Tensor Methods for Nonlinear Matrix Completion

TL;DR

This paper introduces a novel tensor-based approach for nonlinear matrix completion, extending low-rank assumptions to algebraic varieties and demonstrating improved performance over traditional methods.

Contribution

It proposes a tensorization-based algorithm for nonlinear matrix completion that outperforms existing methods and provides theoretical guarantees for its success.

Findings

Outperforms state-of-the-art methods in union of subspaces scenarios.

Provides mathematical bounds and guarantees for tensorized data rank.

Successfully handles cases where traditional LRMC fails.

Abstract

In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a second-order tensorized representation is formed by taking the Kronecker product of each column with itself, and we consider higher order tensorizations as well. This approach will succeed in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We provide a formal mathematical justification for the success of our method. In particular, we give bounds of the rank of these data in the tensorized representation, and we prove sampling requirements to guarantee uniqueness of the solution. We also provide experimental results showing that the new approach outperforms existing state-of-the-art methods for matrix completion under a union of subspaces model.