# 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.

## References

SHOWING 1-10 OF 34 REFERENCES

### Power Series Kernels

- Mathematics, Computer Science
- 2009

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…

### Countable Tensor Products of Hermite Spaces and Spaces of Gaussian Kernels

- MathematicsJ. Complex.
- 2022

### Stable Evaluation of Gaussian Radial Basis Function Interpolants

- Computer ScienceSIAM J. Sci. Comput.
- 2012

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

- Computer Science, Mathematics
- 2010

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

- Mathematics
- 2016

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

- Mathematics
- 2012

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…

### High-dimensional integration on Rd, weighted Hermite spaces, and orthogonal transforms

- MathematicsJ. Complex.
- 2015

### Separability of reproducing kernel spaces

- Mathematics
- 2015

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

- MathematicsProceedings 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

- Computer Science, MathematicsStat. Comput.
- 2020

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.