Orthonormal Expansions for Translation-Invariant Kernels
This provides a theoretical tool for kernel methods in machine learning, but it is incremental as it builds on existing Fourier analysis and kernel theory.
The paper tackled the problem of constructing orthonormal basis expansions for translation-invariant kernels by developing a Fourier analytic technique, resulting in explicit expansions for Matérn, Cauchy, and Gaussian kernels using specific functions like Laguerre, rational, and Hermite functions.
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) Matérn 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.