# Gaussian kernel quadrature at scaled Gauss–Hermite nodes

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## 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.

## Keywords

Numerical integration Kernel quadrature Gaussian quadrature Mercer eigendecomposition## Mathematics Subject Classification

45C05 46E22 47B32 65D30 65D32## 1 Introduction

*f*using a

*kernel quadrature rule*(we reserve the term

*cubature*for rules on higher dimensions) based on the Gaussian kernel

*nodes*\(x_1,\ldots ,x_N\), the kernel quadrature rule is an approximation of the form

*weights*\(w_k = (w_{k,1},\ldots ,w_{k,N}) {\in } \mathbb {R}^N\) solved from the linear system of equations

*N*kernel translates \(k(x_1,\cdot ),\ldots ,k(x_N,\cdot )\) are integrated exactly by the quadrature rule. Kernel quadrature rules can be interpreted as best quadrature rules in the reproducing kernel Hilbert space (RKHS) induced by a positive-definite kernel [20], integrated kernel (radial basis function) interpolants [5, 35], and posteriors to \(\mu (f)\) under a Gaussian process prior on the integrand [7, 21, 29].

Recently, Fasshauer and McCourt [12] have developed a method to circumvent the well-known problem that interpolation with the Gaussian kernel becomes often numerically unstable—in particular when \(\ell \) is large—because the condition number of *K* tends to grow with an exponential rate [33]. They do this by truncating the Mercer eigendecomposition of the Gaussian kernel after *M* terms and replacing the interpolation basis \(\{k(x_n,\cdot )\}_{n=1}^N\) with the first *M* eigenfunctions. In this article we show that application of this method with \(M=N\) to kernel quadrature yields, when the nodes are selected by a suitable and fairly natural scaling of the nodes of the classical Gauss–Hermite quadrature rule, an accurate, explicit, and numerically stable approximation to the Gaussian kernel quadrature weights. Moreover, the proposed nodes appear to be a good and natural choice for the Gaussian kernel quadrature.

*N*Mercer eigenfunctions of the Gaussian kernel and uses the nodes

*N*-point Gauss–Hermite quadrature rule. We argue that these weights are a good approximation to \(w_{k}\) and accordingly call them

*approximate Gaussian kernel quadrature weights*. Although we derive no bounds for the error of this weight approximation, numerical experiments in Sect. 5 indicate that the approximation is accurate and that it appears that \({\widetilde{w}}_{k} \rightarrow w_{k}\) as \(N \rightarrow \infty \). In Sect. 4 we extend the weight approximation for

*d*-dimensional Gaussian tensor product kernel cubature rules of the form

We are not aware of any work on efficient selection of “good” nodes in the setting of this article. The Gauss–Hermite nodes [29, Section 3] and random points [31] are often used, but one should clearly be able to do better, while computation of the optimal nodes [28, Section 5.2] is computationally demanding. As such, given the desirable properties, listed below, of the resulting kernel quadrature rules, the nodes \(\tilde{x}_n\) appear to be an excellent heuristic choice. These nodes also behave naturally when \(\ell \rightarrow \infty \); see Sect. 2.5.

Numerical experiments in Sect. 5.3 suggest that both \(w_{k,n}\) (for the nodes \(\tilde{x}_n\)) and \({\widetilde{w}}_{k,n}\) are positive for any \(N \in \mathbb {N}\) and every \(n = 1,\ldots , N\). Besides the optimal nodes, the weights for which are guaranteed to be positive when the Gaussian kernel is used [28, 32], there are no node configurations that give rise to positive weights as far as we are aware of.

Numerical experiments in Sects. 5.1 and 5.3 demonstrate that computation of the approximate weights is numerically stable. Furthermore, construction of these weights only incurs a quadratic computational cost in the number of points, as opposed to the cubic cost of solving \(w_k\) from Eq. (1.2). See Sect. 2.6 for more details. Note that to obtain a numerically stable method it is not necessary to use the nodes \(\tilde{x}_n\) as the method in [12] can be applied in a straightforward manner for any nodes. However, doing so one forgoes a closed form expression and has to use the QR decomposition.

In Sects. 3 and 4 we show that slow enough growth with

*N*of \(\sum _{n=1}^N \left|\widetilde{w}_{k,n}\right|\) (numerical evidence indicates this sum converges to one) guarantees that the approximate Gaussian kernel quadrature rule—as well as the corresponding tensor product version—converges with an exponential rate for functions in the RKHS of the Gaussian kernel. Convergence analysis is based on analysis of magnitude of the remainder of the Mercer expansion and rather explicit bounds on Hermite polynomials and their roots. Magnitude of the nodes \(\tilde{x}_n\) is crucial for the analysis; if they were further spread out the proofs would not work as such.We find the connection to the Gauss–Hermite weights and nodes that the closed form expression for \({\widetilde{w}}_k\) provides intriguing and hope that it can be at some point used to furnish, for example, a rigorous proof of positivity of the approximate weights.

## 2 Approximate weights

This section contains the main results of this article. The main contribution is derivation, in Theorem 2.2, of the weights \({\widetilde{w}}_k\), that can be used to approximate the kernel quadrature weights. We also discuss positivity of these weights, the effect the kernel length-scale \(\ell \) is expected to have on quality of the approximation, and computational complexity.

### 2.1 Eigendecomposition of the Gaussian kernel

*k*admits an absolutely and uniformly convergent eigendecomposition

*k*and orthonormal in \(L^2(\nu )\). Moreover, \(\sqrt{\lambda _n} \varphi _n\) are \(\mathscr {H}\)-orthonormal. If the support of \(\nu \) is not compact, the expansion (2.1) converges absolutely and uniformly on all compact subsets of \(\mathbb {R}\times \mathbb {R}\) under some mild assumptions [37, 38]. For the Gaussian kernel (1.1) and measure the eigenvalues and eigenfunctions are available analytically. For a collection of explicit eigendecompositions of some other kernels, see for instance [11, Appendix A]

### Lemma 2.1

### Proof

### 2.2 Approximation via QR decomposition

^{1}yields the approximations \(K \approx \varPhi \varLambda \varPhi ^\mathsf {T}\) and \(k_\mu \approx \varPhi \varLambda \varphi _\mu \), where \({[\varPhi ]_{ij} \,{{:}{=}}\, \varphi _{j-1}^\alpha (x_i)}\) is an \(N \times M\) matrix, the diagonal \(M \times M\) matrix \([\varLambda ]_{ii} \,{{:}{=}}\, \lambda _{i-1}\) contains the eigenvalues in appropriate order, and \({[\varphi _\mu ]_i \,{{:}{=}}\, \mu (\varphi _{i-1})}\) is an

*M*-vector. The kernel quadrature weights \(w_k\) can be therefore approximated by

Unfortunately, using the QR decomposition does not provide an attractive closed form solution for the approximate weights \({\widetilde{w}}_k^M\) for general *M*. Setting \(M = N\) turns \(\varPhi \) into a square matrix, enabling its direct inversion and formation of an explicit connection to the classical Gauss–Hermite quadrature. The rest of the article is concerned with this special case.

### 2.3 Gauss–Hermite quadrature

*N*-point

*Gaussian quadrature rule*is the unique

*N*-point quadrature rule that is exact for all polynomials of degree at most \(2N-1\). We are interested in

*Gauss–Hermite quadrature rules*that are Gaussian rules for the Gaussian measure \(\mu \):

*N*th Hermite polynomial \(\mathrm {H}_N\) and the weights \(w_1^{\mathrm{GH}}, \ldots , w_N^{\mathrm{GH}}\) are positive and sum to one. The nodes and the weights are related to the eigenvalues and eigenvectors of the tridiagonal Jacobi matrix formed out of three-term recurrence relation coefficients of normalised Hermite polynomials [13, Theorem 3.1].

We make use of the following theorem, a one-dimensional special case of a more general result due to Mysovskikh [27]. See also [8, Section 7]. This result also follows from the Christoffel–Darboux formula (2.12).

### Theorem 2.1

Let \(\nu \) be a measure on \(\mathbb {R}\). Suppose that \(x_1,\ldots ,x_N\) and \(w_1,\ldots ,w_N\) are the nodes and weights of the unique Gaussian quadrature rule. Let \(p_0,\ldots ,p_{N-1}\) be the \(L^2(\nu )\)-orthonormal polynomials. Then the matrix \([P]_{ij} \,{{:}{=}}\, \sum _{n=0}^{N-1} p_n(x_i) p_n(x_j)\) is diagonal and has the diagonal elements \([P]_{ii} = 1/w_i\).

### 2.4 Approximate weights at scaled Gauss–Hermite nodes

*N*first eigenfunctions \(\varphi _0^\alpha ,\ldots ,\varphi _{N-1}^\alpha \). For the Gaussian kernel, we are in a position to do much more. Recalling the form of the eigenfunctions in Eq. (2.5), we can write \(\varPhi = \sqrt{\beta } E^{-1} V\) for the diagonal matrix \([E]_{ii} \,{{:}{=}}\, {{\,\mathrm{e}\,}}^{\delta ^2 x_i^2}\) and the Vandermonde matrix

*V*defined in Eq. (2.8) is precisely the Vandermonde matrix of the normalised Hermite polynomials and \(V V^\mathsf {T}\) is the matrix

*P*of Theorem 2.1. Let \(W_{\mathrm{GH}}\) be the diagonal matrix containing the Gauss–Hermite weights. It follows that \(V^{-\mathsf {T}} = W_{\mathrm{GH}} V\) and

### Theorem 2.2

*N*-point Gauss–Hermite quadrature rule. Define the nodes

*N*-point quadrature rule

Since the weights \(\widetilde{w}_{k}\) are obtained by truncating of the Mercer expansion of *k*, it is to be expected that \(\widetilde{w}_k \approx w_k\). This motivates our calling of these weights the *approximate Gaussian kernel quadrature weights*. We do not provide theoretical results on quality of this approximation, but the numerical experiments in Sect. 5.2 indicate that the approximation is accurate and that its accuracy increases with *N*. See [12] for related experiments.

### 2.5 Effect of the length-scale

*flat limit*in scattered data approximation literature where it has been proved

^{2}that the kernel interpolant associated to an isotropic kernel with increasing length-scale converges to (i) the unique polynomial interpolant of degree \(N-1\) to the data if the kernel is infinitely smooth [22, 24, 34] or (ii) to a polyharmonic spline interpolant if the kernel is of finite smoothness [23]. In our case, \(\ell \rightarrow \infty \) results in

When it comes to node placement, the length-scale is having an intuitive effect if the nodes are selected according to Eq. (2.10). For small \(\ell \), the nodes are placed closer to the origin where most of the measure is concentrated as integrands are expected to converge quickly to zero as \(\left|x\right| \rightarrow \infty \), whereas for larger \(\ell \) the nodes are more—but not unlimitedly—spread out in order to capture behaviour of functions that potentially contribute to the integral also further away from the origin.

### 2.6 On computational complexity

Because the Gauss–Hermite nodes and weights are related to the eigenvalues and eigenvectors of the tridiagonal Jacobi matrix [13, Theorem 3.1] they—and the points \({\tilde{x}}_n\)—can be solved in quadratic time (in practice, these nodes and weights can be often tabulated beforehand). From Eq. (2.11) it is seen that computation of each approximate weight is linear in *N*: there are approximately \((N-1)/2\) terms in the sum and the Hermite polynomials can be evaluated on the fly using the three-term recurrence formula \(\mathrm {H}_{n+1}(x) = x \mathrm {H}_n(x) - n \mathrm {H}_{n-1}(x)\). That is, computational cost of obtaining \({\tilde{x}}_n\) and \({\widetilde{w}}_{k,n}\) for \(n=1,\ldots ,N\) is quadratic in *N*. Since the kernel matrix *K* of the Gaussian kernel is dense, solving the exact kernel quadrature weights from the linear system (1.2) for the points \({\tilde{x}}_n\) incurs a more demanding cubic computational cost. Because computational cost of a tensor product rule does not depend on the nodes and weights after these have been computed, the above discussion also applies to the rules presented in Sect. 4.

## 3 Convergence analysis

In this section we analyse convergence in the reproducing kernel Hilbert space \(\mathscr {H}\subset C^\infty (\mathbb {R})\) induced by the Gaussian kernel of quadrature rules that are exact for the Mercer eigenfunctions. First, we prove a generic result (Theorem 3.1) to this effect and then apply this to the quadrature rule with the nodes \(\tilde{x}_n\) and weights \({\widetilde{w}}_{k,n}\). If \(\sum _{n=1}^N \left|{\widetilde{w}}_{k,n}\right|\) does not grow too fast with *N*, we obtain exponential convergence rates.

*worst-case error*

*e*(

*Q*) of a quadrature rule \(Q(f) = \sum _{n=1}^N w_n f(x_n)\) is

*N*-point quadrature rules

*convergent*if \(e(Q_N) \rightarrow 0\) as \(N \rightarrow \infty \). For given nodes \(x_1,\ldots ,x_N\), the weights \(w_k = (w_{k,1},\ldots ,w_{k,N})\) of the kernel quadrature rule \(Q_k\) are unique minimisers of the worst-case error:

*e*(

*Q*) also applies to \(e(Q_k)\).

A number of convergence results for kernel quadrature rules on compact spaces appear in [5, 7, 14]. When it comes to the RKHS of the Gaussian kernel, characterised in [25, 36], Kuo and Woźniakowski [19] have analysed convergence of the Gauss–Hermite quadrature rule. Unfortunately, it turns out that the Gauss–Hermite rule converges in this space if and only if \(\varepsilon ^2 < 1/2\). Consequently, we believe that the analysis below is the first to establish convergence, under the assumption (supported by our numerical experiments) that the sum of \(\left|{\widetilde{w}}_{k,n}\right|\) does not grow too fast, of an explicitly constructed sequence of quadrature rules in the RKHS of the Gaussian kernel with any value of the length-scale parameter. We begin with two simple lemmas.

### Lemma 3.1

### Proof

^{3}Thus

### Lemma 3.2

### Proof

### Theorem 3.1

*N*-point quadrature rule \(Q_N\) satisfy

- 1.
\(\sum _{n=1}^N \left|w_{n}\right| \le W_N\) for some \(W_N \ge 0\);

- 2.
\(Q_N(\varphi _n^\alpha ) = \mu (\varphi _n^\alpha )\) for each \(n = 0,\ldots ,M_N-1\) for some \(M_N \ge 1\);

- 3.
\(\sup _{1\le n \le N}\left|x_{n}\right| \le 2 \sqrt{M_N} / \beta \).

*N*and \(Q_N\), and \(0< \eta < 1\) such that

### Proof

### Remark 3.1

From Lemma 3.2 we observe that the proof does not yield \(\eta < 1\) (for every \(\ell \)) if the assumption \(\sup _{1\le n \le N}\left|x_{n}\right| \le 2 \sqrt{M_N} / \beta \) on placement of the nodes is relaxed by replacing the constant 2 on the right-hand side with \(C > 2\).

*N*-point approximate Gaussian kernel quadrature rule \({{\widetilde{Q}}_{k,N} = \sum _{n=1}^N {\widetilde{w}}_{k,n} f(\tilde{x}_n)}\) whose nodes and weights are defined in Theorem 2.2 and set \(\alpha = 1/\sqrt{2}\). The nodes \(x_n^{\mathrm{GH}}\) of the

*N*-point Gauss–Hermite rule admit the bound [2]

*N*eigenfunctions, \(M_N = N\). Hence the assumption on placement of the nodes in Theorem 3.1 holds. As our numerical experiments indicate that the weights \({\widetilde{w}}_{k,n}\) are positive and \(\sum _{n=1}^N \left|{\widetilde{w}}_{k,n}\right| \rightarrow 1\) as \(N \rightarrow \infty \), it seems that the exponential convergence rate of Theorem 3.1 is valid for \({\widetilde{Q}}_{k,N}\) (as well as for the corresponding kernel quadrature rule \(Q_{k,N}\)) with \(M_N = N\). Naturally, this result is valid whenever the growth of the absolute weight sum is, for example, polynomial in

*N*.

### Theorem 3.2

Another interesting case are the *generalised Gaussian quadrature rules*^{4} for the eigenfunctions. As the eigenfunctions constitute a complete Chebyshev system [17, 30], there exists a quadrature rule \(Q^*_N\) with positive weights \(w_1^*,\ldots ,w_N^*\) such that \(Q_N^*(\varphi _n) = \mu (\varphi _n)\) for every \(n=0,\ldots ,2N-1\) [3]. Appropriate control of the nodes of these quadrature rules would establish an exponential convergence result with the “double rate” \(M_N = 2N\).

## 4 Tensor product rules

*tensor product rule*on the Cartesian grid \({X \,{{:}{=}}\, X_1 \times \cdots \times X_d \subset \mathbb {R}^d}\) is the cubature rule

*d*-variate standard Gaussian measure

### Proposition 4.1

*separable*, this result can be used in constructing kernel cubature rules out of kernel quadrature rules. We consider

*d*-dimensional separable Gaussian kernels

*d*-dimensional kernel cubature rule \(Q_k^d\) at the nodes \(X = X_1 \times \cdots \times X_d\) is a tensor product of the univariate rules:

*i*th kernel are assigned an analogous subscript. Furthermore, use the notation

### Theorem 4.1

### Theorem 4.2

- 1.
\(\sup _{1\le i \le d} \sum _{n=1}^{N_i} \left|w_{n}^i\right| \le W_{\mathscr {N}}\) for some \(W_{\mathscr {N}} \ge 1\);

- 2.
\(Q_{i,N_i}(\varphi _n^\alpha ) = \mu (\varphi _n^\alpha )\) for each \(n = 0,\ldots ,M_{N_i}-1\) and \(i=1,\ldots ,d\) for some \(M_{N_i} \ge 1\);

- 3.
\(\sup _{1\le n \le {N_i}}\left|x_{i,n}\right| \le 2 \sqrt{M_{N_i}} / \beta \) for each \(i=1,\ldots ,d\).

*C*and \(\eta \) appear in Eq. (4.10).

### Proof

A multivariate version of Theorem 3.2 is obvious.

## 5 Numerical experiments

- 1.
Computation of the approximate weights in Eq. (2.11) is numerically stable.

- 2.
The weight approximation is quite accurate, its accuracy increasing with the number of nodes and the length-scale, as predicted in Sect. 2.5.

- 3.
The weights \(w_{k,n}\) and \({\widetilde{w}}_{k,n}\) are positive for every

*N*and \(n=1,\ldots ,N\) and their sums converge to one exponentially in*N*. - 4.
The quadrature rule \({\widetilde{Q}}_{k}\) converges exponentially, as implied by Theorem 3.2 and empirical observations on the behaviour of its weights.

- 5.
In numerical integration of specific functions, the approximate kernel quadrature rule \({\widetilde{Q}}_{k}\) can achieve integration accuracy almost indistinguishable from that of the corresponding Gaussian kernel quadrature rule \(Q_{k}\) and superior to some more traditional alternatives.

### 5.1 Numerical stability and distribution of weights

### 5.2 Accuracy of the weight approximation

*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 Fig. 2 until the first

*N*such that \(\left|1-w_{k,N}/w_{k,1}\right| > 10^{-6}\), ordering of the nodes being from smallest to the largest so that \(w_{k,N} = w_{k,1}\) in absence of numerical errors.

### 5.3 Properties of the weights

### 5.4 Worst-case error

*e*(

*Q*) of a quadrature rule \(Q(f) = \sum _{n=1}^N w_n f(x_n)\) in a reproducing kernel Hilbert space induced by the kernel

*k*is explicitly computable:

### 5.5 Numerical integration

## Footnotes

- 1.
Low-rank approximations (i.e., \(M < N\)) are also possible [12, Section 6.1].

- 2.
- 3.
In particular, the factor \(n^{-1/6}\) could be added on the right-hand side. This would make little difference in convergence analysis of Theorem 3.1.

- 4.
Note that the cited results are for kernels and functions on compact intervals. However, generalisations for the whole real line are possible [15, Chapter VI].

## Notes

### Acknowledgements

Open access funding provided by Aalto University.

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