Likelihood Geometry of Correlation Models

TL;DR

This paper explores the geometric structure of correlation models, focusing on maximum likelihood estimation, symmetry submodels, and convex loss functions, providing insights into their optimization and general affine covariance models.

Contribution

It offers a detailed geometric analysis of correlation models, including the behavior of convex loss functions and the extension to general affine covariance models.

Findings

Maximum likelihood estimation geometry for correlation models

Convexity and uniqueness of Stein's loss minimization

Extension of results to general affine covariance models

Abstract

Correlation matrices are standardized covariance matrices. They form an affine space of symmetric matrices defined by setting the diagonal entries to one. We study the geometry of maximum likelihood estimation for this model and linear submodels that encode additional symmetries. We also consider the problem of minimizing two closely related functions of the covariance matrix: the Stein's loss and the symmetrized Stein's loss. Unlike the Gaussian log-likelihood these two functions are convex and hence admit a unique positive definite optimum. Some of our results hold for general affine covariance models.