An Alternating Direction Algorithm for Matrix Completion with Nonnegative Factors

TL;DR

This paper presents an alternating direction algorithm for nonnegative matrix completion that effectively combines nonnegativity and low-rank constraints, demonstrating superior performance in image recovery tasks.

Contribution

It introduces a novel algorithm based on the augmented Lagrangian method for nonnegative matrix completion, integrating two related problems for improved results.

Findings

Produces similar quality factorizations with half the data compared to existing methods.

Achieves better image recovery quality than recent algorithms that ignore nonnegativity.

Preliminary convergence properties are established.

Abstract

This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of M are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just using one of the two properties. We propose to solve the non-convex constrained least-squares problem using an algorithm based on the classic alternating direction augmented Lagrangian method. Preliminary convergence properties of the algorithm and numerical simulation results are presented. Compared to a recent algorithm for nonnegative matrix factorization, the proposed algorithm produces factorizations of similar quality using only about half of the matrix entries. On tasks of recovering incomplete grayscale and hyperspectral images, the proposed algorithm yields overall better qualities than those produced by two recent matrix-completion algorithms that do not exploit nonnegativity.