Generalized Singular Value Thresholding [article]

Canyi Lu, Changbo Zhu, Chunyan Xu, Shuicheng Yan, Zhouchen Lin
2018 arXiv   pre-print
This work studies the Generalized Singular Value Thresholding (GSVT) operator Prox_g^σ(·), Prox_g^σ(B)=_X∑_i=1^mg(σ_i(X)) + 1/2||X-B||_F^2, 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,
more » ... <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.
arXiv:1412.2231v2 fatcat:vce6u4i3cvho7eohu3pwg4m66a