An efficient numerical method for condition number constrained covariance matrix approximation

TL;DR

This paper introduces an explicit solution and efficient algorithms for approximating covariance matrices under condition number constraints, ensuring numerical stability and positive definiteness in high-dimensional data analysis.

Contribution

It provides a novel explicit solution for the condition number constrained covariance matrix approximation problem and develops efficient algorithms leveraging matrix decomposition techniques.

Findings

The proposed algorithms are computationally efficient.

The method guarantees positive definiteness and numerical stability.

Numerical experiments confirm the effectiveness of the algorithms.

Abstract

In the high-dimensional data setting, the sample covariance matrix is singular. In order to get a numerically stable and positive definite modification of the sample covariance matrix in the high-dimensional data setting, in this paper we consider the condition number constrained covariance matrix approximation problem and present its explicit solution with respect to the Frobenius norm. The condition number constraint guarantees the numerical stability and positive definiteness of the approximation form simultaneously. By exploiting the special structure of the data matrix in the high-dimensional data setting, we also propose some new algorithms based on efficient matrix decomposition techniques. Numerical experiments are also given to show the computational efficiency of the proposed algorithms.