Guarantees of Riemannian Optimization for Low Rank Matrix Completion

TL;DR

This paper proves that Riemannian optimization methods can reliably recover low rank matrices from partial entries with high probability, providing improved sampling complexity bounds and demonstrating near-minimal measurement recovery in simulations.

Contribution

It introduces new theoretical guarantees for Riemannian gradient and conjugate gradient descent in low rank matrix completion, with improved sampling complexity bounds and a novel initialization analysis.

Findings

Algorithms converge linearly under certain sampling conditions.

Sampling complexity bounds are improved to near-optimal levels.

Numerical simulations confirm recovery from minimal measurements.

Abstract

We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume $m$ entries of an $n\times n$ rank $r$ matrix are sampled independently and uniformly with replacement. We first prove that with high probability the Riemannian gradient descent and conjugate gradient descent algorithms initialized by one step hard thresholding are guaranteed to converge linearly to the measured matrix provided \begin{align*} m\geq C_\kappa n^{1.5}r\log^{1.5}(n), \end{align*} where $C_\kappa$ is a numerical constant depending on the condition number of the underlying matrix. The sampling complexity has been further improved to \begin{align*} m\geq C_\kappa nr^2\log^{2}(n) \end{align*} via the resampled Riemannian gradient descent initialization. The analysis of the new initialization procedure relies on an asymmetric restricted isometry property of the sampling operator and the curvature of the low rank matrix manifold. Numerical simulation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements.