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

TL;DR

The work studies low-rank matrix completion and graph realization problems in the finite field ${ m Z}_2$, linking diagonal perturbations of matrices to topological realizability on surfaces. It develops an algebraic framework around the diagonal-perturbation operator $R(M)$, establishes algorithmic procedures with $O(n^4)$ and $O(n^3)$ complexities, and derives rank lower bounds for matrices with combinatorial relations, including higher-dimensional generalizations ${[m]\choose l}$. A key contribution is the $R(M)$-based criterion for weak realizability of hieroglyphs (and hence graphs) on surfaces via the overlap matrix $M(H)$, bridging matrix completion with modulo-2 embeddings. Additional results include lower bounds for ranks of structured relation matrices and a canonical classification of symmetric bilinear forms over ${ m Z}_2$, collectively providing a unified algebraic toolkit for graph realizability and related topological problems.

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).

Figures (6)

• Figure 1: Disk with ribbons corresponding to the hieroglyph $aabbcc$ (left) and $aabcbc$ (right)
• Figure 2: Realization of nonplanar graphs
• Figure 3: Resolving intersection by adding a handle
• Figure 4: A 'non-general position even drawing' of $K_5$ in the plane. The drawings (i. e., the images of) every two non-adjacent edges intersect at an even number of points.
• Figure 5: A transverse intersection and a non-transverse intersection

Theorems & Definitions (26)

• Proposition 1.1: Bi20
• Proposition 1.2
• proof
• Theorem 1.3: Ko21
• proof : Sketch of a proof (see the details in §\ref{['s:lowr']})
• Theorem 1.4: Ko21, proved in §\ref{['s:lowr']}
• Lemma 2.3
• Lemma 3.2: subadditivity of rank
• proof
• proof : Proof of Theorem \ref{['main']}.a