TY - JOUR

T1 - Hilbert space methods for reduced-rank Gaussian process regression

AU - Solin, Arno

AU - Särkkä, Simo

PY - 2019

Y1 - 2019

N2 - This paper proposes a novel scheme for reduced-rank Gaussian process regression. The method is based on an approximate series expansion of the covariance function in terms of an eigenfunction expansion of the Laplace operator in a compact subset of Rd. On this approximate eigenbasis, the eigenvalues of the covariance function can be expressed as simple functions of the spectral density of the Gaussian process, which allows the GP inference to be solved under a computational cost scaling as O(nm2) (initial) and O(m3) (hyperparameter learning) with m basis functions and n data points. Furthermore, the basis functions are independent of the parameters of the covariance function, which allows for very fast hyperparameter learning. The approach also allows for rigorous error analysis with Hilbert space theory, and we show that the approximation becomes exact when the size of the compact subset and the number of eigenfunctions go to infinity. We also show that the convergence rate of the truncation error is independent of the input dimensionality provided that the differentiability order of the covariance function increases appropriately, and for the squared exponential covariance function it is always bounded by ∼ 1 / m regardless of the input dimensionality. The expansion generalizes to Hilbert spaces with an inner product which is defined as an integral over a specified input density. The method is compared to previously proposed methods theoretically and through empirical tests with simulated and real data.

AB - This paper proposes a novel scheme for reduced-rank Gaussian process regression. The method is based on an approximate series expansion of the covariance function in terms of an eigenfunction expansion of the Laplace operator in a compact subset of Rd. On this approximate eigenbasis, the eigenvalues of the covariance function can be expressed as simple functions of the spectral density of the Gaussian process, which allows the GP inference to be solved under a computational cost scaling as O(nm2) (initial) and O(m3) (hyperparameter learning) with m basis functions and n data points. Furthermore, the basis functions are independent of the parameters of the covariance function, which allows for very fast hyperparameter learning. The approach also allows for rigorous error analysis with Hilbert space theory, and we show that the approximation becomes exact when the size of the compact subset and the number of eigenfunctions go to infinity. We also show that the convergence rate of the truncation error is independent of the input dimensionality provided that the differentiability order of the covariance function increases appropriately, and for the squared exponential covariance function it is always bounded by ∼ 1 / m regardless of the input dimensionality. The expansion generalizes to Hilbert spaces with an inner product which is defined as an integral over a specified input density. The method is compared to previously proposed methods theoretically and through empirical tests with simulated and real data.

KW - Eigenfunction expansion

KW - Gaussian process regression

KW - Laplace operator

KW - Pseudo-differential operator

KW - Reduced-rank approximation

UR - http://www.scopus.com/inward/record.url?scp=85070221375&partnerID=8YFLogxK

U2 - 10.1007/s11222-019-09886-w

DO - 10.1007/s11222-019-09886-w

M3 - Article

AN - SCOPUS:85070221375

JO - STATISTICS AND COMPUTING

JF - STATISTICS AND COMPUTING

SN - 0960-3174

ER -