svt: Singular value thresholding in MATLAB

Abstract

Many statistical learning methods such as matrix completion, matrix regression, and multiple response regression estimate a matrix of parameters. The nuclear norm regularization is frequently employed to achieve shrinkage and low rank solutions. To minimize a nuclear norm regularized loss function, a vital and most time-consuming step is singular value thresholding, which seeks the singular values of a large matrix exceeding a threshold and their associated singular vectors. Currently MATLAB lacks a function for singular value thresholding. Its built-in svds function computes the top r singular values/vectors by Lanczos iterative method but is only efficient for sparse matrix input, while aforementioned statistical learning algorithms perform singular value thresholding on dense but structured matrices. To address this issue, we provide a MATLAB wrapper function svt that implements singular value thresholding. It encompasses both top singular value decomposition and thresholding, handles both large sparse matrices and structured matrices, and reduces the computation cost in matrix learning algorithms.