The Algebraic Combinatorial Approach for Low-Rank Matrix Completion

TL;DR

This paper introduces an algebraic combinatorial framework for low-rank matrix completion, leveraging algebraic geometry and matroid theory to analyze entry recoverability and develop probabilistic algorithms.

Contribution

It presents a new algebraic combinatorial theory for matrix completion, including algorithms for entry completion and error estimation, with insights into phase transitions.

Findings

Algorithms decide if an entry can be completed with probability one

Methods to complete entries from few others are described

Analysis of sampling assumptions and phase transitions in matrix completion

Abstract

We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the treatment of single entries in a closed theoretical and practical framework. More specifically, apart from introducing an algebraic combinatorial theory of low-rank matrix completion, we present probability-one algorithms to decide whether a particular entry of the matrix can be completed. We also describe methods to complete that entry from a few others, and to estimate the error which is incurred by any method completing that entry. Furthermore, we show how known results on matrix completion and their sampling assumptions can be related to our new perspective and interpreted in terms of a completability phase transition.