Alternating minimization for square root principal component pursuit

TL;DR

This paper introduces a tuning-free alternating minimization algorithm for the square root principal component pursuit problem, offering efficient, robust solutions with theoretical and practical advantages.

Contribution

It develops a novel, tuning-free AltMin algorithm with closed-form solutions for SRPCP, enhancing computational efficiency and robustness.

Findings

The proposed algorithms are faster than existing methods.

They demonstrate robustness across various numerical experiments.

The methods achieve accurate matrix recovery in practice.

Abstract

Recently, the square root principal component pursuit (SRPCP) model has garnered significant research interest. It is shown in the literature that the SRPCP model guarantees robust matrix recovery with a universal, constant penalty parameter. While its statistical advantages are well-documented, the computational aspects from an optimization perspective remain largely unexplored. In this paper, we focus on developing efficient optimization algorithms for solving the SRPCP problem. Specifically, we propose a tuning-free alternating minimization (AltMin) algorithm, where each iteration involves subproblems enjoying closed-form optimal solutions. Additionally, we introduce techniques based on the variational formulation of the nuclear norm and Burer-Monteiro decomposition to further accelerate the AltMin method. Extensive numerical experiments confirm the efficiency and robustness of our algorithms.