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arXiv:2606.19205

Discovering a well-conditioned analytic continuation problem via dictionary learning

Thomas Chuna, Phil-Alexander Hofmann, Alexander Benedix Robles, Tobias Dornheim

Published Jun 17, 2026• 0 citations
arXivHTMLPDF

Abstract

Many fields of physics use quantum Monte Carlo (QMC) simulations to simulate quantum systems in imaginary-time $τ$ and estimate imaginary-time correlation functions (ITCF). However, extracting dynamic $ω$-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.

DOI
10.48550/arxiv.2606.19205

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Many fields of physics use quantum Monte Carlo (QMC) simulations to simulate quantum systems in imaginary-time τττ and estimate imaginary-time correlation functions (ITCF). However, extracting dynamic ωωω-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.
Stochastic analytic continuation (SAC) of quantum Monte Carlo (QMC) imaginary-time correlation function data is a valuable tool in connecting many-body models to experimentally measurable dynamic response functions. Recent developments of the SAC method have allowed for spectral functions with sharp features, e.g., narrow peaks and divergent edges, to be resolved with unprecedented fidelity. Often it is not known what exact sharp features, if any, are present a priori, and, due to the ill-posed nature of the analytic continuation problem, multiple spectral representations may be acceptable. In this work we borrow from the machine learning and statistics literature and implement a cross validation technique to provide an unbiased method to identify the most likely spectrum among a set obtained with different spectral parametrizations and imposed constraints. We demonstrate the power of this method with examples using imaginary-time data generated by QMC simulations and synthetic data generated from artificial spectra. Our procedure, which can be considered a form of model selection, can be applied to a variety of numerical analytic continuation methods, beyond just SAC.
A data-science approach to solving the ill-conditioned inverse problem for analytical continuation is proposed. The root of the problem lies in the fact that even tiny noise of imaginary-time input data has a serious impact on the inferred real-frequency spectra. By means of a modern regularization technique, we eliminate redundant degrees of freedom that essentially carry the noise, leaving only relevant information unaffected by the noise. The resultant spectrum is represented with minimal bases and thus a stable analytical continuation is achieved. This framework further provides a tool for analyzing to what extent the Monte Carlo data need to be accurate to resolve details of an expected spectral function.
We present a machine-learning approach to a long-standing issue in quantum many-body physics, namely, analytic continuation. This notorious ill-conditioned problem of obtaining spectral function from an imaginary time Green's function has been a focus of new method developments for past decades. Here we demonstrate the usefulness of modern machine-learning techniques including convolutional neural networks and the variants of a stochastic gradient descent optimizer. The machine-learning continuation kernel is successfully realized without any ``domain knowledge,'' which means that any physical ``prior'' is not utilized in the kernel construction and the neural networks ``learn'' the knowledge solely from ``training.'' The outstanding performance is achieved for both insulating and metallic band structure. Our machine-learning-based approach not only provides the more accurate spectrum than the conventional methods in terms of peak positions and heights, but is also more robust against the noise which is the required key feature for any continuation technique to be successful. Furthermore, its computation speed is 104–105{10}^{4}\text{--}{10}^{5}104–105 times faster than the maximum entropy method.
The presence of dynamical information is a feature distinguishing a finite-temperature quantum Monte Carlo simulation from a classical one. We now discuss numerical methods for extracting this information that use techniques and concepts borrowed from an area of probability theory called Bayesian statistical inference. The use of these techniques and concepts provided a solution to the very difficult problem of analytically continuing imaginary-time Green's functions, estimated by a quantum Monte Carlo simulation, to the real-time axis. Baym and Mermin (1961) proved that a unique mapping between these functions exists. However, executing this mapping numerically, with a simulation's incomplete and noisy data, transforms the problem into one without a unique solution and thus into a problem of finding a “best” solution according to some reasonable criterion. Instead of executing the analytic continuation between imaginary- and real-time Green's functions, thereby obtaining real-time dynamics, we instead estimate the experimentally relevant spectral density function these Green's functions share. We present three “best” solutions and emphasize that making the simulation data consistent with the assumptions of the numerical approach is a key step toward finding any of these best solutions. Preliminary comments The title of this chapter, “Analytic Continuation,” is unusual in the sense that it describes the task we wish to accomplish instead of the method we use to accomplish it. If we used the name of the method, the title would be something like “Bayesian Statistical Inference Using an Entropic Prior.” A shorter title would be “The Maximum Entropy Method.”We hope by the end of the chapter the reader will agree that using the short title is perhaps too glib and the longer one has meaningful content.
Inverse problems are encountered in many domains of physics, with analytic continuation of the imaginary Green's function into the real frequency domain being a particularly important example. However, the analytic continuation problem is ill defined and currently no analytic transformation for solving it is known. We present a general framework for building an artificial neural network (ANN) that solves this task with a supervised learning approach. Application of the ANN approach to quantum Monte Carlo calculations and simulated Green's function data demonstrates its high accuracy. By comparing with the commonly used maximum entropy approach, we show that our method can reach the same level of accuracy for low-noise input data, while performing significantly better when the noise strength increases. The computational cost of the proposed neural network approach is reduced by almost three orders of magnitude compared to the maximum entropy method.
Path integral Monte Carlo (PIMC) simulations are a cornerstone method for studying quantum many-body systems, such as warm dense matter and ultracold atoms. The analytic continuation needed to estimate dynamic quantities from these simulations amounts to an inverse Laplace transform, which is an ill-conditioned problem. If this challenging problem were surmounted, dynamical observables such as the dynamic structure factor (DSF) S ( q , ω ) —a key property e.g. in x-ray and neutron scattering experiments—could be extracted from the imaginary-time correlation functions F ( q , τ ) estimates. Although of fundamental importance, the analytic continuation problem remains challenging due to its ill-posedness, and state-of-the-art techniques continue to deliver unsatisfactory results. To address this challenge, we express the DSF as a linear combination of kernel functions with known Laplace transforms that have been tailored to satisfy its physical constraints, e.g. , detailed balance. Then we employ least-squares optimization regularized with a Bayesian prior estimate to determine the coefficients of this linear combination. We explore various regularization term, such as the commonly used entropic regularizer, as well as the uncommon L 2 -distance and CDF L 2 -distance. We also explore techniques for setting the regularization weight. A key outcome and contribution is the open-source package PyLIT ( Py thon L aplace I nverse T ransform), which leverages Numba for C-level performance, unifying the presented formulations. PyLIT’s core functionality is kernel construction and optimization. In our applications, we find that PyLIT’s DSF estimates share many qualitative features with other more established methods. Drawing from our insights, we identify three key findings. Firstly, independent of the regularization choice, utilizing non-uniform grid point distributions reduced the number of unknowns and thus reduced our space of possible solutions. Secondly, the L 2 -distance and CDF L 2 -distance, previously unexplored regularizers, benefit from their linear gradients and perform about as well as entriopic regularization. Thirdly, future work can meaningfully combine regularized and stochastic optimization.