Generalized Maximum Entropy Methods as Limits of the Average Spectrum Method

TL;DR

This paper demonstrates that the average spectrum method (ASM) is equivalent to maximizing Re9nyi entropies in the continuum limit, generalizing the maximum entropy approach to extract more spectral structure.

Contribution

It establishes a connection between ASM and Re9nyi entropy maximization, introducing a generalized framework for spectral analysis beyond Shannon entropy.

Findings

ASM corresponds to Re9nyi entropy maximization for e9e9=1

Sharper spectral peaks relate to e9e9<1

Proposes a modified Re9nyi entropy with a non-trivial e9e9b0e9e9 limit

Abstract

We show that in the continuum limit, the average spectrum method (ASM) is equivalent to maximizing R\'enyi entropies of order $\eta$, of which Shannon entropy is the special case $\eta=1$. The order of R\'enyi entropy is determined by the way the spectra are sampled. Our derivation also suggests a modification of R\'enyi entropy, giving it a non-trivial $\eta\to0$ limit. We show that the sharper peaks generally obtained in ASM are associated with entropies of order $\eta<1$. Our work provides a generalization of the maximum entropy method that enables extracting more structure than the traditional method.