Multivariate Gaussians, Semidefinite Matrix Completion, and Convex Algebraic Geometry

TL;DR

This paper explores the intersection of statistics and convex algebraic geometry by analyzing maximum likelihood estimation in multivariate normal models with linear inverse covariance constraints, focusing on determinant maximization over spectrahedra.

Contribution

It introduces a convex algebraic geometric framework to characterize maximum likelihood estimation problems involving linear constraints on inverse covariance matrices.

Findings

Characterization of the image of the positive definite cone under linear projections

Analysis of determinant maximization over spectrahedra

Insights into the algebraic structure of constrained Gaussian models

Abstract

We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron, and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection. These problems at the interface of statistics and optimization are here examined from the perspective of convex algebraic geometry.