Robust PCA by Manifold Optimization

TL;DR

This paper introduces manifold optimization algorithms for robust PCA, providing guarantees for convergence and improved theoretical dependence on matrix condition numbers, with demonstrated effectiveness on simulations and real data.

Contribution

It proposes two novel manifold optimization algorithms for robust PCA with convergence guarantees and reduced dependence on matrix condition numbers.

Findings

Algorithms converge linearly with proper initialization

Reduced dependence on the condition number compared to previous methods

Confirmed effectiveness through simulations and real data

Abstract

Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on manifold optimization. It is shown that, with a proper designed initialization, the proposed algorithms are guaranteed to converge to the underlying low-rank matrix linearly. Compared with a previous work based on the Burer-Monterio decomposition of low-rank matrices, the proposed algorithms reduce the dependence on the conditional number of the underlying low-rank matrix theoretically. Simulations and real data examples confirm the competitive performance of our method.