A Statistically Identifiable Model for Tensor-Valued Gaussian Random Variables

TL;DR

This paper introduces a statistically identifiable tensor-valued Gaussian distribution that models multilinear interactions in multi-dimensional data, providing an analytical maximum likelihood estimator and demonstrating its consistency, with applications in climate modeling.

Contribution

It proposes a novel tensor-valued Gaussian distribution with identifiable parameters and an analytical maximum likelihood estimator, extending to joint distributions of multiple tensor variables.

Findings

Likelihood function maximization yields explicit estimators.

Estimator consistency validated through simulations.

Models applicable to climate data analysis.

Abstract

Real-world signals typically span across multiple dimensions, that is, they naturally reside on multi-way data structures referred to as tensors. In contrast to standard ``flat-view'' multivariate matrix models which are agnostic to data structure and only describe linear pairwise relationships, we introduce the tensor-valued Gaussian distribution which caters for multilinear interactions -- the linear relationship between fibers -- which is reflected by the Kronecker separable structure of the mean and covariance. By virtue of the statistical identifiability of the proposed distribution formulation, whereby different parameter values strictly generate different probability distributions, it is shown that the corresponding likelihood function can be maximised analytically to yield the maximum likelihood estimator. For rigour, the statistical consistency of the estimator is also demonstrated through numerical simulations. The probabilistic framework is then generalised to describe the joint distribution of multiple tensor-valued random variables, whereby the associated mean and covariance exhibit a Khatri-Rao separable structure. The proposed models are shown to serve as a natural basis for gridded atmospheric climate modelling.