A Combinatorial Algebraic Approach for the Identifiability of Low-Rank Matrix Completion

TL;DR

This paper introduces a novel algebraic and combinatorial framework to determine when low-rank matrices can be uniquely reconstructed from partial entries, providing new theoretical insights and practical algorithms that outperform existing methods.

Contribution

It establishes the first necessary and sufficient combinatorial conditions for low-rank matrix identifiability, linking algebraic geometry, combinatorics, and graph theory.

Findings

Derived tight combinatorial conditions for matrix identifiability

Developed algorithms that outperform state-of-the-art methods

Validated conditions and algorithms on practical matrix sizes

Abstract

In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of arbitrary rank to be identifiable from a set of matrix entries, yielding theoretical constraints and new algorithms for the problem of matrix completion. We conclude by algorithmically evaluating the tightness of the given conditions and algorithms for practically relevant matrix sizes, showing that the algebraic-combinatoric approach can lead to improvements over state-of-the-art matrix completion methods.