$\ell_0$ factor analysis

TL;DR

This paper introduces a novel factor analysis method that decomposes a covariance matrix into low-rank and sparse parts using an optimization approach with nuclear and $\\ell_0$ norms, solved via an alternating minimization algorithm.

Contribution

It formulates a new optimization framework for factor analysis employing $\\ell_0$ norm for sparsity and nuclear norm for low-rankness, with an efficient algorithm for solution.

Findings

Algorithm effectively recovers low-rank and sparse components.

Method performs well on synthetic datasets.

Method demonstrates practical utility on real data.

Abstract

Factor Analysis is about finding a low-rank plus sparse additive decomposition from a noisy estimate of the signal covariance matrix. In order to get such a decomposition, we formulate an optimization problem using the nuclear norm for the low-rank component, the $\ell_0$ norm for the sparse component, and the Kullback-Leibler divergence to control the residual in the sample covariance matrix. An alternating minimization algorithm is designed for the solution of the optimization problem. The effectiveness of the algorithm is verified via simulations on synthetic and real datasets.