Low-Rank Matrix Completion using Nuclear Norm with Facial Reduction

TL;DR

This paper introduces a facial reduction approach to improve nuclear norm minimization for low-rank matrix completion, enabling more efficient solutions especially for large-scale problems with known target rank.

Contribution

It applies facial reduction to nuclear norm minimization, reducing problem size and computational complexity while ensuring low-rank solutions, even in noisy scenarios.

Findings

Facial reduction significantly reduces problem size.

Method effectively handles noisy and exact data.

Achieves low-rank solutions with guaranteed optimality.

Abstract

Minimization of the nuclear norm is often used as a surrogate, convex relaxation, for finding the minimum rank completion (recovery) of a partial matrix. The minimum nuclear norm problem can be solved as a trace minimization semidefinite programming problem, (SDP). The SDP and its dual are regular in the sense that they both satisfy strict feasibility. Interior point algorithms are the current methods of choice for these problems. This means that it is difficult to solve large scale problems and difficult to get high accuracy solutions. In this paper we take advantage of the structure at optimality for the minimum nuclear norm problem. We show that even though strict feasibility holds, the facial reduction framework can be successfully applied to obtain a proper face that contains the optimal set, and thus can dramatically reduce the size of the final nuclear norm problem while guaranteeing a low-rank solution. We include numerical tests for both exact and noisy cases. In all cases we assume that knowledge of a target rank is available.