Fully Symmetric Kernel Quadrature

Abstract

Kernel quadratures and other kernel-based approximation methods typically suffer from prohibitive cubic time and quadratic space complexity in the number of function evaluations. The problem arises because a system of linear equations needs to be solved. In this article we show that the weights of a kernel quadrature rule can be computed efficiently and exactly for up to tens of millions of nodes if the kernel, integration domain, and measure are fully symmetric and the node set is a union of fully symmetric sets. This is based on the observations that in such a setting there are only as many distinct weights as there are fully symmetric sets and that these weights can be solved from a linear system of equations constructed out of row sums of certain submatrices of the full kernel matrix. We present several numerical examples that show feasibility, both for a large number of nodes and in high dimensions, of the developed fully symmetric kernel quadrature rules. Most prominent of the fully symmetric kernel quadrature rules we propose are those that use sparse grids.

Bayesian Probabilistic Numerical Methods

“…The full distribution B(µ, a) was examined in Briol et al (2016), who established contraction to the exact value of the integral under certain smoothness conditions on the Gaussian covariance function and on the integrand. See also Kanagawa et al (2016); Karvonen and Särkkä (2017).…”

Symmetry exploits for Bayesian cubature methods

“…Third, although we focussed solely on computational aspects, the important statistical question of how to ensure Bayesian cubature methods produce output that is well-calibrated remains to some extent unresolved. 5 As discussed in [31], it appears that symmetry exploits do not easily lend themselves to selection of kernel parameters, for instance via cross-validation or maximisation of marginal likelihood. 6 A potential, though somewhat heuristic, way to proceed might be to exploit the concentration of measure phenomenon [37] or low effective dimensionality of the integrand [62] in order to identify a suitable data subset on which kernel parameters can be calibrated more easily or a priori.…”

“…The results presented in this article, and those originally described in [31], rely on the assumption that the measure ν is fully symmetric (see Section 2.2.2). This is a strong restriction; most measures are not fully symmetric.…”

“…For each q, the proposal distribution ν * was a zero-mean Gaussian with diagonal covariance σ 2 I I I 6 for σ 2 set to the mean of the diagonal elements of the Σ Σ Σ i . For Bayesian cubature we used the Gaussian kernel (27) with a length-scale = 0.8 and the Gauss-Hermite sparse grid [31,Section 4.2] with the mid-point removed. Note that the resulting point sets are not nested for different N .…”

Gaussian kernel quadrature at scaled Gauss–Hermite nodes

“…The truncation length M was selected based on machine precision; see [12, Section 4.2.2] for details. Yet even this does not work for large enough N. Because kernel quadrature rules on symmetric point sets have symmetric weights [16,28,Section 5.2.4], breakdown in symmetricity of the computed kernel quadrature weights was used as a heuristic proxy for emergence of numerical instability: for each length-scale, relative errors are presented in Figure 5.2 until the first N such that |1 − w k,N /w k,1 | > 10 −6 , ordering of the nodes being from smallest to the largest so that w k,N = w k,1 in absence of numerical errors. Figure 5.3 shows the minimal weights min n=1,...,N w k,n and convergence to one of ∑ N n=1 | w k,n | for a number of different length-scales.…”