Communications in Mathematical Sciences

Volume 15 (2017)

Number 3

Transformed Schatten-1 iterative thresholding algorithms for low rank matrix completion

Pages: 839 – 862

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n3.a12

Authors

Shuai Zhang (Department of Mathematics, University of California at Irvine)

Penghang Yin (Department of Mathematics, University of California at Irvine)

Jack Xin (Department of Mathematics, University of California at Irvine)

Abstract

We study a non-convex low-rank promoting penalty function, the transformed Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty, as a matrix quasi-norm defined on its singular values, interpolates the rank and the nuclear norm through a nonnegative parameter $a \in (0, + \infty)$. We consider the unconstrained TS1 regularized low-rank matrix recovery problem and develop a fixed point representation for its global minimizer. The TS1 thresholding functions are in closed analytical form for all parameter values. The TS1 threshold values differ in subcritical (supercritical) parameter regime where the TS1 threshold functions are continuous (discontinuous). We propose TS1 iterative thresholding algorithms and compare them with some state-of-the-art algorithms on matrix completion test problems. For problems with known rank, a fully adaptive TS1 iterative thresholding algorithm consistently performs the best under different conditions, where ground truth matrices are generated by multivariate Gaussian, $(0,1)$ uniform and Chi-square distributions. For problems with unknown rank, TS1 algorithms with an additional rank estimation procedure approach the level of IRucL-$q$ which is an iterative reweighted algorithm, non-convex in nature and best in performance.

Keywords

transformed Schatten-1 penalty, fixed point representation, closed form thresholding function, iterative thresholding algorithms, matrix completion

2010 Mathematics Subject Classification

90C26, 90C46

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Published 24 February 2017