🤖 AI Summary
This paper addresses graph-structured matrix completion, aiming to overcome three key limitations of conventional graph Laplacian regularization: (i) modeling only local similarity, (ii) sensitivity to spurious edges, and (iii) lack of statistical and computational complexity guarantees. To this end, we propose GSGD—a nonconvex optimization algorithm based on preconditioned projected gradient descent—incorporating higher-order spectral graph propagation to capture long-range dependencies among variables. We establish, for the first time under a nonconvex setting, theoretical guarantees of linear convergence rate and near-optimal sample complexity, while significantly enhancing robustness to graph noise. Experiments on both synthetic and real-world datasets demonstrate superior recovery accuracy and scalability, surpassing the performance ceiling of classical graph-regularized paradigms.
📝 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.