Weighted Singular Value Thresholding and its Application to Background Estimation

TL;DR

This paper introduces a weighted singular value thresholding method combining nuclear and weighted Frobenius norms, with an algorithm that outperforms traditional RPCA in background estimation tasks.

Contribution

It formulates and analyzes a novel weighted singular value thresholding problem and demonstrates its effectiveness in background estimation, offering an alternative to computationally intensive RPCA.

Findings

WSVT outperforms RPCA in background estimation

The algorithm converges reliably

Weighted Frobenius norm can replace $\, ext{l}_1$ norm in RPCA

Abstract

Singular value thresholding (SVT) plays an important role in the well-known robust principal component analysis (RPCA) algorithms which have many applications in computer vision and recommendation systems. In this paper, we formulate and study a weighted singular value thresholding (WSVT) problem, which uses a combination of the nuclear norm and a weighted Frobenius norm. We present an algorithm to numerically solve WSVT and establish the convergence of the algorithm. As a proof of concept, we apply WSVT with a simple choice of weight learned from the data to the background estimation problem in computer vision. The numerical experiments show that our method can outperform RPCA algorithms. This indicates that instead of tackling the computationally expensive $\ell_1$ norm employed in RPCA, one may switch to the weighted Frobenius norm and achieve about the same or even better performance.