On a parametrization of positive semidefinite matrices with zeros

TL;DR

This paper investigates parametrizations of positive semidefinite matrices with prescribed zeros, characterizing when these maps are surjective and providing a semi-algebraic description for certain graph structures, with applications in Gaussian models.

Contribution

It introduces a polynomial parametrization framework for zero-constrained positive semidefinite matrices and characterizes surjectivity conditions based on graph properties, especially for chordless cycles.

Findings

Maps are surjective only for chordal graphs and their clique complexes.

Provides a semi-algebraic description of the image for chordless cycles.

Connects parametrizations to Gaussian models with hidden variables.

Abstract

We study a class of parametrizations of convex cones of positive semidefinite matrices with prescribed zeros. Each such cone corresponds to a graph whose non-edges determine the prescribed zeros. Each parametrization in this class is a polynomial map associated with a simplicial complex supported on cliques of the graph. The images of the maps are convex cones, and the maps can only be surjective onto the cone of zero-constrained positive semidefinite matrices when the associated graph is chordal and the simplicial complex is the clique complex of the graph. Our main result gives a semi-algebraic description of the image of the parametrizations for chordless cycles. The work is motivated by the fact that the considered maps correspond to Gaussian statistical models with hidden variables.