Robust Identification of "Sparse Plus Low-rank" Graphical Models: An Optimization Approach

TL;DR

This paper introduces a robust optimization method for the "Sparse Plus Low-rank" graphical model decomposition, addressing uncertainties in covariance matrix estimation to improve practical applicability.

Contribution

It proposes an alternative optimization approach that enhances robustness in the "Sparse Plus Low-rank" decomposition of covariance matrices affected by data uncertainty.

Findings

The new method improves robustness against covariance estimation errors.

The variational analysis provides insights into the optimization problem.

The approach is suitable for practical scenarios with noisy data.

Abstract

Motivated by graphical models, we consider the "Sparse Plus Low-rank" decomposition of a positive definite concentration matrix -- the inverse of the covariance matrix. This is a classical problem for which a rich theory and numerical algorithms have been developed. It appears, however, that the results rapidly degrade when, as it happens in practice, the covariance matrix must be estimated from the observed data and is therefore affected by a certain degree of uncertainty. We discuss this problem and propose an alternative optimization approach that appears to be suitable to deal with robustness issues in the "Sparse Plus Low-rank" decomposition problem.The variational analysis of this optimization problem is carried over and discussed.