Sparse and Low-Rank Covariance Matrices Estimation

TL;DR

This paper introduces a convex optimization method for estimating covariance matrices that are both sparse and low-rank, providing theoretical guarantees and an efficient algorithm with practical validation.

Contribution

It proposes a novel convex optimization approach combining $l_1$ and nuclear norms for sparse, low-rank covariance estimation, with proven convergence and performance guarantees.

Findings

Estimation error rate of $O(\sqrt{s(\log r)/n})$ under mild conditions

An efficient ADMM algorithm with global convergence

Numerical simulations demonstrating the method's effectiveness

Abstract

This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices. We first benefit from a convex optimization which develops $l_1$-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property. For the proposed estimator, we then prove that with large probability, the Frobenious norm of the estimation rate can be of order $O(\sqrt{s(\log{r})/n})$ under a mild case, where $s$ and $r$ denote the number of sparse entries and the rank of the population covariance respectively, $n$ notes the sample capacity. Finally an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem, and meantime merits of the approach are also illustrated by practicing numerical simulations.