Orthonormal Expansions for Translation-Invariant Kernels

@article{Tronarp2022OrthonormalEF,
  title={Orthonormal Expansions for Translation-Invariant Kernels},
  author={Filip Tronarp and Toni Karvonen},
  journal={ArXiv},
  year={2022},
  volume={abs/2206.08648}
}
We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of L 2 ( 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. 

Figures from this paper

References

SHOWING 1-10 OF 34 REFERENCES

Power Series Kernels

We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable

Stable Evaluation of Gaussian Radial Basis Function Interpolants

We provide a new way to compute and evaluate Gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter, i.e., for “flat” kernels. This

Some Properties of Gaussian Reproducing Kernel Hilbert Spaces and Their Implications for Function Approximation and Learning Theory

We give several properties of the reproducing kernel Hilbert space induced by the Gaussian kernel, along with their implications for recent results in the complexity of the regularized least square

An Introduction to the Theory of Reproducing Kernel Hilbert Spaces

Reproducing kernel Hilbert spaces have developed into an important tool in many areas, especially statistics and machine learning, and they play a valuable role in complex analysis, probability,

Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs

Given a compact metric space X and a strictly positive Borel measure ν on X, Mercer’s classical theorem states that the spectral decomposition of a positive self-adjoint integral operator

Separability of reproducing kernel spaces

We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel

Associated Laguerre and Hermite polynomials

  • R. AskeyJ. Wimp
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 1984
Synopsis Explicit orthogonality relations are found for the associated Laguerre and Hermite polynomials. One consequence is the construction of the [n − 1/n] Padé approximation to Ψ(a + 1, b; x)/Ψ(a,

Hilbert space methods for reduced-rank Gaussian process regression

TLDR
The method is compared to previously proposed methods theoretically and through empirical tests with simulated and real data, and shows that the approximation becomes exact when the size of the compact subset and the number of eigenfunctions go to infinity.