Fast Randomized Singular Value Thresholding for Nuclear Norm Minimization by Tae-Hyun Oh, Yasuyuki Matsushita, Yu-Wing Tai, In So Kweon

Rank minimization can be boiled down to tractable surrogate problems, such as Nuclear Norm Minimization (NNM) and Weighted NNM (WNNM). The problems related to NNM (or WNNM) can be solved iteratively by applying a closed-form proximal operator, called Singular Value Thresholding (SVT) (or Weighted SVT), but they suffer from high computational cost of computing Singular Value Decomposition (SVD) at each iteration. We propose a fast and accurate approximation method for SVT, that we call fast randomized SVT (FRSVT), where we avoid direct computation of SVD. The key idea is to extract an approximate basis for the range of a matrix from its compressed matrix. Given the basis, we compute the partial singular values of the original matrix from a small factored matrix. In addition, by adopting a range propagation technique, our method further speeds up the extraction of approximate basis at each iteration. Our theoretical analysis shows the relationship between the approximation bound of SVD and its effect to NNM via SVT. Along with the analysis, our empirical results quantitatively and qualitatively show that our approximation rarely harms the convergence of the host algorithms. We assess the efficiency and accuracy of our method on various vision problems, e.g., subspace clustering, weather artifact removal, and simultaneous multi-image alignment and rectification.

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