Generalized Singular Value Thresholding

12/06/2014

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by   Canyi Lu, et al.

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This work studies the Generalized Singular Value Thresholding (GSVT) operator _g^σ(·), _g^σ()=_∑_i=1^mg(σ_i()) + 1/2||-||_F^2, associated with a nonconvex function g defined on the singular values of . We prove that GSVT can be obtained by performing the proximal operator of g (denoted as _g(·)) on the singular values since _g(·) is monotone when g is lower bounded. If the nonconvex g satisfies some conditions (many popular nonconvex surrogate functions, e.g., ℓ_p-norm, 0<p<1, of ℓ_0-norm are special cases), a general solver to find _g(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.