Gaussian kernel quadrature at scaled Gauss–Hermite nodes

Abstract

This article derives an accurate, explicit, and numerically stable approximation to the kernel quadrature weights in one dimension and on tensor product grids when the kernel and integration measure are Gaussian. The approximation is based on use of scaled Gauss-Hermite nodes and truncation of the Mercer eigendecomposition of the Gaussian kernel. Numerical evidence indicates that both the kernel quadrature and the approximate weights at these nodes are positive. An exponential rate of convergence for functions in the reproducing kernel Hilbert space induced by the Gaussian kernel is proved under an assumption on growth of the sum of absolute values of the approximate weights.

Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures

“…Given a probability measure µ in the (conceptually univariate) space X , constructing a kernel quadrature for µ ⊗d with respect to k ⊗d is a natural multivariate extension of kernel quadrature that is widely studied in the literature [47,32,3,33], and corresponds to high-dimensional QMCs [18].…”

“…Arguably the most famous applications concerns the case when X is a subset of R d and F is the linear space of polynomials up to a certain degree, that is F is spanned by monomials up to a certain degree. However, more recent applications include the case when X is a space of paths and F is spanned by iterated Ito-Stratonovich integrals [39], or kernel quadrature [33,27] where X is a set that carries a positie definite kernel and F is a subset of the associated reproducing kernel Hilber space that is spanned by eigenfunctions of the integral operator induced by a kernel.…”

“…• Kernel quadrature [33,27]: µ is a probability measure on set X that carries a positive definite kernel k and F d,m is spanned by the eigenfunctions (down to some eigenvalue) of the integral operator g → k ⊗d (•, x)g(x) dµ ⊗d (x), where k ⊗d is a tensor product kernel.…”

On the positivity and magnitudes of Bayesian quadrature weights

“…Analysis of the weights and their positivity naturally reduces to that of the reproduced classical rule. -There is convincing numerical evidence that the weights are positive if the nodes for the Gaussian kernel and measure on R are selected by suitable scaling the classical Gauss-Hermite nodes [35]. -Uniform weighting (i.e., w BQ X,i = 1/n) can be achieved when certain quasi-Monte Carlo point sets and shift invariant kernels are used [27].…”

Worst-case optimal approximation with increasingly flat Gaussian kernels

“…See[9] for an interpolation method based on a closely related basis derived from a Mercer eigendecomposition of the Gaussian kernel and[13] for an explicit construction of weights similar to w φ,ℓ in the case L is the Gaussian integral.…”