Tensor Regression Meets Gaussian Processes

TL;DR

This paper explores the connection between low-rank tensor regression and Gaussian processes, revealing that tensor regression can be viewed as learning a multi-linear kernel within a GP framework, with implications for understanding its success and limitations.

Contribution

It establishes a theoretical link between low-rank tensor regression and Gaussian processes, including proofs of oracle inequality and learning curves for the combined model.

Findings

Low-rank tensor regression corresponds to learning a multi-linear kernel in GPs.

The success of tensor regression depends on eigenvalues of covariance functions.

The paper derives the average case learning curve for the equivalent GP model.

Abstract

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.