Analytical continuation of imaginary axis data using maximum entropy

TL;DR

This paper improves the maximum entropy method for analytical continuation by reducing statistical errors through data batching and iterative refinement, balancing errors for more accurate spectral data reconstruction.

Contribution

It introduces a batching and iterative approach to optimize the MaxEnt method, reducing statistical errors and improving spectral data accuracy.

Findings

Batching data reduces statistical error in MaxEnt.

Iterative use of MaxEnt can worsen results due to increased statistical error.

Linearized analysis provides insights into error balancing in MaxEnt.

Abstract

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 result 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 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.