Deformed logarithms. Associated entropic divergences. Applications to linear inverse problems. Inversion algorithms

TL;DR

This paper explores the use of deformed logarithms and associated entropic divergences within linear inverse problems, proposing new divergence functions based on statistical physics entropy concepts to improve solution methods.

Contribution

It introduces divergence functions derived from deformed algebra and entropy, extending traditional inverse problem frameworks with novel mathematical tools.

Findings

Development of divergence functions based on deformed logarithms

Application of these divergences to linear inverse problems

Potential improvements in inverse problem resolution methods

Abstract

This work belongs to the framework of inverse problems with linear model. The resolution of this type of problem consists in minimizing (possibly under constraints) a function of discrepancy between the measurements and a physical model of the considered phenomenon. We will here extend this study to divergence functions based on forms of entropy developed in the field of Statistical Physics. These various forms of entropy are based on "the deformed algebra" [9] and in particularly on the notion of "Deformed Logarithm" [15][9].