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

arXiv: https://arxiv.org/abs/1512.03401 · HTML: https://mlsanity.com/paper/1512.03401/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2015.

## Built on by (most cited first, 1 of 1)

- [A new perspective on low-rank optimization](https://mlsanity.com/paper/2105.05947/) (2021)

## Abstract (arXiv)

A famous result of Lieb establishes that the map (A,B) mapsto trleft[K^* A1-t K Btright] is jointly concave in the pair (A,B) of positive definite matrices, where K is a fixed matrix and t ∈ [0,1]. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational t ∈ [0,1]. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.