Generic optimization methods such as accelerated proximal gradient algorithm, majorization-minorization (MM) algorithm, and alternating direction method of multipliers (ADMM) have been invoked to solve optimization problem (1). The nuclear norm plays the same role in low-rank matrix approximation that the ℓ 1 norm plays in sparse regression. Where ℓ is a relevant loss function, B ∈ ℝ m × n is a matrix parameter, ‖ B ‖ * = ∑ i σ i ( B ) = ‖ σ ( B ) ‖ 1 (sum of singular values of B) is the nuclear norm of B, and λ is a positive tuning parameter that balances the trade-off between model fit and model parsimony. This leads to a general optimization problem In these matrix estimation problems, the nuclear norm regularization is often employed to achieve a low rank solution and shrinkage simultaneously. 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. Thus it requires a regression coefficient array of same size to completely capture the effects of matrix predictors. In matrix regression ( Zhou and Li 2014), the predictors are two dimensional arrays such as images or measurements on a regular grid. The problem has sparked intensive research in recent years and is enjoying a broad range of applications such as personalized recommendation system ( ACM SIGKDD and Netflix 2007) and imputation of massive genomics data ( Chi, Zhou, Chen, Del Vecchyo, and Lange 2013). Matrix completion ( Candès and Recht 2009 Mazumder, Hastie, and Tibshirani 2010) aims to recover a large matrix of which only a small fraction of entries are observed. Examples include matrix completion, regression with matrix covariates, and multivariate response regression. Many modern statistical learning problems concern estimating a matrix-valued parameter. 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.

To address this issue, we provide a MATLAB wrapper function svt that implements 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. Currently MATLAB lacks a function for singular value thresholding. 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. The nuclear norm regularization is frequently employed to achieve shrinkage and low rank solutions. Many statistical learning methods such as matrix completion, matrix regression, and multiple response regression estimate a matrix of parameters.

Matlab vector code for threshold