Column $\ell_{2,0}$-norm regularized factorization model of low-rank matrix recovery and its computation

TL;DR

This paper introduces a column $\,\ell_{2,0}$-norm regularized factorization model for low-rank matrix recovery, proposing novel algorithms with proven convergence that outperform existing methods in accuracy and efficiency.

Contribution

It develops an alternating majorization-minimization method with extrapolation for nonconvex optimization in low-rank matrix recovery, and provides convergence analysis and practical improvements.

Findings

Outperforms nuclear-norm and max-norm models in accuracy.

Achieves lower error and rank in less time.

Effective on synthetic and real data.

Abstract

This paper is concerned with the column $\ell_{2,0}$-regularized factorization model of low-rank matrix recovery problems and its computation. The column $\ell_{2,0}$-norm of factor matrices is introduced to promote column sparsity of factors and low-rank solutions. For this nonconvex discontinuous optimization problem, we develop an alternating majorization-minimization (AMM) method with extrapolation, and a hybrid AMM in which a majorized alternating proximal method is proposed to seek an initial factor pair with less nonzero columns and the AMM with extrapolation is then employed to minimize of a smooth nonconvex loss. We provide the global convergence analysis for the proposed AMM methods and apply them to the matrix completion problem with non-uniform sampling schemes. Numerical experiments are conducted with synthetic and real data examples, and comparison results with the nuclear-norm regularized factorization model and the max-norm regularized convex model show that the column $\ell_{2,0}$-regularized factorization model has an advantage in offering solutions of lower error and rank within less time.