Asymptotic Convergence Rate of Alternating Minimization for Rank One Matrix Completion

TL;DR

This paper analyzes the asymptotic convergence rate of alternating minimization in rank-one matrix completion, providing bounds based on eigenvalues and graph properties, advancing understanding of algorithm efficiency.

Contribution

It introduces a novel analysis linking convergence rate to eigenvalues and graph structure, offering a polynomial bound for rank-one matrix completion.

Findings

Convergence rate is bounded by eigenvalues of a consensus problem.

Asymptotic rate depends polynomially on number of nodes and maximum degree.

Provides theoretical insights into the efficiency of alternating minimization.

Abstract

We study alternating minimization for matrix completion in the simplest possible setting: completing a rank-one matrix from a revealed subset of the entries. We bound the asymptotic convergence rate by the variational characterization of eigenvalues of a reversible consensus problem. This leads to a polynomial upper bound on the asymptotic rate in terms of number of nodes as well as the largest degree of the graph of revealed entries.