Fast singular value thresholding without singular value decomposition

Abstract

We are interested in solving the following minimization problem Dτ (Y ) := arg min X∈Rm×n 1 2 ∥Y −X∥F + τ∥X∥∗, where Y ∈ Rm×n is a given matrix, and ∥ ⋅ ∥F is the Frobenius norm and ∥ ⋅ ∥∗ the nuclear norm. This problem serves as a basic subroutine in many popular numerical schemes for nuclear norm minimization problems, which arise from low rank matrix recovery such as matrix completion. As Dτ (Y ) has an explicit expression which shrinks the singular values of Y and keeps the singular vectors, Dτ is referred to singular value thresholding (SVT) operator in the literature. Conventional approaches for Dτ (Y ) first find the singular value decomposition (SVD) of Y and then shrink the singular values. However, such approaches are time consuming under some circumstances, especially when the rank of Dτ (Y ) is not low compared to the matrix dimension or is completely unpredictable. In this paper, we propose a fast algorithm for directly computing Dτ (Y ) without using SVDs. Numerical experiments show that the proposed algorithm is much more efficient than the approach using the full SVD.