Hilbert space methods for reduced-rank Gaussian process regression

• Published in: Statistics and computing Vol. 30; no. 2; pp. 419 - 446
• Main Authors: Solin, Arno, Särkkä, Simo
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2020; Springer Nature B.V
• Subjects: Artificial Intelligence; Basis functions; Computer simulation; Covariance; Data points; Density; Eigenvalues; Eigenvectors; Empirical analysis; Error analysis; Gaussian process; Hilbert space; Learning; Mathematics and Statistics; Operators (mathematics); Probability and Statistics in Computer Science; Regression analysis; Series expansion; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; Truncation errors; Laplace operator; Eigenfunction expansion; Gaussian process regression; Reduced-rank approximation; Pseudo-differential operator
• ISSN: 0960-3174; 1573-1375; 1573-1375
• DOI: 10.1007/s11222-019-09886-w

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 R d . 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 ( n m 2 ) (initial) and O ( m 3 ) (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.