Matrix Completion with Graph Information: A Provable Nonconvex Optimization Approach

TL;DR

This paper introduces GSGD, a novel nonconvex optimization algorithm for matrix completion that leverages graph information, capturing higher-order correlations, and providing theoretical guarantees for recovery accuracy and efficiency.

Contribution

The paper proposes GSGD, a graph regularized matrix completion method with provable convergence and robustness, addressing limitations of existing graph Laplacian-based approaches.

Findings

GSGD achieves linear convergence to the global optimum.

GSGD demonstrates superior recovery accuracy on synthetic and real data.

Theoretical guarantees are established for sample complexity and recovery quality.

Abstract

We consider the problem of matrix completion with graphs as side information depicting the interrelations between variables. The key challenge lies in leveraging the similarity structure of the graph to enhance matrix recovery. Existing approaches, primarily based on graph Laplacian regularization, suffer from several limitations: (1) they focus only on the similarity between neighboring variables, while overlooking long-range correlations; (2) they are highly sensitive to false edges in the graphs and (3) they lack theoretical guarantees regarding statistical and computational complexities. To address these issues, we propose in this paper a novel graph regularized matrix completion algorithm called GSGD, based on preconditioned projected gradient descent approach. We demonstrate that GSGD effectively captures the higher-order correlation information behind the graphs, and achieves superior robustness and stability against the false edges. Theoretically, we prove that GSGD achieves linear convergence to the global optimum with near-optimal sample complexity, providing the first theoretical guarantees for both recovery accuracy and efficacy in the perspective of nonconvex optimization. Our numerical experiments on both synthetic and real-world data further validate that GSGD achieves superior recovery accuracy and scalability compared with several popular alternatives.