Scalable Log Determinants for Gaussian Process Kernel Learning

By Kun Dong, David Eriksson, Hannes Nickisch, David S. Bindel, Andrew Gordon Wilson. 2017

109 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistical and Nonlinear Physics

Citations in the corpus

Invertible Residual Networks (2018, background): “The approximation of spectral sums like the log-determinant is of broad interest for many machine learning problems such as Gaussian Process regression (Dong et al., 2017).”
Scalable Global Optimization via Local Bayesian Optimization (2019, methodology): “GPyTorch follows Dong et al. [11] to solve linear systems using the conjugate gradient (CG) method and approximates the log-determinant via the Lanczos process.”
When Gaussian Process Meets Big Data: A Review of Scalable GPs (2018, background): “To have sensible uncertainty, a diagonal correlation akin to that of FITC has been considered [100], [102].”
GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration (2018, methodology): “Additionally, BBMM performs better than the GPU-accelerated version of [13] on SKI.”
Stochastic Chebyshev Gradient Descent for Spectral Optimization (2018, methodology): “We benchmark GP regression under natural sound dataset used in [32, 8] and Szeged humid data [6].”