Convergence of the majorized PAM method with subspace correction for low-rank composite factorization model

TL;DR

This paper proves the convergence of a modified PAM method with subspace correction for low-rank matrix factorization, demonstrating its effectiveness over PALM in one-bit matrix completion tasks.

Contribution

It extends the convergence analysis of the PAM method to low-rank composite models with subspace correction, a challenging task not previously addressed.

Findings

Full convergence of the PAM method is established under the KL property.

Numerical results show PAM with subspace correction achieves lower error faster than PALM.

The method is effective for one-bit matrix completion problems.

Abstract

This paper focuses on the convergence certificates of the majorized proximal alternating minimization (PAM) method with subspace correction, proposed in \cite{TaoQianPan22} for the column $\ell_{2,0}$-norm regularized factorization model and now extended to a class of low-rank composite factorization models from matrix completion. The convergence analysis of this PAM method becomes extremely challenging because a subspace correction step is introduced to every proximal subproblem to ensure a closed-form solution. We establish the full convergence of the iterate sequence and column subspace sequences of factor pairs generated by the PAM, under the KL property of the objective function and a condition that holds automatically for the column $\ell_{2,0}$-norm function. Numerical comparison with the popular proximal alternating linearized minimization (PALM) method is conducted on one-bit matrix completion problems, which indicates that the PAM with subspace correction has an advantage in seeking lower relative error within less time.