Reproduce “Geodesic Exponential Kernels: When Curvature and Linearity Conflict” on OpenResearch

About this paper

Aasa Feragen, Francois Lauze, Søren Hauberg

This research from the University of Copenhagen and DTU Compute establishes the conditions under which geodesic exponential kernels maintain positive definiteness on general geodesic metric spaces and Riemannian manifolds. It demonstrates that geodesic Gaussian kernels are positive definite only on flat or Euclidean-isometric spaces, while geodesic Laplacian kernels are positive definite if the geodesic distance is conditionally negative definite, as observed in spaces like spheres.

Key insights

• Geodesic Gaussian kernels (q=2) are positive definite only if the underlying metric space is flat in the sense of Alexandrov or isometric to a Euclidean space, severely limiting their applicability on truly curved manifolds.
• Geodesic Laplacian kernels (q=1) are positive definite if and only if the geodesic distance itself is conditionally negative definite, a property that holds for specific curved spaces like spheres but not universally.
• The inherent linearity required for positive definite kernel embeddings often conflicts with the curvature of non-Euclidean data spaces, meaning a straightforward generalization from Euclidean distance is often insufficient.

Summary by alphaXiv.