Hilbert Space Methods for Reduced-Rank Gaussian Process Regression (1401.5508v3)

Abstract: 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 $\mathbb{R}^d$. 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 $\mathcal{O}(nm^2)$ (initial) and $\mathcal{O}(m^3)$ (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 is increases appropriately, and for the squared exponential covariance function it is always bounded by ${\sim}1/m$ 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.

• The paper introduces a novel reduced-rank GP regression model using an eigenfunction expansion of the Laplace operator.
• The method decouples basis functions from covariance hyperparameters, reducing training complexity to O(nm²) and hyperparameter learning to O(m³).
• Empirical and theoretical analyses validate its convergence and efficiency, outperforming traditional GP methods on large datasets.

Hilbert Space Methods for Reduced-Rank Gaussian Process Regression

The paper by Solin and Särkkä introduces an innovative approach for reduced-rank Gaussian process (GP) regression, leveraging Hilbert space methods to enhance computational efficiency. This development is pertinent to addressing the computational challenges inherent in Gaussian processes, especially with large datasets, where traditional implementations result in prohibitive computational and memory costs scaling with the cubic power of the number of data points.

Key Methodological Advancements

The authors propose utilizing an approximate series expansion of the covariance function via an eigenfunction expansion of the Laplace operator, specifically within a compact subset of $\mathbb{R}^d$. This expansion allows the spectral density of the Gaussian process to directly inform the eigenvalues of the covariance function, transforming the GP inference problem into a form amenable to reduced-rank approximations.

Computational Efficiency

The reduced-rank GP regression method achieves a computational complexity of $\mathcal{O}(nm^2)$ for initial training and $\mathcal{O}(m^3)$ for hyperparameter learning, where $m$ denotes the number of basis functions and $n$ represents the number of data points. Notably, this method decouples the basis functions from the hyperparameters of the covariance function, significantly enhancing the efficiency of hyperparameter learning.

Theoretical Contributions

The paper explores rigorous error analysis using Hilbert space theory. It demonstrates that the approximation converges exactly as both the compact subset's size and the number of eigenfunctions increase indefinitely. Importantly, for the widely used squared exponential covariance function, this convergence rate is shown to be independent of input dimensionality, bounded by $\sim 1/m$.

This framework not only provides a strong theoretical underpinning but also highlights the adaptability of the approach to various domains and input densities, extending beyond typical grid constraints in spatial statistics.

Empirical Validation

Through empirical tests on simulated and real datasets, the method is shown to perform favorably compared to existing reduced-rank GP methods, such as Nyström-based approximations and sparse spectrum techniques. The evaluations underscore both the computational advantages and the robust modeling capacity of the proposed approximation.

Implications and Future Directions

The implications of this research are multifaceted. Practically, it opens new avenues for efficient Bayesian inference in large-scale datasets across diverse fields, from geostatistics to machine learning applications. Theoretically, it contributes to the growing body of work seeking to reconcile the computational demands of Gaussian processes with modern big data requirements.

Potential future developments include extending the theoretical analysis to non-standard domains, such as spherical surfaces, which are common in climatology and cosmology. Additionally, this method's integration with advanced inference techniques like Hamiltonian Monte Carlo could further augment its applicability in complex models with hierarchical or latent structures.

In summary, the paper offers a significant advancement in the methodology for Gaussian process regression, providing both theoretical insights and practical solutions for scaling GPs to larger datasets while maintaining accuracy and reliability.