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+arXiv:2301.13439v1  [hep-ph]  31 Jan 2023
+USTC-ICTS/PCFT-23-04
+January 2023
+Rare W-boson decays into a vector meson and lepton pair
+Dao-Neng Gao†
+Interdisciplinary Center for Theoretical Study, University of Science and Technology of China,
+Hefei, Anhui 230026 China
+Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026 China
+Abstract
+We have presented a theoretical study of exclusive rare W-boson decays, W → V ℓ¯νℓ
+with V denoting a neutral vector meson and ℓ = e or µ, in the standard model.
+The leading-order contributions to these processes are given by W → γ∗ℓ¯νℓ with the
+subsequent γ∗ → V transition. Branching fractions of these decay modes, for V = ρ,
+ω, φ, and J/Ψ, respectively, have been calculated and predicted around 10−6 ∼ 10−7,
+which are surprisingly larger than those of two-body hadronic radiative decays W ± →
+M±γ with M denoting a pseudoscalar or vector meson. Thus it is expected that rare
+W decays into a neutral vector meson plus lepton pair may be the promising channels
+in future experimental facilities with a large number of W-boson events produced.
+† E-mail address: [email protected]
+
+Exclusive rare W-boson decays, which contain hadronic final states, could provide inter-
+esting probes to increase our understanding of the properties of the fundamental weak gauge
+boson as well as offer some deep insights into quantum chromodynamics [1, 2, 3, 4, 5]. Experi-
+mentally, no such processes have been observed so far, and only upper limits on the branching
+fractions of three exclusive modes: B(W ± → D±
+s γ) < 1.3×10−3, B(W ± → π±γ) < 7×10−6,
+and B(W ± → π+π−π±) < 1.01 × 10−6, were set at 95% confidence level [6]. On the other
+hand, a huge number of W events, about O(1011), will be expectedly accumulated in the
+high-luminosity Large Hadron Collider (LHC) [3]. This may significantly facilitate the ex-
+perimental studies of rare W-boson decay channels, which can be very helpful both to test
+the standard model (SM) and to search for new physics beyond the SM.
+Our main focus in the present paper is on another types of rare W-boson decays: W →
+V ℓ¯νℓ with V denoting the neutral vector particle including heavy quarkonium J/Ψ or light
+mesons ρ, ω, and φ etc.
+ℓ is the lepton with ℓ = e or µ.
+The leading-order Feynman
+diagrams contributing to these processes in the SM have been shown in Figure 1, in which
+the transitions can proceed through W → γ∗ℓ¯νℓ, followed by γ∗ → ¯qq → V . This is similar
+to the case of Z → V ℓ+ℓ+ decays, which have been studied in Refs. [7, 8, 9].
+First let us go into the decay amplitude of W − → V ℓ−¯νℓ. Using the standard vertices
+Wℓ¯νℓ, γq¯q, and γWW, one can carry out the direct calculation for Figure 1, which gives
+M(W − → V ℓ−¯νℓ)
+=
+−e2gQV fV
+2
+√
+2mV
+ǫµ(p)ǫ∗
+ν(q)¯u(k1)
+�2kν
+1γµ + γνq/γµ
+q2 + 2k1 · q
+−(2k + q)νγµ + 2qµγν − 2q/gµν
+q2 + 2k · q
+�
+(1 − γ5)v(k2),
+(1)
+where p, q, k1, and k2 represent the momenta of W − and the final particles including V , ℓ−,
+and ¯νℓ, respectively. k = k1 + k2 denotes the momentum sum of lepton pair. e is the QED
+coupling constant and g is the weak SU(2)L coupling constant. fV is the decay constant of
+the vector meson, which is defined by
+⟨V (p, ǫ)|¯qγνq|0⟩ = fV mV ǫ∗
+ν.
+(2)
+Here ǫ∗
+ν is polarization vector of V , and the value of fV can be extracted from the measured
+V → e+e− width. As shown in Ref. [3], it has been already given that, fρ = 216.3 ± 1.3
+MeV, fω = 194.2 ± 2.1 MeV, fφ = 223.0 ± 1.4 MeV, and fJ/Ψ = 403.3 ± 5.1 MeV. QV is the
+quantity related to the electric charge of the quark inside V with Qρ = 1/
+√
+2, Qω = 1/3
+√
+2,
+Qφ = −1/3, and QJ/Ψ = 2/3. Note that the use of the relation (2) in deriving eq. (1) also
+fulfills the hadronization of the electromagnetic current ¯qγνq into the final state particle V .
+Next, by squaring the decay amplitude (1), summing or averaging the polarizations of
+final or initial particles, the differential decay rate of W − → V ℓ−¯νℓ can be expressed as
+dΓ
+dsV dsℓ
+= mW
+256π3
+1
+3
+�
+spins
+|M(W − → V ℓ−¯νℓ)|2.
+(3)
+Consequently, we get
+dΓ
+dsV dsℓ
+= α2
+emQ2
+V g2f 2
+V
+384πmWr2
+V
+IV ,
+(4)
+1
+
+*
+*
+*
+(a)
+(b)
+W
+W
+W
+V
+V
+�
+�
+�
+�
+��
+�
+�
+_
+_
+Figure 1: The lowest-order Feynman diagrams for W → V ℓ¯νℓ decays.
+where αem = e2/4π, rV = mV /mW, and the lepton mass has been neglected in the calcula-
+tion. The explicit expression of the dimensionless quantity IV is a little tedious, which will
+be shown in the Appendix. The Lorentz invariant dimensionless kinematical variables are
+defined as
+sV ≡ (p − q)2/m2
+W,
+sℓ ≡ (p − k1)2/m2
+W,
+(5)
+and the phase space can be given by
+0 ≤ sV ≤ (1 − sℓ)(1 − r2
+V /sℓ),
+r2
+V ≤ sℓ ≤ 1.
+(6)
+Meanwhile, it is easy to compute the leading-order contribution to the width of pure
+leptonic W-boson decay for ℓ = e or µ, which reads
+Γ(W − → ℓ−¯νℓ) = g2mW
+48π
+= GFm3
+W
+6
+√
+2π ≡ Γ0,
+(7)
+where GF is the Fermi constant given by GF/
+√
+2 = g2/8m2
+W.
+Then one can choose to
+normalize the decay rate of W − → V ℓ−ℓ¯νℓ to Γ0, which leads to
+1
+Γ0
+dΓ
+dsV dsℓ
+= α2
+emQ2
+V f 2
+V
+8m2
+V
+IV .
+(8)
+By further defining
+YV ≡
+�
+IV dsV dsℓ
+(9)
+with the integral bound is given in eq. (6), one can get
+Γ(W − → V ℓ−¯νℓ)
+Γ0
+= α2
+emQ2
+V f 2
+V
+8m2
+V
+YV .
+(10)
+As mentioned above, the decay constants (fV ) of the neutral vector mesons have been
+extracted by the authors of Ref. [3] from the experimental data, and
+Γ(V → e+e−) = 4πQ2
+V f 2
+V
+3mV
+α2
+em(mV )
+(11)
+2
+
+V
+mV (GeV)
+Γ(V → e+e−)(keV)
+YV
+Γ(W − → V ℓ−¯νℓ)/Γ0
+ρ
+0.775
+7.04 ± 0.06
+194.91
+(5.28 ± 0.04) × 10−5
+ω
+0.782
+0.60 ± 0.02
+193.94
+(4.44 ± 0.15) × 10−6
+φ
+1.019
+1.27 ± 0.04
+166.32
+(6.18 ± 0.19) × 10−6
+J/Ψ
+3.097
+5.53 ± 0.10
+74.53
+(3.97 ± 0.07) × 10−6
+Table 1: Decay rates of W − → V ℓ−¯νℓ normalized to Γ(W − → ℓ−¯νℓ) for ℓ = e or µ. The
+values of Γ(V → e+e−) are taken from Ref. [6].
+has been used. Therefore, after integrating over IV in eq. (9) to get YV , one can easily
+predict the decay rates of W → V ℓ¯νℓ for V = ρ, ω, φ, and J/Ψ, respectively.
+On the other hand, note that the scale of the electromagnetic coupling αem in eq. (8)
+should also be at mV since, in Figure 1, this electromagnetic transition is via γ∗ → V .
+Therefore, combing eq. (10) with eq. (11), one will obtain
+Γ(W − → V ℓ−¯νℓ)
+Γ0
+=
+3YV
+32πmV
+Γ(V → e+e−),
+(12)
+which means that we can get Γ(W − → V ℓ−¯νℓ)/Γ0 using the experimental data of Γ(V →
+e+e−) given by Particle Data Group [6] directly. Numerical results have been listed in Table
+1, and the errors of the predictions in the fifth column are due to the uncertainties of the
+measured widths of Γ(V → e+e−) only. To transform them into the branching fractions of
+W → V ℓ¯νℓ, one may use the experimental data of B(W → ℓ¯νℓ), which can be found in Ref.
+[6] that
+B(W − → e−¯νe) = (10.71 ± 0.16)%,
+B(W − → µ−¯νµ) = (10.63 ± 0.15)%.
+(13)
+For our numerical analysis, we take
+B(W − → ℓ−¯νℓ) = (10.67 ± 0.16)%
+(14)
+by simply averaging over the electron and muon modes. Thus, it is straightforward to obtain
+the branching fractions of rare W-boson decays into a vector meson and lepton pair, for ℓ = e
+or µ, which read
+B(W − → ρℓ−¯νℓ) = (5.64 ± 0.10) × 10−6,
+(15)
+B(W − → ωℓ−¯νℓ) = (4.74 ± 0.17) × 10−7,
+(16)
+B(W − → φℓ−¯νℓ) = (6.60 ± 0.23) × 10−7,
+(17)
+B(W − → J/Ψℓ−¯νℓ) = (4.24 ± 0.10) × 10−7.
+(18)
+Here the quoted errors of our theoretical results show the uncertainties from the experimental
+values of Γ(V → e+e−) in the third column of Table 1, and also B(W − → ℓ−¯νℓ) in eq. (14).
+It is found that branching ratios of W → V ℓ¯νℓ decays obtained in the present work are
+quite larger than those of the hadronic radiative decays W ± → M±γ (M is a pseudoscalar
+3
+
+W
+W
+J/�
+�
+�
+�
+_
+c
+s
+c
+_
+*
+Figure 2: The Feynman diagram contributing to W → J/Ψℓ¯νℓ decays via W → J/ΨW ∗
+transition.
+or vector meson such as π, K, ρ, K∗, and Ds etc), which are maximally around 10−8 or even
+smaller, predicted by the authors of Ref. [3]. Naively, one may expect that Γ(W → V ℓ¯νℓ)
+should be smaller than Γ(W ± → M±γ) since the former rate is suppressed by a power of
+αem compared to the latter rate. However, careful observation can tell us this expectation is
+not correct. As given in Ref. [3], we know
+Γ(W ± → M±γ) ∼ αemf 2
+M
+192mW
+.
+(19)
+Comparing with eq. (4), one will find a relevant factor m2
+W/m2
+V in the formula of Γ(W − →
+V ℓ−¯νℓ), which could significantly counteract the suppression of αem if the mass of vector
+meson is very small relative to the W mass. Obviously, the appearance of this factor is
+actually due to the virtual photon propagator of γ∗ → V transition in Figure 1.
+Similar situation also occurs in rare Z-boson decays. In particular, it has been shown in
+Ref. [8] that the dominant contribution to Z → V ℓ+ℓ− comes from Z → γ∗ℓ+ℓ− with the
+subsequent transition γ∗ → V , since, in comparison, the radiative decays Z → V γ are quite
+suppressed. One can thus neglect the contribution from Z → V γ∗ → V ℓ+ℓ− although it is
+of the same order of αem as the dominant part.
+Analogous to Z → V γ∗ → V ℓ+ℓ−, the rare charged weak gauge boson decays considered
+in the present paper could happen through W → V W ∗ → V ℓ¯νℓ. The Feynman diagram
+has been displayed in Figure 2, and we take V = J/Ψ as an explicit example. As a good
+approximation for the leading order calculation, the momenta of the quark (c) and anti-
+quark (¯c) are taken to be one half of J/Ψ momentum q, so the strange quark propagator in
+this diagram is proportional to 1/(k + q
+2)2, which is of order 1/m2
+W. By contrast, the virtual
+photon propagator in the diagrams of Figure 1 is of order 1/m2
+J/Ψ. This means that the
+contribution from Figure 2, relative to that from Figure 1, is strongly suppressed, which can
+be safely neglected.
+Furthermore, recall that the differential decay rate of W − → V ℓ−¯νℓ has been given in
+eq. (4). Now one can rewrite
+sV = 1 + r2
+V − 2EV /mW,
+sℓ = 1 − 2Eℓ/mW,
+(20)
+where EV is the vector meson energy and Eℓ is the lepton energy in the rest frame of W
+4
+
+0
+10
+20
+30
+40
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+EJ (GeV)
+1/
+�
+d
+�
+/dEJ (GeV-1)
+0
+10
+20
+30
+40
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Eℓ
+(GeV)
+1/Γ dΓ/dEℓ (GeV-1)
+Figure 3: The normalized energy spectrum of W → J/Ψℓ−¯νℓ decays with respect to J/Ψ
+energy EJ (left plot), and with respect to the lepton energy Eℓ (right plot).
+boson. In terms of EV and Eℓ, we have
+dΓ
+dEV dEℓ
+= α2
+emQ2
+V g2f 2
+V
+96πm3
+Wr2
+V
+IV .
+(21)
+Thus the energy spectrum of the rare decays can be obtained by integrating over EV or Eℓ.
+The normalized energy distributions of W − → J/Ψℓ−¯νℓ with respect to EJ and Eℓ have
+been plotted in Figure 3, respectively. The peak of the distribution is corresponding to the
+small J/ψ energy or large lepton energy region. Since we have neglected the lepton mass in
+the calculation, the spectrum in left plot does not go to zero for EJ at ∼ mW/2. We are not
+going to display the plots for the differential rate of W − → V ℓ−¯νℓ decays when V is the light
+vector meson (ρ, ω, and φ) because it is believed that one will achieve the similar behavior
+as above.
+To summarize, we have presented the analysis of exclusive rare W-boson decays into a
+vector meson and lepton pair. In the SM, the leading order contributions to these processes
+come from W → γ∗ℓ¯νℓ, followed by γ∗ → V . Using the measured widths of Γ(V → e+e−)
+given in [6], we have determined the branching fractions of W − → V ℓ−¯νℓ for V = ρ, ω,
+φ, and J/Ψ, respectively, as shown in eqs.
+(15) – (18).
+It is surprising that branching
+fractions of these three-body decays, although they are suppressed by a power of αem, are
+quite larger than those of two-body hadronic radiative decays W ± → M±γ, which have been
+predicted by the authors of Ref. [3] already. Furthermore, note that the γWW vertex, as
+shown in Figure 1(b), is involved in the transition, thus both experimental and theoretical
+investigations of W → V ℓ¯νℓ decays may be also helpful to test triple gauge couplings.
+Our experimentalists have been trying to search for exclusive rare W-boson processes
+containing hadronic final states. Unfortunately, so far no such decays have been observed.
+Theoretical predictions on branching fractions of W → V ℓ¯νℓ in the present paper are around
+10−6 ∼ 10−7. Experimentally, the heavy quarkonium J/ψ is in general reconstructed via
+leptonic decays with their rates: B(J/Ψ → ℓ+ℓ−) = (5.971±0.032)% [6]; while for light vector
+mesons, ρ decays almost exclusively to π+π−, ω and φ have a large rate into π+π−π− and
+K+K−, respectively, in the event construction. Therefore, our analysis seems to indicate that
+5
+
+these exclusive rare W decay modes could be the promising candidates in future experimental
+machines, for instance, in the high-luminosity LHC, where large amount of W bosons about
+O(1011) events will be produced. We eagerly await some dedicated searches for such decays
+at these facilities.
+Acknowledgments
+This work was supported in part by the National Natural Science Foundation of China
+under Grants No. 11575175, No. 12047502, and No. 12247103, and by National Research
+and Development Program of China under Contract No. 2020YFA0406400.
+Appendix: Explicit expression of IV
+After squaring the W − → V ℓ−¯νℓ decay amplitude and summing/averaging spins of all par-
+ticles, we get the differential decay rate of eq. (4) as
+dΓ
+dsV dsℓ
+= α2
+emQ2
+V g2f 2
+V
+384πmWr2
+V
+IV .
+The full expression of IV , in terms of dot product of the relevant four-momenta, can be given
+by
+IV = I1 + I2 + I3,
+(A1)
+where
+I1 =
+8
+(q2 + 2k1 · q)2
+�
+(2k1 · qk2 · q − q2k1 · k2) + 2p · k2
+m2
+W
+(2k1 · qp · q − q2k1 · p)
+�
+,
+(A2)
+I2 =
+8
+(q2 + 2k1 · q)(q2 + 2k · q)
+�
+2k1 · k2(4k1 · k2 + 2k · q + 4q2) − 4k1 · qk2 · q
+− 1
+m2
+W
+[(2k1 · q + 4k · q + 3q2)(2k1 · qk2 · q − q2k1 · k2)
+−4p · k2(q2 + 2k2 · q)k1 · k2 + 4p · q(q2 + 2k1 · q)k1 · k2]
+�
+,
+(A3)
+and
+I3 =
+8
+(q2 + 2k · q)2
+�
+12k1 · qk2 · q − ((2k + q)2 + 10q2)k1 · k2
+−
+1
+2m2
+W
+(2k + q)2(2k1 · qk2 · q − q2k1 · k2) + 4(p · q)2
+m2
+W
+�
+.
+(A4)
+On the other hand, one can easily get
+k1 · k2 = m2
+W
+2 sV ,
+p · q = m2
+W
+2 (1 + r2
+V − sV ),
+6
+
+k1 · q = m2
+W
+2 (1 − sV − sℓ),
+k2 · q = m2
+W
+2 (sℓ − r2
+V ),
+k1 · p = m2
+W
+2 (1 − sℓ),
+k2 · p = m2
+W
+2 (sV + sℓ − r2
+V ).
+For on-shell initial and final states particles, we could take p2 = m2
+W, q2 = m2
+V , and k2
+1 =
+k2
+2 = 0 (lepton masses are set to be zero already). This shows that IV can be in terms of the
+kinematical variables sV and sℓ completely.
+References
+[1] L. Arnellos, W. J. Marciano, and Z. Parsa, Nucl. Phys. B196, 378 (1982).
+[2] M. Mangano and T. Melia, Eur. Phys. J. C 75, 258 (2015), arXiv:1410.7475 [hep-ph].
+[3] Y. Grossman, M. K¨onig, and M. Neubert, J. High Energy Phys. 04 (2015) 101,
+arXiv:1501.06569 [hep-ph].
+[4] Y.Y. Keum and X.Y. Pham, Mod. Phys. Lett. A 9, 1545 (1994), hep-ph/9303300.
+[5] S. Ishaq, Y. Jia, X. Xiong, and D.-S. Yang, Phys. Rev. D 100, 054027 (2019),
+arXiv:1903.12627 [hep-ph]; F. Feng, Y. Jia, and W.-L. Sang, arXiv:1902.11288 [hep-
+ph].
+[6] R.L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01
+(2022).
+[7] L. Bergstr¨om and R.W. Robinett, Phys. Lett. B 245, 249 (1990).
+[8] S. Fleming, Phys. Rev. D 48, R1914 (1993), hep-ph/9304270; S. Fleming, Phys. Rev.
+D 50, 5808 (1994), hep-ph/9403396.
+[9] CMS Collaboration, A.M. Sirunyan et al., Phys. Rev. Lett. 121, 141801 (2018),
+arXiv:1806.04213 [hep-ex].
+7
+

-NFQT4oBgHgl3EQf6jZ7/content/tmp_files/load_file.txt

@@ -0,0 +1,261 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf,len=260
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='13439v1  [hep-ph]  31 Jan 2023  USTC-ICTS/PCFT-23-04  January 2023  Rare W-boson decays into a vector meson and lepton pair  Dao-Neng Gao†  Interdisciplinary Center for Theoretical Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Anhui 230026 China  Peng Huanwu Center for Fundamental Theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Anhui 230026 China  Abstract  We have presented a theoretical study of exclusive rare W-boson decays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' W → V ℓ¯νℓ  with V denoting a neutral vector meson and ℓ = e or µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  The leading-order contributions to these processes are given by W → γ∗ℓ¯νℓ with the  subsequent γ∗ → V transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Branching fractions of these decay modes, for V = ρ,  ω, φ, and J/Ψ, respectively, have been calculated and predicted around 10−6 ∼ 10−7,  which are surprisingly larger than those of two-body hadronic radiative decays W ± →  M±γ with M denoting a pseudoscalar or vector meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Thus it is expected that rare  W decays into a neutral vector meson plus lepton pair may be the promising channels  in future experimental facilities with a large number of W-boson events produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  † E-mail address: gaodn@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='cn  Exclusive rare W-boson decays, which contain hadronic final states, could provide inter-  esting probes to increase our understanding of the properties of the fundamental weak gauge  boson as well as offer some deep insights into quantum chromodynamics [1, 2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Experi-  mentally, no such processes have been observed so far, and only upper limits on the branching  fractions of three exclusive modes: B(W ± → D±  s γ) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='3×10−3, B(W ± → π±γ) < 7×10−6,  and B(W ± → π+π−π±) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='01 × 10−6, were set at 95% confidence level [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' On the other  hand, a huge number of W events, about O(1011), will be expectedly accumulated in the  high-luminosity Large Hadron Collider (LHC) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' This may significantly facilitate the ex-  perimental studies of rare W-boson decay channels, which can be very helpful both to test  the standard model (SM) and to search for new physics beyond the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Our main focus in the present paper is on another types of rare W-boson decays: W →  V ℓ¯νℓ with V denoting the neutral vector particle including heavy quarkonium J/Ψ or light  mesons ρ, ω, and φ etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  ℓ is the lepton with ℓ = e or µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  The leading-order Feynman  diagrams contributing to these processes in the SM have been shown in Figure 1, in which  the transitions can proceed through W → γ∗ℓ¯νℓ, followed by γ∗ → ¯qq → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' This is similar  to the case of Z → V ℓ+ℓ+ decays, which have been studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  First let us go into the decay amplitude of W − → V ℓ−¯νℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Using the standard vertices  Wℓ¯νℓ, γq¯q, and γWW, one can carry out the direct calculation for Figure 1, which gives  M(W − → V ℓ−¯νℓ)  =  −e2gQV fV  2  √  2mV  ǫµ(p)ǫ∗  ν(q)¯u(k1)  �2kν  1γµ + γνq/γµ  q2 + 2k1 · q  −(2k + q)νγµ + 2qµγν − 2q/gµν  q2 + 2k · q  �  (1 − γ5)v(k2),  (1)  where p, q, k1, and k2 represent the momenta of W − and the final particles including V , ℓ−,  and ¯νℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' k = k1 + k2 denotes the momentum sum of lepton pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' e is the QED  coupling constant and g is the weak SU(2)L coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' fV is the decay constant of  the vector meson, which is defined by  ⟨V (p, ǫ)|¯qγνq|0⟩ = fV mV ǫ∗  ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (2)  Here ǫ∗  ν is polarization vector of V , and the value of fV can be extracted from the measured  V → e+e− width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3], it has been already given that, fρ = 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='3  MeV, fω = 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='1 MeV, fφ = 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='4 MeV, and fJ/Ψ = 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='3 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' QV is the  quantity related to the electric charge of the quark inside V with Qρ = 1/  √  2, Qω = 1/3  √  2,  Qφ = −1/3, and QJ/Ψ = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Note that the use of the relation (2) in deriving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (1) also  fulfills the hadronization of the electromagnetic current ¯qγνq into the final state particle V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Next, by squaring the decay amplitude (1), summing or averaging the polarizations of  final or initial particles, the differential decay rate of W − → V ℓ−¯νℓ can be expressed as  dΓ  dsV dsℓ  = mW  256π3  1  3  �  spins  |M(W − → V ℓ−¯νℓ)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (3)  Consequently, we get  dΓ  dsV dsℓ  = α2  emQ2  V g2f 2  V  384πmWr2  V  IV ,  (4)  1 (a)  (b)  W  W  W  V  V  �  �  �  �  ��  �  �  _  _  Figure 1: The lowest-order Feynman diagrams for W → V ℓ¯νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  where αem = e2/4π, rV = mV /mW, and the lepton mass has been neglected in the calcula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The explicit expression of the dimensionless quantity IV is a little tedious, which will  be shown in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The Lorentz invariant dimensionless kinematical variables are  defined as  sV ≡ (p − q)2/m2  W,  sℓ ≡ (p − k1)2/m2  W,  (5)  and the phase space can be given by  0 ≤ sV ≤ (1 − sℓ)(1 − r2  V /sℓ),  r2  V ≤ sℓ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (6)  Meanwhile, it is easy to compute the leading-order contribution to the width of pure  leptonic W-boson decay for ℓ = e or µ, which reads  Γ(W − → ℓ−¯νℓ) = g2mW  48π  = GFm3  W  6  √  2π ≡ Γ0,  (7)  where GF is the Fermi constant given by GF/  √  2 = g2/8m2  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Then one can choose to  normalize the decay rate of W − → V ℓ−ℓ¯νℓ to Γ0, which leads to  1  Γ0  dΓ  dsV dsℓ  = α2  emQ2  V f 2  V  8m2  V  IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (8)  By further defining  YV ≡  �  IV dsV dsℓ  (9)  with the integral bound is given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (6), one can get  Γ(W − → V ℓ−¯νℓ)  Γ0  = α2  emQ2  V f 2  V  8m2  V  YV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (10)  As mentioned above, the decay constants (fV ) of the neutral vector mesons have been  extracted by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3] from the experimental data, and  Γ(V → e+e−) = 4πQ2  V f 2  V  3mV  α2  em(mV )  (11)  2  V  mV (GeV)  Γ(V → e+e−)(keV)  YV  Γ(W − → V ℓ−¯νℓ)/Γ0  ρ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='775  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='06  194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='91  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04) × 10−5  ω  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='782  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='02  193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='94  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='15) × 10−6  φ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='019  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04  166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='32  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='19) × 10−6  J/Ψ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='097  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='10  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='53  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='07) × 10−6  Table 1: Decay rates of W − → V ℓ−¯νℓ normalized to Γ(W − → ℓ−¯νℓ) for ℓ = e or µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The  values of Γ(V → e+e−) are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Therefore, after integrating over IV in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (9) to get YV , one can easily  predict the decay rates of W → V ℓ¯νℓ for V = ρ, ω, φ, and J/Ψ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  On the other hand, note that the scale of the electromagnetic coupling αem in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (8)  should also be at mV since, in Figure 1, this electromagnetic transition is via γ∗ → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Therefore, combing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (10) with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (11), one will obtain  Γ(W − → V ℓ−¯νℓ)  Γ0  =  3YV  32πmV  Γ(V → e+e−),  (12)  which means that we can get Γ(W − → V ℓ−¯νℓ)/Γ0 using the experimental data of Γ(V →  e+e−) given by Particle Data Group [6] directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Numerical results have been listed in Table  1, and the errors of the predictions in the fifth column are due to the uncertainties of the  measured widths of Γ(V → e+e−) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' To transform them into the branching fractions of  W → V ℓ¯νℓ, one may use the experimental data of B(W → ℓ¯νℓ), which can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [6] that  B(W − → e−¯νe) = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='16)%,  B(W − → µ−¯νµ) = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='15)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (13)  For our numerical analysis, we take  B(W − → ℓ−¯νℓ) = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='16)%  (14)  by simply averaging over the electron and muon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Thus, it is straightforward to obtain  the branching fractions of rare W-boson decays into a vector meson and lepton pair, for ℓ = e  or µ, which read  B(W − → ρℓ−¯νℓ) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='10) × 10−6,  (15)  B(W − → ωℓ−¯νℓ) = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='17) × 10−7,  (16)  B(W − → φℓ−¯νℓ) = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='23) × 10−7,  (17)  B(W − → J/Ψℓ−¯νℓ) = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='10) × 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (18)  Here the quoted errors of our theoretical results show the uncertainties from the experimental  values of Γ(V → e+e−) in the third column of Table 1, and also B(W − → ℓ−¯νℓ) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  It is found that branching ratios of W → V ℓ¯νℓ decays obtained in the present work are  quite larger than those of the hadronic radiative decays W ± → M±γ (M is a pseudoscalar  3  W  W  J/�  �  �  �  _  c  s  c  _ Figure 2: The Feynman diagram contributing to W → J/Ψℓ¯νℓ decays via W → J/ΨW ∗  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  or vector meson such as π, K, ρ, K∗, and Ds etc), which are maximally around 10−8 or even  smaller, predicted by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Naively, one may expect that Γ(W → V ℓ¯νℓ)  should be smaller than Γ(W ± → M±γ) since the former rate is suppressed by a power of  αem compared to the latter rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' However, careful observation can tell us this expectation is  not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' As given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3], we know  Γ(W ± → M±γ) ∼ αemf 2  M  192mW  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (19)  Comparing with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (4), one will find a relevant factor m2  W/m2  V in the formula of Γ(W − →  V ℓ−¯νℓ), which could significantly counteract the suppression of αem if the mass of vector  meson is very small relative to the W mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Obviously, the appearance of this factor is  actually due to the virtual photon propagator of γ∗ → V transition in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Similar situation also occurs in rare Z-boson decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' In particular, it has been shown in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [8] that the dominant contribution to Z → V ℓ+ℓ− comes from Z → γ∗ℓ+ℓ− with the  subsequent transition γ∗ → V , since, in comparison, the radiative decays Z → V γ are quite  suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' One can thus neglect the contribution from Z → V γ∗ → V ℓ+ℓ− although it is  of the same order of αem as the dominant part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Analogous to Z → V γ∗ → V ℓ+ℓ−, the rare charged weak gauge boson decays considered  in the present paper could happen through W → V W ∗ → V ℓ¯νℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The Feynman diagram  has been displayed in Figure 2, and we take V = J/Ψ as an explicit example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' As a good  approximation for the leading order calculation, the momenta of the quark (c) and anti-  quark (¯c) are taken to be one half of J/Ψ momentum q, so the strange quark propagator in  this diagram is proportional to 1/(k + q  2)2, which is of order 1/m2  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' By contrast, the virtual  photon propagator in the diagrams of Figure 1 is of order 1/m2  J/Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' This means that the  contribution from Figure 2, relative to that from Figure 1, is strongly suppressed, which can  be safely neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Furthermore, recall that the differential decay rate of W − → V ℓ−¯νℓ has been given in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Now one can rewrite  sV = 1 + r2  V − 2EV /mW,  sℓ = 1 − 2Eℓ/mW,  (20)  where EV is the vector meson energy and Eℓ is the lepton energy in the rest frame of W  4  0  10  20  30  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='06  EJ (GeV)  1/  �  d  �  /dEJ (GeV-1)  0  10  20  30  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='06  Eℓ  (GeV)  1/Γ dΓ/dEℓ (GeV-1)  Figure 3: The normalized energy spectrum of W → J/Ψℓ−¯νℓ decays with respect to J/Ψ  energy EJ (left plot), and with respect to the lepton energy Eℓ (right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' In terms of EV and Eℓ, we have  dΓ  dEV dEℓ  = α2  emQ2  V g2f 2  V  96πm3  Wr2  V  IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (21)  Thus the energy spectrum of the rare decays can be obtained by integrating over EV or Eℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  The normalized energy distributions of W − → J/Ψℓ−¯νℓ with respect to EJ and Eℓ have  been plotted in Figure 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The peak of the distribution is corresponding to the  small J/ψ energy or large lepton energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Since we have neglected the lepton mass in  the calculation, the spectrum in left plot does not go to zero for EJ at ∼ mW/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' We are not  going to display the plots for the differential rate of W − → V ℓ−¯νℓ decays when V is the light  vector meson (ρ, ω, and φ) because it is believed that one will achieve the similar behavior  as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  To summarize, we have presented the analysis of exclusive rare W-boson decays into a  vector meson and lepton pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' In the SM, the leading order contributions to these processes  come from W → γ∗ℓ¯νℓ, followed by γ∗ → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Using the measured widths of Γ(V → e+e−)  given in [6], we have determined the branching fractions of W − → V ℓ−¯νℓ for V = ρ, ω,  φ, and J/Ψ, respectively, as shown in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (15) – (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  It is surprising that branching  fractions of these three-body decays, although they are suppressed by a power of αem, are  quite larger than those of two-body hadronic radiative decays W ± → M±γ, which have been  predicted by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3] already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Furthermore, note that the γWW vertex, as  shown in Figure 1(b), is involved in the transition, thus both experimental and theoretical  investigations of W → V ℓ¯νℓ decays may be also helpful to test triple gauge couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Our experimentalists have been trying to search for exclusive rare W-boson processes  containing hadronic final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Unfortunately, so far no such decays have been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Theoretical predictions on branching fractions of W → V ℓ¯νℓ in the present paper are around  10−6 ∼ 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Experimentally, the heavy quarkonium J/ψ is in general reconstructed via  leptonic decays with their rates: B(J/Ψ → ℓ+ℓ−) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='971±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='032)% [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' while for light vector  mesons, ρ decays almost exclusively to π+π−, ω and φ have a large rate into π+π−π− and  K+K−, respectively, in the event construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Therefore, our analysis seems to indicate that  5  these exclusive rare W decay modes could be the promising candidates in future experimental  machines, for instance, in the high-luminosity LHC, where large amount of W bosons about  O(1011) events will be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' We eagerly await some dedicated searches for such decays  at these facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Acknowledgments  This work was supported in part by the National Natural Science Foundation of China  under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 11575175, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 12047502, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 12247103, and by National Research  and Development Program of China under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 2020YFA0406400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Appendix: Explicit expression of IV  After squaring the W − → V ℓ−¯νℓ decay amplitude and summing/averaging spins of all par-  ticles, we get the differential decay rate of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (4) as  dΓ  dsV dsℓ  = α2  emQ2  V g2f 2  V  384πmWr2  V  IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  The full expression of IV ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' in terms of dot product of the relevant four-momenta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' can be given  by  IV = I1 + I2 + I3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (A1)  where  I1 =  8  (q2 + 2k1 · q)2  �  (2k1 · qk2 · q − q2k1 · k2) + 2p · k2  m2  W  (2k1 · qp · q − q2k1 · p)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (A2)  I2 =  8  (q2 + 2k1 · q)(q2 + 2k · q)  �  2k1 · k2(4k1 · k2 + 2k · q + 4q2) − 4k1 · qk2 · q  − 1  m2  W  [(2k1 · q + 4k · q + 3q2)(2k1 · qk2 · q − q2k1 · k2)  −4p · k2(q2 + 2k2 · q)k1 · k2 + 4p · q(q2 + 2k1 · q)k1 · k2]  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (A3)  and  I3 =  8  (q2 + 2k · q)2  �  12k1 · qk2 · q − ((2k + q)2 + 10q2)k1 · k2  −  1  2m2  W  (2k + q)2(2k1 · qk2 · q − q2k1 · k2) + 4(p · q)2  m2  W  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (A4)  On the other hand, one can easily get  k1 · k2 = m2  W  2 sV ,  p · q = m2  W  2 (1 + r2  V − sV ),  6  k1 · q = m2  W  2 (1 − sV − sℓ),  k2 · q = m2  W  2 (sℓ − r2  V ),  k1 · p = m2  W  2 (1 − sℓ),  k2 · p = m2  W  2 (sV + sℓ − r2  V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  For on-shell initial and final states particles, we could take p2 = m2  W, q2 = m2  V , and k2  1 =  k2  2 = 0 (lepton masses are set to be zero already).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' This shows that IV can be in terms of the  kinematical variables sV and sℓ completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
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+arXiv:2301.00743v1  [cs.SC]  2 Jan 2023
+1
+Computing square roots in quaternion algebras
+Przemysław Koprowski
+Institute of Mathematics
+University of Silesia in Katowice
+ul. Bankowa 14, 40-007 Katowice, Poland
+[email protected]
+Abstract. We present an explicit algorithmic method for computing square roots in quaternion
+algebras over global fields of characteristic different from 2.
+Keywords:
+square root computation, quaternion algebra, number fields, global fields
+1.
+Introduction
+The computation of square roots is one of the most basic operations in mathematics. Effective methods
+for computing square roots are among the oldest algorithms in the realm of computational mathemat-
+ics. In fact, Heron’s method for a numerical approximation of a square root of a real number is two
+thousand years old and preceded by the Euclidean algorithm (wildly believed to be the oldest mathe-
+matical algorithm) by only about three to four centuries (for an in-depth discussion on the chronology
+see [1]). Although numerous methods for computing square roots in various algebraic structures are
+known nowadays, some important omissions prevail. Among them are general quaternion algebras.
+Computation of square roots in the algebra of Hamilton quaternions H =
+� −1,−1
+R
+�
+is well-known (see
+[2]) and very simple as for every quaternion
+∈ H there is a subfield K ∼= C of H containing , and
+so the computation the square root in H can be reduced to the computation of the square root in C. It
+is no longer so in a general quaternion algebra Q =
+� α,β
+K
+�
+for an arbitrary field K and two elements
+α, β ∈ K×. To the best of our knowledge, no algorithm for computing quaternionic square roots ex-
+ists in the literature. One possible explanation for this (quite surprising) fact is that in the commutative
+Address for correspondence: Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland
+
+2
+P. Koprowski / Computing square roots in quaternion algebras
+case when one considers a field extension L/K, a typical way to compute a square root of an element
+a ∈ L is to factor the polynomial x2 − a in L[x]. However, for quaternion algebras, there are no
+known polynomial factorization algorithms.
+The sole purpose of this paper is to correct this evident omission and present an explicit algorithm
+for computing square roots in quaternion algebras over arbitrary global fields of characteristic different
+from 2.
+2.
+Notation
+Throughout this paper, K will denote an arbitrary global field of characteristic char K ̸= 2. Hence, K
+is either a number field, i.e. a finite extension of Q (then its characteristic is just 0) or a global function
+field, that is, a finite extension of a rational function field over a finite field Fq, where q is a power of
+an odd prime. The set of nonzero elements of K is denoted K×.
+Recall that a quaternion algebra Q =
+� α,β
+K
+�
+over K is a 4-dimensional K-algebra with a basis
+{1, i, j, k} and a multiplication gathered by the rules:
+i2 = α,
+j2 = β,
+ij = k = −ji.
+As usual, we shall identify the field K with the subfield K · 1 of Q, which is known to coincide with
+the center Z(Q) of Q. We refer the reader to [3, 4] for a comprehensive presentation of the theory of
+quaternion algebras.
+A quaternion
+is called pure (see e.g., [4, Definition 5.2.1]) if
+∈ spanK{i, j, k}. Every quater-
+nion
+∈ Q can be uniquely expressed as a sum
+= a +
+0 of a scalar a ∈ K and a pure quaternion
+0. We write
+:= a −
+0 for the conjugate of . The map that sends a quaternion to its conjugate is an
+involution.
+If x is an element of either a quadratic field extension L = K
+�√α
+�
+of K or a quaternion algebra
+Q =
+� α,β
+K
+�
+over K, we write N (x) := xx and call it the norm of x. If the domain is not clear from
+the context, we write NL/K or NQ/K.
+Remark 2.1. When Q is a quaternion algebra, the norm of in the above sense should not be confused
+with the determinant of the endomorphism of Q defined by the multiplication by , which is often also
+called the norm. For this reason, in [3, 4] the map
+�→
+is called the reduced norm and denoted nrd.
+In that manner, our terminology in the present paper agrees with the one used by Lam in [5] but not
+with the one used by Vigneras in [3] and Voight in [4].
+Equivalency classes of valuations on K are called places. Throughout this paper, places are de-
+noted using fraktur letters p, q, r. Every place of a global field is either archimedean, when it extends
+the standard absolute value on Q (then the field K is necessarily a number field) or non-archimedean.
+Over a global function field, every place is non-archimedean. To avoid monotonous repetitions, non-
+archimedean places will also be called primes (or finite primes when we want to emphasize the fact
+that they are non-archimedean). The completion of K with respect to a place p is denoted Kp. If p is
+a finite prime, we write ordp : K → Z to denote the corresponding (normalized) discrete valuation
+
+P. Koprowski / Computing square roots in quaternion algebras
+3
+on K. The prime p is called dyadic if ordp 2 ̸= 0. The map ordp induces a natural map K×/K×2 → Z/2Z
+on the group of square classes of K that is again denoted ordp.
+If p is an archimedean place, then the completion Kp is isomorphic either to C or to R. The places
+of the second kind are called real. We write sgnr a for the sign of a ∈ K with respect to for the unique
+ordering of K induced by a real place r.
+Given some nonzero elements a1, . . . , an ∈ K we denote by ⟨a1, . . . , an⟩ the quadratic form
+a1x2
+1 + · · · + anx2
+n. Further, if p is a place and a, b ∈ K× we write (a, b)p for the Hilber symbol of a
+and b at p, that is
+(a, b)p :=
+�
+1
+if
+�a,b
+K
+� ∼= M2Kp,
+−1
+otherwise.
+For a quadratic form ξ = ⟨a1, . . . , an⟩ we define its Hasse invariant spξ at p by the formula (see e.g.,
+[5, Definition V.3.17]):
+spξ :=
+�
+i<j
+(ai, aj)p.
+Finally, abusing the notation harmlessly, by log-1 we will denote the (unique) isomorphism from the
+multiplicative group {±1} to the additive group {0, 1} with addition modulo 2.
+3.
+Square roots of non-central elements
+Let us begin by writing down the explicit formula for a square in quaternion algebra so that we can
+easily reference it in the discussion that follows.
+Observation 3.1. If
+= q0 + q1i + q2j + q3k ∈ Q is a quaternion, then
+2 = (q2
+0 + q2
+1α + q2
+2β − q2
+3αβ) + 2q0q1i + 2q0q2j + 2q0q3k
+= (2q2
+0 − N( )) + 2q0 · (q1i + q2j + q3k).
+(1)
+An immediate consequence of the previous observation is the following rather well-known fact.
+Corollary 3.2. If
+∈ Q is a pure quaternion, then
+2 ∈ Z(Q) = K.
+Another direct consequence of Eq. (1) is the following observation that may be treated as a partial
+converse of Corollary 3.2.
+Observation 3.3. Let
+∈ Q be a square root of some element a ∈ K. Then
+is either pure or
+∈ K.
+Proof:
+Let
+= q0 + q1i + q2j + q3k. If
+2 = a ∈ K then by Eq. (1) we have
+2q0q1 = 2q0q2 = 2q0q3 = 0.
+Therefore, if
+is not pure, that is if q0 ̸= 0, then q1 = q2 = q3 = 0, hence
+∈ K.
+⊓⊔
+
+4
+P. Koprowski / Computing square roots in quaternion algebras
+Combining Corollary 3.2 with Observation 3.3 we see that for computing the square roots in
+quaternion algebras it is crucial to distinguish between the case when one computes a quaternionic
+square root of an element in K (i.e., in the center of Q) and the case when the argument comes from
+Q \Z(Q). It turns out that the latter case is, in fact, trivial and requires nothing more than high-school
+mathematics.
+Algorithm 1. Let Q =
+� α,β
+K
+�
+be a quaternion algebra over a field K of characteristic char K ̸= 2.
+Given a quaternion
+= q0 + q1i + q2j + q3k ∈ Q \ Z(Q), this algorithm outputs its square root or
+reports a failure when
+is not a square.
+1. Check if the norm N( ) of
+is a square in K.
+(a) If it is not, then report a failure and quit.
+(b) If it is, let d be an element of K such that d2 = N( ).
+2. Check if any of the following two elements is a square in K:
+a+ := q0 + d
+2
+,
+a− := q0 − d
+2
+.
+3. If neither of them is a square, then report a failure and quit.
+4. Otherwise, fix r0 such that either r2
+0 = a+ or r2
+0 = a−.
+5. Set
+r1 := q1
+2r0
+,
+r2 := q2
+2r0
+,
+r3 := q3
+2r0
+.
+6. Output r = r0 + r1i + r2j + r3k.
+Proof of correctness:
+Since the norm N : Q → K is multiplicative, it is obvious that if N( ) /∈ K2, then
+cannot be a
+square in Q. This fact justifies the early exit in step (1a) of the algorithm. Assume that N( ) = d2 and
+let r = r0 + r1i + r2j + r3k be the sought square root of , if it exists. By Eq. (1) we have
+q1 = 2r0r1,
+q2 = 2r0r2,
+q3 = 2r0r3.
+It is, thus, clear that it suffices to find r0. Again by Eq. (1) we may write
+q0 = r2
+0 + r2
+1α + r2
+2β − r2
+3αβ = r2
+0 +
+� q1
+2r0
+�2
+α +
+� q2
+2r0
+�2
+β −
+� q3
+2r0
+�2
+αβ.
+The above formula can be rewritten in the form of a bi-quadratic equation:
+4r4
+0 − 4q0r2
+0 +
+�
+q2
+1α + q2
+2β − q2
+3αβ
+�
+= 0.
+
+P. Koprowski / Computing square roots in quaternion algebras
+5
+If we treat the left-hand-side as a quadratic equation in r2
+0, then its discriminant equals 16 · N( ) =
+(4d)2, hence
+r2
+0 = q0 ± d
+2
+= a±.
+It follows that the sought quaternion r exists if and only if either a+ or a− is a square in K. This
+proves the correctness of the algorithm.
+⊓⊔
+Remark 3.4. In the above proof, we constructed the square root r of a quaternion
+∈ Q \ Z(Q) by
+solving a bi-quadratic equation. Such equations in general, may have four roots. Hence, one may
+suspect that there are four distinct quaternions r such that r2 = . It is not the case. It is clear from
+the above proof that
+∈ Q \ Z(Q) has only finitely moan square roots in Q. Now, if r2 = Q, then r
+is a root of a quaternionic polynomial x2 − . But [6, Theorem 5] asserts that a quadratic polynomial
+over Q which has more than two zeros must have infinitely many of them. This way, we conclude
+that
+has just two square roots. Notice that for hamiltonian quaternions this fact has been observed
+already 80 years ago by Niven in [2].
+4.
+Square roots of central elements. Split case
+It is evident from the preceding section that the only non-trivial case that must be considered is how
+to compute a quaternionic square root of an element of the base field K, which is not a square in K.
+In contrast to the previous case (cf. Remark 3.4), in general, an element a ∈ K = Z(Q) may have
+infinitely many square roots in Q. Once again, for hamiltonian quaternions it has been observed
+already by Niven.
+First, we need, however, to introduce an auxiliary algorithm that is not specific to quaternions, as
+it deals with an arbitrary quadratic form. Recall that a quadratic form is called isotropic (see e.g., [5,
+Definition I.3.1]) if it represents zero non-trivially. It is well known (see, e.g., [5, Theorem I.3.4]) that
+every isotropic form represents all elements of K.
+Algorithm 2. Let ξ be an isotropic quadratic form of dimension n over a field K of characteristic
+char K ̸= 2. Given an element a ∈ K and a vector V ∈ Kn such that ξ(V ) = 0, this algorithm
+outputs a vector W ∈ Kn satisfying the condition ξ(W) = a.
+1. Find a vector U ∈ Kn such that U and V are linearly independent.
+2. Set b := ξ(U) and c := 1/2 ·
+�
+ξ(U + V ) − ξ(U)
+�
+.
+3. Output
+W := U + a − b
+2c
+· V.
+Proof of correctness:
+
+6
+P. Koprowski / Computing square roots in quaternion algebras
+Just compute:
+ξ(W) = ξ
+�
+U + a − b
+2c
+· V
+�
+= ξ(U) + a − b
+2c
+·
+�
+ξ(U + V ) − ξ(U) − ξ(V )
+�
++ (a − b)2
+4c2
+ξ(V )
+= b + a − b
+2c
+· 2c + 0 = a
+⊓⊔
+Recall that a quaternion algebra Q =
+� α,β
+K
+�
+is said to split (see e.g., [4, Definition 5.4.5]) if Q
+is isomorphic to the matrix ring M2K. It is well known (see e.g., [4, Theorem 5.4.4] or [5, Theo-
+rem III.2.7]) that Q is split if and only if the quadratic form ⟨−α, −β, αβ⟩ is isotropic. If it is the case,
+the preceding algorithm combined with Eq. (1) lets us compute the quaternionic square root of any
+element of the base field. In particular, when K is a global field, char K ̸= 2, then the computation
+of the square root of a ∈ K in a split quaternion algebra boils down to solving a norm equation in
+a quadratic extension of K. Algorithms for the latter task are well known. They can be found in
+[7, 8, 9, 10, 11].
+Algorithm 3. Let Q =
+�α,β
+K
+�
+be a split quaternion algebra over a global field K of characteristic
+char K ̸= 2. Given a nonzero element a ∈ K, this algorithm outputs a pure quaternion
+∈ Q such
+that
+2 = a.
+1. Check if α is a square in K. If there is c ∈ K× such that c2 = α, then set V := (0, c, 1).
+2. Otherwise, if α /∈ K×2, then:
+(a) Construct a quadratic field extension L = K
+�√α
+�
+of K.
+(b) Solve the norm equation
+NL/K (x) = −α
+β
+and denote the solution by λ = b + c√α.
+(c) Set V := (1, b, c).
+3. Let ξ := ⟨−α, −β, αβ⟩ be the pure subform of the norm form of Q. Execute Algorithm 2 with
+the input (−a, V, ξ) to construct a vector W = (w1, w2, w3) such that ξ(W) = −a.
+4. Output
+= 0 + w1i + w2j + w3k.
+Proof of correctness:
+We claim that the vector V constructed either in step (1) or in step (2) of the algorithm is an isotropic
+vector for ξ. First, suppose that α is a square in K. Say α = c2 for some c ∈ K×. Then
+−α · 02 − β · c2 + αβ · 12 = 0.
+
+P. Koprowski / Computing square roots in quaternion algebras
+7
+Conversely, assume that α /∈ K×2 and so L = K
+�√α
+�
+is a proper extension of K. Let λ = b + c√α
+be an element of L such that N (λ) = −α/β. Then
+−α
+β = λλ = b2 − αc2.
+It follows that
+−α · 12 − β · b2 + αβ · c2 = 0.
+Hence, in both cases V is an isotropic vector of ξ, as claimed. Consequently, executing Algorithm 2
+in step (3) we obtain a vector W satisfying the condition ξ(W) = −a. Now, by Eq. (1) the square of
+the quaternion
+outputted by the algorithm equals
+2 = −N( ) = −ξ(W) = a.
+Thus, to conclude the proof, we only need to show that the norm equation in step (2b) is solvable. But
+this follows immediately from the fact that Q is split. Hence ξ is isotropic. Indeed, if V = (v1, v2, v3)
+is an isotropic vector of ξ, then
+−α · v2
+1 − β · v2
+2 + αβ · v2
+3 = 0.
+Observe that v1 must be nonzero since otherwise, α would be a square. It follows that
+−α
+β =
+�v2
+v1
+�2
+− α
+�v3
+v1
+�2
+= NL/K
+�v2
+v1
++ v3
+v1
+√α
+�
+.
+Therefore, the norm equation is solvable, as claimed.
+⊓⊔
+Remark 4.1. The construction of the isotropic vector V in steps (1–2) of Algorithm 3 is equivalent
+to establishing an explicit isomorphism Q ∼= M2K. For details, see [5, Chapter III]. Of course, if
+the quaternion algebra Q is fixed, the vector V should be computed only once and cached between
+successive computations of square roots.
+Remark 4.2. If the isomorphism Q ∼= M2K is a priori known explicitly, then the computation of the
+quaternionic square root of any a ∈ K× trivializes, as we have the identity
+�
+0
+a
+1
+0
+�2
+=
+�
+a
+0
+0
+a
+�
+.
+5.
+Square roots of central elements. Non-split case
+Now the only case left to be dealt with is when a ∈ K× but Q is not split. Here we have to solve
+not one but two norm equations (see Algorithm 5 below). First, however, we need to introduce the
+following auxiliary algorithm that constructs an element simultaneously represented by two binary
+
+8
+P. Koprowski / Computing square roots in quaternion algebras
+forms. Recall (see e.g., [5, Definition I.2.1]) that for a given quadratic form ξ of dimension d, we
+denote the set of nonzero elements of K represented by ξ by the symbol
+DK(ξ) :=
+�
+ξ(V ) | V ∈ Kd and ξ(V ) ̸= 0
+�
+.
+Let P be any finite set of primes of K. Recall that an element a ∈ K× is called P-singular if
+ordp a ≡ 0 (mod 2) for all finite primes p /∈ P. The set of all P-singular elements forms a subgroup
+of the group K× containing K×2. Thus, the notion of P-singularity generalizes naturally to the square
+classes. Define the set
+EP :=
+�
+aK×2 | a is P-singular
+�
+of P-singular square classes. It is a subgroup of the group K×/K×2 of square classes of K, hence a
+vector space over F2. It is known that the dimension of this vector space is finite. In fact it equals (see
+e.g., [12, p. 607])
+dimF2 EP = |P| + dimF2 CP/C2
+P,
+where CP is the P-class group of K. There is a number of know algorithms to construct a basis of
+this vector space. For details see e.g., [13, 14, 15].
+Algorithm 4. Let K be a global field of characteristic char K ̸= 2. Given two binary quadratic
+forms ξ = ⟨x0, x1⟩ and ζ = ⟨z0, z1⟩ over K with x0, x1, z0, z1 ̸= 0, this algorithm outputs a nonzero
+element d ∈ K× such that d ∈ DK(ξ) ∩ DK(ζ) or reports a failure if there is no such d.
+1. If −x0x1 is a square in K, then output z0 and quit.
+2. Likewise, if −z0z1 is a square in K, then output x0 and quit.
+3. Check (using e.g., [16, Algorithm 5]) whether the form
+ξ ⊥ (−ζ) = ⟨x0, x1, −z0, −z1⟩
+is isotropic. If it is not, then report a failure and quit.
+4. Construct a set P consisting of all dyadic places of K (if there are any) and of all these non-
+dyadic primes of K where at least one of the elements x0, x1, z0, z1 has an odd valuation.
+5. If K is a formally real number field, then:
+(a) Construct the set R of all the real places of K, where either ξ or ζ is definite and denote
+its cardinality by r, i.e.
+R =
+�
+r | sgnr x0x1 = 1 or sgnr z0z1 = 1
+�
+,
+r = |R|.
+(b) [Notation only] Let r1, . . . , rr be all the elements of R.
+(c) Construct a vector W = (w1, . . . , wr) ∈ {0, 1}r setting
+wi =
+�
+log-1 sgnri x0
+if sgnri x0x1 = 1,
+log-1 sgnri z0
+if sgnri x0x1 = −1.
+
+P. Koprowski / Computing square roots in quaternion algebras
+9
+Otherwise, if the field K is non-real, set R := ∅, r = 0 and W := ().
+6. Repeat the following steps until the sought element d is found:
+(a) [Notation only] Let p1, . . . , ps be all the elements of P.
+(b) Construct a basis B = {β1, . . . , βk} of the group EP of P-singular square classes.
+(c) Construct vectors U = (u1, . . . , us) and V = (v1, . . . , vs) setting
+ui = log-1(x0, x1)pi
+and
+vi = log-1(z0, z1)pi.
+(d) Construct matrices A = (aij) and B = (bij), with k = |B| columns and s = |P| rows,
+setting
+aij = log-1(−x0x1, βj)pi
+and
+bij = log-1(−z0z1, βj)pi.
+(e) If R ̸= ∅ construct a matrix C = (cij) with k columns and r = |R| rows, setting
+cij = log-1 sgnri βj.
+Otherwise, when R = ∅, set C = ().
+(f) Check if the following system of F2-linear equations has a solution
+
+
+
+A
+B
+C
+
+
+ ·
+
+
+
+
+x1
+...
+xk
+
+
+
+ =
+
+
+
+U
+V
+W
+
+
+
+(Ϙ)
+(g) If it does, denote the solution by (ε1, . . . , εk) ∈ {0, 1}k. Output d = βε1
+1 · · · βεk
+k and quit.
+(h) If the system (Ϙ) has no solution, then append a new prime p to P (see Remark 5.1 below)
+and reiterate the loop.
+Proof of correctness:
+First, suppose that −x0x1 is a square in K. This means that the form ξ is isotropic (see, e.g., [5,
+Theorem I.3.2]). Hence, by [5, Theorem I.3.4] it represents every element of K. In particular, it
+represents z0. Since ζ also represents z0 (trivially), step (1) of the algorithm outputs the correct result.
+The same argument also applies to step (2), when it is the form ζ that is isotropic. It is also clear that
+the sets DK(ξ) and DK(ζ) of elements represented by ξ and ζ, intersect if and only if ξ ⊥ (−ζ) is
+isotropic. This justifies the test in step (3). Therefore, without loss of generality, for the remainder of
+the proof, we may assume that ξ ⊥ (−ζ) is isotropic while both forms ξ and ζ are anisotropic.
+We will first show that the algorithm terminates. Let W = (w0, w1, w2, w3) ∈ K4 be an isotropic
+vector of ξ ⊥ (−ζ). Denote e := ξ(w0, w1) = ζ(w2, w3). Further, let R and P be the sets of
+places (real and non-archimedean, respectively) constructed in steps (4–5) of the algorithm. Now, [17,
+Lemma 2.1] asserts that there exists a finite prime p0 of K and an element d ∈ K× such that:
+i. ordp d = 0 for every finite prime p /∈ P ∪ {p0};
+
+10
+P. Koprowski / Computing square roots in quaternion algebras
+ii. d ≡ e (mod p1+ordp 4) for every p ∈ P;
+iii. ordp0 d = 1;
+iv. sgnr d = sgnr e for every real places r of K.
+Let B = {β1, . . . , βk} be a basis of the group EP∪{p0} of
+�
+P ∪ {p0}
+�
+-singular square classes.
+The element d is
+�
+P ∪ {p0}
+�
+-singular, hence it can be expressed in the form
+d = βε1
+1 · · · βεk
+k ,
+where ε1, . . . , εk ∈ F2 are the coordinates of d with respect to B.
+Fix a real place ri ∈ R. First, suppose that sgnri x0x1 = 1, so the form ξ ⊗ Kp is definite. Then
+sgnri(−d) = sgnri(−e) = sgnri x0 since ⟨−e, x0, x1⟩ is isotropic. But this implies that
+k
+�
+j=1
+(−1)cijεj =
+k
+�
+j=1
+sgnri βεj
+j = sgnri d = sgnri x0 = (−1)wi.
+Consequently
+ci1ε1 + · · · + cikεk = wi.
+(2)
+Conversely, assume that ξ ⊗ Kri is indefinete, hence ζ ⊗ Kri must be definete. Applying the same
+arguments to the form ζ instead of ξ, we show that Eq. (2) also holds in this case.
+Now fix a finite prime pi ∈ P. Observe that by the local square theorem (see, e.g., [5, The-
+orem VI.2.19]) condition (ii) implies that the local squares classes dK×2
+pi
+and eK×2
+pi
+coincide. It
+follows that the form
+⟨−d, x0, x1⟩ ⊗ Kpi ∼= ⟨−e, x0, x1⟩ ⊗ Kpi
+is isotropic. Now, [5, Proposition V.3.22] asserts that the Hasse invariant of ⟨−d, x0, x1⟩ ⊗ Kpi equals
+spi⟨−d, x0, x1⟩ = (−1, x0x1 · d)pi.
+This can be rewritten as
+(−x0x1, d)pi = (x0, x1)pi.
+Substituting βε1
+1 · · · βεk
+k for d we obtain
+k
+�
+j=1
+(−x0x1, βj)εj
+pi = (x0, x1)pi.
+Now, (x0, x1)pi = (−1)ui and (−x0x1, βj)pi = (−1)aij, where ui, aij ∈ {0, 1} are the elements
+constructed in steps (6c–6d). Therefore, the last condition can be expressed as a linear equation over
+F2:
+ai1ε1 + · · · + aikεk = ui.
+(3)
+
+P. Koprowski / Computing square roots in quaternion algebras
+11
+Finally, we will show that the above equation also holds for the index i = 0, that is for the prime p0
+appended to P. This fact follows from Hilbert reciprocity law (see, e.g., [5, Theorem VI.5.5]). We
+already know that for every i ∈ {1, . . . , s} we have
+(−x0x1, d)pi = (x0, x1)pi.
+The same also holds for primes not in P. Indeed, if q /∈ P ∪ {p0} then q is non-dyadin and all three
+elements x0, x1 and d have even valuations at q. Consequently, by [5, Corollary VI.2.5] one obtains
+(−x0x1, d)q = (x0, x1)q = 1.
+Now, by Hilbert reciprocity law, we can write
+1 =
+�
+p
+(−x0x1, d)p ·
+�
+p
+(x0, x1)p
+= (−x0x1, d)p0(x0, x1)p0 ·
+�
+p∈P
+�
+(−x0x1, d)p(x0, x1)p
+�
+·
+�
+q/∈P∪{p0}
+�
+(−x0x1, d)p(x0, x1)p
+�
+= (−x0x1, d)p0(x0, x1)p0.
+Hence, in the same way as above, we show that Eq. (3) also holds for i = 0. Applying the same
+arguments to the form ζ, we obtain
+bi1ε1 + · · · + bikεk = vi,
+(4)
+for all i ∈ {0, 1, . . . , s}.
+All in all, we have proved that Eq. (Ϙ) has a solution in EP∪{p0}. Now, for every P′ ⊇ P ∪ {p0}
+we have EP∪{p0} ⊆ EP′, hence once the prime p0 is appended to P the algorithm terminates (see also
+Remark 5.1 w below).
+Now, when we have proved that the algorithm stops, we must show that it outputs a correct result.
+To this end, we will show that the forms ⟨−d, x0, x1⟩ and ⟨−d, z0, z1⟩ are locally isotropic in every
+completion of K. The assumptions are symmetric with respect to both forms, except in real places.
+Hence it generally suffices to prove the isotropy of one of them.
+Both forms are trivially isotropic in all complex completions of K (provided that there are any)
+and in all real completions Kr for r /∈ R. Fix now a real place ri ∈ R. First, assume that the form
+⟨x0, x1⟩ ⊗ Kri is definite. From the preceding part we know that the element d = βε1
+1 · · · βεk
+k , con-
+structed by the algorithm, satisfies the condition sgnri d = sgnri x0. Therefore the form ⟨−d, x0, x1⟩⊗
+Kri is isotropic. Now, the form ξ ⊥ (−ζ) is isotropic because otherwise, the execution of the algo-
+rithm would have been interrupted already in step (3). Thus, either sgnri z0 = sgnri x0 = sgnri d or
+sgnri z1 = sgnri x0 = sgnri d. In both cases, we have that the form ⟨−d, z0, z1⟩ ⊗ Kri is isotropic, as
+well. Conversely, assume that ξ ⊗ Kri is indefinite, and so it is ζ ⊗ Kri that must be definite. Then,
+⟨−d, x0, x1⟩⊗Kri is trivially isotropic and to the form ⟨−d, z0, z1⟩⊗Kri we apply the some argument
+as to the form ⟨−d, x0, x1⟩ ⊗ Kri in the previous case.
+We may now concentrate on finite primes. Fix a prime p. Suppose p is not among the primes
+constituting P (here, we allow P to have been already enlarged during the execution of the algorithm).
+
+12
+P. Koprowski / Computing square roots in quaternion algebras
+In that case, p is certainly non-dyadic, and all three elements x0, x1, and d have even valuations at p.
+Hence, [5, Corollary VI.2.5] asserts that ⟨−d, x0, x1⟩ ⊗ Kp is isotropic. On the other hand, we know
+from the first part of the proof that if p = pi ∈ P, then d satisfies the condition (−x0x1, d)p =
+(x0, x1)p, which is equivalent to sp⟨−d, x0, x1⟩ = (−1, x0x1 · d)p. The later condition implies that
+⟨−d, x0, x1⟩ ⊗ Kp is isotropic, again by [5, Proposition V.3.22]. The very same arguments may be
+applied to the form ⟨−d, z0, z1⟩ ⊗ Kp.
+All in all, we have shown that the forms ⟨−d, x0, x1⟩ and ⟨−d, z0, z1⟩ are locally isotropic in every
+completion of K. Thus, they are isotropic over K by the Hasse–Minkowski principle (see e.g., [5,
+Theorem VI.3.1]). This means that the forms ξ and ζ represent d over K by [5, Corollary I.3.5].
+⊓⊔
+Remark 5.1. To rigorously prove that Algorithm 4 terminates, we must show that the new primes
+may be appended to the set P in such a way that after finitely many iterations the set will contain the
+prime p0 specified in the proof of correctness. If K is a number field, let p1, p2, p3, . . . = 2, 3, 5, . . .
+be the (strictly increasing) sequence of all prime numbers. On the other hand, if K is a global function
+field, i.e., a finite extension of a rational function field Fq(x), let p1, p2, p3, . . . be a sequence of all
+the irreducible polynomials from Fq[x] ordered in such a way that deg pj ≤ deg pj+1 for every j.
+Now, appending new primes to P, we can first use the places of K that extend p1, then the ones that
+extend p2, then p3, and so on. It is clear that this is an exhaustive method, so eventually, we will
+append p0 and consequently terminate the algorithm. This proves that the algorithm stops but does
+not present a complete picture. The existence of the prime p0 follows from [17, Lemma 2.1], which in
+turn uses Chebotarev’s density theorem. In particular, if P0 denotes the set of primes of K such that
+appending any of them makes the algorithm stop, then the set P0 has positive density. This means that
+in practice, one can just append primes to P at random with exponentially diminishing probability
+that the system (Ϙ) fails to be solvable.
+We are now in a position to present an algorithm that computes a square root of a scalar in a
+non-split quaternion algebra.
+Algorithm 5. Let Q =
+� α,β
+K
+�
+be a non-split quaternion algebra over a global field of characteristic
+char K ̸= 2. Given a nonzero element a ∈ K this algorithm outputs a quaternion
+∈ Q such that
+2 = a or reports a failure if a is not a square in Q.
+1. Check if a is a square in K. If there is c ∈ K× such that a = c2, then output
+= c+0i+0j+0k
+and quit.
+2. Check if aα is a square in K. If there is c ∈ K× such that aα = c2, then output
+= 0+(c/α)i+
+0j + 0k and quit.
+3. Check if aβ is a square in K. If there is c ∈ K× such that aβ = c2, then output
+= 0 + 0i +
+(c/β)j + 0k and quit.
+4. Execute Algorithm 4 with input ξ = ⟨a, −α⟩ and ζ = ⟨β, −αβ⟩. If it fails, then report a failure
+and quit. Otherwise, let d ∈ K× denote the outputted element represented by these two binary
+forms.
+
+P. Koprowski / Computing square roots in quaternion algebras
+13
+5. Construct two quadratic extensions of K:
+L := K
+�√α
+�
+and
+M := K
+�√aα
+�
+.
+6. Solve the following two norm equations:
+d
+β = NL/K (x)
+and
+d
+a = NM/K (y) .
+Denote the solutions by
+λ = l0 + l1
+√α
+and
+µ = m0 + m1
+√aα,
+respectively.
+7. Output
+= 0 + a · m1
+m0 i + l0
+m0 j + l1
+m0 k.
+Proof of correctness:
+The correctness of the results outputted in step (1) is obvious as is the correctness of output of steps
+(2–3). Indeed, if aα = c2 for some c ∈ K× and
+= (c/α)i, then
+2 = α · c2/α2 = a. In the remainder
+of the proof, we can, thus, assume that neither a nor aα is a square in K. Likewise, α is not a square,
+either, since otherwise, the quaternion algebra Q would split. Therefore, L and M are proper quadratic
+extensions of K. It follows from Eq. (1) that a is a square of some pure quaternion
+= q1i+q2j+q3k
+if and only if
+a · 12 − α · q2
+1 = β · q2
+2 − αβ · q2
+3.
+This equality is equivalent to the condition that the sets of elements of K represented by the binary
+forms ξ = ⟨a, −α⟩ and ζ = ⟨β, −αβ⟩ have a non-empty intersection. This proves the correctness of
+step (4). Now, assume that Algorithm 4 returned some element d ∈ DK(ξ) ∩ DK(ζ). Then there are
+l0, l1, m0, m1 ∈ K such that
+�
+d = am2
+0 − α(am1)2 = a · NM/K (m0 + m1
+√aα)
+d = βl2
+0 − αβl2
+1 = β · NL/K (l0 + l1
+√α) .
+Rearranging the terms we have
+a = α
+�am1
+m0
+�2
++ β
+� l0
+m0
+�2
+− αβ
+� l1
+m0
+�2
+.
+Now, the right-hand-side is nothing else but the square of the quaternion
+constructed in step (7). This
+proves that the algorithm is correct.
+⊓⊔
+
+14
+P. Koprowski / Computing square roots in quaternion algebras
+References
+[1] Heath T. A history of Greek mathematics. Vol. I. Dover Publications, Inc., New York, 1981. ISBN
+0-486-24073-8. From Thales to Euclid, Corrected reprint of the 1921 original.
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+https://doi.org/10.2307/2303134.
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+Springer, Berlin, 1980. ISBN 3-540-09983-2.
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+Quaternion algebras, volume 288 of Graduate Texts in Mathematics.
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+S0025-5718-97-00761-8.
+https://doi.org/10.1090/S0025-5718-97-00761-8,
+URL
+https://doi.org/10.1090/S0025-5718-97-00761-8.
+[9] Fincke U, Pohst M. A procedure for determining algebraic integers of given norm. In: Computer al-
+gebra (London, 1983), volume 162 of Lecture Notes in Comput. Sci., pp. 194–202. Springer, Berlin,
+1983.
+doi:10.1007/3-540-12868-9\ 103.
+https://doi.org/10.1007/3-540-12868-9_103, URL
+https://doi.org/10.1007/3-540-12868-9_103.
+[10] Garbanati
+DA.
+An
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+for
+finding
+an
+algebraic
+number
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+rational
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+Math.,
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+316:1–13.
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+10.1515/crll.1980.316.1.
+https://doi.org/10.1515/crll.1980.316.1,
+URL
+https://doi.org/10.1515/crll.1980.316.1.
+[11] Simon
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+in
+relative
+number
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+us-
+ing
+S-units.
+Math.
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+71(239):1287–1305.
+doi:10.1090/
+S0025-5718-02-01309-1.
+https://doi.org/10.1090/S0025-5718-02-01309-1,
+URL
+https://doi.org/10.1090/S0025-5718-02-01309-1.
+[12] Czogała A.
+Witt rings of Hasse domains of global fields.
+J. Algebra, 2001. 244(2):604–630. doi:
+10.1006/jabr.2001.8918. URL https://doi.org/10.1006/jabr.2001.8918.
+[13] Cannon J, Bosma W, Fieker C, (eds) AS. Handbook of Magma Functions, 2.26-4 edition, 2021.
+[14] Koprowski P.
+Computing singular elements modulo squares.
+Fund. Inform., 2021.
+179(3):227–
+238.
+doi:10.3233/fi-2021-2022.
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+URL
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+P. Koprowski / Computing square roots in quaternion algebras
+15
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+doi:10.1016/0022-314X(92)90098-A. URL https://doi.org/10.1016/0022-314X(92)90098-A.
+

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+arXiv:2301.03299v1  [math.NA]  9 Jan 2023
+ASYMPTOTIC ERROR ANALYSIS FOR THE DISCRETE ITERATED
+GALERKIN SOLUTION OF URYSOHN INTEGRAL EQUATIONS WITH
+GREEN’S KERNELS
+GOBINDA RAKSHIT
+ABSTRACT. Consider a Urysohn integral equation x − K(x) = f, where f and the integral
+operator K with kernel of the type of Green’s function are given. In the computation of
+approximate solutions of the given integral equation by Galerkin method, all the integrals
+are needed to be evaluated by some numerical integration formula. This gives rise to the
+discrete version of the Galerkin method. For r ≥ 1, a space of piecewise polynomials
+of degree ≤ r − 1 with respect to a uniform partition is chosen to be the approximating
+space. For the appropriate choice of a numerical integration formula, an asymptotic series
+expansion of the discrete iterated Galerkin solution is obtained at the above partition points.
+Richardson extrapolation is used to improve the order of convergence. Using this method
+we can restore the rate of convergence when the error is measured in the continuous case. A
+numerical example is given to illustrate this theory.
+1 Introduction
+Let X = L∞[0, 1]. Consider the problem of solving Urysohn integral equation
+x(s) −
+� 1
+0
+κ(s, t, x(t)) dt = f(s), s ∈ [0, 1],
+(1.1)
+where f ∈ X and κ ∈ C ([0, 1] × [0, 1] × R) are given. Let the Urysohn integral operator
+K : L∞[0, 1] → C[0, 1] be defined by
+(1.2)
+K(x)(s) =
+� 1
+0
+κ(s, t, x(t)) dt,
+x ∈ X , s ∈ [0, 1].
+Since the kernel κ is continuous, K is compact operator on X . Denoting the equation (1.1)
+by
+(1.3)
+x − Kx = f.
+We assume that the above equation has a solution, say ϕ. We also assume that K is twice
+Frechét differentiable and 1 is not an eigenvalue of the compact linear operator K′(ϕ). This
+gives us that ϕ is an isolated solution of (1.3). See [12], [14]. We are looking for Galerkin
+approximations of ϕ.
+For r ≥ 1, consider the approximating space Xn as a space of piecewise polynomials of
+degree ≤ r − 1 with respect to a uniform partition, say ∆(n), of [0, 1] with n subintervals
+Date: January 10, 2023.
+2020 Mathematics Subject Classification. 45G10, 65B05, 65J15, 65R20.
+Key words and phrases. Urysohn integral operator, Green’s kernel, Galerkin method, Nyström approxima-
+tion, Richardson extrapolation.
+1
+
+2
+G. RAKSHIT
+each of length h = 1
+n. Let πn be the restriction to L∞[0, 1] of the orthogonal projection from
+L2[0, 1] to Xn. Then the Galerkin solution ϕG
+n satisfies the following integral equation
+ϕG
+n − πnK(ϕG
+n ) = πnf.
+Galerkin method for Urysohn integral equation has been studied extensively in research
+literature. See [5], [12], [13], [14]. The iterated Galerkin solution is defined by
+ϕS
+n = K(ϕG
+n ) + f.
+In [5], the following orders of convergence are also obtained.
+∥ϕG
+n − ϕ∥∞ = O (h) ,
+∥ϕS
+n − ϕ∥∞ = O
+�
+h2�
+,
+if r = 1,
+and
+∥ϕG
+n − ϕ∥∞ = O (hr) ,
+∥ϕS
+n − ϕ∥∞ = O
+�
+hr+2�
+,
+if r ≥ 2.
+It is also shown that the order of convergence of ϕS
+n at the points of partition ∆(n), is h2r.
+If an asymptotic expansion for the error exists, one can apply a well-known techniques
+to obtain more accurate approximations. Richardson extrapolation one such method for
+application. In [24], an asymptotic expansion for the iterated Galerkin solution of Urysohn
+integral equation with Green’s function type of kernel, is obtained at the above mentioned
+partition points. Then, by [11] and using Richardson extrapolation, an approximate solution
+with order of convergence h2r+2 can be obtained.
+In the computation of of above approximations, various integrals are involved. There is
+an integral in the definition of the Urysohn integral operator K. In the definition of the
+orthogonal projection πn, the standard inner product on L2[0, 1] comes into picture. In
+practice, it is necessary to replace all these integrals by a numerical quadrature formula.
+This gives rise to the discrete versions of the projection methods. The discrete versions of
+the Galerkin methods for Urysohn integral with Green’s kernel, are investigated in [6], [4].
+Whereas, in [17], a different version of discrete projection method is discussed.
+In this article, we consider the Urysohn integral equation with Green’s kernel, and discrete
+Galerkin method is applied for approximations. Then, an asymptotic expansion for the
+discrete iterated Galerkin solution is obtained.
+We choose a fine partition of [0, 1] with m subintervals each of length ˜h = 1
+m and define a
+composite numerical quadrature formula. Replacing the integrals in the definition of K and
+πn, we define the Nyström operator Km and the discrete orthogonal projection Pn. Then the
+discrete Galerkin and the discrete iterated Galerkin equations are given by
+zG
+n − PnKm(zG
+n ) = Pnf and zS
+n − Km(PnzS
+n) = f
+respectively. If ϕ ∈ Cr+2[0, 1], then from [6] and [17], we have
+(1.4)
+��zG
+n − ϕ
+��
+∞ = O
+�
+max
+�
+hr, ˜h2��
+(1.5)
+��zS
+n − ϕ
+��
+∞ =
+
+
+
+O
+�
+max
+�
+h2, ˜h2��
+,
+r = 1,
+O
+�
+max
+�
+hr+2, ˜h2��
+, r ≥ 2.
+
+Section 2. Preliminaries
+3
+In this article, first we find an asymptotic error expansion due to the discrete orthogonal
+projection. Then using this, the following asymptotic expansion is obtained:
+(1.6)
+zS
+n(ti) = ϕ(ti) + γ(ti)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where the function γ is independent of h. If we choose m such that ˜h ≤ h2r+2, then using
+the Richardson extrapolation, an approximation of ϕ of the order of h2r+2 could be obtained.
+See [11].
+This article is organized as follows. Definitions, notations and some preliminary results
+are given in section 2. In Section 3, a quadrature rule is defined, and using it the discrete
+orthogonal projection and the Nyström approximations of the integral operators are defined.
+Section 4 contains the asymptotic error analysis for the approximations. Numerical example
+is given in Section 5.
+2 Preliminaries
+For an integer α ≥ 0, let Cα[0, 1] denotes the space of all real valued α-times continuously
+differentiable functions on [0, 1] with the norm
+∥x∥α,∞ = max
+0≤j≤α
+��x(j)��
+∞ ,
+where x(j) is the jth derivative of the function x, and
+���x(j)���
+∞ = sup
+0≤t≤1
+|x(j)(t)|. Define
+∥κ∥α,∞ =
+max
+0≤i+j+k≤α
+���D(i,j,k)κ(s, t, u)
+���
+∞ ,
+where
+D(i,j,k)κ(s, t, u) =
+∂i+j+kκ
+∂si∂tj∂uk (s, t, u).
+2.0 Green’s function type kernel
+Let r ≥ 1 be an integer and assume that the kernel κ has the following properties.
+(1) For i = 1, 2, 3, 4, the functions κ, ∂iκ
+∂ui ∈ C(Ω), where C(Ω) denotes the space of all
+real valued continuous function on Ω = [0, 1] × [0, 1] × R.
+(2) Let Ω1 = {(s, t, u) : 0 ≤ t ≤ s ≤ 1, u ∈ R} and Ω2 = {(s, t, u) : 0 ≤ s ≤ t ≤
+1, u ∈ R}. There are two functions κj ∈ Cr(Ωj), j = 1, 2, such that
+κ(s, t, u) =
+�
+κ1(s, t, u),
+(s, t, u) ∈ Ω1,
+κ2(s, t, u),
+(s, t, u) ∈ Ω2.
+(3) Denote ℓ(s, t, u) = ∂κ
+∂u(s, t, u) and λ(s, t, u) = ∂2κ
+∂u2(s, t, u), (s, t, u) ∈ Ω. The partial
+derivatives of ℓ(s, t, u) and λ(s, t, u) with respect to s and t have jump discontinu-
+ities on s = t.
+(4) There are functions ℓj, λj ∈ Cr(Ωj), j = 1, 2, with
+ℓ(s, t, u) =
+�
+ℓ1(s, t, u),
+(s, t, u) ∈ Ω1,
+ℓ2(s, t, u),
+(s, t, u) ∈ Ω2,
+λ(s, t, u) =
+�
+λ1(s, t, u),
+(s, t, u) ∈ Ω1,
+λ2(s, t, u),
+(s, t, u) ∈ Ω2.
+
+4
+G. RAKSHIT
+Under the above assumptions, the operator K is four times Fréchet differentiable, and its
+Fréchet derivatives at x ∈ X are given by
+K′(x)v1(s) =
+� 1
+0
+∂κ
+∂u (s, t, x(t)) v1(t) dt,
+K(i)(x)(v1, . . . , vi)(s) =
+� 1
+0
+∂iκ
+∂ui (s, t, x(t)) v1(t) · · ·vi(t) dt,
+i = 2, 3, 4,
+where
+∂iκ
+∂ui (s, t, x(t)) = ∂iκ
+∂ui (s, t, u)|u=x(t),
+i = 1, 2, 3, 4
+and v1, v2, v3, v4 ∈ X . Note that K′(x) : X → X is linear and K(i)(x) : X i → X are multi-
+linear operators, where X i is the cartesian product of i copies of X . See [25]. The norms of
+these operators are defined by
+��K(i)(x)
+�� = sup
+���K(i)(x)(v1, . . . , vi)
+��
+∞ : ∥vj∥∞ ≤ 1, j = 1, . . . , i
+�
+for i = 1, 2, 3, 4. It follows that
+��K(i)(x)
+��
+≤
+sup
+0≤s,t≤1
+����
+∂iκ
+∂ui (s, t, x(t))
+���� ,
+i = 1, 2, 3, 4.
+Note that, if f ∈ Cα[0, 1] for any positive integer α, then ϕ ∈ Cα[0, 1]. See [5, Corollary
+3.2], [6, Corollary 4.2].
+3 Discretization of Integrals by numerical quadrature
+Rule
+In this section, first we consider a numerical integration formula. We replace the integral
+in the standard inner product of L2[0, 1] (i.e. ⟨x , y⟩ =
+� 1
+0 x(t)y(t)dt) by the quadrature rule
+and define a discrete inner product. Subsequently, the corresponding discrete orthogonal
+projection is defined. After that, an asymptotic error expansion for the discrete orthogonal
+projection is obtained. Next we define the Nyström approximations of the integral operator
+K and its Fréchet derivatives.
+Consider a basic numerical integration formula by
+(3.1)
+� 1
+0
+x(t)dt ≈
+ρ
+�
+q=1
+wq x(µq),
+which is exact at least for polynomials of degree ≤ 3r. If r = 0, then it is assumed that the
+quadrature rule is exact atleast for linear polynomials. It follows that �ρ
+q=1 wq = 1.
+Let n ∈ N and consider the following uniform partition of [0, 1] :
+(3.2)
+∆(n) :
+0 < 1
+n < · · · < n − 1
+n
+< 1.
+Define tj = j
+n, ∆j = [tj−1, tj] and h = tj − tj−1 = 1
+n, j = 1, . . . , n. Define the subspace
+Cα
+∆(n)[0, 1] = {x ∈ X : x ∈ Cα[tj−1, tj], j = 1, 2, 3, . . ., n} . For r ≥ 1, the approximating
+space
+Xn =
+�
+x ∈ X : x|∆j is a polynomial of degree ≤ r − 1
+�
+.
+
+Section 3. Discretization of Integrals by numerical quadrature Rule
+5
+Let p be a positive integer and m = pn. Consider the following uniform partition of [0, 1] :
+(3.3)
+∆(m) :
+0 < 1
+m < · · · < m − 1
+m
+< 1.
+Let ˜h = 1
+m
+and
+si = i
+m,
+i = 0, . . . , m.
+Note : As our goal to find the equation (1.6), where the higher order term is max
+�
+h2r+2, ˜h2�
+,
+we choose the partition ∆(m) such that ˜h2 ≤ hr.
+A composite integration rule with respect to the partition (3.3) is then defined as
+� 1
+0
+x(t) dt
+=
+m
+�
+i=1
+� si
+si−1
+x(t) dt ≈ ˜h
+m
+�
+i=1
+ρ
+�
+q=1
+wq x(si−1 + µq˜h).
+Thus,
+� tj
+tj−1
+x(t) dt =
+� jh
+(j−1)h
+x(t) dt =
+p
+�
+ν=1
+� (j−1)h+ν˜h
+(j−1)h+(ν−1)˜h
+x(t) dt.
+Since h = p˜h,
+� tj
+tj−1
+x(t)dt
+=
+p
+�
+ν=1
+�
+(j−1)p+ν
+p
+h
+(j−1)p+ν−1
+p
+h
+x(t) dt.
+Substituting t =
+(j−1)p+ν−1
+p
+h + ˜hσ =
+(j−1)p+ν−1+σ
+p
+h in the above equation, we obtain
+� tj
+tj−1
+x(t) dt
+=
+h
+p
+p
+�
+ν=1
+� 1
+0
+x
+�(j − 1)p + ν − 1 + σ
+p
+h
+�
+dσ
+=
+h
+p
+p
+�
+ν=1
+� 1
+0
+x
+�
+tj−1 + ν − 1 + σ
+p
+h
+�
+dσ.
+Note that ν−1+σ
+p
+∈ [0, 1]. Now using the numerical quadrature formula (3.1), we obtain
+� tj
+tj−1
+x(t) dt
+≈
+h
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq x
+�
+tj−1 + ν − 1 + µq
+p
+h
+�
+.
+Let
+µqν = ν − 1 + µq
+p
+,
+q = 1, 2, . . . , ρ; ν = 1, 2, . . . , p.
+Then,
+� tj
+tj−1
+x(t) dt
+≈
+h
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq x(tj−1 + µqνh).
+(3.4)
+We prove the following lemma which will be used to find an asymptotic error expansion for
+the discrete orthogonal projection.
+Lemma 3.1. Let Lη be the Legendre polynomial of degree η ∈ {0, 1, . . . , r − 1} defined on
+[0, 1]. Then for any k = 1, 2, . . . , 2r − 1,
+1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(µqν)Lη(τ)(µqν − τ)k
+k!
+=
+� 1
+0
+Λr(τ, s)(s − τ)k
+k!
+ds,
+
+6
+G. RAKSHIT
+where
+r−1
+�
+η=0
+Lη(τ)Lη(s) = Λr(τ, s), for τ, s ∈ [0, 1].
+Proof. Since Lη is a polynomial of degree 0 ≤ η ≤ r − 1,
+1
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη
+�ν − 1 + µq
+p
+�
+= 1
+p
+p
+�
+ν=1
+� 1
+0
+Lη
+�ν − 1 + t
+p
+�
+dt
+=
+p
+�
+ν=1
+�
+ν
+p
+ν−1
+p
+Lη(s) ds
+=
+� 1
+0
+Lη(s) ds.
+Since the basic quadrature formula (3.1) is exact for polynomials of degree ≤ 3r,
+1
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη
+�ν − 1 + µq
+p
+� �ν − 1 + µq
+p
+− τ
+�k
+=
+� 1
+0
+Lη(s) (s − τ)k ds.
+It follows that
+1
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(µqν) (µqν − τ)k
+k!
+=
+� 1
+0
+Lη(s) (s − τ)k
+k!
+ds,
+where µqν = ν − 1 + µq
+p
+. This gives
+1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(τ)Lη(µqν) (µqν − τ)k
+k!
+=
+r−1
+�
+η=0
+Lη(τ)
+� 1
+0
+Lη(s) (s − τ)k
+k!
+ds.
+Let
+r−1
+�
+η=0
+Lη(τ)Lη(s) = Λr(τ, s),
+τ, s ∈ [0, 1].
+Then,
+1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(τ)Lη(µqν) (µqν − τ)k
+k!
+=
+� 1
+0
+Λr(τ, s) (s − τ)k
+k!
+ds.
+Hence the required result follows.
+□
+3.1 Discrete Orthogonal Projection
+Let j ∈ {1, 2, . . . , n} and x, y ∈ C(∆j). Define a discrete inner product on ∆j by
+(3.5)
+⟨x , y⟩∆j,m = ˜h
+p
+�
+ν=1
+ρ
+�
+q=1
+wq x(tj−1 + µqνh) y(tj−1 + µqνh).
+Note that, this is an indefinite inner product. For more details on indefinite inner product
+spaces, see [8]. However, the properties which we need to define a discrete orthogonal
+projection, hold true for (3.5). For η = 0, 1, . . . , r−1, let Lη denote the Legendre polynomial
+of degree η on [0, 1]. For j = 2, . . . , n, and for η = 0, 1, . . . , r − 1, define
+ϕj,η(t)
+=
+� �
+1
+hLη
+�
+t−tj−1
+h
+�
+,
+t ∈ (tj−1, tj],
+0,
+otherwise
+
+3.1
+Discrete Orthogonal Projection
+7
+and, ϕ1,η(t) =
+�
+1
+hLη
+� t−t0
+h
+�
+if t ∈ [t0, t1] and 0 otherwise. Note that
+(3.6)
+ϕj,η (tj−1 + µqνh) = h− 1
+2Lη(µqν)
+for all j = 1, 2, . . . , n.
+Note that {ϕj,η : j = 1, . . . , n, η = 0, 1, . . . , r − 1} be a set of orthonormal basis for Xn,
+where ϕj,η is the Legendre polynomial of degree η defined on [tj−1, tj]. Since the basic
+numerical integration (3.1) has degree of precision 3r, the set {ϕj,η} is also orthonormal
+with respect to the discrete inner product (3.5). Let Pr,∆j be the space of polynomials of
+degree ≤ r − 1 on ∆j. Define the discrete orthogonal projection Pn,j : C[tj−1, tj] → Pr,∆j
+as follows:
+(3.7)
+Pn,jx =
+r−1
+�
+η=0
+⟨x , ϕj,η⟩∆j ϕj,η.
+See [4], [6] for more details. A discrete orthogonal projection Pn : C[0, 1] → Xn is defined
+by
+Pnx =
+n
+�
+j=1
+Pn,jx.
+(3.8)
+It follows that Pnx(t) = Pn,jx(t), for all t ∈ [tj−1, tj]. We also have the following error
+bound:
+∥Pn∥ < ∞ and also, if x ∈ Cr[tj−1, tj], then
+(3.9)
+∥x − Pn,jx∥∆j,∞ ≤ C1
+��x(r)��
+∆j,∞ hr,
+if x ∈ Cr[0, 1], then
+(3.10)
+∥x − Pnx∥∆j,∞ ≤ C1
+��x(r)��
+∞ hr,
+where ∥x∥∆j,∞ =
+sup
+t∈[tj−1,tj]
+|x(t)| and, C1 is a constant independent of h. For details see
+[17].
+In (3.10) we have a error bound for the discrete orthogonal projection. But, by the follow-
+ing lemma we obtain an asymptotic error expansion for the discrete orthogonal projection,
+which is more stronger result than (3.10).
+Lemma 3.2. Let Pn be the discrete orthogonal projection defined by (3.7) - (3.8). Let
+x ∈ C2r+2
+∆(n) [0, 1] and t = tj−1 + τh with τ ∈ [0, 1]. Then
+Pnx(t) − x(t) =
+2r+1
+�
+k=1
+Jk(τ) x(k)(tj−1 + τh) hk + O
+�
+h2r+2�
+,
+where Jk(τ) =
+� 1
+0
+Λr(τ, s)(s − τ)k
+k!
+ds,
+k = 1, 2, . . . , 2r + 1.
+Proof. Define a function vj : [tj−1, tj] → R by
+vj(t) = 1,
+t ∈ [tj−1, tj].
+For τ ∈ [0, 1], let t = tj−1 + hτ ∈ [tj−1, tj]. From (3.7) it is easy to see that
+Pn,jvj = vj.
+
+8
+G. RAKSHIT
+It follows that
+r−1
+�
+η=0
+⟨vj , ϕj,η⟩∆j ϕj,η(t) = 1.
+Since ϕj,η is a polynomial of degree 0 ≤ η ≤ r − 1 on [tj−1, tj],
+⟨vj , ϕj,η⟩∆j =
+� tj
+tj−1
+ϕj,η(s) ds
+=
+h
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq ϕj,η(tj−1 + µqνh).
+Thus for any function x : [tj−1, tj] → R, we have
+x(t) = x(t) h
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq ϕj,η(tj−1 + µqνh) ϕj,η(t).
+It follows that
+Pn,jx(t) − x(t)
+= h
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq ϕj,η(tj−1 + µqνh) [x(tj−1 + µqνh) − x(t)] ϕj,η(t),
+where t = tj−1 + hτ ∈ [tj−1, tj] and τ ∈ [0, 1]. From (3.6), we have
+(3.11)
+Pn,jx(t) − x(t) = 1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(µqν) [x(tj−1 + µqνh) − x(t)] Lη(τ).
+Since x ∈ C2r+2[tj−1, tj], using Taylor series expansion we obtain
+x(tj−1 + µqνh) − x(tj−1 + hτ) =
+2r+1
+�
+k=1
+x(k)(tj−1 + τh) (µqν − τ)k
+k!
+hk + O
+�
+h2r+2�
+.
+Thus
+Pnx(t) − x(t) = Pn,jx(t) − x(t)
+=
+2r+1
+�
+k=1
+x(k)(tj−1 + τh) hk
+�
+1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wqLη(µqν)Lη(τ)(µqν − τ)k
+k!
+�
++ O
+�
+h2r+2�
+.
+Let
+Jk(τ) = 1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(µqν) Lη(τ) (µqν − τ)k
+k!
+,
+τ ∈ [0, 1].
+By Lemma 3.1, we can write Jk(τ) =
+� 1
+0 Λr(τ, s) (s−τ)k
+k!
+ds. Hence
+Pnx(t) − x(t) =
+2r+1
+�
+k=1
+Jk(τ) x(k)(tj−1 + τh) hk + O
+�
+h2r+2�
+.
+The result follows.
+□
+
+3.2
+Approximation of the Integral Operator
+9
+Let
+L = (I − K′(ϕ))−1 K′(ϕ).
+Then L is a compact linear integral operator with kernel ˜ℓ. Note that the smoothness of ˜ℓ is
+same as the kernel ℓ. See [26], [5, Lemma 5.1] for details. It follows that
+LPnx(s) − Lx(s) =
+n
+�
+j=1
+� tj
+tj−1
+˜ℓ(s, t) (Pnx(t) − x(t)) dt
+∀x ∈ X .
+Then using Lemma 3.2, and following the proofs of [22, Theorem 5.1] and [23, Theorem
+3.2], it can be shown that
+(3.12)
+L(I − Pn)ϕ(ti) = E2r(ϕ)(ti)h2r + O
+�
+h2r+2�
+,
+i = 0, 1, . . . , n,
+where
+E2r(ϕ)(ti) = ¯b2r,2r
+� 1
+0
+˜ℓ(ti, t)(t) ϕ(2r)(t) dt
++
+2r−1
+�
+p=1
+¯b2r,p
+� �� ∂
+∂t
+�2r−p−1 �
+˜ℓ(ti, t)ϕ(p)(t)
+��t=1
+t=0
+−
+�� ∂
+∂t
+�2r−p−1 �
+˜ℓ(ti, t)ϕ(p)(t)
+��t=ti+
+t=ti−
+�
+with
+¯b2r,p =
+� 1
+0
+� 1
+0
+Λr(τ, s)(τ − s)p
+p!
+B2r−p(s)
+(2r − p)! dτ ds
+and Bk is the Bernoulli polynomial of degree k ≥ 0.
+3.2 Approximation of the Integral Operator
+Let x ∈ X . Recall that
+K(x)(s) =
+� 1
+0
+κ(s, t, x(t)) dt,
+s ∈ [0, 1].
+Replacing the above integral by the numerical quadrature rule (3.4), we define the Nyström
+approximation of K by
+Km(x)(s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq κ(s, tj−1 + µqνh, x(tj−1 + µqνh)),
+s ∈ [0, 1].
+Let {µj
+qν = tj−1 + µqνh : j = 1, 2, . . . , n; q = 1, 2, . . . , ρ; ν = 1, 2, . . . , p} denotes the set
+of all quadrature nodes in [0, 1]. Then
+Km(x)(s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq κ
+�
+s, µj
+qν, x
+�
+µj
+qν
+��
+,
+s ∈ [0, 1].
+The Nyström method for solving (1.1) is to find the element xm for which
+xm − Km(xm) = f.
+For sufficiently large m, the above equation has a unique solution ϕm in a neighborhood
+B(ϕ, ǫ) of ϕ, and
+∥ϕ − ϕm∥∞
+≤
+C2 ∥K(ϕ) − Km(ϕ)∥∞ = O
+�
+˜h2�
+,
+(3.13)
+
+10
+G. RAKSHIT
+where C2 is a constant independent of m. See [2, Theorem 4]. We write
+(I − Pn) ϕm = (I − Pn) (ϕm − ϕ) + (I − Pn) ϕ.
+Then from (3.10), (3.13), we have
+(3.14)
+(I − Pn) ϕm = O
+�
+max
+�
+hr, ˜h2��
+.
+Let v1, v2 ∈ X and x ∈ B(ϕ, ǫ). Then the Fréchet derivatives of Km at x are given by
+K′
+m(x)v1(s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq D(0,0,1)κ
+�
+s, µj
+qν, x
+�
+µj
+qν
+��
+v1
+�
+µj
+qν
+�
+,
+s ∈ [0, 1],
+K′′
+m(x) (v1, v2) (s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq
+∂2κ
+∂u2
+�
+s, µj
+qν, x
+�
+µj
+qν
+��
+v1
+�
+µj
+qν
+�
+v2
+�
+µj
+qν
+�
+.
+It follows that
+∥K′′
+m(x) (v1, v2)∥∞ ≤
+
+
+
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+����
+∂2κ
+∂u2(s, t, u)
+����
+
+
+ ∥v1∥∞ ∥v2∥∞ .
+This implies
+∥K′′
+m(x)∥ < ∞
+Similarly, it can be shown that
+���K(3)
+m (x)
+��� ≤
+
+
+
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+����
+∂3κ
+∂u3 (s, t, u)
+����
+
+
+ = C3 < ∞
+Lemma 3.3. Let x1, x2 ∈ B(ϕ, ǫ). If D(0,0,3)κ ∈ C(Ω) then
+∥K′′
+m(x1) − K′′
+m(x2)∥ ≤ C3 ∥x1 − x2∥∞ ,
+where C3 is constant independent of n.
+Proof. For v1, v2 ∈ X , we have
+[K′′
+m(x1) − K′′
+m(x2)] (v1, v2)(s)
+= h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq
+�∂2κ
+∂u2
+�
+s, µj
+qν, x1
+�
+µj
+qν
+��
+− ∂2κ
+∂u2
+�
+s, µj
+qν, x2
+�
+µj
+qν
+���
+v1
+�
+µj
+qν
+�
+v2
+�
+µj
+qν
+�
+for all s ∈ [0, 1]. Since D(0,0,3)κ ∈ C(Ω), applying mean value theorem on ∂2κ
+∂u2 with respect
+to its third variable u, we obtain
+∂2κ
+∂u2
+�
+s, µj
+qν, x1
+�
+µj
+qν
+��
+−∂2κ
+∂u2
+�
+s, µj
+qν, x2
+�
+µj
+qν
+��
+=
+�
+x1
+�
+µj
+qν
+�
+− x2
+�
+µj
+qν
+�� ∂3κ
+∂u3
+�
+s, µj
+qν, ζj
+qν
+�
+,
+where ζj
+qν lies in the line segment joining the points x1
+�
+µj
+qν
+�
+and x2
+�
+µj
+qν
+�
+. Then
+����
+∂2κ
+∂u2
+�
+s, µj
+qν, x1
+�
+µj
+qν
+��
+− ∂2κ
+∂u2
+�
+s, µj
+qν, x2
+�
+µj
+qν
+������ ≤ C3 ∥x1 − x2∥∞ .
+Hence
+∥[K′′
+m(x1) − K′′
+m(x2)] (v1, v2)∥∞ ≤ C3 ∥x1 − x2∥∞ ∥v1∥∞ ∥v2∥∞ ,
+which follows the result.
+□
+
+Section 4. Asymptotic Error Analysis
+11
+We will now quote some error estimates for the Nyström approximations.
+For α ≥ 0, if v1, v2 ∈ Cα
+∆(m)[0, 1], then from [27] or [4, Corollary 1], we obtain the
+following errors for numerical integration.
+∥[K′
+m(ϕ) − K′(ϕ)] v1∥∞ = O
+�
+˜h2�
+,
+(3.15)
+∥[K′′
+m(ϕ) − K′′(ϕ)] (v1, v2)∥∞ = O
+�
+˜h2�
+.
+Also from [18, Proposition 3.3], we have
+∥K′
+m(ϕm) − K′
+m(ϕ)∥ ≤ C4 ∥ϕm − ϕ∥∞ = O
+�
+˜h2�
+.
+(3.16)
+Therefore combining (3.15) and the above equation, we obtain
+∥[K′
+m(ϕm) − K′(ϕ)] v∥∞ = O
+�
+˜h2�
+,
+for all v ∈ Cν
+∆m[0, 1].
+(3.17)
+Similarly,
+∥[K′′
+m(ϕm) − K′′(ϕ)] (v1, v2)∥∞ = O
+�
+˜h2�
+,
+∀v1, v2 ∈ Cν
+∆m[0, 1],
+(3.18)
+for all v1, v2, v3 ∈ Cν
+∆m[0, 1] implies
+���
+K(3)
+m (ϕm) − K(3)(ϕ)
+�
+(v1, v2, v3)
+��
+∞ = O
+�
+˜h2�
+.
+(3.19)
+4 Asymptotic Error Analysis
+Replacing K by Km and πn by Pn in the Galerkin equation x − πnK(x) = πnf, the
+discrete Galerkin equation is defined by zG
+n − PnKm(zG
+n ) = Pnf, where zG
+n is the discrete
+Galerkin solution. Then the discrete iterated Galerkin solution is defined by
+zS
+n = Km(zG
+n ) + f.
+Note that PnzS
+n = zG
+n . From the equations ϕm − Km(ϕm) = f and zS
+n − Km(PnzS
+n) = f,
+we obtain the following error term.
+zS
+n − ϕm = [I − K′
+m(ϕm)]−1 �
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+− Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+− Lm(I − Pn)K′
+m(ϕm)(zG
+n − ϕm)
+− Lm(I − Pn)ϕm,
+(4.1)
+where
+Lm = [I − K′
+m(ϕm)]−1 K′
+m(ϕm).
+Using the Resolvent Identity, we get
+(I − K′
+m(ϕm))−1 − (I − K′(ϕ))−1
+= (I − K′(ϕ))−1 [K′
+m(ϕm) − K′(ϕ)] (I − K′
+m(ϕm))−1
+
+12
+G. RAKSHIT
+Therefore
+(I − K′
+m(ϕm))−1 = (I − K′(ϕ))−1
++ (I − K′(ϕ))−1 [K′
+m(ϕm) − K′
+m(ϕ)] (I − K′
+m(ϕm))−1
++ (I − K′(ϕ))−1 [K′
+m(ϕ) − K′(ϕ)] (I − K′
+m(ϕm))−1
+(4.2)
+Now, we will analyze each of the terms appearing in the RHS of the equation (4.1). Error
+estimates for each of the said terms will be obtained by the following propositions.
+Proposition 4.1. Let {ti : i = 0, 1, . . . , n} be the set of partition points of [0, 1] defined by
+(3.2), then
+Lm(I − Pn)ϕm(ti) = E2r(ϕ)(ti)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where E2r is defined by (3.12).
+Proof. It can be easily verified that (using (4.2))
+Lm(I − Pn)ϕm =
+�
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
+=
+�
+I − K′(ϕ)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
++
+�
+I − K′(ϕ)
+�−1 �
+K′
+m(ϕm) − K′
+m(ϕ)
+� �
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
++
+�
+I − K′(ϕ)
+�−1 �
+K′
+m(ϕ) − K′(ϕ)
+� �
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm.
+(4.3)
+Consider the first term of the above equation, we have
+(I − K′(ϕ))−1K′
+m(ϕm)(I − Pn)ϕm
+= (I − K′(ϕ))−1 K′(ϕ)(I − Pn)ϕ
++ (I − K′(ϕ))−1 K′(ϕ)(I − Pn)(ϕm − ϕ)
++ (I − K′(ϕ))−1 [K′
+m(ϕm) − K′(ϕ)] (I − Pn)ϕm.
+Using (3.12), (3.13) and (3.17), we obtain
+(4.4)
+(I − K′(ϕ))−1 K′
+m(ϕm)(I −Pn)ϕm(ti) = E2r(ϕ)(ti)h2r +O
+�
+max
+�
+h2r+2, ˜h2��
+.
+Note that
+∥K′
+m(ϕm)∥ ≤
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+|κu(s, t, u)|
+and from [18, Proposition 4.2], we have
+��(I − K′
+m(ϕm))−1�� < ∞. Thus, from (3.15) and
+(3.16), we have the followings
+���
+�
+I − K′(ϕ)
+�−1 �
+K′
+m(ϕm) − K′
+m(ϕ)
+� �
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
+���
+∞ = O
+�
+˜h2�
+,
+���
+�
+I − K′(ϕ)
+�−1 �
+K′
+m(ϕm) − K′
+m(ϕ)
+� �
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
+���
+∞ = O
+�
+˜h2�
+.
+Hence the required result follows from (4.3), (4.4) and the above two estimates.
+□
+Before each of the following propositions, we prove lemmas and its corollaries which are
+used to prove next propositions.
+
+Section 4. Asymptotic Error Analysis
+13
+Lemma 4.1. Let Pn be the discrete orthogonal projection defined by (3.8). If D(0,0,3)κ ∈
+C(Ω) and v ∈ Cr+2([0, 1]), then for r ≥ 1
+K′′(ϕ)(Pnv − v)2 = T(v)h2r + O(h2r+2),
+(4.5)
+where
+T(v) =
+�� 1
+0
+Jr(τ)2 dτ
+�
+K′′(ϕ)
+�
+v(r)�2 .
+Furthermore, when r = 1, then
+(4.6)
+K(3)(ϕ)(Pnv − v)3 = O
+�
+h4�
+.
+Proof. The proofs of (4.5) and (4.6) follows from Lemma 3.2, [16, Lemma 2.4] and [16,
+Remark 2.4] respectively.
+□
+Given that (I − K′(ϕ))−1 is a bounded linear operator. Let
+M = (I − K′(ϕ))−1 K′′(ϕ).
+Note that M is a compact bi-linear integral operator. Also the smoothness of the kernel of
+M, is same as the kernels of K′′(ϕ). See [4], [26].
+As a consequence of the above lemma, we get the following result.
+Corollary 4.1. For r ≥ 1,
+M(Pnv − v)2 = T (v)h2r + O(h2r+2),
+(4.7)
+where
+T (v) =
+�� 1
+0
+Jr(τ)2 dτ
+�
+M
+�
+v(r)�2
+.
+Lemma 4.2. If ϕ ∈ Cr+2([0, 1]), then for r ≥ 1,
+K′′
+m(ϕm)(zG
+n − ϕm)2 = T(ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where T is defined in Lemma 4.1.
+Proof. Note that
+zG
+n − ϕm = PnzS
+n − Pnϕm − ϕm + Pnϕm + ϕ
+= Pn
+�
+zS
+n − ϕm
+�
+− (I − Pn) ϕm
+(4.8)
+Thus,
+K′′
+m(ϕm)(zG
+n − ϕm)2
+= K′′
+m(ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+��2 − 2K′′
+m(ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+�
+, (I − Pn) ϕm
+�
++ K′′
+m(ϕm) ((I − Pn) ϕm)2
+(4.9)
+Since ∥K′′
+m(ϕm)∥ < ∞ and ∥Pn∥ < ∞, from (1.4) it is easy to see that
+(4.10)
+���K′′
+m(ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+��2���
+∞ = O
+�
+max
+�
+h2r+4, ˜h4��
+,
+(4.11)
+��K′′
+m(ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+�
+, (I − Pn) ϕm
+���
+∞ = O
+�
+˜h2 max
+�
+hr+2, ˜h2��
+.
+
+14
+G. RAKSHIT
+Now we write
+K′′
+m(ϕm) ((I − Pn) ϕm)2 =
+�
+K′′
+m(ϕm) − K′′(ϕ)
+�
+((I − Pn) ϕm)2
++ K′′(ϕ) ((I − Pn) ϕm)2
+= K′′(ϕ) ((I − Pn) ϕ)2 + K′′(ϕ) ((I − Pn) (ϕ − ϕm))2
++
+�
+K′′
+m(ϕm) − K′′(ϕ)
+�
+((I − Pn) ϕm)2 .
+Since ∥K′′
+m(ϕm)∥ < ∞ and ∥Pn∥ < ∞, from (3.13), (3.18) and the above estimate, we
+obtain
+K′′
+m(ϕm) ((I − Pn) ϕm)2 = K′′(ϕ) ((I − Pn) ϕ)2 + O
+�
+˜h2�
+.
+Therefore, from (4.5) we obtain
+K′′
+m(ϕm) ((I − Pn) ϕm)2 = T(ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+.
+Hence, the required result follows from (4.9), (4.10), (4.11) and the above equation.
+□
+From the above lemma, we obtain
+(4.12)
+(I − K′(ϕ))−1 K′′
+m(ϕm)(zG
+n − ϕm)2 = T (ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where T is defined by (4.7).
+Lemma 4.3. If ϕ ∈ Cr+2([0, 1]), then
+[I − K′
+m(ϕm)]−1 K(3)
+m (ϕm)(zG
+n − ϕm)3 =
+
+
+
+O
+�
+max
+�
+h4, ˜h2��
+,
+r = 1,
+O
+�
+max
+�
+h3r, ˜h6��
+, r ≥ 2.
+Proof. First, we consider the case when r ≥ 2.
+Since
+��[I − K′
+m(ϕm)]−1�� < ∞ and
+���K(3)
+m (ϕm)
+��� < ∞, from (1.4) we obtain
+���[I − K′
+m(ϕm)]−1 K(3)
+m (ϕm)(zG
+n − ϕm)3���
+∞ = O
+�
+max
+�
+h3r, ˜h6��
+.
+Now consider the case, when r = 1. We rewrite (4.8) as
+(zG
+n − ϕm)3 =
+�
+Pn
+�
+zS
+n − ϕm
+�
+− (I − Pn) ϕm
+�3 .
+Thus
+K(3)
+m (ϕm)(zG
+n − ϕm)3 = K(3)
+m (ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+��3 − K(3)
+m (ϕm) ((I − Pn) ϕm)3
+− K(3)
+m (ϕm)
+��
+Pn
+�
+zS
+n − ϕm
+��2 , (I − Pn) ϕm
+�
++ K(3)
+m (ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+�
+, ((I − Pn) ϕm)2�
+.
+(4.13)
+Since
+���K(3)
+m (ϕm)
+��� < ∞ and ∥Pn∥ < ∞, from (1.5) and (3.14) we obtain
+K(3)
+m (ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+��3 = O
+�
+h6�
+,
+K(3)
+m (ϕm)
+��
+Pn
+�
+zS
+n − ϕm
+��2 , (I − Pn) ϕm
+�
+= O
+�
+h5�
+,
+K(3)
+m (ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+�
+, ((I − Pn) ϕm)2�
+= O
+�
+h4�
+.
+
+Section 4. Asymptotic Error Analysis
+15
+Note that, we have used the fact ˜h ≤ h in the above three expressions. On the other hand,
+from (3.13), (3.19) we have
+K(3)
+m (ϕm) ((I − Pn) ϕm)3 = K(3)(ϕ) ((I − Pn) ϕ)3 + O
+�
+˜h2�
+.
+From (4.6), it follows that
+(4.14)
+K(3)
+m (ϕm) ((I − Pn) ϕm)3 = O
+�
+max
+�
+h4, ˜h2��
+.
+Now, combining the results (4.13) - (4.14), we obtain
+K(3)
+m (ϕm)(zG
+n − ϕm)3 = O
+�
+max
+�
+h4, ˜h2��
+.
+Therefore
+[I − K′
+m(ϕm)]−1 K(3)
+m (ϕm)(zG
+n − ϕm)3 = O
+�
+max
+�
+h4, ˜h2��
+.
+Hence follows the result.
+□
+Proposition 4.2. Let ϕ ∈ Cr+2([0, 1]). Then for r ≥ 1,
+[I − K′
+m(ϕm)]−1 �
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+(s)
+= 1
+2T (ϕ)(s)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+for all s ∈ [0, 1].
+Proof. Applying the generalized Taylor’s series expansion of Km about ϕm in the neigh-
+bourhood B(ϕ, ǫ), we obtain
+(4.15)
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+= 1
+2K′′
+m(ϕm)(zG
+n − ϕm)2 + 1
+6K(3)
+m (ϕm)(zG
+n − ϕm)3 + R4,m
+�
+zG
+n − ϕm
+�
+,
+where
+R4,m
+�
+zG
+n − ϕm
+�
+=
+� 1
+0
+(1 − θ)3
+3!
+K(4)
+m
+�
+ϕm + θ(zG
+n − ϕm)
+�
+(zG
+n − ϕm)4 dθ.
+Note that for any x ∈ B(ϕ, ǫ), v ∈ X ,
+K(4)
+m (x)v4(s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq
+∂4κ
+∂u4
+�
+s, µj
+qν, x
+�
+µj
+qν
+��
+v4�
+µj
+qν
+�
+,
+s ∈ [0, 1].
+It follows that
+��K(4)
+m (x)v4��
+∞ ≤
+
+
+
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+����
+∂4κ
+∂u4 (s, t, u)
+����
+
+
+ ∥v∥4
+∞ = C5 ∥v∥4
+∞ .
+Since ϕm and zG
+n ∈ B(ϕ, ǫ), ϕm + θ(zG
+n − ϕm) ∈ B(ϕ, ǫ) and therefore
+��K(4)
+m
+�
+ϕm + θ(zG
+n − ϕm)
+�
+(zG
+n − ϕm)4��
+∞ ≤ C5
+��zG
+n − ϕm
+��4
+∞ = O
+�
+h4r�
+.
+It follows that
+R4,m
+�
+zG
+n − ϕm
+�
+= O
+�
+h4r�
+.
+
+16
+G. RAKSHIT
+Using the resolvent identity (3.17) and (4.2), we obtain
+[I − K′
+m(ϕm)]−1 K′′
+m(ϕm)(zG
+n −ϕm)2 = [I − K′(ϕ)]−1 K′′
+m(ϕm)(zG
+n −ϕm)2 +O
+�
+˜h2�
+.
+By (4.12), it follows that
+[I − K′
+m(ϕm)]−1 K′′
+m(ϕm)(zG
+n − ϕm)2 = T (ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+.
+From the Lemma 4.3, we have
+(4.16)
+[I − K′
+m(ϕm)]−1 K(3)
+m (ϕm)(zG
+n − ϕm)3 =
+
+
+
+O
+�
+max
+�
+h4, ˜h2��
+,
+r = 1,
+O
+�
+max
+�
+h3r, ˜h6��
+, r ≥ 2.
+Combining the results from (4.15) to (4.16), we obtain for r ≥ 1,
+[I − K′
+m(ϕm)]−1 �
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+= 1
+2T (ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+which completes the proof.
+□
+Lemma 4.4. If v ∈ X , then for r ≥ 1,
+max
+0≤i≤n |K′
+m(ϕm)(I − Pn)v(ti)|
+≤
+C7 ∥(I − Pn)v∥∞ hr,
+where C7 is a constant independent of h.
+Proof. We write
+(4.17)
+K′
+m(ϕm)(I − Pn)v = K′
+m(ϕ)(I − Pn)v + [K′
+m(ϕm) − K′
+m(ϕ)](I − Pn)v.
+For fixed s ∈ [0, 1], let
+ℓ∗,s(t) = ℓ∗(s, t) = ℓ(s, t, ϕ(t)) = ∂κ
+∂u(s, t, ϕ(t)),
+t ∈ [0, 1].
+From the definition of K′
+m(ϕ) and the discrete inner product, we have
+K′
+m(ϕ)(I − Pn)v(s)
+=
+n
+�
+j=1
+⟨ℓ∗,s , (I − Pn,j)v⟩∆j,m .
+Since Pn,j is self-adjoint on C(∆j), so as I − Pn,j. Therefore
+K′
+m(ϕ)(I − Pn)v(s)
+=
+n
+�
+j=1
+⟨(I − Pn,j)ℓ∗,s , (I − Pn,j)v⟩∆j,m .
+Note that, if s = ti for some i ∈ {0, 1, . . . , n}, then ℓ∗,s ∈ Cr[tj−1, tj] for all j = 1, . . . , n.
+Hence from (3.9),
+∥(I − Pn,j)ℓ∗,ti∥∆j,∞
+≤
+C1
+�
+sup
+t∈[tj−1,tj]
+|D(0,r)ℓ∗(ti, t)|
+�
+hr
+Thus,
+max
+0≤i≤n |K′
+m(ϕ)(I − Pn)v(ti)| ≤
+n
+�
+j=1
+∥(I − Pn,j)ℓm,s∥∆j,∞ ∥(I − Pn,j)v∥∆j,∞h
+≤ C6 ∥(I − Pn)v∥∞ hr,
+
+Section 4. Asymptotic Error Analysis
+17
+where C6 is a constant independent of h. Now, from (3.16), (4.17) and the above estimate,
+we obtain
+max
+0≤i≤n |K′
+m(ϕm)(I − Pn)v(ti)| ≤ C7 ∥(I − Pn)v∥∞ max
+�
+hr, ˜h2�
+,
+where C7 = C4 + C6. Since hr ≥ ˜h2, the result follows.
+□
+Recall that Lm = [I − K′
+m(ϕm)]−1 K′
+m(ϕm). Therefore, the proof of the following result
+is similar to that of the above lemma.
+Corollary 4.2. If v ∈ X , then for r ≥ 1,
+max
+0≤i≤n|Lm(I − Pn)v(ti)|
+≤
+C8 ∥(I − Pn)v∥∞ hr,
+where C8 is a constant independent of h.
+Lemma 4.5. Let v ∈ X . If r = 1, that is, when the range of Pn is the space of piecewise
+polynomials of degree zero, then
+∥(I − Pn)K′′
+m(ϕ)(v, v)∥∞ ≤ C9h ∥v∥2
+∞ ,
+where C9 is a constant independent of h.
+Proof. Given that Xn is the space of piecewise constant functions with respect to the parti-
+tion (3.2). Note that
+∥(I − Pn)K′′
+m(ϕ)(v, v)∥∞
+=
+max
+1≤j≤n
+sup
+s∈[tj−1,tj]
+|(I − Pn,j)K′′
+m(ϕ)(v, v)(s)|.
+Let s ∈ [tj−1, tj]. Since the Legendre polynomial of zero L0(t) = 1 for all t ∈ [0, 1], we
+have from (3.11),
+(4.18)
+(I − Pn)K′′
+m(ϕ)(v, v)(s)
+= 1
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq [K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(tj−1 + µqνh)] .
+We also have
+K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(tj−1 + µqνh)
+= h
+p
+n
+�
+k=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq
+�∂2κ
+∂u2
+�
+s, µk
+qν, ϕ
+�
+µk
+qν
+��
+− ∂2κ
+∂u2
+�
+µj
+qν, µk
+qν, ϕ
+�
+µk
+qν
+���
+v2�
+µk
+qν
+�
+,
+where µj
+qν = tj−1 + µqνh. For fixed s ∈ [0, 1], let
+λ∗,s(t) = λ∗(s, t) = λ(s, t, ϕ(t)) = ∂2κ
+∂u2 (s, t, ϕ(t)),
+t ∈ [0, 1].
+Then, by (3.5)
+K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(µj
+qν) =
+n
+�
+k=1
+�
+λ∗,s − λ∗,µiqν, v2�
+∆k,m
+=
+n
+�
+k=1
+k̸=j
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆k,m +
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆j,m .
+(4.19)
+
+18
+G. RAKSHIT
+First consider the case when k ̸= j. Applying Mean Value Theorem on the first component
+of λ∗(·, ·) in the interval [s, µj
+qν], we obtain
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆k,m = (s − µj
+qν)
+�
+D(1,0)λ∗(θj
+qν, ·) , v2�
+∆k,m ,
+for some θj
+qν ∈ (tj−1, tj), and the function D(1,0)λ∗(s, t), t ∈ [tk−1, tk] is given by
+D(1,0)λ∗(s, t) =
+�
+D(1,0)λ1,∗(s, t) =
+∂
+∂sλ1(s, t, ϕ(t)),
+0 ≤ t ≤ s ≤ 1,
+D(1,0)λ2,∗(s, t) =
+∂
+∂sλ2(s, t, ϕ(t)),
+0 ≤ s ≤ t ≤ 1.
+Therefore, for k ̸= j,
+����
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆k,m
+����
+≤
+��s − µj
+qν
+��
+�
+sup
+s̸=t
+��D(1,0)λ∗(s, t)
+��
+�
+∥v∥2
+∞ h
+≤
+�
+sup
+s̸=t
+��D(1,0)λ∗(s, t)
+��
+�
+∥v∥2
+∞ h2,
+where
+sup
+s̸=t
+|D(1,0)λ∗(s, t)|
+= max
+�
+sup
+0≤t<s≤1
+|D(1,0)λ1,∗(s, t)| ,
+sup
+0≤s<t≤1
+|D(1,0)λ2,∗(s, t)|
+�
+= max
+
+
+
+
+
+sup
+0≤t<s≤1
+|u|≤∥ϕ∥∞
+|D(1,0,2)κ1(s, t, u)| ,
+sup
+0≤s<t≤1
+|u|≤∥ϕ∥∞
+|D(1,0,2)κ2(s, t, u)|
+
+
+
+
+
+.
+On the other hand,
+|
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆j,m| ≤ 2
+�
+sup
+0≤s,t≤1
+|λ∗(s, t)|
+�
+∥v∥2
+∞ h,
+where
+sup
+0≤s,t≤1
+|λ∗(s, t)| =
+sup
+0≤s,t≤1
+|u|≤∥ϕ∥∞
+|D(0,0,2)κ(s, t, u)|. Then, from (4.19) we obtain
+|K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(µj
+qν)|
+≤
+
+
+
+n
+�
+k=1
+k̸=j
+�
+sup
+s̸=t
+|D(1,0)λ∗(s, t)|
+�
+∥v∥2
+∞ h2
+
+
+ + 2
+�
+sup
+0≤s,t≤1
+|λ∗(s, t)|
+�
+∥v∥2
+∞ h.
+It follows that
+|K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(µj
+qν)| ≤ C9 ∥v∥2 h,
+where C9 =
+�
+sup
+s̸=t
+|D(1,0)λ∗(s, t)|
+�
++ 2
+�
+sup
+0≤s,t≤1
+|λ∗(s, t)|
+�
+.
+The result now follows from (4.18) and the above estimate.
+□
+Corollary 4.3. Let v ∈ X . If r = 1, that is, when the range of Pn is the space of piecewise
+polynomials of degree zero, then
+∥(I − Pn)K′
+m(ϕ)v∥∞ ≤ C10h ∥v∥∞ ,
+where C10 is a constant independent of h.
+
+Section 4. Asymptotic Error Analysis
+19
+Proof. The proof is similar to that of Lemma 4.5.
+□
+Proposition 4.3. Let ti be any point of the partition ∆(n) defined by (3.2). Then
+Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+(ti)
+=
+
+
+
+O (h4) ,
+r = 1,
+O
+�
+max
+�
+h3r, hr˜h4��
+,
+r ≥ 2.
+Proof. Generalized Taylor’s series expansion gives
+(4.20)
+Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+= 1
+2Lm(I − Pn)K′′
+m(ϕm)(zG
+n − ϕm)2 + Lm(I − Pn)R3,m
+�
+zG
+n − ϕm
+�
+,
+where
+R3,m
+�
+zG
+n − ϕm
+�
+=
+� 1
+0
+(1 − θ)2
+2!
+K(3)
+m
+�
+ϕm + θ(zG
+n − ϕm)
+�
+(zG
+n − ϕm)3 dθ.
+It follows that
+��R3,m
+�
+zG
+n − ϕm
+���
+∞ ≤ 1
+6
+
+
+
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+����
+∂3κ
+∂u3 (s, t, u)
+����
+
+
+
+��zG
+n − ϕm
+��3
+∞ .
+Therefore, by (1.4)
+��R3,m
+�
+zG
+n − ϕm
+���
+∞ = O
+�
+max
+�
+h3r, ˜h6��
+.
+Since ∥I − Pn∥ ≤ 1 + ∥Pn∥ < ∞, from Corollary 4.2, it is easy to see that
+Lm(I − Pn)R3,m
+�
+zG
+n − ϕm
+�
+(ti) = O
+�
+max
+�
+h4r, hr˜h6��
+.
+(4.21)
+First consider the case r ≥ 2. Since ∥K′′
+m(ϕm)∥ < ∞ and ∥I − Pn∥∞ < ∞, by (1.4) and
+the Corollary 4.2, we have
+1
+2Lm(I − Pn)K′′
+m(ϕm)(zG
+n − ϕm)2(ti) = O
+�
+max
+�
+h3r, hr˜h4��
+,
+r ≥ 2.
+When r = 1, we write
+Lm(I − Pn)K′′
+m(ϕm)(zG
+n − ϕm)2
+= Lm(I − Pn)
+�
+K′′
+m(ϕm) − K′′
+m(ϕ)
+�
+(zG
+n − ϕm)2 + Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2.
+By (1.4), (3.13) and the Lemma 3.3, we have
+Lm(I − Pn) [K′′
+m(ϕm) − K′′
+m(ϕ)] (zG
+n − ϕm)2 = O
+�
+h4�
+.
+On the other hand
+Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2(ti) =
+n
+�
+j=1
+�
+ℓm,ti , (I − Pn,j)K′′
+m(ϕ)(zG
+n − ϕm)2�
+∆j,m .
+
+20
+G. RAKSHIT
+Since I − Pn,j is self-adjoint,
+Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2(ti)
+=
+n
+�
+j=1
+�
+(I − Pn,j)ℓm,ti , (I − Pn,j)K′′
+m(ϕ)(zG
+n − ϕm)2�
+∆j,m .
+It follows that
+max
+0≤i≤n
+��Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2(ti)
+��
+≤
+n
+�
+j=1
+∥(I − Pn,j)ℓm,ti∥∆j,∞
+��(I − Pn,j)K′′
+m(ϕ)(zG
+n − ϕm)2��
+∆j,∞ h.
+By Corollary 4.2 and Lemma 4.5, we obtain
+max
+0≤i≤n|Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2(ti)| = O
+�
+h4�
+,
+r = 1.
+Therefore
+1
+2Lm(I − Pn)K′′
+m(ϕm)(zG
+n − ϕm)2(ti) =
+
+
+
+O (h4) ,
+r = 1,
+O
+�
+max
+�
+h3r, hr˜h4��
+,
+r ≥ 2.
+(4.22)
+Then combing (4.20), (4.21) and (4.22), we obtain
+Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+(ti)
+=
+
+
+
+O (h4) ,
+r = 1,
+O
+�
+max
+�
+h3r, hr˜h4��
+,
+r ≥ 2.
+This follows the result.
+□
+We quote the following result from By [17, Proposition 1, Proposition 6], which will be
+used in the next proposition.
+(4.23)
+∥(I − Pn)K′
+m(ϕ)(I − Pn)ϕ∥∞ =
+�
+O (h3) ,
+r = 1,
+O (hr+2) ,
+r ≥ 2.
+Proposition 4.4. If ϕm and zG
+n are respectively the Nyström and the discrete Galerkin ap-
+proximation of ϕ, then
+Lm(I − Pn)K′
+m(ϕm)(zG
+n − ϕm)(ti) = O
+�
+max
+�
+h2r+2, ˜h2��
+.
+Proof. Adding and subtracting K′
+m(ϕ), we have
+Lm(I − Pn)K′
+m(ϕm)(zG
+n − ϕm)
+= Lm(I − Pn) [K′
+m(ϕm) − K′
+m(ϕ)] (zG
+n − ϕm) + Lm(I − Pn)K′
+m(ϕ)(zG
+n − ϕm).
+Then, using (1.4), (3.16) and the Corollary 4.2, we obtain for r ≥ 1,
+(4.24)
+��Lm(I − Pn) [K′
+m(ϕm) − K′
+m(ϕ)] (zG
+n − ϕm)(ti)
+��
+≤ C4C8 (1 + ∥Pn∥) hr˜h2 �
+max
+�
+hr, ˜h2��
+
+Section 4. Asymptotic Error Analysis
+21
+Note that
+zG
+n − ϕm = PnzS
+n − ϕm = PnzS
+n − Pnϕ + Pnϕ − ϕ + ϕ − ϕm
+= Pn
+�
+zS
+n − ϕ
+�
+− (I − Pn) ϕ + (ϕ − ϕm) .
+Then
+Lm(I − Pn)K′
+m(ϕ)(zG
+n − ϕm) = Lm(I − Pn)K′
+m(ϕ)Pn
+�
+zS
+n − ϕ
+�
+− Lm(I − Pn)K′
+m(ϕ) (I − Pn) ϕ
++ Lm(I − Pn)K′
+m(ϕ) (ϕ − ϕm) .
+By the Corollary 4.2
+��Lm(I − Pn)K′
+m(ϕ)Pn
+�
+zS
+n − ϕ
+�
+(ti)
+�� ≤ C8
+��(I − Pn) K′
+m(ϕ)Pn
+�
+zS
+n − ϕ
+��� hr,
+then by (1.5) and Corollary 4.3, we obtain
+��Lm(I − Pn)K′
+m(ϕ)Pn
+�
+zS
+n − ϕ
+�
+(ti)
+�� = O
+�
+max
+�
+h2r+2, ˜h2��
+,
+for r ≥ 1.
+Also, the Corollary 4.2 and (4.23) implies
+Lm(I − Pn)K′
+m(ϕ) (I − Pn) ϕ(ti) = O
+�
+h2r+2�
+,
+for r ≥ 1.
+It is easy to see (from (3.13) and Corollary 4.2) that
+Lm(I − Pn)K′
+m(ϕ) (ϕ − ϕm) = O
+�
+hr˜h2�
+,
+for r ≥ 1.
+Therefore, for r ≥ 1,
+Lm(I − Pn)K′
+m(ϕ)(zG
+n − ϕm)(ti) = O
+�
+max
+�
+h2r+2, ˜h2��
+.
+Hence, the result follows from (4.24) and the above equation.
+□
+We prove the main theorem as follows.
+Theorem 4.1. Let K be the Urysohn integral operator with Green’s function type kernel κ,
+defined by (1.2). Let ϕ be the unique solution of the equation (1.1). Assume that 1 is not
+an eigenvalue of K′(ϕ). Let Xn be the space of piecewise polynomials of degree ≤ r − 1
+with respect to the partition ∆(n) := 0 = t0 < t1 < · · · < tn = 1 defined by (3.2). Let
+Pn : L∞[0, 1] → Xn be the discrete orthogonal projection defined by (3.8) and ˜zS
+n be the
+discrete iterated Galerkin approximation of ϕ. Then
+�
+zS
+n − ϕ
+�
+(ti) =
+�
+E2r(ϕ)(ti) + 1
+2T (ϕ)(ti)
+�
+h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where the operators E2r and T are respectively defined by (3.12) and (4.5).
+Proof. We have from (4.1)
+zS
+n − ϕ = [I − K′
+m(ϕm)]−1 �
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+− Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+− Lm(I − Pn)K′
+m(ϕm)(zG
+n − ϕm)
+− Lm(I − Pn)ϕm
++ ϕm − ϕ.
+
+22
+G. RAKSHIT
+The result now follows from (3.13), Proposition 4.1, Proposition 4.2, Proposition 4.3 and
+Proposition 4.4.
+□
+We now apply Richardson extrapolation to obtain an approximation of ϕ with higher
+order of convergence. Define
+zEX
+n
+= 24rzS
+2n − zS
+n
+24r − 1
+.
+We choose the partitions ∆(m) and ∆(n) such that m2 ≥ n2r+2. Then, it is easy to see from
+the Theorem 4.1, that
+(4.25)
+�
+zEX
+n
+− ϕ
+�
+(ti) = O
+�
+h2r+2�
+,
+for all i = 1, 2, . . . , n.
+5 Numerical results
+For the numerical results, we consider the following example from [19]. Consider
+(5.1)
+ϕ(s) −
+� 1
+0
+κ(s, t) [ψ (t, ϕ(t))] dt = f(s),
+0 ≤ s ≤ 1,
+where
+κ(s, t) =
+1
+γ sinh γ
+�
+sinh γs sinh γ(1 − t),
+0 ≤ t ≤ s ≤ 1,
+γ(1 − s) sinh γt,
+0 ≤ s ≤ t ≤ 1,
+with γ =
+√
+12, and
+ψ(t, ϕ(t)) = γ2ϕ(t) − 2 (ϕ(t))3 ,
+t ∈ [0, 1].
+We have f(s) =
+1
+sinh γ
+�
+2 sinh γ(1 − s) + 2
+3 sinh γs
+�
+. The exact solution of (5.1) is given
+by
+ϕ(s) =
+2
+2s + 1,
+s ∈ [0, 1].
+Let Xn be the space of piecewise constant functions with respect to the uniform partition
+∆(n) of the interval [0, 1]. Let Pn : L∞[0, 1] → Xn be the discrete orthogonal projection
+defined by (3.8).
+Let ti = i−1
+20 , i = 1, 2, . . . , 21 be the partition points with step size h =
+1
+20. The numerical
+quadrature is chosen to be the composite 2 point Gaussian quadrature rule with respect to
+partition ∆(m) with m = n2 subintervals. Then ˜h = h2. Therefore, it is expected from the
+Theorem 4.1 and equation (4.25), that
+ǫS
+n(ti) = |ϕ(ti) − zS
+n(ti)| = O
+�
+h2�
+and ǫEX
+n
+(ti) = |ϕ(ti) − zEX
+n
+(ti)| = O
+�
+h4�
+,
+where
+zEX
+n
+(ti) = 4zS
+2n(ti) − zS
+n(ti)
+3
+.
+Let δS and δEX be respectively the orders of convergence of zS
+n and zEX
+n
+at the partition
+points. We expect δS = 2 and δEX = 4.
+
+Section 5. Numerical results
+23
+Table 1
+ti
+ǫS
+n(ti) : n = 20
+δS
+ǫEX
+n
+(ti) : n = 20
+δEX
+0.05
+8.6 × 10−3
+2.00
+2.98 × 10−6
+3.99
+0.1
+7.56 × 10−3
+2.00
+2.23 × 10−6
+3.99
+0.15
+6.79 × 10−3
+2.00
+1.59 × 10−6
+3.99
+0.2
+6.22 × 10−3
+2.00
+1.09 × 10−6
+3.97
+0.25
+5.78 × 10−3
+2.00
+7.13 × 10−7
+3.96
+0.3
+5.45 × 10−3
+2.00
+4.46 × 10−7
+3.94
+0.35
+5.19 × 10−3
+2.00
+2.7 × 10−7
+3.91
+0.4
+4.98 × 10−3
+2.00
+1.69 × 10−7
+3.86
+0.45
+4.82 × 10−3
+2.00
+1.3 × 10−7
+3.83
+0.5
+4.68 × 10−3
+2.00
+1.41 × 10−7
+3.85
+0.55
+4.55 × 10−3
+2.00
+1.91 × 10−7
+3.89
+0.6
+4.44 × 10−3
+2.00
+2.72 × 10−7
+3.93
+0.65
+4.33 × 10−3
+2.00
+3.75 × 10−7
+3.95
+0.7
+4.22 × 10−3
+2.00
+4.95 × 10−7
+3.97
+0.75
+4.10 × 10−3
+2.00
+6.26 × 10−7
+3.98
+0.8
+3.98 × 10−3
+2.00
+7.6 × 10−7
+3.99
+0.85
+3.84 × 10−3
+2.00
+8.94 × 10−7
+3.99
+0.9
+3.69 × 10−3
+2.00
+1.02 × 10−6
+3.99
+0.95
+3.52 × 10−3
+2.00
+1.14 × 10−6
+4
+From the above table, it is clear that the obtained orders of convergence match well with
+the theoretical orders of convergence. Also the order of convergence of the extrapolated
+solution improves upon the discrete iterated Galerkin solution.
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+[19] R. P. Kulkarni and A. S. Rane. Asymptotic expansions for approximate solutions of Hammerstein integral
+equations with Green’s function Type Kernels. Mathematical Modelling and Analysis, 19(1), 127-143 (2014).
+[20] Q. Lin, I. H. Sloan and R. Xie. Extrapolation of the Iterated–Collocation Method for Integral Equations
+of the Second Kind. SIAM Journal on Numerical Analysis, 27(6), 1535-1541 (1990).
+[21] P. Linz. Theoretical Numerical Analysis: Introduction to Advanced Techniques, Courier Dover Publica-
+tions, (2019).
+[22] W. McLean. Asymptotic error expansions for numerical solutions of integral equations. IMA Journal of
+Numerical Analysis, 9(3), 373-384 (1989).
+[23] G. Rakshit and A. S. Rane. Asymptotic expansion of iterated Galerkin solution of Fredholm integral
+equations of the second kind with Green’s kernel. Journal of Integral Equations and Applications, 32(4),
+495-507 (2020).
+[24] G. Rakshit, A. S. Rane and K. Patil. Richardson extrapolation for the iterated Galerkin solution of
+Urysohn integral equations with Green’s kernels. International Journal of Computer Mathematics, 19(8),
+1538 - 1556 (2022).
+[25] L. B. Rall, Computational solution of nonlinear operator equations, Wiley New York (1969).
+[26] F. Riesz, and B. S. Nagy. Functional Analysis. Courier Corporation, (2012).
+[27] L. Schumaker. Spline functions: Basic Theory. Cambridge University Press, (2007).
+[28] I. H. Sloan. Improvement by iteration for compact operator equations. Mathematics of Computation,
+30(136), 758-764 (1976).
+DEPARTMENT OF MATHEMATICAL SCIENCES, RAJIV GANDHI INSTITUTE OF PETROLEUM TECHNOL-
+OGY, JAIS CAMPUS, UTTAR PRADESH 229304, INDIA., ORCID ID : 0000-0002-5813-4656
+Email address: [email protected]
+

69AyT4oBgHgl3EQfpvg-/content/tmp_files/2301.00530v1.pdf.txt

@@ -0,0 +1,927 @@
+Test Reuse Based on Adaptive Semantic Matching across Android Mobile Applications
+Shuqi Liu1, Yu Zhou1,∗, Tingting Han2, and Taolue Chen2
+1College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China
+2Department of Computer Science and Data Science, Birkbeck, University of London, UK
+{liushuqi, zhouyu}@nuaa.edu.cn, {t.han, t.chen}@bbk.ac.uk
+*corresponding author
+Abstract—Automatic test generation can help verify and de-
+velop the behavior of mobile applications. Test reuse based on
+semantic similarities between applications of the same category
+has been utilized to reduce the manual effort of Graphical User
+Interface (GUI) testing. However, most of the existing studies fail
+to solve the semantic problem of event matching, which leads to
+the failure of test reuse. To overcome this challenge, we propose
+TRASM (Test Reuse based on Adaptive Semantic Matching), a
+test reuse approach based on adaptive strategies to find a better
+event matching across android mobile applications. TRASM
+first performs GUI events deduplication on the initial test set
+obtained from test generation, and then employs an adaptive
+strategy to find better event matching, which enables reusing
+the existing test. Preliminary experiments with comparison to
+baseline methods on 15 applications demonstrate that TRASM
+can improve the precision of GUI event matching while reducing
+the failure of test reuse and the running time required for test
+reuse.
+Keywords—adaptive semantic matching; android mobile appli-
+cations; GUI event; test reuse; oracle generation
+I. INTRODUCTION
+Graphical User Interface (GUI) testing is commonly em-
+ployed to verify and develop the behaviors of applications by
+designing and executing test cases of GUI applications [1].
+However, with the ever increasing functionalities in mobile
+applications, it takes more effort for developers to manually
+design GUI test cases (GUI test in short) [2–4], which in turn
+decreases the efficiency of testing processes.
+Considering the necessity of reducing time consumption,
+many researchers have conducted a series of investigation on
+automatic test generation [5–15] in GUI testing. Recently,
+some researchers observed that test reuse [16–24] could
+be achieved by exploiting the semantic similarity of GUIs
+between similar applications to generate tests automatically.
+Figure 1 shows a simple example in which the existing test (a)
+of application To-Do List is successfully reused to application
+Minimal, and the reused test (b) is obtained. As Figure 1
+shows, events em
+1 , em
+2 , em
+3 , and em
+4 in test (b) are similar to
+et
+1, et
+2, et
+3, and et
+4 in test (a), respectively.
+Existing research mainly focuses on how to accurately select
+specific characteristics of widgets in GUIs such as ‘text’
+and ‘resource-id’. Combining the selected characteristics, they
+design the semantic similarity calculation method between
+widgets to generate meaningful tests. They attempt to select
+widgets with high similarity in a similar application for match-
+ing each event of the existing test. However, little attention
+has been paid to optimizing the matching process. Taking
+Figure 1 as an example, the widget wt
+3 of To-Do List and
+the correctly similar widget wm
+3
+of the application Minimal
+are laid out differently in the GUI. When reusing test (b) to
+the application To-Do List, adopting the existing approach may
+always incorrectly match the widget wm
+3 with other widgets.
+This may cause subsequent events to match incorrectly or even
+result in failed test reuse. In cases where the existing methods
+do not work well, it is necessary to adopt other corresponding
+measures. The lower the similarity of the generated event,
+the more likely the match is inappropriate. Hence, mining
+such events and exploring other widgets with more similar
+semantics to form events for substitution is considered.
+In addition, in Figure 1, the application To-Do List needs
+to skip the boot page that the application Minimal does not
+before entering the home page. And it is assumed that the
+event step is et
+0 = click(wt
+0). Under this assumption, the
+widget wt
+0 will match the widget with the highest similarity
+on the home page of the application Minimal. Obviously, the
+event produced by this step is redundant in the generated test.
+This simple example explains that we need to solve the event
+redundancy issue in the process of test reuse caused by some
+particular functionality in the existing test.
+(a) The existing test for To-Do List
+(b) The reused test for Minimal
+Figure 1. A simple example of test reuse. The test (b) is obtained by reusing
+the existing test (a).
+Inspired by the above observation, in this paper, we propose
+a novel approach TRASM (Test Reuse based on Adaptive
+Semantic Matching) to reuse the existing tests across android
+arXiv:2301.00530v1  [cs.SE]  2 Jan 2023
+
+自7:36
+47:36
+7:37
+7:37
+三
+To-Do List
+Q
+ ＋
+To-Do List
+三
+To-Do List
+Q+"
+：
+Q
+CLICK+FOR NEW LIST
+CLICK+FORNEWLIST
+No Deadline
+New To-Do task
+e’ = exist(Test')(w')
+e, = fill(w)
+Welcome!
+Test
+8
+scription
+菌 Deadline
+@Reminder
+Notasks available
+Progress:
+0%
+Thisappdoesnotuseanypermissions
+Priority:
+Medium
+List:
+Click to select!
+CANCEL
+OKAY
+e, = click(w')
+e. = click(w²)
++
++
+ADDNEWTASK>
+×
+ADDNEWTASK>
+sKIP
+NEXT日7:35
+47:35
+日7:36
+ Minimal
+X
+Minimal
+itl
+Test
+e" = exist(Test')(w")
+e’ = fill(wz)
+You don't have any todos
+" = click(w"mobile applications. In addition, we carry out comparative
+experiments with the-state-of-the-art baseline approaches to
+evaluate our work. Overall, our main contributions are as
+follows:
+1) We propose a novel approach TRASM, which utilizes an
+adaptive strategy to reuse more existing tests. TRASM
+can get more semantic matches in the generated test.
+2) TRASM includes a GUI events deduplication method,
+which could eliminate duplicated events caused by
+reusing particular functionality contained in the existing
+test to improve the quality of the generated test.
+3) We carry out extensive experiments which confirm that
+TRASM improves the accuracy of GUI event matching
+while reducing test reuse failures and reduces the run-
+ning time required for test reuse.
+The rest of this paper is organized as follows. Section
+II introduces related work. Section III describes the main
+idea and the proposed approach in detail. Section IV carries
+out experimental evaluation. Finally, Section V concludes the
+paper and outlines future research.
+II. RELATED WORK
+A. Test Generation
+In order to improve the efficiency of developers, based on
+different exploration strategies, several studies on automatic
+test generation have been proposed, which has laid a solid
+foundation.
+Sapienz [5] combined random fuzzing, systematic and
+search-based exploration, exploiting seeding and multi-level
+instrumentation to explore and optimize test sequences auto-
+matically. Gu et al. [7] dynamically abstracted the model by
+leveraging decision tree-based representation and updated the
+model by utilizing the evolution mechanism, which balances
+the accuracy and size of the model. ConmboDroid [15] ob-
+tained the use cases for verifying the unique functions of the
+application and then systematically enumerates and combines
+them to generate higher quality input. The advantage of their
+work is that they can mine more hidden bugs or achieve as high
+coverage as possible. Nevertheless, the test generated by their
+method is seldom standardized for verifying the application’s
+functionality.
+Different from their purpose and inspired by their explo-
+ration method, we focus on generating more meaningful tests
+based on semantic information.
+B. Test Reuse
+Test reuse, as an alternative method to automatically gener-
+ate GUI test, makes full use of existing resources to provide
+convenience for verifying the application’s behavior.
+Lin et al. proposed CRAFTDROID [19], an approach of
+test transfer across applications, which utilizes the GUI model
+extracted by static analysis to match event sequences similar to
+the semantics of the existing test in order. They realized the
+successful transfer of GUI and oracle events, which guides
+for improving test transfer. To more accurately express the
+similarity of widgets in test events, Mao et al. [20] raised a
+semantic-based event fuzzy mapping strategy when matching
+candidate widgets to generate target events. They always
+greedily preferentially explore and match the widgets with the
+highest similarity. Unfortunately, when their similarity calcu-
+lation method does not work well, the correctness of event
+matching will be threatened. Considering that the success
+of test reuse heavily depends on the semantic matching of
+test events, there is still space for improvement by adopting
+appropriate strategies to increase the quality of reused tests
+from the perspective of application functionality.
+Leonardo et al. [21] conducted extensive research and
+pointed out that some attributes representing widgets play
+a negative role and how designing the semantic matching
+process is the most influential component to matched results.
+Their key findings point to an entry point for better reuse of
+test. Up to now, there is still no effective method to solve
+semantic problems [25, 26]. Trying to optimize the generated
+test sequence to ensure the quality of reused tests should be
+an optional strategy.
+III. OUR APPROACH
+Figure 2 shows an overview of the proposed test reuse
+approach TRASM. Based on semantic matching of events,
+TRASM considers the test (source test) of the existing appli-
+cation (source app), and the new application (target app) as
+inputs and outputs target test. TRASM employs two significant
+phases to implement test reuse: preliminary preparation and
+source test reuse. For the former, the existing data is processed
+through test augmentation and model extraction to facilitate
+the implementation of source test reuse. For the latter, the
+processed data obtained by the former is used together to reuse
+the source test on the target app.
+Test augmentation and model extraction are preliminary
+preparation steps that follow existing work [19], and we will
+only briefly introduce them. In detail, we focus on introducing
+our main contributions.
+A. Test augmentation
+The main task of test augmentation is to extract semantic
+information of widgets during the execution of collected
+source tests. The semantically represented widgets, together
+with actions, compose augmented tests, which are used to
+match widgets in the GUI of the target app.
+After the source app executes each event, the adb tool1 is
+used to extract the semantic information of the corresponding
+widget in the executed event according to the reached GUI
+state. Multiple attributes (including class, resource-id, text,
+content-desc, clickable, password, parent text, sibling text,
+activity, and package) uniquely represent a widget in the GUI.
+These non-empty attributes and their values constitute the
+widget’s semantic information. For example, for widget wt
+1
+of test (a) in Figure 1, the collected semantic information is
+shown in Table 1.
+1https://developer.android.com/studio/command-line/adb.html
+
+Figure 2.
+The overview of TRASM.
+Table 1. The semantic information of widget wt
+1.
+Attribute
+Value
+class
+android.widget.ImageButton
+resource-id
+fab new task
+clickable
+true
+password
+false
+activity
+.view.MainActivity
+package
+org.secuso.privacyfriendlytodolist
+B. Model extraction
+The model extraction aims to statically analyze the source
+code obtained by the target app to obtain the window transition
+graph (WTG). Following with existing work, we employ tool2
+to obtain the WTG. And the steps of constructing WTG can
+refer to literature [19].
+The WTG can visually represent the interaction between
+application activities and is composed of node sets and edges.
+Among WTG, the node represents the activity of the applica-
+tion, and the directed edge represents the activity transition
+that the event can trigger. For example, for the test (b)
+of application Minimal in Figure 1, the window transition
+triggered by the execution of the event is shown in Figure 3,
+where the nodes Main and AddToDo respectively represent
+the two activities of the application Minimal. By triggering
+the event on edge, state transition occurs between activities.
+The obtained WTG can provide matching candidate widgets
+for widgets in the source test. However, the WTG obtained
+may be incomplete. More fully, we adopt updating the WTG
+based on the feedback running information when executing
+the application.
+C. Test generation
+The purpose of test generation is to generate the initial test
+on the target app according to the semantic information of the
+augmented test and the WTG of the target app.
+Every event in the augmented test iteratively matches the
+corresponding events in the target app. The candidate widgets
+are first obtained by similarity calculation to match the event.
+2https://drive.google.com/file/d/1HEFS96c5nNKnzBPkWlRdwBiunOHgOs-/
+view?usp=sharing
+Figure 3.
+Window transition triggered by test (b).
+Similarity calculation. For each widget, the semantic in-
+formation is captured from the current GUI page of the
+target app to build a word list of attributes. For example,
+the attribute ‘resource-id’ of widget wt
+2 and widget wm
+2
+are ‘et new task name’ and ‘userToDoEditText’ respec-
+tively. After preprocessing [10, 19], we get two word lists
+a=[‘edit’, ‘text’, ‘new’, ‘task’, ‘name’] and a′=[‘user’, ‘todo’,
+‘edit’, ‘text’]. For any word w ∈ a and w′
+∈ a′, the
+highest similarity max sim(w, w′) between words w and w′
+is synthesized as the similarity of attributes:
+sim(a, a′) =
+�
+w∈a maxw′∈a′ sim(w, w′)
+|a|
+(1)
+where a and a′ represent the attributes corresponding to
+widget S in source test and widget T in target app respectively,
+sim(w, w′) expresses the cosine distance of the word vectors
+−→
+Vw and −→
+Vw′, obtained by the Word2Vec model[27]:
+sim(w, w′) =
+−→
+Vw · −→
+Vw′
+|−→
+Vw||−→
+Vw′|
+(2)
+Based on Equation (1), we get the similarity between widget
+S and widget T:
+sim(S, T) =
+�
+a∈S sim(a, a′) ∗ wg(a)
+|S|
+(3)
+where, wg(a) represents the weight of attribute a among
+all attributes. Based above calculation, we build candidate
+
+Source Test
+Preliminary preparation
+② Source test reuse
+Candidate
+Widgets
+口
+A. Test
+E. Test
+adaptation
+augmentation
+口
+APP
+Augmented
+Processed Test
+Source Test
+Target Test
+Similarity
+Source App
+Window
+calculation
+Transition Graph
+B. Model
+APP
+C. Test
+D. GUI events
+extraction
+generation
+deduplication
+Target App
+-
+Initial Test= click(w") em = fill(w²)
+em
+m
+= click(wm) em = exist(Test')(wm)
+m
+m
+Main
+AddToDo
+m
+mwidgets by selecting several widgets in the target app with
+high similarity.
+We identify a reachable widget based on the obtained
+candidate widgets and assign an action to form an event.
+All the paths from the current activity to the activity of
+each candidate widget can be queried from the WTG. These
+paths are executed to identify the reachable widget and return
+leading events. In addition, to avoid repeated path exploration,
+we adopt a strategy to preserve the path that has been explored
+and the corresponding leading events. For example, when
+matching event et
+2 in test (a), the application Minimal reaches
+the AddToDo activity after executing event em
+1 as shown in
+Figure 3, and then candidate widgets on the current page are
+collected. From the stored explored paths, it is found that there
+is a reachable path between activity AddToDo and activity
+AddToDo. Widget wm
+2 is located in the reachable path, which
+is identified as a reachable widget. Finally, according to the
+source event, the action is allocated to the widget wm
+2 .
+D. GUI events deduplication
+Invalid repeated events will increase the complexity of test
+execution. Although repeated events in the test will not affect
+the triggering of the behavior of an application, GUI events
+deduplication intends to reduce the time consumption occupied
+by such events.
+Since the GUIs of the two applications are different, the
+target app may not have the special functionality contained
+in the source test. As explained in Section
+I, the reuse of
+special functionality in test (a), that is, the matching of event
+et
+0 on application Minimal, is meaningless. To remove such
+GUI events in the test sequence, deduplication is performed.
+However, it is a challenge to identify the meaningless events
+in the test. We take the operation of detecting and deleting
+duplicate events unrelated to the generated initial test. Con-
+sidering the variety of possible duplicate event patterns, we
+set two rules to distinguish them. First, if only a single event
+is repeated in the initial test, we delete the repeated events
+at the beginning of the test sequence. Second, if the test
+sequence starts with ⟨en0, en1⟩ and also contains ⟨en1, en0⟩
+such events, we delete the pair of events. After this operation,
+to maintain the correctness, we check whether the test after
+deduplication, that is, the processed test, can maintain the
+functionality as the initial test. If not, we will give up the
+GUI events deduplication.
+E. Test adaptation
+The goal of test adaptation is to explore whether there is a
+better test sequence than the processed test using the designed
+adaptation strategy.
+Test generation always prioritizes the widget with the high-
+est similarity for matching. When the method of calculating
+similarity does not work well, it may not be possible to
+distinguish the best widgets to match, which will affect the
+accuracy of the result. The design idea of test adaptation
+is to find indexes that may have more semantically similar
+events in the processed test and then rematch them. However,
+determining such indexes in the sequences of the processed
+test is a challenging task. In this paper, we first record the
+indexes for which widgets in the processed test have higher
+similarity to another widget in the augmented test, except
+for the current matching event. Then, we choose the indexes
+with the lowest similarity of matching events in the processed
+test, which tries to mine the event with the incorrect match.
+After these two processing stages, we obtain the index sets
+of events that can perform rematching. Based on the above,
+we successively rematch the events of each index set from
+the candidate widgets obtained by Section III-C. We set the
+early termination condition to obtain a new test sequence that
+is more semantically similar than the original ones.
+We explain how test adaptation solves the problem of
+reusing test (b) to application To-Do List in Figure
+1. As
+mentioned in Section
+I, different GUI designs make the
+similarity between the correct widget and the source widget
+low, resulting in the incorrect match of event em
+3 . Through the
+strategies mentioned above, we get the index of event em
+3 to be
+rematched. Then, combined with the WTG obtained from the
+model extraction, the correct reachable widget wt
+3 is searched
+again from the obtained candidate widgets on this index to
+form event et
+3. Finally, the process ends after the oracle event
+et
+4 matching.
+IV. EXPERIMENTAL EVALUATION
+We implement our approach TRASM as a tool. Moreover,
+we compared TRASM with the baseline approach CRAFT-
+DROID [19], a test transfer method across mobile applications
+through semantic mapping, to verify the effectiveness and
+efficiency of TRASM. In this section, we introduce the exper-
+imental setup and experimental results to evaluate TRASM.
+A. Experimental setup
+For consistency, we reused the dataset3 of [19] to evaluate
+the proposed TRASM. Following the steps of the baseline, we
+conducted reuse tests on 15 applications in three categories,
+including browser, Tip Calculator, and To-Do List. These
+applications come from Google Play and F-Droid, which are
+often used in the GUI testing field to explore the functionalities
+of application [5, 24, 25]. Concretely, Table 2 details the
+category, name (version), and source of each application.
+Specifically, for each application category, two typical func-
+tionalities are selected, and the corresponding tests of each
+application are collected according to the functionalities. To
+achieve the goal of verifying the implemented functionality,
+the last event of each test case is set as an oracle. In general,
+there are six functionalities in three categories of applications,
+as shown in Table 3. Table 3 lists the number of test cases for
+each functionality and the average number of GUI and oracle
+events.
+Our experiment was implemented on a Nexus 5X Emulator
+running Android 6.0 (API 23) installed on a Ubuntu desktop
+with a 3.4 GHz Intel Core i7 CPU and 32 GB RAM.
+3https://sites.google.com/view/craftdroid/
+
+Table 2. The specific information of applications.
+Category
+Application (version)
+Source
+a1-Browser
+a11-Lightning (4.5.1)
+F-Droid
+a12-Browser for Android (6.0)
+Google Play
+a13-Privacy Browser (2.10)
+F-Droid
+a14-FOSS Browser (5.8)
+F-Droid
+a15-Firefox Focus (6.0)
+Google Play
+a2-Tip Calculator
+a21-Tip Calculator (1.1)
+Google Play
+a22-Tip Calc (1.11)
+Google Play
+a23-Simple Tip Calculator (1.2)
+Google Play
+a24-Tip Calculator Plus (2.0)
+Google Play
+a25-Free Tip Calculator (1.0.0.9)
+Google Play
+a3-To Do List
+a31-Minimal (1.2)
+F-Droid
+a32-Clear List (1.5.6)
+F-Droid
+a33-To-Do List (2.1)
+F-Droid
+a34-Simply Do (0.9.1)
+F-Droid
+a35-Shopping List (0.10.1)
+F-Droid
+Table 3. Tests for the typical functionalities.
+Functionality
+Test
+Avg
+Avg
+Cases
+GUIs
+Oracles
+b11-Access website by URL
+5
+3.4
+1
+b12-Back button
+5
+7.4
+3
+b21-Calculate total bill with tip
+5
+3.8
+1
+b22-Split bill
+5
+4.8
+1
+b31-Add task
+5
+4
+1
+b32-Remove task
+5
+6.8
+2
+Total
+30
+5.1
+1.5
+B. Experimental results
+This
+subsection
+presents
+the
+experimental
+results
+of
+TRASM and the baseline approach CRAFTDROID under
+the same evaluation metrics. For each functionality of each
+category, we reuse the test of one application on the remaining
+four applications respectively, and the total number of test
+reuse is 5(test cases) × 4(target applications) = 20. This paper
+shows the average result of 20 different test reuses. In order
+to avoid randomness, for each test reuse, we take the average
+of the multiple results recorded.
+Effectiveness. By comparison, the tests reused by TRASM
+perform higher usability than CRAFTDROID. The following
+two aspects, including the evaluation of successful reuse and
+the evaluation of matching events, can support the usability of
+the TRASM approach reuse test.
+Regarding reuse success rate, TRASM has significantly
+improved test reuse in 3 of the six functionalities, as shown
+in the last column of Table 4. For functionalities b21 and
+b22, successfully reused tests achieved a 10% increase. For
+functionality b32, successful reuse also increased from 20%
+to 25%. In addition, 2 of the six functionalities, namely b11
+and b12, have shown the highest successful reuse, i.e., 100%,
+no matter whether it is approach CRAFTDROID or TRASM.
+For the evaluation of matching events, the third and fourth
+columns of Table 4 list the precision and recall of GUI
+events and oracle events, respectively. As shown in the table,
+compared with CRAFTDROID, TRASM improves the preci-
+sion of the GUI by 5% to 15% in different functionalities.
+Unfortunately, while improving the precision, the recall rate
+of GUI events for functionalities b22 and b32 has decreased
+slightly by 2% and 3%, respectively. The success of the reused
+test depends on whether the match of the last oracle event
+in the test sequence is correct. Therefore, the improvement
+of successful reuse also represents the increase in the recall
+rate of oracle events, as listed in Table 4. Among them, the
+most significant is that for functionalities b21 and b22, the
+recall rate is improved by 10%. In general, the improvement
+in the precision and the recall of oracle events shows that
+the proposed TRASM indeed increases the availability of the
+reused test.
+Efficiency. Figure
+4 lists the average test reuse time on
+each functionality. It is obvious that the average time spent on
+reuse testing of TRASM is less than that of CRAFTDROID
+for each functionality. Even the most significant effect is that
+for functionality b21, the average reuse test of CRAFTDROID
+takes 2581 seconds (43 minutes), while our TRASM only takes
+890 seconds (15 minutes), which is close to 35% of the time
+of CRAFTDROID. In summary, the results are attributed to
+two factors. One is that the storage of explored paths avoids
+repeated time consumption, and the other is that the adaptive
+strategy improves the efficiency of widget matching. The
+above results prove that we can break through the limitation
+on efficiency in CRAFTDROID.
+Figure 4. The average test reuse time on each functionality.
+While evaluating the efficiency of TRASM, an important
+finding is that implementing an adaptive strategy can improve
+the success of some reused tests. Inevitably, the potential
+drawback is that more suitable event matching can not be
+found will bring additional time consumption. We need to
+address this crucial point further to balance efficiency and
+effectiveness.
+V. CONCLUSION
+Test reuse as an alternative method of test generation
+can help developers verify the behavior of applications. In
+this paper, a novel test reuse approach has been proposed
+to alleviate the challenge of semantic problems in event
+matching. From the initial test set, we have extended GUI
+events deduplication and test adaptation to build up target tests.
+The experimental results indicate that our proposed approach
+achieves better performance than the baseline approaches with
+increased usability of the reused tests.
+
+12000
+CRAFTDROID
+TRASM
+10000
+(sec)
+8000
+time
+reuse
+6000
+Average
+4000
+2000
+0
+b11
+b12
+b21
+b22
+b31
+b32
+FunctionalityTable 4. Effectiveness and Efficiency Evaluation
+Functionality
+Approach
+GUI Event
+Oracle Event
+Successful Reuse
+Precision
+Recall
+Precision
+Recall
+b11
+CRAFTDROID
+79%
+100%
+100%
+100%
+20/20(100%)
+TRASM
+100%
+100%
+100%
+100%
+20/20(100%)
+b12
+CRAFTDROID
+85%
+100%
+100%
+100%
+20/20(100%)
+TRASM
+100%
+100%
+100%
+100%
+20/20(100%)
+b21
+CRAFTDROID
+82%
+100%
+100%
+80%
+16/20(80%)
+TRASM
+93%
+100%
+100%
+90%
+18/20(90%)
+b22
+CRAFTDROID
+80%
+100%
+100%
+65%
+13/20(65%)
+TRASM
+85%
+98%
+100%
+75%
+15/20(75%)
+b31
+CRAFTDROID
+78%
+100%
+85%
+100%
+17/20(85%)
+TRASM
+87%
+100%
+85%
+100%
+17/20(85%)
+b32
+CRAFTDROID
+69%
+100%
+85%
+80%
+11/20(55%)
+TRASM
+81%
+97%
+93%
+81%
+12/20(60%)
+We believe that matching events are a promising direction,
+and we plan to study how to improve the matching strategy
+further in the future. In addition, we plan to verify the
+generalization of the method and further explore the effect
+of test reuse on more applications.
+ACKNOWLEDGMENTS
+The work is partially supported by the National Natural
+Science Foundation of China (No. 61972197), the Natural Sci-
+ence Foundation of Jiangsu Province (No. BK20201292), and
+the Collaborative Innovation Center of Novel Software Tech-
+nology and Industrialization. T. Chen is partially supported
+by an oversea grant from the State Key Laboratory of Novel
+Software Technology, Nanjing University (KFKT2022A03),
+Birkbeck BEI School Project (EFFECT), National Natural
+Science Foundation of China (NSFC) under Grants (No.
+62072309, 62272397).
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+

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+Performance of the r2SCAN functional in transition metal oxides
+S. Swathilakshmi1, Reshma Devi1, and Gopalakrishnan Sai Gautam1,*
+1Department of Materials Engineering, Indian Institute of Science, Bengaluru, 560012,
+India
+*Email: [email protected]
+Abstract
+We assess the accuracy and computational efficiency of the recently developed meta-generalized gra-
+dient approximation (metaGGA) functional, the restored regularized strongly constrained and appropri-
+ately normed (r2SCAN), in transition metal oxide (TMO) systems and compare its performance against
+SCAN. Specifically, we benchmark the r2SCAN-calculated oxidation enthalpies, lattice parameters, on-
+site magnetic moments, and band gaps of binary 3d TMOs against the SCAN-calculated and experimental
+values. Additionally, we evaluate the optimal Hubbard U correction required for each transition metal
+(TM) to improve the accuracy of the r2SCAN functional, based on experimental oxidation enthalpies,
+and verify the transferability of the U values by comparing against experimental properties on other
+TM-containing oxides. Notably, including the U -correction to r2SCAN increases the lattice parameters,
+on-site magnetic moments and band gaps of TMOs, apart from an improved description of the ground
+state electronic state in narrow band gap TMOs. The r2SCAN and r2SCAN+U calculated oxidation en-
+thalpies follow the qualitative trends of SCAN and SCAN+U, with r2SCAN and r2SCAN+U predicting
+marginally larger lattice parameters, smaller magnetic moments, and lower band gaps compared to SCAN
+and SCAN+U, respectively. We observe that the overall computational time (i.e., for all ionic+electronic
+steps) required for r2SCAN(+U ) to be lower than SCAN(+U ). Thus, the r2SCAN(+U ) framework can
+offer a reasonably accurate description of the ground state properties of TMOs with better computational
+efficiency than SCAN(+U ).
+1
+Introduction
+Density functional theory (DFT [1]) calculations are the bedrock of modern computational materials
+science in terms of predicting thermodynamic and kinetic properties, with such property predictions being
+put to use in subsequent materials discovery [2–7] and understanding underlying physical phenomena. [8–12]
+In recent years, machine learning has been used to augment DFT in property predictions, thereby reducing
+computational cost and accelerating materials discovery. [13–17] Note that a key approximation within
+DFT is the exchange-correlation (XC) functional, the exact form of which is unknown. However, several
+approximations for the XC functional have been proposed over the years, which can be categorized into
+different classes depending on the degree of sophistication and accuracy, and visually represented as rungs
+on the Jacob’s ladder. [1, 2, 18, 19] As with most computational tools, the higher the accuracy (higher up
+Jacob’s ladder) higher is the computational cost.
+1
+arXiv:2301.00535v1  [cond-mat.mtrl-sci]  2 Jan 2023
+
+Most DFT calculations for “large” solid systems (10s to 100s of atoms) are performed using the Perdew-
+Burke-Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA) XC functional,
+[20] as it offers fair accuracy at reasonable computational cost for a wide variety of materials. [21–23]
+Specifically, GGAs include the local electron density as well as the gradient of the electron density in
+describing the XC. As a semilocal functional of electron density, PBE captures short range interactions but
+fails to capture medium and long-range dispersions and also exhibits large electronic self-interaction errors
+(SIEs), especially in highly correlated systems. [24, 25] Also, PBE typically underestimates the formation
+energies [26,27] and semiconductor band gaps of crystalline solids, [26,28] while overestimating their lattice
+volumes. [26,29]
+As we move higher in the Jacob’s ladder, [19] we obtain metaGGA functionals, which may account for
+medium range dispersions and exhibit lower SIEs. Some metaGGAs consider orbital kinetic energy density
+in addition to the local electron density and its gradient, such as the recently developed strongly constrained
+and appropriately normed (SCAN [30]) functional, which offers better numerical accuracy than PBE and
+satisfies all 17 known constraints for a XC functional (namely, 6 for exchange, 6 for correlation, and 5
+for both). The iso-orbital indicator (α), which includes the kinetic energy density in SCAN, distinguishes
+various bonding environments in a given material and consequently improves the accuracy of SCAN over
+GGA. However, SCAN suffers from numerical instability during self-consistent-field (SCF) calculations [31]
+wherein denser k-grids (than PBE) are required for accurate and consistent predictions. [31–33] Thus it is
+computationally expensive (per SCF step) compared to PBE. [21]
+To overcome the numerical instability and reduce the computational cost of SCAN, Bartok and Yates [34]
+developed regularized SCAN (rSCAN), which satisfies 13 out of the 17 known constraints. The authors
+replaced the non-analytical switching α interpolation function in SCAN with a simple polynomial function,
+which improves computational speed. [35] However, subsequent investigations showed a significant drop in
+numerical accuracy with rSCAN (compared to SCAN), which is attributed to the failure of the polynomial
+α function to fully recover the uniform gas limit. [31, 32] Subsequently, Furness et al. [32] introduced the
+restored regularized SCAN (or r2SCAN), wherein the constraints broken by rSCAN were restored except
+the fourth order gradient expansion constraint for exchange (or GE4X). Furness et al. claimed that the new
+r2SCAN functional combines the numerical accuracy of SCAN and computational speed of rSCAN as the
+smooth polynomial α function of rSCAN is modified to satisfy the uniform gas limit in r2SCAN. [32] Recently,
+Kingsbury et al. [36] demonstrated that r2SCAN functional indeed delivers robust numerical accuracy (i.e.,
+similar to SCAN) and better computational performance (faster and numerically stable) by comparing
+r2SCAN and SCAN for solids using a high-throughput computational workflow. Specifically, the authors [36]
+reported that while r2SCAN predicts a smaller band gap (for most of the strongly-bound materials) and
+larger lattice volumes than SCAN, the mean atomization error with r2SCAN is ∼15-20% lower for most
+solids. However, the performance of r2SCAN in correlated electron systems, i.e., transition metal oxides
+(TMOs) containing open-shell d electrons, remains to be seen and forms the main focus of this work.
+Despite the accuracy of SCAN, it still has shortcomings in TMOs, which can be mitigated by adding an
+on-site Hubbard U correction term for the transition metal (TM) under consideration. [37,38] This approach
+is similar to the one followed to mitigate the SIEs of PBE in TMOs. [39,40] However, the magnitude of the
+U correction required is not known a priori, and there are both theory-based approaches such as density
+functional perturbation theory, [41] linear response theory, [42–44] embedded Hartree-Fock method, [45,46]
+and machine learning based Bayesian optimisation, [47] and experimental-data-based approaches to identify
+the appropriate U values. For example, Gautam et al. [37, 38] used the experimental oxidation enthalpies
+2
+
+among binary TMOs to identify optimal U values across various oxidation states of 3d TMs. A similar
+experimental-data-based Hubbard U correction scheme can be developed in conjunction with r2SCAN as
+well, resulting in a r2SCAN+U framework, in case r2SCAN exhibits similar SIEs as SCAN in TMOs. We
+explore the usefulness of such a r2SCAN+U framework also in this work.
+Here, we verify the numerical accuracy and computational efficiency of the r2SCAN and r2SCAN+U
+frameworks in comparison to SCAN and SCAN+U, respectively, in describing material properties such as
+lattice parameters, on-site magnetic moments, and band gaps of binary 3d TMOs, including Ti, V, Cr,
+Mn, Fe, Co, Ni, and Cu. As necessary, we evaluate the optimal Hubbard U correction with r2SCAN for
+each TM by using the experimental-data-based approach employed in previous works. [37,38] We find that
+r2SCAN predicts marginally larger lattice constants and smaller on-site magnetic moments than SCAN for
+most of the TMOs considered. On addition of the U -correction to both SCAN and r2SCAN, we observe an
+increase in the calculated lattice constants, on-site magnetic moments and band gaps. In the case of narrow
+band gap TMOs, SCAN+U and r2SCAN+U generally estimate a non-zero band gap, with r2SCAN+U ’s
+band gap in better agreement with experiments. Also, we perform transferability checks for the optimal U
+values derived in this work for each TM, by benchmarking various properties in oxides that were not used
+in obtaining the U values. Finally, we compare the computational performance of r2SCAN/r2SCAN+U
+relative to SCAN/SCAN+U to explore the accuracy-cost trade-off. We report that r2SCAN/r2SCAN+U
+is computationally less expensive than SCAN and SCAN+U, when all required ionic and electronic steps
+are taken into account for convergence during structure relaxations. We hope that our work will provide a
+foundational basis for further studies on understanding material behavior and computationally discovering
+new materials in the near future.
+2
+Methods
+2.1
+Computational Methods
+We used the Vienna ab-initio simulation package (VASP 6.2.1) [48–50] for all the spin-polarized DFT
+calculations, where the frozen-core PBE-based projector augmented wave (PAW) [51] potentials employed
+were identical to previous work. [37, 38] The plane waves for each system were expanded up to a kinetic
+energy of 520 eV, with each structure converged until the total energy differences and atomic forces became
+<0.01 meV and <|0.01| eV/˚A, respectively.
+We adopted a Γ-centered Monkhorst-Pack [52] grid with a
+density of 48 k-points per ˚A for all systems.
+The conjugate gradient algorithm was used to relax the
+structures (i.e., cell shapes, volumes, and ionic positions), without preserving any underlying symmetry. An
+‘accurate’ level of precision was maintained while projecting the wavefunctions in the reciprocal space. The
+Fermi surface of each system was integrated with a Gaussian smearing of partial occupancies, with a width
+of 0.05 eV. In terms of DFT+U calculations, we used the Dudarev framework [53] for adding a effective
+U correction on the d orbitals of TM atoms. All U values used in SCAN+U calculations were taken from
+previous work (see Table S1 of the Supporting Information –SI). [37,38] Since we used different computing
+systems to perform our structure relaxations for different systems, we normalized the computational time
+with the number of cores used in each calculation to compare the computational efficiency of the different
+XC functionals considered.
+For calculating band gaps, GGA functionals typically use the Kohn Sham potential as a multiplicative
+term, which typically underestimates the band gap of solids even at the SCAN level. [54, 55] Here, we
+3
+
+use the generalized Kohn Sham technique to determine the band gaps by calculating the density of states
+(DOS) for all systems considered. For each DOS calculation, we used the optimized structure and the initial
+charge density from a previous structure relaxation. Subsequently, we introduced a set of zero-weighted k-
+points, corresponding to a density of 96 k-points per ˚A, where the k-points that were used for the structure
+relaxation retained their original weights (as determined by VASP). Finally, we performed a single-SCF
+calculation where the DOS was sampled between energies of -20 to 20 eV in steps of 0.005 eV.
+2.2
+Structures and magnetic configurations
+We considered the binary oxides of each TM, i.e., Ti, V, Cr, Mn, Fe, Co, Ni, and Cu with different
+oxidation states, similar to previous studies. [37, 38] The main criteria in selection of these metal oxides
+are the availability of reliable thermodynamic data (i.e., formation energies [56–58]) and the experimentally-
+determined ground-state structures that are compiled in the inorganic crystal structure database (ICSD) [59]
+Note that the structures from the ICSD were the initial structures in all our DFT structure relaxations,
+including the systems used as transferability checks. In the case of Ni oxides, we chose NiO and LiNiO2
+(similar to previous work, [38]), as reliable thermodynamic data is not available for higher-oxidation-state
+binary Ni oxides (e.g., Ni2O3 and NiO2). The TM in all oxides, except select Co and Ni compounds, was
+initialized in its high-spin configuration (e.g., high-spin configuration of Fe3+ consists of five unpaired d
+electrons). A detailed description of all structures utilised in this work is provided in the SI, under the
+’Crystal Structures’ section, with the magnetic configurations depicted in Figure S1.
+The magnetic configuration of each TMO considered (see Figure S1) was initialized to its appropriate
+(in several cases, experimentally-known) ground state configuration during the structural relaxation. For
+example, we considered the ferromagnetic (FM) ground state configuration for CrO2 and VO2, given that
+CrO2 is metallic [60] and VO2 undergoes a metal-to-insulator transition (MIT) below 341 K. [61] The
+rocksalt (RS) TMOs, namely, VO, MnO, FeO, CoO, and NiO were initialized with their experimentally-
+known type-II antiferromagnetic (AFM) configuration. [62–67] Each Ni’s spin in NiO was initialized with
+two unpaired d electrons (i.e., its high-spin configuration). In CuO, we arranged the magnetic moments of
+Cu2+ antiferromagnetically along the Cu-O-Cu chains in the [10¯1] direction. [68,69]
+We initialized α-Mn2O3 (bixbyite structure) in a FM configuration as this configuration was found to
+be the most stable in previous work. [37] AFM configurations were utilized for rutile-MnO2, [70], and the
+other TM2O3 oxides, namely, V2O3, Fe2O3, Ti2O3, and Cr2O3.
+Note that V2O3 becomes AFM below
+its MIT temperature, [71–73] while Fe2O3 displays an AFM configuration with the magnetic moment of
+Fe alternating every two consecutive layers along the c-axis. [74] Cr2O3 and Ti2O3 exhibit ↑↓↑↓ and ↑↓↓↑
+magnetic configurations, respectively, on the TM centers along the a-axis. [75,76]
+In case of spinels, we used different ferrimagnetic (FIM) configurations, as per experimental observations.
+For example, spinel-Fe3O4 contains both Fe3+ and Fe2+, with up-spin Fe3+ occupying tetrahedral sites and
+down-spin Fe3+ occupying half the octahedral sites. The remaining octahedral sites in Fe3O4 are occupied
+by up-spin Fe2+. [77, 78] In Co3O4, no-spin Co3+ occupies octahedral sites, while high-spin Co2+ (three
+unpaired d electrons) occupies tetrahedral sites in an AFM configuration. [79–81] For Mn3O4, we adopted
+the ”FIM6” configuration, as this was found to be the ground state in previous work. [37] TiO2, CrO3,
+and V2O5 are diamagnetic, since they contain TMs with empty 3d orbitals. Similarly, Cu2O is diamagnetic
+owing to the completely-filled 3d orbitals of Cu.
+4
+
+2.3
+Determining U
+We determined the required U value, with r2SCAN, for each binary TMO oxidation reaction (e.g., Ti3+ →
+Ti4+ in 2Ti2O3 + O2 → 4TiO2) by comparing the experimental enthalpy (per mole of O2) with the calculated
+(r2SCAN+U ) value that minimizes the error against the experimental value. Note that U = 0 eV in our
+data simply reflects a r2SCAN calculation. In order to obtain the experimental oxidation enthalpy, standard
+enthalpy of formation for all the considered TMOs were taken from the Wagman and/or Kubaschewski
+tables, [56,57] thus ignoring the p − V and entropic contributions, similar to previous works. [37,38,82] The
+overall optimal U value for each TM was obtained by taking the average of the required U for each of the
+available oxidation reactions. In the case of Ni oxides, oxidation of NiO to LiNiO2 by 2Li2O + 4NiO + O2
+→ 4LiNiO2 was considered as a proxy for the Ni2+ → Ni3+ oxidation reaction. [38]
+3
+Results
+3.1
+Oxidation energetics
+Figure 1 displays the variation of the enthalpy of different oxidation reactions among binary TMOs, as a
+function of applied U in the r2SCAN+U framework, for all TMs considered except Cr and Cu. Solid lines
+in each panel of Figure 1 represent DFT-calculated oxidation enthalpies, with each color corresponding to
+different oxidation reactions for the TM. For instance in V oxides (Figure 1b), the solid black line corresponds
+to the oxidation reaction, VO → V2O3, while the solid red and green lines indicate V2O3 → VO2 and VO2 →
+V2O5, respectively. Similarly, the experimental enthalpy of each oxidation reaction is represented by dashed
+horizontal line of the same color. For example, the black dashed line in Figure 1b indicates the experimental
+oxidation enthalpy (-7.36 eV) of VO → V2O3. Also, dotted vertical line of a given color highlights the
+required U value to minimize the error between DFT-calculated and experimental value for the oxidation
+reaction enthalpy indicated by the same color. The dotted blue line in each panel signifies the overall optimal
+U for the TM that is averaged across all available oxidation reactions.
+We report an optimal U value of 2.3, 1.0, 1.8, 3.1, 1.8, and 2.1 eV, respectively, for Ti, V, Mn, Fe,
+Co, and Ni oxides, within the r2SCAN+U framework (Figure 1). Notably, the optimal U obtained with
+r2SCAN is less than that reported previously for SCAN functional (Table S1) for all 3d TMs considered
+(except V and Fe), which can be attributed to better accuracy of r2SCAN compared to SCAN, as observed
+in non-TMOs. [36] For V oxides, the required U value for VO2 → V2O5, V2O3 → VO2, VO → V2O3 is
+0.0, 0.7, and 2.2 eV, respectively. Thus, the optimal U value for V is 1.0 eV (average of the three required
+U values), which is identical to the U correction required with SCAN. [38] The decreasing required U with
+increasing oxidation state of V in V oxides is expected due to the decrease in the strength of exchange
+interactions among the d electrons as oxidation state increases. In the case of Fe, FeO → Fe2O3 and FeO
+→ Fe3O4 reactions require a U of 2.9 and 3.3 eV, respectively, resulting in an optimal U of 3.1 eV, which
+is also identical to the optimal U with SCAN. [37] Moreover, we obtain the highest optimal U of 3.1 eV
+for Fe, among all TMs considered in this work, which is consistent with the fact that Fe3+ has the highest
+number of unpaired d electrons resulting in the strongest exchange interactions.
+For Ti and Ni, we observe a marginal improvement in the U -value for r2SCAN when compared to SCAN.
+Specifically, we obtain an optimal U of 2.3 eV and 2.1 eV for Ti and Ni, respectively, versus 2.5 eV for
+both elements with SCAN. We find an optimal U value of 1.8 eV for both Mn (2.7 eV with SCAN) and
+Co (3.0 eV with SCAN). In Mn-oxides, the required U for the oxidation of Mn2O3 → MnO2, and MnO →
+5
+
+Mn2O3 are 1.5 and 2.1 eV, respectively. The optimal U for Mn is transferable to other Mn oxides as well,
+indicated by the robust agreement between r2SCAN+U -calculated and experimental oxidation enthalpy for
+MnO → Mn3O4 (green lines in Figure 1c).
+For Cr and Cu oxides, we obtain reasonable agreement with experimental data without a U correction
+(Figure S2), similar to our observation with SCAN. [38] In fact, for Cu, introducing U -correction worsens the
+error in the calculated oxidation enthalpy for Cu2O → CuO versus experiment, similar to our observation
+with SCAN(+U ) as well, which can be attributed to PAW potentials derived at the PBE-level. [38] However,
+the magnitude of error (versus experiment) is smaller with r2SCAN (≈13.1%) than with SCAN (≈25.7%).
+In case of Cr, the oxidation reaction of CrO2 → CrO3 requires U ∼ 0.9 eV, but introducing a U correction
+worsens any agreement with experiment for Cr2O3 → CrO2 (where required U = 0 eV). Thus, the optimal
+U for Cr oxides is 0.45 eV (<0.5 eV), which only provides a marginal improvement in describing oxidation
+enthalpies. Hence, we recommend using only r2SCAN for calculating any Cr oxide framework.
+0
+1
+2
+3
+4
+U(eV)
+-9
+-8
+-7
+-6
+-5
+-4
+Reaction Enthalpy (eV per O2)
+FeO/Fe2O3
+FeO/Fe3O4
+Experimental
+2.9 eV
+3.3 eV
+3.1 eV
+(d)
+(e)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+U(eV)
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+Reaction Enthalpy (eV per O2)
+CoO/Co3O4
+Experimental
+1.8 eV
+0
+0.5
+1
+1.5
+2
+2.5
+U(eV)
+-4
+-3.5
+-3
+-2.5
+-2
+-1.5
+Reaction Enthalpy (eV per O2)
+NiO/LiNiO2
+Experimental
+2.1 eV
+(f)
+(a)
+0
+0.5
+1
+1.5
+2
+2.5
+U (eV)
+-8.4
+-8.2
+-8
+-7.8
+-7.6
+-7.4
+Reaction Enthalpy (eV per O2)
+Ti2O3/TiO2
+Experimental
+2.3 eV
+(b)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+U (eV)
+-8
+-6
+-4
+-2
+0
+2
+Reaction Enthalpy (eV per O2)
+VO/V2O3
+V2O3/VO2
+VO2/V2O5
+Experimental
+0.0 eV
+0.7 eV
+1.0 eV
+2.2 eV
+(c)
+0
+0.5
+1
+1.5
+2
+2.5
+U(eV)
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+Reaction Enthalpy (eV per O2)
+MnO/Mn2O3
+Mn2O3/MnO2
+Experimental
+MnO/Mn3O4
+1.5 eV
+1.8 eV
+2.1 eV
+Figure 1: Calculated oxidation enthalpy versus the magnitude of U correction within r2SCAN+U framework
+for (a) Ti, (b) V, (c) Mn, (d) Fe, (e) Co, and (f) Ni oxides. Solid, dashed, and dotted lines of a given color
+indicate calculated, experimental, and required U values for a given oxidation reaction. Optimal U for each
+TM is indicated by the dotted blue line in each panel.
+3.2
+Lattice parameters
+All r2SCAN(+U ) and SCAN(+U ) calculated lattice parameters, on-site magnetic moments, and band
+gaps for each TMO are tabulated in Table S2. Additionally, the calculated lattice volumes by the four XC
+functionals are plotted against experimental data in Figure 2a for all oxides. Generally, both SCAN (green
+squares in Figure 2a) and r2SCAN (blue symbols) offer < 2.8% deviation from the experimental lattice
+parameters for all the TMOs considered, except VO, FeO, CuO, and LiNiO2, indicating robust agreement
+with experiments for both functionals. In VO, SCAN and r2SCAN overestimate (by ∼8%) the experimental
+lattice constants, while the deviation in FeO and CuO is ∼3-4% and ∼8-10%, respectively.
+In LiNiO2,
+6
+
+SCAN’s β angle evaluation is ∼4.1% different from experiment.
+Notably, SCAN and r2SCAN do show qualitative differences in their calculated lattice parameters (when
+compared against experiments) across TMOs. For instance, both functionals overestimate the experimental
+lattice constants in TiO2, Ti2O3, and VO, while they underestimate in CrO2, CrO3, MnO2, and Fe3O4.
+There are also examples (MnO and Mn2O3) where SCAN underestimates the experimental lattice constants
+while r2SCAN overestimates. Overall, there are cases where SCAN’s errors in lattice parameter estimations
+are lower versus experiments (e.g., Cr2O3, CoO), r2SCAN’s errors are lower (e.g., CrO2, CrO3, MnO2,
+Fe3O4), and both functionals exhibit identical errors (e.g., TiO2, Co3O4, NiO, Cu2O), signifying that both
+functionals offer similar performance in terms of geometrical properties.
+Comparing r2SCAN and SCAN, we find that r2SCAN’s lattice constants are generally larger than SCAN
+across TMOs (e.g., Ti2O3, Cr2O3, CrO3, VO2, etc.). As a range, r2SCAN estimates lattice constants that
+are a maximum of ∼1.5% larger than SCAN (in CrO3) and a minimum of ∼0.1% larger than SCAN (in
+Mn2O3). Having said that, there are instances where r2SCAN’s lattice constant evaluations are lower than
+SCAN (VO, CoO, CuO, and LiNiO2) and cases where both functionals are identical (TiO2, Co3O4, NiO,
+and Cu2O). In specific TMOs, SCAN and r2SCAN calculate an identical (individual) lattice constant, while
+the other lattice constants with r2SCAN are larger than SCAN. For example, a and c lattice constants with
+r2SCAN are higher than SCAN in V2O5 while both functionals estimate b = 3.55 ˚A.
+On introducing the optimal U correction, an increase in the value of calculated lattice constants is ob-
+tained for both SCAN and r2SCAN functionals for all TMOs. The lattice constants computed by r2SCAN+U
+(yellow symbols in Figure 2a) is up to 1.3% higher than r2SCAN, except FeO (∼4.2% higher). Similar to the
+comparison of r2SCAN vs. SCAN, there are systems where r2SCAN+U predicts larger, smaller, and identical
+lattice constants compared to SCAN+U (red triangles). For example, r2SCAN+U calculates larger lattice
+constants than SCAN+U in VO2, V2O5, MnO, Mn2O3 and Fe3O4 (maximum of ∼0.5% higher in V2O5),
+while for Ti2O3, CoO and NiO, r2SCAN+U ’s estimations are smaller than SCAN+U (maximum deviation
+of ∼2.1% in Ti2O3). Both SCAN+U and r2SCAN+U functionals evaluate identical lattice parameters for
+TiO2, Co3O4 and LiNiO2.
+Overall, lattice constants calculated by SCAN+U and r2SCAN+U deviate <∼3.3% from experiments
+for all TMOs, except VO and VO2 where deviations of ∼8.5% and ∼4.6% are observed, respectively. Adding
+U improves the agreement with experiment for both SCAN and r2SCAN in Co3O4, while r2SCAN+U gives
+the best estimate of the lattice parameters in FeO (< 1% deviation vs. experiments) compared to SCAN,
+SCAN+U and r2SCAN. Notably, all functionals break the rocksalt symmetry of VO, MnO, and FeO, while
+the cubic symmetry of Fe3O4 is retained only by SCAN. In Ti2O3, the hexagonal symmetry is broken by
+SCAN but the symmetry is preserved by the other frameworks. In summary, we find that the differences in
+lattice parameter estimations to be minimal across the four functionals on average, with notable exceptions
+of a few systems.
+3.3
+On-site magnetic moments
+On-site magnetic moments of the TMOs (Figure 2c and Table S2) computed by SCAN and r2SCAN
+generally underestimate experimental values, with the exception of MnO2, Mn2O3, CrO2, and VO2. Note
+that larger magnetic moments typically indicate stronger localization of d electrons. Comparing r2SCAN
+and SCAN calculations, we find that r2SCAN typically estimates smaller magnetic moments than SCAN
+but with several exceptions, such as MnO, MnO2, Mn2O3, Cr2O3, and VO2. Thus, on average, SCAN’s
+magnetic moment predictions are in better agreement with experiments. However, in terms of magnitude,
+7
+
+Ti2O3
+TiO2
+VO
+V2O3
+VO2
+V2O5
+Cr2O3
+CrO2
+CrO3
+MnO
+Mn3O4
+Mn2O3
+MnO2
+Fe2O3
+Fe3O4
+FeO
+CoO
+Co3O4
+NiO
+LiNiO2
+Cu2O
+CuO
+SCAN
+SCAN+U
+r2SCAN
+r2SCAN+U
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+On-site magnetic moment
+difference (
+B)
+Ti2O3
+TiO2
+VO
+V2O3
+VO2
+V2O5
+Cr2O3
+CrO2
+CrO3
+MnO
+Mn3O4
+Mn2O3
+MnO2
+Fe2O3
+Fe3O4
+FeO
+CoO
+Co3O4
+NiO
+LiNiO2
+Cu2O
+CuO
+SCAN
+SCAN+U
+r2SCAN
+r2SCAN+U
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+On-site magnetic moment
+difference (
+B)
+SCAN
+SCAN+U
+r2SCAN
+r2SCAN+U
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+Band gap difference (eV)
+(a)
+(c)
+(b)
+0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+Experimental lattice volume (Å3)
+0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+Predicted lattice volume (Å3)
+SCAN
+SCAN+U
+r2SCAN
+r2SCAN+U
+Figure 2: (a) Comparison of calculated and experimental lattice volume (in ˚A3) of all TMOs considered.
+(b) Violin plot capturing the difference between the experimental and computed band gap (in eV) across
+TMO systems using the four XC frameworks. The empty circle and horizontal line in the inner box plot
+corresponds to the mean and median of the calculated band gaps, respectively. (c) Heat map representation
+of the differences between the experimental and calculated on-site magnetic moments (in µB) using the
+four XC functionals and across all TMOs. A value of zero indicates perfect consistency, while red (blue)
+colors indicate overestimation (underestimation) of magnetic moments. Hatched boxes either correspond to
+experimentally undetermined magnetic moments (VO) or calculations not executed with U frameworks (Cr
+and Cu oxides).
+moments predicted by r2SCAN deviate by < 3% from SCAN’s estimates, except CuO (∼ 6.8% deviation),
+CrO2 (∼ 3.5%), and MnO2 (∼ 3.5%), highlighting that the differences in the predictions are marginal.
+Adding optimal U to both SCAN and r2SCAN increases the magnitude of the calculated on-site magnetic
+moments for all TMOs (except VO2, which is predicted to be metallic by all functionals), consistent with
+the expectation that the U correction facilitates d electron localization. r2SCAN+U -calculated data are
+similar to the corresponding SCAN+U values (< 2.3% variation), except LiNiO2 (∼6.3% variation), and
+Ti2O3 (∼3.8%). Similar to r2SCAN versus SCAN, r2SCAN+U estimates smaller magnetic moments than
+SCAN+U, with notable exceptions being VO2, Mn2O3, MnO2 and FeO. Overall, we observe the accuracy
+in calculated on-site magnetic moments versus experiments to follow the order SCAN+U > r2SCAN+U >
+SCAN > r2SCAN for several TMOs. However, there are specific cases where specific XC frameworks offer
+better accuracy in calculating magnetic moments, such as SCAN in CrO2, Mn2O3, MnO2, Fe3O4 and CuO,
+8
+
+r2SCAN in Mn3O4 and Cr2O3, and r2SCAN+U in V2O3. Given the numerically marginal deviations in
+calculated magnetic moments across the XC frameworks (∼10% deviation), we expect an increase/decrease
+in accuracy to be marginal amongst the XC frameworks considered.
+3.4
+Band gaps
+The differences between calculated and experimental band gaps of all TMOs considered are visualized
+as violin plots for SCAN (green violin), SCAN+U (red), r2SCAN (blue), and r2SCAN+U in Figure 2b.
+The top and bottom ends of the individual violins mark the highest and lowest differences in the respective
+calculated data. Note that the mean values (white empty circles) are similar for SCAN and r2SCAN, and
+in turn are lower than their U -corrected versions. In other words, addition of the U-correction reduces
+the error of calculated band gaps compared to experimental values, which is expected given that semi-local
+DFT typically underestimates band gaps.
+Also, we find that SCAN+U displays the lowest mean band
+gap difference among the XC functionals considered, indicating that on-average SCAN+U provides better
+computed band gaps.
+We present calculated electronic DOS of select TMOs, namely CoO (panels a and b), V2O3 (c and d),
+and Mn2O3 (e and f), in Figure 3, to illustrate qualitative trends in computed band gaps. The DOS for
+the remaining TMOs, calculated by the four XC frameworks, are compiled in Figures S3-S19 of the SI. In
+each DOS panel, solid orange and solid green lines correspond to the 2p-states of O and the 3d-states of the
+TM, respectively. Dashed black lines represent Fermi levels in metallic compounds. Dotted vertical lines
+represent valence and conduction band edges in semiconducting/insulating compounds, with the band gaps
+indicated by the text annotation near the conduction band minimum (CBM). The zero of the energy scale
+is set to the valence band maximum (VBM) for TMOs with a band gap and to the Fermi level in metallic
+TMOs.
+We observe that r2SCAN generally calculates a smaller band gap than SCAN for most TMOs (maximum
+of ∼66% lower in MnO2, see Table S2), as illustrated by the case of CoO in panels a and b of Figure 3.
+Notable exceptions do exist to this observation, such as V2O5 (∼1.7% larger), CrO3 (∼3.2%), MnO (∼4.3%),
+and Fe2O3 (∼1.7%), where r2SCAN calculated band gaps are marginally larger than SCAN. Both SCAN
+and r2SCAN incorrectly describe the ground state electronic configuration of narrow band gap TMOs (i.e.,
+experimental band gaps < 1 eV), including Ti2O3 (Figure S4), V2O3(Figure 3c and S3c), VO2 (Figure S7)
+and Fe3O4 (Figure S15) to be metallic, with the exception of MnO2 where both SCAN and r2SCAN estimate
+a narrow gap (Figures S12a and S12c). Additionally, both functionals also calculate the wrong electronic
+structure in the case of a non-narrow-gap semiconductor, Mn2O3 (Figure S3), which exhibits an experimental
+gap of 1.2-1.3 eV. [83,84] However, SCAN and r2SCAN qualitatively describe the right electronic structure in
+the case of wide band gap TMOs such as FeO (Figure S13), Fe2O3 (Figure S14), and NiO (Figure S17), with
+a significant quantitative underestimation of the experimental gaps. In any case, the differences in electronic
+structure predictions between SCAN and r2SCAN in TMOs are minimal, with SCAN being marginally better
+in accuracy.
+Introducing a U correction to SCAN and r2SCAN widens or opens the band gap, especially in narrow
+band gap TMOs, as illustrated by the case of V2O3 (panels c and d in Figure 3). The opening of band
+gap with U correction is expected since localization of d electrons, which form the VBM and/or CBM in
+3d-TMOs, is faciliated with U addition, in turn resulting in a larger gap. However, in the case of VO2
+(Figure S7), adding U does not capture the MIT that occurs at low temperatures (< 341 K [61]) with either
+SCAN or r2SCAN, causing the erroneous prediction of metallic behavior. Generally, SCAN+U calculates
+9
+
+0.36 eV
+(a)
+0.91 eV
+(b)
+0.61 eV
+(c)
+(d)
+0.24 eV
+(e)
+(f)
+Figure 3: DOS for CoO calculated using (a) SCAN and (b) r2SCAN, DOS for V2O3 computed using (c)
+r2SCAN and (d) r2SCAN+U, and DOS for Mn2O3 estimated using (e) SCAN+U and (f) r2SCAN+U.
+a larger band gap than r2SCAN+U (Table S2), as highlighted by the case of Mn2O3 (panels e and f in
+Figure 3). In fact, SCAN+U is the only framework (among those considered) to estimate a band gap in
+10
+
+6
+/eV
+(states)
+4
+2
+Density of states (
+-6
+-4-3　-2　-1
+2
+3
+4
+5
+Energy (eV)6
+/eV
+(states)
+4
+Density of states (
+-6
+.5
+-4　-3-2
+0
+2
+3
+4
+Energy (eV)/eV)
+60
+Mnd
+(states)
+40
+20
+MWN
+Density of states
+-20
+40
+-60
+-6
+-5
+4
+¥-3
+¥-2-101
+2
+3
+4
+Energy (eV)60
+/eV)
+Mnd
+(states)
+40
+20
+Density of states
+-20
+40
+-60
+-6
+-5
+4
+2
+3
+4
+Energy (eV)/ev
+10
+(states)
+5
+Density of states
+0
+-10
+-6
+-5
+-2
+0
+2
+3
+4
+Energy (eV)/eV
+10
+(states)
+Density of states
+Wi
+-10
+-6
+-5
+¥1
+2
+3
+4
+Energy (eV)Mn2O3, which is consistent with experiment. Moreover, SCAN+U ’s evaluations of larger band gaps results
+in better (poorer) quantitative agreement with experiments in wide (narrow) gap materials, such as MnO
+and FeO (V2O3 and MnO2).
+Note that SCAN+U and r2SCAN+U do underestimate the experimental band gaps, similar to SCAN and
+r2SCAN, in wide gap TMOs. The only exception to this observation is CoO, where SCAN+U overestimates
+the band gap versus experiment (Figure S3a and Table S2), as also observed in our previous work. [38] In select
+TMOs, including Fe2O3 and V2O5, r2SCAN+U ’s band gap is larger than SCAN+U, but the magnitude of
+difference (≤ 0.2 eV) is meagre. Thus, for electronic structure predictions, we expect SCAN+U to provide the
+best qualitative and quantitative band gaps across TMOs, among the functionals considered here, especially
+for wide gap semiconductors/insulators. However, the qualitative trends provided by r2SCAN+U are quite
+robust as well and in small gap semiconductors (< 1 eV gap), r2SCAN+U ’s quantitative accuracy is often
+better than SCAN+U.
+3.5
+Transferability checks
+To examine the transferability of the optimal U values determined in this work (with r2SCAN), to oxide
+systems not used for obtaining the values, we perform checks on systems with different oxidation state and/or
+coordination environment for each TM. We compare calculated values against available experimental data,
+such as structural, electronic, magnetic, and/or electrochemical properties. Specifically, we choose Ba2TiO4
+as a check for Ti, BiVO4 for V, K3MnO4, K2MnO4, and Mn2O7 for Mn, SrFeO3 for Fe, LiCoO2-CoO2 for
+Co, and LiNiO2-NiO2 for Ni. Data related to transferability checks are compiled in Figure 4, Table 1, and
+Table S3.
+In the case of Ba2TiO4, we compare the calculated lattice parameters with experimental values (see
+Table S3 and lattice voliume differences plotted in Figure 4). Ba2TiO4 crystallizes in a monoclinic structure
+(space group P21/n) at low temperatures, where the unit cell is composed of four formula units. [85, 86]
+Ti atoms are present in distorted tetrahedra composed of neighbouring oxygen atoms (TiO4) within the
+Ba2TiO4 lattice, which is different from the octahedral environments sampled in TiO2 and Ti2O3. Upon
+structure relaxation, we observe that both r22SCAN and r22SCAN+U functionals marginally overestimate
+(by ∼2%) experimental lattice parameters (Figure 4 and Table S3). Similar to trends observed in Table S2,
+adding U to r2SCAN increases the calculated lattice parameters in Ba2TiO4 (by ∼0.03 ˚A), thereby marginally
+reducing the agreement with experiment.
+We benchmark both structural and electronic properties of BiVO4 as a transferability check for V-based
+systems. Note that BiVO4 transforms from tetragonal (I 41/a) to a monoclinic (I 2/b) ‘scheelite’ phase below
+∼ 528 K, [87, 88] which is a reversible second order ferroelastic transition driven by soft optical phonon
+modes. The BiVO4 unit cell possesses four formula units, with tetrahedrally coordinated V ions, which is
+different from the coordination environments of V in VO, V2O3, VO2, and V2O5. Importantly, monoclinic-
+BiVO4 spontaneously transforms to the tetragonal structure upon structure relaxation with r22SCAN and
+r22SCAN+U, similar to the observation by Liu et al [87] with GGA and hybrid functionals. Thus, neither
+r22SCAN nor r22SCAN+U predict the correct ground state structure. Additionally, BiVO4 possess a band
+gap of 2.4–2.48 eV [89] and is a candidate photocatalyst. [87] Both r22SCAN and r2SCAN+U provide similar
+band gap predictions (2.01-1.98 eV), which is in good qualitative agreement with experiment. Surprisingly,
+r2SCAN+U evaluates a marginally lower band gap than r2SCAN (see panels a and b in Figure 4). However,
+both r22SCAN and r2SCAN+U predict similar states occupying the valence band (Op) and conduction band
+(Vd) edges.
+11
+
+Ba2TiO4
+BiVO4
+K3MnO4
+K2MnO4
+Mn2O7
+SrFeO3
+r2SCAN
+r2SCAN+U
+20
+10
+0
+10
+20
+Lattice volume
+difference (Å3)
+(a)
+1.98 eV
+(b)
+(c)
+2.013 eV
+Figure 4: DOS for BiVO4 calculated using (a) r2SCAN and (b) r2SCAN+U. (c) Difference between experi-
+mental and calculated lattice volumes (using r2SCAN and r2SCAN+U ), plotted as a heatmap, for various
+systems. Red (blue) squares indicate overestimated (underestimated) calculated lattice volumes versus ex-
+periment.
+The rationale behind the choice of K3MnO4, K2MnO4, and Mn2O7 as checks for Mn-based systems is to
+explore the higher, unsampled oxidation states of Mn, namely +5, +6, and +7 in K3MnO4, K2MnO4, and
+Mn2O7, respectively. Also, Mn resides in tetrahedral coordination in these compounds, which is different from
+the octahedral coordination observed in MnO, Mn2O3, and MnO2. Although Mn2+ resides in tetrahedral
+sites in spinel-Mn3O4, we had not used in the spinel structure to obtain our optimal U. We benchmark the
+calculated lattice parameters versus experiments for all Mn-based transferability checks.
+Mn2O7 is a volatile liquid at 298 K and solidifies to a monoclinic crystal structure (P21/c) below ∼ 279 K,
+with the unit cell consisting of 8 formula units of corner sharing tetrahedral MnO4 pairs. [90, 91] Upon
+structural relaxation, both r2SCAN and r2SCAN+U underestimate the lattice constants of monoclinic-
+Mn2O7 by ∼1-3% (Figure 4 and Table S3). In the case of K3MnO4, the tetragonal symmetry (I 42m) [92]
+is broken with r2SCAN functional resulting in an orthorhombic structure, while the symmetry is preserved
+by r2SCAN+U (see Figure
+4 and Table S3). Nonetheless, both r2SCAN and r2SCAN+U significantly
+underestimate the c parameter (by ∼ 13.5%) and overestimate the a or b parameter (∼ 10.2%). K2MnO4 is
+an orthorhombic crystal (Pnma) with four formula units per unit cell. [93] Here, r2SCAN and r2SCAN+U
+predict identical lattice parameters, which marginally underestimate experimental values (by ∼ 0.4-1%, see
+Figure 4 and Table S3).
+The choice of SrFeO3, a cubic perovskite, as a check for Fe is largely motivated by the 4+ oxidation
+state exhibited by Fe in the structure, which is not sampled in FeO, Fe2O3, or Fe3O4. Both r2SCAN and
+r2SCAN+U preserve the cubic symmetry during structure relaxation, with r2SCAN+U ’s lattice parameters
+12
+
+(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bip
+Bi
+Density of states (
+-6
+-5
+-4 -3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)identical to experiments and r2SCAN’s parameters being a slight underestimation (∼ 0.5%, see Figure 4
+and Table S3). In terms of magnetic configuration of Fe in SrFeO3, Takeda et al. [94] reported a helical spin
+structure via their neutron diffraction experiments, with competing FM and AFM interactions. However,
+Shein et al. [95] found a FM metallic state to be the ground state of SrFeO3, over a wide range of pressures,
+based on their first principles calculations, which they attributed to stronger FM than AFM interactions. We
+considered a FM configuration of Fe atoms in the SrFeO3 unit cell, and the on-site magnetic moments on Fe
+calculated by both r2SCAN (3.375 µB, Table 1) and r2SCAN+U (3.819 µB) overestimate the experimental
+value (2.7±0.4 µB [94]). However, our calculated magnetic moments do indicate a localization of ∼4 electrons
+on the d orbitals of Fe, consistent with its +4 oxidation state.
+We choose CoO2 (R3m or ‘O3‘ polymorph [96]), and NiO2 (P1m1 or ‘O1’ [97]), both layered structures,
+as transferability checks for Co and Ni, respectively, owing to the unsampled 4+ oxidation states of each
+TM. In terms of experimental property to benchmark, we choose the average Li intercalation voltage in these
+structures, i.e., LiCoO2-CoO2, and LiNiO2-NiO2 pairs, since they have been measured with high precision.
+The reader is referred to previous works on calculating and benchmarking average ‘topotactic’ intercalation
+voltages. [98,99] r2SCAN underestimates the experimental average voltage [96,99–103] in LiNiO2-NiO2 (by
+∼ 8%), while it overestimates the average voltage in LiCoO2-CoO2 (by ∼ 1.7%), similar to trends observed
+with SCAN. [99] The addition of U to r2SCAN leads to an improvement in agreement with the experimental
+voltage in the Ni-system (deviation of ∼ 1.8%), while it worsens the agreement in the Co-system (deviation
+of ∼ 4.4%). Nevertheless, r2SCAN+U does overestimate the average voltage in both Co and Ni systems,
+similar to the behavior of SCAN+U. [99]
+Table 1: Voltage and magnetic moments calculated by r2SCAN, and r2SCAN+U compared against experi-
+mental values (denoted by ‘Expt.’). The U values used with r2SCAN+U are the corresponding optimal U
+values obtained for each TM (from Figure 1).
+Composition
+Source
+Voltage
+Magnetic moment
+(space group)
+(V)
+(µB)
+LiCoO2-CoO2
+Expt.
+4.05
+-
+(R¯3m)
+r2SCAN
+4.12
+-
+r2SCAN+U
+4.23
+-
+LiNiO2-NiO2
+Expt.
+3.85
+-
+(P1m1)
+r2SCAN
+3.54
+-
+r2SCAN+U
+3.92
+-
+SrFeO3
+Expt.
+-
+2.7±0.4
+(Pm¯3m)
+r2SCAN
+-
+3.375
+r2SCAN+U
+-
+3.819
+4
+Discussion
+In this work, we evaluated the performance of the r2SCAN functional among binary TMOs consisting
+of 3d-TMs by calculating the oxidation enthalpies, lattice parameters, on-site magnetic moments, and band
+gaps. Additionally, for each TM-O2 system considered, we calculated the optimal Hubbard-U corrections
+to be used in a r2SCAN+U framework, based on experimental oxidation enthalpies. Although theoretical
+approaches exist to derive U values, [41–47] using oxidation enthalpies nominally gives an “average” correc-
+tion that is suitable across several oxidation states of a given TM. Specifically, our optimal U values are 2.3,
+13
+
+1.0, 1.8, 3.1, 1.8, and 2.1 eV for Ti, V, Mn, Fe, Co, and Ni, respectively, while we don’t deem a U correction
+necessary for Cr and Cu oxides. Interestingly, the optimal U corrections needed with r2SCAN are lower in
+magnitude compared to SCAN for Ti, Mn, Co, and Ni oxides (while the corrections are identical for V and
+Fe oxides), indicating that r2SCAN exhibits lower errors with oxidation enthalpies and possibly lower SIEs
+than SCAN. However, this is not reflected in other physical properties. On an average, we find the accuracy,
+versus experimental values, to be similar for r2SCAN compared to SCAN, and for r2SCAN+U compared to
+SCAN+U, respectively, in lattice parameter, on-site magnetic moment, and band gap evaluations as seen in
+Figure 2.
+The general trends in lattice parameter, magnetic moment, and band gap predictions, across the XC
+frameworks considered, can be summarized as follows. We observe that r2SCAN generates larger lattice
+constants than SCAN and on addition of the U correction to both functionals, the lattice constants further
+increase. Thus, in systems where SCAN underestimates experimental lattice constants (e.g., CrO2, CrO3,
+MnO2), shifting to r2SCAN improves agreement (e.g., error in r2SCAN in CrO3 is 0.8% versus 2.3% with
+SCAN). Also, there are instances where the ground state symmetry of the TMO is not preserved by some
+or all of the XC frameworks considered (i.e., in VO, MnO, FeO, Fe3O4, and Ti2O3), highlighting systematic
+issues in the XC treatment across the four frameworks considered. The calculated on-site magnetic moments
+by r2SCAN (and r2SCAN+U ) are marginally lower than SCAN (SCAN+U ), with the U correction nom-
+inally increasing the calculated moments calculated by r2SCAN and SCAN. However, calculated magnetic
+moments across the four XC frameworks differ by < 10% (except LiNiO2), signifying marginal differences
+in accuracy. Both SCAN and r2SCAN underestimate band gaps across all TMOs (except MnO2), with
+band gaps calculated by r2SCAN typically being lower than SCAN, and adding the U opens/widens the
+gap. Thus, SCAN+U offers the best quantitative accuracy versus experimental band gaps, especially for
+wide gap semiconductors. Note that the qualitative trends from r2SCAN+U are consistent with the trends
+exhibited by SCAN+U and should be reliable in electronic structure predictions in other TM-based oxide
+systems.
+r2SCAN adopts the smooth polynomial interpolation function of rSCAN to maintain numerical stability
+during SCF calculations. Additionally, the reformed gradient expansion for correlation introduced in r2SCAN
+(partially) negates the error introduced to the slowly varying density by the non-vanishing interpolation
+function, [32] which largely accounts for the observed variation in accuracy of r2SCAN versus SCAN. Based
+on our data, we observe that r2SCAN is not systematically more accurate than SCAN across all TMOs
+and for all property predictions. For example, we have lower optimal U values indicating lower SIEs with
+r2SCAN versus SCAN, but also lower on-site magnetic moments (except Mn and Cr oxides) signifying poorer
+d-electron localization with r2SCAN. Further, the smaller band gaps with r2SCAN (versus SCAN) may be
+caused by the residual SIEs, resulting in an underestimation of the CBM across TMOs. Hence, usage of
+r2SCAN(+U ) in TM-based systems must be done with care and efforts should be made to benchmark as
+many available experimental properties as possible before performing “true” computational predictions.
+We considered the transferability of the U values estimated in this work, with r2SCAN, by examining
+systems for each TM with oxidation states and/or coordination environments not sampled while calculating
+the optimal U. In general, we find that r2SCAN or its Hubbard U corrected version estimate similar lattice
+parameters and hence yield similar accuracies on structural properties. Analogously, the calculated on-site
+magnetic moments in SrFeO3 and the band gaps in BiVO4 are similar between r2SCAN and r2SCAN+U. In
+case of electrochemical properties, we do find tangible variations in the calculated average voltages of r2SCAN
+and r2SCAN+U, with r2SCAN+U exhibiting overall lower errors across the Co and Ni systems. Thus, we
+14
+
+SCAN+U
+r2SCAN
+r2SCAN+U
+Ti
+V
+Cr
+Mn
+Fe
+Co
+Ni
+Cu
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Ti
+V
+Cr
+Mn
+Fe
+Co
+Ni
+Cu
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+(a)
+Ti
+V
+Cr
+Mn
+Fe
+Co
+Ni
+Cu
+0.6
+0.8
+1
+1.2
+(c)
+(b)
+Rela�ve overall 
+computa�on �me
+Rela�ve computa�on �me
+per ionic step
+Rela�ve computa�on �me
+per electronic step
+Figure 5: (a) Overall computational time (electronic+ionic steps) (b) computational time per ionic step and
+(c) computational time per electronic loop taken for each TM-O2 binary system with SCAN+U, r2SCAN,
+and r2SCAN+U frameworks relative to SCAN. Values greater (smaller) than 1 in each panel indicates that
+a given calculation is slower (faster) than SCAN.
+15
+
+find the optimal U values obtained in this work to be transferable across oxide frameworks not sampled a
+priori. Nevertheless, more benchmarking studies to compare the performance of r2SCAN+U with r2SCAN
+(and experiments) will help in quantifying the reliability and errors associated with using r2SCAN+U.
+Given that r2SCAN(+U ) is not systematically more or less accurate than SCAN(+U ), the computational
+performance and numerical stability of r2SCAN(+U ) is critical in determining its utility in property pre-
+dictions across materials. Thus, we have quantified the computational time of r2SCAN(+U ) and SCAN+U
+relative to SCAN for each TM-O2 system considered in Figure S1. Specifically, panels a, b, and c of Fig-
+ure 5 plot the overall (electronic+ionic steps), per ionic step, and per electronic step computational time,
+respectively, taken by the SCAN+U (blue bars), r2SCAN (red), and r2SCAN+U (yellow) frameworks, rel-
+ative to the computational time taken by the SCAN functional (dotted black lines), for each TM-based
+set of oxides. Details on calculating the computational times used by the functionals is described in the
+‘Computational time’ section of the SI. Note that our objective is not to provide a rigorous quantification of
+computational resources required for each XC framework, but to provide a qualitative understanding of the
+relative computational costs across the frameworks considered.
+For each electronic step, r2SCAN(+U ) is typically faster than SCAN (Figure 5), signifying better numer-
+ical stability than SCAN, with Mn, Ni, and Cu oxides being marginal exceptions. In contrast, on a per-ionic
+step basis, r2SCAN and r2SCAN+U is slower than SCAN, by ∼1.05-1.78× and ∼1.1-1.31×, respectively,
+highlighting that r2SCAN(+U ) takes more electronic steps to converge per ionic step. Importantly, the over-
+all computational time (ionic+electronic steps, Figure 5) required for structural relaxation of TMOs using
+r2SCAN and r2SCAN+U is lower than SCAN, by ∼12.1-61.2% and ∼1.9-34.5%, respectively, except in Fe
+oxides, indicating that r2SCAN(+U ) takes lower number of ionic steps to converge, which possibly indicates
+a better description of atomic forces. The higher overall computation time in Fe oxides with r2SCAN(+U )
+than SCAN is primarily due to the difficulty in converging Fe3O4 with r2SCAN(+U ). Comparing r2SCAN
+and r2SCAN+U, we find that r2SCAN+U takes a higher overall computational time to converge, except
+in Fe and Ni oxides.
+Thus, we expect r2SCAN(+U ) to provide good utility in property predictions in
+TM-containing systems given its better computational performance and reasonable accuracy compared to
+SCAN(+U ).
+5
+Conclusion
+3d-TMs and their compound phases find applications in several fields such as energy storage, solar
+cells, catalysts, thermochemical water splitting, etc., and it is imperative to predict their properties such
+as lattice constants, magnetic moments, reaction enthalpies, and band gaps accurately using DFT-based
+techniques for designing better materials. Recently, the r2SCAN metaGGA XC functional was proposed
+to exhibit the accuracy of its predecessor, SCAN, and the computational performance of rSCAN in main-
+group compounds, but the accuracy of r2SCAN was not rigorously tested on TM-based systems.
+Here,
+we assessed the numerical accuracy and computational performance of r2SCAN in binary 3d-TMOs, in
+calculating the lattice parameters, on-site magnetic moments, binary oxidation enthalpies, and band gaps
+against experimental data. Notably, we observed that r2SCAN exhibited similar qualitative trends as that
+of SCAN, with marginally larger estimations of lattice parameters than SCAN, while the on-site magnetic
+moments and band gap calculations are marginally smaller than SCAN. While both r2SCAN and SCAN
+underestimated the band gaps in wide gap TMOs, with SCAN offering slightly better accuracy, they failed
+to predict the correct ground state electronic configurations of narrow band gap TMOs (e.g., Mn2O3).
+16
+
+On analysing the addition of Hubbard U -correction to improve the accuracy of the r2SCAN functional,
+we observed that a lower optimal U value, based on experimental oxidation enthalpies, was required in
+a r2SCAN+U framework for Ti, Mn, Co and Ni oxides, when compared to a SCAN+U framework. The
+optimal U values were identical in both r2SCAN+U and SCAN+U frameworks for V and Fe oxides, while we
+did not observe the need for a U correction in Cr and Cu oxides with r2SCAN, similar to SCAN. Moreover,
+introducing the U -correction to SCAN and r2SCAN increased the calculated lattice parameters, on-site
+magnetic moments and the band gaps of the TMOs.
+r2SCAN+U and SCAN+U successfully opened a band gap for narrow gap TMOs (except VO2 and
+Mn2O3 with r2SCAN+U ). Upon testing the optimal U values with r2SCAN+U on oxides with different
+oxidation states and/or coordination environments, we found that the U values derived in this work are in
+general transferable to other TM-containing oxides as well. Furthermore, we observed that r2SCAN(+U )
+took less overall computational time (ionic+electronic steps) to converge when compared to SCAN, which
+indicated that r2SCAN(+U ) was computationally more efficient than SCAN(+U ). Since r2SCAN+U offers
+a reasonably accurate prediction of material properties at a lower computational expense than SCAN+U, we
+observe that r2SCAN+U can be used in high-throughput materials discovery, after adequate benchmarking
+tests are done in each new chemical space explored.
+Acknowledgments
+G.S.G. acknowledges the Indian Institute of Science (IISc) Seed Grant, SG/MHRD/20/0020 and SR/MHRD/20/0013
+and the Science and Engineering Research Board (SERB) of the Department of Science and Technology, Gov-
+ernment of India, under sanction numbers SRG/2021/000201 and IPA/2021/000007 for financial support.
+R.D. thanks the Ministry of Human Resource Development, Government of India, for financial assistance.
+S.S. acknowledges financial support from SERB under IPA/2021/000007. All the authors acknowledge the
+computational resources provided by the Supercomputer Education and Research Centre, IISc, for enabling
+some of the density functional theory calculations showcased in this work.
+Author Contributions
+G.S.G. envisioned and designed the work. S.S. and R.D. performed the calculations. All authors con-
+tributed in data analysis and writing the paper.
+Conflicts of Interest
+The authors declare no competing financial or non-financial interests.
+Availability of data
+The data that support the findings of this study are openly available at https://github.com/sai-mat-
+group/r2SCAN-U-benchmarking.
+Supplementary Materials
+Electronic Supporting Information is available online at , with details on the crystal structures used for
+calculations, oxidation energetics of Cr and Cu oxides, densities of states of all systems not showcased in the
+17
+
+main text, and details on computational time calculations.
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+Astronomy & Astrophysics manuscript no. main
+©ESO 2022
+2022-12-31
+Stellar Karaoke: Deep Blind Separation of Terrestrial Atmospheric
+Effects out of Stellar Spectra by Velocity Whitening
+Nima Sedaghat1, J. Bryce Kalmbach1, Brianna M. Smart1, and Erin L. Howard1
+DIRAC Institute and the Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA
+e-mail: [email protected]
+2022-12-31
+ABSTRACT
+We exploit the statistical independence of stellar features and atmospheric adversarial effects in stellar spectra, to remove the latter
+from observed signals using a fully unsupervised data-driven approach. Concretely, we first increase the inter-observation entropy of
+telluric absorption lines by imposing a random, virtual radial velocity to the observed spectrum. This novel “trick” results in a non-
+standard form of “whitening” in the atmospheric components of the spectrum, decorelating them across multiple observations. Then
+we use deep convolutional auto-encoders, to learn a feature-space in which the two “sources” of information, stellar and atmospheric,
+are easily separable, leading to removal of the latter. We apply the process on spectra from two different data collections: ~250,000
+HARPS spectra and ~660,000 from SDSS.
+We compare and analyze the results across datasets, as well as with existing tools, and discuss directions for utilizing the introduced
+method as a fast and more reliable tool in the future.
+Key words. stellar spectra – telluric lines – deep learning – unsupervised – whitening – decorrelation – source separation
+1. Introduction
+Throughout this paper, we introduce and elaborate a method
+based on signal processing tricks combined with a fully unsuper-
+vised deep neural network to remove the non-linear, time-variant
+effect of telluric lines out of observed spectra without any need
+for manual modeling and tuning. Concretely, we show how look-
+ing at an enormous dataset of spectra reveals new insights into
+their features, allowing for novel solutions to existing, difficult
+problems.
+In ground-based observations of stellar spectra, it is not un-
+usual for the source’s spectrum to reach our detectors in an
+altered state. The spectrum may be affected by a number of
+events in space (redshift/blueshift, emission, absorption, etc.).
+When finally reaching the Earth, the photons in the spectrum
+will undergo additional transformations as they travel through
+the Earth’s atmosphere and are collected by telescopes. Tele-
+scope effects are often well characterized and constrained (CCD
+noise, fringing, mirror defects, etc.). However, other effects are
+usually more difficult to model and correct.
+Particularly, passing through the terrestrial molecules present
+in the Earth’s atmosphere results in absorption lines in the sensed
+spectrum. These absorption lines, known as telluric lines, are the
+results of interaction of photons with molecules due to a num-
+ber of electron, rotational, and vibrational transitions, and are a
+persistent source of contamination and loss of information in the
+observed spectra. The majority of the molecules responsible for
+absorption are O2, H2O, CO2, CH4, O3, N2O, and the Chappuis
+ozone absorption bands. The many of these absorption features
+lie in the near-infrared and ultraviolet part of the electromagnetic
+spectrum, with weaker absorption features from ozone, oxygen,
+and water present in optical wavelengths. Additional emission
+features are also added to the spectrum as photons are emitted
+from the molecules. The emission features are relatively straight-
+4000
+4500
+5000
+5500
+6000
+6500
+Wavelength (Å)
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Normalized Flux (-)
+Input Spectrum
+Reconstructed Spectrum
+Fig. 1. We exploit the statistical properties of stellar spectra in large
+datasets, pass them through a convolutional auto-encoder, and get tel-
+luric lines rejected with close to zero effort.
+forward to handle by observing source-free portions of the sky.
+Observed emission lines in "empty" regions provide an easily ac-
+cessible template for emission removal. The absorption features
+are more complicated, scaling non-linearly, and are affected by
+atmospheric conditions at the time of observation.
+Most of the traditional methods used for the removal of tel-
+luric lines consider each spectrum as a single independent en-
+tity, discarding the set of observed spectra as a whole – e.g. see
+Hrudková & Harmanec (2005). One group of such methods use
+standard stars to assist with atmospheric line removal. They do
+Article number, page 1 of 10
+arXiv:2301.00313v1  [astro-ph.SR]  1 Jan 2023
+
+A&A proofs: manuscript no. main
+this by observing standard stars with relatively featureless spec-
+tra and using them as atmospheric templates. This is often done
+by observing A0V or G-type stars, though this is predominantly
+used at the near-infrared and IR wavelengths (Vacca et al. 2003;
+Artigau et al. 2014). The reference star should be ideally lo-
+cated near the target star and observed close in time to measure
+the atmospheric conditions as accurately as possible. The target
+source’s spectrum is then divided by the telluric template. How-
+ever, this method is limited if there are no such standard stars
+available during an observing run or if observing conditions re-
+sult in a poor spectrum. Atmospheric absorption can also vary
+rapidly, on the order of minutes, requiring significant time and
+resources dedicated to observing the template stars. The division
+is an iterative process requiring wavelength and intensity scaling
+adjustments to match the atmospheric effects.
+Another common method is to model the Earth’s atmosphere
+and create a synthetic spectrum which precisely models the at-
+mospheric absorption. These models solve the radiative trans-
+fer equation of the Earth’s atmosphere using numerical models
+– e.g. Allart et al. (2022). These methods are dependent on at-
+mospheric conditions measurements from the night’s observa-
+tions. This is commonly done with programs such as molecfit
+(Smette et al. 2015) and telfit (Gullikson et al. 2014). The
+radiative transfer code retrieves the temperature, pressure, and
+humidity from the time of observation and uses a database of
+molecular parameters to create a fit for the telluric absorption.
+While this technique is relatively successful, it may perform
+poorly if there are a large number of intrinsic features, little or
+no continuum, low signal-to-noise ratio (S/N), or large airmass
+observations with high water vapor content. Moreover, current
+implementations of this technique suffer from rather slow perfor-
+mance to the extent that fitting a model to a single spectrum may
+take up to several minutes on today’s computers. See Ulmer-
+Moll et al. (2019) for a comparison of the above methods.
+More recent methods, however, have incorporated implicit
+modeling of the spectra by looking at a set of examples, poten-
+tially eliminating the need for manual, explicit modeling. Such
+methods, loosely dubbed data-driven approaches, have gained
+good momentum in the past couple of decades. Particularly in
+astronomy, a data-driven look at spectra has been mainly incor-
+porated in the broad context of dimensionality reduction.
+Arguably the most popular dimensionality reduction method
+used in astronomy has been the Principal Component Analy-
+sis (PCA – see Jolliffe & Cadima (2016) for a review), which
+has been used as a dimensionality reduction tool on astronom-
+ical spectra for many years now. Connolly et al. (1995) used
+it to classify galaxy spectra with only the first two principal
+components while Bailer-Jones et al. (1998) showed that PCA
+can compress stellar spectra by a factor of over 30 and classify
+anomalous and non-stellar spectra in the data set. Furthermore,
+Bailer-Jones et al. (1998) found that the compression removed
+noise as well as bogus features in the spectra such as dust ap-
+pearing on the plate during plate scanning. Some works such as
+Artigau et al. (2014), even attempt to remove telluric lines of a
+few specific objects in the HARPS dataset using PCA decom-
+position and demonstrate improved radial velocity measurement
+accuracy. However, their method does not operate solely on the
+stellar spectra and still requires observations of telluric standard
+stars. Our approach solely requires a set of input stellar spectra
+without specific telluric standard stars.
+Moreover, the very important, but often overlooked point
+about applicability of PCA for such applications, is its linear na-
+ture, making it unable to account for modeling non-linear phe-
+nomena by definition. E.g. in the case of telluric lines, a method
+simply based on PCA, may not be able to tell the difference
+between a telluric and stellar line, in crowded regions of the
+wavelength. This inability has been well observed as early as
+in Bailer-Jones et al. (1998). In this work, we show how trans-
+formation to a more sophisticated feature space is necessary for
+such a task.
+There is a closely related family of neural networks known
+as encoder-decoder networks (fig. 2). The contraction part com-
+presses the input to an often low-dimensional representation,
+also known as "latent representation" or simply "code". The ex-
+pansion part is then used to decode this low-dimensional repre-
+sentation up to the desired output. The transformation used for
+coding (and decoding) is learned during training, according to
+the task at hand. A special case of such networks, auto-encoders,
+is an unsupervised network trained to reconstruct the input with-
+out the need for labeled input data. As described in Hinton &
+Salakhutdinov (2006) these networks can be used to reduce the
+dimensionality of input data in a non-linear generalization of
+PCA.
+In Yang & Li (2015) the authors use a classical (non-
+convolutional) auto-encoder to transform 3000-dimensional
+spectra from the Sloan Digital Sky Survey (SDSS) Data Re-
+lease 7 (Abazajian et al. 2009) to lower-dimensional features,
+which are later used for estimation of atmospheric parame-
+ters. Wang et al. (2016a) also used classical (non-convolutional)
+auto-encoders for feature learning on astronomical spectra. They
+compare spectral classification based on their learned features
+with PCA and locally linear embedding (LLE), an alternate
+non-linear dimensionality reduction method. They find that their
+auto-encoder approach performs the best when classifying spec-
+tra among a data set of F, G and K-type stars.
+One problem with the mere use of auto-encoders, especially
+for applications in (astro-)physics, is the entangled mapping of
+concepts into the reduced signal – the latent space. The Varia-
+tional AutoEncoder (VAE: Kingma & Welling 2014) has shown
+to mitigate this effect to some extent. Portillo et al. (2020) used
+a VAE to reconstruct SDSS galaxy spectra resampled to an input
+spectrum with 1000 components. They found that when limiting
+the dimensionality of the latent spaces to 2, 4, 6 or 10 compo-
+nents (≥ 99% compression) the VAEs and traditional autoen-
+coders both reconstructed SDSS galaxy spectra with a lower re-
+construction error than PCA and non-negative matrix factoriza-
+tion (NMF) applied with the same number of components. The
+authors then demonstrated that different galaxy classes occupied
+different areas of the latent space enabling classification.
+Classical auto-encoders are merely composed of fully-
+connected layers and thus suffer from a lack of scalability to
+deeper networks and high-dimensional data – usually the case
+in modern astronomical spectroscopy. We address this drawback
+by incorporating convolutional (and up-convolutional) layers for
+transformation of data to the desired feature space; an idea bor-
+rowed from computer vision. In Zhang et al. (2004) the authors
+map images of people’s faces to low-dimensional manifolds, and
+reconstruct them back. They find a correlation between the face
+pose and the low-dimensional manifold, independently of the
+person in the image. Wang et al. (2016b) shows comprehensive
+experiments on dimensionality reduction using auto-encoders,
+and studies the effects of different latent dimensions using syn-
+thetic and real images. Note that in case of spectra, the networks
+need to be updated to use 1D (up-)convolutional layers, as op-
+posed to 2D in case of images.
+Perhaps the closest to our work, both in terms of the used
+network architecture, as well as the data-driven unsupervised
+concept is Sedaghat et al. (2021), where a convolutional VAE
+Article number, page 2 of 10
+
+Sedaghat et al.: Stellar Karaoke
+Fig. 2. A simple auto-encoder trained to reconstruct stellar spectra, can decompose physically meaningful components out of the input, when the
+compression is high enough and the number of convolutional kernels is limited. Blue is the input spectrum and orange is the reconstructed version.
+The region annotated by the red circle indicates a high density of such reconstruction rejections.
+is used to extract knowledge from a large number of HARPS
+spectra, in a fully unsupervised manner. We borrow and use the
+exact same neural architecture for our work. However, we addi-
+tionally utilize the conceptual decomposition of stellar features
+and atmospheric effects into two different spectra; an idea that
+turns out to have been touched as early as in Hadrava (1997),
+however without a data-driven perspective.
+Our Contributions
+• We process a huge number of very high-dimensional spec-
+tra as a whole, letting statistical properties emerge in them,
+allowing to be treated as random processes.
+• We model the telluric components in stellar spectra as inde-
+pendent stars and impose a virtual radial velocity on them to
+achieve statistical whitening/decorrelation.
+• We incorporate a convolutional auto-encoder, that automati-
+cally acts as a source separation tool, rejecting telluric lines
+in a fully unsupervised fashion, with zero explicit modeling.
+2. Problem Formulation
+We seek to clean adversarial atmospheric effects out of an ar-
+bitrary observed signal, x. The observed signal in our case is a
+stellar spectrum, and so is a function of wavelength, λ, letting us
+denote it as x(λ).
+We use the below formulation to model the various phenom-
+ena affecting a stellar spectrum, before it is captured by our sen-
+sors:
+x(λ) =
+�
+s(λ) × t(λ)
+�
+∗ h(λ) + n(λ)
+(1)
+where s is the stellar spectrum, incorporating line-of-sight ef-
+fects from relative stellar velocity and the interstellar medium. t
+is an imaginary signal representing the telluric lines affecting the
+spectrum. x is the observed signal: a single spectrum that takes
+on different values at each wavelength depending on the flux at
+that wavelength. So we use the notation, x(λ), throughout this
+article.
+We use h to denote what we call the observation transfer
+function, and models the changes the signal goes through dur-
+ing the sensing process, and includes, but is not limited to, the
+line spread function, etc. n models the additive noise which is
+not modeled in the transfer function, h.
+We work with a large ensemble of N observations, mostly
+coming from different sources. We use the subscript, i, to differ-
+entiate between various observations:
+xi(λ) =
+�
+si(λ) × ti(λ)
+�
+∗ h(λ) + ni(λ)
+i ∈ {1, 2, . . . , N}
+(2)
+Note that the spectrum, s, depends on i, as we work with various
+objects at the same time. The atmospheric conditions are also
+time-variant, an important point that is reflected in dependence
+of t on i. The same is true for noise, n. Without loss of generality,
+and for the sake of simplicity, we assume that h is constant across
+the whole set of observations.
+We seek to eliminate the effect of telluric lines from the
+observed signal, which in this model is equal to extracting
+si(λ) ∗ h(λ) for every observation – note that removal of the ob-
+servation effect, h, is not part of the objective here.
+We also model the effect of the radial velocity, v, of the target
+object on the observed spectrum, s as:
+si(λ) = V {˚si(λ), vi}
+= ˚si
+�
+λ�1 − vi
+c
+��
+(3)
+where ˚si would be the observed spectrum, if the radial velocity
+was zero – i.e. no Doppler shift. We call ˚si the static spectrum
+hereafter. The V{.} operator represents the effect of the radial
+velocity, vi, on the spectrum, as expanded in the second line of
+the above equation, and c is the speed of light. The physical units
+of vi and c can be arbitrarily chosen, as long as they are kept the
+same.
+ti on the other hand, and by definition, do not have any de-
+pendence on vi, and therefore can be modeled as:
+ti(λ) = V
+�
+˚ti(λ), 0
+�
+(4)
+2.1. Continuous vs. Discrete
+All the signals discussed above are of continuous nature up until
+the point they are sensed by the detector. Sensing by the detector
+Article number, page 3 of 10
+
+A&A proofs: manuscript no. main
+is a process that involves sampling as one of its steps, converting
+a spectrum into a series of real-valued samples. Therefore, the
+data we work with in our experiments is the discrete representa-
+tion of xi(λ), namely Xl
+i – we use l as the discrete-valued index
+for the sampled pixels.
+However, for the reasons below, keeping the formulation in
+the continuous representation is safe – and clearer. First, the
+signals modeled so far are all the constituent elements of the
+pre-sampling signal, xi(λ), and so the continuous models hold.
+Secondly, the only part of our method which explicitly modifies
+the signal along the wavelength axis, the V {, } operator (sec-
+tion 3.1), does its job by regridding the interpolated version of
+the signal; practically converting the discrete signal back to its
+continuous version, applying the transformation and sampling it
+back again to the discrete space. Therefore mathematical defini-
+tion of the operator in the continuous space is valid.
+2.2. Signals as Random Processes
+The process of sensing a signal, xi, from an arbitrarily chosen
+object in the sky, which is affected by many non-deterministic
+phenomena along the way, can be seen as one realization of a
+random process. In other words, the set of xi, or their discrete
+representation, Xi, for various i, represent an ensemble of real-
+izations of a random process, {X} 1. Note that in this particular
+application, the index set of the random process is sampled from
+wavelengths, λ – a bit counter-intuitive, as it is usually of a tem-
+poral nature in typical applications.
+Therefore each Xl represents a random variable, with out-
+comes Xl
+iii. Similarly, we can model {S }, { ˚S } and {T} as discrete-
+index random processes of the same type. {S } is then a genera-
+tor of various clean spectra, S i, while { ˚S } generates the static ˚S l
+i
+spectra. As we will see in the next section, this non-deterministic
+view on signals allows us to explain the statistical operations and
+properties of the components more clearly.
+3. Method
+The method is composed of two key steps:
+a) Increasing the entropy of telluric components across obser-
+vations and,
+b) transforming the spectra into a space where the stellar and
+telluric components are separable, then removing the non-
+dominant one.
+Median Normalization As a pre-processing step, we normalize
+each spectrum in the dataset based on its median, to mitigate the
+effect of different distances in sources, which otherwise results
+in extreme inter-sample flux range variations.
+3.1. Velocity Whitening
+In the default conditions, the spectroscopic observations are in
+the so-called topocentric reference frame where the telluric lines,
+if they exist, take on the same wavelengths – or pixels locations.
+What it means from a statistical point of view though is this:
+Let us assume, for simplicity, that all the observed objects
+have the same static spectra, ˚si(λ). The resulting spectra, si(λ)
+1 The term random in this context is not in contradiction with the struc-
+ture in stellar spectra. The structure is encoded in the basic parameters
+of the constituent set of random variables, such as the expected value
+and covariance matrix.
+would then only be different based on their radial velocities,
+vi, which, by definition, are modeled as scaling transformations
+along the wavelength axis (eq. (3)). Note that since vi are out-
+comes of a random variable, si(λ) are consequently still highly
+independent and uncorrelated.
+ti on the other hand, are composed of a set of absorption lines
+with different strengths, but all happening at pre-defined specific
+wavelengths. This feature makes them highly correlated across
+multiple observations and easy to fit for any model – fig. 3, left
+column.
+To decrease the existing correlation between realizations of
+the telluric signals, we increase the entropy across ti by random-
+izing them using an emulated radial velocity,
+t′
+i(λ) = V
+�
+˚ti(λ), v′
+i
+�
+(5)
+where each v′
+i is a random velocity and is drawn from
+V′ : R → R. The above effect can be implemented by sim-
+ply contracting or expanding the wavelength axis, according to
+eq. (3). Note though that this is a purely artificial phenomenon,
+and although it is chosen to have the same effect as the “real”
+radial velocity of stars, it has no particular meaning for telluric
+lines 2.
+In practice, however, we only have access to the observed
+signal, x, and not its forming components. Hence the proposed
+randomization cannot be applied directly on t alone, and the
+whole observed signal gets affected. But in the below, we show
+that it can still have the desired effect:
+λ −→ λ′
+i = λ × (1 − v′
+i
+c )
+(6)
+x′
+i(λ) = V �xi(λ), v′
+i
+�
+= V
+��
+si(λ) × ti(λ)
+�
+∗ h(λ) + ni(λ), v′
+i
+�
+= V
+��
+si(λ) × ti(λ)
+�
+∗ h(λ), v′
+i
+�
++ V �ni(λ), v′
+i
+�
+(7)
+which, according to the proof provided in appendix A becomes:
+x′
+i(λ) = (1 − v′
+i
+c )V �si(λ) × ti(λ), v′
+i
+� ∗ V �h(λ), v′
+i
+� + V �ni(λ), v′
+i
+�
+= (1 − v′
+i
+c )
+�
+si(λ′
+i) × ti(λ′
+i)
+�
+∗ h(λ′
+i) + ni(λ′
+i)
+(8)
+Now since
+�
+1 −
+v′
+i
+c
+�
+is, for each x′
+i, a constant coefficient and
+is normalized out before being passed to the next step of the
+method, we can see that the emulated randomized velocity has
+founds its way from xi down to ti. In other words, we have
+achieved the required velocity randomization in ti by modifying
+xi. The two main components in the tweaked signal can now be
+rewritten as:
+Telluric:
+Stellar:
+V
+�
+˚ti(λ), v′
+i
+�
+V
+�
+˚si(λ), vi + v′
+i
+�
+So, we have achieved two components which are virutally mov-
+ing independently of each other, when going over various obser-
+vations. Figure 3 illustrates this effect in practice.
+2 A similar deterministic effect occurs when taking stellar spectra to
+the barycentric frame.
+Article number, page 4 of 10
+
+Sedaghat et al.: Stellar Karaoke
+6265
+6270
+6275
+6280
+topo-centric
+6265
+6270
+6275
+6280
+ADP.2014-09-16T11:03:35.377
+randomized
+6265
+6270
+6275
+6280
+6265
+6270
+6275
+6280
+ADP.2014-09-16T11:03:36.737
+6265
+6270
+6275
+6280
+Wavelength (Å)
+6265
+6270
+6275
+6280
+Wavelength (Å)
+ADP.2014-09-16T11:03:38.307
+Fig. 3. On the left a set of exemplar spectra are depicted. Stellar lines are
+unaligned due to different radial velocities. But telluric lines are aligned,
+even though they may have different shapes due to their inherent time
+dependence (the dotted vertical line indicates the location of a specific
+telluric line in all plots). On the right the same spectra after velocity
+randomization are depicted. Telluric lines are now unaligned too, but
+with a pattern different to that of stellar lines!
+From a statistical point of view, what we achieve with the
+above velocity randomization is a degree of “whitening” of the
+tellurics random process, {T}. More precisely, we decrease the
+mutual correlation between every pair, (T l, T m), pushing the sta-
+tistical behavior of the telluric component toward white noise (Li
+& Zhang 1998; Eldar & Oppenheim 2003; Kessy et al. 2018).
+Note again that (T l, T m), the random variables we try to decor-
+relate, are each defined on a single pixel of the spectrum, and
+their outcomes vary along with different observations. A visual-
+ization of the results of the above whitening process is illustrated
+in section 5.
+3.2. Deep Feature Space
+We borrow and use the exact architecture of the 1D convolu-
+tional auto-encoder introduced by Sedaghat et al. (2021) – Fig-
+ure 2. The convolutional layers of the encoder transform the in-
+put spectrum down to a pre-defined number of “latent variables”
+at the bottleneck of the network. This low-dimensional repre-
+sentation of the input, also known as the “code”, is the most
+compressed version of the input spectrum. The dimensionality
+of this vector is chosen based on the desired compression rate in
+various experiments. Note though that in the VAE version of our
+networks, which is the case in most of the experiments of this
+work, the latent nodes are implemented probabilistically, each
+being modeled with a pair of scalars: mean and std of a normal
+distribution. On the other side of the bottleneck, the decoder re-
+Fig. 4. Visual illustration of the covariance matrix of the ensemble of
+signals, before and after whitening. On the left, the covariance ma-
+trix of the [6275, 6285]Å region for some subset of size 90 of the
+observations is visualized. On the right, the same is done after veloc-
+ity whitening, where vi were sampled from the uniform distribution:
+V ∼ U(−30km/s, 30km/s). The covariance matrix is closer to the iden-
+tity matrix now, confirming achievement of some degree of whiten-
+ing/decorrelation.
+ceives the compressed code and takes it step-by-step up to the
+same dimension as the original input (218 + 216 for HARPS). We
+keep using the same per-pixel L1 end-to-end loss function, as in-
+troduced in the original work, to achieve acceptable pixel-level
+accuracy.
+This ∼typical architecture has proven to be able to transform
+the input to a space where noise-like components of the signal
+are easily separable. As we show with our experiments, the sta-
+tistical trick developed in the previous section pushes the telluric
+components farther from the stellar features and closer to the
+noise, in the learned feature space, letting them be rejected as
+easily as noise.
+We of course need to constrain the reconstruction abilities of
+the network with a high compression rate as well as a variational
+loss at the bottleneck, not to have too strong of a network capable
+of fitting the two independent components at the same time!
+4. Data
+We use data from two publicly available colletions: HARPS and
+SDSS. Our main experiments are run on HARPS, while SDSS is
+used as a difficult test-bench.
+4.1. HARPS
+HARPS (High Accuracy Radial-velocity Planet Searcher; Mayor
+et al. 2003) is an instrument on the 3.6m La Silla Telescope.
+It is a fibre-fed high-resolution echelle spectrograph dedicated
+to the discovery of exoplanets, with a spectral resolution of
+R = 115, 000 and covers the spectral range 378–691nm 3. The
+data used has been instrument-corrected and detector-corrected,
+as well as sky-subtracted.
+Our downloaded dataset initially consisted of 272376 total
+spectra, which after automatic removal of corrupted files, NaNs
+and noise-like ones, was reduced to 267361 “stable” ones. This
+collection is mainly composed of stellar spectra. However, there
+are some random contaminant objects too, such as SUN, MOON,
+etc., which we allowed to enter our training set on purpose,
+to increase robustness of the learned features. We trimmed, re-
+gridded and homogenized all spectra before being passed to the
+3 http://archive.eso.org/wdb/wdb/adp/phase3_main/form
+Article number, page 5 of 10
+
+A&A proofs: manuscript no. main
+Fig. 5. Results Qualitative illustration of the results on an exemplar selection of HARPS spectra. Each row depicts one spectrum, with its HARPS
+ID written on the right, while columns focus on different regions of interest. Interpretation hint: reconstruction (orange) should follow the pseudo-
+truth (molecfit: green). The left-most column covers the whole spectrum, as is fed into the network, and illustrates the robustness of the network
+to different characteristics of the spectra (continuum, noise level, etc.). Major stellar lines such as Hα can be easily spotted in the less ‘busy’
+examples. In the middle column we zoom in on a potentially complex region where narrow stellar lines, when existing, can collide with telluric
+lines of similar shapes. E.g. in the second and 4th row there are examples of such cases, where the network rejects the telluric component, while
+still preserving the stellar part very well – thus rejecting the hypothesis that it might be simply rejecting narrow lines by a applying a moving
+average. In the third column, we focus on a region where similarly narrow stellar lines occur in some of the spectra, but not in the others. This way
+we reject the hypothesis that the network might have ’memorized’ the locations of the lines. Note that the network (orange) is even outperforming
+the pseudo-truth (green) in some cases, illustrating a smoother, more robust reconstruction in abrupt changes.
+network in exactly the same way as is done by Sedaghat et al.
+(2021).
+Note that the pipeline the HARPS spectra go through is set
+up such that the spectra are automatically transformed to the
+barycentric reference frame and re-gridded before being stored
+in the archive. The originally captured version, in the topocen-
+tric reference frame, is also not preserved. Therefore we had to
+transform them back to the topocentric frame for our experi-
+Article number, page 6 of 10
+
+pseudo-truth
+input
+reconstruction
+Normalized Flux (-)
+ADP.2014-09-16T11:05:10.627
+0
+4000
+5000
+6000
+6278
+6279
+6280
+6281
+6282
+5565.05567.55570.05572.5
+Normalized Flux (-)
+ADP.2014-09-16T11:03:31.413
+1
+4000
+5000
+6000
+6278
+6279
+6280
+6281
+6282
+5565.05567.55570.05572.5
+Normalized Flux (-)
+ADP.2014-09-16T11:03:31.673
+1
+0
+4000
+5000
+6000
+6278
+6279
+6280
+6281
+6282
+5565.05567.55570.05572.5
+Normalized Flux (-)
+ADP.2014-10-01T10:22:48.507
+1
+0
+4000
+5000
+6000
+6278
+6279
+6280
+6281
+6282
+5565.05567.55570.05572.5
+Wavelength (A)
+Wavelength (A)
+Wavelength (A)Sedaghat et al.: Stellar Karaoke
+ments. Although this is touching the core concept of our pre-
+sented method, it turned out to be a safe procedure: transforma-
+tion to the barycentric frame exerts added randomness on the
+radial velocity, which is perfectly compatible with our method.
+In fact, our method has been inspired by observing traces of the
+above-mentioned fact in our initial experiments.
+4.2. SDSS
+Our second dataset consisted of spectra from the SEGUE (Yanny
+et al. 2009) and SEGUE-2 spectroscopic surveys (Rockosi et al.
+2022) that were part of the larger Sloan Digital Sky Survey
+(SDSS). The SEGUE surveys both used the 2.5m Sloan Foun-
+dation Telescope (Gunn et al. 2006) located at Apache Point Ob-
+servatory with the two original Sloan Digital Sky Survey fiber
+spectrographs (Smee et al. 2013). The SDSS spectrographs have
+a resolution of R ∼ 1800 and together have 640 fibers (320 each)
+that plug into aluminum "plugplates" for each observation. In
+each plate 32 plugs are reserved for blank sky observations and
+16 for spectrophotometric standard stars in the field. Each spec-
+trograph has a red and a blue channel that collect data on separate
+CCDs with the blue wavelength range from approximately 3800
+Å to 6100 Å and the red wavelengths spanning approximately
+5900 Å to 9200 Å. Sources in the original SEGUE survey sam-
+pled Milky Way stars at a variety of distances, colors and metal-
+licities while the SEGUE-2 targets focused on stars in the Milky
+Way halo.
+In our experiments we used the ~660,000 uncalibrated spec-
+tra from the red CCD of one of the spectrographs (labeled as ’r1’
+in the SDSS data archive) provided with SDSS Data Release 17
+(Abdurro’uf et al. 2022) and accessible via the DR17 FITS web-
+site4. The uncalibrated spectra consist of multiple 10-30 minute
+exposures of each source. Since the SEGUE surveys imaged
+each source multiple times to create coadded spectra this means
+we have multiple spectra in our dataset for the same source. For
+each plate we exclude the fibers that were intentionally pointed
+at empty patches of sky and labelled "SKY" in the data.
+The uncalibrated spectra used are labelled ‘spFrame‘ in the
+SDSS data model5 and according to the details of the SDSS spec-
+troscopic pipeline (Stoughton et al. 2002) they are flat-fielded but
+not flux-calibrated. The flux calibration spectra (‘spFluxCalib‘
+in the data model) include telluric absorption calculated by the
+spectroscopic pipeline based upon the spectrophotometric stan-
+dard stars that are included in the observation set of each plate.
+These calibration are what we use as a "pseudo-truth" for com-
+parisons of our network results.
+5. Experiments and Results
+We train and evaluate multiple networks based on various com-
+binations of the below (hyper-)parameters:
+• Latent-space dimensionality
+• Velocity randomization level
+For our main experiments, that include hyper-parameter
+sweeping and controlled tests, we mainy use the HARPS dataset.
+Other datasets are used for comparative testing of the method in
+extreme conditions.
+4 https://data.sdss.org/sas/dr17/
+5 https://data.sdss.org/datamodel/
+5.1. Main Results
+Figure 5 illustrates a few examples of how the method manages
+to reject the telluric lines and preserve the stellar features. The
+example spectra and regions are specifically chosen to rule out
+potential naive hypotheses for how the network rejects telluric
+lines. In other words, the results confirm that the network has
+learned a semantic representation of the constituent components
+and is separating the sources in that feature space, as opposed
+to e.g. simply having “missed” narrower lines (simple averag-
+ing, low-pass filtering). In the caption of fig. 5 a few other such
+hypotheses are elaborated and rejected.
+5.2. Quantitative Evaluation
+Various (hyper-)parameters, such as the degree of compression
+during dimensionality reduction, or the resolution of the input
+data, may influence the capability of the network in rejecting
+the telluric lines. Precisely, there is always a trade off between
+telluric-rejection and stellar-reconstruction. Therefore, we de-
+velop and incorporate the below mutual metrics to quantify both
+aspects at the same time:
+Qt =
+� �
+Mt ��� ˆS − ˜G
+���
+�
+� Mt
+(9)
+where Qt is a proxy for the quality of rejection of telluric lines.
+ˆS represents the reconstructed spectrum (i.e. the direct output of
+the network) and ˜G is the pseudo ground truth. ˜G is a corrected
+version of the output of molecfit in case of HARPS data, and a
+modified version of the publicly available calibrated spectra in
+case of SDSS. Mt is a binary mask; a vector of the same size
+of the input, having ones at all pixels containing known telluric
+lines, and zeros everywhere else. Note that the subscript i, index-
+ing each observed spectrum, is omitted in Qt, ˆS , ˜G and Mt for
+the sake of simplicity, and the summation is calculated over all
+the pixels of each spectrum.
+We similarly define the dual metric for measuring the recon-
+struction (preservation) of the stellar features as follows:
+Qs =
+�
+�������Ms
+�������1 −
+ˆS
+˜G
+�������
+�������
+� Ms
+(10)
+where Qs is, conversely, a proxy for the quality of stellar recon-
+struction. Note that both of the above are metrics of a distance
+nature, and hence, the lower the better.
+5.3. The Effect of Compression
+It is reasonable to expect, and is supported by our experiments
+that, with a lower dimensionality at the information bottleneck
+of the network, the network’s capacity for preserving many con-
+current features during compression diminishes. This results in a
+higher rejection of details in the reconstructed spectra, which can
+be naively interpreted as an improvement in telluric line rejection
+– a decrease in Qt. Note however that stellar reconstruction qual-
+ity can get worse at the same time, and so the two metrics need
+to be considered at the same time.
+Table 1 compares results of various runs with different di-
+mensionalities at the latent space on a fixed subset of HARPS
+spectra. Note how the stellar reconstruction quality starts to de-
+cay by increasing the compression (lower latent dimensions). On
+Article number, page 7 of 10
+
+A&A proofs: manuscript no. main
+8800
+8900
+9000
+9100
+telluric
+8100
+8200
+8300
+8400
+telluric
+6400
+6500
+6600
+6700
+6800
+stellar: H-alpha
+8400
+8600
+8800
+104_3110_spFrame-r1-00055323_230
+stellar
+6000
+7000
+8000
+9000
+Normalized Flux (-)
+8800
+8900
+9000
+9100
+Wavelength (Å)
+8100
+8200
+8300
+8400
+Wavelength (Å)
+6400
+6500
+6600
+6700
+6800
+Wavelength (Å)
+8400
+8600
+8800
+Wavelength (Å)
+104_3114_spFrame-r1-00056165_144
+6000
+7000
+8000
+9000
+Wavelength (Å)
+Normalized Flux (-)
+input
+tellurics profile
+reconstruction
+input
+tellurics profile
+reconstruction
+Fig. 6. Test results on a low-resolution data collection: SDSS. On the left, the whole spectrum is depicted, while the other columns depict various
+regions of interest along the wavelength axis, so the qualitative performance of the method can be assessed on specific stellar and telluric features.
+Note that since pseudo-truth data is not easily obtainable in this case, we over-plot a "tellurics profile" in red, just for visual comparison. It should
+be interpreted differently from the pseudo-truth curves in other figures, as it does follow the telluric lines. Stellar features are preserved very well,
+even in bad signal-to-noise conditions. Some of the wider telluric lines, however, are not fully rejected in this case due to the significantly low
+resolution as compared to HARPS. Please refer to the main text for a more detailed discussion.
+Latent Dim.↓
+Telluric
+Stellar
+8
+0.053
+0.015
+32
+0.055
+0.012
+128
+0.063
+0.014
+1024
+0.176
+0.012
+Table 1. Quality of stellar line reconstruction vs. telluric line rejection
+on HARPS, for different configurations of the network. Latent Dim. is
+the number of dimensions of the code, or the latent representation. A
+lower number is better in both columns.
+the other hand, by increasing the number of latent nodes, the
+network starts to become too powerful, managing to reconstruct
+both components and losing the source separation capability.
+5.4. The Effect of Velocity-Whitening
+To demonstrate the effect of the whitening step, we compare the
+results of two experiments in controlled conditions. The first ex-
+periment is run on HARPS spectra in topocentric frame, where
+telluric lines are all aligned – zero velocity randomization. Then
+we apply velocity whitening, sampling vi from the uniform dis-
+tribution: V ∼ U(−30km/s, 30km/s). All other configurations of
+the test are kept fixed. Table 2 shows a clear difference in the
+telluric rejection performance of the two runs.
+HARPS
+Telluric
+Stellar
+Topocentric
+0.199
+0.018
+Rand. Velocity
+0.063
+0.014
+Table 2. Comparison of two experiments with zero velocity randomiza-
+tion (‘Topocentric’) and V ∼ U(−30km/s, 30km/s) (‘Rand. Velocity’).
+5.5. HARPS vs. SDSS: High- vs. Low-Resolution
+Figure 6 depicts the results of applying our method on SDSS
+spectra. The extremely low pixel resolution we used for this
+dataset (1.66Å as opposed to 0.01Å for HARPS), makes this
+in practice a stress test for our method, since many of the tel-
+luric lines get merged, appearing as wide artifacts. However, the
+method performs at an acceptable level. E.g. the middle column
+of fig. 6 shows how the method misses the very wide telluric ar-
+tifact while managing to reject the neighboring telluric lines. In
+general, it seems our method can “hunt” telluric lines up until
+the point the resolution goes so low that merging/blending of the
+lines converts them to “slow” artifacts.
+6. Conclusions and Future Directions
+We presented a method that, by incorporating a Big Data-
+inspired view at stellar spectra, exploits the statistical indepen-
+dence of the radial velocity of stars with telluric lines in their
+observed spectra, reinforces it using a novel trick, and utilizes
+a fully unsupervised convolutional neural network to reject the
+undesireable part.
+The method is superior to existing, traditional telluric line
+removal tools in terms of preparation effort, performance, and
+accuracy. The fact that it is fully unsupervised, obviates the need
+for any kind of model parameter tuning – which is the case in
+e.g. molecfit, where a list of wavelengths should be manually
+specified to initialize the model. Training the model, in practice,
+calls for merely passing a large number of spectra through the
+network – and it will do the rest.
+Training one network suffices for one whole data collection.
+Once trained, it processes each spectrum in a fraction of a second
+– depending on the size of the spectrum and, consequently, the
+size of the network. But the speed-up does not sacrifice accuracy;
+as seen in section 5, it can even detect and suppress hard-to-
+locate lines which are missed by molecfit – the opposite may
+happen too, though in rarer situations.
+Article number, page 8 of 10
+
+Sedaghat et al.: Stellar Karaoke
+Nevertheless, the current version is still a demonstration of
+a research product, showcasing the strengths of a fully unsuper-
+vised approach. But for Stellar Karaoke to become a ready-to-
+use package in every application, more work is required. No-
+tably, the decision of which component to keep and which one
+to reject (stellar vs. telluric), should not be left to the network. It
+can be enforced by a minimal supervision.
+Acknowledgements. Some of the experiments demonstrated in this work have
+been run on compute servers provided by ESO, during NS’s collaboration with
+the ESCAPE project between 2019 and 2021.
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+Article number, page 9 of 10
+
+A&A proofs: manuscript no. main
+Appendix A: Wavelength Transformation and
+Convolution
+x(λ) = z(λ) ∗ h(λ)
+=
+�
+w
+z(w)h(λ − w)dw
+(A.1)
+x(λ′
+i) = x(aλ)
+=
+�
+w
+z(w)h(aλ − w)dw
+(A.2)
+Let w = au,
+(A.3)
+x(λ′
+i) = a
+�
+u
+z(au)h(aλ − au)du
+(A.4)
+Let z(au) = z′(u), h(au) = h′(u)
+(A.5)
+=⇒ x(λ′
+i) = a
+�
+u
+z′(u)h′(λ − u)du
+= a z′(λ) ∗ h′(λ)
+= a z(aλ) ∗ h(aλ)
+= a z(λ′
+i) ∗ h(λ′
+i)
+(A.6)
+The same could be shown in a rather shorter way using
+the Fourier transform. However, to avoid confusion between the
+terms frequency and wavelength used in two different domains,
+here we use the direct expansion of the convolution operator.
+Article number, page 10 of 10
+

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+Observation of anisotropic superfluid density in an artificial crystal
+J. Tao,∗ M. Zhao,∗ and I. B. Spielman
+Joint Quantum Institute, University of Maryland and National Institute
+of Standards and Technology, College Park, Maryland 20742, USA
+(Dated: January 4, 2023)
+We experimentally and theoretically investigate the anisotropic speed of sound of an atomic
+superfluid (SF) Bose-Einstein condensate in a 1D optical lattice. Because the speed of sound derives
+from the SF density, this implies that the SF density is itself anisotropic. We find that the speed
+of sound is decreased by the optical lattice, and the SF density is concomitantly reduced. This
+reduction is accompanied by the appearance of a normal fluid in the purely Bose condensed phase.
+The reduction in SF density—first predicted [A. J. Leggett, Phys.
+Rev.
+Lett.
+25 1543–1546
+(1970)] in the context of supersolidity—results from the coexistence of superfluidity and density
+modulations, but is agnostic about the origin of the modulations. We additionally measure the
+moment of inertia of the system in a scissors mode experiment, demonstrating the existence of
+rotational flow. As such we shed light on some supersolid properties using imposed, rather than
+spontaneously formed, density-order.
+Superfluidity and Bose-Einstein condensation (BEC)
+are deeply connected.
+In dilute atomic BECs, the
+superfluid (SF) and condensate densities are generally
+equal [1, 2]. By contrast, SF 4He can be a nearly pure SF,
+with only about 14 % condensate fraction [3], and infinite
+2D Berezinskii–Kosterlitz–Thouless (BKT) SFs have no
+condensate at all [4, 5]. In 1970 Tony Leggett showed
+that supersolids—systems spontaneously forming both
+SF and crystalline order (i.e., density modulations)—
+exhibit the reverse behavior: SF density far below the
+condensate density [6].
+Here we observe this effect in
+a nearly pure atomic BEC with artificial crystal order
+imprinted by an optical lattice.
+The
+complex-valued
+order
+parameter
+φ(r)
+=
+�
+ρsf exp[iϕ(r)],
+describing a SF with number den-
+sity ρsf and phase ϕ(r), gives rise to two hallmark
+SF properties: dissipationless supercurrents associated
+with spatial gradients in ϕ(r) and (Bogoliubov [2])
+sound described by traveling waves in ϕ(r).
+Because
+supercurrents arise from phase gradients, they are locally
+irrotational; in liquid 4He, the resulting non-classical
+rotational inertia appears below the SF transition
+temperature Tc.
+Supersolids are more exotic systems
+spontaneously forming crystalline order while exhibiting
+SF transport properties.
+Recent experiments with
+dipolar BECs of Dy and Er are suggestive of these
+properties [7, 8].
+Leggett argued that the modulated
+density ρ(r) of a supersolid leads to an unavoidable
+reduction in ρsf, and derived an upper bound for ρsf [6].
+This reduction results from the 3D density distribution,
+and as such is masked in tight binding descriptions such
+as the Bose-Hubbard model, which makes the unrelated
+prediction of vanishing ρsf at the superfluid to Mott
+insulator transition [9, 10].
+We created an artificial SF crystal by imprinting pe-
+riodic density modulations into an atomic BEC using a
+1D optical lattice as in Fig. 1(a).
+While these modu-
+lations do not form spontaneously, Leggett’s result still
+applies, making this an ideal system for understanding
+crystalline SFs without the added complexity of spon-
+taneously broken symmetries. We experimentally mea-
+sured an anisotropic speed of sound via Bragg spec-
+troscopy of the phonon mode. This implies the existence
+of an effective anisotropic superfluid density—which can
+be expressed as a second rank tensor ρsf
+ij—and we find
+that it saturates Leggett’s bound, in agreement with
+Gross-Pitaveskii equation (GPE) simulations.
+We also
+determined an associated anisotropic suppression of the
+moment of inertia in terms of the scissor-mode frequen-
+cies [11, 12].
+Anisotropic superfluids—Here we consider pure 3D
+BECs whose condensate mode ψ(r) = |ψ(r)| exp[iϑ(r)] is
+well described by the Gross-Pitaveskii equation (GPE).
+An optical lattice potential V (r) = (U0/2) cos(2krx) peri-
+odically modulates the condensate density ρ(r) = |ψ(r)|2
+with unit cell (UC) size a = π/kr [Fig. 1(b)-i]. By con-
+trast, the SF order parameter φ(r) is a coarse grained
+quantity describing system properties on a scale ≫ a,
+giving the nominally uniform density in Fig. 1(c)-i.
+Even disregarding potential differences in ρsf(r) and
+ρ(r), we argue that φ(r) is not simply equal to ψ(r) av-
+eraged over some scale large compared to a. The fun-
+damental origin of this effect can be understood by con-
+sidering a 1D system of size L with periodic boundary
+conditions in which both the condensate phase ϑ and SF
+phase ϕ wind by an integer multiple N of 2π [Fig. 1(b,c)-
+ii], yielding a metastable quantized supercurrent [13].
+To satisfy the steady-state continuity equation, the mi-
+croscopic current J(x) = ρ(x) [ℏ∂xϑ(x)/m] must be in-
+dependent of x [Fig. 1(b)-ii], however, the periodically
+modulated density ρ(x) > 0 implies the local velocity
+v(x) = ℏ∂xϑ(x)/m has oscillatory structure and conse-
+quently ϑ(x) follows a staircase pattern [Fig. 1(b)-iii, iv].
+From macroscopic considerations the superfluid cur-
+rent is J = ρsf [ℏ∂xϕ(x)/m] = 2πNℏρsf/(mL). Equating
+the currents obtained from considering the condensate
+arXiv:2301.01258v1  [physics.atom-ph]  3 Jan 2023
+
+2
+(a)
+−1
+0
+1
+(b) BEC
+−1
+0
+1
+(c) SF
+Position x/a
+0.0
+2.5
+ρ/¯ρ
+i.
+0
+1
+2
+3
+J (arb.)
+ii.
+0
+1
+ϑ/(2π)
+iii.
+0
+5
+v/¯v
+iv.
+ρsf/¯ρ
+i.
+J (arb.)
+ii.
+ϕ/(2π)
+iii.
+v/¯v
+iv.
+FIG. 1.
+Concept. (a) A BEC is confined in a harmonic trap
+superimposed with a 1D optical lattice (along ex, green), spa-
+tially modulating the condensate density (red). The dashed
+and dotted lines call out a region of nominally constant mean
+density and the left and right columns indicate the (b) state
+of the condensate and (c) SF in the presence of a current.
+These were computed for a 5Er deep lattice and plot: i. den-
+sity (red), ii. current (green), iii. phase (orange), and iv. local
+velocity (blue). The red dashed line plots the mean density
+¯ρ.
+mode and the SF order parameter and integrating over
+a unit cell yields Leggett’s equation [6]
+ρsf =a
+��
+UC
+dx
+ρ(x)
+�−1
+, as well as ϕ = 1
+a
+�
+UC
+ϑ(x)dx. (1)
+This implies that ρsf ≤ ¯ρ, where ¯ρ is the spatial average
+of the condensate density over a UC, and as we discuss
+below the remaining density ρn = ¯ρ − ρsf behaves as a
+pseudo-normal fluid. In the more general context where
+the GPE is inapplicable, the Leggett expression for ρsf
+is an upper bound for the SF density in systems with
+crystalline order [6].
+In a 3D system, the current Ji = ρsf
+ij [ℏ∂jϕ/m] derives
+from a SF density tensor. For systems with rectangular
+symmetry [14] ρsf
+ij is diagonal, and the analogs to Eq. (1)
+for each of the three elements use a 1D density integrated
+along the transverse directions. In our experiments this
+implies that the superfluid density is only reduced along
+the direction of the optical lattice, so ρsf
+yy = ρsf
+zz = ¯ρ.
+Experiment—We used 87Rb BECs with N ≈ 2 × 105
+atoms in the |F = 1, mF = 1⟩ hyperfine ground state. A
+1064 nm trapping laser with an elliptical cross-section,
+traveling along ex provided strong vertical confinement
+with frequency ωz/(2π) = 220 Hz; the in-plane frequen-
+cies, from ωx,y/(2π) = (34, 51) Hz to (56, 36) Hz, were
+optimized for our different experiments. We created a
+1D optical lattice using a retro-reflected λ = 532 nm
+laser traveling along ex, giving an a = 266 nm lattice pe-
+riod, comparable to the ξ = 170(20) nm minimum heal-
+ing length. The optical lattice was linearly ramped on
+in 100 ms to a final depth ≤ 10 Er, with single pho-
+ton recoil energy and momentum Er = ℏ2k2
+r /(2m), and
+ℏkr = 2πℏ/λ respectively [15]. For Bragg experiments
+the final state was measured using resonant absorption
+imaging after a 15 ms time of flight (TOF); scissors mode
+measurements were performed in-situ using partial trans-
+fer absorption imaging [16].
+Anisotropic speed of sound—The speed of sound for di-
+agonal ρsf
+ij is predicted to result from c2
+i = f sf
+ii /(κm) in
+terms of the superfluid fractions f sf
+ii = ρsf
+ij/¯ρ, the com-
+pressibility κ = ¯ρ−1 (∂¯ρ/∂µ), and the chemical potential
+µ. This reduces to the well-known value c2 = µ/m for
+an isotropic homogeneous system (See [17] for the full
+dispersion beyond the linear approximation). The sound
+speed ratio
+c2
+x
+c2y
+= ρsf
+xx
+ρsf
+yy
+= f sf
+xx,
+(2)
+provides direct access to the different components of the
+superfluid density [see [17] for a Josephson sum rule [18]
+argument]. Because the density is y-independent, Eq. (1)
+implies ρsf
+yy = ¯ρ.
+We Bragg-scattered the BEC off a weak sinusoidal po-
+tential with reciprocal lattice vector δk slowly moving
+with velocity v by patterning a laser beam with a dig-
+ital micro-mirror device (DMD [19]) and measured the
+scattered fraction p. This results from what are effec-
+tively two interfering laser beams driving two-photon
+transitions with difference-wavevector δk and angular fre-
+quency δω = δk v. We applied this potential for ≈ 5 ms.
+Bragg transitions ensued when the difference energy and
+momentum were resonant with the BEC’s Bogoliubov
+dispersion, and Fig. 2(a) shows data in the linear regime.
+The width of this spectral feature is limited by our BEC’s
+inhomogeneous density profile; the resonance (vertical
+dashed line) obtained from a Lorentzian fit (solid curve)
+therefore reflects an average speed of sound [20].
+A series of such fits lead to phonon dispersion relations
+with Bragg-lattice period from 2.25 µm to 8.5 µm. Repre-
+sentative dispersions taken along ex and ey are shown in
+Fig. 2(b), and we obtain the phonon speed of sound using
+linear fits. Figure 2(c) summarizes these data showing
+the speed of sound decreasing along the lattice direction
+ex, but slightly increasing along ey (resulting from the
+increased atomic density in the individual lattice sites).
+Finally Fig. 2(d) shows our main result: the normalized
+superfluid density obtained from these data using Eq. (2)
+decreases as a function of U0.
+We compared these data to GPE simulations in two
+ways, we:
+(1) used the Bogoliubov-de Gennes (BdG)
+equations to obtain cx and cy and (2) directly evaluated
+Eq. (1) from the GPE ground state density. The solid
+curves in Fig. 2(c) plot the sound speed obtained from
+
+3
+0
+500
+1000
+δω/2π (Hz)
+0.0
+0.1
+0.2
+0.3
+p
+(a)
+0.0
+0.2
+0.4
+δk/2π (µm−1)
+0
+200
+400
+600
+800
+δω/2π (Hz)
+(b)
+0
+2
+4
+6
+8
+10
+U0/Er
+0
+1
+2
+3
+c (mm/s)
+cx
+cy
+(c)
+0
+2
+4
+6
+8
+10
+U0/Er
+0.00
+0.25
+0.50
+0.75
+1.00
+ρsf
+xx/¯ρ
+(d)
+FIG. 2.
+Bragg spectroscopy. Black and red symbols mark excitations created along ex and ey respectively. (a) Transferred
+population fraction p as a function of frequency difference δω with wavevetor δk/2π = 0.26 µm−1 and lattice depth U0 = 5.7Er.
+The solid curve is a Lorentzian fit giving the resonance frequency marked by the vertical dashed line. (b) Phonon dispersion
+obtained from Bragg spectra. The bold symbols resulted from (a) and the linear fit (with zero intercept) gives the speed of
+sound. (c) Anisotropic speed of sound. The bold symbols are derived from (b) and the solid curves are from BdG simulations
+(no free parameters [17]). (d) SF density obtained from speed of sound measurements (blue markers, error bars mark single-
+sigma statistical uncertainties). We compare with two models: the red dashed curve plots a homogeneous gas BdG calculation,
+and the solid black curve plots the result of Eq. (1). The simulations used our calibrated experimental parameters.
+solving the 1D BdG [21], and the red dashed curve in (d)
+is the ratio of these speeds. To compare with Leggett’s
+prediction, we found the ground state of the 2D GPE
+for our experimental parameters and evaluated Eq. (1)
+throughout our inhomogeneous system. The black curve
+in Fig. 2 plots the resulting weighted average. Remark-
+ably the BdG results are in near-perfect agreement with
+Leggett’s expression.
+Scissors mode—The single-valued nature of the SF or-
+der parameter greatly affects rotational properties such
+as the moment of inertia I. For highly anisotropic traps,
+the scissors mode [11] describes a fixed density distribu-
+tion pivoting by a small angle θ about the trap center
+with frequency ωsc. Scissors mode experiments are remi-
+niscent of torsional balance experiments in 4He [22] which
+give access to the non-classical rotational inertia [6].
+It is suggestive to quantify these dynamics in terms of
+the Lagrangian L = I ˙θ2/2 − V (θ), for moment of inertia
+0
+0.25 0.5 0.75
+1
+fsf
+xx
+0.00
+0.25
+0.50
+0.75
+1.00
+ωsc/ωsc,0
+(a)
+0
+2
+4
+6
+8 10
+U0/Er
+20
+40
+60
+ωd/2π Hz
+ωx,d
+ωy,d
+0
+0.25 0.5 0.75
+1
+fsf
+xx
+−0.2
+0.0
+0.2
+0.4
+I/Ic
+(54, 36) Hz
+(36, 50) Hz
+(b)
+FIG. 3.
+Moment of inertia from scissors mode.
+(a-inset)
+Measured dipole mode frequencies (markers) along with fits
+(curves) where the frequency at U0 is the only free param-
+eter for each curve. (a) Scissors mode frequency. Blue and
+green correspond to U = 0 trap frequencies (34, 51) Hz and
+(54, 36) Hz respectively. (b) Moment of inertia in units of Ic.
+Symbols are the data computed as described in the text, and
+the solid curves are GPE predictions using I = ∂Ω⟨Lz⟩, with
+angular frequency Ω.
+I and potential energy V (θ). For small θ the potential
+can be expanded as V (θ) ≈ Iω2
+scθ2/2 with
+I
+Ic
+=
+(ω2
+x − ω2
+y)2
+ω2sc(ω2x + ω2y)
+(3)
+in terms of the classical moment of inertia Ic and in agree-
+ment with Ref. [11] for isotropic superfluids. Although
+this interpretation is highly intuitive, it does not survive
+careful consideration.
+The anisotropic superfluid den-
+sity couples radial and azimuthal flow and as a result
+a single parameter Lagrangian is insufficient to describe
+rotational dynamics.
+Instead the superfluid hydrodynamic equations pre-
+dict a moment of inertia scaled by a factor of (f sf
+xxω2
+x −
+f sf
+yyω2
+y)/(ω2
+x − ω2
+y) (see [17]) compared to Eq. (3). There-
+fore we expect ωsc, in conjunction with the superfluid
+density will give I/Ic as a function of lattice depth.
+The inset to Fig. 3(a) plots the dipole mode frequen-
+cies ωx,d and ωy,d for a trap with frequencies (54, 36) Hz.
+The frequency reduction is also related to ρsf via f sf =
+(ωx,d/ωx)2 along the lattice direction [17].
+This ratio
+can also be expressed in terms of an increased effective
+mass that converges to the predictions of single-particle
+band structure [23] when the lattice period falls below
+the healing length; in our case the value computed per-
+turbatively from the GPE differs by about 20 % from the
+band structure prediction. The result of this modeling is
+shown by the solid curves.
+We excited the scissors mode using our DMD to tilt the
+harmonic potential by 50 to 140 mrad for ≈ 1 ms (shorter
+than the trap periods) and let the BEC evolve in the orig-
+inal trap for a variable time. We measured the resulting
+dynamics in-situ and extracted the angle by fitting the re-
+sulting density profile to a rotated Gaussian. Figure 3(a)
+shows scissor mode frequency normalized to the expected
+frequency [24] of ω2
+sc,0 = f sf
+xxω2
+x + f sf
+yyω2
+y for a trap elon-
+gated either along ex [with frequencies (56, 36) Hz, blue]
+
+4
+I/Ic
+I/Ic
+−20
+0
+20
+x (µm)
+−20
+0
+20
+y (µm)
+(a)
+−20
+0
+20
+x (µm)
+−20
+0
+20
+y (µm)
+(b)
+-1.0
+-0.5
+0.0
+0.5
+1.0
+Lz(r) (arb. units)
+−0.3
+0.0
+0.3
+(c)
+0
+0.25 0.5 0.75
+1
+fsf
+xx
+0.0
+0.3
+0.6
+0
+1
+(d)
+0
+0.25 0.5 0.75
+1
+fsf
+xx
+0
+1
+FIG. 4.
+Moment of inertia in rotating systems computed
+using 2D GPE simulations. The left column indicates sim-
+ulations in which the lattice is static while in the right col-
+umn the lattice co-rotates with the confining potential. (a,b)
+Angular momentum density for trap frequencies 2π×(56,36)
+and U0 = 10Er. (c, d) Total momentum of inertia in traps
+with frequencies 2π×(56,36) (top, green) and 2π ×(36, 56) Hz
+(bottom, blue). Dashed curves plot Isf/Ic and the solid curve
+plots I/Ic.
+or along ey [with frequencies (36, 50) Hz, green]. In both
+cases ωsc is about 5 % in excess of the simple predic-
+tion, perhaps from finite temperature or anharmonicities
+in the ODT.
+We combine these observations in Fig. 3(b) to ob-
+tain I/Ic; the data (symbols) and our 2D GPE simu-
+lations (curves) are in agreement. For traps elongated
+along ex (green) I/Ic unexpectedly changes sign when
+ωx,d = ωy,d. To understand the physical origin of this
+effect we now turn our attention to rotating systems.
+Rotation—Thus far we focused exclusively on the su-
+perfluid density, while avoiding questions about any as-
+sociated normal fluid. We can deduce the existence of a
+normal fluid component by considering two thought ex-
+periments in a 1D ring geometry (with radius R) and
+quantify both in terms of the resulting angular momen-
+tum [25]. In case (i), we consider an Aharonov-Bohm ge-
+ometry and slowly thread the ring with a single quanta
+of magnetic flux (see Ref. [26] for an artificial gauge field
+proposal). The process is equivalent to imprinting a 2π
+phase winding (of the type discussed on page 1), giving
+angular velocity Ω = ℏ/(mR2) and angular momentum
+Lz/ℏ = 2πRρsf. In case (ii), we consider a complimen-
+tary experiment in which the lattice is very slowly accel-
+erated to a final angular velocity Ω; this is best under-
+stood by transforming into the frame co-rotating with
+the lattice. This leads to a lab frame angular momentum
+Lz/ℏ = 2πR(¯ρ−ρsf) which we interpret as resulting from
+the normal fluid co-moving with the lattice.
+With this insight we extended our 2D numerical sim-
+ulations to analogous cases for rotating harmonically
+trapped systems where : (i) the lattice is static in the
+lab frame (as in scissors mode experiments) or (ii) it co-
+rotates with the confining potential. In both cases we
+use the coarse graining defined in Eq. (1) to obtain the
+superfluid density and phase. In this way we compute
+the total moment of inertia I from ψ(r, t), the superfluid
+component Isf from φ(r, t), and we define the normal
+component as the difference In = I − Isf.
+Case (i): as in our 1D thought experiment only the SF
+component responds. Then although ∇ϕ is manifestly
+irrotational, because ρsf
+xx ̸= ρsf
+yy the superfluid current
+can be rotational. In this case, the relative magnitude
+of the co- and counter-rotating contributions vary with
+the lattice depth, leading to regions of negative angular
+momentum density L(r) along the BEC’s semi-minor axis
+[Fig. 4(a)]. The superfluid moment of inertia computed
+from these simulations [Fig. 4(c)] is in full agreement with
+the scissor mode simulation, and as expected for a static
+lattice Isf = I (no normal flow).
+When the lattice is along the semi-minor axis, as pic-
+tured in (a) and the green curve in (c), the counter-
+rotating contribution increases with U0, until the dipole
+mode frequencies along ex and ey invert, after which
+point, I/Ic becomes negative.
+The reverse is the case
+when the lattice is along the semi-major axis and I/Ic
+increases monotonically. This novel observation confirms
+the negative kinetic energy resulting from ˙θ.
+Case (ii): In contrast, the angular momentum density
+is strictly positive [Fig. 4(b)] for both lattice orientations
+and I/Ic increases with lattice depth [Fig. 4(d)]. In this
+case the normal fluid to co-rotate with the trap giving
+the current Jn = (−ρn
+xxy, ρn
+yyx) ˙θ. The total I/Ic is then
+the sum of the superfluid [17] and normal contribution
+I
+Ic
+=
+(f sf
+xxω2
+x − f sf
+yyω2
+y)2
+(f sf
+xxω2x + f sf
+yyω2y)(ω2x + ω2y) + f n
+xxω2
+x + f n
+yyω2
+y
+ω2x + ω2y
+. (4)
+This result, along with our 2D GPE simulations, are
+plotted in Fig. 4(d). The dashed curve plots the super-
+fluid contribution to Isf/Ic in agreement with the coarse-
+grained GPE (crosses). The solid curve and the triangles
+plot the corresponding total moment of inertia, in excess
+of the SF contribution. This implies the appearance of
+normal fluid flow.
+This agreement confirms that the superfluid contribu-
+tion derives from gradients of the coarse-grained phase
+ϕ, while the normal contribution stems from variations
+of ϑ within each lattice site.
+Discussion and outlook—Our inability to obtain I/Ic
+from scissors mode measurements without detailed mod-
+eling reinforces similar conclusions in dipolar gases [27].
+In both cases the simple argument fails because ˙θ cou-
+ples to more internal degrees of freedom than Lz alone.
+In this context Ref. [27] concluded that the scissors mode
+
+5
+does yield the moment of inertia when 1D density mod-
+ulations comove with the oscillatory motion: this is con-
+sistent with our findings comparing motion in static and
+rotating lattices. Our GPE simulations indicate that the
+analytical relations generalize to lattices with period in
+excess of the healing length.
+Although we conclude that a normal fluid exists, it is
+inseparable from the optical lattice and lacks any internal
+dynamics of its own, i.e., it is not described by a dynami-
+cal equation of motion. In contrast, both the superstripe
+phase in spin-orbit coupled BECs [28–32] and supersolid
+phases of dipolar gases [33–35], support dynamical den-
+sity modulations. Leggett’s expression applies to both
+of these systems implying a reduced superfluid density,
+which in this case could exhibit dynamics, as expected
+for a system described by a two-fluid model [11, 12].
+This leaves open questions regarding nature the nor-
+mal fluid of spin-orbit coupled systems where an interplay
+between single-particle physics and interactions govern
+supersolid-like properties.
+In addition, ρsf is expected
+to be reduced outside of the superstripe phase [31, 32]
+where the density is uniform (making Leggett’s expres-
+sion inapplicable), but the BEC’s spin vector is spatially
+periodic.
+Note: During the early stages of manuscript prepara-
+tion we become aware of a related work, using a long
+period 1D lattice applied to a homogeneously confined
+2D BEC.
+The authors thank S. Stringari for suggesting this line
+of investigation and to both S. Stringari and S. Roccuzzo
+for stimulating discussions. In addition W. D. Phillips
+and S. Mukherjee carefully read the manuscript. This
+work was partially supported by the National Institute
+of Standards and Technology, and the National Science
+Foundation through the Physics Frontier Center at the
+Joint Quantum Institute (PHY-1430094) and the Quan-
+tum Leap Challenge Institute for Robust Quantum Sim-
+ulation (OMA-2120757).
+∗ These two authors contributed equally
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+Improvement of Computational Performance of Evolutionary AutoML in a Heterogeneous
+Environment
+Nikolay O. Nikitin, Sergey Teryoshkin, Valerii Pokrovskii, Sergey Pakulin, Denis Nasonov
+ITMO University, Saint-Petersburg, Russia
+Abstract
+Resource-intensive computations are a major factor that limits the effectiveness of automated machine learning solutions. In the
+paper, we propose a modular approach that can be used to increase the quality of evolutionary optimization for modelling pipelines
+with a graph-based structure. It consists of several stages - parallelization, caching and evaluation. Heterogeneous and remote
+resources can be involved in the evaluation stage. The conducted experiments confirm the correctness and effectiveness of the
+proposed approach. The implemented algorithms are available as a part of the open-source framework FEDOT.
+Keywords: AutoML, heterogeneous infrastructure, evolutionary optimization, caching
+1. Introduction
+Nowadays,
+automated machine learning (AutoML) is
+widely used in science, and industry [16, 33]. The major prob-
+lem of solving real-world tasks with AutoML is the high com-
+putational cost of the search for an optimal modelling pipeline.
+During the evaluation of the candidate pipelines’ quality, many
+machine learning models are trained. This task is very resource-
+intensive, so it can take a considerable amount of time to
+achieve the appropriate result. It can be considered a bottle-
+neck for any existing AutoML solution. This issue raises vari-
+ous problems from different fields: from integration of AutoML
+to business processes [31] to carbon emission and sustainability
+concerns[35].
+There are many approaches for improving computational
+performance that are used in state-of-the-art (SOTA) AutoML
+solutions [8]. First of all, almost all solutions support parallel
+execution. Some of them also support caching of evaluated can-
+didates [22]. Also, the graphics processing unit can be used to
+reduce the training time [15].
+There is a variety of open-source tools that could improve
+the efficiency of certain steps of machine learning pipelines.
+For instance, involving various MLOps tools like MLFlow [38],
+task-specific databases [37] and scaling tools like Ray [18] al-
+lows the effectiveness of ML applications to be notably in-
+creased.
+However, the optimal design of the computational strategy
+depends on the infrastructure and the underlying AutoML al-
+gorithm. The SOTA AutoML solutions are based on differ-
+ent optimization methods: random search, Bayesian optimiza-
+tion, genetic algorithms, and meta-learning [8, 2]. The struc-
+tural patterns used in modelling pipelines can also be different:
+linear pipelines or ensembling techniques (stacking, blending,
+Email address: [email protected] (Nikolay O. Nikitin)
+and boosting) [39]. The most complicated case is a composite
+pipeline represented as a directed acyclic graph [21]. At the
+same time, there is no ready-to-use solution for improving the
+computational performance of an automated open-ended search
+for pipelines in the composite AI field.
+In the paper, we want to propose a adaptive approach
+to reduce the computational cost of AutoML for compos-
+ite pipelines.
+Several techniques are implemented: pipeline
+caching, parallelization of the fitness function evaluation, com-
+putation with hybrid (GPU and CPU) systems, and integration
+with remote distributed systems.
+This approach differs from existing solutions since it can be
+configured for automated machine learning in various computa-
+tional environments (including distributed and heterogeneous).
+Also, the caching procedure can be effectively used for vari-
+ous pipeline designs (linear, weighted ensembles, multi-layer
+ensembles, etc).
+To confirm the effectiveness of the proposed approach in
+empirical way, we conducted a set of numerical experiments us-
+ing set of open datasets of various sizes (described in Table 1).
+The results presented in Section 6 allow us to conclude that a
+larger number of pipelines can be evaluated and better qual-
+ity metrics can be achieved by AutoML using this approach.
+The software implementation is available in the open-source
+AutoML framework FEDOT.
+The paper is organized as follows: Section 2 describes the
+computational strategies used in state-of-the-art AutoML tools.
+Section 3 provides the problem statement for AutoML perfor-
+mance improvement. Section 4 proposes a set of novel im-
+provements for the composite evolutionary AutoML. Section 5
+describes the software implementation of these techniques in an
+open-source framework. Section 6 provides the experimental
+evaluation of the proposed techniques for different case studies.
+Finally, Sec. 7 provides an analysis of the obtained results and
+possible extensions of the research.
+Preprint submitted to Algorithms
+January 13, 2023
+arXiv:2301.05102v1  [cs.LG]  12 Jan 2023
+
+2. Related Works
+There are dozens of open-source AutoML solutions that can
+be used for designing modelling pipelines. The first frame-
+works that became well-known are H2O [15], TPOT [14] and
+Auto-sklearn [4].
+As more novel AutoML solutions, Auto-
+Gluon [3] and LAMA [36] can be noted. Also, there are a lot
+of other AutoML tools with various specific features [17].
+There are different strategies for performance improve-
+ment used in the noted frameworks. In the TPOT framework,
+pipeline caching is implemented [14].
+TPOT-SH [26] uses
+the concept of Successive Halving to explore the search space
+faster, especially for larger datasets.
+Various techniques are
+used to evaluate the pipelines on different subsets of training
+data (e.g. layering [6]).
+A widely-used parallelization tool is the joblib library im-
+plemented in Python. However, there are more advanced frame-
+works for parallelization that can be noted. For example, Ray
+[18] can be used to scale AI and Python applications in dis-
+tributed environments. It provides various instruments for dis-
+tributed data preprocessing, distributed training of ML models
+and scalable hyperparameter tuning.
+Improving the computational performance for evolutionary
+algorithms outside AutoML is also discussed in the literature.
+As an example, parallel GPU-based evaluation of the fitness
+function can be used [24] to solve the expensive problems re-
+lated to big data [23]. There are various techniques that im-
+prove the performance of evolutionary algorithms in concurrent
+mode [7]. The tensor-based computational model can be used
+to achieve cross-platform hardware acceleration [12].
+Also,
+platform-specific open-source solutions are presented in this
+field (e.g. scikit-learn-intelex 1).
+One of the widely used techniques to avoid fitness evalua-
+tion bottlenecks in evolutionary algorithms is caching [29]. Fi-
+nal values of the fitness evaluation can be cached [13] as well
+as partial results [30, 11, 10].
+Moreover, a number of solutions exist that can perform
+remote/distributed training (e.g.
+Auto-sklearn, H2O, TPOT,
+LAMA). These AutoML frameworks use different frameworks
+for distributed computing. Autosklearn and TPOT use Dask2,
+LAMA uses Apache Spark3. H2O uses its own Apache Spark
+modification called Sparkling Water4. Distributed computing
+frameworks allow the processing of large datasets spread over
+the nodes of a cluster system.
+We can conclude that there is a large number of techniques
+and solutions that can reduce the resource consumption for Au-
+toML and EA. However, there is still no well-developed ap-
+proach that can be used to identify graph-based pipelines in
+the heterogeneous computational environment in composite AI
+problems. For this reason, we decided to formulate the problem
+statement specific to composite AutoML and propose possible
+solutions.
+1https://github.com/intel/scikit-learn-intelex
+2https://dask.org
+3https://spark.apache.org
+4https://h2o.ai/products/h2o-sparkling-water
+3. Problem Statement
+We want to design multi-task and multi-modal pipelines for
+various tasks using a single flexible instrument. Consequently,
+it becomes necessary to implement the framework’s architec-
+ture more abstractly to separate the pipeline search process
+from the top-level API. The modelling pipeline is represented
+as a directed acyclic graph in this case. Each node (modelling
+or data transformation operation) is described by the operation’s
+name and set of hyper-parameters. If necessary, different data
+sources (tables, time series, images, texts) can be involved in
+the pipeline. Also, metadata is attached to the data flow, making
+it possible to change the task several times during the pipeline
+evaluation (e.g., solve a classification task and then - a regres-
+sion task).
+The drawback of this approach is the increased search space
+that should be explored during the optimization.
+In auto-
+mated modelling, we want to control the balance between open-
+endedness [25] and local search. The simplest way is to apply
+the of direct constraints (e.g. limit to the pipeline size). Also, it
+can be more effective to apply the regularization and sensitivity
+analysis procedures [21] and adaptive optimisation strategies
+to control the convergence of optimisation. At the same time,
+avoiding over-complicated pipelines and reckless spending of a
+limited time budget is also essential. It makes the effectiveness
+of the computational part even more critical for open-ended Au-
+toML.
+It pushes us to compromise between pipeline complexity
+and training time. However, if we can improve computing ef-
+ficiency, the framework will probably be able to build more
+complicated models with higher quality while consuming the
+same training time.
+There are many approaches to improv-
+ing evolutionary algorithms’ computing performance, such as
+parallelization, caching, etc. These approaches can be divided
+into single-machine optimizations and horizontal scaling tech-
+niques. Single-machine optimizations aim to improve comput-
+ing performance only on the machines performing the compu-
+tations. Horizontal scaling allows involving additional servers
+to speed up computing. Both techniques can be used separately
+or combined.
+Evolutionary algorithms’ computational time mainly de-
+pends on the population size and the number of generations.
+Increasing population size leads to an increased probability of
+getting better individuals. A more significant number of gener-
+ations means more attempts to grow better individuals based on
+the best previous generation.
+From a computational point of view, we have several iter-
+ations, each of which requires the results from the previous it-
+eration. So, it is complicated to scale computations over the
+iterations, and the total computation time is Equation 1.
+Ttotal =
+n
+�
+i=1
+Ti
+(1)
+where n is the number of iterations and Ti - the computa-
+tion time of generation i. Inside one generation, all individuals
+are processed independently from one another, allowing us to
+2
+
+scale these computations according to the available computa-
+tional resources. The population training time can be estimated
+with Equation 2.
+Ttotal =
+n
+�
+i=1
+argmax
+di
+j∈Di
+(argmin(τ(di
+j,
+rj∈Ri/{rj−1,..,r1}
+ω(Di−1,..,D1), rj))
+(2)
+where Di is the set of individuals in the population i and di
+j is
+individual j, Ri is the set of available resources on iteration i,
+ω is a cache function with pre-calculated elements on iterations
+i − 1, i − 2, ..., 1, and τ is a function that returns the calculation
+time considering caching and evaluation of individuals Di.
+Due to the Equations 1 and 2, we should increase the popu-
+lation size as much as possible. It allows to speed up the algo-
+rithm convergence and improve its result using horizontal scal-
+ing. For this purpose, we can use both remote computing on
+production servers and distributed computing using homoge-
+neous and heterogeneous computing clusters.
+Remote computing allows model training to be delegated to
+a remote infrastructure. This approach is justified if the local
+machine computational resources are insufficient to train a set
+of models in a reasonable amount of time. Remote computing
+may take place on a dedicated computing server or a cluster of
+servers that accepts tasks to train models using REST API, RPC
+or message queues.
+The main challenge in the investigated problem is to pro-
+pose the performance improvement strategy for AutoML that
+is adaptive to various types of computational infrastructures.
+In Figure 1, five classes are noted: shared memory system,
+multi-node cluster with distributed memory, complex homo-
+geneous and heterogeneous supercomputer environments, and
+hybrid systems. The system with structure (a) can execute par-
+allel tasks in a straightforward way. In the systems (b)-(e), the
+remote nodes are involved (homogeneous and heterogeneous).
+For the system (e), the structure is hybrid since various remote
+nodes have different computational performance and connec-
+tions overheads. For this reason, the adaptability of the compu-
+tational strategy is especially important.
+There are various ways can be uses to adapt the compu-
+tational strategy to specific infrastructure.
+For example, the
+empirical performance models [9] can be used to choose the
+optimal infrastructure for evaluating specific pipelines. Sim-
+ple pipelines with low fitting time can be assigned to low-
+performance computational nodes.
+Otherwise, complicated
+pipelines with high fitting time can be assigned to high-
+performance nodes. It makes it necessary to develop a modular
+approach that effectively utilizes all available resources.
+Our main motivation is to develop an approach that can be
+used at the computational layer of AutoML. It should be pos-
+sible to adapt this layer to the specified infrastructure (local or
+remote) in a frame of the same AutoML approach. This solution
+should be high-level, modular and flexible to allow integrating
+it with different AutoML tools. Also, it should support the dif-
+ferent types of pipelines (from simplest linear pipeline to the
+multi-level ensembles).
+4. Proposed Improvements
+This section is devoted to various aspects of the proposed
+approach for improving the computational performance of evo-
+lutionary AutoML. The high-level scheme of the approach is
+presented in Figure 2. Four main aspects are considered: (1)
+parallelization of the fitness function evaluation; (2) partial
+caching of evaluated individuals; (3) combining CPU and GPU
+to accelerate the processing of individuals (4) integration with
+remote infrastructure for a complex task. Algorithmic-based
+improvements (e.g.
+surrogate-assisted optimization) are not
+considered here.
+The detailed implementation of the proposed approach is
+described in Alg. 1. In this notation, graph represents the struc-
+ture of the composite pipeline. The details of the evolutionary
+optimisation are hidden to make the proposed improvements
+more clear.
+Algorithm 1 High-level pseudocode of the evaluation dispatch-
+ing algorithm implemented in the proposed approach. Paral-
+lelization, caching and evaluation stages are demonstrated for
+processing one generation of the evolutionary algorithm.
+1: procedure ProcessPopulation
+2:
+Input:
+inds (set of non-evaluated individuals),
+objective (objective function that calculates the fitness
+of an individual),
+n (number of parallel jobs)
+timer (timer-like object)
+infrastructure (description of setup)
+3:
+Output: evaluated inds
+4:
+do in parallel(n)
+5:
+if timer.enough time( ) then
+6:
+graph ← inds[i].graph ▷ get structure of each ind.
+7:
+if infrastructure.is remote( ) then
+8:
+cache ← DistributedCache()
+9:
+sync cache
+▷ sync cache database
+10:
+task id ← create task(graph)
+11:
+wait task id
+12:
+inds[i].fitness ← request result(graph)
+13:
+else
+14:
+prepare graph
+▷ assign CPU and GPU to
+nodes
+15:
+cache ← LocalCache()
+16:
+load cache
+▷ init cache database
+17:
+if cache.exists(graph)( ) then
+18:
+fit from cache(graph)
+19:
+inds[i].fitness ← obj(graph, cache)
+20:
+fit(graph)
+▷ Fit nodes that are not in cache
+21:
+save cache
+▷ preserve updated cache
+22:
+if not inds[i].is valid then
+23:
+delete inds[i]
+▷ for unsuccessful evaluation
+24:
+else
+25:
+delete inds[i]
+▷ not enough time, skipping
+26:
+return inds
+▷ candidates for selection
+3
+
+Figure 1: Different types of computational infrastructures that can be used in AutoML: (a) shared memory (SM) system (b) multi-node cluster with distributed
+memory (c) complex homogeneous supercomputer system with spatially distributed infrastructure (d) supercomputer system with heterogeneous distributed infras-
+tructure.
+Figure 2: Workflow of the proposed approach for the improvement of compu-
+tational performance for composite AutoML
+4.1. Parallelization
+Parallelizing evolutionary algorithms is not a novel idea.
+There are a lot of papers and open-source solutions devoted
+to this problem. However, parallelization in AutoML has its
+specifics. For example, various computationally efficient strate-
+gies of parallel evolution can be used [9].
+We are considering an evolutionary algorithm for search-
+ing for the best solution in the space of pipelines that can be
+represented as directed acyclic graphs. The classic approach
+to parallelizing evolutionary optimization is evaluating all indi-
+viduals in the population concurrently [9]. It works because of
+the nature of the evolutionary algorithm. There are no depen-
+dencies between individuals in a generation. Other approaches
+suggest dividing populations into isolated parts [32] or using
+co-evolutionary algorithms to divide tasks into subtasks [5].
+The proposed algorithm considers the maximum evaluation
+time length for each pipeline evaluation to resolve the possi-
+ble evaluation time anomalies caused by the stochastic nature
+of data-driven model training. If the training process does not
+converge at least in one cross-validation fold, the time required
+for corresponding fitness evaluation can be increased signifi-
+cantly. So, the individuals that spend excess time on evaluation
+are skipped to preserve the overall performance of the evolu-
+tionary optimizer.
+4.2. Caching
+The existing caching approaches are aimed at preserving
+and reusing fitted pipelines [22]. However, separate nodes of
+composite pipelines can be cached individually [21]. It makes
+it possible to reuse the fitted models and reduce the fitness func-
+tion’s evaluation time. The optimizer can share the in-memory
+cache across the populations, and individuals [9]. However, it
+raises the problems of memory consumption.
+After analyzing existing solutions, we focused on the rela-
+tional database approach for pipeline caching. More specifi-
+cally, the sqlite3 library was used to implement it. First of all,
+it provides only one output file, which is not guaranteed for
+non-relational databases - e.g. shelve. Secondly, all concurrent
+save-load operations can be fully processed during the parallel
+evaluation of the fitness functions without direct usage of syn-
+chronization primitives, atomic variables and other instruments
+necessary for simultaneous access to data.
+Finally, this approach allows extracting several operations
+simultaneously, which helps to improve the overall perfor-
+mance of caching. Also, the set of operations can be saved
+to the database taking into account the existence of cache items
+with the same primary key.
+The caching procedure for the multi-layer ensemble
+pipelines should take into account that the cached model/-
+operations are suitable only for the specific configuration of
+previous nodes and edges in the modelling pipelines.
+So,
+the key contains the recursive description of the structure
+4
+
+a)Shared memory
+b) Cluster with
+distributed memory
+system (SM)
+c)Homogeneoussupercomputer
+environment
+Core
+Core
+Core
+Main
+1
+N
+Node
+Sheduler
+Node
+Node
+Node
+1
+N2
+Node
+Node
+Node
+1
+..
+N
+Node
+Node
+Node
+Node
+Node
+Node
+1
+N1
+d) Heterogeneous
+1
+N3
+supercomputerenvironment
+e) Hybrid environment
+Sheduler
+SM
+System
+Low-speedchannel
+High-speed channel
+Low-speed channel Mid speed channel High-speed channel
+SM
+Node
+Node
+Node
+SM
+Clust.
+Clust.
+Embed.
+Embed.
+Syst.
+1
+N
+Syst.
+system
+systemCandidate
+ML pipeline
+Population K
+0000
+N
+Individuals
+不
+Parallelization
+Evaluation
+Caching
+Local
+Remote
+CPU
+GPUof previous nodes and edges.
+Also, the identifier of cross-
+validation fold is specified. The following notation is used:
+(/[node name] [hparams];)/[node name] [hparams]...”, where
+/ denotes the beginning of the node name and round brackets
+represent the nested edges. The caching details are presented in
+Figure 3.
+Figure 3: Interaction between operations’ cache and the modelling pipeline that
+should be fitted
+4.3. GPU
+Evaluating ML models with GPUs is a well-developed fea-
+ture in many solutions. For example, the RAPIDS library [34]
+contains the CuML module that allows training classification,
+regression and clustering models with GPUs. To adapt this so-
+lution to composite pipelines, we should consider a setup in
+which only a part of the nodes can be evaluated with GPUs. In
+this situation, the pipeline should be fitted in a heterogeneous
+way.
+The proposed approach makes it possible to use both CPUs
+and GPUs for fitting by separating the ML model type and its
+implementation. The same model (e.g. random forest) can have
+several implementations (CPU-based and GPU-based).
+Figure 4 shows an example of a computationally hetero-
+geneous composite model structure. Data transfer between the
+GPU-based nodes (yellow) is performed within the video mem-
+ory, and the models themselves in the nodes are trained on
+graphics processing units (GPUs). Other nodes are executed
+on CPUs.
+Due to the multiple software limitations set by RAPIDS li-
+braries (CUDA-compatible GPU driver, restricted set of sup-
+ported operation systems), it is practical to conduct the compu-
+tations within Docker-based containers.
+4.4. Remote evaluation
+Remote evaluation can be integrated into the evolutionary
+optimiser in various ways. Both dataset folds and population
+parts can be distributed across several computational nodes to
+satisfy time or memory limits. Since evolutionary algorithms
+do not always require processing large datasets, we have fo-
+cused on the parallelism aspect of remote computing. The pro-
+posed implementation relies on Kubernetes. The REST API
+Figure 4: The structure of a pipeline that can be evaluated in a heterogeneous
+(CPU - blue and GPU - yellow) way
+service inside the Kubernetes cluster is used to run computa-
+tions via HTTP requests. The client implements a wrapper for
+requests.
+During the population training, the evaluator uses the
+client’s methods to process individuals on the Kubernetes clus-
+ter. Then, after starting processing all individuals, the evaluator
+waits for computations to be completed via client methods. The
+run request contains the container image, resources limit for the
+container, mount paths and model parameters. The REST API
+service creates the requested container and keeps monitoring it.
+The client uses requests to the REST API service to get actual
+containers’ statuses to see if it is still running, completed or
+failed.
+Finally, the client downloads the fitted pipeline when the
+training is completed. We wrap the result into a compressed
+archive to reduce the amount of data transferred over the net-
+work. Then, the files are sent to the client. This process scheme
+is presented in Figure 5.
+Figure 5: Communication between AutoML and remote cluster
+This way, we can divide the population training process into
+three stages: (1) requests to evaluate individuals, (2) computing
+and waiting for the completion and (3) fetching the results.
+5. Software Implementation
+The proposed approach can be used as a part of the archi-
+tecture that includes:
+5
+
+Preprocessor cache
+Nodes cache
+data_description 1: fitted preprocessor 1
+node uid 1: fitted node 1
+data description 2: fitted preprocessor 2
+node uid 2: fitted node 2
+Data fold 1
+preprocessor
+Node 1
+uid=node 4/
+evaluation
+(node1;node2;node3)
+Fitness
+Data
+Data fold 2
+Node 2
+Node 4
+Node 3
+Data fold 3
+Pipeline
+8
+Evo. opt
+88
+N
+N+1
+pop.
+pop.Decision
+Tree
+(GPU)
+SVC
+(GPU)
+Random
+Scaling
+Forest
+(GPU)
+Logit
+(CPU)
+GPU memory
+(CPU)
+XGBoost
+(CPU)LocalAutoML
+Kubernetes
+RemoteEvaluatol
+Client
+AutoML
+Create individual
+Individual-1
+create task
+RESTAPI
+AutoML
+Wait until ready
+get task status
+Service
+Individual-2
+AutoML
+Fetch individual
+download result
+Individual-N• The model repository block, which provides storage and
+selection of various implementations of predictive mod-
+els and data processing blocks. One model can contain
+several implementations (e.g., for CPU and GPU);
+• The block of the generative design of composite mod-
+els, which implements the creation of models with spec-
+ified properties by evolutionary algorithms. The proper-
+ties of the models are determined by the target function
+passed to the optimizer. If there is more than one tar-
+get function specified (as an example, the training time
+and modelling error can be used together as objectives
+for AutoML), then the multi-criteria formulation of the
+optimization problem is implemented, where the result
+of the model design is a Pareto front containing various
+compromising solutions. The genotype is represented in
+graph form, and the crossing and mutation operators are
+implemented accordingly.
+• The pipeline execution block on a given computational
+infrastructure. It allows individual pipeline execution on
+the given computational nodes.
+This architecture is implemented in the core of the open-
+source FEDOT framework. Different aspects of its implemen-
+tation are already detailed in a series of papers: [19] describes
+the main schemes and the implementation of the evolutionary
+operators, [28] is devoted to the multi-objective modification of
+this approach, and [20] provides an extended description of the
+various aspects of the evolutionary design for composite mod-
+elling pipelines. The tuning strategy of the pipeline hyperpa-
+rameters is based on Bayesian optimization.
+Custom models can be put inside this node. Search space
+for hyperparameters and initial approximations for the models
+should be specified manually if necessary. It makes it possi-
+ble to involve the infrastructure-specific implementation of the
+model in AutoML.
+The example below demonstrates the AutoML workflow
+from input data processing to obtaining prediction.
+api = Fedot(problem=’classification ’,
+seed =42, timeout =30, preset=’gpu’)
+api.fit(features=x_train , target=
+y_train)
+predictions = api.predict(features=
+x_test)
+Figure 6 provides the UML class diagram for the imple-
+mentation of various evaluation strategies that allow combining
+CPU- and GPU-based nodes in a single modelling pipeline. A
+high-level modelling method (e.g. Support Vector Classifica-
+tion) can be implemented using different algorithms: a CPU-
+optimised implementation of SVC can be obtained from the
+scikit-learn library [27]. In contrast, a GPU-optimised imple-
+mentation is available in the CUML library. The proposed ar-
+chitecture makes it possible to hide these details inside the spe-
+cific modelling pipeline and use the same optimisation logic for
+different implementations of the algorithms.
+Figure 6: The class diagram for the implementation of a modelling pipeline that
+consists of several operations. The Operation class represents a high-level mod-
+elling strategy that is used inside the operation. EvaluationStrategy is a base
+class for the algorithmic implementation of this strategy. SklearnEvaluation-
+Strategy represents the implementation obtained from the scikit-learn library
+and CumlEvaluationStrategy represents the implementation from the CUML
+library.
+The optimizer operates on individual models as a black box
+with input, output and fit/predict methods. The following build-
+ing blocks can be used for pipelines: models (Bernoulli Naive
+Bayes classifier, logistic regression, multilayer perceptron, ran-
+dom forest, gradient boosting, k-nearest classifier, QDA, LDA,
+decision tree) and data transformation operations (scaling, nor-
+malization, polynomial features transformation, principal com-
+ponent analysis, independent component analysis, isolation for-
+est, resampling).
+6. Experimental Studies
+We conducted a series of experiments to confirm the cor-
+rectness and effectiveness of the proposed approach.
+It can
+be divided into experiments with local and remote infrastruc-
+ture. As benchmarks, various classification datasets from the
+OpenML base [1] and synthetic datasets were used (the full list
+is presented in Table 1). A description of the computational
+infrastructure is provided for each experiment.
+The following methodology was used for experimental
+studies: each experiment started with dividing samples into two
+groups: ‘learning’ and ‘validation’ samples in the ratio 70% to
+30% to avoid data leaks. Then, the learning sample was trans-
+ferred to the evolutionary optimizer. During the optimisation,
+the 5-fold cross-validation procedure was applied to estimate
+the values of the fitness function.
+The experiment is repeated three times for each dataset to
+take the stochasticity of the optimizer into account. The quality
+metrics are averaged over these iterations.
+6.1. Local infrastructure
+For experiments with the local infrastructure, we configured
+a server based on Xeon Cascadelake (2900MHz) with 12 cores
+6
+
+EvaluationStrategy
+operation_type
+fito
+Operation
+predictO
+operation_type
+A
+strategy
+define_strategy0
+exectute_strategy0
+SklearnEvaluationStrategy
+operation_implementation
+fito
+predicto
+CUMLEvaluationStrategy
+Pipeline
+operation_implementation
+fito
+predict0Table 1: The properties of OpenML datasets that were used during the exper-
+iments. The random forest model is used as a baseline for the training time
+estimation.
+Dataset
+name
+Rows,
+10ˆ3
+Feat.
+Total
+elem.,
+10ˆ3
+Base.
+train.
+time,
+sec
+Num.
+of
+clas
+ses
+adult
+49
+14
+684
+12.5
+2
+amazon
+employee
+access
+33
+9
+295
+1.5
+2
+australian
+0.69
+15
+10
+0.2
+2
+bank-
+marketing
+45
+17
+769
+6.9
+2
+blood-
+transfusion
+0.75
+5
+4
+0.1
+2
+car
+1,8
+7
+12
+0.5
+4
+cnae-9
+1,1
+857
+926
+14.7
+9
+jungle chess
+2pcs
+45
+7
+314
+48.0
+3
+numerai28
+96
+22
+2119
+8.4
+2
+phoneme
+54
+6
+32
+0.12
+2
+sylvine
+51
+21
+108
+0.5
+2
+volkert
+58
+181
+10554
+128.4
+10
+synthetic
+blobs
+100
+10
+1000
+6.2
+2
+synthetic
+moons
+1
+2
+2
+0.12
+2
+and 24Gb memory.
+As our approach claims to increase the number of evaluated
+pipelines during fitting due to caching, it will be correct to com-
+pare this metric with and without the caching option. For that
+reason, we created the benchmark considering different compu-
+tational setups for AutoML. It utilizes a dataset for classifica-
+tion present using the FEDOT framework as a test bench.
+In Figure 7 the comparison of cache-based and cache-free
+configurations is provided. For the first one, both the pipelines
+cache and data preprocessing cache are activated. The number
+of parallel jobs used during optimization is one.
+During evolutionary optimization, a lot of candidate solu-
+tions (pipelines) are evaluated. We repeated the experiment for
+different timeouts that limit the execution time for the entire Au-
+toML run since they affect the number of evaluated pipelines.
+Also, an additional time limit is applied to the entire pipeline (to
+process the fit time anomalies for large pipelines). It is specified
+as 1/4 of the total timeout.
+Because of the stochastic nature of the optimization-based
+experiments, each run was repeated three times, and the ob-
+tained metrics were averaged.
+The results presented in Figure 8 are obtained with the
+n jobs hyperparameter value equal to 12.
+Table 2 summarises the averaged metrics of the experiments
+with single-process and multi-process caching. The average
+performance was increased by 14 %, which empirically con-
+Figure 7: The dependence between the number of pipelines and the usage of
+cache (averaged for ten runs). Single-processing is used.
+Figure 8: The dependence between the number of pipelines and the usage of
+cache (averaged for ten runs). 8 parallel jobs are used.
+firms the effectiveness of the proposed approach.
+The next stage of the experiment is devoted to the analysis
+of the evolutionary algorithm’s performance in multiprocessing
+mode. We compared algorithm performance with the number
+of processes equal to 1 and 8 with a timeout set to 10 minutes.
+The optimization of the pipeline structure was repeated three
+times with no seed and with five cross validation folds to take
+stochasticity into account.
+The dependency of correctly evaluated pipelines on a speci-
+fied number of jobs for a single dataset is presented in Figure 9.
+It can be seen that near-linear improvement in parallel speedup
+is achieved. Figure 10 demonstrates the dependency of the best
+fitness calculated using cross-validation on the timestamp from
+the configuration. Launches with 8 processes find a better solu-
+tion faster than launches with one process.
+Table 3 summarises the averaged results of experiments in
+single-process and multiprocessing modes. The fitness score
+calculated using cross-validation increases linearly with the
+number of evaluated pipelines. It confirms the effectiveness of
+the local parallelization of evolutionary AutoML.
+7
+
+350
+without cache
+with cache
+5
+actual time for optimization in minutes
+300
+correctly evaluated pipelines
+4
+250
+200
+3
+150
+2
+100
+1
+50
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+timeout in minuteswithout cache
+with cache
+5
+1200
+actual time for optimization in minutes
+correctly evaluated pipelines
+4
+1000
+800
+3
+600
+2
+400
+1
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+timeout in minutesTable 2: The results of experiments with caching of pipeline nodes and data preprocessing operations. The first column indicates whether the cache has been used,
+and the second column represents the number of parallel processes. The next three columns represent different metric values (the number of evaluated pipelines,
+ROC AUC for validation sample, and ROC AUC for cross-validation of training sample).
+Dataset
+Configuration
+Pipelines count
+ROC-AUC final
+ROC-AUC cross-validation
+Cache
+Number of processes
+adult
+on
+1
+27
+0,92
+0,9117
+off
+23
+0,9213
+0,913
+on
+8
+190
+0,921
+0,9131
+off
+170
+0,922
+0,9137
+amazon employee access
+on
+1
+85
+0,8447
+0,8346
+off
+78
+0,8497
+0,8376
+on
+8
+416
+0,8507
+0,8356
+off
+369
+0,849
+0,8398
+australian
+on
+1
+879
+0,9313
+0,9432
+off
+838
+0,9283
+0,9401
+on
+8
+6354
+0,928
+0,9411
+off
+6199
+0,934
+0,9442
+bank-marketing
+on
+1
+38
+0,93
+0,931
+off
+30
+0,9313
+0,93
+on
+8
+205
+0,931
+0,932
+off
+211
+0,932
+0,931
+blood-transfusion
+-service-center
+on
+1
+2175
+0,748
+0,75
+off
+2064
+0,7383
+0,759
+on
+8
+13943
+0,745
+0,761
+off
+13834
+0,749
+0,7659
+car
+on
+1
+812
+0,921
+0,933
+off
+728
+0,9233
+0,9319
+on
+8
+4856
+0,922
+0,935
+off
+4608
+0,92
+0,934
+cnae-9
+on
+1
+214
+0,995
+0,9939
+off
+195
+0,995
+0,9939
+on
+8
+1100
+0,995
+0,9942
+off
+1161
+0,995
+0,9953
+jungle chess 2pcs raw
+endgame complete
+on
+1
+30
+0,9671
+0,9637
+off
+44
+0,9667
+0,9627
+on
+8
+89
+0,969
+0,9631
+off
+99
+0,9713
+0,9649
+numerai28
+on
+1
+3
+0.508
+0,51
+off
+6
+0,511
+0,5182
+on
+8
+20
+0,527
+0,528
+off
+23
+0,5273
+0,528
+phoneme
+on
+1
+354
+0,9599
+0,951
+off
+325
+0,9597
+0,9515
+on
+8
+1954
+0,9631
+0,955
+off
+1885
+0,963
+0,9547
+sylvine
+on
+1
+221
+0,9852
+0,9809
+off
+206
+0,9853
+0,9806
+on
+8
+932
+0,9878
+0,981
+off
+827
+0,9877
+0,9829
+volkert
+on
+1
+4
+0,932
+0,9298
+off
+6
+0,9393
+0,9344
+on
+8
+20
+0,9313
+0,934
+off
+21
+0,9317
+0,9273
+synthetic blobs
+on
+1
+31
+1
+1
+off
+27
+1
+1
+on
+8
+235
+1
+1
+off
+224
+1
+1
+synthetic moons
+on
+1
+1124
+1
+1
+off
+1026
+1
+1
+on
+8
+12356
+1
+1
+off
+12227
+1
+1
+8
+
+Table 3: The results of experiments with parallelization of evolution. The first two rows for each dataset (1 and 8 jobs) represent the results obtained with a fit time
+limit for pipelines, while the ”without limit” row contain the results obtained without limits with 8 jobs.
+Dataset
+Number of processes
+Pipelines
+count
+ROC-AUC
+final
+ROC-AUC
+cross-validation
+adult
+1 (with limit)
+23
+0,9213
+0,913
+8 (with limit)
+170
+0,922
+0,9137
+8 (without limit)
+126
+0,9217
+0,9141
+amazon employee access
+1 (with limit)
+78
+0,8497
+0,8376
+8 (with limit)
+369
+0,849
+0,8398
+8 (without limit)
+329
+0,849
+0,8387
+australian
+1 (with limit)
+838
+0,9283
+0,9401
+8 (with limit)
+6199
+0,934
+0,9442
+8 (without limit)
+10009
+0,9307
+0,9444
+bank-marketing
+1 (with limit)
+30
+0,9313
+0,93
+8 (with limit)
+211
+0,932
+0,931
+8 (without limit)
+184
+0,9313
+0,93
+blood-transfusion-service-center
+1 (with limit)
+2064
+0,7383
+0,759
+8 (with limit)
+13834
+0,749
+0,7659
+8 (without limit)
+8329
+0,716
+0,7658
+car
+1 (with limit)
+728
+0,9233
+0,9319
+8 (with limit)
+4608
+0,92
+0,934
+8 (without limit)
+4612
+0,925
+0,9345
+cnae-9
+1 (with limit)
+195
+0,995
+0,9939
+8 (with limit)
+1161
+0,995
+0,9953
+8 (without limit)
+1168
+0,995
+0,9949
+jungle chess 2pcs raw endgame complete
+1 (with limit)
+44
+0,9667
+0,9627
+8 (with limit)
+99
+0,9713
+0,9649
+8 (without limit)
+144
+0,9723
+0,9661
+numerai28
+1 (with limit)
+6
+0,511
+0,5182
+8 (with limit)
+23
+0,5273
+0,528
+8 (without limit)
+22
+0,5253
+0,5266
+phoneme
+1 (with limit)
+325
+0,9597
+0,9515
+8 (with limit)
+1885
+0,963
+0,9547
+8 (without limit)
+1917
+0,963
+0,9536
+sylvine
+1 (with limit)
+206
+0,9853
+0,9806
+8 (with limit)
+827
+0,9877
+0,9829
+8 (without limit)
+816
+0,986
+0,9821
+volkert
+1 (with limit)
+6
+0,9373
+0,9329
+8 (with limit)
+21
+0,9393
+0,9344
+8 (without limit)
+21
+0,9317
+0,9273
+synthetic blobs
+1 (with limit)
+8
+1
+1
+8 (with limit)
+37
+1
+1
+8 (without limit)
+30
+1
+1
+synthetic moons
+1 (with limit)
+1026
+1
+1
+8 (with limit)
+12227
+1
+1
+8 (without limit)
+12569
+1
+1
+9
+
+Figure 9: The detailed analysis of dependency of number of pipelines from the
+number of jobs (dataset blood-transfusion-service-center)
+Figure 10: The dependency of the best fitness values on the optimization time.
+The intervals represent the stochasticity of the optimization runs.
+6.2. Heterogeneous infrastructure
+In the next series of experiments, we aim to estimate the
+efficiency of heterogeneous infrastructure for large datasets.
+Computation experiments were performed in a supercomputer
+environment configured based on two DGX-1 clusters. Each
+cluster contains eight Tesla V100 graphics cards and 128 GB of
+video memory. The number of graphics cores is 40960.
+The first experiment compares AutoML performance for the
+CPU-only and the hybrid infrastructures for various tasks. The
+aim of the experiment is to estimate the decreasing of fitting
+time after involvement of GPU-based nodes.
+To reduce the
+computational complexity of experiments, we decided not to
+use the full set of datasets from Table 1. In this experiment, four
+synthetic binary classification datasets with 10 features and dif-
+ferent number of rows (10000, 100000, 200000 and and 300000
+rows).
+Both single-model (SVC) and multi-model pipelines
+(consisting of SVC, Logistic Regression, and Random Forest)
+are considered to estimate the overhead for data flow transfer
+between models in the pipeline. The results are presented in
+Table 4: The training time of the pipelines on synthetic data under different
+conditions (single SVC classifier and composite pipeline with several models
+are considered). Averaged efficiency estimations are presented for homoge-
+neous (one server with multi-core CPU) and heterogeneous (CPU and GPU)
+computing environments.
+Rows, 103
+Fitting time, sec
+Improvement,
+%
+Single
+model
+Comp.
+pipeline
+Single
+model
+Comp.
+pipeline
+CPU
+CPU+
+GPU
+CPU
+CPU+
+GPU
+10
+0.3
+2.2
+0.7
+2.4
+-
+-
+100
+11.4
+1.9
+8.3
+1.6
+91
+88
+200
+39.3
+2.8
+21
+3.2
+94
+85
+300
+76.0
+4.2
+37.8
+5.5
+95
+86
+Table 4.
+The results confirmed that the overhead could exceed the
+performance gain for a small amount of data. However, the
+proposed hybrid approach to pipeline evaluation is reasonably
+practical for large datasets.
+6.3. Remote infrastructure
+The next series of experiments uses a homogeneous cluster
+of 20 nodes under Kubernetes control. Each node has 40 CPU
+cores and 256 Gb RAM. We have trained populations of 50,
+100 and 200 individuals. Each population has been trained four
+times, then the estimated time values were averaged. The aim
+of this experiment is to analyze the structure of computing time
+for remote evaluation and confirm that remote evaluation can
+be viable for large datasets regardless of existing overheads. To
+make the results of the experiment more compact, we focused
+on an analysis of a single synthetic binary classification dataset
+with 300000 rows and 10 features.
+Figure 11a presents training time depending on population
+size without results fetching. The evaluation of each individual
+has no CPU and system memory limits. Also, we have drawn
+a linear fit time as reference line. The initial point for this line
+is the time for the population size of 50. We found that training
+time increases almost linearly with the increase in population.
+To explain the near-linear time growth, we can consider the
+operations that took the most time during the computing and
+the overheads they have. Since requests overheads are nearly 0
+seconds, they are not presented in Figure 12.
+Figure 12 shows that computing time and overheads for
+each individual are the same and are independent of popula-
+tion size. It means that the performance bottleneck is not on the
+cluster side. The fit stages timeline (including results fetching)
+is presented in Figure 13.
+This timeline shows that the most time is spent on results
+fetching. Result fetching consists of zip-file downloading over
+the network, unpacking and then deserializing the model. Since
+the results are fetched concurrently, a large number of individu-
+als lead to a high network and drive load on the local machine.
+Moreover, even though the overhead for the request to run
+one individual is less than 1 sec, a large number of requests also
+10
+
+8 process
+0.855
+1process
+0.850
+0.845
+score
+ROC-AUC
+0.840
+0.835
+0.830
+0.825
+0
+500
+1000
+1500
+2000
+2500
+3000
+Time in secondsDependency ofthe bestfitness from numberof jobs
+14000
+8
+minutes
+12000
+7
+correctly evaluated pipelines
+6
+actual time for optimization in 
+10000
+8000
+4
+6000
+m
+4000
+2
+1
+2000
+1
+2
+3
+4
+5
+6
+7
+8
+numberof jobs(a) Without limits (fast calculations)
+(b) CPU limit = 0.2 core (heavy calculations emulation)
+Figure 11: The dependence of the total fit time on individual numbers in different computational setups. The orange line represents linear acceleration; the blue line
+represents the observed values of fit time.
+Figure 12: Overheads and computing time during experiments with remote infrastructure
+Figure 13: The explanation of remote training timeline with remote infrastruc-
+ture.
+consume a significant amount of time. The time range between
+the last request and the last completed individual is less than 20
+seconds, and it falls within the 75-percentile of computing time.
+It means that the computing cluster is underutilized because it
+is not running the individuals in parallel as well as expected
+because it is waiting for requests from the client.
+To sum up, it is not reasonable to use remote training if we
+have lightweight and fast computations. Overheads in the form
+of requests and results fetching will be significantly larger than
+the payload. To prove this assumption, we have repeated the
+same experiment, but we have artificially introduced CPU limit-
+ing for each individual (up to 0.2 CPU core) to emulate ”heavy”
+computing. The results for heavy-weight tasks are presented in
+Figure 11b.
+We can conclude that remote computing provides a signif-
+icant speedup for expensive computations. However, the over-
+head for small datasets should be taken into account.
+7. Conclusions and Discussions
+In the paper, we propose a modular approach that improves
+the efficiency of evolutionary AutoML in a heterogeneous en-
+11
+
+remote fit time
+linear fit timeremote fit time
+linear fit time主vironment. The proposed approach differs from existing solu-
+tions since it can be configured for automated machine learning
+in various computational environments. It makes it possible
+to parallelize and distribute the computational tasks across hy-
+brid and/or remote computational systems. Also, caching al-
+gorithms are implemented to increase the optimization perfor-
+mance for composite pipelines.
+The AutoML-based experimental setup consisted of (1) the
+estimation of parallel speedup for a different number of pro-
+cesses; (2) an analysis of the efficiency of the cache; (3) an anal-
+ysis of the GPU computations efficiency; (4) optimisation runs
+with remote infrastructure involved. The experiments confirm
+the proposed approach’s efficiency. It allows achieving signifi-
+cant improvements in the number of evaluated individuals and
+in the fitness function.
+There are several ways to improve remote computing per-
+formance aimed at different bottlenecks that can be used sepa-
+rately or combined:
+1. Efficient cluster resources utilization requires a custom
+scheduler and additional plugins for batch workload such
+as Volcano5.
+2. Refuse to request to run each individual. Better to use
+one request to run a batch of individuals. This way, the
+number of requests will be reduced to one independent
+request for the all population. Also, we can apply specu-
+lative computing mode when the number of rest individ-
+uals is small;
+3. Provide a cluster file system mount on the local machine.
+This will reduce the number of requests for downloading
+results, and the client will also skip zip file unpacking.
+Instead, the client will read the results from the mounted
+file system. If it is impossible, then we have to implement
+not only batch run requests but also batch download re-
+quests;
+4. Perform model validation using remote infrastructure
+too.
+This way, we also have to provide a validation
+dataset to the remote system. Remote computing will
+validate individuals and save the score. It will make it
+unnecessary to fetch the trained models, and the calcu-
+lated score will be enough for further decisions;
+5. Heterogeneous environment.
+We can use a heteroge-
+neous environment not only on the cluster layer but on
+the client-server layer. For example, the client can per-
+form lightweight calculations locally, heavy calculations
+at the same time will be sent to the cluster, and the heavi-
+est calculations may be sent to the most powerful cluster
+nodes (e.g. special GPU nodes).
+Another direction of improvement is the support of large
+dataset processing. It can be based on the implementation of
+the distributed evaluation of different folds of the data set. The
+caching system can also be implemented in a distributed way.
+5https://volcano.sh
+8. Code and Data Availability
+The software implementation of all described methods and
+algorithms is available in the open repository https://gith
+ub.com/ITMO-NSS-team/fedot-performance-improve
+ment-benchmark.
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+13
+

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@@ -0,0 +1,734 @@
+UNIFYSPEECH: A UNIFIED FRAMEWORK FOR ZERO-
+SHOT TEXT-TO-SPEECH AND VOICE CONVERSION
+Haogeng Liu1, Tao Wang1, Ruibo Fu2, Jiangyan Yi2, Zhengqi Wen2
+Chinese Academy of Science
+Beijing, China
+Jianhua Tao1
+Department of Automation, Tsinghua University
+School of Artificial Intelligence, University of Chinese Academy of Sciences, China
+Beijing, China
+ABSTRACT
+Text-to-speech (TTS) and voice conversion (VC) are two different tasks both aim-
+ing at generating high quality speaking voice according to different input modality.
+Due to their similarity, this paper proposes UnifySpeech, which brings TTS and
+VC into a unified framework for the first time. The model is based on the assump-
+tion that speech can be decoupled into three independent components: content
+information, speaker information, prosody information. Both TTS and VC can be
+regarded as mining these three parts of information from the input and completing
+the reconstruction of speech. For TTS, the speech content information is derived
+from the text, while in VC it’s derived from the source speech, so all the remaining
+units are shared except for the speech content extraction module in the two tasks.
+We applied vector quantization and domain constrain to bridge the gap between
+the content domains of TTS and VC. Objective and subjective evaluation shows
+that by combining the two task, TTS obtains better speaker modeling ability while
+VC gets hold of impressive speech content decoupling capability.
+Index Terms: decoupling, zero-shot learning, text-to-speech, voice conversion,
+vector quantization
+1
+INTRODUCTION
+Cloning the voice of the target speaker is an attractive technology, which can be applied to various
+scenes (Sisman et al., 2020), such as entertainment creation, personalized mobile assistants, security
+field, etc. The most ideal voice cloning operation is to give only one relatively short speech of the
+unseen target speaker as a reference and then any speech of the target speaker can be synthesized,
+which is called zero-shot voice clone. In the speech research community, text-to-speech (TTS) and
+voice conversion (VC) are two mainstream ways to realize voice clone (Sisman et al., 2020). There-
+fore, a variety of techniques for zero-shot TTS and VC have been proposed recently (Gorodetskii &
+Ozhiganov, 2022; Tang et al., 2021).
+However, although TTS and VC techniques are two important ways of voice clone with same output
+form, the research of TTS and VC is more or less independent. There isn’t much interaction between
+them. But they are both speech synthesis task. In terms of speech generation, we categorize the
+information of the target speaker’s speech into three kinds of information: (1) speech content, the
+characters of phonemes or phonetic posteriorgram (PPG) in voice conversion, represents the content
+of the speech. (2) speaker information, which represents the characteristics of speakers, is related
+to the speaker’s articulation organ. (3) prosody information, which covers the intonation, stress,
+and rhythm of speech. According to FastSpeech2 (Ren et al., 2020), pitch, energy and duration
+information can reflected them certainly. As TTS extracts speech content directly from phonemes,
+it is easier to obtain content information irrelevant to speaker than VC. As VC encounters more
+speakers, it’s possible to obtain more robust speaker modeling ability. So, by integrating TTS and
+1
+arXiv:2301.03801v1  [cs.SD]  10 Jan 2023
+
+VC into a unified framework and combining their training data, it can help the model learn these
+three kinds of information better.
+Unfortunately, investigating TTS and VC in the same framework is challenging, as the speech
+content are extracted from different modality. Specifically, the speech content in TTS is obtained
+through phoneme information while the phonemes and speech in TTS are unequal sequences, need-
+ing attention mechanism to align them. However, the attention mechanism is often affected by the
+speaker’s information, so it is impossible to learn the speech content representation completely ir-
+relevant to the speaker. While in VC, the source speech and target speech are aligned in speech
+content, so the speech content can be extracted directly from the source speech, which is different
+from the TTS. In contrast to speech content information, speaker information and prosody infor-
+mation can be modeled using the same network in TTS and VC. Therefore, the difficulty here is
+to keep speech content for TTS and VC consistent. With the development of speech synthesis, the
+recently proposed Adaspeech2 (Yan et al., 2021) can combine text information with speech infor-
+mation, which can effectively decouple the speech content from the input. The unified framework
+becomes possible.
+Overall, the main contributions of this paper are:
+• We propose UnifySpeech, a unified framework for TTS and VC. VC enables unlabeled data
+to join training process, making TTS encounters more speakers. TTS enhance the ability
+for voice content decoupling in VC. Thus, both pipeline benefits from the other one.
+• We apply vector quantization and domain constrain to bridge the gap between the content
+domains of TTS and VC. Ablation experiment shows this method’s effectiveness.
+• We perform extensive experiments: zero-shot TTS, zero-shot VC. Results proves that
+jointly trained TTS outperformes StyleSpeech (Min et al., 2021)and jointly trained VC
+gains better speech decoupling ability.
+Demos for this paper are available at https://liuhaogeng.github.io/UnifySpeech/.
+2
+BACKGROUND
+In this section, we will briefly review the background of this work, including neural TTS and VC
+models, and the zero-shot learning for TTS and VC tasks.
+2.1
+TEXT-TO-SPEECH TASK
+TTS task is to model the mapping between text and speech, which is a modeling problem between
+two unequal length sequences. According to the alignment mechanism (Battenberg et al., 2020)
+between text and speech in the model, the end-to-end TTS model can be divided into two categories:
+1) Using a neural network to learn the alignment information between text and speech, such as
+local sensitive attention in Tacotron (Wang et al., 2017b). Various improvements to the attention
+mechanism have been proposed. In addition, inspired by CTC (Kim et al., 2017) in ASR, glow-
+TTS (Kim et al., 2020), VITS (Kim et al., 2021) can automatically learn the alignment information.
+2) By introducing the duration information (Ren et al., 2019) of phonemes as prior knowledge, the
+text is upsampled to achieve alignment (McAuliffe et al., 2017). Since the upsampled information
+based on text is independent of the speaker, the speech content can be well separated from the
+speech. Therefore, we introduce the duration information to build the TTS model in this paper.
+2.2
+VOICE CONVERSION TASK
+Voice conversion can be seen as two steps (Sisman et al., 2020).
+Firstly, extract the speaker-
+independent speech content information from the source speech, and then embed the target speaker
+information to the speech content to reconstruct the speech of the target speaker. According to the
+way of extracting speech content, the VC model can be divided into two categories: (1) text-based
+approach. (2) Information bottleneck approach. The first approach requires an additional pre-trained
+ASR model. Since the ASR is trained in a supervised manner, it demands a lot of paired text and
+speech. Additionally, pipeline modeling is easy to accumulate errors and affects the performance of
+2
+
+the system. Therefore, a lot of research work is focused on the latter. By adding some restrictions
+to the information bottleneck, different kinds of information can be decoupled. However, if the in-
+formation bottleneck can not be decoupled well, the performance of the model will be significantly
+reduced.
+2.3
+ZERO SHOT LEARNING FOR TTS AND VC
+The research of zero-shot learning based on TTS and VC focused on how to extract effective speaker
+information and then embed it into TTS or VC model for joint training or segmented training.
+Typical speaker features include i-vector (Wang et al., 2017a), d-vector (Variani et al., 2014), x-
+vector (Snyder et al., 2018) and so on. In addition, the modules that extract speaker-style informa-
+tion through the specially designed network structure, such as GST (Wang et al., 2018), VAE (Van
+Den Oord et al., 2017), can also achieve good results.
+3
+UNIFYSPEECH
+In this section, we introduce the details of UnifySpeech for zero-shot TTS and VC tasks. We first
+give the key idea of UnifySpeech: speech factorization, and then introduce the formulation of Uni-
+fySpeech. Finally, we will describe the model structure of UnifySpeech.
+Figure 1: Structure of UnifySpeech.
+3.1
+SPEECH REPRESENTATION DISENTANGLEMENT
+The core of the controllable and migratable speech generation task is to decouple the components of
+the generated speech first, and then control and transfer each component. Although some end-to-end
+models can directly model the relationship between text and speech (TTS) or speech-to-speech task
+(VC), due to the mutual coupling of various components of the end-to-end model, it brings great
+difficulties to the transfer learning of the model. Therefore, we first decouple the speech generation
+task into three independent components and then input them into the decoder to synthesize speech.
+This idea is also the main architecture of UnifySpeech. The three components and their sources will
+be described in detail below.
+Speech Content: To generate intelligible speech signals, it is important to model accurate speech
+content information. Speech content is linguistic information, which is irrelevant to the speaker.
+Due to the different types of input signals of TTS and VC, the ways of extracting speech content are
+different. For TTS, the source of speech content is text. Firstly, to learn the context information of
+the text, a text encoder is used to encode the text to obtain the context representation. The context
+representation is up-sampled according to the phoneme’s duration information to obtain the speech
+content information. For VC, since the source speech is aligned with the target speech, we directly
+use a content encoder to extract speech content information from the source speech.
+Speaker: The speaker information includes the speaker’s characteristics, such as the timbre, volume,
+etc. We can extract the speaker information from the given speech of the target speaker. This is
+3
+
+Decoder
+Decoder
+Pitch
+Pitch
+ Pitch predictor
+Pitch predictor
+Prosody information Speaker embedding
+Speech content
+Speech content
+Speaker embedding Prosody information
+VQ
+VQ
+Prosody encoder
+Speaker encoder
+Speaker encoder
+Prosody encoder
+Text encoder
+ Content encoder
+Pitch
+Reference mel
+Phonemes
+Speech
+Reference mel
+Pitchcommon for TTS and VC tasks, so the speaker extraction network can be shared in the two tasks.
+Since the speaker extraction network can directly extract information from speech without text, so a
+large number of data without text annotation in the VC task can be used for training, which can help
+to improve the transfer learning ability of the model.
+Prosody: The prosody information represents how the speaker says the content information. It
+is independent of the speaker information and related to the way of expression. Since the pitch
+information can reflect the rhythm of speech, therefore, the pitch information is used to extract
+prosody information. The prosody information, like speaker information, can be shared by both TTS
+and VC. In addition, in the training process, we can obtain pitch information from the ground truth
+speech, but there is no ground truth in the inference stage. Therefore, a pitch prediction module (Ren
+et al., 2020) is proposed in the training stage, which takes the speech content information and speaker
+information as the input to predict the pitch information.
+3.2
+SPEECH CONTENT WITH VECTOR QUANTIZATION
+Since there are different ways to obtain the speech content in TTS and VC, it is very easy to de-
+viate between the two domains. If this deviation occurs, it will cause devastating damage to some
+downstream shared modules (such as pitch predictor and decoder). Therefore, to ensure that the
+consistency of the two speech content domains as close as possible, we reconstruct them from two
+aspects:
+• First, we use the shared codebook to quantify the continuous feature space.
+• Second, we use the labeled data to narrow two discrete feature spaces.
+The detailed process is described below. Suppose that the vector obtained by the text encoder in
+TTS pipeline is Cp = (C1
+p, C2
+p, · · · , CT
+p ) with length T, the vector obtained by the content encoder
+in VC pipeline is Cs = (C1
+s, C2
+s, · · · , CT
+′
+s ) with length T
+′. It should be noted that we add a length
+regulator module in the text encoder to solve the problem of length mismatch between the text and
+speech sequence, which is introduced in FastSpeech (Ren et al., 2019). Therefore, if text and speech
+are paired, T
+′ = T. The vector Cp and Cs is a sequence of continuous vector in Eq. Due to the
+large representation range of continuous features, Cp and Cs are difficult to match. We borrow
+the discretization method for latent variables from Vector Quantized Variational AutoEncoder (Van
+Den Oord et al., 2017). Specifically, for each time step t, the continuous latent representations Ct
+p in
+Cp can be mapped into C
+t
+p by finding the nearest pre-defined discretized embedding in the dictionary
+as:
+C
+t
+p = ek,
+k = argminj
+��Ct
+p − ej
+��
+2
+(1)
+where ej is the j-th embedding in the codebook dictionary, and j ∈ 1, 2, · · · , V . Since selecting
+the entry with the minimum distance will cause the operation to be non-differentiable, the straight-
+through gradient estimator can be used to approximate the gradient, which can be expressed as:
+¯ht = ht + ev − sg (ht) ,
+v = arg min
+k
+∥ht − ek∥2
+(2)
+where sg(·) is the stop-gradient (Van Den Oord et al., 2017) operation that treats its input as constant
+during back-propagation.
+After vector quantization, the quantized sequence Cp
+=
+(C
+1
+p, C
+2
+p, · · · , C
+T
+p ) and Cs
+=
+(C
+1
+s, C
+2
+s, · · · , C
+T
+s ) can be obtained. It should be noted that when Cp and Cs are quantized into
+Cp and Cs, they share the same codebook e. The advantage of this is that since the speech content
+features Cp in the TTS pipeline are independent of the speaker, sharing the same codebook can
+help learn the speech content features Cs independent of the speaker in the VC pipeline, which is
+essential for the VC task.
+Although both pipeline use the same codebook for coding, there is no guarantee that there is no
+deviation between the two fields after coding. Therefore, to further eliminate the deviation between
+the two domains, we use the labeled speech in the TTS pipeline to supervise the training of the two
+domains. Specifically, for paired text and speech data, we constrain the feature distance between the
+quantized sequence Cp and Cs:
+4
+
+Lpair =
+��Cp − Cs
+��2
+2
+(3)
+In this way, we can efficiently minimize the domain discrepancy by using limited labeled data.
+Figure 2: Structure of vector quantized operation.
+3.3
+UNIFYSPEECH PIPELINE
+An overview of our proposed UnifySpeech architecture is illustrated in Fig. 2. It consists of a
+sequence-to-sequence TTS, and a sequence-to-sequence VC. The key idea is to share most of the
+module parameters (speaker encoder, prosody encoder, decoder and pitch predictor) and map the
+speech content in TTS and VC to the same space. As mentioned above, the UnifySpeech allows
+us to train the model on the concatenation of both the labeled and unlabeled data. For supervised
+training with labeled data, both models can be trained independently by minimizing the loss between
+their predicted speech and the ground truth. For unsupervised training with unlabeled data, the VC
+pipeline can be trained, and the parameters are shared with TTS.
+To further clarify the training process, we unrolled the framework as follows.
+3.3.1
+TTS PIPELINE
+Denote the text and speech sequence pair (x, y, F0) ∈ D, where D is the paired text and speech
+corpus. Each element in the text sequence x represents a phoneme or character, while each element
+in the speech sequence y represents a frame of speech. F0 is the pitch information of y. The
+representation obtained after three encoders are speech content C, speaker S and prosody P.
+Then, the three parts are added and input into a decoder to obtain the predicted speech y′. In addition,
+to obtain the pitch information in the interference stage, we use the content information and speaker
+information to predict the F0. These processes can be expressed as:
+y′ = decoder(C + S + P)
+(4)
+F0′ = pitch predictor(C + S)
+(5)
+Therefore, the reconstruction loss in TTS process includes two parts:
+LV C
+rec = MSE(y, y′) + MSE(F0, F0′)
+(6)
+5
+
+C
+Look up min
+Look up min
+distance
+distance
+vector
+vector
+1
+V
+L2
+L2
+distance
+ev
+distance
+Ct
+Cs
+Text encoder
+Content encoderIn addition, we use the content encoder in the VC pipeline to extract the content representation Cs
+for the training speech in TTS, and close the distance between the two domains by calculating the
+distance loss of Cs and Cp, which is explained in Sec. 3.2.
+Lpair =
+��Cp − Cs
+��2
+2
+(7)
+The loss of TTS pipeline can be expressed as:
+LT T S = LT T S
+rec
++ Lpair
+(8)
+where MSE denotes the mean squared errors.
+3.3.2
+VC PIPELINE
+Denote all the unlabeled or labeled speech (y, F0) ∈ Y . We first extract three information from
+(y, F0), which are speech content Cs, speaker Ss and prosody Ps.
+Then, similar to the TTS pipeline, the shared decoder and pitch predictor module is used to predict
+the speech signal y′ and F0′, which is similar to the Eq.1 and Eq.2.
+The loss of VC pipeline only includes reconstruction loss, which can be expressed as:
+LV C = MSE(Y, Y ′) + MSE(F0, F0′)
+(9)
+where MSE denotes the mean squared errors.
+3.3.3
+TRAINING PROCESS
+With such a unified framework, TTS and VC can learn from each other through joint training. The
+details of the algorithm can be found below.
+Procedure 1 UnifySpeech training algorithm
+1: Input: Paired speech and text dataset (x, y), speech only dataset y
+′
+2: repeat
+3:
+# A. TTS pipeline with speech-text data pairs
+1.
+Sample paired speech and text (x, y)
+2.
+Extract speech content information from x for domain loss
+3.
+Generate the predict speech y, pitch F0 and speech content from text
+4.
+Calculate the loss for TTS LT T S
+rec
+4:
+# B. VC pipeline with speech-only data
+1.
+Sample paired speech and text (x, y)
+2.
+Extract speech content information from y for domain loss
+3.
+Calculate the domain loss for the two pipeline Lpair
+4.
+Sample speech y
+′ in speech only dataset
+5.
+Generate the predict speech y, pitch F0 and speech content from speech y
+′
+6.
+Calculate the loss for VC LV C
+5:
+# C. Loss combination:
+1.
+Combine all loss (LT T S
+rec , Lpair, LV C) into a single loss variable
+2.
+Calculate TTS and VC parameters gradient
+3.
+Update TTS and VC parameters with gradient descent optimization
+6: until convergence
+4
+EXPERIMENTS AND RESULTS ANALYSIS
+In this section, we conduct experiments to evaluate our proposed methods. The experiments are
+carried out from two aspects: zero-shot TTS, zero-shot VC.
+6
+
+4.1
+DATASETS
+Two datasets are used to simulate labeled data and unlabeled data, respectively. VCTK dataset, an
+English language dataset containing 44 hours of speech and 109 speakers is used as labeled data.
+Each speaker has approximately 400 sentences. LibriTTS (Zen et al., 2019) are used as unlabeled
+data, which consists of 585 hours of speech data from 2484 speakers. We only use the speech data in
+LibriTTS and discard the text for unsupervised training. In this way, it can simulate the scene where
+a large number of speech that we can obtain are unlabeled. We use a 16-bit, 22050 Hz sampling rate
+for all experiments. The 80-dim Mel spectrogram is extracted with Hann windowing, frame shift
+of 12.5-ms, frame length of 50-ms, and 1024-point Fourier transform. In this experiment, we use
+hifigan (Kong et al., 2020) as vocoder.
+4.2
+MODEL DETAILS
+Figure 3: Structure of each module in UnifySpeech. FFT (Ren et al., 2019) means feed-forward
+Transformer.
+The detail of each module in our proposed method is shown in the Fig. 3. Specifically, to make
+the sequence of speech content extracted from the text encoder and content encoder equal, a length
+regulator is added to the text encoder, which is inspired by the FastSpeech (Ren et al., 2019). The
+structure of the duration predictor is the same as that in FastSpeech. The structure of the decoder
+and content encoder is similar, but the dimensions of input and output are opposite. For the prosody
+encoder, we first quantize F0 of each frame to 32 possible values and encode them into a learnable
+embedding vector according to the value. And we change the output of the pitch predictor into a
+distribution. By this way, pitch prediction becomes a classification task, reducing the difficulty of
+frame-level pitch prediction. We use the speaker encoder in StyleSpeech and fuse features with Style
+Adaptive Layer Norm(SALN) (Min et al., 2021) method to make the comparision fairly.
+The number of feed-forward Transformer (Vaswani et al., 2017) (FFT) blocks in the text encoder is
+4 and it is 6 in the decoder module. In each FFT block, the dimension of hidden states is 256. The
+kernel sizes of all the 1D-convolution are set to 3. The dropout rate is set to 0.5. The dimension of the
+last linear layer in the decoder is 256. The dimension of last linear layer in encoders (text encoder,
+pitch encoder, content encoder) is 256. An Adam optimizer (Kingma & Ba, 2014) is used to update
+the parameters. The initial learning rate is 0.001 and the learning rate decreased exponentially.
+7
+
+Linear layer
+Linear layer
+Duration
+Length regulator
+predictor
+m block FFT
+不
+不
+LN + Dropout
+n block FFT
+不
+Speaker
+Speaker
+Embedding
+ConvlD + ReLU
+Phoneme
+Embedding
+Embedding
+Linear layer
+Linear layer
+Pooling
+个
+LN + Dropout
+LN + Dropout
+FFT
+ConvlD + ReLU
+ConvlD + ReLU
+LN + Dropout
+LN + Dropout
+不
+Pitch Embedding
+ConvlD + ReLU
+ConvlD + ReLU
+Randomly select a clip
+LSTM
+Quantify to [0, 32]
+from target speaker4.3
+ZERO-SHOT TTS
+We first carried out zero-shot TTS task. We choose four speakers from VCTK that are not used
+during training process as target speakers. For each speaker, we randomly select about 20 sentences
+to be our target. Then we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek,
+1993) (Mel-cepstrum distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR
+(correlation factor of F0) between synthesized speech and ground truth speech as the objective met-
+rics.
+Table 1: Objective evaluation results for zero-shot TTS.
+Model
+F0 RMSE (Hz)
+MCD (db)
+V/UV
+F0 CORR
+UnifySpeech
+17.84
+2.51
+16.9%
+0.93
+StyleSpeech
+19.02
+2.63
+18.06%
+0.92
+Subjective evaluation was also conducted to compare the speech’s quality and similarity. We choose
+mean opinion scores(MOS) for naturalness and similarity mean opinion scores(SMOS) for similar-
+ity. Both metrics are rated in 1-to-5 scale and reported with the 95% confidence intervals (CI).
+Table 2: Mean opinion score (MOS) of the models. With VC means the model is jointly trained.
+Model
+MOS
+SMOS
+GT
+4.32 ± 0.15
+−
+GT mel + Vocoder
+4.09 ± 0.15
+−
+StyleSpeech
+3.52 ± 0.13
+3.82 ± 0.13
+UnifySpeech-TTS (with VC)
+3.76 ± 0.12
+3.95 ± 0.13
+To better show our method’s effectiveness, we demonstrate the t-SNE projection of the speaker
+embedding vectors from speakers in both VCTK and LibriTTS. (Van der Maaten & Hinton, 2008)
+Fig. 4 shows the speaker visualization. For the seen speakers (x) and unseen speakers (o), the
+corresponding speaker embedding form a cluster and distinct from others. The boundary between
+different speakers is clear. This shows that the speaker encoder performs well.
+Figure 4: Speaker visualization of generated speeches,where the circle and triangle indicate unseen
+speaker,while x and square indicate seen speaker.
+4.4
+ZERO-SHOT VC
+We carried out zero-shot VC task, using unseen speakers voice to be the reference speech. As we
+are lack of parallel corpus, we only conduct subjective evaluation. But VC and TTS shares the same
+8
+
+30
+p225
+p226
+p227
+20
+p228
+p279
+X
+p274
+10
+p307
+p311
+8012
+0
+￥5181
+3885
+2961
+-10
+1089
+7021
+20
+KX
+-20
+15
+-10
+-5
+0
+5
+10
+15speaker encoder, Fig. 4 in zero-shot TTS can also be a reference.
+Just as in zero-shot TTS task, we choose mean opinion scores(MOS) for naturalness and similarity
+mean opinion scores(SMOS) for similarity. Both metrics are rated in 1-to-5 scale and reported with
+the 95% confidence intervals (CI).
+Table 3: Mean opinion score (MOS) of the VC models. With TTS means the model is jointly
+trained.
+Model
+MOS
+SMOS
+GT
+4.32 ± 0.15
+−
+GT mel + Vocoder
+4.09 ± 0.15
+−
+UnifySpeech-VC(without TTS)
+3.63 ± 0.13
+1.31 ± 0.06
+UnifySpeech-VC(with TTS)
+3.58 ± 0.12
+3.31 ± 0.13
+It can be found that when VC pipeline is trained alone, its performance is poor. In other words, it
+doesn’t have the ability to discriminate. But jointly training with TTS improves its speech decou-
+pling ability, indirectly improving the speaker modeling ability, which we will analysis later.
+4.5
+ABLATION EXPERIMENT
+To figure out whether jointly training is effective, we separately train the TTS and VC parts, carrying
+out objective evaluation on them. We also remove the VQ parts and test the model. For TTS,
+we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek, 1993) (Mel-cepstrum
+distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR (correlation factor of F0)
+between synthesized speech and ground truth speech as the objective metrics.
+Table 4: Objective evaluation results for zero-shot TTS.
+Model
+F0 RMSE (Hz)
+MCD (db)
+V/UV
+F0 CORR
+UnifySpeech-TTS(without VC)
+19.31
+2.55
+16.9%
+0.91
+UnifySpeech-TTS-novq(with VC)
+19.41
+2.58
+17.2%
+0.92
+UnifySpeech-TTS(with VC)
+17.84
+2.51
+16.9%
+0.93
+For VC, we separately calculate the Average Cosine Similarity (ACS) (Lei et al., 2022) for the
+embedding from same (S-ACS) and different speakers (D-ACS). And then their ratio is used as an
+evaluation metric.
+Table 5: Ratio of ACS from same and different speaker. Unseen or seen means whether the speaker
+is from test set. With or without TTS means whether the VC pipeline is trained with TTS pipeline.
+Model
+S−ACS
+D−ACS (unseen)
+S−ACS
+D−ACS (seen)
+UnifySpeech-VC (with TTS)
+2.8
+6.0
+UnifySpeech-VC (without TTS)
+1.0
+1.0
+To validate whether VQ can bridge the gap between the speech content parts in TTS and VC, we
+caculate the L2 distance between the phoneme representation from TTS and VC. As in VC there
+are many frames represent same phoneme, so we choose their clustering center as the correspond-
+ing phoneme representation. We randomly select 4 sentences from validation set to carry out the
+experiment.
+Table 6: L2 distance between the phoneme representation in TTS and VC. S1 means sentence1.
+Model
+S1
+S2
+S3
+S4
+average
+UnifySpeech-VC(without VQ)
+0.464
+0.476
+0.474
+0.553
+0.492
+UnifySpeech-VC(with VQ)
+0.152
+0.205
+0.208
+0.205
+0.193
+9
+
+4.6
+RESULT ANALYSIS
+Above results show that jointly training actually improves TTS’s speaker modeling ability and VC’s
+speech decoupling ability. And VQ enhances the consistency of representation of the same content
+from phonemes and speeches, ensuring the model’s correctly working.
+For TTS, sharing modules with VC enables unlabeled data to participate in its training process.
+Along with the richer speaker style pattern, the speaker style modeling capability has been enhanced.
+For VC, when trained alone, it doesn’t have the speaker modeling ability. As the self-supervised
+training process aims at reducing the reconstruction loss of source audio, the reference audio of the
+target speaker isn’t crucial in the process. Only the content encoder and the decoder are enough for
+the process, that’s possibly why the speaker embeddings are all similar, though they are from differ-
+ent speakers. When jointly trained, the text loss plays a role of regularization factor, resulting that the
+content encoder just extracting the content information from the source speech (speaker information
+is reserved and others are discarded). This makes the reference speech with rich speaker information
+become indispensable for the reconstruction process. Thus the model’s speaker modeling ability has
+been improved.
+5
+CONCLUSIONS
+In this paper, we propose UnifySpeech, a unified framework for TTS and VC. Both task benefits
+from the other one. Due to training with large amounts of unlabeled data, their few-shot modeling
+ability makes progress as well as the synthesized speech’s quality. In the future, further improving
+the synthesized speech’s quality and making the generated speech’s style more similar to target
+speaker will be our endeavor.
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+11
+

GNE2T4oBgHgl3EQfTQeG/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf,len=465
+page_content='UNIFYSPEECH: A UNIFIED FRAMEWORK FOR ZERO-  SHOT TEXT-TO-SPEECH AND VOICE CONVERSION  Haogeng Liu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Tao Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Ruibo Fu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Jiangyan Yi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Zhengqi Wen2  Chinese Academy of Science  Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' China  Jianhua Tao1  Department of Automation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Tsinghua University  School of Artificial Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' China  Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' China  ABSTRACT  Text-to-speech (TTS) and voice conversion (VC) are two different tasks both aim-  ing at generating high quality speaking voice according to different input modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Due to their similarity, this paper proposes UnifySpeech, which brings TTS and  VC into a unified framework for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The model is based on the assump-  tion that speech can be decoupled into three independent components: content  information, speaker information, prosody information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Both TTS and VC can be  regarded as mining these three parts of information from the input and completing  the reconstruction of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For TTS, the speech content information is derived  from the text, while in VC it’s derived from the source speech, so all the remaining  units are shared except for the speech content extraction module in the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  We applied vector quantization and domain constrain to bridge the gap between  the content domains of TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Objective and subjective evaluation shows  that by combining the two task, TTS obtains better speaker modeling ability while  VC gets hold of impressive speech content decoupling capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Index Terms: decoupling, zero-shot learning, text-to-speech, voice conversion,  vector quantization  1  INTRODUCTION  Cloning the voice of the target speaker is an attractive technology, which can be applied to various  scenes (Sisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020), such as entertainment creation, personalized mobile assistants, security  field, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The most ideal voice cloning operation is to give only one relatively short speech of the  unseen target speaker as a reference and then any speech of the target speaker can be synthesized,  which is called zero-shot voice clone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In the speech research community, text-to-speech (TTS) and  voice conversion (VC) are two mainstream ways to realize voice clone (Sisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' There-  fore, a variety of techniques for zero-shot TTS and VC have been proposed recently (Gorodetskii &  Ozhiganov, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  However, although TTS and VC techniques are two important ways of voice clone with same output  form, the research of TTS and VC is more or less independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' There isn’t much interaction between  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' But they are both speech synthesis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In terms of speech generation, we categorize the  information of the target speaker’s speech into three kinds of information: (1) speech content, the  characters of phonemes or phonetic posteriorgram (PPG) in voice conversion, represents the content  of the speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' (2) speaker information, which represents the characteristics of speakers, is related  to the speaker’s articulation organ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' (3) prosody information, which covers the intonation, stress,  and rhythm of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' According to FastSpeech2 (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020), pitch, energy and duration  information can reflected them certainly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As TTS extracts speech content directly from phonemes,  it is easier to obtain content information irrelevant to speaker than VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As VC encounters more  speakers, it’s possible to obtain more robust speaker modeling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' So, by integrating TTS and  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='03801v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='SD]  10 Jan 2023  VC into a unified framework and combining their training data, it can help the model learn these  three kinds of information better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Unfortunately, investigating TTS and VC in the same framework is challenging, as the speech  content are extracted from different modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Specifically, the speech content in TTS is obtained  through phoneme information while the phonemes and speech in TTS are unequal sequences, need-  ing attention mechanism to align them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' However, the attention mechanism is often affected by the  speaker’s information, so it is impossible to learn the speech content representation completely ir-  relevant to the speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' While in VC, the source speech and target speech are aligned in speech  content, so the speech content can be extracted directly from the source speech, which is different  from the TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In contrast to speech content information, speaker information and prosody infor-  mation can be modeled using the same network in TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, the difficulty here is  to keep speech content for TTS and VC consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' With the development of speech synthesis, the  recently proposed Adaspeech2 (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021) can combine text information with speech infor-  mation, which can effectively decouple the speech content from the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The unified framework  becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Overall, the main contributions of this paper are:  We propose UnifySpeech, a unified framework for TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' VC enables unlabeled data  to join training process, making TTS encounters more speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' TTS enhance the ability  for voice content decoupling in VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Thus, both pipeline benefits from the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  We apply vector quantization and domain constrain to bridge the gap between the content  domains of TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Ablation experiment shows this method’s effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  We perform extensive experiments: zero-shot TTS, zero-shot VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Results proves that  jointly trained TTS outperformes StyleSpeech (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021)and jointly trained VC  gains better speech decoupling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Demos for this paper are available at https://liuhaogeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='io/UnifySpeech/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2  BACKGROUND  In this section, we will briefly review the background of this work, including neural TTS and VC  models, and the zero-shot learning for TTS and VC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1  TEXT-TO-SPEECH TASK  TTS task is to model the mapping between text and speech, which is a modeling problem between  two unequal length sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' According to the alignment mechanism (Battenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020)  between text and speech in the model, the end-to-end TTS model can be divided into two categories:  1) Using a neural network to learn the alignment information between text and speech, such as  local sensitive attention in Tacotron (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Various improvements to the attention  mechanism have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In addition, inspired by CTC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017) in ASR, glow-  TTS (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020), VITS (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021) can automatically learn the alignment information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2) By introducing the duration information (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019) of phonemes as prior knowledge, the  text is upsampled to achieve alignment (McAuliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Since the upsampled information  based on text is independent of the speaker, the speech content can be well separated from the  speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, we introduce the duration information to build the TTS model in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2  VOICE CONVERSION TASK  Voice conversion can be seen as two steps (Sisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Firstly, extract the speaker-  independent speech content information from the source speech, and then embed the target speaker  information to the speech content to reconstruct the speech of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' According to the  way of extracting speech content, the VC model can be divided into two categories: (1) text-based  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' (2) Information bottleneck approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The first approach requires an additional pre-trained  ASR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Since the ASR is trained in a supervised manner, it demands a lot of paired text and  speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Additionally, pipeline modeling is easy to accumulate errors and affects the performance of  2  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, a lot of research work is focused on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' By adding some restrictions  to the information bottleneck, different kinds of information can be decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' However, if the in-  formation bottleneck can not be decoupled well, the performance of the model will be significantly  reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3  ZERO SHOT LEARNING FOR TTS AND VC  The research of zero-shot learning based on TTS and VC focused on how to extract effective speaker  information and then embed it into TTS or VC model for joint training or segmented training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Typical speaker features include i-vector (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017a), d-vector (Variani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2014), x-  vector (Snyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2018) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In addition, the modules that extract speaker-style informa-  tion through the specially designed network structure, such as GST (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2018), VAE (Van  Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017), can also achieve good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3  UNIFYSPEECH  In this section, we introduce the details of UnifySpeech for zero-shot TTS and VC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We first  give the key idea of UnifySpeech: speech factorization, and then introduce the formulation of Uni-  fySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Finally, we will describe the model structure of UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Figure 1: Structure of UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1  SPEECH REPRESENTATION DISENTANGLEMENT  The core of the controllable and migratable speech generation task is to decouple the components of  the generated speech first, and then control and transfer each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Although some end-to-end  models can directly model the relationship between text and speech (TTS) or speech-to-speech task  (VC), due to the mutual coupling of various components of the end-to-end model, it brings great  difficulties to the transfer learning of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, we first decouple the speech generation  task into three independent components and then input them into the decoder to synthesize speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  This idea is also the main architecture of UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The three components and their sources will  be described in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Speech Content: To generate intelligible speech signals, it is important to model accurate speech  content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Speech content is linguistic information, which is irrelevant to the speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Due to the different types of input signals of TTS and VC, the ways of extracting speech content are  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For TTS, the source of speech content is text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Firstly, to learn the context information of  the text, a text encoder is used to encode the text to obtain the context representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The context  representation is up-sampled according to the phoneme’s duration information to obtain the speech  content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For VC, since the source speech is aligned with the target speech, we directly  use a content encoder to extract speech content information from the source speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Speaker: The speaker information includes the speaker’s characteristics, such as the timbre, volume,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We can extract the speaker information from the given speech of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' This is  3  Decoder  Decoder  Pitch  Pitch  Pitch predictor  Pitch predictor  Prosody information Speaker embedding  Speech content  Speech content  Speaker embedding Prosody information  VQ  VQ  Prosody encoder  Speaker encoder  Speaker encoder  Prosody encoder  Text encoder  Content encoder  Pitch  Reference mel  Phonemes  Speech  Reference mel  Pitchcommon for TTS and VC tasks, so the speaker extraction network can be shared in the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Since the speaker extraction network can directly extract information from speech without text, so a  large number of data without text annotation in the VC task can be used for training, which can help  to improve the transfer learning ability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Prosody: The prosody information represents how the speaker says the content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' It  is independent of the speaker information and related to the way of expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Since the pitch  information can reflect the rhythm of speech, therefore, the pitch information is used to extract  prosody information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The prosody information, like speaker information, can be shared by both TTS  and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In addition, in the training process, we can obtain pitch information from the ground truth  speech, but there is no ground truth in the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, a pitch prediction module (Ren  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020) is proposed in the training stage, which takes the speech content information and speaker  information as the input to predict the pitch information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2  SPEECH CONTENT WITH VECTOR QUANTIZATION  Since there are different ways to obtain the speech content in TTS and VC, it is very easy to de-  viate between the two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' If this deviation occurs, it will cause devastating damage to some  downstream shared modules (such as pitch predictor and decoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, to ensure that the  consistency of the two speech content domains as close as possible, we reconstruct them from two  aspects:  First, we use the shared codebook to quantify the continuous feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Second, we use the labeled data to narrow two discrete feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  The detailed process is described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Suppose that the vector obtained by the text encoder in  TTS pipeline is Cp = (C1  p, C2  p, · · · , CT  p ) with length T, the vector obtained by the content encoder  in VC pipeline is Cs = (C1  s, C2  s, · · · , CT  ′  s ) with length T  ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' It should be noted that we add a length  regulator module in the text encoder to solve the problem of length mismatch between the text and  speech sequence, which is introduced in FastSpeech (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, if text and speech  are paired, T  ′ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The vector Cp and Cs is a sequence of continuous vector in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Due to the  large representation range of continuous features, Cp and Cs are difficult to match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We borrow  the discretization method for latent variables from Vector Quantized Variational AutoEncoder (Van  Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Specifically, for each time step t, the continuous latent representations Ct  p in  Cp can be mapped into C  t  p by finding the nearest pre-defined discretized embedding in the dictionary  as:  C  t  p = ek,  k = argminj  ��Ct  p − ej  ��  2  (1)  where ej is the j-th embedding in the codebook dictionary, and j ∈ 1, 2, · · · , V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Since selecting  the entry with the minimum distance will cause the operation to be non-differentiable, the straight-  through gradient estimator can be used to approximate the gradient, which can be expressed as:  ¯ht = ht + ev − sg (ht) ,  v = arg min  k  ∥ht − ek∥2  (2)  where sg(·) is the stop-gradient (Van Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017) operation that treats its input as constant  during back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  After vector quantization, the quantized sequence Cp  =  (C  1  p, C  2  p, · · · , C  T  p ) and Cs  =  (C  1  s, C  2  s, · · · , C  T  s ) can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' It should be noted that when Cp and Cs are quantized into  Cp and Cs, they share the same codebook e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The advantage of this is that since the speech content  features Cp in the TTS pipeline are independent of the speaker, sharing the same codebook can  help learn the speech content features Cs independent of the speaker in the VC pipeline, which is  essential for the VC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Although both pipeline use the same codebook for coding, there is no guarantee that there is no  deviation between the two fields after coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, to further eliminate the deviation between  the two domains, we use the labeled speech in the TTS pipeline to supervise the training of the two  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Specifically, for paired text and speech data, we constrain the feature distance between the  quantized sequence Cp and Cs:  4  Lpair =  ��Cp − Cs  ��2  2  (3)  In this way, we can efficiently minimize the domain discrepancy by using limited labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Figure 2: Structure of vector quantized operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3  UNIFYSPEECH PIPELINE  An overview of our proposed UnifySpeech architecture is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' It consists of a  sequence-to-sequence TTS, and a sequence-to-sequence VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The key idea is to share most of the  module parameters (speaker encoder, prosody encoder, decoder and pitch predictor) and map the  speech content in TTS and VC to the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As mentioned above, the UnifySpeech allows  us to train the model on the concatenation of both the labeled and unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For supervised  training with labeled data, both models can be trained independently by minimizing the loss between  their predicted speech and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For unsupervised training with unlabeled data, the VC  pipeline can be trained, and the parameters are shared with TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  To further clarify the training process, we unrolled the framework as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1  TTS PIPELINE  Denote the text and speech sequence pair (x, y, F0) ∈ D, where D is the paired text and speech  corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Each element in the text sequence x represents a phoneme or character, while each element  in the speech sequence y represents a frame of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' F0 is the pitch information of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The  representation obtained after three encoders are speech content C, speaker S and prosody P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Then, the three parts are added and input into a decoder to obtain the predicted speech y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In addition,  to obtain the pitch information in the interference stage, we use the content information and speaker  information to predict the F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' These processes can be expressed as:  y′ = decoder(C + S + P)  (4)  F0′ = pitch predictor(C + S)  (5)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' the reconstruction loss in TTS process includes two parts:  LV C  rec = MSE(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' y′) + MSE(F0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' F0′)  (6)  5  C  Look up min  Look up min  distance  distance  vector  vector  1  V  L2  L2  distance  ev  distance  Ct  Cs  Text encoder  Content encoderIn addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' we use the content encoder in the VC pipeline to extract the content representation Cs  for the training speech in TTS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' and close the distance between the two domains by calculating the  distance loss of Cs and Cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' which is explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Lpair =  ��Cp − Cs  ��2  2  (7)  The loss of TTS pipeline can be expressed as:  LT T S = LT T S  rec  + Lpair  (8)  where MSE denotes the mean squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2  VC PIPELINE  Denote all the unlabeled or labeled speech (y, F0) ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We first extract three information from  (y, F0), which are speech content Cs, speaker Ss and prosody Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Then, similar to the TTS pipeline, the shared decoder and pitch predictor module is used to predict  the speech signal y′ and F0′, which is similar to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  The loss of VC pipeline only includes reconstruction loss, which can be expressed as:  LV C = MSE(Y, Y ′) + MSE(F0, F0′)  (9)  where MSE denotes the mean squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3  TRAINING PROCESS  With such a unified framework, TTS and VC can learn from each other through joint training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The  details of the algorithm can be found below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Procedure 1 UnifySpeech training algorithm  1: Input: Paired speech and text dataset (x, y), speech only dataset y  ′  2: repeat  3:  # A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' TTS pipeline with speech-text data pairs  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Sample paired speech and text (x, y)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Extract speech content information from x for domain loss  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Generate the predict speech y, pitch F0 and speech content from text  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Calculate the loss for TTS LT T S  rec  4:  # B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' VC pipeline with speech-only data  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Sample paired speech and text (x, y)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Extract speech content information from y for domain loss  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Calculate the domain loss for the two pipeline Lpair  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Sample speech y  ′ in speech only dataset  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Generate the predict speech y, pitch F0 and speech content from speech y  ′  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Calculate the loss for VC LV C  5:  # C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Loss combination:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Combine all loss (LT T S  rec , Lpair, LV C) into a single loss variable  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Calculate TTS and VC parameters gradient  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Update TTS and VC parameters with gradient descent optimization  6: until convergence  4  EXPERIMENTS AND RESULTS ANALYSIS  In this section, we conduct experiments to evaluate our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The experiments are  carried out from two aspects: zero-shot TTS, zero-shot VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1  DATASETS  Two datasets are used to simulate labeled data and unlabeled data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' VCTK dataset, an  English language dataset containing 44 hours of speech and 109 speakers is used as labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Each speaker has approximately 400 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' LibriTTS (Zen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019) are used as unlabeled  data, which consists of 585 hours of speech data from 2484 speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We only use the speech data in  LibriTTS and discard the text for unsupervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In this way, it can simulate the scene where  a large number of speech that we can obtain are unlabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We use a 16-bit, 22050 Hz sampling rate  for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The 80-dim Mel spectrogram is extracted with Hann windowing, frame shift  of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='5-ms, frame length of 50-ms, and 1024-point Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In this experiment, we use  hifigan (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020) as vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2  MODEL DETAILS  Figure 3: Structure of each module in UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' FFT (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019) means feed-forward  Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  The detail of each module in our proposed method is shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Specifically, to make  the sequence of speech content extracted from the text encoder and content encoder equal, a length  regulator is added to the text encoder, which is inspired by the FastSpeech (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The  structure of the duration predictor is the same as that in FastSpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The structure of the decoder  and content encoder is similar, but the dimensions of input and output are opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For the prosody  encoder, we first quantize F0 of each frame to 32 possible values and encode them into a learnable  embedding vector according to the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' And we change the output of the pitch predictor into a  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' By this way, pitch prediction becomes a classification task, reducing the difficulty of  frame-level pitch prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We use the speaker encoder in StyleSpeech and fuse features with Style  Adaptive Layer Norm(SALN) (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021) method to make the comparision fairly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  The number of feed-forward Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017) (FFT) blocks in the text encoder is  4 and it is 6 in the decoder module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In each FFT block, the dimension of hidden states is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The  kernel sizes of all the 1D-convolution are set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The dropout rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The dimension of the  last linear layer in the decoder is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The dimension of last linear layer in encoders (text encoder,  pitch encoder, content encoder) is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' An Adam optimizer (Kingma & Ba, 2014) is used to update  the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The initial learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='001 and the learning rate decreased exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  7  Linear layer  Linear layer  Duration  Length regulator  predictor  m block FFT  不  不  LN + Dropout  n block FFT  不  Speaker  Speaker  Embedding  ConvlD + ReLU  Phoneme  Embedding  Embedding  Linear layer  Linear layer  Pooling  个  LN + Dropout  LN + Dropout  FFT  ConvlD + ReLU  ConvlD + ReLU  LN + Dropout  LN + Dropout  不  Pitch Embedding  ConvlD + ReLU  ConvlD + ReLU  Randomly select a clip  LSTM  Quantify to [0, 32]  from target speaker4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3  ZERO-SHOT TTS  We first carried out zero-shot TTS task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We choose four speakers from VCTK that are not used  during training process as target speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For each speaker, we randomly select about 20 sentences  to be our target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Then we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek,  1993) (Mel-cepstrum distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR  (correlation factor of F0) between synthesized speech and ground truth speech as the objective met-  rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 1: Objective evaluation results for zero-shot TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  F0 RMSE (Hz)  MCD (db)  V/UV  F0 CORR  UnifySpeech  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='51  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='9%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='93  StyleSpeech  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='63  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='06%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='92  Subjective evaluation was also conducted to compare the speech’s quality and similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We choose  mean opinion scores(MOS) for naturalness and similarity mean opinion scores(SMOS) for similar-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Both metrics are rated in 1-to-5 scale and reported with the 95% confidence intervals (CI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 2: Mean opinion score (MOS) of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' With VC means the model is jointly trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  MOS  SMOS  GT  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='15  −  GT mel + Vocoder  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='15  −  StyleSpeech  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  UnifySpeech-TTS (with VC)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  To better show our method’s effectiveness, we demonstrate the t-SNE projection of the speaker  embedding vectors from speakers in both VCTK and LibriTTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' (Van der Maaten & Hinton, 2008)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 4 shows the speaker visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For the seen speakers (x) and unseen speakers (o), the  corresponding speaker embedding form a cluster and distinct from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The boundary between  different speakers is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' This shows that the speaker encoder performs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Figure 4: Speaker visualization of generated speeches,where the circle and triangle indicate unseen  speaker,while x and square indicate seen speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='4  ZERO-SHOT VC  We carried out zero-shot VC task, using unseen speakers voice to be the reference speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As we  are lack of parallel corpus, we only conduct subjective evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' But VC and TTS shares the same  8  30  p225  p226  p227  20  p228  p279  X  p274  10  p307  p311  8012  0  ￥5181  3885  2961  10  1089  7021  20  KX  20  15  10  5  0  5  10  15speaker encoder, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 4 in zero-shot TTS can also be a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Just as in zero-shot TTS task, we choose mean opinion scores(MOS) for naturalness and similarity  mean opinion scores(SMOS) for similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Both metrics are rated in 1-to-5 scale and reported with  the 95% confidence intervals (CI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 3: Mean opinion score (MOS) of the VC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' With TTS means the model is jointly  trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  MOS  SMOS  GT  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='15  −  GT mel + Vocoder  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='15  −  UnifySpeech-VC(without TTS)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='06  UnifySpeech-VC(with TTS)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  It can be found that when VC pipeline is trained alone, its performance is poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In other words, it  doesn’t have the ability to discriminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' But jointly training with TTS improves its speech decou-  pling ability, indirectly improving the speaker modeling ability, which we will analysis later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='5  ABLATION EXPERIMENT  To figure out whether jointly training is effective, we separately train the TTS and VC parts, carrying  out objective evaluation on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We also remove the VQ parts and test the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For TTS,  we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek, 1993) (Mel-cepstrum  distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR (correlation factor of F0)  between synthesized speech and ground truth speech as the objective metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 4: Objective evaluation results for zero-shot TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  F0 RMSE (Hz)  MCD (db)  V/UV  F0 CORR  UnifySpeech-TTS(without VC)  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='31  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='55  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='9%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='91  UnifySpeech-TTS-novq(with VC)  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='41  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='58  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='92  UnifySpeech-TTS(with VC)  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='51  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='9%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='93  For VC, we separately calculate the Average Cosine Similarity (ACS) (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2022) for the  embedding from same (S-ACS) and different speakers (D-ACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' And then their ratio is used as an  evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 5: Ratio of ACS from same and different speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Unseen or seen means whether the speaker  is from test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' With or without TTS means whether the VC pipeline is trained with TTS pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  S−ACS  D−ACS (unseen)  S−ACS  D−ACS (seen)  UnifySpeech-VC (with TTS)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='0  UnifySpeech-VC (without TTS)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='0  To validate whether VQ can bridge the gap between the speech content parts in TTS and VC, we  caculate the L2 distance between the phoneme representation from TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As in VC there  are many frames represent same phoneme, so we choose their clustering center as the correspond-  ing phoneme representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We randomly select 4 sentences from validation set to carry out the  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 6: L2 distance between the phoneme representation in TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' S1 means sentence1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  S1  S2  S3  S4  average  UnifySpeech-VC(without VQ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='464  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='476  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='474  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='553  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='492  UnifySpeech-VC(with VQ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='152  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='205  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='208  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='205  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='193  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='6  RESULT ANALYSIS  Above results show that jointly training actually improves TTS’s speaker modeling ability and VC’s  speech decoupling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' And VQ enhances the consistency of representation of the same content  from phonemes and speeches, ensuring the model’s correctly working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  For TTS, sharing modules with VC enables unlabeled data to participate in its training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Along with the richer speaker style pattern, the speaker style modeling capability has been enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  For VC, when trained alone, it doesn’t have the speaker modeling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As the self-supervised  training process aims at reducing the reconstruction loss of source audio, the reference audio of the  target speaker isn’t crucial in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Only the content encoder and the decoder are enough for  the process, that’s possibly why the speaker embeddings are all similar, though they are from differ-  ent speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' When jointly trained, the text loss plays a role of regularization factor, resulting that the  content encoder just extracting the content information from the source speech (speaker information  is reserved and others are discarded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' This makes the reference speech with rich speaker information  become indispensable for the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Thus the model’s speaker modeling ability has  been improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  5  CONCLUSIONS  In this paper, we propose UnifySpeech, a unified framework for TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Both task benefits  from the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Due to training with large amounts of unlabeled data, their few-shot modeling  ability makes progress as well as the synthesized speech’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In the future, further improving  the synthesized speech’s quality and making the generated speech’s style more similar to target  speaker will be our endeavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  REFERENCES  Eric Battenberg, RJ Skerry-Ryan, Soroosh Mariooryad, Daisy Stanton, David Kao, Matt Shannon,  and Tom Bagby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Location-relative attention mechanisms for robust long-form speech synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In  ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing  (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+page_content=' Glow-tts: A generative flow for  text-to-speech via monotonic alignment search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+page_content='  Jaehyeon Kim, Jungil Kong, and Juhee Son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Suyoun Kim, Takaaki Hori, and Shinji Watanabe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Joint ctc-attention based end-to-end speech recog-  nition using multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In 2017 IEEE international conference on acoustics, speech and  signal processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 4835–4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+page_content=' De-  tailed information about the two captured datasets is provided  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' iV2V covers sidelink communication scenarios between  Automated Guided Vehicles (AGVs), while iV2I+ is conducted  at an industrial setting where an autonomous cleaning robot is  connected to a private cellular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The combination of dif-  ferent communication technologies, together with a common mea-  surement methodology, provides insights that can be exploited  by Machine Learning (ML) for tasks such as fingerprinting,  line-of-sight detection, prediction of quality of service or link  selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Moreover, the datasets are labelled and pre-filtered for  fast on-boarding and applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The corresponding testbeds  and measurements are also presented in detail for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Index Terms—Measurement data, QoS prediction, AGV, drive  tests, V2X, campus networks, wireless communications  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' INTRODUCTION  It is anticipated that the next generation of wireless commu-  nication systems (5G and beyond) will bring about an upsurge  in the number of new services and applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' each of which  demanding for a specific Quality of Service (QoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In parallel,  there is a resurgence of interest in promoting the concept of  predictive Quality of Service (pQoS), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', QoS estimation for a  given time instance in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This can be done in different  prediction horizons, ranging from milliseconds to hours or  even days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' pQoS can pave the way to satisfy a very demanding  set of QoS requirements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', very low latency, minimum  Signal-to-Noise Ratio (SNR), delay, packet error rate, or huge  Uplink (UL) or Downlink (DL) throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  pQoS can be particularly important for wireless networks  in the industrial domain, where communication needs to be  highly reliable due to, among other reasons, its integration into  This work was supported by the Federal Ministry of Education and Re-  search (BMBF) of the Federal Republic of Germany as part of the AI4Mobile  project (16KIS1170K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The authors alone are responsible for the content of  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  control loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Wireless links are especially relevant in mobile  setups, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', with one or more Automated Guided Vehicles  (AGVs) connected in a Vehicle-to-vehicle (V2V), Vehicle-to-  infrastructure (V2I) or Vehicle-to-everything (V2X) manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In this regard, some datasets are available for automotive sce-  narios to train and test Machine Learning (ML) algorithms and  thus enhance such schemes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' However, the availability of  datasets from industrial and indoor measurement campaigns,  such as [2], is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' With proper knowledge of the upcoming  QoS conditions, pQoS can facilitate the proper operation  of industrial applications to guarantee human-machine safe  interaction or robot cooperation to fulfill a common task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Other use cases may include tele-operated driving, high-  density platooning, and High Definition (HD) map collecting  and sharing for optimal route selection [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In this manner, we can see a growing tendency toward  applying deep learning algorithms for pQoS applications, such  as [5]–[8], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A consolidated overview of the ML-  enabled throughput prediction scenarios is presented in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In the same vein, [6] investigates a ML-model to predict the  throughput in a non-standalone 5G network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  ML has been a very active research area in the past few  years and there is ample literature around it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' however, its real-  world implementation or validation has remained elusive for  industrial communication due to its high dependency on avail-  able datasets to test, validate and generalize the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Therefore, creating a reference dataset from experimental  testbeds or practical simulations is paramount to evaluate the  underlying theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In this paper, we will describe the industrial Vehicle-  to-vehicle (iV2V) dataset and the industrial Vehicle-to-  infrastructure plus Sensor (iV2I+) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' These two datasets  aim to pave the way for future advancement in the experi-  mentation of mobile networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The measurement campaigns  that were conducted here are part of a bigger measurement  framework and procedure that is described in detail in [9]  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='03364v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='NI]  20 Dec 2022  with some first results being reported in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In  Section II and Section III, we describe the iV2V and iV2I+  testbed and datasets and we elaborate on their details and  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We conclude with an overview of possible future  research directions in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' THE IV2V TESTBED AND DATASET  In this section, we present the first of the two collected  datasets and the considerations that have been taken into  account for its measurement campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We give a brief in-  troduction to the sidelink technology, continue with a detailed  description of the testbed, and finally describe the processing  and resulting data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Figure 1a depicts one of the AGVs, carrying the measure-  ment and communication hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The AGVs communicate  directly in a V2V manner, using the sidelink technology as  introduced by 3rd Generation Partnership Project (3GPP) in  Release 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the sidelink setup, every AGV acts both as  transmitter and sender (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Testbed Components  1) Sidelink: Sidelink has been standardized in 3GPP during  4G and 5G mobile networks to define a framework where  communication is possible with and without network coverage  and with varying degrees of interaction between the devices  and the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Two modes of resource allocation are defined  [11]:  Network-based resource allocation (Mode 1 in 5G  sidelink and Mode 3 in 4G sidelink): This mode is only  available when all the devices are in network coverage,  and the network selects the resources and other transmit  parameters used by the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Autonomous resource allocation (Mode 2 in 5G sidelink  and Mode 4 in 4G sidelink): This mode offers a com-  pletely decentralized solution in which the User Equip-  ments (UEs) autonomously select the resources and other  transmit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Overall, network-based resource allocation can outperform  the autonomous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This is due to the network controlling  the resources to be used by each of the UEs involving UL  signaling from the UE to the network to obtain a grant for  transmissions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  On the other hand, autonomous resource allocation is mainly  useful when there is no possibility of having network coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The basic operation of autonomous resource selection involves  the device performing sensing within a pre-configured resource  pool, detecting which resources are not in use by other devices  with higher-priority traffic, and choosing some of these free  resources for its transmissions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The autonomous resource  allocation is more prone to collisions while also suffering from  hidden node and half duplex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Solutions have been  considered in 3GPP to mitigate these issues [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  For the measurement campaign, we use a full stack,  software-based, standard-compliant and open implementation  of the 3GPP Release 14 PC5 Mode 4 standard [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The plat-  form allows research concepts and standard features to be val-  idated in hardware testbeds and it provides interfaces and tools  for recording measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Changes and adjustments are  possible at every layer, which allows a realistic verification of  new features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The sidelink software (all layers incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' baseband  processing) can be run on standard general purpose computing  hardware in connection with suitable Software Defined Radio  (SDR) hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We opted for a full stack implementation,  thus providing a standard based IP to IP (one to all) interface  for any application, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', all protocols on OSI layer 3 and higher  can be transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The hardware setup is shown in Figure 1c  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' [14] for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  2) Localization: Precise position of the communicating  devices is required to link the environmental conditions with  the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' For that purpose, the position information  provided by the AGVs, carrying the communication entities,  was recorded during the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Two types of local-  ization methods are used by different AGVs in the testbed,  namely, marker/track-based, and Simultaneous Localization  and Mapping (SLAM)-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the former method, the AGVs  follow a track on the floor with help of an onboard camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Additionally, Radio-frequency identification (RFID) tags are  placed on the track to provide the exact position information  to the AGVs, when they pass over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Between the RFID tags,  the AGVs estimate their position by using odometry, which  describes a method of estimating the position and orientation  of a mobile system using data from its propulsion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Wheel-driven systems use the measurement of the wheel rota-  tions for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In combination with dead reckoning, odometry  is a basic navigation method for ground-based vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Since  the AGVs do not leave the track, the transversal error is in the  order of few mm, while the longitudinal error depends on the  separation distance between the RFID tags and the positional  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  For our testbed, the longitudinal error was in the order of  few cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the latter localization method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', SLAM-based,  the AGVs are equipped with a laser scanner to detect and  estimate the distance to landmarks (reference points) in the  testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' These landmarks are also defined in the map of the  AGV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Hence, the AGVs can estimate their position in  the map through a combination of information from several  landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the testbed, we achieve a position accuracy in  the order of few cm with the SLAM-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The reported AGV position was timestamped during the  measurements, so that a combination with other measured data  is possible during post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Unless otherwise stated,  the measurements scenarios presented below consider that the  SLAM-based AGVs were static and the marker/track-based  AGVs were moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Since the moving AGV is guided by an optical line, the real  lateral position is better than ±2 mm (3σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The along track  (longitudinal) error while passing an RFID tag has a timing  uncertainty of up to 30 ms, which gives an error depending  on the actual speed (up to 30 mm at 1m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Dead reckoning  results in additional errors being displayed due to the route  (a) AGV testbed in industrial environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' (b) Schematic illustration of sidelink mea-  surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  (c) Sidelink SDR platform with all components inte-  grated in 19 inch case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 1: iV2V testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  and steering angle sensors not being perfectly adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The  longitudinal error increases with the length of the unsupported  route driven without an RFID tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The repeatability of the  position information is less than ±2 mm transversally and less  than +2 cm longitudinally at the driven speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  3) Time Synchronization: To enable accurate evaluation  of network latency and other QoS properties, a proper time  synchronization is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The time was synchronized across  sidelink devices by running NTP over Ethernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The error is  typically in the order of several µs with a worst case error of  1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' However, precisely quantifying this is difficult without  specialized measurement equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  4) Controlled Packet Generation: To ensure a highly pre-  cise packet generation, we used a network packet generator  tool based on Real-time User Datagram Protocol (UDP) Data  Emitter (RUDE) & Collector for RUDE (CRUDE) which  is able to produce heterogeneous UDP network traffic for  realistic network workloads [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' It consists of two main  modules: RUDE generates traffic to the network, which is then  received and logged by the other module of the network with  CRUDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We extended the packet generator tool by enabling  the capability to log all channel information for successfully  received packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  5) Automatic Gain Control: In order to be able to assess  the received signal quality, which is an elementary quantity for  assessing the QoS, the function of the Automatic Gain Control  (AGC) in a receiver must be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' AGC is a feature in an  RF receiver signal path that is used to keep the received signal  magnitude at a suitable level for subsequent signal processing  so that signals are not clipped and the receive path with good  sensitivity is operated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The function of the AGC is technically  realized by controllable amplifiers in the received signal path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  For this purpose, the signal level is determined in the signal  processing during special preambles at the beginning of a  defined data frame and regulated within a specified range  by setting the AGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A criterion for the regulation can be  the evaluation of the preamble of the Orthogonal Frequency-  Division Multiplexing (OFDM)-based signal obtained from  the I/Q samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' However, the corresponding values at the  antenna input, namely Received Signal Strength Indicator  (RSSI) and Reference Signal Received Power (RSRP), are of  interest for signal evaluation and a corresponding indication of  comparable level values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' With knowledge of the amplification  and attenuation of individual components in the front-end and  the AGC setting, these can be determined from the measured  values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' from the total gain between the antenna input and  the evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The AGC calculations have already been  performed before the output and the values supplied for pre-  processing are the correct values and can be used as is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Measurement Scenarios  We collected data for roughly 10 hours over the course  of two days to acquire almost 50 GB communication data  between up to three industrial AGVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A schematic of the test  area and its surroundings is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The dotted  gray line depicts the track used by AGV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Several obstacles,  depicted in different blue tones in the figure, were located  within the test area to achieve different radio propagation  conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', Line-of-Sight (LOS) and Non-Line-of-Sight  (NLOS) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The obstacles were rearranged during the  measurement campaign to create two scenarios, A and B, with  different N/LOS characteristics, as marked in light blue in the  figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The Dataset  1) Captured Sidelink Data: For each scenario illustrated  in Figure 2, we capture the sidelink channel parameters for  every transmitter/receiver pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The selected sidelink channel  parameters and their description are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The  parameters in the table are obtained/estimated from the De-  modulation Reference Signal (DMRS) of the Physical Sidelink  Shared Channel (PSSCH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The AGV localization data is provided as x and y coordi-  nates in a local coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  2) Dataset Pre-processing: In this section, we describe the  pre-processing of captured sidelink data, and we present a  dataset constructed with the pre-processed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The dataset  is constructed in a tabular format where each row represents  a sample and the columns contain the value of the mea-  sured sidelink channel parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Note that the side channel  RF Frontend  3)  ise  Shecker  SAt  BasebandPC  SDR46cm  X  Wall Height ~3,10  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='5m  22m  65cm  1m  123cm  (0,0)  AGV  2  170cm  AGV  3  263cm  251cm  2  3  AGV  1  1  AGV Test Track  Glass door/window  Foam wall  Metallic wall  Foam wall in scenario B only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Replaced by metallic wall  in scenario A  Foam wall in scenario A only  2  3  1  AGV 1 moves along  the test track  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 2: Illustration of measurement scenarios A & B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  TABLE I: Selected iV2V Data Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Parameter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Description  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='SNR [dB]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Derived from noise and power estimations of DMRS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='RSRP [dBm]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Average energy per carrier/RE for DMRS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='RSSI [dBm]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Signal power over the whole band  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Noise Power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Estimated on DMRS in decoded subframe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Time [sec]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Receive time of first IQ-Sample of decoded subframe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Frame Number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='System frame number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Subframe Number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='System subframe number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='UHD Rx Gain [dB]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Receive antenna gain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='SCI FRL N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Starting subchannel of decoded PSSCH  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='SCI FRL L  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Number of used subchannels for PSSCH  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='RLC SN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Sequence number of radio link control header  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Location  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Local x and y coordinates of AGV 1 on the track  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='parameters and AGV location are measured independently  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='and simultaneously in different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' With this setting,  each measuring device embeds its own timestamp into the  measured parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Since the processing of the received  signal in different devices requires different lengths of time,  the embedded timestamps in these devices also have some  differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We align the timestamps between the location  data and the sidelink data as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Given the timestamp  tloc  n corresponding to the measured location Ln = [xn, yn] of  the AGV at nth time step, we find the timestamp tsl  n from the  measured sidelink such that  ��tloc  n − tsl  n  �� ≤ γ, where γ denotes  the alignment tolerance, and we use γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='005s to construct  the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In addition, we present indicators Mn ∈ Z and Sn ∈ Z,  where Mn indicates the measurement scenario (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', the place-  ment of obstacles), and Sn denotes the source of the received  sidelink signal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', AGV1 receives a signal from AGV2 or  from AGV3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' With the parameters described in Table I, the nth  row of the tabular dataset is denoted by Rn, and it contains the  parameters as Rn =  �  tloc  n , tsl  n,  ��tloc  n − tsl  n  �� , Ln, Pn, Sn, Mn  �  ,  where Pn ∈ RK denotes the K measured parameters of  sidelink channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' THE IV2I+ TESTBED AND DATASET  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Testbed Components  The testbed for the measurements is located in an industrial  co-working space in Berlin, with a layout as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The hall had a gateway, which allowed the AGV to drive  outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  1) The AGV: The AGV used in the testbed is an au-  tonomous cleaning robot from the company Enway, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' They are specially designed for use under the  operating conditions of the manufacturing industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The sweeper has all the necessary navigation data saved on  a digital map, and drives over the cleaning area autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Thanks to high-performance sensors and control software  from Enway, the AGV navigates the environment completely  independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Using a combination of laser distance mea-  surement and cameras, the robot captures the environment  in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This 360-degree view enables very safe  navigation between people, complex production lines, and  overhanging systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The AGV immediately detects obstacles  that suddenly appear along the route, and drives around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Additional equipment such as floor markings, QR codes, or  magnetic tracks are not required for navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the event  that the AGV encounters an unsolvable situation, the robot  stops and reports automatically to Enway headquarters via a  data connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The remote team monitors every movement  of the device around the clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The specialists can end  autonomous journeys at any time, and can take control from a  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Collisions with people, production systems, vehicles,  and stored goods are thus avoided at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The machine  can also be navigated manually by the operating personnel  on site, if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Because the AGV is a retrofitted ride-  on sweeper, it can be controlled from the driver’s seat in  the traditional sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Ongoing use of the autonomous sweeper  can be monitored, controlled, and then evaluated using mobile  devices such as smartphones, laptops, or tablets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The software  completely logs the cleaning trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  2) Cellular Network: The mobile network used for the  measurements in this test bed corresponded to a standardized  4G campus network with TDD medium access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The bandwidth  was 20 MHz in the frequency band between 3700 and 3800  MHz approved by the Federal Network Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A correspond-  ing frequency assignment was applied for the period of the  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The hardware consisted of a server running  the LTE core and a radio base station with integrated antennas  connected to the server via Gbit LAN and powered via Power  over Ethernet (PoE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The location of the base station is marked  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3c with a yellow circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A Mini-PC with the Linux  operating system was used as the UE to carry out QoS-relevant  measurements on the AGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A Quectel RM500Q-GL card was  used as the radio device, which was connected to the Mini-PC  via USB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' External antennas were connected to the radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The server provides a service interface to which the appli-  cations required for the measurements can be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The  stationary applications can communicate with mobile applica-  tions running in the Mini-PC via the service interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This  (a) The Autonomous Cleaning Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  (b) Architecture of the iV2I+ Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' (c) Map of the Environment as captured by  the LIDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Walls and AGV tracking route  are shown with red and black, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The yellow circle is the location of the base  station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3: iV2I+ testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  way, the location-dependent data rate and latency parameters  relevant for evaluating the QoS can be determined at the Mini-  PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  3) Time Synchronization: The AGV and the server were  time synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3b visualizes the communication set-  up, where the dotted arrow shows the wireless connection  and the solid line the cable connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Both the server and  the Mini-PC on the AGV were connected to a GPS receiver,  allowing accurate time synchronization by using Pulse-per-  Second (PPS) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The maximum error is typically in µs-  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' However, the AGV only had consistent GPS reception  at the start point and the outdoor area, which could lead to  inaccuracies the ms-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  4) LTE Modem Access: We used Mobile Insight, an open-  source cross-platform application for mobile network moni-  toring and analytics to capture mobile network data at the  Mini-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' It collects mobile network information across several  cellular protocols, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Radio Resource Control (RRC) or  EPS Mobility Management (EMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' During the measurement  campaign, the available information was logged every 40 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Additionally, a Python script that accesses the modem  via a virtual serial interface, a few radio parameters (RSSI,  RSRP, Reference Signal Received Quality (RSRQ), Signal-  to-Interference-plus-Noise Ratio (SINR)) were logged every  200 ms by the modem and written to a file with a time stamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The data can then be linked to other data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' the location,  via the common time stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  5) Controlled Packet Generation: We used the application  iperf3 on the Mini-PC and the server to generate UDP traffic  in either UL or DL direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Apart from the generated iperf3  log files, tcpdump was used both on the Mini-PC and server  to capture all incoming and outgoing packets at the respective  network interfaces, allowing a more detailed evaluation on a  packet level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Measurement Scenarios  For the measurement campaign, one AGV, equipped with  the sensors and measurement devices described in the prior  section, was driving through the testbed area over the course  of three days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Overall, 16 hours of data were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  UL and DL communication was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Two types of  packet flows were established to generate high and medium  throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In DL, high throughput measurements were con-  ducted with a throughput target of 80 Mbps, in UL with  25 Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Approximately twice as many high throughput  measurements as medium throughput measurements were col-  lected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The route contained a diverse set of radio conditions,  namely: LOS and NLOS situations, coverage loss as well  as indoor and outdoor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The measured path is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The Dataset  For each scenario, data from the described network compo-  nents and the sensors from the AGV was collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A subset  of the captured data is presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  TABLE II: Selected iV2I+ Data Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Parameter  Description  SINR [dB]  Derived from noise and power estimations of DMRS  RSRP [dBm]  Average energy per carrier/RE for DMRS  RSSI [dBm]  Signal power over the whole band  Throughput  Acquired throughput in respective link direction  Ping [ms]  Time in ms until a ping reply was received  Jitter  Delay variation measured over 1s  Odometry  Fused position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' orientation and speed of the AGV  Map static elevation  Single pre-computed map of the whole area  Near/far map obstacles  36 m2/400 m2 obstacle map around the AGV  LIDAR  3D point cloud with obstacles  1) AGV-Sensor Data: The AGV delivers a series of sensor  data via its Robot Operating System (ROS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' including the last  fourth rows in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Except for Light Detection and Rang-  ing (LIDAR), all ROS topics shown here are obtained through  MENWAY  IENWAY  21178  ITENWAY  100% AUTONOMOUS | 100% ELECTRIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='sensor fusion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', through techniques such as Extended  Kalman Filter (EKF)) and provide the relevant information  from a wireless perspective: position, orientation and speed  of the AGV and location of walls and obstacles in different  formats (2D, 3D, offline and online, near and far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The raw  input for the sensor fusion comes from sources including pure  wheel odometry, drive commands, an Inertial Measurement  Unit (IMU) and the already mentioned LIDAR, all of them  also available in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Pre-Processing  Similar to what was described in Section II-C2 the data was  pre-processed to further simplify the work with the collected  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Moreover, and since the position of the base station is  known and fixed, the distance and clearance of the wireless  link can be easily inferred from the sensor data and is provided  as part of the pre-processed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  We have merged the GPS logs, the LTE stack measurements  and the throughput measurement together with the sensor-  based link distance and link clearance into a single dataframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  As these data streams have different sampling frequencies, we  re-sampled as needed to 1 second before the final merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' CONCLUSION  In this paper, we have describe industrial Vehicle-to-vehicle  (iV2V) and industrial Vehicle-to-infrastructure plus Sensor  (iV2I+), two testbeds for wireless communications in indus-  trial settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We have provided detailed information about the  components of the testbeds, together with the initial concept  and captured scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' As mentioned before, the described  datasets are publicly available for ML research, what we  consider a valuable contribution to the available industrial  datasets both in terms of size and quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  iV2V and iV2I+ contain extensive and complete data that  we believe to be highly useful to answer questions regarding  the use and generalisation of ML for mobile use cases in  industrial environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Indeed, both datasets can be used  to train and evaluate methods for pQoS, which is a crucial  enabler for high reliability in wireless industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Moreover, pQoS in general, and our data in particular can be  used as an ingredient for ML algorithms that optimize the  network itself, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', by performing proactive radio resource  management (RRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This type of new network capabilities  will be an important part of the evolution of wireless com-  munications, such as the 6G cellular evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Finally, the  addition of AGV sensor data and localization opens the gate  to advanced techniques like fingerprinting or channel charting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  REFERENCES  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+page_content=' Morgenroth, and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+page_content=' Li, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Bizon, “High-Resolution Radio Environment  Map Data Set for Indoor Office Environment,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Available:  https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='21227/waxd-9525  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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JNE4T4oBgHgl3EQfhg0d/content/tmp_files/2301.05125v1.pdf.txt

@@ -0,0 +1,1586 @@
+Adaptive Dynamic Global Illumination
+SAYANTAN DATTA, McGill Unviversity, Canada
+NEGAR GOLI, Huawei/AMD, Canada
+JERRY ZHANG, Huawei, Canada
+𝑡 = start
+𝑡 = start + 4𝑠
+𝑡 = start + 8𝑠
+Direct only
+Direct + Indirect
+Tunnel interior (D + I)
+SSIM/MSE:
+0.989/0.007
+SSIM/MSE:
+0.974/0.008
+SSIM/MSE:
+0.970/0.008
+SSIM/MSE:
+0.957/0.008
+SSIM/MSE:
+0.942/0.009
+SSIM/MSE:
+0.939/0.010
+SSIM/MSE:
+0.993/0.000
+SSIM/MSE:
+0.953/0.003
+SSIM/MSE:
+0.932/0.004
+SSIM/MSE:
+0.837/0.003
+SSIM/MSE:
+0.826/0.009
+SSIM/MSE:
+0.875/0.010
+Left: [email protected]
+Right: [email protected]
+Fig. 1. Our technique demonstrated on a modified Bistro Exterior scene containing 192 × 64 × 192 probes.
+The third row shows the changes inside the tunnel as the gate opens over time. Our techniques responds faster
+to a dynamic stimuli and offers 1.7-times higher performance compared to the Q-DDGI implementation even
+with large probe grid containing excess of 2.3 million probes. Q-DDGI, detailed in section 5, is an extension of
+vanilla DDGI making it more competitive and comparable against our approach.
+We present an adaptive extension of probe based global illumination solution that enhances the response
+to dynamic changes in the scene while while also enabling an order of magnitude increase in probe count.
+Our adaptive sampling strategy carefully places samples in regions where we detect time varying changes in
+radiosity either due to a change in lighting, geometry or both. Even with large number of probes, our technique
+robustly updates the irradiance and visibility cache to reflect the most up to date changes without stalling
+the overall algorithm. Our bandwidth aware approach is largely an improvement over the original Dynamic
+Diffuse Global Illumination while also remaining orthogonal to the recent advancements in the technique. 1 2
+CCS Concepts: • Computing methodologies → Ray tracing; Rasterization.
+Additional Key Words and Phrases: Adaptive sampling, irradiance probes, global illumination, real-time
+1
+INTRODUCTION
+Global illumination (GI) strikingly improves the realism of a virtual scene, but its high computational
+cost has been a long-standing challenge in its application to real-time rendering [22].
+1Project Page
+2Poster
+1
+arXiv:2301.05125v1  [cs.GR]  12 Jan 2023
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Several real-time GI solutions have been proposed, such as screen space [43] techniques, which
+support fully dynamic scenes but suffer from quality issues due to the limited availability of
+information in screen space. On the other hand, baked texture light-maps only support static
+geometry but remain popular due to their simplicity, low run-time cost, and quality. Precomputed
+Radiance Transfer [51] combined with light probes [31] and light-maps [15] solved some of the issues
+plaguing static light maps; in particular, these approaches support semi-dynamic geometry and self-
+occlusion while adhering to a strict compute budget. The advent of real-time ray-tracing hardware
+set the stage for modern fully dynamic GI. Dynamic real-time GI methods build upon the decades
+of research in sampling, and amortization of shading and visibility across space (pixel/world), angle,
+and time to improve convergence [40]. Adaptations of several offline techniques such as photon
+mapping [17], many-light rendering [20, 62], and radiosity maps [54] have also been explored
+in the context of modern [26, 27] ray-tracing capable hardware. However, presence of noise in
+sampled algorithms require the use of strong denoisers. Machine learning denoisers [6, 66] have
+demonstrable advantages in terms of quality compared to more traditional frequency [32] or
+variance [46] based denoisers. However, the prospect of training a neural network, the added
+complexity of integrating machine learning inference with traditional graphics pipeline, and the
+proprietary nature of machine learning frameworks have stalled the industry-wide adoption of
+these techniques. The recent probe-based algorithm, Dynamic Diffuse Global Illumination (DDGI)
+[28], extending the classic irradiance probes, still remains an excellent choice due to its relative
+simplicity, quality, and cloud streaming capabilities [14, 53]. However, scaling of DDGI in its original
+formulation is limited, and approaches such as multi-grid hierarchy and probe rolling [29] are
+necessary to scale it across large environments. Our adaptive approach focuses on dynamic contents
+in environments containing millions of probes in a single hierarchy.
+We propose Adaptive Dynamic GI (ADGI) algorithm where we trace a few pilot rays per frame to
+scan the environment and build a coarse representative model of the dynamic events. Using Markov-
+Chain sampling, we dynamically allocate resources to the critical areas, improving convergence in
+those regions. While DDGI allocates a fixed number of samples per probe and uniformly distributes
+samples across directions, ADGI non-uniformly samples the joint spatio-angular domain of the
+discretized 5D light-field represented by the probes. Our approach essentially decouples resource
+allocation from the number of probes resulting in a user-controlled performance target (FPS)
+and improved scaling even with millions of probes. Additionally, our approach results in faster
+convergence in static and dynamic environments given equal render time. Our approach is drop-in
+compatible with the original implementation and its several other extensions such as probe rolling
+and probe volume hierarchies [29].
+We achieve these objectives by formulating a guided function approximation technique, which is
+purposefully accurate in specific regions highlighted by our guiding function and thus eliminates
+the need for uniform resource allocation. Furthermore, we develop a sampling methodology based
+on temporal Markov-chain, which adapts naturally to a dynamic environment while also enabling
+scaling across large number of probes. Finally, we discuss memory and bandwidth preserving color
+compression schemes tailored specifically for our purpose.
+2
+RELATED WORK
+Probe-base approaches: Modern games rely extensively on light probes for static and dynamic
+global illumination due to their ease of integration into the game engine pipeline at low run-time
+cost. Some advocate a uniform grid probe placement due to their simplicity while others have
+proposed non-unform probes due to their efficiency. Probe based techniques are usually prone to
+light leakage. As such, uniform grid approaches [28, 31] use additional information, stored in the
+probes to determine whether a probe is visible from a shade point. Non-uniform approaches may
+2
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Guiding function
+ℎ(𝑥) = 𝑝 (𝑥) × 𝑙 (𝑥)
+𝑝 (𝑥)
+𝑙 (𝑥)
+𝑥
+𝑥
+State of
+Environment
+Sample
+Feedback
+(a) Construct the guide ℎ(𝑥).
+𝑥
+Metropolis sampling
+𝑥𝑖
+ℎ(𝑥)
+(b) Sample 𝑥𝑖 ∼ ℎ(𝑥).
+𝑥
+𝑥𝑖
+𝑔(𝑥)
+(c) Evaluate objective 𝑔(𝑥𝑖).
+𝑥
+𝑥𝑖
+𝐸𝑟
+𝑔
+^𝑔
+^𝑔(𝑥)
+(d) ^𝑔(𝑥) - Reconstruct 𝑔(𝑥)
+from 𝑔(𝑥𝑖). 𝐸𝑟 indicates the
+reconstruction error.
+Fig. 2. Figure showing the steps in our adaptive-sampling strategy. We define a guiding function ℎ(𝑥) that
+highlights (in yellow) the interesting regions of the domain. The samples 𝑥𝑖 obtained from ℎ(𝑥) are used to
+evaluate the objective 𝑔(𝑥𝑖). Our goal is to obtain an approximate representation of 𝑔(𝑥), denoted as ^𝑔(𝑥),
+from the (𝑥𝑖,𝑔(𝑥𝑖)) pairs. As more samples are obtained from the highlighted region, the reconstruction error
+is lower in the yellow area, as shown in sub-figure (d).
+use carefully curated probe placement [63] combined with spatial data-structures like octrees to
+determine the visibility of a probe from a surfel. McGuire et al. [28, 31] stores the depth values
+of the surrounding geometry from a probe and use a similar idea as Variance-Shadow-Mapping
+[9] to approximate visibility. However, non-uniform approaches has been mostly limited to static
+geometry due to their high initial construction cost. Some approaches use rasterization [31, 63]
+while other may use ray-tracing [28] to compute the probe content. Probe based techniques also
+differ on how they store the information in the probes. Some use discrete textures [28, 31] while
+other may use a compressed basis representation such as Spherical Harmonics [14, 55]. Spherical
+harmonics implicitly pre-filters the content before storage but may cause light and dark ringing
+issues. Memory bandwidth required for reading and writing from the probes is also a major concern.
+Texture compression [31, 53] is usually the preferred choice to minimize memory bandwidth.
+Bandwidth is also crucial for cloud streaming of probe data. In such scenarios, Spherical Harmonics
+[14] representation may be preferable as they provide excellent compression for low frequency
+data. At run-time, dynamic probe based [28] GI solutions uniformly distributes rays across probes
+to update their content; this quickly becomes a bottleneck as the number of probes increases. Our
+approach on the other hand, focuses on the optimal distribution of resources to maximize visual
+fidelity. Various extensions have also been proposed to increase scalablity [29] of uniform grid
+approaches such as multiple-volume hierarchies and probe rolling. Our approach remains largely
+orthogonal and fully compatible with these extensions.
+Adaptive sampling: Adaptive sampling has been used in the context of screen-space ray-traced
+global illumination where more samples are accumulated in regions with high noise and high
+frequency [12]. Adaptive sampling is also useful for filtering soft shadows [32], where pilot-rays
+model the spatial frequency of shadow-penumbra and provide the number of additional samples
+required at each pixel to improved convergence. Neural versions [13] of adaptive sampling has
+also been proposed where a neural network generates a sampling-map that is tightly coupled to a
+post-process neural-denoiser. Conceptually our approach is similar, but our execution is tailored
+for the problem of temporally coherent sampling of probes. We refer readers to section 8 for an
+extend related work in irradiance-caching, screen-space GI and MCMC techniques
+3
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+3
+OVERVIEW
+We focus on two primary issues with DDGI in its original formulation. First, the technique does
+not allow for the non-uniform allocation of resources, resulting in unnecessary probe updates in
+regions that are not crucial for visual fidelity. Seconds, it does not update the probes quick enough
+to reflect transient changes in the scene environment. Our adaptive strategy involves detecting
+the changes in the environment and allocating resources driven by the detected changes. While
+the detection phase requires allocating additional resources, our empirical evaluations suggest our
+non-uniform adaptive sampling compensates for the lost efficiency in the detection phase. Our
+detection phase also enables fast probe updates for capturing transient changes in the scene. We
+model our technique as guided function approximation where we approximate a continuous function
+(e.g. 5D light-field) using a discrete (e.g. probes) representation driven by a guiding function.
+A naive approach to approximate a continuous function is to discretize the domain and reserve a
+representative sample for each discretization. The strategy is useful when the domain is relatively
+small; however, as the domain gets larger or the number of discretizations increases, it is prohibi-
+tively expensive to update all discretizations in real-time. This is one of the issues plaguing the
+original DDGI technique. In many applications, it is not necessary to update the entire domain
+uniformly; instead, we can tolerate more approximation errors in some regions than others. A
+simple example is foveated rendering, where errors in the periphery are less intrusive than those
+near the gaze center. In our case, we need the most accuracy in probes contributing to final shading.
+We introduce the notion of guiding function, which highlights the regions where a higher
+reconstruction accuracy is desired. We define the guide using a product of terms - the first term
+represents the current state of the environment while the second term is a feedback from the
+sampled cache. We sample the guide using a temporally coherent Markov-chain and use the
+samples to update our approximate representation using a parallel thread-safe approach. Thus our
+approach is summarized in three steps - defining a guiding function, sampling the guide, and using
+the samples to update the approximate representation. We describe these steps in sections 3.2, 3.3
+and 3.4 while we discuss various implementation specific details in section 4. See figure 2.
+Our approach provides two distinct advantages compared to the original DDGI - approximation
+quality and scalability. At any time, we concentrate our resources on a potentially challenging
+area as opposed to the entire domain. Provided our guide correctly identifies the challenging
+regions, the quality is improved due to a higher concentration of resources in the appropriate
+region. Since we sample the guide independent of the number of discretizations, the decoupling
+allows for a high number of statically allocated probes without affecting run-time performance.
+Increased discretizations improve approximation quality while the independence of sampling from
+the number of discretizations improves scalability. More specifically, we transparently increase
+the number of discrete probes without affecting performance. The run-time performance depends
+on the number of samples we generate; the samples are channeled to the appropriate areas by the
+guiding function. Our Markov-chain sampling is highly parallel, temporally coherent, and scalable,
+making it suitable for real-time temporally distributed reconstruction of large probe grids.
+3.1
+Background
+Here we briefly describe the original DDGI algorithm. DDGI consists of a 3D grid of directionally
+resolved irradiance probes that are updated in real-time through hardware ray-tracing. The probes
+also contains visibility information to prevent light leakage. The probe representation has many
+benefits, it performs optimally for diffuse indirect transport and is relatively inexpensive to encode
+and decode information to and from the probes. The algorithm evenly distributes ray-samples
+outwards from the probe center at each active probe in a stochastic rotated spiral pattern. DDGI is
+4
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Uniform
+probe grid
+placement
+Generating &
+tracing rays
+Evenly distribute
+Probe state update
+Update irr &
+vis 2𝐷 atlas
+Shade each
+point
+8 cage probe
+Fig. 3. The figure illustrates the main steps of DDGI algorithm. Algorithm defines a uniform grid of probes
+and trace uniform-random rays in all direction from each probe. Based on the hit information, we compute
+the visibility (vis) and irradiance (irr) and update the 2𝐷 atlas. We also update the probe states based on
+visibility information (back-face hit ratio). Finally, for each shade-point, we query the eight bounding probes
+surrounding it and interpolate them to compute incoming indirect illumination.
+a two step algorithm. First, it updates the shading on the probe texels. Next a screen-space pass
+where the up-to-date probe content is used for shading the camera-pixels. The probe texel values
+are encoded into a spherical-mapped diffuse irradiance-texture with 8 × 8 resolution. Probes also
+captures the average ray-hit distance, and squared distances to the nearest geometry at 16 × 16
+resolution. DDGI temporally filters the probe texels by blending in the new values using a fixed
+hysteresis. The visibility data is used to decide whether a probe is visible at a shade-point and also
+used to infer whether a probe is inside a geometry and deactivated. The probe’s state is not limited
+to on or off and can vary with scenarios [29]. The world-position of the screen-space pixel is used
+as a key to the probe-texture lookup. The lookup interpolates the corresponding eight probes of
+the grid voxel containing the shade-point. The algorithm is illustrated in figure 3. DDGI algorithm
+is suitable for diffuse and slow changing phenomena in time. Therefore DDGI, combined with our
+adaptive-sampling strategy is a reasonable real-time GI approximation for dynamic scenes.
+3.2
+Guiding function
+As summarised in section 3 and figure 2, a guiding function highlights the important areas in
+the domain, i.e., challenging regions where more resources are required. These highlighted areas
+receive more adaptive samples, reducing the approximation error in those regions. Mathematically,
+the domain of the guiding function ℎ : 𝑅𝑑 → 𝑅 is the continuous 5D light field. Upon query, the
+guide function returns a scalar value indicating the importance of a sampled point. In our case,
+𝑑 = 5 as the domain is a 5-dimensional space of world-space positions and directions, and the guide
+encodes the importance of sampling a direction on a probe (texel’s importance).
+We model the guiding function (ℎ) as a product of two terms. The first term, we call 𝑓 : 𝑅𝑑 → 𝑅,
+represents the value in sampling a texel based on our understanding (limited) of whether such a
+texel would contribute towards the final screen-space shading. The second term is the observed
+sampled evidence (a.k.a irradiance cache) as they become available. Initially, the irradiance cache is
+empty but filled progressively through sampling. We define the first term based on some heuristics
+that describes our understanding of the probe-environment:
+• Probes closer to the camera,
+• Probes closer to geometric surfaces,
+• Directions on the probes facing away from geometric surfaces,
+• Directions on the probe with higher incoming irradiance,
+• Directions with temporal change in irradiance and visibility
+We trace pilot rays from the probes to generate the information necessary to quantify the above
+heuristics. We also call it the detection phase where we pre-scan the scene environment for changes.
+We denote the individual heuristics as 𝑓𝑖 : 𝑅𝑑 → 𝑅, and compose them into its final form 𝑓 as
+shown in equation 1, where 𝜙 represents a composition function. The composition function is
+5
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Algorithm 1: Metropolis algorithm
+Input: ℎ: Guide distribution, 𝑀 : No. of iterations
+Input: 𝐾: No. of initial samples to reject
+Output: 𝑥 : Sample
+Ensure: 𝑀 ≥ 2, and 𝐾 < 𝑀
+1 𝑗 ← ShaderInvocationIndex()
+2 𝑥0 ← 𝑆[𝑗]
+// Initialize Markov-chain from memory
+3 while 𝑖 ← 0 to 𝑀 − 1 do
+4
+𝑥𝑖+1 ← RandomWalk (𝑥𝑖,ℎ(𝑥𝑖))
+// Random walk step, algorithm 5
+5
+if 𝑖 > 𝐾 then
+/* Use sample 𝑥𝑖+1 for probe updates, see algorithm 2
+*/
+6 𝑆[𝑗] ← 𝑥𝑖 + 1
+// Save Markov-chain state
+simply a recipe to appropriately combine the individual heuristics. We quantify the individual
+heuristics (𝑓𝑖) in section 4.1 and the composition (𝜙) in section 4.2.
+𝑓 = 𝜙(𝑓0, 𝑓1, ..., 𝑓𝑖).
+(1)
+The second term uses the stored irradiance in the probes, denoted by ^𝑔, to modulate the first
+term. We model the second term as - 𝑒𝑥𝑝 (𝛼 · ^𝑔(𝑥)/𝑓 (𝑥)), where the scalar 𝛼 ∈ [0, ∞) indicates our
+confidence in the irradiance probe content; a higher value indicating greater confidence. Note that a
+stored texel with high irradiance value may or may not have a high contribution to the final shading.
+Example - in a dynamic environment the probe content from the last frame is quickly outdated
+and thus less useful. The parameter 𝛼 models this uncertainty. The term 𝑓 (𝑥) in the denominator
+ensures that we only trust ^𝑔(𝑥) when 𝑓 (𝑥) is low. Finally, we define the guiding function as:
+ℎ(𝑥) = 𝑒𝑥𝑝
+�
+𝛼 · ^𝑔(𝑥)
+𝑓 (𝑥)
+�
+· 𝑓 (𝑥).
+(2)
+3.3
+Sampling the guide
+Next we sample the guiding function (equation 2). Mathematically, given an unnormalized distribu-
+tion ℎ : 𝑅𝑑 → 𝑅, our goal is to obtain samples 𝑥𝑖 from ℎ(𝑥), where 𝑥𝑖 ∈ 𝑅𝑑.
+Our sampling algorithm is straightforward. We use the Metropolis sampling, as shown in al-
+gorithm 1 to sample ℎ. The algorithm randomly initializes a state (𝑥0 ∈ 𝑅𝑑) and moves the state
+forward based on the acceptance of a newly proposed state. We generate the proposed states by
+perturbing the current state with a zero-mean Gaussian noise, also known as Random-walk [5].
+Parallelism: Note that algorithm 1 runs as a shader invocation, meaning several instances of
+the chain run in parallel. Each instance is independent with its own memory to load and store the
+chain state (denoted by S[] in algorithm 1). The instances generate thousands of samples per frame.
+As an input to our algorithm, we explicitly specify the number of chains that run in parallel, thus
+controlling the number of adaptive samples and performance. Contrasting with the original DDGI,
+the number of samples in the original implementation is proportional to the number of probes
+which increases cubically with scene dimensions. As such, it is difficult to scale up when the scene
+gets larger or when using a denser probe grid. Our approach is independent of the discretization
+resolution and scales better to higher probe counts without compromising approximation quality.
+Mixing-time: Initially, a Markov chain requires many iterations for the chain to generate
+samples from the target distribution (here ℎ(𝑥)), a phenomenon known as mixing time. We avoid
+6
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Table 1. List of symbols
+Symbol
+Description
+Remarks
+𝑓
+Heuristics model
+Section 3.2
+ℎ
+Guiding function/Target distribution
+Section 3.2, 3.3
+𝑔
+Objective function
+Symbolic proxy for 𝑔𝑟, 𝑔𝑐.
+Section 3.4
+^𝑔
+Approximation of objective function
+Symbolic proxy for ^𝑔𝑟, ^𝑔𝑐.
+Section 3.4
+𝑔𝑟
+5D Light field
+Section 3.4
+𝑔𝑐
+Chebychev visibility
+Section3.4
+^𝑔𝑟
+Approximation of 5D light field
+(Irradiance cache)
+Section 3.4, 4.5
+^𝑔𝑐
+Approximation of Chebychev visibility
+(Visibility cache)
+Section 3.4, 4.6
+𝑥 or 𝑥𝑖
+Markov-chain samples
+Symbolic proxy for 𝑝𝑖, 𝜔𝑖.
+Section 3.3
+𝑝𝑖
+Positional (∈ 𝑅3) component of 𝑥𝑖
+–
+𝜔𝑖
+Directional (∈ 𝑅2) component of 𝑥𝑖
+–
+this problem by bootstrapping the initial chain state from the last frame. As such, we keep the
+number of iterations per frame small, but over frames, the chain effectively accrues many iterations.
+Distribution stationarity: Markov chain sampling requires the target distribution ℎ(𝑥) remain
+stationary. Due to a dynamic scene environment, the stationarity condition is seemingly violated.
+This may affect the approximation quality of our technique if the distribution changes rapidly
+between frames. However, we have several contingencies to deal with the issue. First, we target high
+frame-rates, which minimizes the change in the target distribution between consecutive frames.
+As an additional margin of safety, we reject initial 𝐾 samples per frame as shown in algorithm 1,
+line 5. This ensures our usable samples are obtained closer to the target distribution. Note that
+the evaluation time for ℎ(𝑥) negligible and thus rejecting few initial samples per frame does not
+significantly impact performance. We also smooth out the target distribution (see section 4.1.4)
+using spatio-temporal convolution to minimize abrupt changes in the target across frames.
+Temporal tracking: Since our target distribution may vary with time, we require the samples
+generated from the Markov-chain to closely follow the distribution to capture the transient changes
+in the environment. We make some crucial modifications to our sampling algorithm to allow for
+fast tracking of the target distribution, which we discuss in detail in section 4.9.
+3.4
+Approximation
+With samples obtained from the highlighted (figure 2(b)) parts of the domain, we focus on using
+the samples to evaluate (figure 2(c)) and reconstruct (figure 2(d)) our objective function. The term
+objective function refers to the quantity we aim to approximate. Mathematically, we denote our
+objective function as 𝑔 : 𝑅𝑑 → 𝑅𝑐, and its approximate reconstruction as ^𝑔. For ADGI, we have
+two objective functions - the light field 𝑔𝑟 : 𝑅5 → 𝑅3, and Chebychev-visibility 𝑔𝑐 : 𝑅5 → 𝑅2
+surrounding the probes. We denote their approximate reconstructions as the irradiance cache ^𝑔𝑟,
+and the visibility cache - ^𝑔𝑐 respectively. See section 4.5 and 4.6 for more details.
+Updating ^g: We evaluate the continuous objective function 𝑔 at collected sample points 𝑥𝑖 and
+store the evaluations - 𝑔(𝑥𝑖) into ^𝑔, as shown in algorithm 2. For ADGI, the evaluation step involves
+7
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Algorithm 2: Approximation algorithm
+Input: 𝑥: Markov-chain samples
+1 function UpdateRepresentation(𝑥):
+2
+𝑣 ← 𝑔(𝑥)
+// Evaluate sample, ray-trace
+3
+AtomicMovingAvg(𝑥, 𝑣)
+// Populate ^𝑔, see algorithm 4
+tracing a ray to query the local light field and visibility. At each Metropolis iteration, the evaluated
+samples update the closest entry in the probes (^𝑔) within a critical section construct.
+Representing ^g: Prior work represent ^𝑔 as either as discrete LUTs [28], continuous Spherical
+Harmonics [14], Neural Networks [36], or any combination. In our case, the choice to use a discrete
+representation is based on several factors. First, multiple parallel streams of Markov-chain samples
+may update the same memory location in ^𝑔. As such, provisions are necessary to prevent race
+conditions. We also need a representation that handles temporal accumulation and quickly update
+itself to reflect any transient changes in the scene. Finally, the representation must be bandwidth
+efficient to improve the read and write performance. We refer to section 4.5 and 4.9 for details.
+3.5
+MCMC analysis
+In this section, we analyze our adaptive sampling algorithm in the context of MCMC (Markov
+Chain Monte Carlo). Note that our goal is not variance reduction through importance sampling;
+rather the focus is guided approximation of the objective function via sampling the target function.
+As such, unlike importance sampling, the sampling function is not necessarily correlated to the
+integrand. With this distinction in mind, we first look at the equation driving importance sampling
+using MCMC and then repurpose it for guided function approximation.
+The following equation shows a typical case of importance sampling where the objective is to
+compute the integral
+∫
+ℎ(𝑥)𝑔(𝑥)𝑑𝑥 and there exists a strategy to sample from h(x). In many typical
+scenarios (e.g. full Bayesian inference), the distribution ℎ(𝑥) is a proper distribution (
+∫
+ℎ(𝑥)𝑑𝑥 = 1)
+but does not have an efficient sampling mechanism. This where Markov Chain MC is useful.
+∫
+ℎ(𝑥)𝑔(𝑥)𝑑𝑥 ≈
+�
+1
+𝑀
+𝑀−1
+∑︁
+𝑖=0
+𝑔(𝑥𝑖)
+� ∫
+ℎ(𝑥)𝑑𝑥, 𝑥𝑖 ∼ ℎ(𝑥).
+(3)
+In contrast, our choice of Markov Chain (Metropolis) is primarily technical - simplicity, GPU
+parallelism and temporal sample tracking. Nevertheless, the same equations provide meaningful
+insight - albeit in a different context of adaptive sampling. In our algorithm, we simply sum the
+samples obtained from the target distribution without taking into account the sample density. This
+is equivalent to computing the following:
+𝐼 = 1
+𝑀
+𝑀−1
+∑︁
+𝑖=0
+𝑔(𝑥𝑖), 𝑥𝑖 ∼ ℎ(𝑥).
+(4)
+While our goal is to estimate
+∫
+Ω 𝑔(𝑥)𝑑𝑥, the expectation of 𝐼 (rearranging equation 3) is:
+E [𝐼] =
+∫
+Ω ℎ(𝑥)𝑔(𝑥)𝑑𝑥
+∫
+Ω ℎ(𝑥)𝑑𝑥
+,
+(5)
+where Ω is the domain of integration. Clearly, the expected value of 𝐼 does not converge to the
+correct estimate -
+∫
+Ω 𝑔(𝑥)𝑑𝑥. However, there are two factors to consider - size of the domain Ω and
+8
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+shape of ℎ(𝑥) in the domain. First consider the limit case where Ω → 0. In this case, the integrals
+collapses to a point evaluation and indeed the expected value of 𝐼 equals the unbiased estimate as
+shown below.
+𝐿.𝐻.𝑆. = lim
+Ω→0
+∫
+Ω ℎ(𝑥)𝑔(𝑥)𝑑𝑥
+∫
+Ω ℎ(𝑥)𝑑𝑥
+=
+∫
+Ω ℎ(𝑥)𝑔(𝑥)𝛿(𝑥 − 𝑥0)𝑑𝑥
+∫
+Ω ℎ(𝑥)𝛿(𝑥 − 𝑥0)𝑑𝑥
+= 𝑔(𝑥0).
+(6)
+𝑅.𝐻.𝑆. = lim
+Ω→0
+∫
+Ω
+𝑔(𝑥)𝑑𝑥 =
+∫
+Ω
+𝑔(𝑥)𝛿(𝑥 − 𝑥0)𝑑𝑥 = 𝑔(𝑥0).
+(7)
+In the above equation, 𝛿 is the Kronecker delta. The result is important as it shows with increasing
+probe resolution, bias is reduced. However, reducing texel size is not always practical as more rays
+and memory are required to populate and store a high resolution probe. Notice how the term ℎ(𝑥)
+is cancelled in equation 6. When the domain of integration is sufficiently small, ℎ(𝑥) is practically
+constant and the term cancels out in the denominator and numerator.
+We now consider the shape of ℎ(𝑥). While the target h(x) varies globally, it is piece-wise constant
+at a local scale due to its tabular nature. More crucially, the target ℎ(𝑥) is stored at a much lower
+resolution compared to the irradiance probe ^𝑔(𝑥). This implies ℎ(𝑥) is practically constant across a
+texel of the irradiance probe. The expected value of 𝐼 for the 𝑘𝑡ℎ texel is thus given by:
+E [𝐼𝑘] =
+∫
+𝑇𝑘 ℎ(𝑥)𝑔(𝑥)𝑑𝑥
+∫
+𝑇𝑘 ℎ(��)𝑑𝑥
+=
+∫
+𝑇𝑘 𝑐𝑘𝑔(𝑥)𝑑𝑥
+∫
+𝑇𝑘 𝑐𝑘𝑑𝑥
+=
+∫
+𝑇𝑘 𝑔(𝑥)𝑑𝑥
+∫
+𝑇𝑘 𝑑𝑥
+,
+(8)
+where 𝑇𝑘 represents the domain of 𝑘𝑡ℎ texel and 𝑐𝑘 represents the piece-wise constant value of
+ℎ(𝑥) when 𝑥 ∈ 𝑇𝑘. The area estimate
+∫
+𝑇𝑘 𝑑𝑥 is fixed for all texels and equivalent to 4𝜋/#𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛.
+Thus, due to the tabular nature of our target function, the estimates of irradiance texels remain
+un-biased. While performing texture filtering over irradiance texels, it is possible to compute an
+unbiased estimate by weighing the texel values with 𝑐𝑘 as follows:
+𝐼 𝑓 𝑖𝑙𝑡𝑒𝑟
+𝑘
+=
+∑︁
+𝑗 ∈N𝑘
+𝑤𝑘−𝑗𝐼𝑘−𝑗, 𝑤𝑖 = 𝑐𝑖/
+∑︁
+𝑗 ∈N𝑘
+𝑐 𝑗,
+(9)
+where N𝑘 represents the texels in the neighbourhood of texel 𝑘. The values 𝑐𝑖 are obtained by
+querying the probes storing ℎ(𝑥). Note that bias is unavoidable as we blend samples temporally
+in a dynamic environment. In a dynamic environment, the objective is evolving and the bias
+manifests itself as temporal lag. Practically however, within a small time window, both ℎ(𝑡) and
+𝑔(𝑡) are assumed constant and the samples can be blended using a windowed moving average.
+Note that windowed moving average requires storing historical information. A cheaper but biased
+approximation to windowed moving average is exponential moving.
+4
+IMPLEMENTATION DETAILS
+This section provides the several implementation details with a brief summary in figure 4.
+4.1
+Heuristics construction
+The section describes the construction of 𝑓 using the heuristics discussed in section 3.2. Our goal is
+to measure and quantify the heuristics that highlight the probes which actively contribute to the
+final shading and require additional resources for faster convergence. We represent the heuristics
+either parametrically (equation 10) or using an explicit LUT representation as shown in figure 5(a).
+The LUT is constructed such that each probe has eight texels corresponding to an octant. We trace
+a ray for each octant; the rays return the hit distance and incoming irradiance at the hit-point.
+From this information, we compute several quantities (equation 11 - 18) and store them in the
+9
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+World space
+Trace 8
+pilot rays
+Octahedral
+Quantify heuristics
+𝜙 (𝑓0, 𝑓1, ..., 𝑓𝑖)
+Model feedback
+𝑒𝑥𝑝 (𝛼 · ^𝑔/𝑓 )
+Guide function
+𝑝 (𝑥) × 𝑙 (𝑥)
+Metropolis sampling
+ℎ(𝑝,𝜔)
+(𝑝𝑖,𝜔𝑖)
+Irradiance
+Visibility
+𝜔𝑖
+−𝜔𝑖
+𝑝𝑖
+𝑝𝑖
+Trace ray
+Update irradiance, visibility
+cache
+Irradiance ( ^𝑔𝑟 )
+8 × 8
+Visibility ( ^𝑔𝑐)
+16 × 16
+Deferred shading
+Fig. 4. This figure illustrates our overall algorithm. We trace 8 pilot rays, one from each octant on the
+probe and approximate the heuristic model 𝑓 (𝑝,𝜔). Using the heuristic and feedback, we define the guide
+ℎ(𝑝,𝜔) and sample it using Metropolis sampling. The sampled (𝑝𝑖,𝜔𝑖) are used to trace more adaptive ray
+samples, gathering hit-distance and irradiance at the sample points. We update the probe-cache (^𝑔) with
+adaptive-samples. The cache is used in the next shader and also looped back as feedback to model the target.
+LUT/texture mapped to the probe octants. We define and evaluate the following heuristics for a
+probe at position 𝑝 and a direction 𝜔.
+4.1.1
+Distance from camera. A probe far away from the camera is less likely to contribute to the
+final shading. We represent this parametrically as described in equation 10, where 𝑝 represents
+probe position, 𝑐 camera position and 𝑘 is a threshold set by the user.
+𝑓𝑐 (𝑝,𝜔) =
+�
+1
+if ||𝑝 − 𝑐|| < 𝑘 ,
+𝑒−( ||𝑝−𝑐 ||−𝑘)
+otherwise.
+(10)
+4.1.2
+Probe visibility. Only the probes encompassing a geometry participates in the deferred
+shading. Thus, probes closer to a geometric surface are more important. Similarly, texels facing
+away from the surface are queried more often for shading. We express both quantities together in
+equation 11, where 𝑝 represents probe location and 𝑡 = 𝑡𝑟𝑎𝑐𝑒(𝑝, −𝜔). The function trace returns
+the distance of the nearest surface hit, and the scalar 𝑠 is the diagonal distance of a grid voxel.
+𝑓𝑣(𝑝,𝜔) = 𝑒−2𝑡/𝑠
+(11)
+4.1.3
+Incoming radiance. We consider directions with high incoming radiance as more impor-
+tant. To identify those directions, we query the radiance along each probe octant and use it as a
+representative for incoming radiance.
+𝑓𝑟 (𝑝,𝜔) = 𝑚𝑖𝑛(𝑟, 𝛽)
+𝛽
+,
+(12)
+where 𝑟 = 𝑙𝑢𝑚(𝑝,𝜔). The function lum returns the incoming luminance using direct illumination
+at the surface hit point. The parameter 𝛽 controls the dynamic range and we set 𝛽 = 5.
+4.1.4
+Probe visibility change . Detection of dynamic geometry is crucial for increased resource
+allocation in regions affected by these changes. We detect dynamic geometry by computing a
+temporal gradient of probe visibility followed by a spatio-temporal smoothing operation.
+𝑓0(𝑝,𝜔) = 𝑓 𝑡
+𝑣 (𝑝,𝜔) − 𝑓 𝑡−1
+𝑣
+(𝑝,𝜔),
+(13)
+where 𝑓 𝑡
+𝑣 , 𝑓 𝑡−1
+𝑣
+represent visibility in the current and last time step respectively. Equation 13
+implicitly states we keep the position and the direction fixed when measuring the time difference
+across frames to avoid noisy gradients. The gradient is passed through a temporal trigger (𝑇𝑟) as:
+10
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+a. Probes storing prior-information (𝑓 ).
+octant-mapped
+pilot-rays
+b. Irradiance cache ( ^𝑔𝑟 ).
+c. Visibility cache ( ^𝑔𝑐).
+Fig. 5. Figure showing various probe-mapped textures and LUT in our technique.
+𝑓1(𝑝,𝜔) = 𝑇𝑟 (𝑓0(𝑝,𝜔),𝜃) ,
+(14)
+where 𝑇𝑟 converts a pulse in time to a decaying signal controlled by the parameter 𝜃 as shown
+in figure 6(a). For simplicity, we drop the time axis from the function 𝑇𝑟. The function minimizes
+temporal discontinuities, thus helping the Markov-chain to closely follow the target distribution
+(ℎ) across frames. Finally, we perform a spatial convolution as follows:
+𝑓Δ𝑣(𝑝,𝜔) =
+∑︁
+𝑖,𝑗
+𝑓1(𝑝 − 𝑝𝑖,𝜔 − 𝜔𝑗).
+(15)
+The convolution step smooths out uncertainties in a single texel and also serves as a weak
+predictor of possible locations of the dynamic geometry in the next frame. We use a 5 × 5 × 5 and
+3 × 3 convolution in space and direction, respectively.
+4.1.5
+Probe radiance change. Similar to the previous section, we detect a change in radiosity
+using a temporal gradient of the probe radiance. We apply the same temporal trigger and spatial
+convolution operator as in the previous section. The corresponding equations are as follows:
+𝑓2(𝑝,𝜔) = 𝑓 𝑡
+𝑟 (𝑝,𝜔) − 𝑓 𝑡−1
+𝑟
+(𝑝,𝜔),
+(16)
+𝑓3(𝑝,𝜔) = 𝑇𝑟 (𝑓2(𝑝,𝜔),𝜃) ,
+(17)
+𝑓Δ𝑟 (𝑝,𝜔) =
+∑︁
+𝑖,𝑗
+𝑓3(𝑝 − 𝑝𝑖,𝜔 − 𝜔𝑗).
+(18)
+4.2
+Heuristics composition
+Now that the individual heuristics are defined, as described in equation 1, we compose them for
+the static and dynamic cases as follows:
+𝑓𝑠 (𝑝,𝜔) =
+𝑠𝑡𝑎𝑡𝑖𝑐
+����
+𝑓𝑐 𝑓𝑣 ,
+(19)
+𝑓𝑑 (𝑝,𝜔) =
+𝑑𝑦𝑛𝑎𝑚𝑖𝑐
+��������������������������������
+𝑓𝑐 𝑓𝑣(𝑓Δ𝑣 + 𝜇𝑓Δ𝑟) .
+(20)
+When the environment is static, we sample according to the camera and probe-to-surface distance
+heuristics denoted by 𝑓𝑐 and 𝑓𝑣 in equation 19. In the dynamic case represented by equation 20, we
+modulate the changes in the environment by the static term 𝑓𝑐 𝑓𝑣. The modulation indicates we are
+more interested in changes close to the camera and geometric surfaces. The factor 𝜇 weighs the
+strength of change in geometry versus change in lighting. We use 𝜇 = 2 in all our experiments.
+11
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+𝑇𝑟 (𝑡; 𝑣,𝜃)
+𝑣
+Temporal-pulse
+𝜃
+Linear decay
+start
+𝑡
+a. Transform temporal-pulse to a decaying signal.
+𝑓𝑐 ≥ 0.75
+𝑓𝑐 ≥ 0.5
+|𝑐𝑙𝑖𝑝𝑥𝑦 | ≤ 1.2
+|𝑐𝑙𝑖𝑝𝑥𝑦 | ≤ 1.4
+b. Defining clip volumes for probes.
+Fig. 6. Figure (a) shows the construction of temporal-trigger𝑇𝑟 (𝑣,𝜃). In figure (b), we call the volume bounded
+by the blue frustum and black boundary as inner volume 𝑉𝑖𝑛. Similarly, outer volume 𝑉𝑜𝑢𝑡 is the volume
+bounded by green frustum and outer grey boundary. All probes in 𝑉𝑜𝑢𝑡 participate in the heuristic modelling,
+as described in section 4.4. Probes inside the blue frustum participate in adaptive sampling as described in
+section 4.8, 4.9. We set the probe state 𝑁 = 16 for all probes outside 𝑉𝑖𝑛 but inside 𝑉𝑜𝑢𝑡, refer section 4.8.
+4.3
+Heuristics storage
+We store the quantities 𝑓𝑣, 𝑓𝑠, 𝑓𝑑 as a 6-10-10 bit encoded 32 bit integer at each octant of the probes.
+The remaining 6 bits are used for other flags. When querying the LUT/texture, we use a mapping
+function that maps the continuous position 𝑝 and direction 𝜔 to the corresponding texel in the
+LUT. We note that 𝑓𝑐 is implicitly defined, hence do not require additional storage.
+4.4
+Improving construction efficiency
+The heuristics construction step is a potential bottleneck if we trace 8 rays per probe for all probes
+in the scene. As such, we restrict the pilot-rays to the probes that are contained within an extended
+camera frustum as shown in figure 6(b). To maximize the efficiency of our algorithm, we further
+reuse the samples collected from the 8 pilot-rays to populate the irradiance ( ^𝑔𝑟) and visibility ( ^𝑔𝑐)
+caches. We change the ray-directions at alternate frames in an AABBCCDD... pattern, improving
+the detection of temporally varying light-field surrounding the probes. We measure the time-delta
+(equation 13 and 16) between two frames with identical set of ray-queries, avoiding noisy gradients.
+However, this effectively halves the detection frequency (frame-rate / 2) but improves the spatial
+awareness. We use a stratified-random ray-direction such that there is always one ray per octant.
+We update the irradiance and visibility cache at each alternate frame.
+4.5
+Probe irradiance cache
+As shown in figure 5(b), the irradiance cache ( ^𝑔𝑟) is represented as a uniform probe grid in space
+where each probe stores the surrounding diffuse irradiance at a 8 × 8 texel resolution using a
+spherical mapping. At each texel, we store the irradiance in a custom RGB encoding with 9-9-8 bits
+for the three channels. The remaining 6 bits (out of 32bit) store the sample accumulation count
+(N), used for computing the moving average (see algorithm 3) of a sample stream in time. We
+take several considerations into account for the choice of our encoding. Our encoding should be
+bandwidth efficient and must support atomic updates on a commodity GPU. We found both DX12
+and GLSL supports atomic operations on 32 bit integers. Finally, our encoding must faithfully
+12
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Algorithm 3: Moving Average algorithm
+Input: 𝑥: Update location, 𝑣 : New sample, 𝑁𝑚𝑎𝑥 : Max sample count
+Output: 𝑉 : Updated value, 𝑁: Sample count
+1 function MovingAvgUpdate(𝑥, 𝑣, 𝑁𝑚𝑎𝑥):
+2
+𝑛 ← ^𝑔[𝑥].𝑁
+// Cumulative sample count
+3
+𝑜 ← ^𝑔[𝑥].𝑉
+// Cumulative value
+4
+𝑉 ←
+𝑣
+𝑛+1 + 𝑛·𝑜
+𝑛+1
+// Update cumulative value
+5
+𝑁 ← 𝑚𝑖𝑛(𝑛 + 1, 𝑁𝑚𝑎𝑥)
+// Increment sample count
+6
+return 𝑉, 𝑁
+encode intensities beyond the standard definition. We apply a non-linear color compression across
+the three color channels, 𝑖 ∈ [0..2] as shown in the equation below.
+𝑢𝑖 = 𝑚𝑖𝑛 (𝑙𝑛(𝛾 · 𝑣𝑖 + 1), 𝛽)
+𝛽
+.
+(21)
+We apply an inverse transform (𝑒𝑥𝑝(𝛽 · 𝑢𝑖) − 1) /𝛾 while decoding where 𝛽 = 5 and 𝛾 = 15.
+More details regarding our choice of compression scheme is provided in appendix B and figure 11.
+4.6
+Probe visibility cache
+As shown in figure 5(c), texels in the visibility probes store the mean distances and mean squared
+distances to the nearest geometry at 16x16 texel resolution. We call this ^𝑔𝑐 - our visibility cache.
+Each texel stores the two channels with 13 bits of precision each while the rest 6 bits are used for
+sample accumulation count. We normalize the distances with probe cage diagonal length. Similar to
+irradiance cache, we apply a logarithmic encoding as per equation 21 for efficient use of available
+precision. We use (𝛽,𝛾) values of (5, 15) and (8, 20) for the linear and squared channels respectively.
+4.7
+Temporal sample accumulation mecahnism
+We use a moving-average accumulation to store the samples in the irradiance and visibility caches. In
+the algorithm 3, we have two parameters 𝑁 and 𝑁𝑚𝑎𝑥 to control the moving-average accumulation.
+As we start accumulating samples, 𝑁 is incremented and the algorithm performs like a true moving
+average. However, as 𝑁 approaches 𝑁𝑚𝑎𝑥 − 1, the algorithm switches to an exponential moving
+average form with hysteresis (𝑁𝑚𝑎𝑥 − 1)/𝑁𝑚𝑎𝑥. Also, note that when the value of 𝑁 is low, the
+cache updates itself quickly, but the stored values may be noisy. As 𝑁 increases, the new samples
+are weighed less in their contribution to the cache. We exploit these parameters to control the
+learning rate and noise in the static and dynamic cases as discussed in the following sections.
+4.8
+Adaptive sampling - static
+We split our adaptive sampling strategy into two stages - static and dynamic. We have two separate
+Markov-chain sets, each focusing on different aspects of capturing the surrounding light-field.
+While the static chain focuses more on the accuracy, the dynamic chain is tuned for capturing the
+transient responses. We discuss the dynamic chain in detail in the next section.
+We set up equation 2 as - ℎ = 𝑒𝑥𝑝(𝑚𝑖𝑛(^𝑔𝑟/𝑓𝑠, 1)) · 𝑓𝑠. The feedback from irradiance cache ^𝑔𝑟 is
+obtained from the previous frame and from a higher mip-level (also used in deferred shader). The
+lowest mip-level ^𝑔𝑟 is continuously updated and thus avoided as feedback due to possible violation
+of stationarity condition within a frame. We use the Metropolis sampling, algorithm 1, to generate
+the samples 𝑥𝑖 ≡ (𝑝𝑖,𝜔𝑖). As summarized in the algorithm, 2, we use the samples to evaluate the
+13
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Algorithm 4: Atomic moving average algorithm
+Input: 𝑥: Update location, 𝑣 : New update value
+Output: Update ^𝑔[𝑥]
+1 function AtomicMovingAvg(𝑥, 𝑣):
+2
+current ← ^𝑔[𝑥]
+/* Repeat until destination value stops changing
+*/
+3
+do
+4
+expected ← current
+5
+next ← MovingAvgUpdate(𝑥, 𝑣, 64)
+6
+InterlockedCompareExchange(^𝑔[𝑥], expected, next, current)
+// Refer HLSL
+7
+while current ≠ expected
+continuous light field 𝑔𝑟, which involves tracing a ray originating at 𝑝𝑖 along the direction 𝜔𝑖.
+We trace an additional shadow-ray per sample to compute the visibility in the opposite direction
+(−𝜔𝑖) as the probe queries in the deferred shader for visibility is exactly 180◦ out of phase w.r.t
+irradiance. Next we store the irradiance and visibility values in the irradiance ( ^𝑔𝑟) and visibility ( ^𝑔𝑐)
+caches using an atomic update rule as presented in the algorithm 4. Atomic updates are required
+as multiple invocations of the chain may update the same location in the irradiance and visibility
+caches. Figure 4 summarizes the overall idea.
+We set the random walk step size, denoted by 𝜎 ∈ 𝑅5 in algorithm 5, proportional to the size of
+discretization in the irradiance and visibility cache. Thus positional step size is proportional to the
+size of a voxel in the probe grid, while angular step size is roughly
+√︁
+𝜋/256. Due to the small step
+size, texels in the cache may accumulate more than one sample per texel, thereby accumulating
+a large sample count over time. We also note that our cache behaves like a true moving average
+between sample count 𝑁 = 0 to 64, which also contributes to better accuracy.
+The static adaptive samples are useful for improving convergence in a static scene and for slow
+changes that are undetected during prior construction. For example, slow changes in lighting such
+as day-night cycles in games. We lower the hysteresis by setting 𝑁 = 16 for all probes in the region
+{𝑉𝑜𝑢𝑡} − {𝑉𝑖𝑛} in figure 6(b). This enables the probe to quickly catch-up to the most recent values.
+4.9
+Adaptive sampling - dynamic
+We run a second set of Markov-chain when dynamic content is detected in the scene. When there
+are dynamic elements, especially moving geometry, we run into two main issues. The generated
+samples are not well distributed in the region of interest i.e. the areas where time varying changes
+are present. When the step size is small, the chain cannot track the target distribution fast enough
+to generate samples from the target, causing the samples to lag the moving target distribution.
+The second problem is noise due to multi-sampling of the irradiance texel. Potentially, this can be
+solved by increasing the hysteresis to improve temporal sample reuse. However, the reduced noise
+comes at the cost of introducing objectionable temporal blur.
+We solve the first issue by increasing the chain step size and by coarsening the target function
+(𝑓𝑑). Practically, this amounts to grouping the heuristics-probes into virtual proxies. In our case, a
+virtual proxy represents a group the 3 × 3 × 3 probes. This virtual probe has 8 directions and each
+direction represents an axis-aligned octant. The value of a texel of the virtual probe is the max of
+all 27 probes it represents along the corresponding direction. We also drop the sampled evidence by
+setting 𝛼 = 0 in equation 2, as the stale irradiance cache ( ^𝑔𝑟) provide little useful information for
+14
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Table 2. Table showing probe grid details for various scenes used in our technique.
+Scene
+Probe Grid
+Probe spacing
+(in meters)
+Irradiance ( ^𝑔𝑟)
+Cache Resolution
+Visibility ( ^𝑔𝑐)
+Cache Resolution
+Bistro - Exterior
+192 × 64 × 192
+0.5 × 0.5 × 0.5
+8 × 8
+16 × 16
+Sponza - Diffuse
+192 × 64 × 192
+0.5 × 0.5 × 0.5
+8 × 8
+16 × 16
+Sponza - Glossy
+192 × 64 × 192
+0.1 × 0.1 × 0.1
+16 × 16
+16 × 16
+Table 3. Table showing probe encoding details for the various techniques we use in our comparison.
+Technique
+Irradiance
+Cache Encoding
+Visibility
+Cache Encoding
+Temporal
+Hysteresis
+Ours
+⌊R9⌋⌊G9⌋⌊B8⌋ − N
+[R13][G13] − N
+Static: 0.98 (𝑁𝑚𝑎𝑥 = 63)
+Dyna: 0.91 (𝑁𝑚𝑎𝑥 = 10)
+Q-DDGI
+⌊R11⌋⌊G11⌋⌊B10⌋ − N
+[R16][G16] − N
+0.94
+Reference
+RGB32f
+RG32f
+N/A
+sampling a time varying region. The chain step size is 3x, and 6x larger for position and directions,
+respectively w.r.t the static case.
+Since each sample from the coarse chain represents an entire octant, we trace 64 rays for the
+octant for all underlying 3x3x3 probes in the group. We make the tracing step more efficient by
+culling probes that are not used in deferred shading. The scheduling of ray-direction is deterministic,
+passing through the center of a texel in the irradiance cache ( ^𝑔𝑟). This solves the problem of sampling
+noise and also affords the opportunity to simplify the atomic updates. Since the rays are not random,
+we do not benefit from multiple shader invocations updating the same octant. As such, the first
+invocation to update the octant marks (atomically) it updated such that other invocations do not
+repeat the same work move to the next.
+We run the dynamic sampling after the static sampling step. During static sampling, if a probe
+has non-zero dynamic component(𝑓𝑑 > 0), we quantize the ray directions to go through the
+irradiance/visibility cache texel center to avoid injecting sampling noise in the texels.
+5
+RESULTS AND COMPARISONS
+We compare our results with Q-DDGI and a reference probe-based implementation in different
+scenarios - static scene (fig. 7), dynamic geometry (fig. 1, 8, 10), and dynamic lighting (fig. 9).
+Q-DDGI: Quantized-DDGI or Q-DDGI is a performance enhanced extension of original DDGI
+[28], achieved without major modifications to the base algorithm. Q-DDGI is equipped with a more
+compact irradiance and visibility cache representation that closely resembles ours. See table 3. We
+also enable camera-frustum culling of probes in Q-DDGI as described in section 4.4 and figure 6.
+These modifications allow Q-DDGI to have similar performance (table 4) at same probe count (table
+2) as ours across different scenes. We believe these modifications make our comparisons more fair.
+We use 32 rays per probe for a total ray budget of 800-1600k (depending on scene) rays per frame.
+Reference: Reference implementation uses a standard FP32 representation for irradiance and
+visibility caches as shown in table 3. We also use a higher resolution 32×32 irradiance and visibility
+cache. Due to memory constraints, we are limited to a smaller probe-grid of size 32 × 32 × 32 using
+same probe spacing (table 2) as other techniques. For each frame, we discard any previous values
+in the probes and accumulate samples using a true-average with 64 rays per texel.
+Ours: We use 4096 instances of static chain invocations and 1024 instances of dynamic chain
+invocations. Overall, we use use between 500-900k (depending on scene) rays per frame.
+15
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Table 4. Performance breakdown of our technique and Q-DDGI. Our probe sampling stage is divided into
+three sub-stages - heuristic construction (P), static adaptive sampling (S), and dynamic adaptive sampling (D).
+Scene
+Ours (in milliseconds)
+Q-DDGI (in milliseconds)
+Probe Sampling
+(P + S + D)
+Deferred
+Total
+Probe sampling
+Deferred
+Total
+Bistro - Exterior
+4.01 + 2.23 + 4.73
+= 11.0
+4.63
+15.6
+22.3
+4.47
+26.8
+Sponza - Diffuse
+1.21 + 1.85 + 3.18
+=6.24
+3.62
+9.86
+9.69
+3.51
+13.2
+Sponza - Glossy
+4.83 + 2.11 + 4.33
+=11.27
+6.44
+19.7
+24.9
+6.71
+31.6
+Figure 1 and 8 shows a large scene (Bistro Exterior), with the tunnel’s entry and exit modified
+with dynamic gates. The tunnel interior walls are illuminated by indirect illumination alone,
+controlled by the direct light bouncing off the floor. The direct illumination on the floor is controlled
+by the dynamic entry gate. The scene tests the tracking capabilities of our algorithm; the dynamic
+Markov-chain should sample the probes close to the moving door. The scene also tests our color
+compression scheme under low-light and moving-average accumulation.
+Figure 9 shows the Sponza scene under dynamic lighting, testing the detection capabilities of
+ADGI in the absence of dynamic geometry. Figure 7 shows a static scene without dynamic geometry
+or lighting, testing the convergence of our static adaptive sampling when no dynamism is detected
+or the dynamic changes are too slow to detect, such as day-night cycles in games.
+Figure 10 shows a dynamic geometry (Stanford Buddha) under glossy indirect illumination
+with ambient lighting as direct component. The scene is stressful as the camera frustum contains
+many times more probes compared to other scenes due to the increased probe density required
+for glossy illumination. This scene tests the transient response of a dynamic geometry on a glossy
+floor. Thus the scene is less forgiving of spatio-temporal blurring.
+We measured the results on a desktop with Nvidia 2080Ti GPU and AMD 5600X CPU at
+1920 × 1080 resolution. The performance numbers cited in table 4 are only for ADGI and Q-
+DDGI algorithms. The GBuffer and direct-illumination passes require an additional 2ms and 3ms,
+respectively.
+6
+LIMITATIONS
+We inherit similar limitations as the vanilla DDGI algorithm. The probe visibility from a shade-point
+is only approximate and requires modifications such as probe movement to minimize light leakage.
+The probe representation is not efficient in capturing glossy light-transport and requires a dense
+spatio-angular discretization of irradiance cache to capture glossy reflections.
+Accurate detection of transient spatio-temporal changes in a scene are difficult. The accuracy of
+detecting dynamic geometry reduces with the distance of the dynamic object from a probe. The
+same is true for dynamic lighting; especially high frequency localized lighting that is far from a
+probe is difficult to detect. Also, for the Markov chain to track the target distribution, the speed
+of motion should be capped comparable to the product of Markov-chain step size and average
+frame-rate. While many game engines keep track of the dynamic objects, facilitating the detection
+of changing in visibility, we still need ray-tracing to detect dynamic radiosity.
+16
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+7
+CONCLUSION
+Our adaptive sampling approach improves upon the efficiency of the original DDGI algorithm. Our
+approach non-uniformly allocates resources in regions with time varying phenomena and captures
+transient localized changes in an environment containing millions of probes. By contrast, DDGI’s
+uniform allocation policy dilutes resource concentration in critical regions, especially when a large
+number of probes are present. These improvements reduce temporal lag and minimizes reliance on
+temporal blur to reduce noise. Our probe encoding scheme minimizes memory requirements by 4x
+(and by extension memory bandwidth) with minimal impact on quality while also enabling millions
+of probes in a scene. Our adaptive sampling stages have a fixed upper bound on the compute
+requirement and also decouples sampling from the number of probes, further reducing memory
+bandwidth requirement. These changes enable improved probe-based rendering while also enabling
+1.5-2x performance improvements.
+8
+RELATED WORK EXTENSION
+Irradiance caching Irradiance caching is another line of techniques attempting to overcome
+the high computation cost of GI. The irradiance caching method assumes that irradiance vary
+smoothly across the scene, and texture detail can be recovered using albedo modulation [64]. The
+interpolation and location of the various cache records is a critical, especially when the assumptions
+on smoothness do not hold. While robust, principled offline solutions exist [16, 24], real-time
+applications often resort to complex heuristics and impose harsh constraints to achieve online
+GI. Compression [56], sparse interpolation [49], pre-convolved environment maps [42, 45], spatial
+hashing [3] and using neural network [37] are instances of advancements in real-time irradiance
+caching. Although these approaches aim for real-time performance, their complexity and constraints
+make them challenging to implement and deploy.
+𝑡 = 32 ms
+64 ms
+96 ms
+128 ms
+32 ms
+64 ms
+96 ms
+128 ms
+Ours
+Q-DDGI
+MSE: 0.072
+0.031
+0.022
+0.014
+SSIM: 0.750
+0.894
+0.910
+0.940
+MSE:0.085
+0.064
+0.049
+0.031
+SSIM:
+0.561
+0.758
+0.828
+0.888
+SSIM: 0.732
+0.895
+0.908
+0.914
+SSIM: 0.524
+0.745
+0.838
+0.874
+MSE : 0.076
+0.038
+0.025
+0.020
+MSE : 0.087
+0.067
+0.052
+0.041
+Green: Diminished luminance
+Red: Excess luminance
+Fig. 7. Comparing the convergence of our technique over time on a static Bistro Exterior scene. The figure
+demonstrates the effectiveness of our static adaptive sampling step. The two rows measure the difference in
+luminance w.r.t reference and highlight the error in red and green color.
+17
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+𝑡 = start
+𝑡 = start + 4𝑠
+𝑡 = start + 8𝑠
+Direct + Indirect
+Tunnel interior (D + I)
+SSIM/MSE:
+0.971/0.008
+SSIM/MSE:
+0.974/0.008
+SSIM/MSE:
+0.982/0.007
+SSIM/MSE:
+0.951/0.010
+SSIM/MSE:
+0.951/0.009
+SSIM/MSE:
+0.967/0.009
+SSIM/MSE:
+0.931/0.004
+SSIM/MSE:
+0.948/0.004
+SSIM/MSE:
+0.994/0.000
+SSIM/MSE:
+0.913/0.009
+SSIM/MSE:
+0.871/0.005
+SSIM/MSE:
+0.804/0.004
+Left: [email protected]
+Right: [email protected]
+Fig. 8. Our technique compared with Q-DDGI on a modified Bistro Exterior scene augmented with a
+moving door. The scene has 192 × 64 × 192 probes and shows the convergence of the two techniques near
+a dynamic area in the scene. The second row shows the changes inside the tunnel as the door closes over
+time. Our technique is better able to allocate the resources closer to the dynamic areas resulting in faster
+convergence and higher performance.
+Path tracing The flexibility and generality offered by path tracing [18] is highly desirable for
+real-time rendering. However, path tracing has been out of reach for real-time applications due to its
+substantial computational requirements. Even with the advent of hardware-accelerated ray tracing
+[23], it is only possible to trace a few tens of rays at each pixel in real-time. Therefore, effective
+sampling strategies and high-quality denoising algorithms [38, 46, 47] are essential. Many sampling
+methods try to learn the representation of incident illumination during rendering [1, 8, 34, 44, 60].
+While these approaches can provide substantial error reduction, constructing these structures in
+parallel on a GPU incurs a significant overhead that seem unsuitable for real-time applications.
+Recently proposed ReSTIR GI [41] provides an efficient real-time sampling strategy by reusing the
+paths spatially and temporally but the algorithm becomes complicated after second bounce and still
+requires denoising for the final stage. Deep learning has also been applied to path guiding, including
+work by [35, 36]. These approaches demonstrated a substantial reduction in error due to more
+effective path sampling, though their performance remain insufficient for real-time applications.
+Screen space approaches: Approximating physically plausible illumination at real-time frame
+rates with screen space methods is popular in games. Screen space methods are fast, GPU-friendly,
+and simple to implement. Screen space ambient occlusion (SSAO) [2, 33] is part of many real-time
+rendering engines. Following SSAO, screen Space Directional Occlusion (SSDO) [43] is used for
+near-field direct and indirect diffuse lighting. Sousa et al. [52] proposed Screen Space Reflections
+(SSR) using a 2D ray-tracing approach directly in screen space to obtain the indirect specular
+component. Recently Screen-Space Global Illumination (SSGI) [43, 50, 52] methods offer a viable
+solution to real-time GI. However, these methods are limited by the information visible from the
+observer’s position, thus making it difficult to engineer a robust solution.
+Importance sampling and Bayesian modeling: Importance sampling provides a tool to
+reduce the cost of brute force integration by selectively evaluating elements of the integrand based
+on prior knowledge, i.e. an educated guess. Previous works in importance sampling proposed
+different methods to apply importance sampling to various Monte-Carlo integration existing in
+rendering equations [21, 48, 57]. Although Markov Chain Monte Carlo(MCMC) methods have been
+18
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+𝑡 = start
+𝑡 = start + 2𝑠
+𝑡 = start + 3𝑠
+Direct only
+Indirect - Ours
+Indirect - Q-DDGI
+SSIM/MSE: 0.920/0.005
+SSIM/MSE: 0.966/0.006
+SSIM/MSE:0.957/0.007
+SSIM/MSE: 0.868/0.012
+SSIM/MSE: 0.763/0.023
+SSIM/MSE: 0.794/0.021
+Difference in luminance w.r.t reference
+Error - Ours
+Error - Q-DDGI
+Green: Diminished luminance
+Red: Excess luminance
+[email protected]
+[email protected]
+Fig. 9. Figure comparing the convergence of our technique under dynamic lighting controlled by the direct
+component shown in the first row. The last two rows measure the difference in luminance w.r.t reference and
+highlight the error in red and green color.
+used in Bayesian learning from the early days of neural networks [39], and Stochastic-Gradient
+MCMC has been proposed [65] with various applications [25], our approach is neither Monte
+Carlo-based nor Neural-network learning. We exploit Bayesian inference and Markov Chains as our
+mathematical means to sample the important texels on the probe, by defining our guide function
+(prior), likelihood, and posterior.
+Markov Chain: Markov Chains are used broadly in Monte Carlo path-tracing. For example,
+Veach and Guibas [58] used Metropolis Sampling to explore the space of all possible paths. Kelemen
+et al. [19] later applied the exact sampling in the space of random numbers, i.e., in Primary Sample
+19
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+𝑡 = start
+𝑡 = start + 10𝑠
+𝑡 = start + 15𝑠
+Glossy indirect - Ours
+Glossy indirect without texture
+Ours
+Q-DDGI
+SSIM/MSE:0.996/0.017
+SSIM/MSE:0.995/0.019
+SSIM/MSE:0.992/0.018
+SSIM/MSE:0.990/0.019
+SSIM/MSE:0.989/0.020
+SSIM/MSE:0.987/0.028
+Ours
+Reference
+Q-DDGI
+Ours
+Reference
+Q-DDGI
+Ours
+Reference
+Q-DDGI
+SSIM: 0.979
+0.981
+SSIM: 0.995
+0.993
+SSIM: 0.998
+0.992
+MSE : 0.013
+0.028
+MSE : 0.016
+0.034
+MSE : 0.014
+0.058
+[email protected]
+[email protected]
+Fig. 10. Figure comparing glossy indirect reflection on a scene lit by ambient lighting. The scene tests transient
+response due to the moving Buddha geometry over a glossy floor.
+Space. The most recent work by Bitterli et al. [4] combines a simple path tracing integrator with
+MCMC by using the random seeds of high variance paths as starting points for the Markov Chains.
+Although Markov Chains are encountered extensively beneficial in solving Monte Carlo sampling,
+our point of view on sampling and employing the Markov Chain to draw samples from the guide
+function is distinct.
+Bayesian inference: Bayesian modeling is a widespread methodology in computer vision and
+graphics. Brouillat et al. [5] and Marques et al. [30] pioneered the use of Bayesian Monte Carlo
+(BMC) [11] in light transport simulation. In contrast, [59] keep the efficient classic, frequentist
+MC approach and apply Bayesian modeling to optimize their sampling distributions for direct
+illumination estimates across the scene. Similar approach is used by Vorba et al. [61], who employ
+a maximum a posteriori (MAP) formulation to regularize training of parametric mixture models for
+optimized indirect illumination sampling. Our approach uses Bayesian modeling in the context of
+light-probes to detect important probes and directions based on sampled evidence.
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+A
+METROPOLIS-HASTINGS
+Markov Chain Monte Carlo (MCMC) allows sampling from the posterior without computing
+the marginal. [10]. Metropolis-Hastings (Metropolis), which we exploit in this work, is a specific
+implementation of MCMC [7]. The Metropolis–Hastings algorithm can draw samples from any
+probability distribution with probability density 𝑃(𝑥), provided a function ℎ(𝑥) proportional to
+the density 𝑃(𝑥). The Metropolis algorithm works by generating a sequence of sample values so
+that, as more samples are produced, the distribution of samples more closely approximates the
+desired distribution. These sample values are produced iteratively, meaning the next sample being
+dependent on the current sample (thus making the sequence of samples into a chain). Let ℎ(𝑥)
+be a function that is proportional to the desired probability density function 𝑃(𝑥) (a.k.a. a target
+distribution). The Metropolis Markov Chain algorithm with random walk is defined as follows:
+Algorithm 5: Random-walk algorithm
+Input: 𝑥𝑖: Current state, 𝑦𝑖 : Probability of current state
+Input: 𝜎 : Step size or std-dev of Gaussian noise
+Output: 𝑥𝑖+1: Next state, 𝑦𝑖+1 : Probability of next state
+1 function RandomWalk(𝑥𝑖, 𝑦𝑖):
+2
+𝑥𝑖+1 ← 𝑥𝑖 + N (𝜎)
+// Propose a new state
+3
+𝑦𝑖+1 ← ℎ(𝑥𝑖+1)
+4
+𝜇 ← min
+�
+𝑦𝑖+1
+𝑦𝑖 , 1
+�
+// Compute acceptance ratio
+5
+𝜖 ∼ 𝑈 (0, 1)
+// Sample uniform distribution
+6
+if 𝜖 > 𝜇 then
+/* Reject proposed state
+*/
+7
+𝑥𝑖+1 ← 𝑥𝑖
+8
+𝑦𝑖+1 ← 𝑦𝑖
+9
+return 𝑥𝑖+1,𝑦𝑖+1
+Initialization: Choose an arbitrary point 𝑥𝑖−1 as the initial observation in the sample-space and
+choose an arbitrary probability density N (𝑥𝑖 | 𝑥𝑖−1) that suggests the next sample candidate 𝑥𝑖,
+given the previous sample value 𝑥𝑖−1. In our work, N is assumed to be symmetric. A usual choice
+is to let N (𝑥𝑖 | 𝑥𝑖−1) be a Gaussian distribution centered at 𝑥𝑖−1, so that points closer to 𝑥𝑖−1 are
+more likely to be visited next, making the sequence of samples resemble a random walk [7]. The
+random walk algorithm is described in algorithm 5.
+B
+PROBE COMPRESSION
+We tested several 26-bit encoding and settled on a non-linear RGB encoding represented by
+⌊R9⌋⌊G9⌋⌊B8⌋ −N in figure 11. In this encoding, the RGB color is first passed through a logarithmic
+non-linearity as per equation 21 such that the quantization errors are distributed evenly across
+intensities. We perform a round-to-lowest-integer (⌊⌋) quantization for all channels, although round-
+to-nearest-integer ([ ] ) is more accurate. Our quantization scheme ensures the moving-average
+updates produce dark colors when the intensity of new samples are low. In a round-to-nearest set-
+ting, due to a round-up error, the colors may never go to zero. Interestingly, YCbCr encoding allows
+round-to-lowest for the Y channel and round round-to-nearest for Cb and Cr channels, however,
+they perform poorly in both luminance and color preservation metrics as shown in figure 11.
+24
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Intensity:
+[0 - 0.25]*
+[0.25 - 1.0]
+[1.0 - 5.0]*
+Decoded RGB
+Error
+Decoded RGB
+Error
+Decoded RGB
+Error
+⌊Y8⌋[ Cb9] [ Cr9] − N
+⌊Y8⌋[ Cb9] [ Cr9]
+⌊R9⌋ ⌊G9⌋ ⌊B8⌋ − N
+⌊R9⌋ ⌊G9⌋ ⌊B8⌋
+LUM: 0.0024
+LUM: 0.0025
+LUM: 0.0025
+LUM: 0.0004
+LUM: 0.0011
+LUM: 0.0043
+LUM: 0.0045
+LUM: 0.0049
+LUM: 0.0050
+LUM: 0.0007
+LUM: 0.0021
+LUM: 0.0086
+COR: 0.9972
+COR: 0.9999
+COR: 1.0000
+COR: 1.0000
+COR: 1.0000
+COR: 1.0000
+COR: 0.9944
+COR: 0.9997
+COR: 1.0000
+COR: 1.0000
+COR: 1.0000
+COR: 1.0000
+Fig. 11. Figure comparing 26-bit color encodings on slices of the 3D color-space with dynamic range. We
+compare the reconstruction error measured in Luminance and Color Correlation with RGB32f reference. The
+log-non-linear encodings marked with - N suffix shifts the bit error from lower to higher intensities - which
+are less frequent in indirect illumination. ⌊⌋ and [ ] denotes round-low and round-nearest quantizations
+respectively. * Color map visualizations are normalized.
+The parameters in equation 21 are obtained by performing a grid search minimizing the recon-
+struction error w.r.t RGB32f reference across various color and intensity combinations as shown in
+figure 11. Luminance error is the r.m.s. value of the difference between the two color-maps. Color
+accuracy is measured using a normalized dot product between the two flattened color-maps.
+25
+