Matrix completion with deterministic pattern - a geometric perspective

TL;DR

This paper investigates the matrix completion problem with deterministic observed patterns, providing a geometric perspective and a verifiable condition for local uniqueness and stability of solutions, along with methods for rank determination.

Contribution

It introduces a well-posedness condition for local uniqueness in matrix completion with deterministic patterns and analyzes its necessity, stability, and genericity from a geometric viewpoint.

Findings

The well-posedness condition ensures local uniqueness of solutions.

Low-rank approximation methods are more stable than matrix completion under certain conditions.

A sequential statistical testing procedure effectively determines the true rank from observed entries.

Abstract

We consider the matrix completion problem with a deterministic pattern of observed entries. In this setting, we aim to answer the question: under what condition there will be (at least locally) unique solution to the matrix completion problem, i.e., the underlying true matrix is identifiable. We answer the question from a certain point of view and outline a geometric perspective. We give an algebraically verifiable sufficient condition, which we call the well-posedness condition, for the local uniqueness of MRMC solutions. We argue that this condition is necessary for local stability of MRMC solutions, and we show that the condition is generic using the characteristic rank. We also argue that the low-rank approximation approaches are more stable than MRMC and further propose a sequential statistical testing procedure to determine the "true" rank from observed entries. Finally, we provide numerical examples aimed at verifying validity of the presented theory.