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Fast randomized tensor singular value thresholding for low‐rank tensor optimization

Low‐rank tensor optimization can be converted to a convex optimization problem, which minimizes a convex surrogate to the tensor tubal rank. This problem can be solved iteratively by applying a closed‐form proximal operator, called tensor singular value thresholding (t‐SVT). But they suffer from high computational cost of tensor singular value decomposition (t‐SVD) at each iteration. In this article, we propose a fast randomized algorithm for t‐SVT, which is called as FR t‐SVT. The key idea is to extract an approximate basis for the range of each frontal slice of a third‐order tensor in the Fourier domain from its compressed part. Our theoretical analysis shows the relationship between the approximation bounds of t‐SVD and its effect to the problem via t‐SVT. Numerical examples illustrate the effectiveness of our model.