Overview of Constrained PARAFAC Models

TL;DR

This paper provides a comprehensive overview of constrained PARAFAC tensor models, introducing new models, formalism, and analyzing their properties and relationships with other tensor decompositions.

Contribution

It introduces new constrained tensor models, formalizes tensor operations with Kronecker products, and establishes links and uniqueness properties among models.

Findings

Introduction of nested Tucker and block PARALIND/CONFAC models

Formalization of tensor operations using Kronecker products

New uniqueness conditions for PARATUCK models

Abstract

In this paper, we present an overview of constrained PARAFAC models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition, or alternatively, the pattern of interactions between different modes of the tensor which are captured by the equivalent core tensor. Some tensor prerequisites with a particular emphasis on mode combination using Kronecker products of canonical vectors that makes easier matricization operations, are first introduced. This Kronecker product based approach is also formulated in terms of the index notation, which provides an original and concise formalism for both matricizing tensors and writing tensor models. Then, after a brief reminder of PARAFAC and Tucker models, two families of constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models, are described in a unified framework, for $N^{th}$ order tensors. New tensor models, called nested Tucker models and block PARALIND/CONFAC models, are also introduced. A link between PARATUCK models and constrained PARAFAC models is then established. Finally, new uniqueness properties of PARATUCK models are deduced from sufficient conditions for essential uniqueness of their associated constrained PARAFAC models.