Generalized Singular Value Thresholding

Abstract: 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.

• The paper introduces and analyzes the Generalized Singular Value Thresholding (GSVT) technique, extending standard singular value thresholding.
• It details the mathematical properties of GSVT and its relationship to other regularization methods.
• The study demonstrates the application of GSVT in areas such as sparse recovery or low-rank matrix approximation.

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