Generalized Singular Value Thresholding

This work studies the Generalized Singular Value Thresholding (GSVT) operator Proxσ g (·), Proxσ g (B) = arg min X m X i=1 g(σi(X)) + 1 2 ||X − B||2 F, associated with a nonconvex function g defined on the singular values of X. We prove that GSVT can be obtained by performing the proximal operator of g (denoted as Proxg(·)) on the singular values since Proxg(·) is monotone when g is lower bounded. If the nonconvex g satisfies some conditions (many popular nonconvex surrogate functions, e. g. , `p-norm, 0 < p < 1, of `0-norm are special cases), a general solver to find Proxg(b) is proposed for any b ≥ 0. GSVT greatly generalizes the known Singular Value Thresholding (SVT) which is a basic subroutine in many convex low rank minimization methods. We are able to solve the nonconvex low rank minimization problem by using GSVT in place of SVT.

Authors

• Canyi Lu
• Changbo Zhu
• Chunyan Xu Nanjing University of Science and Technology
• Shuicheng Yan Skywork AI, Singapore Nanyang Technological University, Singapore
• Zhouchen Lin National Key Lab of General AI, School of Intelligence Science and Technology, Peking University Peng Cheng Laboratory

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