Hadamard-Hitchcock decompositions: identifiability and computation

TL;DR

This paper introduces Hadamard-Hitchcock decompositions for multidimensional arrays, proving their generic identifiability and providing an algorithm for their computation, with applications to probabilistic graphical models like restricted Boltzmann machines.

Contribution

It establishes the generic identifiability of Hadamard-Hitchcock decompositions and proposes a flexible algorithm for their computation.

Findings

Proved generic identifiability using reshaped Kruskal criterion.

Developed a flexible algorithm leveraging existing tensor decomposition methods.

Numerical experiments demonstrate computational efficiency and accuracy.

Abstract

A Hadamard-Hitchcock decomposition of a multidimensional array is a decomposition that expresses the latter as a Hadamard product of several tensor rank decompositions. Such decompositions can encode probability distributions that arise from statistical graphical models associated to complete bipartite graphs with one layer of observed random variables and one layer of hidden ones, usually called restricted Boltzmann machines. We establish generic identifiability of Hadamard-Hitchcock decompositions by exploiting the reshaped Kruskal criterion for tensor rank decompositions. A flexible algorithm leveraging existing decomposition algorithms for tensor rank decomposition is introduced for computing a Hadamard-Hitchcock decomposition. Numerical experiments illustrate its computational performance and numerical accuracy.