The Kernel Manifold: A Geometric Approach to Gaussian Process Model Selection

TL;DR

Gaussian Process performance hinges on kernel choice, but exploring a combinatorially large kernel space with hyperparameters is costly. The authors propose a geometry-driven BO framework that treats kernels as points on the kernel manifold, constructing a distance matrix between GP priors via expected divergences and embedding it with MDS into a Euclidean space where BO can operate efficiently. They show that divergences like the sqrt{JS} distance yield Euclidean embeddability, enabling a kernel-of-kernels surrogate (potentially multi-scale) and a BO procedure that selects kernels by maximizing the log marginal likelihood $LML$. Across synthetic benchmarks, real time-series, and an AM melt-pool case study, the approach outperforms LLM-guided search and random baselines, delivering faster convergence, improved uncertainty calibration, and robust predictive performance. The work provides a reusable, geometry-centric framework for automated kernel discovery with broad applicability to GP modeling and beyond.

Abstract

Gaussian Process (GP) regression is a powerful nonparametric Bayesian framework, but its performance depends critically on the choice of covariance kernel. Selecting an appropriate kernel is therefore central to model quality, yet remains one of the most challenging and computationally expensive steps in probabilistic modeling. We present a Bayesian optimization framework built on kernel-of-kernels geometry, using expected divergence-based distances between GP priors to explore kernel space efficiently. A multidimensional scaling (MDS) embedding of this distance matrix maps a discrete kernel library into a continuous Euclidean manifold, enabling smooth BO. In this formulation, the input space comprises kernel compositions, the objective is the log marginal likelihood, and featurization is given by the MDS coordinates. When the divergence yields a valid metric, the embedding preserves geometry and produces a stable BO landscape. We demonstrate the approach on synthetic benchmarks, real-world time-series datasets, and an additive manufacturing case study predicting melt-pool geometry, achieving superior predictive accuracy and uncertainty calibration relative to baselines including Large Language Model (LLM)-guided search. This framework establishes a reusable probabilistic geometry for kernel search, with direct relevance to GP modeling and deep kernel learning.

Figures (22)

• Figure 1: Conceptual overview of BO for kernel selection on the kernel‑of‑kernels manifold. Discrete kernel candidates are embedded into a continuous space, and BO traverses the log‑marginal‑likelihood landscape from an initial kernel toward an optimized structure.
• Figure 2: Geodesic (great-circle) distances between points on a sphere. These distances arise from a curved Riemannian manifold and cannot be embedded exactly into Euclidean space.
• Figure 3: Chordal-mapping correction for spherical geometry. Left: reconstruction error from MDS collapses to zero at $k=3$. Right: eigenvalue spectrum shows elimination of negative eigenvalues after the transformation, confirming Euclidean embeddability.
• Figure 4: Comparison of MDS reconstruction error before and after log-warping. The log-transformed distances produce significantly lower error and a smoother Euclidean embedding.
• Figure 5: Eigenvalue spectra of the double-centered Gram matrix before (left) and after (right) log-warping. Log-warping removes large negative eigenvalues, substantially reducing curvature.