[edit]

# Scalable High-Order Gaussian Process Regression

*Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics*, PMLR 89:2611-2620, 2019.

#### Abstract

While most Gaussian processes (GP) work focus on learning single-output functions, many applications, such as physical simulations and gene expressions prediction, require estimations of functions with many outputs. The number of outputs can be much larger than or comparable to the size of training samples. Existing multi-output GP models either are limited to low-dimensional outputs and restricted kernel choices, or assume oversimplified low-rank structures within the outputs. To address these issues, we propose HOGPR, a High-Order Gaussian Process Regression model, which can flexibly capture complex correlations among the outputs and scale up to a large number of outputs. Specifically, we tensorize the high-dimensional outputs, introducing latent coordinate features to index each tensor element (i.e., output) and to capture their correlations. We then generalize a multilinear model to a hybrid of a GP and latent GP model. The model is endowed with a Kronecker product structure over the inputs and the latent features. Using the Kronecker product properties and tensor algebra, we are able to perform exact inference over millions of outputs. We show the advantage of the proposed model on several real-world applications.