Elliptically-Contoured Tensor-variate Distributions with Application to Improved Image Learning

TL;DR

This paper introduces a family of elliptically contoured tensor-variate distributions, extending traditional tensor-normal models, and demonstrates their advantages in robust image classification and analysis of tensor data with heavy tails.

Contribution

It develops a comprehensive framework for EC tensor distributions, including estimation, classification, and regression methods, improving robustness over tensor-normal models.

Findings

EC distributions outperform TVN in heavy-tailed data scenarios.

EC-based classification improves accuracy on image datasets.

Tensor-on-tensor regression with EC errors better captures data variability.

Abstract

Statistical analysis of tensor-valued data has largely used the tensor-variate normal (TVN) distribution that may be inadequate when data comes from distributions with heavier or lighter tails. We study a general family of elliptically contoured (EC) tensor-variate distributions and derive its characterizations, moments, marginal and conditional distributions, and the EC Wishart distribution. We describe procedures for maximum likelihood estimation from data that are (1) uncorrelated draws from an EC distribution, (2) from a scale mixture of the TVN distribution, and (3) from an underlying but unknown EC distribution, where we extend Tyler's robust estimator. A detailed simulation study highlights the benefits of choosing an EC distribution over the TVN for heavier-tailed data. We develop tensor-variate classification rules using discriminant analysis and EC errors and show that they better predict cats and dogs from images in the Animal Faces-HQ dataset than the TVN-based rules. A novel tensor-on-tensor regression and tensor-variate analysis of variance (TANOVA) framework under EC errors is also demonstrated to better characterize gender, age and ethnic origin than the usual TVN-based TANOVA in the celebrated Labeled Faces of the Wild dataset.