This work studies the Generalized Singular Value Thresholding (GSVT) operator ${ \text{ Prox}}_{ g}^{ { \sigma}}(\cdot)$, \begin{ equation*} { \text{ Prox}}_{ g}^{ { \sigma}}(B)=\arg\min\limits_{ X}\sum_{ i=1}^{ m}g(\sigma_{ i}(X)) + \frac{ 1}{ 2}||X-B||_{ F}^{ 2}, \end{ equation*} 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 $\text{ Prox}_g(\cdot)$) on the singular values since $\text{ Prox}_g(\cdot)$ is monotone when $g$ is lower bounded. If the nonconvex $g$ satisfies some conditions (many popular nonconvex surrogate functions, e.g., $\ell_p$-norm, $0<p<1$, of $\ell_0$-norm are special cases), a general solver to find $\text{ Prox}_g(b)$ is proposed for any $b\geq0$. 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.
Categories: cs.CV cs.LG cs.NA math.NA