Lieb's concavity theorem, matrix geometric means, and semidefinite optimization

TL;DR

This paper provides an explicit semidefinite programming formulation for Lieb's concavity theorem involving matrix functions, utilizing semidefinite representations of weighted matrix geometric means, with practical implementation in MATLAB.

Contribution

It introduces a novel semidefinite programming formulation for Lieb's concavity theorem applicable to rational parameters, connecting it with matrix geometric means.

Findings

Explicit SDP formulation for Lieb's function at rational t

Connection between Lieb's theorem and matrix geometric means

MATLAB implementation of the proposed formulations

Abstract

A famous result of Lieb establishes that the map $(A,B) \mapsto \text{tr}\left[K^* A^{1-t} K B^t\right]$ is jointly concave in the pair $(A,B)$ of positive definite matrices, where $K$ is a fixed matrix and $t \in [0,1]$. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational $t \in [0,1]$. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.