Symmetric Matrix Completion with ReLU Sampling

TL;DR

This paper investigates symmetric positive semi-definite low-rank matrix completion using ReLU sampling, revealing non-globally benign landscapes but providing conditions and initialization strategies for successful gradient descent convergence.

Contribution

It introduces a theoretical analysis of ReLU sampling in matrix completion, proving local strong convexity and proposing an initialization method that ensures convergence to the global minimum.

Findings

Gradient descent often converges to stationary points that are not globally optimal.

Under mild assumptions, the nonconvex objective is strongly convex near the true solution.

The proposed initialization consistently leads to convergence to the global minimum in experiments.

Abstract

We study the problem of symmetric positive semi-definite low-rank matrix completion (MC) with deterministic entry-dependent sampling. In particular, we consider rectified linear unit (ReLU) sampling, where only positive entries are observed, as well as a generalization to threshold-based sampling. We first empirically demonstrate that the landscape of this MC problem is not globally benign: Gradient descent (GD) with random initialization will generally converge to stationary points that are not globally optimal. Nevertheless, we prove that when the matrix factor with a small rank satisfies mild assumptions, the nonconvex objective function is geodesically strongly convex on the quotient manifold in a neighborhood of a planted low-rank matrix. Moreover, we show that our assumptions are satisfied by a matrix factor with i.i.d. Gaussian entries. Finally, we develop a tailor-designed initialization for GD to solve our studied formulation, which empirically always achieves convergence to the global minima. We also conduct extensive experiments and compare MC methods, investigating convergence and completion performance with respect to initialization, noise level, dimension, and rank.