Faster Kernel Interpolation for Gaussian Processes

Abstract

A key challenge in scaling Gaussian Process (GP) regression to massive datasets is that exact inference requires computation with a dense n × n kernel matrix, where n is the number of data points. Significant work focuses on approximating the kernel matrix via interpolation using a smaller set of m “inducing points”. Structured kernel interpolation (SKI) is among the most scalable methods: by placing inducing points on a dense grid and using structured matrix algebra, SKI achieves per-iteration time of O(n + m log m) for approximate inference. This linear scaling in n enables approximate inference for very large data sets; however, the cost is per-iteration, which remains a limitation for extremely large n. We show that the SKI per-iteration time can be reduced to O(m log m) after a single O(n) time precomputation step by reframing SKI as solving a natural Bayesian linear regression problem with a fixed set of m compact basis functions. For a fixed grid, our new method scales to truly massive data sets: after the initial linear time pass, all subsequent computations are independent of n. We demonstrate speedups in practice for a wide range of m and n and for all the main GP inference tasks. With per-iteration complexity independent of the dataset size n for a fixed grid, our method scales to truly massive data sets. We demonstrate speedups in practice for a wide range of m and n and apply the method to GP inference on a three-dimensional weather radar dataset with over 100 million points.