Typical and Generic Ranks in Matrix Completion

TL;DR

This paper investigates the geometric properties of low-rank matrix completion, defining and analyzing typical and generic ranks over real and complex fields for various entry patterns.

Contribution

It introduces formal definitions of typical and generic ranks and provides inequalities and exact results for different matrix completion patterns.

Findings

Unique generic rank over complex numbers for fixed patterns.

Multiple typical ranks can occur over real numbers.

Provides bounds and exact values for ranks in various patterns.

Abstract

We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends on the values of the known entries. If the entries of the matrix are complex numbers, then for a fixed pattern of locations of specified and unspecified entries there is a unique completion rank which occurs with positive probability. We call this rank the generic completion rank. Over the real numbers there can be multiple ranks that occur with positive probability; we call them typical completion ranks. We introduce these notions formally, and provide a number of inequalities and exact results on typical and generic ranks for different families of patterns of known and unknown entries.