A General Kernel Framework for Non-CND Distance Measures Using |D|-Dimensional Sparse Landmark Embeddings
Proposes the Sparse Landmark Embedding kernel, guaranteeing PSD kernels for arbitrary distances like geodesic and Wasserstein without CND requirements.
The paper introduces the Sparse Landmark Embedding (SLE) kernel, which embeds inputs via compactly supported bump functions at all |D| training points so any standard PSD kernel applies, removing the Hilbertian (CND) distance requirement that fails on manifolds and distribution spaces. Compact support controls sparsity, keeping kernel matrices well-conditioned despite high dimensionality. The authors prove PSD, sparsity, stability, and universal approximation guarantees, and show SLE matches or exceeds domain-specific baselines using geodesic and Wasserstein distances on accuracy and uncertainty quantification.
- Removes CND distance requirement for Gaussian Process kernels
- Bump-function embeddings yield provably PSD kernels for arbitrary distances
- Validated on geodesic and Wasserstein distances against domain-specific baselines
- Products
- Sparse Landmark Embedding
From the original article
1 min read at arxiv.orgKernel 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.
By Marcus M. Noack · arxiv.org
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