Splitting numerical integration for matrix completion

TL;DR

This paper introduces a novel splitting numerical integration method for low-rank matrix approximation and completion, leveraging Riemannian manifold optimization and demonstrating scalability and accuracy in large-scale problems.

Contribution

It develops a splitting numerical integration algorithm for matrix completion based on Riemannian manifold optimization, with proven convergence and applicability to large-scale problems.

Findings

Monotonically decreasing Frobenius norm error

Good scalability for large-scale matrix completion

Satisfactory accuracy in experiments

Abstract

Low rank matrix approximation is a popular topic in machine learning. In this paper, we propose a new algorithm for this topic by minimizing the least-squares estimation over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of classical gradient descent within the framework of optimization on manifolds. In particular, we reformulate an unconstrained optimization problem on a low-rank manifold into a differential dynamic system. We develop a splitting numerical integration method by applying a splitting integration scheme to the dynamic system. We conduct the convergence analysis of our splitting numerical integration algorithm. It can be guaranteed that the error between the recovered matrix and true result is monotonically decreasing in the Frobenius norm. Moreover, our splitting numerical integration can be adapted into matrix completion scenarios. Experimental results show that our approach has good scalability for large-scale problems with satisfactory accuracy