Don't Get Your Kroneckers in a Twist: Gaussian Processes on High-Dimensional Incomplete Grids
TL;DR
CUTS-GPR enables Bayesian modelling of high-dimensional PESs - a longstanding challenge in computational chemistry by running full GPR calculations, including hyperparameter optimization, on synthetic datasets with up to 447 data points and 500 dimensions.
Abstract
We introduce CUTS-GPR, a new method for performing numerically exact 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. The scalability of the matrix-vector product is verified by benchmarks with billions of data points and thousands of dimensions. We demonstrate the end-to-end scalability of CUTS-GPR by running full GPR calculations, including hyperparameter optimization, on synthetic datasets with up to $N = 4\,494\,001$ and $D = 500$. As a realistic and challenging test, we finally apply CUTS-GPR to a set of ten potential energy surfaces (PESs) with $N = 447\,265$ and $D = 24$. The calculations are completed in a matter of hours, showing that CUTS-GPR enables Bayesian modelling of high-dimensional PESs - a longstanding challenge in computational chemistry.
