# Tensor-on-tensor regression

arXiv: https://arxiv.org/abs/1701.01037 · HTML: https://mlsanity.com/paper/1701.01037/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jan 2017.

## Built on by (most cited first, 22 of 22)

- [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022)
- [Factor Augmented Tensor-on-Tensor Neural Networks](https://mlsanity.com/paper/2405.19610/) (2024)
- [Low-Rank Robust Subspace Tensor Clustering for Metro Passenger Flow Modeling](https://mlsanity.com/paper/2404.04403/) (2024)
- [Computational and Statistical Guarantees for Tensor-on-Tensor Regression with Tensor Train Decomposition](https://mlsanity.com/paper/2406.06002/) (2024)
- [Tensor-on-tensor Regression Neural Networks for Process Modeling with High-dimensional Data](https://mlsanity.com/paper/2510.05329/) (2025)
- [Near-Universal Multiplicative Updates for Nonnegative Einsum Factorization](https://mlsanity.com/paper/2602.02759/) (2026)
- [Multiway sparse distance weighted discrimination](https://mlsanity.com/paper/2110.05377/) (2021)
- [Efficient Decision Trees for Tensor Regressions](https://mlsanity.com/paper/2408.01926/) (2024)
- [Coupled Support Tensor Machine Classification for Multimodal Neuroimaging Data](https://mlsanity.com/paper/2201.07683/) (2022)
- [Elliptically-Contoured Tensor-variate Distributions with Application to Improved Image Learning](https://mlsanity.com/paper/2211.06940/) (2022)
- [Decomposable Sparse Tensor on Tensor Regression](https://mlsanity.com/paper/2212.05024/) (2022)
- [Few-shot Personalized Saliency Prediction Based on Interpersonal Gaze Patterns](https://mlsanity.com/paper/2307.02799/) (2023)
- [Tensor-based process control and monitoring for semiconductor manufacturing with unstable disturbances](https://mlsanity.com/paper/2401.17573/) (2024)
- [Integrative Variational Autoencoders for Generative Modeling of an Image Outcome with Multiple Input Images](https://mlsanity.com/paper/2402.02734/) (2024)
- [Generalized Nonnegative Structured Kruskal Tensor Regression](https://mlsanity.com/paper/2509.19900/) (2025)
- [Guaranteed Noisy CP Tensor Recovery via Riemannian Optimization on the Segre Manifold](https://mlsanity.com/paper/2510.00569/) (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)
- [Combinatorial Bandit Bayesian Optimization for Tensor Outputs](https://mlsanity.com/paper/2602.00640/) (2026)
- [Cellwise Outliers](https://mlsanity.com/paper/2604.14182/) (2026)
- [Dual-Channel Tensor Neural Networks: Finite-Sample Theory and Conformal Structure Selection](https://mlsanity.com/paper/2605.19122/) (2026)
- [Beyond Additive Decompositions: Interpretability Through Separability](https://mlsanity.com/paper/2605.31200/) (2026)
- [Structured Adaptive Tensor Prediction for Streaming Data](https://mlsanity.com/paper/2606.10085/) (2026)

## Abstract (arXiv)

We propose a framework for the linear prediction of a multi-way array (i.e., a tensor) from another multi-way array of arbitrary dimension, using the contracted tensor product. This framework generalizes several existing approaches, including methods to predict a scalar outcome from a tensor, a matrix from a matrix, or a tensor from a scalar. We describe an approach that exploits the multiway structure of both the predictors and the outcomes by restricting the coefficients to have reduced CP-rank. We propose a general and efficient algorithm for penalized least-squares estimation, which allows for a ridge (L_2) penalty on the coefficients. The objective is shown to give the mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for inference. We illustrate the approach with an application to facial image data. An R package is available at https://github.com/lockEF/MultiwayRegression .