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Orthonormal Expansions for Translation-Invariant Kernels (2206.08648v4)
Published 17 Jun 2022 in math.CA, cs.LG, cs.NA, math.NA, and stat.ML
Abstract: We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of $\mathscr{L}_2(\mathbb{R})$. This allows us to derive explicit expansions on the real line for (i) Mat\'ern kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.