Statistics Papers

Document Type

Journal Article

Date of this Version

2016

Publication Source

Electronic Journal of Statistics

Volume

10

Issue

1

Start Page

1493

Last Page

1525

DOI

10.1214/16-EJS1147

Abstract

Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly when the underlying sampling scheme is non-uniform. In this paper we propose and analyze a max-norm constrained empirical risk minimization method for noisy matrix completion under a general sampling model. The optimal rate of convergence is established under the Frobenius norm loss in the context of approximately low-rank matrix reconstruction. It is shown that the max-norm constrained method is minimax rate-optimal and yields a unified and robust approximate recovery guarantee, with respect to the sampling distributions. The computational effectiveness of this method is also discussed, based on first-order algorithms for solving convex optimizations involving max-norm regularization.

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.ejs/1464710239#info

Keywords

Compressed sensing, low-rank matrix, matrix completion, max-norm constrained minimization, minimax optimality, nonuniform sampling, sparsity

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Date Posted: 27 November 2017

This document has been peer reviewed.