Matrix Completion using Kronecker Product Approximation

TL;DR

This paper proposes a novel matrix completion method based on low Kronecker rank, offering a more flexible and parsimonious alternative to traditional low-rank approaches, with proven consistency and practical aggregation techniques.

Contribution

Introduces a low Kronecker rank matrix completion framework, including configuration identification, consistency proof, and aggregation methods for complex missing patterns.

Findings

Effective configuration identification using MSE and cross-validation

Proven consistency of the method under certain conditions

Demonstrated superior performance through numerical studies

Abstract

A matrix completion problem is to recover the missing entries in a partially observed matrix. Most of the existing matrix completion methods assume a low rank structure of the underlying complete matrix. In this paper, we introduce an alternative and more general form of the underlying complete matrix, which assumes a low Kronecker rank instead of a low regular rank, but includes the latter as a special case. The extra flexibility allows for a much more parsimonious representation of the underlying matrix, but also raises the challenge of determining the proper Kronecker product configuration to be used. We find that the configuration can be identified using the mean squared error criterion as well as a modified cross-validation criterion. We establish the consistency of this procedure under suitable conditions on the signal-to-noise ratio. A aggregation procedure is also proposed to deal with special missing patterns and complex underlying structures. Both numerical and empirical studies are carried out to demonstrate the performance of the new method.