2017/12/14 by Ingo Steinwart, Johanna F. Ziegel, Steinwart, Ingo +1 · 4 citations
Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Machine Learning (stat.ML) #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1712.05279
openalex publication_date 2017/12/14 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
Strictly proper kernel scores are well-known tool in probabilistic\nforecasting, while characteristic kernels have been extensively investigated in\nthe machine learning literature. We first show that both notions coincide, so\nthat insights from one part of the literature can be used in the other. We then\nshow that the metric induced by a characteristic kernel cannot reliably\ndistinguish between distributions that are far apart in the total variation\nnorm as soon as the underlying space of measures is infinite dimensional. In\naddition, we provide a characterization of characteristic kernels in terms of\neigenvalues and -functions and apply this characterization to the case of\ncontinuous kernels on (locally) compact spaces. In the compact case we further\nshow that characteristic kernels exist if and only if the space is metrizable.\nAs special cases of our general theory we investigate translation-invariant\nkernels on compact Abelian groups and isotropic kernels on spheres. The latter\nare of particular interest for forecast evaluation of probabilistic predictions\non spherical domains as frequently encountered in meteorology and climatology.\n