A Max-Norm Constrained Minimization Approach to 1-Bit Matrix Completion

TL;DR

This paper introduces a max-norm constrained maximum likelihood approach for noisy 1-bit matrix completion under non-uniform sampling, establishing optimal convergence rates and discussing algorithms and performance.

Contribution

It proposes a novel max-norm constrained estimation method for 1-bit matrix completion and derives its optimal convergence rate under general sampling conditions.

Findings

Established the rate of convergence for the proposed estimator.

Derived minimax lower bounds matching the upper bounds.

Discussed computational algorithms and numerical results.

Abstract

We consider in this paper the problem of noisy 1-bit matrix completion under a general non-uniform sampling distribution using the max-norm as a convex relaxation for the rank. A max-norm constrained maximum likelihood estimate is introduced and studied. The rate of convergence for the estimate is obtained. Information-theoretical methods are used to establish a minimax lower bound under the general sampling model. The minimax upper and lower bounds together yield the optimal rate of convergence for the Frobenius norm loss. Computational algorithms and numerical performance are also discussed.