Kernel methods and Gaussian processes typically rely on a Hilbertian distance measure to ensure the kernel matrix is positive semi-definite (PSD). As the authors note, this condition is violated for many natural input spaces, including smooth manifolds and spaces of probability distributions, limiting the applicability of standard approaches.
The new framework, introduced by Marcus M. Noack, Maher B. Alghalayini, and Mark D. Risser, removes that requirement. Their Sparse Landmark Embedding (SLE) kernel maps each input to a sparse feature vector using 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, while the compact support keeps kernel matrices well-conditioned and computationally tractable.
The paper reports theoretical guarantees covering PSD, sparsity, stability, and universal approximation. In experiments, the SLE kernel matched or substantially exceeded domain-specific baselines in predictive accuracy and uncertainty quantification, using geodesic and Wasserstein distances as test cases.