Scalable Log Determinants for Gaussian Process Kernel Learning

TL;DR

This paper introduces fast, scalable methods for computing log determinants and their derivatives for large positive definite matrices, enabling efficient Gaussian process kernel learning.

Contribution

It proposes novel $ ext{O}(n)$ stochastic approximation techniques using Chebyshev, Lanczos, and surrogate models for log determinants from matrix-vector multiplications.

Findings

Lanczos outperforms Chebyshev in kernel learning tasks.

Surrogate models achieve high efficiency and accuracy with common kernels.

Methods enable scalable Gaussian process kernel learning for large datasets.

Abstract

For applications as varied as Bayesian neural networks, determinantal point processes, elliptical graphical models, and kernel learning for Gaussian processes (GPs), one must compute a log determinant of an $n \times n$ positive definite matrix, and its derivatives - leading to prohibitive $\mathcal{O}(n^3)$ computations. We propose novel $\mathcal{O}(n)$ approaches to estimating these quantities from only fast matrix vector multiplications (MVMs). These stochastic approximations are based on Chebyshev, Lanczos, and surrogate models, and converge quickly even for kernel matrices that have challenging spectra. We leverage these approximations to develop a scalable Gaussian process approach to kernel learning. We find that Lanczos is generally superior to Chebyshev for kernel learning, and that a surrogate approach can be highly efficient and accurate with popular kernels.