Robust PCA via Nonconvex Rank Approximation

TL;DR

This paper introduces a nonconvex rank approximation method for robust PCA that provides a tighter estimate of the matrix rank, leading to improved accuracy and efficiency over existing nuclear norm-based approaches.

Contribution

It proposes a novel nonconvex rank approximation and an efficient optimization algorithm, addressing limitations of nuclear norm methods in real-world applications.

Findings

Outperforms state-of-the-art algorithms in accuracy

Achieves higher efficiency in matrix recovery

Provides a closer approximation to the true matrix rank

Abstract

Numerous applications in data mining and machine learning require recovering a matrix of minimal rank. Robust principal component analysis (RPCA) is a general framework for handling this kind of problems. Nuclear norm based convex surrogate of the rank function in RPCA is widely investigated. Under certain assumptions, it can recover the underlying true low rank matrix with high probability. However, those assumptions may not hold in real-world applications. Since the nuclear norm approximates the rank by adding all singular values together, which is essentially a $\ell_1$-norm of the singular values, the resulting approximation error is not trivial and thus the resulting matrix estimator can be significantly biased. To seek a closer approximation and to alleviate the above-mentioned limitations of the nuclear norm, we propose a nonconvex rank approximation. This approximation to the matrix rank is tighter than the nuclear norm. To solve the associated nonconvex minimization problem, we develop an efficient augmented Lagrange multiplier based optimization algorithm. Experimental results demonstrate that our method outperforms current state-of-the-art algorithms in both accuracy and efficiency.