Efficient Algorithms for Robust and Stable Principal Component Pursuit Problems

TL;DR

This paper introduces efficient algorithms with proven complexity bounds for solving robust and stable principal component pursuit problems, enabling accurate recovery of low-rank matrices from highly corrupted data in large-scale applications.

Contribution

The paper presents novel algorithms with theoretical guarantees for RPCP and SPCP, improving computational efficiency and scalability over existing methods.

Findings

Algorithms handle millions of variables efficiently

Achieve competitive accuracy and speed

Effective in applications like video foreground extraction

Abstract

The problem of recovering a low-rank matrix from a set of observations corrupted with gross sparse error is known as the robust principal component analysis (RPCA) and has many applications in computer vision, image processing and web data ranking. It has been shown that under certain conditions, the solution to the NP-hard RPCA problem can be obtained by solving a convex optimization problem, namely the robust principal component pursuit (RPCP). Moreover, if the observed data matrix has also been corrupted by a dense noise matrix in addition to gross sparse error, then the stable principal component pursuit (SPCP) problem is solved to recover the low-rank matrix. In this paper, we develop efficient algorithms with provable iteration complexity bounds for solving RPCP and SPCP. Numerical results on problems with millions of variables and constraints such as foreground extraction from surveillance video, shadow and specularity removal from face images and video denoising from heavily corrupted data show that our algorithms are competitive to current state-of-the-art solvers for RPCP and SPCP in terms of accuracy and speed.