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arXiv:1810.06147

Suppression of Overfitting in Extraction of Spectral Data from Imaginary Frequency Green Function Using Maximum Entropy Method

Enzhi Li

Published Oct 15, 2018• 0 citations
arXivHTMLPDF

Abstract

Although maximum entropy method (maxEnt method) is currently the standard algorithm for extracting real frequency information from imaginary frequency Green function, still this method is beset with overfitting problem, which manifests itself as the spurious spikes in the resultant spectral functions. To address this issue and motivated by the regularization techniques widely used in machine learning and statistics, here we propose to add one more regularization term into the original maxEnt loss function to suppress these redundant spikes. The essence of this extra regularization term is to demand that the resultant spectral functions should pay a price for being spiky. We test our algorithm with both artificial and real data, and find that spurious spikes in the resultant spectral functions can be effectively suppressed by this method.

DOI
10.48550/arxiv.1810.06147

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Although maximum entropy method (maxEnt method) is currently the standard algorithm for extracting real frequency information from imaginary frequency Green function, still this method is beset with overfitting problem, which manifests itself as the spurious spikes in the resultant spectral functions. To address this issue and motivated by the regularization techniques widely used in machine learning and statistics, here we propose to add one more regularization term into the original maxEnt loss function to suppress these redundant spikes. The essence of this extra regularization term is to demand that the resultant spectral functions should pay a price for being spiky. We test our algorithm with both artificial and real data, and find that spurious spikes in the resultant spectral functions can be effectively suppressed by this method.
An important problem in many-body physics is to reconstruct the spectral density from the imaginary-time domain Green's function. Typically, the imaginary-time Green's function is generated by Monte Carlo methods. As the one-point fermionic kernel diverges exponentially for large frequencies, numerical noise generically causes instabilities. We use a convolutional neural network to obtain the spectral density for a given imaginary time Green's function. The network is trained by data which we generate using random Gaussians. We improve the training data set available by including collision centers for the Gaussians rather than employing uniformly distributed Gaussians. Our network is constructed in such a way that its output fulfills positive semidefiniteness. We compare the results of our network with results of the Maximum Entropy method (MaxEnt), a standard method for the same reconstruction problem for the spectral density. This comparison is performed for three different cases, namely our Gaussian based test data as well as two physical models, the 1d Hubbard model showing spin-charge separation, and the two-dimensional SSH model in the self-consistent Born approximation. We find that the network outperforms MaxEnt when presented data close to the training set. For the physical models considered, MaxEnt recognizes physical features more precisely as compared to our network prediction. While it is hard to improve MaxEnt, the quality of the network depends on the training data set which can be systematically enhanced and improved.
We study the maximum entropy (MaxEnt) approach for analytical continuation of spectral data from imaginary times to real frequencies. The total error is divided in a statistical error, due to the noise in the input data, and a systematic error, due to deviations of the default function, used in the MaxEnt approach, from the exact spectrum. We find that the MaxEnt approach in its classical formulation can lead to a nonoptimal balance between the two types of errors, leading to an unnecessary large statistical error. The statistical error can be reduced by splitting up the data in several batches, performing a MaxEnt calculation for each batch and averaging. This can outweigh an increase in the systematic error resulting from this approach. The output from the MaxEnt calculation can be used as a default function for a new MaxEnt calculation. Such iterations often lead to worse results due to an increase in the statistical error. By splitting up the data in batches, the statistical error is reduced and the increase resulting from iterations can be outweighed by a decrease in the systematic error. Finally we consider a linearized version to obtain a better understanding of the method.
An outstanding problem in the simulation of condensed matter phenomena is how to obtain dynamical information. We consider the numerical analytic continuation of imaginary time Quantum Monte Carlo data to obtain real frequency spectral functions. We suggest an image reconstruction approach which has been widely applied to data analysis in experimental research, the Maximum Entropy Method (MaxEnt). We report encouraging preliminary results for the Fano-Anderson model of an impurity state in a continuum. The incorporation of additional prior information, such as sum rules and asymptotic behavior, can be expected to significantly improve results. We also compare MaxEnt to alternative methods. 17 refs., 4 figs.
In this proceeding, an idea is outlined suggesting a generic treatment on any type of input data for a numerical analytic continuation problem, which is needed when dynamical information is to be extracted from a calculationally convenient one particle imaginary time Green function. The quality of the resulting spectral function will rely only on the data to be treated, viz, data-driven. This is different from the Maximum Entropy or the Stochastic method which relies on an entropy term to guide convergence of the resulting spectral function.
Imaginary-time correlation functions calculated by quantum Monte Carlo (QMC) are analyzed using the maximum entropy method (MaxEnt) to determine the ground-state energy and spectral overlap function. In contrast to earlier applications of MaxEnt, the data is obtained from importanced-sampled zero-temperature quantum Monte Carlo simulations. The analysis includes two steps. First, that spectral overlap function and ground state energy which maximizes the entropy and agrees with the QMC correlation functions is obtained. Then the errors in the energy are evaluated by averaging over all the possible images (average MaxEnt method), the multidimensional integrals being computed using the Metropolis algorithm. The central feature of this approach is that all the information present in the correlation functions is used in the only way consistent with fundamental probabilistic hypotheses. This allows us to fully exploit the information contained in the correlation functions at small imaginary times, thus avoiding large statistical fluctuations associated with large imaginary times. In addition, the computed errors include both the statistical errors and systematic extrapolation errors. The method is illustrated with a harmonic oscillator and the four-electron LiH molecule.
We investigate the possibility to assist the numerically ill-posed calculation of spectral properties of interacting quantum systems in thermal equilibrium by extending the imaginary-time simulation to a finite Schwinger-Keldysh contour. The effect of this extension is tested within the standard maximum entropy approach to analytic continuation. We find that the inclusion of real-time data improves the resolution of structures at high energy, while the imaginary-time data are needed to correctly reproduce low-frequency features such as quasiparticle peaks. As a nonequilibrium application, we consider the calculation of time-dependent spectral functions from retarded Green function data on a finite time interval, and compare the maximum entropy approach to direct Fourier transformation and a method based on Padé approximants.