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.
Highlights
Many modern statistical learning problems concern estimating a matrix-valued parameter
We develop a MATLAB wrapper function svt for the singular value thresholding (SVT) computation
For a non-sparse, structured matrix, we can use the same function handle for singular value thresholding
Summary
Many modern statistical learning problems concern estimating a matrix-valued parameter. Another example is regression with multiple responses (Yuan, Ekici, Lu, and Monteiro 2007; Zhang, Zhou, Zhou, and Sun 2017), which involves a matrix of regression coefficients instead of a regression coefficient vector In these matrix estimation problems, the nuclear norm regularization is often employed to achieve a low rank solution and shrinkage simultaneously. The current implementation of svds is efficient only for sparse matrix input, while the matrix estimation algorithm involves singular value thresholding of dense but structured matrices.
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