A Note on Alternating Minimization Algorithm for the Matrix Completion Problem

TL;DR

This paper analyzes two variants of the Alternating Minimization algorithm for low-rank matrix completion, proving their effectiveness under specific conditions and demonstrating the superior performance of a message passing-based approach through simulations.

Contribution

The paper provides theoretical guarantees for two Alternating Minimization variants in low-rank matrix completion with rank 1 matrices and bounded entries, and compares their empirical performance.

Findings

Both algorithms successfully reconstruct matrices under specified conditions.

The message passing-based algorithm outperforms the other variant in simulations.

Reconstruction is achieved in polynomial time from arbitrary initialization.

Abstract

We consider the problem of reconstructing a low rank matrix from a subset of its entries and analyze two variants of the so-called Alternating Minimization algorithm, which has been proposed in the past. We establish that when the underlying matrix has rank $r=1$, has positive bounded entries, and the graph $\mathcal{G}$ underlying the revealed entries has bounded degree and diameter which is at most logarithmic in the size of the matrix, both algorithms succeed in reconstructing the matrix approximately in polynomial time starting from an arbitrary initialization. We further provide simulation results which suggest that the second algorithm which is based on the message passing type updates, performs significantly better.