Nonconvex Matrix Completion with Linearly Parameterized Factors

TL;DR

This paper introduces a unified nonconvex optimization framework for matrix completion with linearly parameterized factors, providing geometric analysis and empirical validation for various structured matrix completion problems.

Contribution

It proposes a novel nonconvex optimization approach with a new condition called Correlated Parametric Factorization, enabling unified analysis across different matrix completion scenarios.

Findings

The framework applies to subspace-constrained and skew-symmetric matrix completion.

Uniform upper bounds for low-rank estimation are established.

Empirical results demonstrate the effectiveness of the proposed method.

Abstract

Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones. In practice including collaborative filtering, prior information and special structures are usually employed in order to improve the accuracy of matrix completion. In this paper, we propose a unified nonconvex optimization framework for matrix completion with linearly parameterized factors. In particular, by introducing a condition referred to as Correlated Parametric Factorization, we can conduct a unified geometric analysis for the nonconvex objective by establishing uniform upper bounds for low-rank estimation resulting from any local minimum. Perhaps surprisingly, the condition of Correlated Parametric Factorization holds for important examples including subspace-constrained matrix completion and skew-symmetric matrix completion. The effectiveness of our unified nonconvex optimization method is also empirically illustrated by extensive numerical simulations.