On the equivalence between low rank matrix completion and tensor rank

TL;DR

This paper demonstrates the fundamental equivalence between low rank matrix completion, rank minimization, and tensor rank determination, unifying these problems in linear algebra and tensor analysis.

Contribution

It establishes that low rank matrix completion, rank minimization, and tensor rank problems are mutually reducible, revealing their underlying theoretical connection.

Findings

The three problems are computationally equivalent.

Reductions between matrix and tensor rank problems are possible.

Unified framework for low rank problems in matrices and tensors.

Abstract

The Rank Minimization Problem asks to find a matrix of lowest rank inside a linear variety of the space of n x n matrices. The Low Rank Matrix Completion problem asks to complete a partially filled matrix such that the resulting matrix has smallest possible rank. The Tensor Rank Problem asks to determine the rank of a tensor. We show that these three problems are equivalent: each one of the problems can be reduced to the other two.