The Kernel Manifold: A Geometric Approach to Gaussian Process Model Selection
This work addresses a critical bottleneck in probabilistic modeling for researchers and practitioners using Gaussian Processes, offering a reusable geometric approach that improves model selection efficiency and performance.
The paper tackles the challenge of selecting covariance kernels in Gaussian Process regression by introducing a Bayesian optimization framework that uses kernel geometry and multidimensional scaling to map kernels into a continuous manifold, achieving superior predictive accuracy and uncertainty calibration on synthetic, real-world, and case study datasets compared to baselines like LLM-guided search.
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.