Hadamard Extensions and the Identification of Mixtures of Product Distributions

TL;DR

This paper investigates the conditions under which Hadamard extensions of matrices have full column rank, which is crucial for identifying mixtures of product distributions in binary variables, advancing theoretical understanding in this area.

Contribution

It provides new results characterizing when Hadamard extensions possess full column rank, aiding the development of identification algorithms for mixture models.

Findings

Hadamard extensions can have full column rank under specific conditions.

Full column rank is necessary for identifying mixture of product distributions.

The paper offers criteria to determine the rank properties of Hadamard extensions.

Abstract

The Hadamard Extension of a matrix is the matrix consisting of all Hadamard products of subsets of its rows. This construction arises in the context of identifying a mixture of product distributions on binary random variables: full column rank of such extensions is a necessary ingredient of identification algorithms. We provide several results concerning when a Hadamard Extension has full column rank.