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    Sobolev Spaces, Kernels and Discrepancies over Hyperspheres

    Hubbert, Simon and Porcu, E. and Oates, C. and Girolami, M. (2023) Sobolev Spaces, Kernels and Discrepancies over Hyperspheres. Transactions on Machine Learning Research , pp. 1-18. ISSN 2835-8856.

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    Abstract

    This work extends analytical foundations for kernel methods beyond the usual Euclidean manifold. Specifically, we characterise the smoothness of the native spaces (reproducing kernel Hilbert spaces) that are reproduced by geodesically isotropic kernels in the hyper- spherical context. Our results are relevant to several areas of machine learning; we focus on their consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms based on Stein’s method. First, we introduce a characterisation of Sobolev spaces on the d-dimensional sphere based on the Fourier–Schoenberg sequences associated with a given kernel. Such sequences are hard (if not impossible) to compute analytically on d-dimensional spheres, but often feasi- ble over Hilbert spheres, where d = ∞. Second, we circumvent this problem by finding a projection operator that allows us to map from Hilbert spheres to finite-dimensional spheres. Our findings are illustrated for selected parametric families of kernel.

    Metadata

    Item Type: Article
    School: Birkbeck Faculties and Schools > Faculty of Business and Law > Birkbeck Business School
    Depositing User: Simon Hubbert
    Date Deposited: 05 May 2023 05:28
    Last Modified: 02 Aug 2023 18:21
    URI: https://eprints.bbk.ac.uk/id/eprint/51156

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