Generalized Singular Value Thresholding

This work studies the Generalized Singular Value Thresholding (GSVT) operator Prox(g)(sigma) (.),Prox(g)(sigma) (B) = arg minX Sigma(m)(i =1) g(sigma(i) (X)) vertical bar 1/2 vertical bar vertical bar X -B vertical bar vertical bar(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 Prox(g) (.)) on the singular values since Prox(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 Prox(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.