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arxiv: 2606.19205 · v1 · pith:M5WBKMUMnew · submitted 2026-06-17 · ⚛️ physics.comp-ph

Discovering a well-conditioned analytic continuation problem via dictionary learning

Thomas Chuna , Phil-Alexander Hofmann , Alexander Benedix-Robles , Tobias Dornheim This is my paper

Pith reviewed 2026-06-26 18:41 UTC · model grok-4.3

classification ⚛️ physics.comp-ph
keywords analytic continuationquantum Monte Carlodictionary learninginverse problemspectral functionsregularized stochastic optimizationimaginary-time correlation functions
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The pith

Regularized stochastic optimization discovers a sparse dictionary that transforms the ill-conditioned analytic continuation problem into a well-conditioned one.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the regularized stochastic optimization method (RSOM) to reformulate analytic continuation as a dictionary learning problem. It shows that a sparse dictionary can be discovered which maps the exponentially ill-conditioned inverse problem from imaginary-time correlation functions to a low-dimensional well-conditioned problem. This approach is tested on synthetic cases and real quantum Monte Carlo data from the finite temperature electron gas, producing competitive spectral functions. The work suggests dictionary learning as a unifying angle for stochastic and regularized methods in analytic continuation.

Core claim

RSOM reformulates analytic continuation as a dictionary learning problem and discovers a sparse dictionary that maps an ill-conditioned inverse problem to a low-dimensional problem that is well-conditioned, yielding accurate results for both synthetic test problems and authentic QMC data.

What carries the argument

The regularized stochastic optimization method (RSOM), which learns a sparse dictionary to represent solutions to the analytic continuation problem.

If this is right

  • Analytic continuation can be solved by reducing it to a well-conditioned low-dimensional problem via the learned dictionary.
  • Dictionary learning provides a new framework that encompasses both stochastic and regularized approaches to analytic continuation.
  • The method produces competitive results on common synthetic test problems and real data from the electron gas.
  • Future AC methods can attack the problem from the angle of dictionary learning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The existence of such a dictionary implies that many existing methods implicitly rely on similar sparse representations.
  • Applying RSOM to other inverse problems in physics could reveal similar well-conditioned mappings.
  • Testing the stability of the learned dictionary across different temperatures or system sizes would strengthen the method.

Load-bearing premise

That a sparse dictionary learned via regularized stochastic optimization on synthetic data will exist, remain stable, and accurately reconstruct spectral functions from real quantum Monte Carlo data.

What would settle it

Applying the learned dictionary from synthetic tests to QMC data from a more complex system, such as a lattice model, and observing large discrepancies in the recovered spectral functions compared to known benchmarks.

Figures

Figures reproduced from arXiv: 2606.19205 by Alexander Benedix-Robles, Phil-Alexander Hofmann, Thomas Chuna, Tobias Dornheim.

Figure 1
Figure 1. Figure 1: This plot demonstrates the RSOM’s performance for the AC of a realistic rho-meson spectrum, comparing with the [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: This plot demonstrates the competitive performance of the RSOM in comparison to entropic methods (which [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: This plot demonstrates RSOM’s performance on the AC of the skew Gaussian test is comparable to that of entropic [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: This plot compares the RSOM’s results for a Gaussian transformed sinusoidal signal are competitive with other [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Top: Heatmaps presenting the analytic continuation results for the dynamic structure factor of the finite temperature [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

Many fields of physics use quantum Monte Carlo (QMC) simulations to simulate quantum systems in imaginary-time $\tau$ and estimate imaginary-time correlation functions (ITCF). However, extracting dynamic $\omega$-dependent quantities from ITCFs is a notoriously difficult task, known as analytic continuation (AC), that amounts to solving an exponentially ill-conditioned inverse problem. Within the AC literature, there are competing stochastic and regularized approaches, as well as an emerging collection of works using parameterized models like neural networks. Here we transcend the traditional divides between the communities, introducing the regularized stochastic optimization method (RSOM). This method reformulates AC as a dictionary learning problem, discovering a sparse dictionary to represent the solution. Our approach is motivated by the astounding results dictionary learning has produced in many scientific fields. Remarkably, RSOM discovers a sparse dictionary that maps an ill-conditioned inverse problem to a low-dimensional problem that is well-conditioned. We demonstrate that the method yields competitive results for common synthetic test problems as well as for authentic QMC data from the finite temperature electron gas. This work exposes that a dictionary exists within all stochastic and regularized methods and that dictionary learning provides a new angle of attack for future AC methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces the regularized stochastic optimization method (RSOM) that recasts analytic continuation of imaginary-time correlation functions as a dictionary-learning task. It claims that RSOM discovers a sparse dictionary mapping the exponentially ill-conditioned kernel to a low-dimensional, intrinsically well-conditioned sub-problem, and reports competitive performance on both synthetic test cases and authentic finite-temperature electron-gas QMC data.

Significance. If the headline claim is substantiated, the work supplies a new conceptual framing that unifies stochastic and regularized AC approaches under dictionary learning and demonstrates that such dictionaries can be learned directly from data. The absence of free parameters in the core construction and the explicit link to existing methods are strengths that would be noteworthy if the conditioning improvement is rigorously verified.

major comments (3)
  1. [Abstract, §4] Abstract and §4 (results): the central assertion that the learned dictionary produces a 'well-conditioned' low-dimensional inverse problem is not supported by any reported condition-number spectra, singular-value distributions, or perturbation-stability tests before versus after projection onto the dictionary. Without these quantities the distinction between genuine conditioning improvement and regularization-induced stability cannot be assessed.
  2. [§3, §5] §3 (method) and §5 (real-data application): no quantitative metrics (MSE, integrated absolute error, error bars, or direct comparisons to MaxEnt, stochastic analytic continuation, or neural-network baselines) are supplied for either the synthetic or the electron-gas QMC examples, making the 'competitive results' claim impossible to evaluate.
  3. [§4.2] §4.2 (synthetic tests): the manuscript does not demonstrate that the recovered spectral functions remain accurate when the input ITCF is corrupted by realistic QMC noise levels rather than the idealized synthetic noise used for validation.
minor comments (2)
  1. [Abstract, §2] Notation for the dictionary atoms and the projection operator should be introduced once and used consistently; several symbols appear without prior definition in the abstract and early sections.
  2. [Figures 2-5] Figure captions lack explicit statements of the regularization parameter schedule and convergence tolerance used in the RSOM runs.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. Below we respond point-by-point to the major comments and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract, §4] Abstract and §4 (results): the central assertion that the learned dictionary produces a 'well-conditioned' low-dimensional inverse problem is not supported by any reported condition-number spectra, singular-value distributions, or perturbation-stability tests before versus after projection onto the dictionary. Without these quantities the distinction between genuine conditioning improvement and regularization-induced stability cannot be assessed.

    Authors: We agree that explicit numerical verification of the conditioning improvement is necessary to distinguish the effect from regularization. In the revised manuscript we will add condition-number spectra, singular-value distributions, and perturbation-stability tests comparing the original kernel to the dictionary-projected sub-problem. revision: yes

  2. Referee: [§3, §5] §3 (method) and §5 (real-data application): no quantitative metrics (MSE, integrated absolute error, error bars, or direct comparisons to MaxEnt, stochastic analytic continuation, or neural-network baselines) are supplied for either the synthetic or the electron-gas QMC examples, making the 'competitive results' claim impossible to evaluate.

    Authors: We acknowledge the absence of these quantitative metrics in the submitted version. The revised manuscript will include MSE, integrated absolute error, error bars, and direct comparisons against MaxEnt, stochastic analytic continuation, and neural-network baselines for both the synthetic tests and the electron-gas QMC data. revision: yes

  3. Referee: [§4.2] §4.2 (synthetic tests): the manuscript does not demonstrate that the recovered spectral functions remain accurate when the input ITCF is corrupted by realistic QMC noise levels rather than the idealized synthetic noise used for validation.

    Authors: The synthetic tests employed controlled noise to isolate the dictionary-learning mechanism. We will add a new set of experiments in the revised §4.2 that inject noise amplitudes matching typical QMC statistical uncertainties to confirm robustness under realistic conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; RSOM presented as empirical dictionary-learning reformulation without self-referential reductions.

full rationale

The paper introduces RSOM as a reformulation of analytic continuation into a dictionary-learning problem solved via regularized stochastic optimization. The central claim—that a learned sparse dictionary maps the ill-conditioned kernel to a low-dimensional well-conditioned subproblem—is supported by reported performance on synthetic test cases and real finite-temperature electron-gas QMC data. No equations, procedures, or self-citations in the abstract or described method reduce the claimed discovery to a fitted parameter, self-definition, or prior author result by construction. The dictionary is learned from data rather than presupposed, and the well-conditioned property is asserted as an observed outcome rather than a definitional tautology. This is the common honest finding for a method paper whose validation rests on external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The central claim implicitly assumes existence of a useful sparse dictionary for the AC operator.

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discussion (0)

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Works this paper leans on

90 extracted references · 5 linked inside Pith

  1. [1]

    W. M. C. Foulkes, L. Mitas, R. J. Needs, G. Rajagopal, Quantum monte carlo simulations of solids, Rev. Mod. Phys. 73 (2001) 33–83.doi:10.1103/RevModPhys.73.33. URLhttps://link.aps.org/doi/10.1103/RevModPhys.73.33

    doi:10.1103/revmodphys.73.33 2001

  2. [2]

    Anderson, Quantum Monte Carlo: Origins, Development, Applications, Oxford University Press, USA, 2007

    J. Anderson, Quantum Monte Carlo: Origins, Development, Applications, Oxford University Press, USA, 2007. URLhttps://books.google.de/books?id=_QUSDAAAQBAJ

    2007

  3. [3]

    D. M. Ceperley, Path integrals in the theory of condensed helium, Rev. Mod. Phys 67 (1995) 279. URLhttps://journals.aps.org/rmp/abstract/10.1103/RevModPhys.67.279

    doi:10.1103/revmodphys.67.279 1995

  4. [4]

    Knechtli, M

    F. Knechtli, M. G¨ unther, M. Peardon, Lattice Quantum Chromodynamics: Practical Essentials, 1st Edition, SpringerBriefs in Physics, Springer, Dordrecht, Netherlands, 2017.doi:10.1007/978-94-024-0999-4. URLhttps://doi.org/10.1007/978-94-024-0999-4

    doi:10.1007/978-94-024-0999-4 2017

  5. [5]

    Jarrell, J

    M. Jarrell, J. Gubernatis, Bayesian inference and the analytic continuation of imaginary-time quantum monte carlo data, Physics Reports 269 (3) (1996) 133–195.doi:https://doi.org/10.1016/0370-1573(95)00074-7. URLhttps://www.sciencedirect.com/science/article/pii/0370157395000747

    doi:10.1016/0370-1573(95)00074-7 1996

  6. [6]

    Boninsegni, N

    M. Boninsegni, N. V. Prokofev, B. V. Svistunov, Worm algorithm and diagrammatic Monte Carlo: A new approach to continuous-space path integral Monte Carlo simulations, Phys. Rev. E 74 (2006) 036701. URLhttps://journals.aps.org/pre/abstract/10.1103/PhysRevE.74.036701

    doi:10.1103/physreve.74.036701 2006

  7. [7]

    Rabani, D

    E. Rabani, D. R. Reichman, G. Krilov, B. J. Berne, The calculation of transport properties in quantum liquids using the maximum entropy numerical analytic continuation method: Application to liquid para-hydrogen, Proceedings of the National Academy of Sciences 99 (3) (2002) 1129–1133.arXiv:https://www.pnas.org/doi/pdf/10.1073/pnas.261540698, doi:10.1073/p...

    doi:10.1073/pnas.261540698 2002

  8. [8]

    Dornheim, Z

    T. Dornheim, Z. Moldabekov, P. Tolias, M. B¨ ohme, J. Vorberger, Physical insights from imaginary-time density–density correlation functions, Matter and Radiation at Extremes 8 (2023) 056601.doi:10.1063/5.0149638. URLhttps://doi.org/10.1063/5.0149638

    doi:10.1063/5.0149638 2023

  9. [9]

    Dornheim, Z

    T. Dornheim, Z. A. Moldabekov, J. Vorberger, Nonlinear density response from imaginary-time correlation functions: Ab initio path integral monte carlo simulations of the warm dense electron gas, The Journal of Chemical Physics 155 (5) (2021) 054110.doi:10.1063/5.0058988. URLhttps://doi.org/10.1063/5.0058988

    doi:10.1063/5.0058988 2021

  10. [10]

    S. Shi, L. Wang, K. Zhou, Rethinking the ill-posedness of the spectral function reconstruction — why is it fundamentally hard and how artificial neural networks can help, Computer Physics Communications 282 (2023) 108547.doi:https: //doi.org/10.1016/j.cpc.2022.108547. URLhttps://www.sciencedirect.com/science/article/pii/S0010465522002661

    doi:10.1016/j.cpc.2022.108547 2023

  11. [11]

    Asakawa, Y

    M. Asakawa, Y. Nakahara, T. Hatsuda, Maximum entropy analysis of the spectral functions in lattice qcd, Progress in Particle and Nuclear Physics 46 (2) (2001) 459–508.doi:https://doi.org/10.1016/S0146-6410(01)00150-8. URLhttps://www.sciencedirect.com/science/article/pii/S0146641001001508

    doi:10.1016/s0146-6410(01)00150-8 2001

  12. [12]

    Burnier, A

    Y. Burnier, A. Rothkopf, Bayesian approach to spectral function reconstruction for euclidean quantum field theories, Phys. Rev. Lett. 111 (2013) 182003.doi:10.1103/PhysRevLett.111.182003. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.111.182003

    doi:10.1103/physrevlett.111.182003 2013

  13. [13]

    Otsuki, M

    J. Otsuki, M. Ohzeki, H. Shinaoka, K. Yoshimi, Sparse modeling approach to analytical continuation of imaginary-time quantum monte carlo data, Phys. Rev. E 95 (2017) 061302.doi:10.1103/PhysRevE.95.061302. URLhttps://link.aps.org/doi/10.1103/PhysRevE.95.061302

    doi:10.1103/physreve.95.061302 2017

  14. [14]

    Otsuki, M

    J. Otsuki, M. Ohzeki, H. Shinaoka, K. Yoshimi, Sparse modeling in quantum many-body problems, Journal of the Physical Society of Japan 89 (1) (2020) 012001.doi:10.7566/JPSJ.89.012001. URLhttps://doi.org/10.7566/JPSJ.89.012001

    doi:10.7566/jpsj.89.012001 2020

  15. [15]

    N. V. Prokof’ev, B. V. Svistunov, Spectral analysis by the method of consistent constraints, JETP letters 97 (11) (2013) 649–653.doi:10.1134/S002136401311009X. URLhttps://doi.org/10.1134/S002136401311009X

    doi:10.1134/s002136401311009x 2013

  16. [16]

    M. Han, H. J. Choi, Parameter-free analytic continuation for quantum many-body calculations, Phys. Rev. B 106 (2022) 245150.doi:10.1103/PhysRevB.106.245150. URLhttps://link.aps.org/doi/10.1103/PhysRevB.106.245150

    doi:10.1103/physrevb.106.245150 2022

  17. [17]

    Chuna, N

    T. Chuna, N. Barnfield, T. Dornheim, M. P. Friedlander, T. Hoheisel, Dual formulation of the maximum entropy method applied to analytic continuation of quantum monte carlo data, Journal of Physics A: Mathematical and Theoretical 58 (33) (2025) 335203.doi:10.1088/1751-8121/adf924

    doi:10.1088/1751-8121/adf924 2025

  18. [18]

    Benedix Robles, P.-A

    A. Benedix Robles, P.-A. Hofmann, T. Chuna, T. Dornheim, M. Hecht, Pylit: Reformulation and implementation of the analytic continuation problem using kernel representation methods, Computer Physics Communications 319 (2026) 109904.doi:https://doi.org/10.1016/j.cpc.2025.109904. URLhttps://www.sciencedirect.com/science/article/pii/S0010465525004059

    doi:10.1016/j.cpc.2025.109904 2026

  19. [19]

    A. W. Sandvik, Stochastic method for analytic continuation of quantum monte carlo data, Phys. Rev. B 57 (1998) 10287– 10290.doi:10.1103/PhysRevB.57.10287. URLhttps://link.aps.org/doi/10.1103/PhysRevB.57.10287

    doi:10.1103/physrevb.57.10287 1998

  20. [20]

    A. S. Mishchenko, N. V. Prokof’ev, A. Sakamoto, B. V. Svistunov, Diagrammatic quantum monte carlo study of the fr¨ ohlich polaron, Phys. Rev. B 62 (2000) 6317–6336.doi:10.1103/PhysRevB.62.6317. URLhttps://link.aps.org/doi/10.1103/PhysRevB.62.6317 15

    doi:10.1103/physrevb.62.6317 2000

  21. [21]

    Vitali, M

    E. Vitali, M. Rossi, L. Reatto, D. E. Galli, Ab initio low-energy dynamics of superfluid and solid 4He, Phys. Rev. B 82 (2010) 174510.doi:10.1103/PhysRevB.82.174510. URLhttps://link.aps.org/doi/10.1103/PhysRevB.82.174510

    doi:10.1103/physrevb.82.174510 2010

  22. [22]

    Saccani, S

    S. Saccani, S. Moroni, M. Boninsegni, Excitation spectrum of a supersolid, Phys. Rev. Lett. 108 (2012) 175301.doi: 10.1103/PhysRevLett.108.175301. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.108.175301

    doi:10.1103/physrevlett.108.175301 2012

  23. [23]

    Shu, H.-T

    H.-T. Shu, H.-T. Ding, O. Kaczmarek, S. Mukherjee, H. Ohno, A stochastic approach to the reconstruction of spectral functions in lattice qcd, arXiv preprint arXiv:1510.02901 (2015)

    Pith/arXiv arXiv 2015

  24. [24]

    F. Bao, Y. Tang, M. Summers, G. Zhang, C. Webster, V. Scarola, T. A. Maier, Fast and efficient stochastic optimization for analytic continuation, Phys. Rev. B 94 (2016) 125149.doi:10.1103/PhysRevB.94.125149. URLhttps://link.aps.org/doi/10.1103/PhysRevB.94.125149

    doi:10.1103/physrevb.94.125149 2016

  25. [25]

    N. S. Nichols, P. Sokol, A. Del Maestro, Parameter-free differential evolution algorithm for the analytic continuation of imaginary time correlation functions, Phys. Rev. E 106 (2022) 025312.doi:10.1103/PhysRevE.106.025312. URLhttps://link.aps.org/doi/10.1103/PhysRevE.106.025312

    doi:10.1103/physreve.106.025312 2022

  26. [26]

    H. Shao, A. W. Sandvik, Progress on stochastic analytic continuation of quantum monte carlo data, Physics Reports 1003 (2023) 1–88.doi:https://doi.org/10.1016/j.physrep.2022.11.002. URLhttps://www.sciencedirect.com/science/article/pii/S0370157322003921

    doi:10.1016/j.physrep.2022.11.002 2023

  27. [27]

    Goulko, A

    O. Goulko, A. S. Mishchenko, L. Pollet, N. Prokof’ev, B. Svistunov, Numerical analytic continuation: Answers to well- posed questions, Phys. Rev. B 95 (2017) 014102.doi:10.1103/PhysRevB.95.014102. URLhttps://link.aps.org/doi/10.1103/PhysRevB.95.014102

    doi:10.1103/physrevb.95.014102 2017

  28. [28]

    Bergeron, A.-M

    D. Bergeron, A.-M. S. Tremblay, Algorithms for optimized maximum entropy and diagnostic tools for analytic continuation, Phys. Rev. E 94 (2016) 023303.doi:10.1103/PhysRevE.94.023303. URLhttps://link.aps.org/doi/10.1103/PhysRevE.94.023303

    doi:10.1103/physreve.94.023303 2016

  29. [29]

    A. W. Sandvik, R. R. P. Singh, High-energy magnon dispersion and multimagnon continuum in the two-dimensional heisenberg antiferromagnet, Phys. Rev. Lett. 86 (2001) 528–531.doi:10.1103/PhysRevLett.86.528. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.86.528

    doi:10.1103/physrevlett.86.528 2001

  30. [30]

    E. Katz, S. Sachdev, E. S. Sørensen, W. Witczak-Krempa, Conformal field theories at nonzero temperature: Operator product expansions, monte carlo, and holography, Phys. Rev. B 90 (2014) 245109.doi:10.1103/PhysRevB.90.245109. URLhttps://link.aps.org/doi/10.1103/PhysRevB.90.245109

    doi:10.1103/physrevb.90.245109 2014

  31. [31]

    Buividovich, J

    P. Buividovich, J. Ostmeyer, A. Troisi, High-precision quantum monte-carlo study of charge transport in a lattice model of molecular organic semiconductors, arXiv preprint arXiv:2411.17460 (2024)

    arXiv 2024

  32. [32]

    Groth, T

    S. Groth, T. Dornheim, J. Vorberger, Ab initio path integral monte carlo approach to the static and dynamic density response of the uniform electron gas, Phys. Rev. B 99 (2019) 235122.doi:10.1103/PhysRevB.99.235122. URLhttps://link.aps.org/doi/10.1103/PhysRevB.99.235122

    doi:10.1103/physrevb.99.235122 2019

  33. [33]

    A. V. Filinov, J. Ara, I. M. Tkachenko, Dynamical response in strongly coupled uniform electron liquids: Observation of plasmon-roton coexistence using nine sum rules, shannon information entropy, and path-integral monte carlo simulations, Phys. Rev. B 107 (2023) 195143.doi:10.1103/PhysRevB.107.195143. URLhttps://link.aps.org/doi/10.1103/PhysRevB.107.195143

    doi:10.1103/physrevb.107.195143 2023

  34. [34]

    Yoon, J.-H

    H. Yoon, J.-H. Sim, M. J. Han, Analytic continuation via domain knowledge free machine learning, Phys. Rev. B 98 (2018) 245101.doi:10.1103/PhysRevB.98.245101. URLhttps://link.aps.org/doi/10.1103/PhysRevB.98.245101

    doi:10.1103/physrevb.98.245101 2018

  35. [35]

    Kades, J

    L. Kades, J. M. Pawlowski, A. Rothkopf, M. Scherzer, J. M. Urban, S. J. Wetzel, N. Wink, F. P. G. Ziegler, Spectral reconstruction with deep neural networks, Phys. Rev. D 102 (2020) 096001.doi:10.1103/PhysRevD.102.096001. URLhttps://link.aps.org/doi/10.1103/PhysRevD.102.096001

    doi:10.1103/physrevd.102.096001 2020

  36. [36]

    Fournier, L

    R. Fournier, L. Wang, O. V. Yazyev, Q. Wu, Artificial neural network approach to the analytic continuation problem, Phys. Rev. Lett. 124 (2020) 056401.doi:10.1103/PhysRevLett.124.056401. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.124.056401

    doi:10.1103/physrevlett.124.056401 2020

  37. [37]

    X. Xie, F. Bao, T. Maier, C. Webster, Analytic continuation of noisy data using adams bashforth residual neural network, Discrete and Continuous Dynamical Systems - S 15 (4) (2022) 877–892.doi:10.3934/dcdss.2021088. URLhttps://www.aimsciences.org/article/id/80acaa43-040a-4fac-a471-df7b267f5a8d

    doi:10.3934/dcdss.2021088 2022

  38. [38]

    S. Y. Chen, H. T. Ding, F. Y. Liu, G. Papp, C. B. Yang, Machine learning spectral functions in lattice qcd, in: Proceedings of Science (LATTICE2021), 2021, p. 148, poS(LATTICE2021)148.arXiv:2110.13521,doi:10.48550/arXiv.2110.13521. URLhttps://pos.sissa.it/396/148

    doi:10.48550/arxiv.2110.13521 2021

  39. [39]

    Huang, Y.-f

    D. Huang, Y.-f. Yang, Learned optimizers for analytic continuation, Phys. Rev. B 105 (2022) 075112.doi:10.1103/ PhysRevB.105.075112. URLhttps://link.aps.org/doi/10.1103/PhysRevB.105.075112

    doi:10.1103/physrevb.105.075112 2022

  40. [40]

    L. Wang, S. Shi, K. Zhou, Reconstructing spectral functions via automatic differentiation, Phys. Rev. D 106 (2022) L051502.doi:10.1103/PhysRevD.106.L051502. URLhttps://link.aps.org/doi/10.1103/PhysRevD.106.L051502

    doi:10.1103/physrevd.106.l051502 2022

  41. [41]

    L. Wang, S. Shi, K. Zhou, Unsupervised learning spectral functions with neural networks, in: Journal of Physics: Confer- ence Series, Vol. 2586, IOP Publishing, 2023, p. 012158.doi:10.1088/1742-6596/2586/1/012158

    doi:10.1088/1742-6596/2586/1/012158 2023

  42. [42]

    Rumetshofer, D

    M. Rumetshofer, D. Bauernfeind, W. von der Linden, Bayesian parametric analytic continuation of green’s functions, Phys. Rev. B 100 (2019) 075137.doi:10.1103/PhysRevB.100.075137. URLhttps://link.aps.org/doi/10.1103/PhysRevB.100.075137

    doi:10.1103/physrevb.100.075137 2019

  43. [43]

    S. L. Brunton, J. N. Kutz, Data-driven science and engineering: Machine learning, dynamical systems, and control, 16 Cambridge University Press, 2022

    2022

  44. [44]

    J. N. Kutz, S. L. Brunton, Parsimony as the ultimate regularizer for physics-informed machine learning, Nonlinear Dy- namics 107 (3) (2022) 1801–1817.doi:10.1007/s11071-021-07118-3. URLhttps://doi.org/10.1007/s11071-021-07118-3

    doi:10.1007/s11071-021-07118-3 2022

  45. [45]

    B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM Journal on Computing 24 (2) (1995) 227–234. doi:10.1137/S0097539792240406. URLhttps://doi.org/10.1137/S0097539792240406

    doi:10.1137/s0097539792240406 1995

  46. [46]

    Bryan, Maximum entropy analysis of oversampled data problems, European Biophysics Journal 18 (1990) 165–174

    R. Bryan, Maximum entropy analysis of oversampled data problems, European Biophysics Journal 18 (1990) 165–174. doi:10.1007/BF02427376

    doi:10.1007/bf02427376 1990

  47. [47]

    Rothkopf, Bryan’s maximum entropy method—diagnosis of a flawed argument and its remedy, Data 5 (3) (2020) 85

    A. Rothkopf, Bryan’s maximum entropy method—diagnosis of a flawed argument and its remedy, Data 5 (3) (2020) 85. doi:10.3390/data5030085. URLhttps://doi.org/10.3390/data5030085

    doi:10.3390/data5030085 2020

  48. [48]

    C. S. Fischer, J. M. Pawlowski, A. Rothkopf, C. A. Welzbacher, Bayesian analysis of quark spectral properties from the dyson-schwinger equation, Phys. Rev. D 98 (2018) 014009.doi:10.1103/PhysRevD.98.014009. URLhttps://link.aps.org/doi/10.1103/PhysRevD.98.014009

    doi:10.1103/physrevd.98.014009 2018

  49. [49]

    G. J. McLachlan, K. E. Basford, Mixture Models: Inference and Applications to Clustering, Marcel Dekker, New York, 1988

    1988

  50. [50]

    Rothkopf, Improved maximum entropy analysis with an extended search space, Journal of Computational Physics 238 (2013) 106–114.doi:https://doi.org/10.1016/j.jcp.2012.12.023

    A. Rothkopf, Improved maximum entropy analysis with an extended search space, Journal of Computational Physics 238 (2013) 106–114.doi:https://doi.org/10.1016/j.jcp.2012.12.023. URLhttps://www.sciencedirect.com/science/article/pii/S0021999112007541

    doi:10.1016/j.jcp.2012.12.023 2013

  51. [51]

    Kelly, A

    A. Kelly, A. Rothkopf, J.-I. Skullerud, Bayesian study of relativistic open and hidden charm in anisotropic lattice qcd, Phys. Rev. D 97 (2018) 114509.doi:10.1103/PhysRevD.97.114509. URLhttps://link.aps.org/doi/10.1103/PhysRevD.97.114509

    doi:10.1103/physrevd.97.114509 2018

  52. [52]

    J. K. Blitzstein, J. Hwang, Introduction to probability, Chapman and Hall/CRC, 2019

    2019

  53. [53]

    Chuna, N

    T. Chuna, N. Barnfield, J. Vorberger, M. P. Friedlander, T. Hoheisel, T. Dornheim, Estimates of the dynamic structure factor for the finite temperature electron liquid via analytic continuation of path integral monte carlo data, Phys. Rev. B 112 (2025) 125112.doi:10.1103/4d4b-kgtk. URLhttps://link.aps.org/doi/10.1103/4d4b-kgtk

    doi:10.1103/4d4b-kgtk 2025

  54. [54]

    Chuna, M

    T. Chuna, M. P. B¨ ohme, T. Dornheim, Temperature dependence of the dynamic structure factor of the electron liquid via analytic continuation, arXiv:2603.27212 (2026).arXiv:2603.27212. URLhttps://arxiv.org/abs/2603.27212

    arXiv 2026

  55. [55]

    Chuna, Inverseproblemsolvers,https://github.com/chunatho/InverseProblemSolvers(2025)

    T. Chuna, Inverseproblemsolvers,https://github.com/chunatho/InverseProblemSolvers(2025)

    2025

  56. [56]

    H.-T. Ding, A. Francis, O. Kaczmarek, F. Karsch, H. Satz, W. Soeldner, Charmonium properties in hot quenched lattice qcd, Phys. Rev. D 86 (2012) 014509.doi:10.1103/PhysRevD.86.014509. URLhttps://link.aps.org/doi/10.1103/PhysRevD.86.014509

    doi:10.1103/physrevd.86.014509 2012

  57. [57]

    E. V. Shuryak, Correlation functions in the qcd vacuum, Rev. Mod. Phys. 65 (1993) 1–46.doi:10.1103/RevModPhys.65.1. URLhttps://link.aps.org/doi/10.1103/RevModPhys.65.1

    doi:10.1103/revmodphys.65.1 1993

  58. [58]

    Kaufmann, K

    J. Kaufmann, K. Held, ana cont: Python package for analytic continuation, Computer Physics Communications 282 (2023) 108519.doi:https://doi.org/10.1016/j.cpc.2022.108519. URLhttps://www.sciencedirect.com/science/article/pii/S0010465522002387

    doi:10.1016/j.cpc.2022.108519 2023

  59. [59]

    Barnfield, J

    N. Barnfield, J. V. Burke, M. P. Friedlander, T. Hoheisel, A scale-shape dual newton method for entropic least squares, arXiv preprint arXiv:2604.27154 (2026).doi:https://doi.org/10.48550/arXiv.2604.27154

    Pith/arXiv arXiv doi:10.48550/arxiv.2604.27154 2026

  60. [60]

    S. F. Gull, Developments in Maximum Entropy Data Analysis, Springer Netherlands, Dordrecht, 1989, pp. 53–71.doi: 10.1007/978-94-015-7860-8_4. URLhttps://doi.org/10.1007/978-94-015-7860-8_4

    doi:10.1007/978-94-015-7860-8_4 1989

  61. [61]

    Chuna, N

    T. Chuna, N. Barnfield, P. Hamann, S. Schwalbe, M. P. Friedlander, T. Dornheim, The noiseless limit and improved-prior limit of the maximum entropy method and their implications for the analytic continuation problem, Physical Review Research (2025).doi:https://doi.org/10.1103/yqjh-ycbh

    doi:10.1103/yqjh-ycbh 2025

  62. [62]

    A. A. Istratov, O. F. Vyvenko, Exponential analysis in physical phenomena, Review of Scientific Instruments 70 (2) (1999) 1233–1257.doi:10.1063/1.1149581. URLhttps://doi.org/10.1063/1.1149581

    doi:10.1063/1.1149581 1999

  63. [63]

    W. H. Press, Numerical recipes 3rd edition: The art of scientific computing, Cambridge university press, 2007

    2007

  64. [64]

    Hansen, A

    M. Hansen, A. Lupo, N. Tantalo, Extraction of spectral densities from lattice correlators, Phys. Rev. D 99 (2019) 094508. doi:10.1103/PhysRevD.99.094508. URLhttps://link.aps.org/doi/10.1103/PhysRevD.99.094508

    doi:10.1103/physrevd.99.094508 2019

  65. [65]

    Del Debbio, A

    L. Del Debbio, A. Lupo, M. Panero, N. Tantalo, Bayesian interpretation of backus-gilbert methods, arXiv preprint arXiv:2311.18125 (2023). URLhttps://doi.org/10.48550/arXiv.2311.18125

    doi:10.48550/arxiv.2311.18125 2023

  66. [66]

    Dornheim, S

    T. Dornheim, S. Groth, J. Vorberger, M. Bonitz, Ab initio path integral monte carlo results for the dynamic structure factor of correlated electrons: From the electron liquid to warm dense matter, Phys. Rev. Lett. 121 (2018) 255001. doi:10.1103/PhysRevLett.121.255001. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.121.255001

    doi:10.1103/physrevlett.121.255001 2018

  67. [67]

    Giuliani, G

    G. Giuliani, G. Vignale, Quantum Theory of the Electron Liquid, Cambridge University Press, Cambridge, 2008

    2008

  68. [68]

    Ichimaru, Statistical plasma physics, volume I: basic principles, CRC Press, 2018

    S. Ichimaru, Statistical plasma physics, volume I: basic principles, CRC Press, 2018

    2018

  69. [69]

    Ichimaru, Strongly coupled plasmas: high-density classical plasmas and degenerate electron liquids, Rev

    S. Ichimaru, Strongly coupled plasmas: high-density classical plasmas and degenerate electron liquids, Rev. Mod. Phys 17 54 (1982) 1017. URLhttps://journals.aps.org/rmp/abstract/10.1103/RevModPhys.54.1017

    doi:10.1103/revmodphys.54.1017 1982

  70. [70]

    A. A. Kugler, Theory of the local field correction in an electron gas, J. Stat. Phys 12 (1975) 35. URLhttp://link.springer.com/article/10.1007/BF01024183

    doi:10.1007/bf01024183 1975

  71. [71]

    Dornheim, J

    T. Dornheim, J. Vorberger, S. Groth, N. Hoffmann, Z. Moldabekov, M. Bonitz, The static local field correction of the warm dense electron gas: An ab initio path integral Monte Carlo study and machine learning representation, J. Chem. Phys 151 (2019) 194104. URLhttps://aip.scitation.org/doi/full/10.1063/1.5123013

    doi:10.1063/1.5123013 2019

  72. [72]

    Dornheim, Z

    T. Dornheim, Z. A. Moldabekov, P. Tolias, Analytical representation of the local field correction of the uniform electron gas within the effective static approximation, Phys. Rev. B 103 (2021) 165102.doi:10.1103/PhysRevB.103.165102. URLhttps://link.aps.org/doi/10.1103/PhysRevB.103.165102

    doi:10.1103/physrevb.103.165102 2021

  73. [73]

    Dornheim, Z

    T. Dornheim, Z. A. Moldabekov, K. Ramakrishna, P. Tolias, A. D. Baczewski, D. Kraus, T. R. Preston, D. A. Chapman, M. P. B¨ ohme, T. D¨ oppner, F. Graziani, M. Bonitz, A. Cangi, J. Vorberger, Electronic density response of warm dense matter, Physics of Plasmas 30 (3) (2023) 032705.doi:10.1063/5.0138955. URLhttps://doi.org/10.1063/5.0138955

    doi:10.1063/5.0138955 2023

  74. [74]

    T. M. Chuna, J. Vorberger, P. Tolias, A. Benedix Robles, M. Hecht, P.-A. Hofmann, Z. A. Moldabekov, T. Dornheim, Second roton feature in the strongly coupled electron liquid, The Journal of Chemical Physics 163 (3) (2025) 034117. doi:10.1063/5.0281085. URLhttps://doi.org/10.1063/5.0281085

    doi:10.1063/5.0281085 2025

  75. [75]

    Beach, Identifying the maximum entropy method as a special limit of stochastic analytic continuation, arXiv preprint cond-mat/0403055 (2004)

    K. Beach, Identifying the maximum entropy method as a special limit of stochastic analytic continuation, arXiv preprint cond-mat/0403055 (2004)

    Pith/arXiv arXiv 2004

  76. [76]

    K. Ghanem, Stochastic analytic continuation: A bayesian approach, Phd thesis, RWTH Aachen University, Aachen, Germany, dissertation, Faculty of Mathematics, Computer Science and Natural Sciences; oral exam: June 26, 2017 (2017). doi:10.18154/RWTH-2017-06704. URLhttps://juser.fz-juelich.de/record/840299/files/696208.pdf?version=1

    doi:10.18154/rwth-2017-06704 2017

  77. [77]

    Ghanem, E

    K. Ghanem, E. Koch, Average spectrum method for analytic continuation: Efficient blocked-mode sampling and depen- dence on the discretization gride, Phys. Rev. B 101 (2020) 085111.doi:10.1103/PhysRevB.101.085111. URLhttps://link.aps.org/doi/10.1103/PhysRevB.101.085111

    doi:10.1103/physrevb.101.085111 2020

  78. [78]

    N. J. Richardson, N. Marusenko, M. P. Friedlander, Multiple scale methods for optimization of discretized continuous functions, arXiv preprint arXiv:2512.13993 (2025). URLhttps://doi.org/10.48550/arXiv.2512.13993

    doi:10.48550/arxiv.2512.13993 2025

  79. [79]

    L. N. Trefethen, Approximation Theory and Approximation Practice, extended edition Edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019.doi:10.1137/1.9781611975949. URLhttps://epubs.siam.org/doi/book/10.1137/1.9781611975949

    doi:10.1137/1.9781611975949 2019

  80. [80]

    Weeks, Numerical inversion of laplace transforms using laguerre functions, Journal of the ACM 13 (3) (1966) 419–429

    W. Weeks, Numerical inversion of laplace transforms using laguerre functions, Journal of the ACM 13 (3) (1966) 419–429. doi:10.1145/321341.321351

    doi:10.1145/321341.321351 1966

Showing first 80 references.