# Bayesian Tensor-on-Tensor Regression with Efficient Computation

arXiv: https://arxiv.org/abs/2210.11363 · HTML: https://mlsanity.com/paper/2210.11363/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2022.

## Built on by (most cited first, 3 of 3)

- [Efficient Decision Trees for Tensor Regressions](https://mlsanity.com/paper/2408.01926/) (2024)
- [Gaussian Process Model with Tensorial Inputs and Its Application to the Design of 3D Printed Antennas](https://mlsanity.com/paper/2407.15877/) (2024)
- [BAMITA: Bayesian Multiple Imputation for Tensor Arrays](https://mlsanity.com/paper/2410.23412/) (2024)

## Abstract (arXiv)

We propose a Bayesian tensor-on-tensor regression approach to predict a multidimensional array (tensor) of arbitrary dimensions from another tensor of arbitrary dimensions, building upon the Tucker decomposition of the regression coefficient tensor. Traditional tensor regression methods making use of the Tucker decomposition either assume the dimension of the core tensor to be known or estimate it via cross-validation or some model selection criteria. However, no existing method can simultaneously estimate the model dimension (the dimension of the core tensor) and other model parameters. To fill this gap, we develop an efficient Markov Chain Monte Carlo (MCMC) algorithm to estimate both the model dimension and parameters for posterior inference. Besides the MCMC sampler, we also develop an ultra-fast optimization-based computing algorithm wherein the maximum a posteriori estimators for parameters are computed, and the model dimension is optimized via a simulated annealing algorithm. The proposed Bayesian framework provides a natural way for uncertainty quantification. Through extensive simulation studies, we evaluate the proposed Bayesian tensor-on-tensor regression model and show its superior performance compared to alternative methods. We also demonstrate its practical effectiveness by applying it to two real-world datasets, including facial imaging data and 3D motion data.