$\ell_0$ Factor Analysis: A P-Stationary Point Theory

TL;DR

This paper introduces a novel $$ norm-based factor analysis method for stationary time series, establishing theoretical properties and providing an ADMM algorithm with convergence guarantees, validated through numerical experiments.

Contribution

It develops a nonconvex $$ norm optimization framework for low-rank plus sparse covariance decomposition, with theoretical analysis and an efficient ADMM algorithm.

Findings

The method effectively decomposes covariance matrices into low-rank and sparse components.

Theoretical convergence of the ADMM algorithm is established.

Numerical experiments show superior performance over existing methods.

Abstract

Factor Analysis is a widely used modeling technique for stationary time series which achieves dimensionality reduction by revealing a hidden low-rank plus sparse structure of the covariance matrix. Such an idea of parsimonious modeling has also been important in the field of systems and control. In this article, a nonconvex nonsmooth optimization problem involving the $\ell_0$ norm is constructed in order to achieve the low-rank and sparse additive decomposition of the sample covariance matrix. We establish the existence of an optimal solution and characterize these solutions via the concept of proximal stationary points. Furthermore, an ADMM algorithm is designed to solve the $\ell_0$ optimization problem, and a subsequence convergence result is proved under reasonable assumptions. Finally, numerical experiments demonstrate the effectiveness of our method in comparison with some alternatives in the literature.