Intelligent Initialization and Adaptive Thresholding for Iterative Matrix Completion; Some Statistical and Algorithmic Theory for Adaptive-Impute

TL;DR

This paper introduces a new matrix completion algorithm that adaptively chooses thresholding parameters, achieving minimax error rates and strong empirical results, while providing theoretical insights into its convergence.

Contribution

It proposes a data-dependent, theoretically-justified thresholding scheme for iterative matrix completion, bridging empirical performance with theoretical convergence analysis.

Findings

Achieves minimax error rate in matrix completion.

Outperforms existing methods empirically.

Provides theoretical analysis of convergence behavior.

Abstract

Over the past decade, various matrix completion algorithms have been developed. Thresholded singular value decomposition (SVD) is a popular technique in implementing many of them. A sizable number of studies have shown its theoretical and empirical excellence, but choosing the right threshold level still remains as a key empirical difficulty. This paper proposes a novel matrix completion algorithm which iterates thresholded SVD with theoretically-justified and data-dependent values of thresholding parameters. The estimate of the proposed algorithm enjoys the minimax error rate and shows outstanding empirical performances. The thresholding scheme that we use can be viewed as a solution to a non-convex optimization problem, understanding of whose theoretical convergence guarantee is known to be limited. We investigate this problem by introducing a simpler algorithm, generalized-\SI, analyzing its convergence behavior, and connecting it to the proposed algorithm.