On Basis Function Selection for Sparse Gaussian Process Regression

arXiv:2609.26624v1 Announce Type: cross Abstract: Sparse Gaussian processes achieve $O(N)$ inference by replacing the kernel with an appropriate expansion in a fixed basis $\{\phi_j\}$ on the input space. Given a compute budget $M \ll N$, practitioners conventionally truncate the basis to its first $M$ entries. Nothing in the formalism, however, prevents one from selecting only those $M$ basis functions that matter for the data at hand. This would avoid spending budget on basis functions where there is no signal, but it requires a criterion for ranking the candidates. We propose three such criteria derived from an information-theoretic view of the basis-function selection problem. Each criterion matches a different state of knowledge at selection time: a no-data state, a no-prior state, and an in-between state. We then study the performance of truncation versus selection strategies on six UCI regression benchmarks across three basis families: Hilbert-space Gaussian processes (HSGP), variational Fourier features (VFF), and variational inducing spherical harmonics (VISH). We observe that the no-data criterion is a safe default, matching or improving on truncation for HSGP, VFF and VISH, with substantial gains for VISH and improvements over a recently developed selection heuristic for that basis family. The data-aware no-prior and in-between criteria provide substantial gains over truncation specifically for HSGP, which is the most broadly used of the three families in practice.

Abstract:Sparse Gaussian processes achieve $O(N)$ inference by replacing the kernel with an appropriate expansion in a fixed basis $\{\phi_j\}$ on the input space. Given a compute budget $M \ll N$, practitioners conventionally truncate the basis to its first $M$ entries. Nothing in the formalism, however, prevents one from selecting only those $M$ basis functions that matter for the data at hand. This would avoid spending budget on basis functions where there is no signal, but it requires a criterion for ranking the candidates. We propose three such criteria derived from an information-theoretic view of the basis-function selection problem. Each criterion matches a different state of knowledge at selection time: a no-data state, a no-prior state, and an in-between state. We then study the performance of truncation versus selection strategies on six UCI regression benchmarks across three basis families: Hilbert-space Gaussian processes (HSGP), variational Fourier features (VFF), and variational inducing spherical harmonics (VISH). We observe that the no-data criterion is a safe default, matching or improving on truncation for HSGP, VFF and VISH, with substantial gains for VISH and improvements over a recently developed selection heuristic for that basis family. The data-aware no-prior and in-between criteria provide substantial gains over truncation specifically for HSGP, which is the most broadly used of the three families in practice.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2609.26624 [stat.ML]

(or arXiv:2609.26624v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.26624 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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On Basis Function Selection for Sparse Gaussian Process Regression

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