Positive Definite Estimation of Large Covariance Matrix Using Generalized Nonconvex Penalties

TL;DR

This paper introduces a new class of positive-definite covariance matrix estimators using generalized nonconvex penalties, along with an efficient algorithm and theoretical analysis, demonstrating improved performance in high-dimensional settings.

Contribution

It proposes a novel positive-definite covariance estimation method with nonconvex penalties, and develops a convergent first-order algorithm for high-dimensional statistical analysis.

Findings

Algorithm converges efficiently and reliably.

Estimators outperform traditional methods in simulations.

Effective in gene clustering for tumor tissues.

Abstract

This work addresses the issue of large covariance matrix estimation in high-dimensional statistical analysis. Recently, improved iterative algorithms with positive-definite guarantee have been developed. However, these algorithms cannot be directly extended to use a nonconvex penalty for sparsity inducing. Generally, a nonconvex penalty has the capability of ameliorating the bias problem of the popular convex lasso penalty, and thus is more advantageous. In this work, we propose a class of positive-definite covariance estimators using generalized nonconvex penalties. We develop a first-order algorithm based on the alternating direction method framework to solve the nonconvex optimization problem efficiently. The convergence of this algorithm has been proved. Further, the statistical properties of the new estimators have been analyzed for generalized nonconvex penalties. Moreover, extension of this algorithm to covariance estimation from sketched measurements has been considered. The performances of the new estimators have been demonstrated by both a simulation study and a gene clustering example for tumor tissues. Code for the proposed estimators is available at https://github.com/FWen/Nonconvex-PDLCE.git.