Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications

TL;DR

This paper introduces a novel partial sum of singular values approach for robust PCA that better utilizes known rank information, improving low-rank recovery especially with limited data in various vision tasks.

Contribution

It proposes replacing nuclear norm minimization with partial singular value sum minimization to incorporate known rank constraints in robust PCA.

Findings

Outperforms nuclear norm minimization with limited samples

Achieves similar results to traditional methods with sufficient data

Improves performance in multiple low-level vision applications

Abstract

Robust Principal Component Analysis (RPCA) via rank minimization is a powerful tool for recovering underlying low-rank structure of clean data corrupted with sparse noise/outliers. In many low-level vision problems, not only it is known that the underlying structure of clean data is low-rank, but the exact rank of clean data is also known. Yet, when applying conventional rank minimization for those problems, the objective function is formulated in a way that does not fully utilize a priori target rank information about the problems. This observation motivates us to investigate whether there is a better alternative solution when using rank minimization. In this paper, instead of minimizing the nuclear norm, we propose to minimize the partial sum of singular values, which implicitly encourages the target rank constraint. Our experimental analyses show that, when the number of samples is deficient, our approach leads to a higher success rate than conventional rank minimization, while the solutions obtained by the two approaches are almost identical when the number of samples is more than sufficient. We apply our approach to various low-level vision problems, e.g. high dynamic range imaging, motion edge detection, photometric stereo, image alignment and recovery, and show that our results outperform those obtained by the conventional nuclear norm rank minimization method.