A General Kernel Framework for Non-CND Distance Measures Using |D|-Dimensional Sparse Landmark Embeddings
Marcus M. Noack, Maher B. Alghalayini, Mark D. Risser
Source abstract
Kernel methods, and Gaussian Processes (GPs) in particular, require a Hilbertian distance measure---one whose square is conditionally negative definite (CND)---to guarantee positive semi-definiteness (PSD) of the kernel matrix; a condition that fails for many natural input spaces, including smooth manifolds and spaces of probability distributions. We propose the Sparse Landmark Embedding (SLE) kernel, which eliminates this requirement entirely. Each input is embedded into a sparse feature vector via compactly supported bump functions centered at all |D| training points; applying any standard PSD kernel in this embedding space yields a kernel that is provably PSD for arbitrary distance measures. The compact support automatically controls embedding sparsity, keeping kernel matrices well-conditioned and computationally tractable despite the high ambient dimension. We provide theoretical guarantees on PSD, sparsity, stability, and universal approximation, and demonstrate, using geodesic and Wasserstein distances, that the SLE kernel matches or substantially exceeds domain-specific baselines in both predictive accuracy and uncertainty quantification.
Evidence graph
No public relationships recorded yet.
Integrity note: This page is a factual metadata record created by deterministic ingestion. It is not a claim that the work moves a mathematical frontier or has been independently verified.