Strong Convexity of Sandwiched Entropies and Related Optimization Problems

TL;DR

This paper investigates the strong convexity properties of sandwiched quasi-relative entropy, providing theoretical results that enable the proof of global linear convergence for gradient projection algorithms in related optimization problems, especially in optimal transport.

Contribution

It introduces new theorems on convexity and differential formulae for sandwiched quasi-relative entropy, advancing the understanding of their optimization landscape and convergence behavior.

Findings

Established strict and strong convexity results for sandwiched quasi-relative entropy.

Derived higher order differential formulae for these entropy functions.

Proved global linear convergence of gradient projection algorithms for related optimization problems.

Abstract

We present several theorems on strict and strong convexity, and higher order differential formulae for sandwiched quasi-relative entropy (a parametrised version of the classical fidelity). These are crucial for establishing global linear convergence of the gradient projection algorithm for optimisation problems for these functions. The case of the classical fidelity is of special interest for the multimarginal optimal transport problem (the $n$-coupling problem) for Gaussian measures.