# Tensor Regression Meets Gaussian Processes

arXiv: https://arxiv.org/abs/1710.11345 · HTML: https://mlsanity.com/paper/1710.11345/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2017.

## Built on by (most cited first, 12 of 12)

- [Local approximate Gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks](https://mlsanity.com/paper/2105.04554/) (2021)
- [Learning Multi-Task Gaussian Process Over Heterogeneous Input Domains](https://mlsanity.com/paper/2202.12636/) (2022)
- [Recent Advances in Data-Driven Wireless Communication Using Gaussian Processes: A Comprehensive Survey](https://mlsanity.com/paper/2103.10134/) (2021)
- [Scalable Multi-Task Gaussian Processes with Neural Embedding of Coregionalization](https://mlsanity.com/paper/2109.09261/) (2021)
- [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)
- [High-Dimensional Uncertainty Quantification via Tensor Regression with Rank Determination and Adaptive Sampling](https://mlsanity.com/paper/2103.17236/) (2021)
- [Lower and Upper Bounds on the VC-Dimension of Tensor Network Models](https://mlsanity.com/paper/2106.11827/) (2021)
- [Targeted Advertising on Social Networks Using Online Variational Tensor Regression](https://mlsanity.com/paper/2208.10627/) (2022)
- [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)
- [Combinatorial Bandit Bayesian Optimization for Tensor Outputs](https://mlsanity.com/paper/2602.00640/) (2026)

## Abstract (arXiv)

Low-rank tensor regression, a new model class that learns high-order correlation from data, has recently received considerable attention. At the same time, Gaussian processes (GP) are well-studied machine learning models for structure learning. In this paper, we demonstrate interesting connections between the two, especially for multi-way data analysis. We show that low-rank tensor regression is essentially learning a multi-linear kernel in Gaussian processes, and the low-rank assumption translates to the constrained Bayesian inference problem. We prove the oracle inequality and derive the average case learning curve for the equivalent GP model. Our finding implies that low-rank tensor regression, though empirically successful, is highly dependent on the eigenvalues of covariance functions as well as variable correlations.