Matrix Completion for the Independence Model

TL;DR

This paper studies the conditions under which partial matrices can be completed to rank-one matrices within the simplex, providing algebraic criteria and optimization methods relevant for testing independence in statistical data.

Contribution

It introduces algebraic equations and inequalities characterizing feasible completions, advancing the understanding of matrix completion in the context of the independence model.

Findings

Characterization of completion conditions via equations and inequalities

Description of the set of valid matrix completions

Optimization techniques for completing matrices in the independence model

Abstract

We investigate the problem of completing partial matrices to rank-one matrices in the standard simplex. The motivation for studying this problem comes from statistics: A lack of eligible completion can provide a falsification test for partial observations to come from the independence model. For each pattern of specified entries, we give equations and inequalities which are satisfied if and only if an eligible completion exists. We also describe the set of valid completions, and we optimize over this set.