Low rank matrix completion and realization of graphs: results and problems

TL;DR

This paper surveys the problem of low rank matrix completion and graph realization, focusing on linear relations among matrix entries and applications to graph embeddings in surfaces.

Contribution

It generalizes matrix completion to linear relations and explores applications to graph embeddings, highlighting open problems and recent results.

Findings

Connections between matrix completion and graph embeddings.

Applications to embeddings with rotation systems and modulo 2.

Identification of open problems in low rank matrix realization.

Abstract

The Netflix problem (from machine learning) asks the following. Given a ratings matrix in which each entry $(i,j)$ represents the rating of movie $j$ by customer $i$, if customer $i$ has watched movie $j$, and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. The remaining entries are predicted so as to minimize the {\it rank} of the completed matrix. In this survey we study a more general problem, in which instead of knowing specific matrix elements, we know linear relations on such elements. We describe applications of these results to embeddings of graphs in surfaces (more precisely, embeddings with rotation systems, and embeddings modulo 2).