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

arXiv: https://arxiv.org/abs/2211.06940 · HTML: https://mlsanity.com/paper/2211.06940/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 80 references that are in the map.
- First posted on arXiv 13 Nov 2022.

## Builds on (most cited first, 10 of 10)

- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Multilinear tensor regression for longitudinal relational data](https://mlsanity.com/paper/1412.0048/) (2014)
- [Tensor-on-tensor regression](https://mlsanity.com/paper/1701.01037/) (2017)
- [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022)
- [A novel approach for fusion of heterogeneous sources of data](https://mlsanity.com/paper/1803.00138/) (2018)
- [AFIF4: Deep Gender Classification based on AdaBoost-based Fusion of Isolated Facial Features and Foggy Faces](https://mlsanity.com/paper/1706.04277/) (2017)
- [Covariate-Adjusted Tensor Classification in High-Dimensions](https://mlsanity.com/paper/1805.04421/) (2018)
- [Reduced-Rank Tensor-on-Tensor Regression and Tensor-variate Analysis of Variance](https://mlsanity.com/paper/2012.10249/) (2020)
- [Classification with the matrix-variate-t distribution](https://mlsanity.com/paper/1907.09565/) (2019)

## Abstract (arXiv)

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.