comprise Multi-level +Monte Carlo Actor-Critic (MAC) algorithm. +• We establish the convergence rate dependence of the +proposed MAC algorithm on the mixing time without +any assumption on its decay rate, which is alleviates +prior exponentially fast mixing conditions. +• Despite the two-timescale nature of MAC, our use of +a modified Adagrad stepsize in the actor allows us to +obtain final sample complexity of ˜O(ϵ−2), instead of +the ˜O(ϵ−2.5) of previous two-timescale analyses. +• We perform initial proof of concept experiments and +observe that MAC outperforms vanilla actor-critic for +settings with sparse rewards. +1.1. Related Works +We provide a brief overview of the related works here. +Please refer to Appendix A for a detailed context. +TD Learning. For discounted TD with Markovian samples, +Bhandari et al. (2018) established finite-time convergence +bounds which scale linearly with mixing time τmix. Dorf- +man & Levy (2022) then improved the rate to be propor- +tional to the √τmix using a multi-level gradient estimator +and adaptive learning rate. Qiu et al. (2021a) studied TD +under the average reward setting, which also imposes expo- +nentially fast mixing that manifests in an additional logarith- +mic term in the sample complexity. These results all hinge +upon imposing restrictive conditions on mixing time. +Policy Gradient. More recently, its sample complexity +has been established for a variety of settings: for tabular +(Bhandari & Russo, 2019; Agarwal et al., 2020) and softmax +policies (Mei et al., 2020), rates to global optimality exist. +For general parameterized policies, early works focused on +“policy improvement” bounds (Pirotta et al., 2013; 2015), +and more recently, rates towards stationarity (Bedi et al., +2022) and local extrema (Zhang et al., 2020) have been +studied, and under special neural architectures, globally +optimal solutions (Wang et al., 2019; Leahy et al., 2022) are +achievable. This topic is an active area of work – we merely +identify that these performance certificates all require the +mixing rate going to null exponentially fast. +Actor-Critic. As previously mentioned, the stability of +actor-critic was initially focused on asymptotics (Borkar & +Konda, 1997). More recently, its non-asymptotic rate has +been derived under i.i.d. assumptions (Kumar et al., 2019; +Wang et al., 2019), and more recently under a variety of +different types of Markovian data – see Table 1. However, +these results impose that any temporal correlation of data +across time vanishes exponentially fast as quantified by the +mixing rate. In this way, we are able to match (Chen & +Zhao, 2022) but without these restrictions. + +Multi-Level Monte Carlo Actor-Critic (MAC) +3 +2. Problem Formulation +We consider a reinforcement learning problem with an av- +erage reward criterion, which can be mathematically de- +fined as a Markov Decision Process (MDP), i.e., a tuple +M := (S, A, p, r). Here, S is a finite state space; A is a fi- +nite action space; p(· | s, a) is a distribution that determines +transition to the next state s′, and r : S ×A → [0, rmax] is a +bounded reward function that informs the merit of selecting +action a when starting in state s. A policy π(· | s) of an MDP +maps the state s to the probability distribution over actions a. +Formally, π : S → △(A), where △(A) is the set of prob- +ability distributions over A. In the average reward setting, +we seek to find a policy π such that the long-term average +reward is given by J(π) := limT →∞ E +� +1 +T +�T +t=0 r(st, at) +� +is maximized. In practice, when the state space is large, it +is difficult to search over a general class of policies since +its parameterization scales with |S|. Therefore, we restrict +focus to the case that π is parameterized by a vector θ ∈ Rd, +where d denotes the parameter dimension, which leads to +the notion of a parameterized policy πθ. Optimizing the +average reward with respect to policy parameters θ is the +main goal of this work, which we formalize as: +max +θ +J(θ) := lim +T →∞ Est+1∼p(·|st,at),at∼πθ(·|st) [RT ] , (1) +where RT := 1 +T +�T +t=0 r(st, at). Denote as dπθ the unique +stationary state distribution induced by policy πθ. Then +we can also write J(θ) = Es∼dπθ ,a∼πθ[r(s, a)]. It turns to +be essential to further algorithm development to define the +action-value (Q) function as +Qπθ(s, a) =E +� ∞ +� +t=0 +[r(st, at) − J(θ)] +� +, +(2) +such that s0 = s, a0 = a, and action a ∼ πθ. This implies +that we can write the state value function as +V πθ(s) =Ea∼πθ(·|s)[Qπθ(s, a)]. +(3) +From (2) and (3), we can write the value of a state s, in terms +of another via Bellman’s Equation as (Puterman, 2014) +V πθ(s) = E[r(s, a) − J(θ) + V πθ(s′)], +(4) +where the expectation is over a ∼ πθ(·|s), s′ ∼ p(·|a, s). +Next, we shift to defining the standard actor-critic frame- +work to solve (1), in order to illuminate its merits and draw- +backs. +2.1. Decay Rates of Mixing Times +It is inherent to RL that the data-generating mechanism is +state-dependent and Markovian, which means that assump- +tions that trajectory data is independent and identically dis- +tributed do not hold (Wang et al., 2019; Kumar et al., 2019; +Qiu et al., 2021b). That is, the noise driving the estimation +error of the algorithm updates is heteroscedastic (variance is +heterogeneous). Because of this challenge, various technical +conditions have been considered to quantify the degree of +correlation in data across time, mostly inherited from the +applied probability literature – see (Levin & Peres, 2017). +Most prior stability and sample complexity results of RL al- +gorithms for the average reward setting are defined in terms +of the mixing time, which is the minimum time at which +the transition dynamics are near the long-term steady-state +distribution induced by a policy πθ, as formalized next. +Definition 2.1 (ϵ-Mixing Time). Let dπθ denote the station- +ary distribution of the Markov chain induced by πθ. Define +Pθ(s′|s) = +� +A p(s′|s, a)πθ(a|s)da. The ϵ-mixing time of +the Markov chain induced by πθ is defined as +τ θ +mix(ϵ) := inf{t : sup +s∈S +∥P t +θ(·|s) − dπθ(·)∥T V ≤ ϵ}, (5) +where ∥ · ∥T V is the total variation distance. The conven- +tional mixing time is defined as τ θ +mix := τ θ +mix(1/4). +Limitations. In all of the earlier works mentioned in Table +1, a crucial and common assumption is regarding the expo- +nentially fast decay rate of the mixing time. Specifically, all +the works assume that there exist ζ > 0 and ρ ∈ (0, 1) such +that, for all θ, it holds that sups∈S ∥P t +θ(·|s)−dπθ∥T V ≤ ζρt. +This stipulates that exponentially fast mixing must hold uni- +formly for all induced Markov chains. Also, to proceed with +the convergence analysis in the works mentioned in Table 1, +knowledge of ζ and ρ is required for the optimal step size +selection, which is usually unknown in practice. Moreover, +there is a wide range of applications where polynomial de- +cay rates have some fundamental role to play in defining +RL algorithms that can generalize well across tasks - see +(Riemer et al., 2021) for a detailed description. +Therefore, in this work, we are interested in going beyond +the exponentially mixing requirements and seek to develop +actor-critic algorithms which do not require access to mixing +time values a priori for optimal performance. We present +our proposed algorithm in the next section. +3. Actor-Critic Method +3.1. Elements of Actor-Critic +We start by providing a quick recap of the standard actor- +critic (AC) algorithm in average reward RL settings. The +AC algorithm operates by alternating updates between the +actor and critic, which are respectively defined in terms +of gradient updates to policy parameters θ and estimates +of the value function V πθ(s) based on the fixed point re- +cursion implied by Bellman’s equation (4). To do so, we +proceed by writing down a gradient ascent iteration for the + +Multi-Level Monte Carlo Actor-Critic (MAC) +4 +maximization in (1) given by +θt+1 = θt + αt∇θJ(θt), +(6) +where αt is the step size. From the Policy Gradient (PG) +Theorem (Williams, 1992; Sutton et al., 1999), it is well- +known that ∇θJ(θt) takes the explicit form: +∇θJ(θ) = E(s,a,s′)∼Γθ [δπθ · ∇θ log πθ(a|s)] , +(7) +with the temporal difference (TD) δπθ defined as (Sutton, +1988): +δπθ := r(s, a) − J(θ) + V πθ(s′) − V πθ(s), +(8) +and Γθ := s ∼ dπθ, a ∼ πθ, s′ ∼ p(·|s, a) is the short +notation for the joint distribution. We note that there are +two parts in the expression of PG in (7): ∇θ log πθ(a|s), the +score function which comes from the policy parameteriza- +tion. The TD term δπθ is defined in terms of rearranging +the V πθ(s) term in (4) to the other side of the expression, +and group expectations. Observe that the differential value +function V πθ(s′) − V πθ(s) distinguishes the PG (7) in the +average-reward case different from the discounted setting. +Critic update: We restrict focus to the case where the +value function V πθ(s) is estimated by the inner product +between a given feature map φ(s) and a weight vector ω, +which can be shown to be exact under some special cases +such as linear MDP where the assumption of realizability +is met (Tsitsiklis & Van Roy, 1997; Bhandari et al., 2018; +Dorfman & Levy, 2022; Qiu et al., 2021a). Hence, we can +write Vω(s) = ⟨φ(s), ω⟩ where Vω(s) denotes the estimator +to V πθ(s) in terms of parameters ω ∈ Rm and feature map +φ : S → Rm of state s to m-dimensional space such that +∥φ(s)∥ ≤ 1 for all s ∈ S. TD learning-style updates are +then used to find ω, which minimizes the error G(ω) defined +as +min +ω∈Ω G(ω) := +� +s∈S +dπθ(V πθ(s) − Vω(s))2. +(9) +The TD(0) update for the critic parameter ω is given as +ωt+1 =ΠΩ +� +ωt − βt +� +r(st, at) − J(θt) + ⟨φ(st+1), ωt⟩ +− ⟨φ(st), ωt⟩ +� +φ(st) +� +, +(10) +where βt is the critic learning rate. We remark that the critic +update in (11) requires knowledge of J(θt) (time-averaged +reward), which is typically not available. We can replace this +unknown quantity with a recursive estimate for the average +reward given by ηt+1 = ηt −γt(ηt −r(st, at)). Putting this +all together, we can write the vanilla actor-critic scheme as +ηt+1 =ηt − γt · ft +(reward tracking) +ωt+1 =ΠΩ +� +ωt − βt · gt +� +, +(critic update) +θt+1 =θt + ηt · δπθt · ht, +(actor update) +(11) +Algorithm 1 Multi-level Monte Carlo Actor-Critic (MAC) +1: Initialize: Policy parameter θ0, actor step size αt, critic +step size βt, average reward tracking step size γt, initial +state s(0) +1 +∼ µ0(·), maximum rollout length Tmax. +2: for t = 0 to T − 1 do +3: +Sample level length jt ∼ Geom(1/2) +4: +for i = 1, . . . , 2jt do +5: +Take action ai +t ∼ πθt(·|si +t) +6: +Collect next state si+1 +t +∼ P(·|si +t, ai +t) +7: +Receive reward ri +t = r(si +t, ai +t) +8: +end for +9: +Compute MLMC gradients f MLMC +t +, +gMLMC +t +, +hMLMC +t +via (13)-(16) +10: +Update parameters as +ηt+1 =ηt − γt · f MLMC +t +(reward tracking) +ωt+1 =ΠΩ +� +ωt − βt · gMLMC +t +� +, +(critic update) +θt+1 =θt + ηt · δπθt · hMLMC +t +, +(actor update) +11: end for +where we have +ft =ηt − r(st, at), +gt = +� +r(st, at) − ηt + ⟨φ(st+1) − φ(st), ωt⟩ +� +φ(st), +ht =δπθt · ∇θ log πθt(at|st), +δπθt =r(st, at) − ηt + ⟨φ(st+1) − φ(st), ωt⟩. +(12) +As previously mentioned, the stability of (11)-(12) can only +be ensured under the exponentially fast mixing condition, +which can preclude sparse-reward or large state space cases. +For this reason, we develop an augmentation of actor-critic +that alleviates this restriction in the following subsection. +3.2. Multi-level Monte Carlo Actor-Critic +Recent work of Dorfman & Levy (2022) has developed the +use of Multi-level Monte Carlo techniques together with +AdaGrad step-size selection to develop a gradient estima- +tor for Markovian data in stochastic optimization settings. +We build upon these techniques in putting forth an MLMC +gradient estimator for the actor, critic, and reward tracking. +In doing so, we also allow for the sampling distribution +for the critic to be Markovian. Specifically, we propose to +replace the stochastic gradients ft, gt, and ht in (11) with +the following MLMC gradients. Letting Jt ∼ Geom(1/2) +and fixing a maximum number Tmax of samples, we collect +a trajectory Tt := {si +t, ai +t, ri +t, si+1 +t +}2Jt +i=1 for each t by inter- +acting with the environment using policy parameter vector +θt. For the policy gradient estimate, for example, we then + +Multi-Level Monte Carlo Actor-Critic (MAC) +5 +construct the MLMC estimate +hMLMC +t += h0 +t + +� +2Jt(hJt +t − hJt−1 +t +), +if 2Jt ≤ Tmax +0, +otherwise +(13) +with hj +t = +1 +2j +�2j +i=1 h(θt; si +t, ai +t) aggregating 2j gradients: +h(θt; si +t, ai +t) = δ +πθt +i +· ∇θ log πθt(ai +t|si +t), +(14) +δ +πθt +i += r(si +t, ai +t) − ηt + ⟨φ(si +t+1) − φ(si +t), ωt⟩. +We can formulate estimates analogous to (13) for the reward +tracking gradient f MLMC +t +and critic gradient gMLMC +t +by +using corresponding versions of (14): +f(ηt; si +t, ai +t) = r(si +t, ai +t) − ηt, +(15) +g(ωt; si +t, ai +t) = δ +πθt +i +· φ(si +t). +(16) +The multi-level gradient in (13) is different from the one +in (11) where we only need one sample (st, at, st+1) to +evaluate the actor and critic updates. Overall, the proposed +multi-level Monte Carlo actor-critic (MAC) takes the form +ηt+1 =ηt − γt · f MLMC +t +(reward tracking) +ωt+1 =ΠΩ +� +ωt − βt · gMLMC +t +� +, +(critic update) +θt+1 =θt + ηt · δπθt · hMLMC +t +, +(actor update) +(17) +We summarize the proposed algorithm in Algorithm 1. +4. Non-asymptotic Convergence Analysis +In this section we provide convergence rate and sample +complexity results for Algorithm 1. We extend the MLMC +analysis of Dorfman & Levy (2022) to the actor-critic set- +ting, where we combine it with the two-timescale finite-time +analysis of Wu et al. (2020) to obtain non-asymptotic con- +vergence guarantees for MAC (cf. Algorithm 1). Salient +features of our approach: (1) it avoids uniform ergodicity +assumptions required in previous finite-time analyses (Zou +et al., 2019; Wu et al., 2020; Chen & Zhao, 2022); (2) it +explicitly characterizes convergence rate dependence on the +mixing times encountered during training; (3) it (i) clarifies +the trade-offs between mixing times and MLMC rollout +length Tmax, and (ii) extends the standard analysis to handle +additional sources of bias in the MLMC estimator, both of +which were missing from the analysis of Dorfman & Levy +(2022); (4) it leverages modified Adagrad stepsizes to avoid +the slower convergence rates of previous two-timescale anal- +yses (Wu et al., 2020) (cf. Theorem 4.8). +The rest of this section is structured as follows. We first out- +line standard assumptions (cf. Sec. 4.1) from the literature +and provide some preliminary results. Second, we analyze +the policy gradient norm (cf. Sec. 4.2) associated with Al- +gorithm 1, which provides a preliminary convergence rate +and characterizes its dependence on the error arising from +the critic estimation procedure, the MLMC bias resulting +from the choice of Tmax and mixing times encountered, and +the bias inherent in using function approximation for the +critic. Third, we analyze the convergence (cf. Sec. 4.3) +of the critic estimation error, characterizing its dependence +on the MLMC bias and its convergence rate. Finally, we +combine the actor and critic analyses to provide our main +convergence rate and sample complexity (cf. Theorem 4.8) +results for MAC. To keep the exposition clear, we provide +simplified versions of our main results and omit proofs in +this section. Mathematically precise statements and detailed +proofs of all results are presented in the appendix. +4.1. Assumptions and Propositions +The algorithmic setting considered in this paper is that of +actor-critic with linear function approximation, where the +critic updates correspond to using TD(0) (Sutton & Barto, +2018) to estimate the state value function. Specifically, we +assume that, for a given critic parameter ω ∈ Rk and state +s, our critic approximator is of the form Vω(s) = φ(s)T ω +[cf. (9), where φ : S → Rk is a given feature mapping that +we assume satisfies sups ∥φ(s)∥ ≤ 1. +As discussed in Ch. 9 of (Sutton & Barto, 2018), for a fixed +policy parameter θ, TD(0) with linear function approxima- +tion will converge to the minimum of the mean squared +projected Bellman error (MSPBE), which satisfies +Aθω = bθ, +(18) +Aθ = Es∼µθ,a∼πθ,s′∼p(·|s,a) +� +φ(s)(φ(s) − φ(s′))T � +, +bθ = Es∼µθ,a∼πθ [(r(s, a) − J(θ))φ(s)] . +In what follows, we will use ω∗(θ) to denote the fixed +point satisfying Eq. (18) for a given θ. We will also use +ω∗ +t = ω∗(θt) to denote the fixed point associated with policy +parameter vector θt at time t. For a given feature mapping +φ, we define the worst-case approximation error to be +Eapp = sup +θ +� +Es∼µθ [φ(s)T ω∗(θ) − V πθ(s)]2, +(19) +which we assume to be finite. Intuitively, Eapp quantifies +the quality of the feature mapping: when the features are +well-designed, Eapp will be small or even 0, while poorly +designed features will tend to have higher worst-case error. +Analyses of TD learning typically assume positive definite- +ness of the matrices Aθ to ensure the solvability of the +MSPBE minimization problem and uniqueness of its so- +lutions (Bhandari et al., 2018; Zou et al., 2019; Qiu et al., +2021b), which we subsequently impose via Assumption 4.1. +Assumption 4.1. There exist λ > 0 such that, for all θ, +the matrix Aθ is positive definite, its eigenvalues are all +bounded and have norm greater than or equal to λ. + +Multi-Level Monte Carlo Actor-Critic (MAC) +6 +As indicated in our description of the algorithm in the previ- +ous section, we execute a projection onto a norm-ball with +radius Rω > 0, denoted by set Ω, in our critic update step +[cf. (17)]. As mentioned in (Wu et al., 2020), given Assump- +tion 4.1, we can simply take Rω = 2R/λ, since ∥bθ∥ ≤ 2R +by the boundedness of rewards, and ∥A−1 +θ ∥ ≤ 1/λ. +In order to establish an ascent-type condition on the policy +gradient, we require some regularity conditions which have +been considered in recent analyses of model-free RL meth- +ods (Papini et al., 2018; Kumar et al., 2019; Zhang et al., +2020; Xu et al., 2020a), as detailed next. +Assumption 4.2. Let {πθ}θ∈Rd denote our parameterized +policy class. There exist B, K, L > 0 such that +1. ∥∇ log πθ(a|s)∥ ≤ B, for all θ ∈ Rd, +2. ∥∇ log πθ(a|s) − ∇ log πθ′(a|s)∥ ≤ K∥θ − θ′∥, for +all θ, θ′ ∈ Rd, +3. |πθ(a|s) − πθ′(a|s)| ≤ L∥θ − θ′∥, for all θ, θ′ ∈ Rd. +Finally, for our last major assumption we impose a con- +dition on the ergodicity coefficients of the family of state +transition kernels {Pθ} induced by the policy class {πθ}, +where Pθ(s′|s) = +� +A πθ(a|s)p(s′|s, a)da. +For a fixed +transition kernel P, defined its ergodicity coefficient to +be κ(P) := sups,s′ ∥P(·|s) − P(·|s′)∥T V (Mitrophanov, +2005). Furthermore, for a given k ∈ N and fixed P, let P k +denote the induced k-step transition kernel. +Assumption 4.3. For every θ, there exists k ∈ N such that +the ergodicity coefficient κ(P k +θ ) satisfies κ(P k +θ ) < 1. +In prior works, related quantities are assumed to go to null +exponentially fast (uniform ergodicity) in finite-time anal- +yses of average-reward actor-critic (Wu et al., 2020; Qiu +et al., 2021b; Chen & Zhao, 2022) and related RL meth- +ods (Melo et al., 2008; Bhandari et al., 2018; Zou et al., +2019) (Theorem 3.1 of (Mitrophanov, 2005) establishes a +correspondence). In our case, we merely require it to be +upper-bounded by a constant, meaning that the degree of +non-stationarity of the transition dynamics cannot be arbi- +trarily large, and at worst has bounded drift with time. This +allows us to better accomodate large state spaces comprised +of distinct regions, which may be defined by seasonality. +We are now ready to provide two important propositions +that will be important in the core analysis to follow. +Proposition 4.4. Under Assumptions 4.1-4.3, there exists +Lω > 0 s.t. ∥ω∗(θ) − ω∗(θ′)∥ ≤ Lω∥θ − θ′∥, for all θ, θ′. +Please refer Lemma D.2 in the appendix for the proof of +Proposition 4.4. The next proposition is a generalization of +Lemma 3.1 from Dorfman & Levy (2022), adapted to our +actor-critic setting, that explicitly characterizes the compu- +tational cost associated with MLMC rollout length Tmax. +Before stating our main results, we first establish a result +characterizing the mean and variance of the MLMC gradient +estimators f MLMC +t +, gMLMC +t +, hMLMC +t +used in the MAC +updates defined in (17). Since the core result is the same for +all three estimators, we formulate and derive the result for +a general MLMC estimator lMLMC +t +. We note that lMLMC +t +can be replaced by any one of f MLMC +t +, gMLMC +t +, hMLMC +t +and the result will hold. To prepare to state the result, let a +policy parameter θt be given and sample Jt ∼ Geom(1/2). +Fix Tmax ∈ N such that Tmax ≥ τ θt +mix. Fix a trajectory +zt = {zi +t = (si +t, ai +t, ri +t, si+1 +t +)}i∈[2Jt] generated by follow- +ing policy πθt starting from s0 +t ∼ µ0(·). Let ∇L(x) := +Ez∼µθt,πθt [l(x, z)] be a gradient that we wish to estimate +over zt where x ∈ K ⊂ Rk is the parameter of the estimator +l, e.g., x could be xt = θt, ηt, or ωt. hence, the MLMC +estimator (cf. (13)) becomes +lMLMC +t += l0 +t + +� +2Jt(lJt +t − lJt−1 +t +), +if 2Jt ≥ Tmax, +0, +otherwise. +(20) +We are ready to present our result for the MLMC estimator +in Proposition 4.5. +Proposition 4.5. Let jmax = ⌊log Tmax⌋. Fix xt measur- +able w.r.t. Ft−1. Assume ∥∇L(x)∥ ≤ GL, for all x ∈ K, +and ∥lN +t ∥ ≤ GL, for all N ∈ [Tmax]. Then +Et−1 +� +lMLMC +t +� += Et−1 +� +ljmax +t +� +, +(21) +E +� +∥lMLMC +t +∥ +2� +≤ � +O +� +G2 +Lτ θt +mix log Tmax +� +. +(22) +We provide the proof of Proposition 4.5 with a detailed +description of the statement in Lemma B.3 in the appendix. +Remark. We note that the corresponding result in Dorfman +& Levy (2022) hides the logarithmic dependence of the +second moment bound (22) on the MLMC rollout length +Tmax, subsuming it into the � +O (·) order notation. When +Tmax is allowed to grow with time, e.g., by setting Tmax = +T as in Dorfman & Levy (2022), the true impact of using +MLMC is not accurately accounted for. Furthermore, a finite +value for Tmax must be used in practice, so it is important +to understand its true effect. We rigorously characterize its +effect with Proposition 4.5. +In addition, Proposition 4.5, its precursor results (see Lem- +mas B.1, B.2 in appendix), and our extensions of it (see Lem- +mas C.1, D.3, C.2, D.4 in appendix) are the critical tools that +allow us to smoothly accommodate Markovian sampling +and reveal the dependence on mixing times encountered in +the analysis. Equation (21) is used at many points in the +analysis to tie the behavior of our MLMC estimates to that +of the lower-bias estimators f jmax +t +, gjmax +t +, hjmax +t +, while equa- +tion (22) renders the dependence on log Tmax and mixing +time explicitly, and allows us to avoid uniform ergodicity + +Multi-Level Monte Carlo Actor-Critic (MAC) +7 +assumptions. These innovations allow us to derive the im- +proved actor and critic convergence analyses presented next. +4.2. Convergence of the Actor +In this section, we take the first step towards establishing +convergence of Algorithm 1 by providing a bound on the +average policy gradient norm. This result explicitly char- +acterizes the actor convergence in terms of its dependence +on the average reward tracking and critic estimation error, +mixing times encountered during training, MLMC rollout +length Tmax, and the function approximation bias Eapp. We +present our first main result in Theorem 4.6. +Theorem 4.6. Assume J(θ) is L-smooth, supθ |J(θ)| ≤ M, +and ∥∇J(θ)∥, ∥hMLMC +t +∥ ≤ GH, for all θ, t. Let αt = +α′ +t/ +��T +t=1 ∥hMLMC +t +∥ +2, where {α′ +t} is an auxiliary step- +size sequence with α′ +t ≤ 1, for all t ≥ 1. Then +1 +T +T +� +t=1 +E +� +∥∇J(θt)∥2� +≤ O +� 1 +√ +T +� ++ O +� +1 +T +T +� +t=1 +E(t) +� ++ � +O +� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ O (Eapp) , +(23) +where E(t) = E +� +∥ηt − η∗ +t ∥2� ++ E +� +∥ωt − ω∗ +t ∥2� +. +We provide a more detailed statement of Theorem 4.6 and +a complete proof in Theorem C.4 in the appendix. In ad- +dition to the O +� +T −1/2� +term and the inherent O (Eapp) +bias term, this bound depends on the average value of +the critic error via E(t) and Markovian sampling through +maxt∈[T ] τ θt +mix +log Tmax +Tmax . As we will see in Theorem 4.7 in +the following subsection, the E(t) term dies to 0 at a favor- +able rate. The presence of the Markovian sampling term, +however, marks the point where our work departs signifi- +cantly from previous work. +Remark. Interestingly, we note that the right-hand side of +(23) no longer depends upon the step size rate as in Wu et al. +(2020, Theorem 4.5) due to the use of our modified Adagrad +stepsize in the actor update. This allows us to derive an +improved overall sample complexity in Theorem 4.8. +An important consequence of Theorem 4.6 is that the level +of bias resulting from Markov sampling can be controlled +by choosing Tmax appropriately. When the maximum mix- +ing time likely to be encountered during training – captured +here by the term maxt∈[T ] τ θt +mix, is small – it makes sense +to choose Tmax to be relatively small as well. When mixing +times are long, on the other hand, choosing Tmax accord- +ingly keeps the Markovian sampling bias manageable. +4.3. Convergence of the Critic +We turn next to characterizing the convergence of the critic +error term arising in bound (4.6) of Theorem 4.6. Sim- +ilar to that theorem, the resulting bound expresses critic +convergence in terms of mixing times encountered during +training as well as MLMC rollout length Tmax. This result +is also where our actor-critic scheme explicitly becomes +two-timescale due to our choice of stepsize sequences. +Theorem +4.7. +Assume +γt += +(1 + t)−ν, α += +α′ +t/ +��t +k=1 ∥hMLMC +t +∥ +2, and α′ +t = (1 + t)−σ, where +0 < ν < σ < 1. Then +1 +T +T +� +t=1 +E(t) ≤O +� +T ν−1� ++ O +� +T −2(σ−ν)� ++ � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +O +� +T −ν� ++ � +O +� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +. +(24) +For the proof of Theorem 4.7, refer to Theorems D.1 and +D.5) in the appendix. Unlike the actor bound (4.6), the only +term in (a) that does not diminish with T is the Markovian +sampling term containing maxt∈[T ] τ θt +mix +log Tmax +Tmax . As in +the actor case, this bias can be controlled via the proper +selection of Tmax. As we will see in the final result of this +section, this Markovian sampling term will ultimately be +absorbed into the analogous term from Theorem 4.6. +4.4. Convergence Rate and Sample Complexity +We now present our main result characterizing the conver- +gence rate of Algorithm 1 in terms of only the total number +of iterations, mixing times encountered and Tmax used dur- +ing training, and the function approximation bias Eapp. We +present the result in Theorem 4.8 next, which follows di- +rectly from Theorems 4.6 and 4.7. +Theorem 4.8. (Convergence Rate) Under the assumptions +of Theorems 4.6 and 4.7 and with selection σ = 0.75 and +ν = 0.5, we have +1 +T +T +� +t=1 +E +� +∥∇J(θt)∥2� +≤O (Eapp) + � +O +�τmix log Tmax +√ +T +� ++ � +O +�τmix log Tmax +Tmax +� +, +(25) +where τmix := maxt∈[T ] τ θt +mix. +The proof of Theorem 4.8 is provided in Appendix E. The +result in Theorem 4.8 provides an explicit dependence of +the final convergence rate on the maximum mixing time +τmix encountered during training as well as rollout length +Tmax. The first term is O (Eapp) on the right-hand side +of (25) is unavoidable due to the use of linear function +approximation for the critic, but can be kept small or even +driven to zero with appropriate feature selection. The second +term shows the dependence on the mixing rate and shows +that we recover the original iid rates if τmix = 1. The last + +Multi-Level Monte Carlo Actor-Critic (MAC) +8 +(a) +(b) +Figure 1. (a) Mean Rewards over 3 million samples with Tmax = 8 for MAC and rollout = 3 for Vanilla AC with 6 × 6 grid. (b) Mean +Rewards over 4 million samples with Tmax = 16 for MAC and rollout = 4 for Vanilla AC with 10 × 10 grid. +term on the right-hand side of (25) is interesting because +that is the final bias we are incurring due to the use of finite +length rollout trajectories Tmax. If we make Tmax = T +as in Dorfman & Levy (2022), we will recover the rate of +O (Eapp) + � +O +� +τmix log Tmax +√ +T +� +. +We present the sample complexity result next. +Corollary 4.9. Let us consider Tmax = +√ +T and Eapp ≤ ϵ. +Absorbing the logarithmic terms in the ˜O notation, it holds +that to achieve min1≤t≤T E +� +∥∇J(θt)∥2� +≤ ϵ, we need +T ≥ ˜O +� +τ 2 +mix +ϵ2 +� +. +The proof of Corollary 4.9 follows directly from the state- +ment of Theorem 4.8. We remark that, even for fast mixing +settings where we can ignore the dependence on τ 2 +mix in +Corollary 4.9, our proposed algorithm achieves sample com- +plexity ˜O +� 1 +ϵ2 +� +, which improves upon the state of the art +result of ˜O +� +1 +ϵ2.5 +� +in Wu et al. (2020). This improvement is +due to the use of Adagrad step size in the actor update. +Remark. It is interesting to note that the analysis pre- +sented in this section recovers results for the simplified +i.i.d. sampling setting: since mixing occurs immediately, +maxt∈[T ] τ θt +mix = 1, so we can simply choose Tmax = 1. +At the other extreme, when mixing is very slow we intu- +itively expect that single- or few-sample estimates of the +policy gradient like those considered in (Wu et al., 2020; +Xu et al., 2020b; Qiu et al., 2021b; Chen & Zhao, 2022) +will be highly inaccurate due to the failure of the fast mix- +ing condition of Assumption 4.2 of (Wu et al., 2020) and +Assumption 2 of (Xu et al., 2020b), for example, making +a larger number of samples imperative. Theorems 4.6, 4.7, +and 4.8 are the first results to shed light on this trade-off. +5. Experiments +In this section, we perform preliminary proof of concept +experiments to evaluate the performance of the proposed +MAC algorithm and compare it against the vanilla actor- +critic. While we concede that numerous enhancements to +actor-critic have been considered, based on Nesterov accel- +eration (Kumar et al., 2019), parallelization (Asynchronous +Advantage Actor-Critic (Mnih et al., 2016)), and offline +processing of prior trajectory information (Soft Actor-Critic +(Haarnoja et al., 2018)), our focus is on revealing the ex- +perimental dependence of actor-critic’s stability on the envi- +ronment’s mixing time. Therefore, for carefully controlled +experimentation, we only compare against Vanilla actor- +critic as detailed in Sec. 3.1. We consider an n×n grid with +a starting position at the top left and a goal at the bottom +right. There are five actions: stay, up, down, left, and right. +An action that results in the goal state gives the agent a +1 +reward and +0 for all other states. In Figure 1, we report +algorithm performance in terms of mean reward returns over +5 trials with 95% confidence intervals. +We compare MAC against Vanilla AC with a standard gradi- +ent estimator. In practice, we use a constant learning rate for +the actor, critic, and reward estimation. For comparison, we +ran the Vanilla AC for 1 million iterations setting its constant +rollout length to the largest integer under the average rollout +length of MAC. For Tmax = 8, the average rollout length +is 3.42, so the rollout length for Vanilla AC is 3. Thus, 3 +million samples were observed for the Vanilla AC. To have +a similar number of observed samples, we ran MAC for +877192 iterations. Similarly when Tmax = 16, the average +rollout length is 4.26. Therefore, we ran MAC for 936768 +iterations. The details table of hyperparameters is provided +in Appendix F. In Figure 1 (a) we set n = 6 and Tmax = 8 + +0.5 +0.4 +Mean Rewards +0.3 +0.2 +0.1 +MAC +VanillaAC +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Iterations +1e60.5 +0.4 +Mean Rewards +0.3 +0.2 +0.1 +MAC +VanillaAC +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Iterations +1e6Multi-Level Monte Carlo Actor-Critic (MAC) +9 +for MAC. For MAC and Vanilla AC, we set the learning rate +for actor, critic, and reward estimation to .01. In Figure 1 +(b), n = 10 and Tmax = 16 and learning rate is .005. We +observe that for both experiments, MAC converges faster to +the maximum reward than Vanilla AC, showing MLMC’s +advantage over a standard gradient estimator. +6. Conclusions and Limitations +In this work, for the first time, we established the explicit +dependence of the convergence rate of the actor-critic +algorithm on the mixing time of the underlying Markov +transitions induced by the policy. 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Advances in +neural information processing systems, 32, 2019. 5, 6, 23 + +Multi-Level Monte Carlo Actor-Critic (MAC) +12 +Appendix +Table of Contents +A Detailed Context of Related Works +12 +B +Preliminaries +13 +B.1 +Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +B.2 +Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +C Convergence Analysis of Actor +14 +D Average Reward Tracking and Critic Error Analyses +19 +D.1 +Average Reward Tracking Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +D.2 +Critic Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +E +Proof of Theorem 4.8 +27 +F +Hyperparametrs for the Experiments +28 +A. Detailed Context of Related Works +Actor-critic by Konda & Tsitsiklis (1999) comprises algorithms that alternate between value function estimation (critic) and +policy search updates (actor), which may be seen as a form of policy iteration (Bertsekas, 2011) that incorporates stochastic +approximation (Borkar & Konda, 1997). We discuss each facet separately, before launching into their fusion. +TD Learning To evaluate the policy update direction, an estimate of the value function is required. To compute this estimate, +stochastic fixed point iterations are considered to solve Bellman’s equation Sutton (1988), whose stability under linear +function approximation was established in Tsitsiklis & Van Roy (1997). Since then, a plethora of works has studied the +stability properties of TD-based policy evaluation. Initially, their asymptotic convergence was prioritized (Tadi´c, 2001), but +more recently, non-asymptotic results have gained salience. For discounted TD with Markovian samples, Bhandari et al. +(2018) established finite-time convergence bounds which scale linearly with mixing time τmix. Dorfman & Levy (2022) +then improved the rate to be proportional to the √τmix using a multi-level gradient estimator and adaptive learning rate. Qiu +et al. (2021a) studied TD under the average reward setting, which also imposes exponentially fast mixing that manifests in +an additional logarithmic term in the sample complexity. These results all hinge upon imposing restrictive conditions on the +mixing time. +Policy Gradient With a value function estimate in hand, one can multiple this quantity together with the gradient of the +log-likelihood of a policy, i.e., the score function, to evaluate an estimate of the policy gradient (Williams, 1992; Sutton +et al., 1999). Then, gradient ascent steps are taken with respect to policy parameters. The convergence of policy gradient +has been studied extensively. Similar to TD, early work (Borkar & Meyn, 2000) focused on asymptotic stability via tools +from dynamical systems (Borkar & Meyn, 2000). More recently, its sample complexity has been established for a variety of +settings: for tabular (Bhandari & Russo, 2019; Agarwal et al., 2020) and softmax policies (Mei et al., 2020), rates to global +optimality exist. For general parameterized policies, early works focused on “policy improvement” bounds (Pirotta et al., +2013; 2015), and more recently, rates towards stationarity (Bedi et al., 2022) and local extrema (Zhang et al., 2020) have +been studied, and under special neural architectures, globally optimal solutions (Wang et al., 2019; Leahy et al., 2022) are +achievable. This topic is an active area of work, and covering all related sub-topics is beyond our scope. We merely identify +that these performance certificates all hinge upon the mixing time of the induced Markov chain going to null exponentially +fast. + +Multi-Level Monte Carlo Actor-Critic (MAC) +13 +Actor-Critic As previously mentioned, the stability of actor-critic was initially focused on asymptotics (Borkar & Konda, +1997). More recently, its non-asymptotic rate has been derived under i.i.d. assumptions (Kumar et al., 2019; Wang et al., +2019), and more recently under a variety of different types of Markovian data – see Table 1. However, these results impose +that any temporal correlation of data across time vanishes exponentially fast as quantified by the mixing rate. In this way, we +are able to match (Chen & Zhao, 2022) but without this restriction. +B. Preliminaries +Before proceeding with our analysis of Algorithm 1, we need some preliminary results and assumptions. +B.1. Preliminary Results +The statements of the results in this section have been adapted from (Dorfman & Levy, 2022) to fit the setting considered +in our paper. Except in the case of Lemma B.3, their proofs follow directly from that work. First, we need the following +concentration bound concerning gradient estimation from Markovian data. +Lemma B.1. Lemma A.5, (Dorfman & Levy, 2022). Fix K, N ∈ N such that N ≥ 2K. Let a policy parameter θt ∈ Θ be +given, and fix a trajectory zt = {zi +t = (si +t, ai +t, ri +t, si+1 +t +)}i∈[N] generated by following policy πθt starting from s0 +t ∼ µ0(·). +Let ∇L(x) be a gradient that we wish to estimate over zt, where Ez∼µθt,πθt [l(x, z)] = ∇L(x), and x ∈ K ⊂ Rk is the +parameter of the estimator l, i.e., xt = θt, ηt, or ωt. Finally, assume that ∥l(x, z)∥, ∥∇L(x)∥ ≤ GL, for all x ∈ K, z ∈ +S × A × R × S. Then, for every δ > Ndmix(K) and every xt ∈ K measurable w.r.t. Ft−1 = σ(θk, ηk, ωk, zk; k ≤ t − 1), +we have +Pt−1 +������ +1 +N +N +� +i=1 +l(xt, zi +t) − ∇L(xt) +����� ≤ 12GL +� +K +N +� +1 + +� +log(K/˜δ) +� ++ 6GK +N +� +≥ 1 − δ, +(26) +where ˜δ = δ − Ndmix(K). +We will use this result to facilitate our analyses of each of the MLMC estimators f MLML +t +, gMLMC +t +, lMLMC +t +used in +Algorithm 1. We also need the following error bound, which follows from Lemma B.1. +Lemma B.2. Lemma A.6, (Dorfman & Levy, 2022). Let ∇L, l, zt be as in Lemma B.1. Define lN +t = 1 +N +�N +i=1 l(xt, zi +t). Fix +Tmax ∈ N and let K = τ θt +max⌈2 log Tmax⌉. Then, for every N ∈ [Tmax] and every xt ∈ K measurable w.r.t. Ft−1, +E +� +∥lN +t − ∇L(xt)∥ +� +≤ O +� +GL +� +log KN +� +K +N +� +, +(27) +E +� +∥lN +t − ∇L(xt)∥ +2� +≤ O +� +G2 +L log(KN)K +N +� +. +(28) +The following important result establishes key properties of MLMC estimators. It is an extension of Lemma 3.1 from +(Dorfman & Levy, 2022), clarifying the effect of using rollout length Tmax in the MLMC estimator. +Lemma B.3. Let ∇L, l, zt be as in Lemma B.1. Let Jt ∼ Geom(1/2). Define the MLMC estimator +lMLMC +t += l0 +t + +� +2Jt(lJt +t − lJt−1 +t +), +if 2Jt ≥ Tmax, +0, +otherwise. +(29) +Let jmax = ⌊log Tmax⌋. Fix xt measurable w.r.t. Ft−1. Assume Tmax ≥ τ θt +mix, ∥∇L(x)∥ ≤ GL, for all x ∈ K, and +∥lN +t ∥ ≤ GL, for all N ∈ [Tmax]. Then +Et−1 +� +lMLMC +t +� += Et−1 +� +ljmax +t +� +, +(30) +E +� +∥lMLMC +t +∥ +2� +≤ � +O +� +G2 +Lτ θt +mix log Tmax +� +. +(31) +Proof. For brevity, let lt := lMLMC +t +. To show (30), we simply recall that lt = l0 +t + 2Jt +� +lJt +t − lJt−1 +t +� +and note that +Et−1 [lt] = Et−1 +� +l0 +t +� ++ +jmax +� +i=1 +P(Jt = j)2jEt−1 +� +lj +t − lj−1 +t +� += Et−1 +� +ljmax +t +� +. +(32) + +Multi-Level Monte Carlo Actor-Critic (MAC) +14 +For (31), first note that by Cauchy-Schwarz and boundedness of lj +t, for all j ∈ [Tmax], we know that +E +� +∥lt∥2� +≤ 2E +� +∥lt − l0 +t ∥ +2� ++ 2G2 +L. +(33) +Now, since lt = l0 +t + 2Jt +� +lJt +t − lJt−1 +t +� +, +E +� +∥lt − l0 +t ∥ +2� += +jmax +� +j=1 +P(Jt = j)E +����2j � +lj +t − lj−1 +t +���� +2� +(34) += +jmax +� +j=1 +2jE +���� +� +lj +t − lj−1 +t +���� +2� +(35) +≤ +jmax +� +j=1 +2j +� +2E +����lj +t − ∇J(θt) +��� +2� ++ 2E +����lj−1 +t +− ∇J(θt) +��� +2�� +(36) +(a) +≤ +jmax +� +j=1 +2j +� +� +O +� 1 +2j G2 +Lτ θt +mix log(Tmax) +�� +(37) += +jmax +� +j=1 +� +O +� +G2 +Lτ θt +mix log Tmax +� +(38) += � +O +� +G2 +Lτ θt +mix log Tmax +� +, +(39) +where (a) follows from Lemma B.2 and (39) holds by the definition of jmax. Combining (33) with (39) gives the result. +Finally, we will use the following result to manipulate the AdaGrad stepsizes in the final result of this section. +Lemma B.4. Lemma 4.2, (Dorfman & Levy, 2022). For any non-negative real numbers {ai}i∈[n], +n +� +i=1 +ai +��i +j=1 aj +≤ 2 +� +� +� +� +n +� +i=1 +ai. +(40) +B.2. Assumptions +We will also need the following assumptions. +Assumption B.5. The objective J(θ) is L-Lipschitz in θ. There exists GH such that ∥∇J(θ)∥ ≤ GH, for all θ. +Assumption B.6. The critic update includes a projection onto the ball of radius Rω about the origin. +Assumption B.7. For each θ, the matrix Aθ = Es∼µθ,a∼πθ,s′∼p(·|s,a) +� +φ(s)(φ(s) − φ(s′))T � +is positive definite. +C. Convergence Analysis of Actor +In this section, we provide a bound on the average policy gradient norm achieved by Algorithm 1, leveraging the MLMC +analysis machinery of (Dorfman & Levy, 2022) to reveal dependence on the worst-case mixing time encountered during +training. Combined with the error analysis of Section D, this forms the core of our analysis of Algorithm 1. The analysis +largely follows that of (Dorfman & Levy, 2022), with key modifications to accommodate the average reward estimation, +critic estimation, and critic function approximation bias inherent in the average-reward actor-critic setting. +As the first step in our actor analysis, we prove a version of Lemma B.2 that incorporates average reward estimation error + +Multi-Level Monte Carlo Actor-Critic (MAC) +15 +and critic error. Before starting the result and its proof, we develop some notation to facilitate the exposition. Let +∇Ji +t = +� +ri +t − ηt + ⟨φ(si+1 +t +), ωt⟩ − ⟨φ(si +t), ωt⟩ +� +∇ log πθt +� +ai +t|si +t +� +, +(41) +∇Ji,η +t += +� +ri +t − η∗ +t + ⟨φ(si+1 +t +), ωt⟩ − ⟨φ(si +t), ωt⟩ +� +∇ log πθt +� +ai +t|si +t +� +, +(42) +∇Ji,η,ω +t += +� +ri +t − η∗ +t + ⟨φ(si+1 +t +), ω∗ +t ⟩ − ⟨φ(si +t), ω∗ +t ⟩ +� +∇ log πθt +� +ai +t|si +t +� +, +(43) +∇Ji,η,V +t += +� +ri +t − η∗ +t + Vθt(si+1 +t +) − Vθt(si +t) +� +∇ log πθt +� +ai +t|si +t +� +, +(44) +where η∗ +t = J(θt) and ω∗ +t is the limiting point of TD(0) applied to evaluating the policy πθt. Notice that +∇Ji +t − ∇J(θt) = +� +∇Ji +t − ∇Ji,η +t +� +�� +� +(a) +� ++ +� +∇Ji,η +t +− ∇Ji,η,ω +t +� +�� +� +(b) +� ++ +� +∇Ji,η,ω +t +− ∇Ji,η,V +t +� +�� +� +(c) +� ++ +� +∇Ji,η,V +t +− ∇J(θt) +� +�� +� +(d) +� +, +(45) +where +(a): ∇Ji +t − ∇Ji,η +t += (η∗ +t − ηt) ∇ log πθt +� +ai +t|si +t +� +(46) +(b): ∇Ji,η +t +− ∇Ji,η,w +t += ⟨φ(si+1 +t +) − φ(si +t), ωt − ω∗ +t ⟩∇ log πθt +� +ai +t|si +t +� +(47) +(c): ∇Ji,η,w +t +− ∇Ji,η,V +t += +�� +⟨φ(si+1 +t +), ω∗ +t ⟩ − Vθt(si+1 +t +) +� +− +� +⟨φ(si +t), ω∗ +t ⟩ − Vθt(si +t) +�� +∇ log πθt +� +ai +t|si +t +� +(48) +and, since Eµθt,πθt +� +∇Ji,η,V +t +� += ∇J(θt), (d) is the error between ∇J(θt) and the ideal policy gradient estimator. Define +Eapp := sup +s,θ +|⟨φ(s), ω(θ) − Vθ(s)⟩|, +C := sup +s,s′ ∥φ(s) − φ(s′)∥, +(49) +and let B > 0 be such that +sup +θ,a,s +∥∇ log πθ(a|s)∥ ≤ B. +(50) +Lemma C.1. Assume ∥∇J(θ)∥, ∥∇Ji,η,V +t +∥ ≤ GH, for all θ, si +t, ai +t. Fix Tmax ∈ N and let K = τ θt +max⌈2 log Tmax⌉. Define +hN +t = 1 +N +�N +i=1 ∇Ji +t, for N ∈ [Tmax]. Then, for all N ∈ [Tmax] and θt measurable w.r.t. Ft−1, +E +� +∥hN +t − ∇J(θt)∥ +� +≤ O +� +GH +� +log KN +� +K +N +� ++ E1(t) + 2BEapp, +(51) +E +� +∥hN +t − ∇J(θt)∥ +2� +≤ O +� +G2 +H log(KN)K +N +� ++ E2(t) + 16B2Eapp, +(52) +where +E1(t) = BE [∥ηt − η∗ +t ∥] + BCE [∥ωt − ω∗ +t ∥] , +(53) +E2(t) = 4B2E +� +∥ηt − η∗ +t ∥2� ++ 4B2C2E +� +∥ωt − ω∗ +t ∥2� +. +(54) +Proof. First notice that +��hN +t − ∇J(θt) +�� ≤ +����� +1 +N +N +� +i=1 +∇Ji,η,V +t +− ∇J(θt) +����� + +����� +1 +N +N +� +i=1 +∇Ji +t − ∇Ji,η +t +����� +(55) ++ +����� +1 +N +N +� +i=1 +∇Ji,η +t +− ∇Ji,η,ω +t +����� + +����� +1 +N +N +� +i=1 +∇Ji,η,ω +t +− ∇Ji,η,V +t +����� +(56) +≤ +����� +1 +N +N +� +i=1 +∇Ji,η,V +t +− ∇J(θt) +����� + B∥ηt − η∗ +t ∥ + BC∥ωt − ω∗ +t ∥ + 2BEapp. +(57) + +Multi-Level Monte Carlo Actor-Critic (MAC) +16 +As a consequence, we also have +��hN +t − ∇J(θt) +��2 ≤ 4 +����� +1 +N +N +� +i=1 +∇Ji,η,V +t +− ∇J(θt) +����� +2 ++ 4B2∥ηt − η∗ +t ∥2 + 4B2C2∥ωt − ω∗ +t ∥2 + 16B2E2 +app. +(58) +Taking expectations and applying Lemma B.2 with xt = θt, l(θt, zi +t) = ∇Ji,η,V +t +, ∇L(θt) = ∇J(θt) yields the result. +We next prove a key result regarding the bias and second moment of our policy gradient estimate. It is a generalization of +Lemma 3.1 in (Dorfman & Levy, 2022) building on our Lemma C.1. +Lemma C.2. Let jmax = ⌊log Tmax⌋ in Algorithm 1. Fix θt measurable w.r.t. Ft−1. Assume Tmax ≥ τ θt +mix, ∥∇J(θ)∥ ≤ +GH, for all θ, and ∥hN +t ∥ ≤ GH, for all N ∈ [Tmax]. Then +Et−1 +� +hMLMC +t +� += Et−1 +� +hjmax +t +� +, +(59) +E +� +∥hMLMC +t +∥ +2� +≤ � +O +� +G2 +Hτ θt +mix log Tmax +� ++ 8 log(Tmax)Tmax +� +E2(t) + 16B2E2 +app +� +. +(60) +Proof. For brevity, let ht := hMLMC +t +. Equation (59) follows directly from Lemma B.3. For (60), first note that by +Cauchy-Schwarz and boundedness of hj +t, for all j ∈ [Tmax], we know that +E +� +∥ht∥2� +≤ 2E +� +∥ht − h0 +t∥ +2� ++ 2G2 +H. +(61) +Now, since ht = h0 +t + 2Jt +� +hJt +t − hJt−1 +t +� +, +E +� +∥ht − h0 +t∥ +2� += +jmax +� +j=1 +P(Jt = j)E +����2j � +hj +t − hj−1 +t +���� +2� +(62) += +jmax +� +j=1 +2jE +���� +� +hj +t − hj−1 +t +���� +2� +(63) +≤ +jmax +� +j=1 +2j +� +2E +����hj +t − ∇J(θt) +��� +2� ++ 2E +����hj−1 +t +− ∇J(θt) +��� +2�� +. +(64) +Next, we can write +E +� +∥ht − h0 +t∥ +2� (a) +≤ +jmax +� +j=1 +2j +� +� +O +� 1 +2j G2 +Hτ θt +mix log(Tmax) +� ++ 4E2(t) + 16B2E2 +app +� +(65) += +jmax +� +j=1 +� +� +O +� +G2 +Hτ θt +mix log Tmax +� ++ 4 · 2j � +E2(t) + 16B2E2 +app +�� +(66) +(b) +≤ log Tmax +� +� +O +� +G2 +Hτ θt +mix log Tmax +� ++ 4Tmax +� +E2(t) + 16B2E2 +app +�� +(67) += � +O +� +G2 +Hτ θt +mix log Tmax +� ++ 4 log(Tmax)Tmax +� +E2(t) + 16B2E2 +app +� +, +(68) +where (a) follows from Lemma C.1 and (b) holds by the definition of jmax. Combining (61) with (68) gives the result. +Before proceeding to the final policy gradient norm bound of our actor analysis, we need one additional auxiliary result. +Lemma C.3. Assume J(θ) is L-smooth. Let ∆t = supθ J(θ) − J(θt) and ∆T +max = maxt∈[T ] ∆t. Then +T +� +t=1 +∥∇J(θt)∥2 ≤ ∆T +max +αT ++ L +2 +T +� +t=1 +α∥hMLMC +t +∥ +2 + +T +� +t=1 +⟨∇J(θt) − hMLMC +t +, ∇J(θt)⟩. +(69) + +Multi-Level Monte Carlo Actor-Critic (MAC) +17 +Proof. Once again, write ht := hMLMC +t +for brevity. We first have +J(θt+1) ≥ J(θt) + αt∇J(θt)T ht − Lα2 +t +2 ∥ht∥2 +(70) += J(θt) + αt∥∇J(θt)∥2 − αt⟨∇J(θt) − ht, ∇J(θt)⟩ − Lα2 +t +2 ∥ht∥2, +(71) +where the first equality holds from the smoothness of J(θ) and the fact that θt+1 = θt + αtht. Rearranging gives +∥∇J(θt)∥2 ≤ J(θt+1) − J(θt) +αt ++ Lαt +2 ∥ht∥2 + ⟨∇J(θt) − ht, ∇J(θt)⟩, +(72) +and summing yields +T +� +t=1 +∥∇J(θt)∥2 ≤ +T +� +t=1 +∆t − ∆t+1 +αt ++ L +2 +T +� +t=1 +αt∥ht∥2 + +T +� +t=1 +⟨∇J(θt) − ht, ∇J(θt)⟩ +(73) +≤ +T +� +t=1 +∆T +max +αT ++ L +2 +T +� +t=1 +αt∥ht∥2 + +T +� +t=1 +⟨∇J(θt) − ht, ∇J(θt)⟩. +(74) +We are now ready to prove the main result of this section. +Theorem C.4. Assume J(θ) is L-smooth, supθ |J(θ)| ≤ M, and ∥∇J(θ)∥, ∥hMLMC +t +∥ ≤ GH, for all θ, t. Let αt = +α′ +t/ +��T +t=1 ∥hMLMC +t +∥ +2, where {α′ +t} is an auxiliary stepsize sequence with α′ +t ≤ 1, for all t ≥ 1. Then +1 +T +T +� +t=1 +E +� +∥∇J(θt)∥2� +≤ � +O +� +(M + L)GH +1 +√ +T +� +max +t∈[T ] τ θt +mix log Tmax +� +(75) ++ 2M + L +T +� +� +� +� +T +� +t=1 +8 log(Tmax)Tmax +� +E2(t) + 16B2E2app +� +(76) ++ � +O +� +G2 +H max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ 1 +T +T +� +t=1 +E2(t) + 16B2E2 +app. +(77) +Proof. Again let ht := hMLMC +t +. We have +T +� +t=1 +∥∇J(θt)∥2 (a) +≤ ∆max +� +� +� +� +T +� +t=1 +∥ht∥2 + L +2 +T +� +t=1 +α′ +t∥ht∥2 +��t +k=1 ∥hk∥2 + +T +� +t=1 +⟨∇J(θt) − ht, ∇J(θt)⟩ +(78) +(b) +≤ ∆max +� +� +� +� +T +� +t=1 +∥ht∥2 + L +2 +T +� +t=1 +∥ht∥2 +��t +k=1 ∥hk∥2 + +T +� +t=1 +⟨∇J(θt) − ht, ∇J(θt)⟩ +(79) +(c) +≤ (∆max + L) +� +� +� +� +T +� +t=1 +∥ht∥2 + +T +� +t=1 +⟨∇J(θt) − ht, ∇J(θt)⟩, +(80) + +Multi-Level Monte Carlo Actor-Critic (MAC) +18 +where (a) follows from Lemma C.3, inequality (b) by the definition of αt, and (c) is by Lemma B.4. This implies that +T +� +t=1 +E +� +∥∇J(θt)∥2� (a) +≤ E +� +�(∆max + L) +� +� +� +� +T +� +t=1 +∥ht∥2 +� +� + +T +� +t=1 +E +� +⟨∇J(θt) − hjmax +t +, ∇J(θt)⟩ +� +(81) +(b) +≤ E +� +�(∆max + L) +� +� +� +� +T +� +t=1 +∥ht∥2 +� +� + +T +� +t=1 +E +� +∥∇J(θt) − hjmax +t +∥ · ∥∇J(θt)∥ +� +(82) +(c) +≤ E +� +�(∆max + L) +� +� +� +� +T +� +t=1 +∥ht∥2 +� +� + +T +� +t=1 +� +E +� +∥∇J(θt) − hjmax +t +∥ +2��1/2 � +E +� +∥∇J(θt)∥2��1/2 +(83) +(d) +≤ E +� +�(∆max + L) +� +� +� +� +T +� +t=1 +∥ht∥2 +� +� + +� T +� +t=1 +E +� +∥∇J(θt) − hjmax +t +∥ +2��1/2 � T +� +t=1 +E +� +∥∇J(θt)∥2��1/2 +, +(84) +where (a) follows from the law of total expectation, the fact that θt, θ∗ +t are deterministic conditioned on Ft−1, and Lemma +C.2, (b) follows by Cauchy-Schwarz, and (b) and (c) by applications of H¨older’s inequality. Define +A(T) = E +� +�(∆max + L) +� +� +� +� +T +� +t=1 +∥ht∥2 +� +� , +(85) +B(T) = 1 +4 +T +� +t=1 +E +� +∥∇J(θt) − hjmax +t +∥ +2� +, +(86) +C(T) = +T +� +t=1 +E +� +∥∇J(θt)∥2� +. +(87) +The foregoing inequality becomes +C(T) ≤ A(T) + 2 +� +B(T) +� +C(T) +(88) +Consider the following chain of implications: +C(T) ≤ A(T) + 2 +� +B(T) +� +C(T) =⇒ +�� +C(T) − +� +B(T) +�2 +≤ A(T) + B(T) +(89) +=⇒ +� +C(T) − +� +B(T) ≤ +� +A(T) + +� +B(T) +(90) +=⇒ +� +C(T) ≤ +� +A(T) + 2 +� +B(T) +(91) +=⇒ C(T) ≤ 2A(T) + 8B(T). +(92) +We therefore have +T +� +t=1 +E +� +∥∇J(θt)∥2� +≤ 2E +� +�(∆max + L) +� +� +� +� +T +� +t=1 +∥ht∥2 +� +� + 2 +T +� +t=1 +E +� +∥∇J(θt) − hjmax +t +∥ +2� +(93) +(94) + +Multi-Level Monte Carlo Actor-Critic (MAC) +19 +Now, +E +� +�(∆max + L) +� +� +� +� +T +� +t=1 +∥ht∥2 +� +� +(a) +≤ (2M + L) +� +� +� +� +T +� +t=1 +E +� +∥ht∥2� +(95) +(b) +≤ (2M + L) +� +� +� +� � +O +� +TG2 +H max +t∈[T ] τ θt +mix log Tmax +� ++ +T +� +t=1 +8 log(Tmax)Tmax +� +E2(t) + 16B2E2app +� +(96) +(c) +≤ � +O +� +(M + L)GH +� +T max +t∈[T ] τ θt +mix log Tmax +� ++ (2M + L) +� +� +� +�8 +T +� +t=1 +log(Tmax)Tmax +� +E2(t) + 16B2E2app +� +, +(97) +where (a) follows by the fact that ∆max ≤ 2M and Jensen’s inequality, (b) is from Lemma C.2, and (c) follows since +√ +a + b ≤ √a + +√ +b. Furthermore, by the second-order bound of Lemma C.1 we have +T +� +t=1 +E +� +∥∇J(θt) − hjmax +t +∥ +2� +≤ � +O +� +TG2 +Hτ θt +mix +log Tmax +Tmax +� ++ +T +� +t=1 +E2(t) + T16B2E2 +app. +(98) +Combining these expressions and dividing by T completes the proof. +D. Average Reward Tracking and Critic Error Analyses +In this section we bound the error arising from the average reward tracking and critic estimation. Combined with the actor +gradient norm bound of Section C, this will complete the analysis of Algorithm 1. Our analysis broadly follows that of +(Wu et al., 2020), with key modifications leveraging our novel MLMC machinery to handle Markovian sampling in a more +streamlined manner. +D.1. Average Reward Tracking Analysis +The main result of this subsection is the following bound on the average reward tracking error. +Theorem D.1. Assume γt = (1 + t)−ν, α = α′ +t/ +��t +k=1 ∥ht∥2, and α′ +t = (1 + t)−σ, where 0 < ν < σ < 1. Furthermore, +assume sups,a |r(s, a)| ≤ R. Then +1 +T +T +� +t=1 +E +� +(ηt − η∗ +t )2� +≤ O +� +T ν−1� ++ O +� +T −2(σ−ν)� +(99) ++ � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +O +� +T −ν� +(100) ++ � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +. +(101) +Proof. First, recall that the average reward tracking update is given by +ηt+1 = ηt − γtft, +(102) +where for brevity we set ft := f MLMC +t +. We can rewrite the tracking error term (ηt+1 − η∗ +t+1)2 as +(ηt+1 − η∗ +t+1)2 = (ηt+1 − η∗ +t + η∗ +t − η∗ +t+1)2 +(103) += (ηt − γtft − η∗ +t + η∗ +t − η∗ +t+1)2. +(104) + +Multi-Level Monte Carlo Actor-Critic (MAC) +20 +Expanding the squares and regrouping terms yields +(ηt+1 − η∗ +t+1)2 = (ηt − η∗ +t )2 − 2γt(ηt − η∗ +t )ft + 2(ηt − η∗ +t )(η∗ +t − η∗ +t+1) +− 2γt(η∗ +t − η∗ +t+1)ft + (η∗ +t − η∗ +t+1)2 + γ2 +t (ft)2 +(105) += (ηt − η∗ +t )2 − 2γt(ηt − η∗ +t )ft + 2(ηt − η∗ +t )(η∗ +t − η∗ +t+1) ++ (η∗ +t − η∗ +t+1 − γtft)2. +(106) +Next, we utilize the bound (a + b)2 ≤ 2a2 + 2b2 to upper bound the last term in the right hand side of (106) to obtain +(ηt+1 − η∗ +t+1)2 ≤ (ηt − η∗ +t )2 − 2γt(ηt − η∗ +t )ft + 2(ηt − η∗ +t )(η∗ +t − η∗ +t+1) ++ 2(η∗ +t − η∗ +t+1)2 + 2(γtft)2. +(107) +Now notice that the function whose gradient we are estimating with ft is simply the strongly convex function F(ηt) = +1 +2 (ηt − η∗ +t )2 = 1 +2 (ηt − J(θt))2. Clearly F ′(ηt) = ηt − J(θt) is Lipschitz in ηt and F has strong convexity parameter +mF = 1. Adding and subtracting 2γt(ηt − η∗ +t )F ′(ηt) in the above expression gives +(ηt+1 − η∗ +t+1)2 ≤ (ηt − η∗ +t )2 − 2γt(ηt − η∗ +t )F ′(ηt) + 2γt(ηt − η∗ +t )(F ′(ηt) − ft) + 2(ηt − η∗ +t )(η∗ +t − η∗ +t+1) ++ 2(η∗ +t − η∗ +t+1)2 + 2(γtft)2. +(108) +From the strong convexity of F with mF = 1, we can write +(ηt+1 − η∗ +t+1)2 ≤ (ηt − η∗ +t )2 − 2γt(ηt − η∗ +t )2 + 2γt(ηt − η∗ +t )(F ′(ηt) − ft) + 2(ηt − η∗ +t )(η∗ +t − η∗ +t+1) ++ 2(η∗ +t − η∗ +t+1)2 + 2(γtft)2 +(109) += (1 − 2γt)(ηt − η∗ +t )2 + 2γt(ηt − η∗ +t )(F ′(ηt) − ft) + 2(ηt − η∗ +t )(η∗ +t − η∗ +t+1) ++ 2(η∗ +t − η∗ +t+1)2 + 2(γtft)2. +(110) +Taking expectations and summing yields +T +� +t=1 +E[(ηt − η∗ +t )2] ≤ +T +� +t=1 +1 +2γt +E[(ηt − η∗ +t )2 − (ηt − η∗ +t )2] +� +�� +� +I1 ++ +T +� +t=1 +E[(ηt − η∗ +t )(F ′(ηt) − ft)] +� +�� +� +I2 ++ +T +� +t=1 +1 +γt +E[(ηt − η∗ +t )(η∗ +t − η∗ +t+1)] +� +�� +� +I3 ++ +T +� +t=1 +1 +γt +E[(η∗ +t − η∗ +t+1)2] +� +�� +� +I4 ++ +T +� +t=1 +γtE[(ft)2] +� +�� +� +I5 +. +(111) +We next provide intermediate bounds for all the terms I1, I2, I3, I4 and I5 in the right hand side of (111). We will +subsequently manipulate these intermediate bounds to obtain the final bound of Theorem D.1. +Bound on I1: By rearranging terms in I1, we get +I1 = +T +� +t=1 +1 +2γt +E[(ηt − η∗ +t )2 − (ηt − η∗ +t )2] += +1 +2γ1 +E[(η1 − η∗ +1)2] + +T +� +t=2 +� 1 +2γt +− +1 +2γt−1 +� +E[(ηt − η∗ +t )2] − +1 +2γT +E[(ηT +1 − η∗ +T +1)2] +(112) +≤ R2 +γT +, +(113) +where we use the fact that (ηt − η∗ +t )2 ≤ 2R2. +Bound on I2: For I2, first notice that ηt, η∗ +t = J(θt) are deterministic conditioned on Ft−1 from Lemma B.1. This means +we can rewrite the expectation in I2 as +I2 = +T +� +t=1 +E[Et−1[(ηt − η∗ +t )(F ′(ηt) − ft)]] = +T +� +t=1 +E[(ηt − η∗ +t )(F ′(ηt) − Et−1[ft])], +(114) + +Multi-Level Monte Carlo Actor-Critic (MAC) +21 +where Et−1[. . .] denotes expectation conditioned on Ft−1. From C.2 we know that Et−1[ft] = Et−1[f jmax +t +], hence we can +write the expression in (114) as +I2 = +T +� +t=1 +E[(ηt − η∗ +t )(F ′(ηt) − Et−1[f jmax +t +)]] = +T +� +t=1 +E[Et−1[(ηt − η∗ +t )(F ′(ηt) − f jmax +t +)]] +(115) += +T +� +t=1 +E[(ηt − η∗ +t )(F ′(ηt) − f jmax +t +)]. +(116) +Taking absolute values, then applying the triangle, Jensen, and Cauchy-Schwarz inequalities, we can upper bound (116) by +|I2| = +���� +T +� +t=1 +E[(ηt − η∗ +t )(F ′(ηt) − f jmax +t +)] +���� ≤ +T +� +t=1 +E +���(ηt − η∗ +t )(F ′(ηt) − f jmax +t +) +�� +� +≤ +T +� +t=1 +E +���(ηt − η∗ +t ) +�� · +��(F ′(ηt) − f jmax +t +) +�� +� +. +(117) +We know that |ηt − η∗ +t | ≤ 2R by assumption, implying +|I2| ≤ 2R +T +� +t=1 +E +���(F ′(ηt) − f jmax +t +) +�� +� +. +(118) +By Lemma B.2 with xt = ηt, ∇L(xt) = ∇F(ηt) and l(xt, zt) = ft, and the fact that the Lipschitz constant of ∇F(ηt) is 1, +we obtain the following upper bound on I2: +|I2| ≤ 2R +T +� +t=1 +� +O +�� +τ θt +mix +log Tmax +Tmax +� +. +(119) +Bound on I3: By H¨older’s inequality, +|I3| = +���� +T +� +t=1 +1 +γt +E[(ηt − η∗ +t )(η∗ +t − η∗ +t+1)] +���� ≤ +� T +� +t=1 +E +� +(ηt − η∗ +t )2� +�1/2 � T +� +t=1 +1 +γ2 +t +E +� +(η∗ +t − η∗ +t+1)2� +�1/2 +. +(120) +Notice that |η∗ +t − η∗ +t+1| = |J(θt) − J(θt+1)| ≤ L|θt − θt+1| ≤ LGHαt due to the Lipschitz continuity of J(θ) in θ and +boundedness of ∥∇J(θ)∥ from Assumption B.5. This implies +|I3| ≤ +� T +� +t=1 +E +� +(ηt − η∗ +t )2� +�1/2 � +L2G2 +H +T +� +t=1 +α2 +t +γ2 +t +�1/2 +. +(121) +Bound on I4: Similarly, due to Assumption B.5 we have +I4 = +T +� +t=1 +1 +γt +E[(η∗ +t − η∗ +t+1)2] ≤ L2G2 +H +T +� +t=1 +α2 +γt +. +(122) +Bound on I5: Finally, by Lemma B.3 and taking GF = 2R without loss of generality, we have +I5 = +T +� +t=1 +γtE[(ft)2] ≤ +T +� +t=1 +γt � +O +� +R2τ θt +mix log Tmax +� +. +(123) + +Multi-Level Monte Carlo Actor-Critic (MAC) +22 +Combining the foregoing and recalling that γt = (1 + t)−ν, α′ +t = (1 + t)−σ, 0 < ν < σ < 1, and αt ≤ α′ +t, we get +T +� +t=1 +E[(ηt − η∗ +t )2] ≤ 2R2(1 + T)ν + 2TR � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +(124) ++ LGH +� T +� +t=1 +E[(ηt − η∗ +t )2] +� 1 +2 � T +� +t=1 +(1 + t)−2(σ−ν) +� 1 +2 +(125) ++ L2G2 +H +T +� +t=1 +(1 + t)(ν−2σ) + � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +T +� +t=1 +(1 + t)−ν +(126) +≤ 2R2(1 + T)ν + +� +L2G2 +H + � +O +� +max +t∈[T ] τ θt +mix log Tmax +�� +T +� +t=1 +(1 + t)−ν +(127) ++ 2TR � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +(128) ++ +� T +� +t=1 +E[(ηt − η∗ +t )2] +� 1 +2 � +L2G2 +H +T +� +t=1 +(1 + t)−2(σ−ν) +� 1 +2 +, +(129) +where the second inequality follows from the fact that ν − 2σ < −ν. +We now manipulate the foregoing inequality to obtain the desired bound. Define +A(T) = +T +� +t=1 +E[(ηt − η∗ +t )2], +(130) +B(T) = L2G2 +H +4 +T +� +t=1 +(1 + t)−2(σ−ν), +(131) +C(T) = 2R2(1 + T)ν + +� +L2G2 +H + � +O +� +max +t∈[T ] τ θt +mix log Tmax +�� +T +� +t=1 +(1 + t)−ν +(132) ++ 2TR � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +(133) +We can thus rewrite the foregoing inequality as +A(T) ≤ C(T) + 2 +� +A(T) +� +B(T). +(134) +This expression is equivalent to +�� +A(T) − +� +B(T) +�2 +≤ C(T) + B(T), +(135) +which in turn gives the following chain of implications: +�� +A(T) − +� +B(T) +�2 +≤ C(T) + B(T) =⇒ +� +A(T) − +� +B(T) ≤ +� +C(T) + +� +B(T) +(136) +=⇒ +� +A(T) ≤ +� +C(T) + 2 +� +B(T) +(137) +=⇒ A(T) ≤ 2C(T) + 4B(T). +(138) + +Multi-Level Monte Carlo Actor-Critic (MAC) +23 +As a result, we have shown that +T +� +t=1 +E[(ηt − η∗ +t )2] ≤ 4R2(1 + T)ν + +� +2L2G2 +H + � +O +� +max +t∈[T ] τ θt +mix log Tmax +�� +T +� +t=1 +(1 + t)−ν +(139) ++ 4TR � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +(140) ++ L2G2 +H +T +� +t=1 +(1 + t)−2(σ−ν). +(141) +Using the bound �T +t=1(1 + t)−ξ ≤ +� t+1 +0 +x−ξdx = (t + 1)1−ξ/(1 − ξ), this implies +T +� +t=1 +E[(ηt − η∗ +t )2] ≤ O(T ν) + � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +O(T 1−ν) + O(T 1−2(σ−ν)) +(142) ++ T � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +(143) +Dividing by T completes the proof. +Notice that, for σ = 0.75 and ν = 0.5, this result becomes +1 +T +T +� +t=1 +E[(ηt − η∗ +t )2] ≤ � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +O +� 1 +√ +T +� ++ � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +. +(144) +D.2. Critic Error Analysis +In this subsection we provide a bound on the critic estimation error term 1 +T +�T +t=1 E +� +∥ωt − ω∗ +t ∥2� +appearing in the main +actor analysis bound in Theorem C.4. To get started, we recall some facts about the TD(0) algorithm (Sutton, 1988). As +discussed in Ch. 9 of (Sutton & Barto, 2018), for a fixed policy parameter, θ, TD(0) with linear function approximation will +converge to the minimum of the mean squared projected Bellman error (MSPBE), which satisfies +Aθω = bθ, +(145) +Aθ = Es∼µθ,a∼πθ,s′∼p(·|s,a) +� +φ(s)(φ(s) − φ(s′))T � +, +(146) +bθ = Es∼µθ,a∼πθ [(r(s, a) − J(θ))φ(s)] . +(147) +The target critic parameter ω∗ +t at iteration t of our Algorithm 1 is thus given by ω∗ +t = A−1 +θt bθt. From the definition of +gMLMC +t +, the critic update ωt+1 = ωt + βtgMLMC +t +is clearly an attempt to use an MLMC estimator to approximately +perform the ideal update ωt+1 = ωt + βt(bθt − Aθtωt). We can thus view ∇G(ωt) = bθt − Aθtωt as the gradient of the +true critic objective G(ωt) corresponding to using least squares minimization to solve the equation Aθω = bθ. +Our task in this section is to characterize the average error that arises when using critic parameters {ωt} generated by +Algorithm 1 to track the ideal parameters {ω∗ +t }. Before we provide the main result of this section, we need three useful +lemmas and an assumption. The first result ensures that the optimal critic parameter is Lipschitz in θ. +Lemma D.2. Define Pθ(s′|s) = +� +A p(s′|s, a)πθ(a|s)da, for each θ. Assume that, for all θ, the ergodicity coefficient κ(Pθ) +of Pθ satisfies κ(Pθ) < 1. Then there exists Lω such that, for all θ, θ′, ω∗(θ) = A−1 +θ bθ and ω∗(θ′) = A−1 +θ′ bθ′ satisfy +∥ω∗(θ) − ω∗(θ′)∥ ≤ Lω∥θ − θ′∥. +Proof. The result follows by applying the same reasoning as that for Lemma A.3 in (Zou et al., 2019) to the bound from +Theorem 3.3 in (Mitrophanov, 2005). +The next result is an extension of Lemma B.2 to our MLMC critic gradient estimator. + +Multi-Level Monte Carlo Actor-Critic (MAC) +24 +Lemma D.3. Assume ∥∇G(ω)∥ ≤ GG, for all ω such that ∥ω∥ ≤ Rω. Define D = sups ∥φ(s)∥. Fix Tmax ∈ N, θt +measurable with respect to Ft−1, and let K = τ θt +max⌈2Tmax⌉. Define gN +t += +1 +N +�N +i=1 δi +tφ(si +t), for N ∈ [Tmax], where +δi +t = ri +t − ηt + (φ(si+1 +t +) − φ(si +t))T ωt. Then, for all N ∈ [Tmax], +E +���gN +t − ∇G(ωt) +��� +≤ O +� +GG +� +log KN +� +K +N +� ++ DE [|ηt − η∗ +t |] , +(148) +E +���gN +t − ∇G(ωt) +��2� +≤ O +� +G2 +G log(KN)K +N +� ++ D2E +� +(ηt − η∗ +t )2� +. +(149) +Proof. Define +δi,η +t += ri +t − η∗ +t + (φ(si+1 +t +) − φ(si +t))T ωt, +(150) +gN,η +t += 1 +N +N +� +i=1 +δi,η +t φ(si +t). +(151) +Clearly +��gN +t − ∇G(ωt) +�� ≤ +���gN +t − gN,η +t +��� + +���gN,η +t +− ∇G(ωt) +��� +(152) += +����� +1 +N +N +� +i=1 +δi +tφ(si +t) − δi,η +t φ(si +t) +����� + +����� +1 +N +N +� +i=1 +δi,η +t φ(si +t) − ∇G(ωt) +����� . +(153) +Notice that the first term can be bounded by D|ηt − η∗ +t | and that Lemma B.2 applies to the second term. The remainder of +the proof is analogous to that of Lemma C.1. +Next, we need a critic version of Lemma B.3. +Lemma D.4. Let jmax = ⌊log Tmax⌋ and fix θt measurable w.r.t. Ft−1. Assume Tmax ≥ τ θt +mix and ∥∇G(ω)∥ ≤ GG, for +all ω such that ∥ω∥ ≤ Rω. Then +Et−1 [gt] = Et−1 +� +gjmax +t +� +(154) +E +� +∥gt∥2� +≤ � +O +� +G2 +Gτ θt +mix log Tmax +� ++ 8 log(Tmax)TmaxD2E +� +(ηt − η∗ +t )2� +. +(155) +Proof. The claim follows from Lemma D.3 by the same argument as that used in the proof of Lemma C.2. +We now provide the main result of this section. The analysis is a modification of that used for the average reward tracking +setting. +Theorem D.5. Assume βt = (1 + t)−ν, αt = α′ +t/ +��t +k=1 ∥ht∥2, and α′ +t = (1 + t)−σ, where 0 < ν < σ < 1. Assume +without loss of generality that αt ≤ α′ +t, for all t. Furthermore, assume that Assumptions B.6 and B.7 hold. Then +1 +T +T +� +t=1 +E +� +∥ωt − ω∗ +t ∥2� +≤ O +� +T ν−1� ++ O +� +T −2(σ−ν)� +(156) ++ � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +O +� +T −ν� +(157) ++ � +O +� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +. +(158) +Proof. By Assumption B.7 and the fact that ∇2G(ω) = −Aθ, G(ω) is strongly concave. Let m denote its strong concavity +parameter, so that ⟨∇G(ω) − ∇G(ω′), ω − ω′⟩ ≤ −m∥ω − ω′∥2, for all ω, ω′. Recall that ωt+1 = ΠRω (ωt + βgt), where +we use gt = gMLMC +t +for brevity. We have +∥ωt+1 − ω∗ +t+1∥2 = ∥ΠRω (ωt + βgt) − ω∗ +t+1∥2 ≤ ∥ωt + βgt − ω∗ +t+1∥2, +(159) + +Multi-Level Monte Carlo Actor-Critic (MAC) +25 +where the inequality holds since ∥ω∗ +t+1∥ ≤ Rω by definition, so projection can only reduce the distance. Furthermore, +∥wt+1 − w∗ +t+1∥2 ≤ ∥wt − βtht − w∗ +t + w∗ +t − w∗ +t+1∥2 +(160) += ∥ωt − ω∗ +t ∥2 + 2βt⟨ωt − ω∗ +t , gt⟩ + 2⟨ωt − ω∗ +t , ω∗ +t − ω∗ +t+1⟩ +(161) ++ 2βt⟨ω∗ +t − ω∗ +t+1, gt⟩ + ∥ω∗ +t − ω∗ +t+1∥2 + β2 +t ∥ht∥2 +(162) +(a) +≤ ∥ωt − ω∗ +t ∥2 + 2βt⟨ωt − ω∗ +t , gt⟩ + 2⟨ωt − ω∗ +t , ω∗ +t − ω∗ +t+1⟩ +(163) ++ 2∥ω∗ +t − ω∗ +t+1∥2 + 2β2 +t ∥ht∥2 +(164) += ∥ωt − ω∗ +t ∥2 + 2βt⟨ωt − ω∗ +t , ∇G(ωt)⟩ + 2βt⟨ωt − ω∗ +t , gt − ∇G(ωt)⟩ +(165) ++ 2⟨ωt − ω∗ +t , ω∗ +t − ω∗ +t+1⟩ + 2∥ω∗ +t − ω∗ +t+1∥2 + 2β2 +t ∥ht∥2 +(166) +(b) +≤ (1 − 2mβt)∥ωt − ω∗ +t ∥2 + 2βt⟨ωt − ω∗ +t , gt − ∇G(ωt)⟩ +(167) ++ 2⟨ωt − ω∗ +t , ω∗ +t − ω∗ +t+1⟩ + 2∥ω∗ +t − ω∗ +t+1∥2 + 2β2 +t ∥ht∥2, +(168) +where (a) follows from completing the square with the last three terms and the fact that (a + b)2 ≤ 2a2 + 2b2, and (b) +follows from the strong concavity of G(ω). +Rearranging, dividing by 2mβt, taking expectations, and summing yields +T +� +t=1 +E[∥wt − w∗ +t ∥2] ≤ +T +� +t=1 +1 +2mβt +E[∥wt − w∗ +t ∥2 − ∥wt − w∗ +t ∥2] +� +�� +� +M1 ++ +T +� +t=1 +1 +mE[⟨wt − w∗ +t , ∇G(wt) − gt⟩] +� +�� +� +M2 ++ +T +� +t=1 +1 +mβt +E[⟨wt − w∗ +t , w∗ +t − w∗ +t+1⟩] +� +�� +� +M3 ++ +T +� +t=1 +1 +mβt +E[∥w∗ +t − w∗ +t+1∥2] +� +�� +� +M4 ++ +T +� +t=1 +βt +mE[∥gt∥2] +� +�� +� +M5 +. +(169) +As in the proof of Theorem D.1, we first provide intermediate bounds on M1, M2, M3, M4, M5, then manipulate the +resulting expressions to obtain the desired, final bound on the critic error. With the exception of M2, the intermediate bounds +follow by the same reasoning as their counterparts in Theorem D.1. +Bound for M1: By the same reasoning as for I1, +M1 ≤ 2R2 +ω +mβt +. +(170) +Bound for M2: Since ωt, ω∗ +t are deterministic given Ft−1, by the law of total expectation and Lemma D.4 we have +M2 = +T +� +t=1 +1 +mE +� +⟨ωt − ω∗ +t , gjmax +t +− ∇G(ωt)⟩ +� +. +(171) +Furthermore, +|M2| +(a) +≤ +T +� +t=1 +1 +mE +� +∥ωt − ω∗ +t ∥ · ∥gjmax +t +− ∇G(ωt)∥ +� +(172) +(b) +≤ +T +� +t=1 +1 +m +� +E +� +∥ωt − ω∗ +t ∥2��1/2 � +E +� +∥gjmax +t +− ∇G(ωt)∥ +2��1/2 +(173) +(c) +≤ +� +1 +m2 +T +� +t=1 +E +� +∥ωt − ω∗ +t ∥2��1/2 � T +� +t=1 +E +� +∥gjmax +t +− ∇G(ωt)∥ +2��1/2 +(174) +(d) +≤ +� +1 +m2 +T +� +t=1 +E +� +∥ωt − ω∗ +t ∥2��1/2 � +T � +O +� +G2 +G max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ D2 +T +� +t=1 +E +� +∥ηt − η∗ +t ∥2��1/2 +, +(175) + +Multi-Level Monte Carlo Actor-Critic (MAC) +26 +where (a) follows by applying the triangle, Jensen’s, and Cauchy-Schwarz inequalities, (b) and (c) follow from H¨older’s +inequality, and (d) results from applying Lemma D.3. +Bound for M3: Since ω∗(θ) is Lω-Lipschitz in θ by Lemma D.2, we have ∥ω∗ +t − ω∗ +t+1∥ ≤ Lω∥θt − θt+1∥ ≤ LωGHαt, +where we recall that supθ ∥∇J(θ)∥ ≤ GH. Thus, by reasoning analogous to I3, +|M3| ≤ +� T +� +t=1 +E +� +∥ωt − ω∗ +t ∥2��1/2 � +L2 +ωG2 +H +m2 +T +� +t=1 +α2 +t +β2 +t +�1/2 +. +(176) +Bound for M4: Similarly, +M4 ≤ L2 +ωG2 +H +m +T +� +k=1 +α2 +t +βt +. +(177) +Bound for M5: Finally, by Lemma D.4 and the fact that |ηt| ≤ R, for all t, +M5 ≤ +T +� +t=1 +βt +m +� +� +O +� +G2 +Hτ θt +mix log Tmax +� ++ 8D2 log(Tmax)TmaxE +� +(ηt − η∗ +t )2�� +(178) +≤ +� +� +O +� +G2 +Hτ θt +mix log Tmax +� ++ 16D2R2 log(Tmax)Tmax +� +T +� +k=1 +βt +m. +(179) +Combining the foregoing and recalling the definitions of βt, αt, α′ +t, we have +T +� +t=1 +E +� +∥ωt − ω∗ +t ∥2� +≤ 2Rω +m (1 + t)ν +(180) ++ +� +1 +m2 +T +� +t=1 +E +� +∥ωt − ω∗ +t ∥2��1/2 � +T � +O +� +G2 +G max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ D2 +T +� +t=1 +E +� +∥ηt − η∗ +t ∥2��1/2 +(181) ++ +� T +� +t=1 +E +� +∥ωt − ω∗ +t ∥2��1/2 � +L2 +ωG2 +H +m2 +T +� +t=1 +(1 + t)−2(σ−ν) +�1/2 +(182) ++ L2 +ωG2 +H +m +T +� +k=1 +(1 + t)ν−2σ +(183) ++ � +O +� +G2 +H max +t∈[T ] τ θt +mix log(Tmax)Tmax +� +T +� +t=1 +(1 + t)−ν. +(184) +Define +Z(T) = +T +� +t=1 +E +� +∥ωt − ω∗ +t ∥2� +, +(185) +F(T) = L2 +ωG2 +H +4m2 +T +� +t=1 +(1 + t)−2(σ−ν), +(186) +G(T) = +1 +16m +� +T � +O +� +G2 +G max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ D2 +T +� +t=1 +E +� +∥ηt − η∗ +t ∥2�� +, +(187) +A(T) = 2Rω +m (1 + t)ν + L2 +ωG2 +H +m +T +� +k=1 +(1 + t)ν−2σ + � +O +� +G2 +H max +t∈[T ] τ θt +mix log(Tmax)Tmax +� +T +� +t=1 +(1 + t)−ν. +(188) + +Multi-Level Monte Carlo Actor-Critic (MAC) +27 +The previous inequality is thus the same as +Z(T) ≤ A(T) + 2 +� +Z(T) +� +F(T) + 2 +� +Z(T) +� +G(T), +(189) +which is in turn equivalent to +�� +Z(T) − +� +F(T) − +� +G(T) +�2 +≤ A(T) + +�� +F(T) + +� +G(T) +�2 +. +(190) +This yields +� +Z(T) − +� +F(T) − +� +G(T) ≤ +� +A(T) + +�� +F(T) + +� +G(T) +�2�1/2 +(191) +≤ +� +A(T) + +� +F(T) + +� +G(T), +(192) +whence +� +Z(T) ≤ +� +A(T) + 2 +� +F(T) + 2 +� +G(T) +(193) +and thus +Z(T) ≤ 2A(T) + 2 +� +2 +� +F(T) + 2 +� +G(T) +�2 +(194) +≤ 2A(T) + 16F(T) + 16G(T). +(195) +Noticing that 2A(T) + 16F(T) = O (T ν) + O +� +T 1+ν−2σ� ++ O +� +T 1−ν� +and using the bound �T +t=1(1 + t)−ξ ≤ (1 + +t)1−ξ/(1 − ξ), we have +T +� +t=1 +E +� +∥ωt − ω∗ +t ∥2� +≤ 1 +m +� +T � +O +� +G2 +G max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ D2 +T +� +t=1 +E +� +∥ηt − η∗ +t ∥2�� +(196) ++ O (T ν) + O +� +T 1+ν−2σ� ++ O +� +T 1−ν� +. +(197) +Dividing by T, combining with Theorem D.1, and absorbing constants into the order notation finishes the proof. +E. Proof of Theorem 4.8 +Proof. From the statement of Theorems 4.6 and 4.7, we have +1 +T +T +� +t=1 +E +� +∥∇J(θt)∥2� +≤ O +� 1 +√ +T +� ++ O +� +1 +T +T +� +t=1 +E(t) +� ++ � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ O (Eapp) , +(198) +and +1 +T +T +� +t=1 +E(t) ≤O +� +T ν−1� ++ O +� +T −2(σ−ν)� ++ � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +O +� +T −ν� ++ � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� +. (199) +utilizing the upper bound in (199) into the right hand side of (198), we get +1 +T +T +� +t=1 +E +� +∥∇J(θt)∥2� +≤O +� 1 +√ +T +� ++ O +� +T ν−1� ++ O +� +T −2(σ−ν)� ++ � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +O +� +T −ν� ++ � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ O (Eapp) . +(200) + +Multi-Level Monte Carlo Actor-Critic (MAC) +28 +For the selection ν = 0.5 and σ = 0.75 (which satisfies the constraint that 0 < ν < σ < 1), we obtain +1 +T +T +� +t=1 +E +� +∥∇J(θt)∥2� +≤O +� 1 +√ +T +� ++ O +� 1 +√ +T +� ++ O +� 1 +√ +T +� ++ � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +O +� 1 +√ +T +� ++ � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ O (Eapp) . +(201) +Therefore, after further simplification, we can write +1 +T +T +� +t=1 +E +� +∥∇J(θt)∥2� +≤ � +O +� +max +t∈[T ] τ θt +mix log Tmax +� +O +� 1 +√ +T +� ++ � +O +�� +max +t∈[T ] τ θt +mix +log Tmax +Tmax +� ++ O (Eapp) . +(202) +completes the proof. +F. Hyperparametrs for the Experiments +We list all the hyperparameters in Table 2 here. +Table 2. This table compares the hyperparameters and performance between the four experiments, each run for five trials. From the table, +we see that given the same learning rates, environment, and the number of samples, MAC and Vanilla AC converge to the same reward +value. +Method +Learning Rate +Grid Size +Tmax +Samples +Limiting +Limiting Policy +Actor +Critic +Reward Estimator +Processed +Mean Reward +Gradient Norm +MAC +.01 +.01 +.01 +6 × 6 +8 +3 · 106 +0.4 +0 +Vanilla AC +.01 +.01 +.01 +6 × 6 +3 +3 · 106 +0.4 +0 +MAC +.005 +.005 +.005 +10 × 10 +16 +4 · 106 +0.5 +0 +Vanilla AC +.005 +.005 +.005 +10 × 10 +4 +4 · 106 +0.5 +0 + diff --git a/kNFLT4oBgHgl3EQfci-Z/content/tmp_files/load_file.txt b/kNFLT4oBgHgl3EQfci-Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4f154380e5212b1b9dfe1f0f89ef43f8ff280ee --- /dev/null +++ b/kNFLT4oBgHgl3EQfci-Z/content/tmp_files/load_file.txt @@ -0,0 +1,1662 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf,len=1661 +page_content='Beyond Exponentially Fast Mixing in Average-Reward Reinforcement Learning via Multi-Level Monte Carlo Actor-Critic Wesley A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' Suttle * 1 Amrit Singh Bedi * 2 Bhrij Patel 2 Brian Sadler 1 Alec Koppel 3 Dinesh Manocha 2 Abstract Many existing reinforcement learning (RL) meth- ods employ stochastic gradient iteration on the back end, whose stability hinges upon a hypoth- esis that the data-generating process mixes expo- nentially fast with a rate parameter that appears in the step-size selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' Unfortunately, this as- sumption is violated for large state spaces or set- tings with sparse rewards, and the mixing time is unknown, making the step size inoperable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' In this work, we propose an RL methodology attuned to the mixing time by employing a multi-level Monte Carlo estimator for the critic, the actor, and the average reward embedded within an actor- critic (AC) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' This method, which we call Multi-level Actor-Critic (MAC), is developed especially for infinite-horizon average-reward set- tings and neither relies on oracle knowledge of the mixing time in its parameter selection nor as- sumes its exponential decay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' it, therefore, is read- ily applicable to applications with slower mixing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' Nonetheless, it achieves a convergence rate comparable to the state-of-the-art AC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' We experimentally show that these alleviated re- strictions on the technical conditions required for stability translate to superior performance in prac- tice for RL problems with sparse rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' Introduction Modern machine learning (ML) techniques have enabled an- alyzing and making predictions from large-scale data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' This is achieved through backpropagation in neural networks (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=', 2006), cloud processing of industrial data sets (McAfee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=', 2012), complex event simulators (Silver Equal contribution 1U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' Army Research Laboratory, Adel- phi, MD, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' 2Department of Computer Science, University of Maryland, College Park, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' 3JP Morgan Chase AI Re- search, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' Correspondence to: Wesley A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFLT4oBgHgl3EQfci-Z/content/2301.12083v1.pdf'} +page_content=' Suttle