# Optimal Monte Carlo integration on closed manifolds

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## Abstract

The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with *n* random points decays as \(n^{-1/2}\). However, the re-weighting of random points, as exemplified in the Bayesian Monte Carlo method, can sometimes be used to improve the convergence order. This paper contributes general theoretical results for Sobolev spaces on closed Riemannian manifolds, where we verify that such re-weighting yields optimal approximation rates up to a logarithmic factor. We also provide numerical experiments matching the theoretical results for some Sobolev spaces on the sphere \({\mathbb {S}}^2\) and on the Grassmannian manifold \({\mathcal {G}}_{2,4}\). Our theoretical findings also cover function spaces on more general sets such as the unit ball, the cube, and the simplex.

## Keywords

Bayesian cubature Covering radius Reproducing kernel## 1 Introduction

*n*random points using (1) in reproducing kernel Hilbert spaces does not decay faster than \(n^{-1/2}\), cf. Brauchart et al. (2014), Breger et al. (2018), Hinrichs (2010), Novak and Wozniakowski (2010), Plaskota et al. (2009) and Gräf (2013), proof of Corollary 2.8. To improve the approximation, it has been proposed to re-weight the random points (Briol et al. 2018; Oettershagen 2017; Rasmussen and Ghahramani 2003; Sommariva and Vianello 2006; Ullrich 2017), which is of particular importance when \(\mu \) can only be sampled (Oates et al. 2017) and evaluating

*f*is rather expensive.

That re-weighting of *deterministic* points can lead to optimal convergence order has been known since the pioneering work of Bakhvalov (1959). For Sobolev spaces on the sphere and more generally on compact Riemannian manifolds, there are numerically feasible strategies to select deterministic points and weights matching optimal worst case error rates, cf. Brandolini et al. (2014), Brauchart et al. (2014), Breger et al. (2017), see also Hellekalek et al. (2016), Hinrichs et al. (2016) and Niederreiter (2003).

The use of *random* points avoids the need to manually specify a point set and can potentially lead to simpler algorithms if the geometry of the manifold \({\mathcal {M}}\) is complicated. For random points, it was derived in Briol et al. (2018) that the optimal rate for \([0,1]^d\), the sphere, and quite general domains in \({\mathbb {R}}^d\) can be matched up to a logarithmic factor if the weights are optimized with respect to the underlying reproducing kernel. Decay rates of the worst case integration error for Sobolev spaces of dominating mixed smoothness on the torus and the unit cube were studied in Oettershagen (2017). Gaussian kernel quadrature is studied in Karvonen and Särkkä (2019). Numerical experiments on the Grassmannian manifold were provided in Ehler and Gräf (2017). We refer to Trefethen (2017a, b), for further related results.

The present paper is dedicated to verify that, for Sobolev spaces on closed Riemannian manifolds, random points with optimized weights yield optimal decay rates of the worst case error up to a logarithmic factor. We should point out that we additionally allow for the restriction to nonnegative weights, a desirable property not considered in Briol et al. (2018). Our findings also transfer to functions defined on more general sets such as the *d*-dimensional unit ball and the simplex.

The paper is structured as follows: First, we bound the worst case error by the covering radius of the underlying points. Second, we use estimates on the covering radius of random points from Reznikov and Saff (2015), see also Brauchart et al. (2018) for the sphere, to establish the optimal approximation rate up to a logarithmic factor. Some consequences for the Bayesian Monte Carlo method are then presented. Numerical experiments for the sphere and the Grassmannian manifold are provided that support our theoretical findings. We also discuss the extension to the unit ball, the cube, and the simplex.

## 2 Preliminaries

*d*, endowed with the normalized Riemannian measure \(\mu \) throughout the manuscript. Prototypical examples for \({\mathcal {M}}\) are the sphere and the Grassmannian

*K*as

The present paper is dedicated to the question if and, as the case may be, how much one can actually improve the error rate in (4) when replacing the equal weights \(\frac{1}{n}\) with weights \(\{ w_j\}_{j=1}^n\) that are customized to the random points \(\{x_j\}_{j=1}^n\). From a practical perspective, the methods studied in this paper require that the integrals appearing in (3) can be analytically evaluated.

### Remark 1

## 3 Bounding the worst case error by the covering radius

*s*, \({\mathcal {M}}\), and

*p*.

The worst case error for the optimized weights is upper bounded by the covering radius:

### Theorem 1

Note that the constant in (8) may depend on \({\mathcal {M}}\), *s*, and *p*.

### Remark 2

*n*and \(\{x_j\}_{j=1}^n\), then any weights \(\{\widetilde{ w}^p_j\}_{j=1}^n\subset {\mathbb {R}}\) with

### Proof of Theorem 1

### Remark 3

*n*points in \({\mathcal {M}}\) is lower bounded by \(\gtrsim n^{-1/d}\), which follows from standard volume arguments. If \(\{x_j\}_{j=1}^n\subset {\mathcal {M}}\) are points with asymptotically optimal covering radius, i.e., \(\rho _n\asymp n^{-1/d}\), then Theorem 1 yields the optimal rate for the worst case integration error

Several point sets on \({\mathbb {S}}^2\) with asymptotically optimal covering radius are discussed in Hardin et al. (2016), see quasi-uniform point sequences therein, and see Breger et al. (2018) for general \({\mathcal {M}}\). The covering radius of random points is studied in Brauchart et al. (2018), Oates et al. (2018) and Reznikov and Saff (2015), which leads to almost optimal bounds on the worst case error in the subsequent section. Although we shall consider independent random points, it is noteworthy that it is verified in Oates et al. (2018) that the required estimates on the covering radius still hold for random points arising from a Markov chain instead of being independent. Note also that results related to Theorem 1 are derived in Mhaskar (2018) for more general spaces \({\mathcal {M}}\).

## 4 Consequences for random points

*s*,

*p*, and \({\mathcal {M}}\). Note that if \(\{x_j\}_{j=1}^n\subset {\mathcal {M}}\) are random points, then the weights \(\{{\widehat{w}}^{\ge 0;\, p}_j\}_{j=1}^n\) are random as well. We shall deduce that Theorem 1 implies that the optimal worst case error rate is (almost) matched in these cases:

### Corollary 1

Note that Corollary 1 yields the optimal rate up to the logarithmic factor \(\log (n)^{s/d}\), cf. (9), and that the constant in (10) may depend on *s*, \({\mathcal {M}}\), *p*, and *r*.

### Proof

*r*. Thus, Theorem 1 implies

### Remark 4

Let \(\nu \) be a probability measure on \({\mathcal {M}}\) that is absolutely continuous with respect to \(\mu \) and its density is bounded away from zero, i.e., \(\nu =f \mu \) with \(f(x)\ge c>0\), for all \(x\in {\mathcal {M}}\). Corollary 1 still holds for independent samples from \(\nu \), where the constant in (10) then also depends on *c*. This is due to \(\approx \frac{2}{c}n\) independent samples from \(\nu \) covering \({\mathcal {M}}\) at least as well as *n* independent samples from \(\mu \).

Corollary 1 yields bounds on the moments of the worst case integration error. The results in Reznikov and Saff (2015) also enable us to derive probability estimates:

### Corollary 2

*s*and

*p*, such that,

### Proof

*s*, and

*p*, such that

*s*and multiplying by

*c*yields the desired result with \(c_2:=c{\tilde{c}}^s_2\). \(\square \)

Our bounds address functions in \(H_p^s({\mathcal {M}})\) exclusively. For bounds beyond Sobolev functions in misspecified settings, we refer to Kanagawa et al. (2019) and references therein.

### Remark 5

## 5 Implications for Bayesian Monte Carlo

Our results have consequences for *Bayesian cubature*, cf. Larkin (1972), an integration method whose output is not a scalar but a distribution. Bayesian cubature enables a statistical quantification of integration error, useful in the context of a wider computational work-flow to measure the impact of integration error on subsequent output, cf. Briol et al. (2018) and Cockayne et al. (2017).

Consider a linear topological space \({\mathcal {L}}\) of continuous functions on \({\mathcal {M}}\) such as a reproducing kernel Hilbert space on \({\mathcal {M}}\). The integrand *f* in Bayesian cubature is treated as a Gaussian random process; that is, \(f : {\mathcal {M}} \times {\varOmega }\rightarrow {\mathbb {R}}\), where \(f(\cdot ,\omega ) \in {\mathcal {L}}\) for each \(\omega \in {\varOmega }\), and the random variables \(\omega \mapsto L f(\cdot ,\omega ) \in {\mathcal {L}}\) are (univariate) Gaussian for all continuous linear functionals *L* on \({\mathcal {L}}\), such as integration (\({\mathcal {I}} f = \int _{{\mathcal {M}}} f(x) \mathrm {d} \mu (x)\)) and point evaluation (\(\delta _x f = f(x)\)) operators, cf. Bogachev (1998). The Bayesian approach is then taken, wherein the process *f* is constrained to interpolate the values \(\{(x_j,f(x_j))\}_{j=1}^n\). Formally, this is achieved by conditioning the process on the data provided through the point evaluation operators \(\delta _{x_j}(f) = f(x_j)\), for \(\{x_j\}_{j=1}^n \subset {\mathcal {M}}\). The conditioned process, denoted \(f_n\), is again Gaussian, cf. Bogachev (1998), and as such the linear functional \({\mathcal {I}} f_n\) is a (univariate) Gaussian; this is the output of the Bayesian cubature method. This distribution, defined on the real line, provides statistical uncertainty quantification for the (unknown) true value of the integral.

*kernel mean embedding*of the distribution \(\mu \) (Muandet et al. 2017).

*K*can be viewed as a reproducing kernel. In particular, the Bessel kernel

*Bayesian Monte Carlo*in Rasmussen and Ghahramani (2003). Therefore, our results in Sect. 4 have direct consequences for Bayesian Monte Carlo. Due to Remark 5 within this Bayesian setting, Corollaries 1 and 2 generalize earlier work of Briol et al. (2018) to a general smooth, connected, closed Riemannian manifold.

## 6 Numerical experiments for the sphere and the Grassmannian

*K*is any positive definite kernel on \({\mathcal {M}}\) that reproduces \(H^s({\mathcal {M}})\) with equivalent norms, then

*b*are as in (12) and (13). To provide numerical experiments for Sobolev spaces on the sphere \({\mathbb {S}}^2\subset {\mathbb {R}}^3\) and on the Grassmannian \({\mathcal {G}}_{2,4}\), we shall specify suitable kernels in the following. We shall consider two kernels \(K_1,K_2\) on the sphere \({\mathbb {S}}^2\) and two kernels \(K_3,K_4\) on the Grassmannian \({\mathcal {G}}_{2,4}\).

The numerical results are produced by taking sequences of random points \(\{x_j\}_{j=1}^n\) with increasing cardinality *n*. We compute each of the three worst case errors \({{\,\mathrm{wce}\,}}(\{(x_j,\tfrac{1}{n})\}_{j=1}^{n},{\mathcal {H}}_{K_i})\), \({{\,\mathrm{wce}\,}}(\{(x_j,{\widehat{w}}^{K_i}_j)\}_{j=1}^{n},{\mathcal {H}}_{K_i})\), and \({{\,\mathrm{wce}\,}}(\{(x_j,{\widehat{w}}^{\ge 0;K_i}_j)\}_{j=1}^{n},{\mathcal {H}}_{K_i})\), for \(i=1,\ldots ,4\), and averaged these results over 20 instantiations of the random points. The constrained minimization problem for the latter two quantities is solved by using the Python CVXOPT library. It should be mentioned that numerical experiments on the sphere for the unconstrained optimizer \({\widehat{w}}^{K_1}\) are also contained in Briol et al. (2018).

*n*. Indeed, we see in Fig. 1a that \({{\,\mathrm{wce}\,}}(\{(x_j,\tfrac{1}{n})\}_{j=1}^{n},{\mathcal {H}}_{K_1})\) for random points matches the error rate \(-\,1/2\) predicted by (4) with \(d=2\). When optimizing the weights, we observe the decay rate \(-\,3/4\) for both optimizations, \(\widehat{ w}^{\ge 0;K}\) in (15) and the unconstrained minimizer \({\widehat{w}}^K\). Hence, the numerical results match the rate predicted by the theoretical findings in (9), (10) with \(p=2\) and \(r=1\). The logarithmic factor in (10) is not visible.

For \(K_3\), we observe in Fig. 2a that the random points with equal weights yield decay rate \(-\,1/2\) and optimizing weights leads to \(-\,7/8\) matching the optimal rate in (9), (10) with \(d=4\). In Fig. 2b, it seems that the worst case error for \({\mathcal {H}}_{K_4}\) decays faster than the \(-\,1/2\) rate when optimizing the weights for random points on the Grassmannian \({\mathcal {G}}_{2,4}\) outperforming the case where weights are equal.

## 7 Beyond closed manifolds

*S*is a topological space and \(h:{\mathcal {M}}\rightarrow S\) is Borel measurable and surjective. We endow

*S*with the push-forward measure \(h_*\mu \) defined by \((h_*\mu )(A)=\mu (h^{-1}A)\) for any Borel measurable subset \(A\subset S\). By abusing notation, let \({{\,\mathrm{dist}\,}}_{{\mathcal {M}}}(A,B):=\inf _{a\in A;\, b\in B} {{\,\mathrm{dist}\,}}_{{\mathcal {M}}}(a,b)\) for \(A,B\subset {\mathcal {M}}\), and we put

### Theorem 2

- (a)
\(\# h^{-1}x\) is finite, for all \(x\in S\),

- (b)
\({{\,\mathrm{dist}\,}}_{{\mathcal {M}}}(\{a\},h^{-1}x) \asymp {{\,\mathrm{dist}\,}}_{{\mathcal {M}}} (h^{-1}h(a),h^{-1}x)\), for all \(a\in {\mathcal {M}}\), \(x\in S\).

Note that (16) is a quasi-metric on *S* if the assumptions in Theorem 2 are satisfied, i.e., the conditions of a metric are satisfied except for the triangular inequality that still holds up to a constant factor.

Conventional integration bounds on standard Euclidean domains *S*, see Kanagawa et al. (2019), Proposition 4, for instance, usually require bounded densities. Theorem 2 also applies to standard Euclidean domains that now inherit the measure and the distance from the closed manifold. The measure can now have unbounded density because there is a potential compensation by the induced distance. This shall be observed for the unit ball, the cube, and the simplex in the present section.

### Proof of Theorem 2

### Remark 6

*S*with covering radius \(\rho _n\) are generated by independent random points \(\{z_j\}_{j=1}^n\) with respect to \(\mu \) on \({\mathcal {M}}\) with \(x_j=h(z_j)\), for \(j=1,\ldots ,n\), the observation

*h*yield reasonable function spaces \(H^s_p(S)_h\), distances \({{\,\mathrm{dist}\,}}_{S,h}\), and measures \(h_*\mu \). For instance, if

*h*is also injective with measurable \(h^{-1}\), then \(H^s(S)_h\) is the reproducing kernel Hilbert space with kernel

*h*is not injective. By using the results in Xu (1998, 2001), we shall determine \(H^s(S)_h\) for

*S*being the unit ball \({\mathbb {B}}^d:=\{x\in {\mathbb {R}}^d : \Vert x\Vert \le 1\}\), the cube \([-1,1]^d\), and the simplex \({\varSigma }^d:=\{x\in {\mathbb {R}}^d: x_1,\ldots ,x_d\ge 0;\; \sum _{i=1}^dx_i\le 1\}\).

*d*coordinates, i.e., \(h(x)=(x_1,\ldots ,x_d)\in {\mathbb {B}}^d\). The push-forward measure \(h_*\mu _{{\mathbb {S}}^d}\) on \({\mathbb {B}}^d\) is given by

### Proposition 1

For related results on approximation on \({\mathbb {B}}^d\), we refer to Petrushev and Xu (2008) and references therein.

### Proof

*spherical harmonics of order*\(\ell \), given by the homogeneous harmonic polynomials in \(d+1\) variables of exact total degree \(\ell \) restricted to \({\mathbb {S}}^d\). Each eigenspace \(E_\ell \) associated to \(\lambda _\ell \) splits orthogonally into \(E_\ell = E_\ell ^{(1)} \oplus E_\ell ^{(2)}\), where

### Example 1

### Example 2

*d*-dimensional torus \({\mathbb {T}}^d:={\mathbb {S}}^1\times \ldots \times {\mathbb {S}}^1\) leads to \(h:{\mathbb {T}}^d \rightarrow [-1,1]^d\) defined by

*h*is

### Proposition 2

### Proof

\(\square \)

### Proposition 3

### Proof

### Remark 7

Our Theorem 2 is an elementary way to transfer results from closed manifolds to more general settings. Our treatment of the unit ball, the cube, and the simplex were based on this transfer. The proof of the underlying Theorem 1 is based on results in Filbir and Mhaskar (2010), and we restricted attention to closed manifolds although the setting in Filbir and Mhaskar (2010) is more general. Alternatively, we could have stated our Theorem 1 in more generality and then attempted to check that the technical requirements in Filbir and Mhaskar (2010) hold. For instance, technical requirements for \([-1,1]\) were checked in Coulhon et al. (2012), and the recent work (Kerkyacharian et al. 2019) covers technical details for the unit ball and the simplex.

## 8 Perspectives

Re-weighting techniques for statistical and numerical integration have attracted attention in different disciplines. Partially complementing findings in Briol et al. (2018) and Oettershagen (2017), we have here established that re-weighting random points can yield almost optimal approximation rates of the worst case integration error for isotropic Sobolev spaces on closed Riemannian manifolds. Our results suggest several directions for future work, for instance, allowing for more general spaces \({\mathcal {M}}\), considering other smoothness classes than \(H^s_p({\mathcal {M}})\), considering other types of point processes such as determinantal point processes (Bardenet and Hardy 2016), and replacing the expected worst case error by alternative error functionals such as the average error, cf. Novak and Wozniakowski (2010) and Ritter (2000).

Our results have direct consequences for the Bayesian Monte Carlo method, as indicated in Sect. 5. Indeed, there has been recent interest in exploiting Bayesian cubature in applications including global illumination in computer vision (Marques et al. 2015), signal processing (Prüher and Šimandl 2016), uncertainty quantification (Oettershagen 2017) and Bayesian computation (Briol et al. 2018). The results in this paper justify the use of a *random* point set in these applications, in situations where a deterministic point set would otherwise need to be explicitly constructed.

## Notes

### Acknowledgements

Open access funding provided by University of Vienna. ME and MG were funded by the Vienna Science and Technology Fund (WWTF) through project VRG12-009. CJO was supported by the Lloyd’s Register Foundation program on data-centric engineering at the Alan Turing Institute, UK. This research was supported by the National Science Foundation, USA, under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the above-named funding bodies and research institutions.

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