Uniqueness of Low Rank Matrix Completion and Schur Complement

TL;DR

This paper investigates the conditions for unique low rank matrix completion using Schur complement techniques, focusing on matrices with a staircase observation pattern and linking uniqueness to submatrix ranks.

Contribution

It establishes necessary and sufficient conditions for global uniqueness in low rank matrix completion with staircase-structured data, utilizing Schur complement methods.

Findings

Uniqueness depends on the rank of corner submatrices.

Complete characterization of when the completion is globally unique.

Excludes local uniqueness, focusing on global solutions.

Abstract

In this paper we study the low rank matrix completion problem using tools from Schur complement. We give a sufficient and necessary condition such that the completed matrix is globally unique with given data. We assume the observed entries of the matrix follow a special "staircase" structure. Under this assumption, the matrix completion problem is either globally unique or has infinitely many solutions (thus excluding local uniqueness). In fact, the uniqueness of the matrix completion problem totally depends on the rank of the submatrices at the corners of the "staircase". The proof of the theorems make extensive use of the Schur complement.