# Reduced-Rank Tensor-on-Tensor Regression and Tensor-variate Analysis of Variance

arXiv: https://arxiv.org/abs/2012.10249 · HTML: https://mlsanity.com/paper/2012.10249/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2020.

## Built on by (most cited first, 10 of 10)

- [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022)
- [Computational and Statistical Guarantees for Tensor-on-Tensor Regression with Tensor Train Decomposition](https://mlsanity.com/paper/2406.06002/) (2024)
- [Robust Low-rank Tensor Train Recovery](https://mlsanity.com/paper/2410.15224/) (2024)
- [Near-Universal Multiplicative Updates for Nonnegative Einsum Factorization](https://mlsanity.com/paper/2602.02759/) (2026)
- [Elliptically-Contoured Tensor-variate Distributions with Application to Improved Image Learning](https://mlsanity.com/paper/2211.06940/) (2022)
- [Higher Order Reduced Rank Regression](https://mlsanity.com/paper/2503.06528/) (2025)
- [A Latent-Variable Formulation of the Poisson Canonical Polyadic Tensor Model: Maximum Likelihood Estimation and Fisher Information](https://mlsanity.com/paper/2511.05352/) (2025)
- [Robust Tensor Regression with Nonconvexity: Algorithmic and Statistical Theory](https://mlsanity.com/paper/2605.07448/) (2026)
- [Structured Adaptive Tensor Prediction for Streaming Data](https://mlsanity.com/paper/2606.10085/) (2026)
- [Heteroskedastic Canonical Polyadic Tensor Decomposition](https://mlsanity.com/paper/2610.00498/) (2026)

## Abstract (arXiv)

Fitting regression models with many multivariate responses and covariates can be challenging, but such responses and covariates sometimes have tensor-variate structure. We extend the classical multivariate regression model to exploit such structure in two ways: first, we impose four types of low-rank tensor formats on the regression coefficients. Second, we model the errors using the tensor-variate normal distribution that imposes a Kronecker separable format on the covariance matrix. We obtain maximum likelihood estimators via block-relaxation algorithms and derive their computational complexity and asymptotic distributions. Our regression framework enables us to formulate tensor-variate analysis of variance (TANOVA) methodology. This methodology, when applied in a one-way TANOVA layout, enables us to identify cerebral regions significantly associated with the interaction of suicide attempters or non-attemptor ideators and positive-, negative- or death-connoting words in a functional Magnetic Resonance Imaging study. Another application uses three-way TANOVA on the Labeled Faces in the Wild image dataset to distinguish facial characteristics related to ethnic origin, age group and gender.