Tensor Regression Meets Gaussian Processes
This work provides theoretical insights into a popular empirical method for multi-way data analysis, but is incremental as it links existing models.
The paper connects low-rank tensor regression to Gaussian processes by showing it learns a multi-linear kernel, and analyzes its theoretical properties including oracle inequality and learning curves.
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.