Improved Maximum Entropy Method with an Extended Search Space

TL;DR

This paper introduces an improved Maximum Entropy Method that extends the search space beyond the traditional SVD-based basis, using a Fourier basis to enhance stability and resolution in spectral reconstructions.

Contribution

The authors propose replacing the SVD-based basis in MEM with a Fourier basis, enabling a more complete search space and more stable, resolution-independent spectral reconstructions.

Findings

Fourier basis improves numerical stability.

Extended search space enhances spectral reconstruction accuracy.

Method is demonstrated on mock lattice data.

Abstract

We report on an improvement to the implementation of the Maximum Entropy Method (MEM). It amounts to departing from the search space obtained through a singular value decomposition (SVD) of the Kernel. Based on the shape of the SVD basis functions we argue that the MEM spectrum for given $N_\tau$ data-points $D(\tau)$ and prior information $m(\omega)$ does not in general lie in this $N_\tau$ dimensional singular subspace. Systematically extending the search basis will eventually recover the full search space and the correct extremum. We illustrate this idea through a mock data analysis inspired by actual lattice spectra, to show where our improvement becomes essential for the success of the MEM. To remedy the shortcomings of Bryan's SVD prescription we propose to use the real Fourier basis, which consists of trigonometric functions. Not only does our approach lead to more stable numerical behavior, as the SVD is not required for the determination of the basis functions, but also the resolution of the MEM becomes independent from the position of the reconstructed peaks.