# Bayesian Tensor Regression

arXiv: https://arxiv.org/abs/1509.06490 · HTML: https://mlsanity.com/paper/1509.06490/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Sep 2015.

## Built on by (most cited first, 21 of 21)

- [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022)
- [Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence](https://mlsanity.com/paper/2104.12031/) (2021)
- [Low Rank Optimization for Efficient Deep Learning: Making A Balance between Compact Architecture and Fast Training](https://mlsanity.com/paper/2303.13635/) (2023)
- [Scalable Spatiotemporally Varying Coefficient Modelling with Bayesian Kernelized Tensor Regression](https://mlsanity.com/paper/2109.00046/) (2021)
- [Interpretable Bayesian Tensor Network Kernel Machines with Automatic Rank and Feature Selection](https://mlsanity.com/paper/2507.11136/) (2025)
- [Bayesian longitudinal tensor response regression for modeling neuroplasticity](https://mlsanity.com/paper/2309.10065/) (2023)
- [High-Dimensional Uncertainty Quantification via Tensor Regression with Rank Determination and Adaptive Sampling](https://mlsanity.com/paper/2103.17236/) (2021)
- [Targeted Advertising on Social Networks Using Online Variational Tensor Regression](https://mlsanity.com/paper/2208.10627/) (2022)
- [An Adaptive Tensor-Train Decomposition Approach for Efficient Deep Neural Network Compression](https://mlsanity.com/paper/2408.01534/) (2024)
- [Efficient Decision Trees for Tensor Regressions](https://mlsanity.com/paper/2408.01926/) (2024)
- [Efficient Tensor Contraction via Fast Count Sketch](https://mlsanity.com/paper/2106.13062/) (2021)
- [Bayesian predictive modeling of multi-source multi-way data](https://mlsanity.com/paper/2208.03396/) (2022)
- [Bayesian Additive Main Effects and Multiplicative Interaction Models using Tensor Regression for Multi-environmental Trials](https://mlsanity.com/paper/2301.03655/) (2023)
- [Heterogeneous Image-based Classification Using Distributional Data Analysis](https://mlsanity.com/paper/2403.07126/) (2024)
- [Maximum likelihood inference for high-dimensional problems with multiaffine variable relations](https://mlsanity.com/paper/2409.03495/) (2024)
- [BAMITA: Bayesian Multiple Imputation for Tensor Arrays](https://mlsanity.com/paper/2410.23412/) (2024)
- [Generalized Nonnegative Structured Kruskal Tensor Regression](https://mlsanity.com/paper/2509.19900/) (2025)
- [Low-rank MMSE filters, Kronecker-product representation, and regularization: a new perspective](https://mlsanity.com/paper/2512.14932/) (2025)
- [Dual-Channel Tensor Neural Networks: Finite-Sample Theory and Conformal Structure Selection](https://mlsanity.com/paper/2605.19122/) (2026)
- [Bayesian Poisson-Randomized Gamma Tensor Factorization with Application to International Trade Flows](https://mlsanity.com/paper/2606.17267/) (2026)
- [Additive Nonparametric Regression with Spatial and Network Objects](https://mlsanity.com/paper/2609.26867/) (2026)

## Abstract (arXiv)

This article proposes a Bayesian approach to regression with a scalar response against vector and tensor covariates. Tensor covariates are commonly vectorized prior to analysis, failing to exploit the structure of the tensor, and resulting in poor estimation and predictive performance. We develop a novel class of multiway shrinkage priors for the coefficients in tensor regression models. Properties are described, including posterior consistency under mild conditions, and an efficient Markov chain Monte Carlo algorithm is developed for posterior computation. Simulation studies illustrate substantial gains over vectorizing or using existing tensor regression methods in terms of estimation and parameter inference. The approach is further illustrated in a neuroimaging application. Keywords: Dimension reduction; multiway shrinkage prior; magnetic resonance imaging (MRI); parafac decomposition; posterior consistency; tensor regression