Don't Get Your Kroneckers in a Twist: Gaussian Processes on High-Dimensional Incomplete Grids

TL;DR

The paper presents CUTS-GPR, a scalable Gaussian process regression method for high-dimensional incomplete grids, enabling efficient Bayesian modeling of complex energy surfaces with billions of data points.

Contribution

Introducing CUTS-GPR, a novel kernel matrix-vector product technique that achieves near-linear scaling, facilitating exact GPR in high-dimensional, large-scale datasets.

Findings

Scalable kernel matrix-vector product demonstrated with billions of data points.

Full GPR computations completed in hours for 447,265 points in 24 dimensions.

Enables Bayesian modeling of high-dimensional potential energy surfaces.

Abstract

We introduce CUTS-GPR, a new method for performing numerically exact Gaussian process regression (GPR) in high-dimensional settings. The key component of CUTS-GPR is an extremely fast kernel matrix-vector product, which exhibits near-linear or even linear scaling with the amount of training data, $N$, and low-order polynomial scaling with dimensionality, $D$. This is obtained by combining an additive kernel with an incomplete grid and exploiting the resulting structure of the kernel matrix. We demonstrate the scalability of the matrix-vector product by running benchmarks with billions of data points and thousands of dimensions. Full GPR calculations, including hyperparameter optimization, are completed in a matter of hours for $N = 447 265$ and $D = 24$. We demonstrate that our CUTS-GPR enables Bayesian modeling of high-dimensional potential energy surfaces - a longstanding challenge in computational chemistry.