Positive semidefinite rank

TL;DR

This paper explores the mathematical properties and geometric interpretations of positive semidefinite rank, a measure related to matrix factorizations with applications in geometry and information theory.

Contribution

It develops and surveys key properties of psd rank, including its geometry, relationships with other ranks, and computational aspects.

Findings

Characterizes geometric interpretations of psd rank

Examines relationships between psd rank and other matrix ranks

Discusses computational and algorithmic challenges

Abstract

Let M be a p-by-q matrix with nonnegative entries. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices $A_i, B_j$ of size $k \times k$ such that $M_{ij} = \text{trace}(A_i B_j)$. The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and information-theoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.