A Proximal Approach for a Class of Matrix Optimization Problems

TL;DR

This paper introduces a Douglas-Rachford based method for matrix optimization problems involving Bregman divergences and spectral regularization, demonstrating effectiveness in sparse covariance estimation and noisy graphical lasso.

Contribution

It develops a unified proximal framework with proximity operators for spectral and arbitrary regularization, and proposes an algorithm for noisy graphical lasso using majorization-minimization.

Findings

Validates the approach on sparse covariance matrix estimation

Shows good numerical performance compared to state-of-the-art methods

Provides convergence conditions for the proposed iterative scheme

Abstract

In recent years, there has been a growing interest in mathematical models leading to the minimization, in a symmetric matrix space, of a Bregman divergence coupled with a regularization term. We address problems of this type within a general framework where the regularization term is split in two parts, one being a spectral function while the other is arbitrary. A Douglas-Rachford approach is proposed to address such problems and a list of proximity operators is provided allowing us to consider various choices for the fit-to-data functional and for the regularization term. Numerical experiments show the validity of this approach for solving convex optimization problems encountered in the context of sparse covariance matrix estimation. Based on our theoretical results, an algorithm is also proposed for noisy graphical lasso where a precision matrix has to be estimated in the presence of noise. The nonconvexity of the resulting objective function is dealt with a majorization-minimization approach, i.e. by building a sequence of convex surrogates and solving the inner optimization subproblems via the aforementioned Douglas-Rachford procedure. We establish conditions for the convergence of this iterative scheme and we illustrate its good numerical performance with respect to state-of-the-art approaches.