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

Weighted Singular Value Thresholding and its Application to Background Estimation