Sobolev Spaces, Kernels and Discrepancies over Hyperspheres
Authors: Simon Hubbert
Emilio Porcu
Chris J. Oates
Mark Girolami
Venue: Accepted by TMLR ()

Abstract: This work extends analytical foundations for kernel methods beyond the usual Euclidean manifold.  Specifically, we characterise the smoothness of the native spaces (reproducing kernel Hilbert spaces) that are reproduced by geodesically isotropic kernels in the hyperspherical context.  Our results are relevant to several areas of machine learning; we focus on their consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms based on Stein's method.  First, we introduce a characterisation of Sobolev spaces on the $d$-dimensional sphere based on the Fourier--Schoenberg sequences associated with a given kernel.  Such sequences are hard (if not impossible) to compute analytically on $d$-dimensional spheres, but often feasible over Hilbert spheres, where $d = \infty$.  Second, we circumvent this problem by finding a projection operator that allows us to map from Hilbert spheres to finite-dimensional spheres.  Our findings are illustrated for selected parametric families of kernel. 

Published in Transactions on Machine Learning Research (04/2023)
Sobolev Spaces, Kernels and Discrepancies over Hyperspheres
Simon Hubbert
s.hubbert uk
Birkbeck, University of London
Emilio Porcu
emilio.poreukku.ccc..
Khalifa University
Trinity College Dublin
Chris J. Oates
chris. wwr.s.ootee@@ncl.c..oo
Newcastle University
The Alan Turing Institute
Mark Girolami
mag920eng.comm wwm.gg..ccmm........m uk
University of Cambridge The Alan Turing Institute
Reviewed OpenReview: https: htttps://ppenreviaa
openreview. net/ forum? ? id= 82hRt
Abstract
This work extends analytical foundations for kernel methods beyond the usual Euclidean manifold. Specifically, characterise the smoothness of the native spaces (reproducing kernel Hilbert spaces) that are reproduced by geodesically isotropic kernels in the hyper- spherical context. Our results are relevant to several areas of machine learning: we focus on their consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms based on Stein's method. First, we introduce a characterisation of Sobolev spaces on the d-dimensional sphere based the Fourier-Schoenberg sequences associated with given kernel. Such sequences hard (if not impossible) to compute analytically ble over Hilbert spheres, where d = xo. Second, circumvent this problem by finding projection operator that allows us to map from Hilbert spheres to finite-dimensional spheres. Our findings are illustrated for selected parametric families of kernel.
d-dimensional spheres, but often feasi-
1 Introduction
This paper precisely characterises the Hilbert spaces reproduced by certain kernels defined over the hyper- sphere gd, the d-dimensional sphere embedded in [R++1 A kernel K gd x gd + R is said to be geodesically isotropic it depends only on the geodesic distance (denoted throughout) between any pair of points located over the spherical shell. Hence, under geodesic isotropy, there exists a mapping : [0, a] - such that, for all &, n Sd. K(G. n) (f(((E n)). Is there a suitable spectral characterisation of Sobolcv-type kernels, K, under the assumption of geodesic isotropy? The literature elusive this aspect.
Several existing contributions to the literature consider the restriction of isotropic kernels from Rd+1 to sd arguing that under the restriction smoothness propertics are retained. This logic is correct, albeit quite unnatural; the restriction of an isotropic kernel in Id++1 to the embedded sphere Sá has drawbacks that have been emphasised by Gneiting (2013) and by Porcu et al. (2018). For instance, choosing a covariance model which depends on the chordal distance to approximate a random field on the sphere will suffer from poor accuracy at large distances (where the deviation between chordal and geodesic distances are significant). The consequences of this somewhat. difficult to assess theoretically, but intuitively it is unnatural to work

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Published in Transactions on Machine Learning Research (04/2023)
under metrics that are not compatible. To make progress, it becomes necessary to properly define Sobolev spaces for geodesically isotropic kernels over hyperspheres; that is the contribution of this work.
1.1 Motivation: Kernels, Cubature, Discrepancies and Beyond
The applications of reproducing kernels are myriad, but the principal motivation for this work was to provide theoretical foundations for the related computational tools of kernel cubature and kernel discrepancy beyond the usual Euclidean manifold. The simplest and perhaps most useful important of a non-Euclidean manifold is the hypersphere $ which is the focus of our discussion, but aspects of the methods developed in this paper are more generally applicable we return to this point in Section 6
Let K : X' X + R be a reproducing kernel defined on a set X, and let NK denote the Hilbert space that is (uniquely) reproduced by this kernel. Let ((r) denote a unit mass at x X. Given probability distribution P on X' and set of locations {zI,..., Lnn C X, kernel cubature associates to each location 2; a scalar weight W,, such that the kernel discrepancy
T DE
Σ w (..(.:).))
n Σ (xf((x))) - / ap II WEs'
sup
is minimised. These topics have received considerable recent interest in statistics, machine learning, and numerical analysis, where kernel cubature has been applicd to such tasks as sampling (Teymur et al. 2021), experimental design (P (Pronzato and Zhigljavsky 2020), model selection (Briol et al. 2019), ; and numerical integration (Jagadeeswaran and Hickernell 2019). The increasing popularity of kernel cubature is due in part to closed-form expressions for the cubature weights w, and to the fact that their error decay is ratc-optimal among all cubature methods for NK. even when the locations 2; are randomly sampled. Further, these tools have received attention in connection with Stein's 's method from applied probability (Stein) 1972 Gorham and Mackey 2015). where Chwialkowski et al. (2016); Liu et al. (2016) introduced kernel discrepancy that can be computed even when P is implicitly defined up to a normalisation constant, addressing problem that is routinely encountered in the Bayesian context.
Specialising the discussion to X = $4, kernel cubature appears in rendering algorithms for glossy surfaces (Marques et al.
2013 2022) and as a criterion by which the performance of rendering algorithms is measured
(Maqques et al. 2019), while kernel cubature has been used in combination with Stein's method to numer- ically approximate postcrior expectations in directional statistics (Barp et al.] 2022). The theory of kernel cubature on Sd is well-developed (e.g. a special case of the general theory of Novak and Wožniakowski 2008), and optimality properties of kernel cubature have been established in the case where the Hilbert space reproduced by the kernel is equivalent to a Sobolev space on gd (Krieg and Sonnleitner 2021). However, applications of kernel cubature on Sd are limited by the availability of kernels that satisfy thcoretical as- sumptions and are computationally practical. For example, the methodology of Barpp et al.] (2022) requires K to reproduce a Hilbert space equivalent to an order-3 Sobolev space and to admit computable expressions for its derivatives in order for Stein's method to be applied. Further, it is desirable from the perspective of empirical performance for the kernel to be intrinsically defined on Sd,, to properly reffect the geometry of Sd. Tf such kernels can be found, then the Riemann- Stein kernel method of [Barp et al. (2022) facilitates approximation of posterior expectations with error O(n O(dd//), improving on the conventional O(n-1) error of Markov chain Monte Carlo whenever 3 > d/2 (i.c. when the Hobolev-Hölder embedding is well-defined). An improved understanding of Sobolev-type kernels therefore has the potential to eliminate the gap bet ween the theory and practice of kernel cubature and kernel discrepancy on $d and to drive the improvement of methodologies in application areas such as graphics rendering and directional statistics where kernel cubature and kernel discrepancies are used.
Beyond cubature and discrepancies, there is a need for Sobolev kernels within other arcas of machine learning. One such example is the use of kernels to approximate the solution of (partial) differential equations (PDEs), where the solution space of the most commonly encountered elliptical PDEs are Sobolev spaces (see e.g. Fasshauer, 2007]. The use of a Sobolev kernel then ensures that the solution of the PDE belongs to the RKHS and, through the use of an appropriate kernel method, can be consistently approximated (see Fuselier
and Wright, 2009 2012 Hubbert et al.) 2015). The recent. interest. within the machine learning community
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in probabilistic numeries (Hennig et al. 2022), in which kernel methods (Gaussian processes) are used to infer the solution of PDFs, has created renewed demand for theoretical analyses of the kind reported in this work (e.g. Chen et al., 2021 Krämer et al. 2022 Pförtner et al. 2022 A second important application is in learning with spatial data; kernel mothods have been popular since Furrer and Nychka (2007) and are becoming even more popular now, thanks to the fusion of classical spatial statistics with machine learning (Cui a 2019 Nikparvar and Thill, 2021). Authors such as Stein (1999) explicitly advocate for the use of Sobolev kernels in this context, as opposed to smoother kernels whose assumptions are unlikely to be satisfied when analysing real-world dataset.
1.2 Contribution
The present article contributes to mathematical understanding of Sobolev-type kernels on s making in particular the following relevant contributions:
Native and Sobolev Spaces. We start with the native spaces (reproducing kernel Hilbert spaces) associ- ated with kernels that are geodesically isotropic over d-dimensional spheres. We then define suitable Sobolev spaces with exponent 3 for the case of d-dimensional spheres. Using Fourier analysis over spheres, we prove that a given kernel, K, belongs to given Sobolev space with exponent / if and only if the related Fourier- Sccieebbrh sequences (see subsequent sections) have precise rate of decay.
Hilbert Spheres and Projections. For d and given, attaining the Schoeerr-c sequence is extremely difficult. After noting that such sequences are more easily attainable on the (infinite dimensional) Hilbert sphere, we prove that there exists projection operator that relates those sequences on the Hilbert sphere with their analogue on finite dimensional sphere.
Euclidean Kernels and Their Restriction. One potential source of geodesically isotropic kernels, alluded to above, is to simply restrict radially isotropic kernel in the ambient Euclidean space Rd+1 lo the d d-dimensional sphere. If one knows in advance the Fourier transform of the radial kernel then one can employ a formula due to Narcowich and Ward (2002) to derive the corresponding Fouricr-Schou Schriir--choebberg sequence of its restriction to the sphere. We remark that we do not implement the restriction directly (i.e., viewing the problem in Rd-1) but instead we take advantage of the fact that the chordal distance can be written in terms of geodesic distance, so that the resulting kernel geodesically isotropic, thus allowing us perform a more natural type of harmonic analysis on the sphere.
Some Parametric Classes of Kernels. The above ingredients will allow us to prove that celebrated classes of kernels for d-dimensional spheres are actually Sobolev kernels, and can be used within the kernel cubature and kernel discrepancy machinery previously described. A complete Sobolev assessment (derivation of Sourier-Schoenerrg sequences complete with precise asymptotic decay rates) for the class of generalised Wendland functions was provided in Hubbert and Jäger (2021) and the present paper adds to this existing knowledge by providing a complete Sobolev assessment for two additional classes, not yet covered in the literature, namely the Matérn class (tein 1999)
and the class Alegría et al. (2021).
Section provides succinct mathematical background. Section 3 starts with expository material native spaces and then propose definition of Sobolev space with given exponent. sequences determine the Sobolev space where the kernel sits. Section sets out a framework which delivers closed form expressions for Fourier-Schoenerrg sequences derived via the two routes described above, i.e., by projection from the Hilbert sphere and also by restricting Euclidean radial function to the sphere. We conclude in Section 5 by presenting three explicit parametric cases. Using the tools developed Section g we give closed form expressions for the Fourier-Saae Sccirierrcho coefficients, we also provide their asymptotic rates of decay and hence conclude by specifying their corresponding Sobolev spaces. A short discussion in Section closes the paper.
The paper proceeds as follows:
We prove that the
3

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2 Background
Let d be a positive integer. We consider the d-dimensional unit sphere S = {E E R2+1 || = 1}, embedded in Rd-1 with - denoting Euclidean norm. We shall also refer to the Hilbert sphere g = { E R |el| = 1}. We equip gd with the great circle (geodesic) distance, defined as 0(C, 17) = arccos n) E [0, m]. for E. 84, where T denotes transpose. A mapping K : S g > is called kernel if is positive definite, that is Lig= (Ea Ej) 2 0, for all finite system {6.
gd and constants al....0y ON R. This paper works with geodesically isotropic kernels, so that K(E, 7) = o' 7)) for some continuous function & : [0, + with (0) = 1, and for o² O. We define Va as the class continuous functions & being the isotropic part of a kernel in Sd. We also define L = N Vd with the inclusion relation
V 3 2/ - ... C V 3 V 0c1
(2)
being strict. Schoenberg 1942) showed that continuous mapping and only if it can be uniquely writlen
[0, T]
IR belongs to the class V if
Pm p(d-1)/2 (cos(0))
(0)
bm.d
0 E [0, m).
(3)
p.d p(d-11/22 (1)
m=0
where Pa denotes the X-Gegenbaner polynomial of degree m (Abramowitz and Stegun 1965, 22.2.3), and
) o is a probability mass sequence. Schoenberg 1942 also showed that & belongs to the class Yoo if and only if
(0) bm (cos(())) (0)) E [0, T),
(4)
with (bm J being again a probability mass sequence. We follow Daley and Porcu 2013 and call the sequence (Dm.lm==o in (3) d-Schoenberg sequence. Analogously, we call f6m | a Schoenberg sequence. Throughout, for a given d and a given element & E V. C Va, we call ((hm m v) and (4bm.a) ) v/) a Schoenberg and a d-Schoenberg pair, respectively. Section 4 proves that these pairs are related through an operator defined therein.
For d 1, it true that (see Gneiting 2013
10. l' (0)dd
2 l" cos(m0)))(0)d0, for m 1, J
1m,1 and
(5)
while for d 2 2 we have
2m + d - (T (1))
r(d- l- (0()) (:(00)pddd))) (0dd1)/22(oos///// (sin 0) d0.
bmnd
(6)
23-d
m
where I(.) denotes the gamma function Abramowitz and Stegun 1965 6.1.1).
The property of strict positive definiteness is described here through the members a of the classes V. By strict we mean that the inequality in the definition positive definiteness becomes strict provided the real numbers C,, On are not all zero; we let V C V denote the class of continuous functions associated with a strictly positive definite kernel on gd. Arguments in [Schoenberg] (1942) prove that if the elements of the d-Schoenberg sequence 1 in (3) are positive for all m 0 then E w This simple condition is sufficient for our purposes but the reader may consult (D. Chen and Sun 2003) for a careful investigation
of the necessary and sufficient conditions.
2.1 Harmonic Analysis on Spheres
We now consider members & from + and invoke arguments in Hubbert et al. (2015 to dig into alternative view of the expansion 3 Specifically, we resort to Fourier expansion through spherical harmonics, that is
Nm.
(EE m) Σ
Σ
(5))
EnnESt
(7)
m== n=
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where {mm.. = i...., Nm,a) ) is real orthonormal basis for the space of spherical harmonics of degree and the collection (Nm,n n = ,,....,,,, Nm,d, m 2 0} forms a real orthonormal basis for (2*)) In addition, {9 Im-n referred to as the sequence of spherical Fourier coefficients for & E V and these are related the aforementioned d-Schoenberg coefficients via. the formula (ef Hubbert al. 2013 Equation 1.33)
r ( (d+1) Nm.dum 2
6m.d
(8)
2T 211
where Nm, denotes the dimension of the space of spherical harmonics of degree given by
( m
d- =-1)
(m + d 2)!
No.d and Nm.d
(9)
(d - 1)!m!
Remark 2.1. The identity proves that the d-Schoenberg (bm, ) and the Fourier fum J==o sequences are linearly related. This justify the vague terminology Fourier-Schoenberg sequences used in the introduction to this paper.
The positive spherical Fourier coefficients of & E V decay at a polynomial rate if there exist positive constants A1; A2 and y such that
A1 m)d+7 S 4m S
A2 (1+ m)dd7
2 0.
(10)
Using Stirling's formula Abbamooitz and Stegun 1965 6.1.39), that is
I(az + b) 22r " (az)as+b
(11)
we deduce that N. 2(m++1)d-1 (d-1)]- from which one also show that there are posilive constants C₁ Cz. independent of m, such that Ci + 1)d-1 < Nm.a S C2(m+ 1)d-1, m 2 0. Thus, in view of 8 we note
that the decay condition (10) on the spherical Fourier coefficients can be recast in terms the d Schoenberg sequence; specifically, there exist constants A1, such that
A (1+ (1+m) S bm,d
A
m 2 0.
(12)
(1 m)+
2.2 Special Functions
Hypergeometric functions will feature heavily in the course of this work and so we briefly remind the reader that a general hypergeometric function is defined by
[a bi
Op by
(a1), (ap) 2"
F
:
(13)
(bi)n ... (b,)n
where
(c)n oc 1)... (c+ n - Inc+n) for n 2 T(c)
(14)
denotes the Pochhammer symbol, with (c)o 1. Throughout, B denotes the Beta function, defined for 0 and y by
B(r,y) ('' Jo 10 1d r())(()) T(x +
(15)
3 Native Spaces on Spheres
For given 1 E V whose spherical Fourier coefficients we assume to be positive, we define the following subspace of (zd))
No
Fm.nl oc).
No = { E :11 =
(16)
Vm
m=0 x=
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where fm.n denote the expansion coefficients associated to the spherical Fourier series representation
20
of
where Jm. |
We observe that || ll norm induced by the inner-product
Nucal
Im
Im..
(f.g)= =
(17)
m
We shall call N₂ the Native space induced by vh. We observe that if we consider the function ti whose spherical Fourier coefficients are given by
Vm = (1+ (1+m) m))7
then the corresponding Native space coincides with the Sobolev space of order 7, that
Nu.a
W3(("))= = { 12( 1111ll = :
Σ m=
Σ (1 m)) < c0
More generally, if the spherical Fourier coefficients of 1 E V satisfy the decay condition (10 or equivalently, if its d-Scohenberg sequence satisfies (12), then the induced Native space Ny norm equivalent to the Sobolev space W (8d) where 3 = (d + 7)/2. That is the two spaces agree as sets and the norms are equivalent since
V 1lfll & I1lllwa I A2llflle.
In particular, if N2 and Nar are norm-equivalent, then their kernel discrepancies equation [ define the same topology on the space of probability distributions on gd. We observe that since 3 > d/2 then, as a consequence of the Sobolev embedding theorem, the Native space of No continuously embedded in C(Sd). the space of continuous functions on $4, and this implics that N₂. is a reproducing kernel Hilbert space. The following result concerning Native spaces adapted from Levesley and Sun (2005) Proposition 3.1.
Lemma 3.1. Let (E () n) (T n) denote kernel induced by s E W having expansion 7 according to {16 is
a Fourier sequence {Wm) 100 m of strictly posilive coefficients. The correspording Nalive space No reproducing kernel Hilbert space with reproducing kernel w.
Remark 3.1. Let vp E V induce a kernel & as in Lemma 3.1 If the sphericol Fourier coefficients of W 10 then V is a reproducing kernel for space that is equivalent to the Sobolev space W ( where
satisfy
d+7
For two mappings S.g: No - R, we say that ((n) g(n) if and only if
lim f(n) gnn) 1.
(18)
A direct implication of 18 is that there exists positive constants A and Ag such that
Aig(n) < f(m) A2g(n), n n20 0.
In view of Remark 3.1 we observe that by establishing asymptotic decay rates for various classes covariance function will be able to establish which order Sobolev space the covariance kernels reproducing for.

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4 Quantifying Smoothness on d-dimensional Spheres
In this paper consider parametric classes of members & of the class M We access these via the two approaches described in Section 1 Specifically we will either take t E v a starting point and consider its projection to & or we will take positive definite radial kernel in R as starting point, and consider its restriction to $ In both cases we will derive closed form expressions for the associated d-Schoenberg sequences and by examining their asymptotic decay we can quantify the smoothness properties which, in turn, determines whether the induced Native space norm equivalent Sobolev space of certain order.
4.1 Projecting * to V.
Many of the well known parametric classes in numerical analysis and statistics are defined through members of the class W i..., the Schoenberg sequence (bm)m=o is known for the representation 4 This is an obstacle in the case where one wants employ such functions d-Schhonberg sequence in order to quantify the smoothness of the kernel and consequently to state whether the induced Native space is norm equivalent to a Sobolev space of a certain order. In order to circumvent this we consider the following projection operator which we will show maps the Schoenberg sequence dh E 2o unique d Z--cchonberg sequence when viewed as member of Va. To the best of our knowledge, this contribution is novel.
finite dimensional sphere, where one requires the
Remark 4.1 (The Projection Operator). We define the operator Ta that acts pointwise on the coefficients bm from Schoenbery sequence {bm through the identily
V
T (m + - 1)
(m + 2n)!
Vm E No, Ya (bm)) = 2m 2T
Σ
bm+25 j122s (m 1
(19)
m!T (m d=y))
)=
where (x); = -Txx [x) denoles the Pochhammer symbbol Abramoowila and Stegun 1965 6. 6.1.22).
Proposition 4.1 (Projection Operator). Let Ya be as defined through 19 Then, Ta maps V ( ) into Va (W That is, let dmd be defined as bm.d = Y (bm), for m E N and for {bm J a Schoemberg sequence. Then, the sequence (bm,d) is a d-Schoenberg sequence.
Proof. According to Bingham 1973
Lemma 1, the following identity holds
m!l' d=1
(m - 2k + d-1 d-2)(m - 2k + d - 2)
p(d-1)/2 1)/2 (cos(f))
(cos 0)
2
m- 2k
(20)
2m (d 2)
Am(m 2T) (m + 111)
pid-11 1)/2 (1)
0<2k5m
This allows us to deduce that
(0) Σ
byn (0)
m m=0
(21)
r
bmm
(m 2k 1
2k d d-2) 2)1 p(d-1)/2
(cos
m m-
(d 2)
2m
k:(m 2k)l (m
d+1)
pld-13/2 1)/2
(1)
m=0
m
m 24
pld 1)/2 ()/2(()s(() (c))(
By inspection, the coefficient of
is given by
pla 13/2

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D (m=²)) (m + 11))
(m + d - 22)
xc
(m 2j)!
b
2m
2
ml(d 2)
j1220r (m + + 111
j=0
r [
(m + d-1
(m 2)
00
(m + 2n}!
0m+2) j122j (m + 211) j=0
(22)
[(d - 1) P (m + d+1)
2m
m!!
2
V
T (m + -4-1)
(m + 2ÿ)!
Um+2j
2m+d-21 (5) mlD (m
=2"
d-1
j122d
(m
d+1 2
jm0
where the final line follows from T(x + 1) = xI(x) and an application the Gamma function identity (Abramowitz and Stegun 1965 6.1.18),
[(22)
22z-1
( ( 1
r
(23)
[(2)
2
4.2 Restricting Radial Kernels to the Sphere
An alternative source of members of V be accessed by choosing radial kernel 6 that is known to be positive definite on Rd+1 and then defining its restriction to the sphere. Specifically, fixed space dimension and we take a parametric family A), À E θ C [P} of radial functions that are positive definite on Rd+1 The chordal distance on gd is connected to the geodesic distance via
suppose that d is a
dcn(E, n) 11 nl v2 cos ()(
EnES..
(24)
Using this we define
(0 = (V2-2coss
cos (0). (0).
(25)
and, by construction, this restricted family belongs to V.. A crucial ingredient for computing the d d-Schoenberg coefficients of the restricted family is prior knowledge of the d 2-dimensional radial Fourier transform of Φb
Definition 4.1. Let o(-) denote continuous real valued function on [0, oo). The d- dimensional radial Fourier transform of 6 is defined by Stein (1999)
3(r) Fab(r) =r 2=3 I-" 22 r 0,
(26)
where J-(-) denotes the Bessel function of the first kind with order v. We note that sufficient condition for ((r) to be well-defined is that ((t))d-1 is absolutely integrable.
In this framework the d-Schoenberg coelficients associated to members of v that are defined via. 25 is
given by the following formula (cf. Narcowich and Ward 2002) Theorem 4.1)
Um,d d (2T)) Kmndd ( 1J2 m+ (1),(1)dl (1)
(27)
d+1
(2m + 1)(m + d 2)!
where
Km,d
2T d+1
I(d)
ml
In the next section we use the results presented here on Hilbert space projections and on spherical restrictions derive closed form expressions for the d d-Shhoenberg coefficients for different classes of parameterised families belonging to V..
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5 Parameterised Families and Native Sobolev Spaces
The parameterised families under consideration in this paper have been chosen for their flexibility. In each case, one paramcter the family dictates the order of the Sobolev space reproduced by the kernel. There are other examples of kernels in the literature which are known to reproduce Soboley spaces, although these tend be less flexibly parametrised. For instance, in the case of S?, it is shown in Brauchart et al. (2014) that the Cui and Freeden kernel defined by
20
P (cos(()) p (1)
1 - cos(O)
KCF (0)= 1+
2log 1 +
m (m + 1)
2
is a reproducing kernel for space that is norm equivalent to W (8²). This is casily verified here using the decay rate of the expansion coefficients and Remark 3.1 Further, in the same paper it is also shown that the adjusted distance kernel, defined by
Kaist (0) - 2VT V2- - c0s,
is another reproducing kernel for W (S²). This provides an example of 'equivalent' kernels whose cubature rules achieve the same order of convergence. Further examples of kernels on hyperspheres are provided in Minh et al. 20006, including the infinitely smooth spherical Gaussian
σ² (-2-2 cos(0)
) o > 0.
Kaauss (0) cxx cxp
Tt shown that the corresponding d-Schoenberg sequence of this kernel exhibits exponentially fast decay and, consequently, its reproducing kernel Hilbert space is a rather small space of infinitely differentiable functions; we do not consider such cases in this paper, since infinite smoothness is rarely suitable assumption for the cubature applications that motivated this work. However, the analytical techniques in this paper could also be applied to analyse smooth kernels of this kind.
For each of the families of geodesically isotropic kernels presented in this section we will provide:
1. closed form expression of their d d-Schoenberg sequence.
2. The asymptotic rate of decay of their d S-Shoonnberg sequence.
3. The Native Sobolev space for which the kernels are reproducing for.
5.1
The Matérn Class of Functions
For a,a > 0, the Matérn class of functions are well-known positive definite radial kernels defined as Stein 1999
2 r(v) (2)xx²
K. (2):
Mu,a
r r
with Kr, modified Bessel function of the second kind of order // (Abrmmoittz and Stegun 1965 [9.6.22 9.6.22). The Matórn class has been especially popular in spatial statistics after Stein (1999). We consider the restriction of this family to the sphere which we define as
(0, A) = Myal (v22 - -2cos(0),, for A (v, 00)
(28)
In order to apply (27 and derive the d Schoenenggg coefficients associated to the family (0, A) we require prior knowledge the radial Fourier transform of Mu. (r). This given in the following result.
Lemma 5.1. Let and be positive real numbers. The d- d-dimensional radial Fourier transform of the Matérn kernel Mr)(r) is given by
2 (v + Mya = FaMra())
(29)
22"I (v) (2) + +22
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A proof of this result can be found in Stein (1999].
Equipped with the expression for Mu,o (r) we can now employ (27 to derive the d- (c-hcbenn coefficients and investigate their asymptotic decay rate. This leads us to the following result.
Proposition 5.2. Let À (v.a) E [0, 20)2 and consider the spherical Matérn family (0, X) given by
28
Then have that
The d- Schoenbery coefficients are given by
(A) = (2T)² 28T (v 2) T(m - M.mdd
v
1F2
2( I(m + 1' + +d)
+ 1 -
+ v + +d' a²
(30)
2x 41
(-1) Kimdd
m m+3 +
1
+
F2
I(v) 7(m +1 11 (m + 2 (2a)
m - V +1 2m + d' a²
2. Purther,
2 ( 1
(
20 [()) m+
3. The nalive space Nom associaled with the Matérn kernel is reproducing kernel Hilbert space with reproducing kernel L (x. = +b y; A). Furthermore, Nym is norm equivalent to the Sobolev space W (8t) where 3 + 4.
Proof. See Appendix A
5.2 The F-Class of Functions
Recently, Alegría et al. (2021) have proposed the FF family by
(b A) B(a,v++))
B(a,v) 2F a + +ut cos(())) A (T,a,t) (0)
(31)
where 0 E m), B is the Beta function defined by (5) and 2F, is defined through
Proposition 5.3. Lel X =E (TT E R2 denote the parameler vector associaled with 31 The corre- sponding Schoenbery sequence is given by bm. = B(a,u+T)) (T)m(a)
B(a,v) (a + + > 0 m 2 0,
(32)
and consequently (0, belongs to the class ttt.
Proof. This follows from the definition of the hypergeometric aFi the parameters of X are positive.
13 The coefficients are positive since
Equipped with the expression for (A) d-Schoenberg coefficients and investigate their asymptotic decay rate. This leads us to the following result.
employ the projection operator 19 to derive the
Proposition 5.4. Let
(2) (0, x= (T,0, T E {
be the Schoenberg pair for the F-family given in Proposition 5.3 Then,
1. The dl d-Schoenberg sequence (2)) m2) is uniquely determined through
atm a-m+1
mimil
(m.) = Cm,alT, a, u)4Fg
2
2
(33)
atvtrtm atut r
d+1
10

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Published in Transactions on Machine Learning Research (04/2023)
where
a I (m + d d-11) Vi
Cm
(34)
2m+d-2
m + d-1 r (
It true that
d + +v) (A) T( +1++ a(++((((+t) T ((a)T())T(T)
(35)
m 1+2
3. The native space (2) associated with Frav E VJ is reproducing kernel Hilbert space with re- producing kernel F Fx,aw(xy Furthermore, N1r(2) is norm equivalent to the Sobolev space W (8a) where B = V
Proof. See Appendix
5.3 The Generalised Wendland Family
The Generalised Wendland family of radial functions are defined as
1 (ra B(2a, + 1) B(a,v+1) 220 (1 (er)3) 2F
[n (t),((t) (t2 (er) dt
(36)
1 ++ 1i (er)2 rE 0.
v+l
where >>> 0, and the constant multiplier is chosen to ensure Φv.a. (0) Here, (r) := -r) with (a)+ denoting the positive part of the real number We note that the functions in this family are compactly supported, where the parameter controls the size of the supporting interval.
Arguments in Chernih and Hubbert (2014) show that (r),.(r) is positive definite on Rd+1 provided that v z d+2 + a and so, under these conditions, we can define their restriction to the sphere gd via
2
vw(0, A) Weae 2 cos(
A= = E) E E
(37)
where 8 E x]. By construction (/w(0, A) belong to v provided v 2 d+2 restricted functions have been investigated in detail in Hubbert and Jäger (2021 and these findings are summarised in the following result.
+ a.
The propertics of these
2
Theorem 5.5. Let
(A)) (w A)); >= (a,,, R²}
be the d Schoenberg pair for the generalised Wendland family 37
Then,
It is true that
22 +
(A) T(2x + d)B (a + 2: d
ed
(38)
mm + 1-1) (m + d 2)! x
m
+
m + d+1 + a
1
3F2
2
m!!
+
d+1 a
mt1 '22"
2. There crist two positive constants Ai < A2 such that
20+1
20+1 (1 22)22
(1 m)242a I bm.d.((A) -
(39)
3. The native space New associated to thw is a reproducing kernel Hilbert space with reproducing kernel d y, >). Furthermore, New is norm equivalent to the Sobolev space W (Sd) where B a +
11

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Published in Transactions on Machine Learning Research (04/2023)
Proof. The expression for the (c-Schoenberg coefficients of the generalised Wendland functions can be derived from the closed form expression of their spherical Fourier coefficients as computed in Hubbert and Jäger h202 h (Theorem 4.7), together with (8) The tight asymptotic bounds follow from Hubbert and Jäger 22021 p (Thoorem 5.8). Lemma 3.1 shows that the native space New possesses the stated reproducing kernel properties. The norm equivalence New to the Sobolev space of order a + 1 + follows from Remark 3.1
and the decay condition (39) on Dm,d.W())
6 Discussion
This paper provides new tools that allow for a precise identification of the Sobolev space associated with given kernel defined over d-dimensional hypersphere. An immediate consequence of our results is improved understanding of kernel cubature, since once Sobolev space associated to kernel has been identified one can determine the rate of convergence of the associated discrepancy (i.e. the worst-case cubature error), using for example the techniques described in Krieg and Sonnleitner (2021) and the references therein. Our results also extend the applicability of the Riemann- Stein cubature method of Barp et al. (2022) used to accelerate posterior computation in the Bayesian context, since this method requires the Sobolev space associated with a kernel to be precisely identified.
Some further extensions of our results might be possible at the expense of additional effort. For instance, we are confident that the extension of the present work to the case of compact two-point homogeneous spaces would apply mulalis mulandis by replacing the Gegenbauer polynomials in the Schoenberg expansion with Jacobi polynomials. Some other extensions might be more challenging. For instance, we are unaware at the moment of how to characterise Sobolev cases on hyperspheres when the kernels is not isotropic, but axially sy symmetrrc only (Jones] 1963). Another interesting case would be that of product spaces involing the hypersphere with any locally compact group. Finally, we would like to mention that the recent tour de force by Wynne al. (2022) opens for considering the present work in the direction of operator valued kernels.
Acknowledgement
EP was supported by FSU-2021-016 at Khalifa University. CJO was supported by EPSRC [EP/ HMP/WW1950//11 MG supported by Royal Academy of Engineering Research Chair and EPSRC [EP/ T00044411 EP/ RR188433// EP EP/P020720/2 EP RP/R034710/1, EP (P/R004889/1].
A Results Associated to the Matérn Kernel
Here present the proof of the 3 statements of Proposition 5.2
associated to the Matérn kernel.
Proposition 5.2 Statement
Proof. Using
27 and 29 we can write
bm.d.M (A) = (2xx) dill
Rm,dd
/o tJj
m=++ +1 ( (tattt)MMuttt
2 2411 d+tt (v d+1
1.2
m+ 1 (1)
(2xx
Kym,d
T(v)
(1 242 dt
(40)
Kmnd
2d+ IT (v d+1)
000
t.2
m+
m+
dt.
r (v) 2v
+2)
+22)
The following formula is adapted from Prudnikov et al.
1981b) 2.12.32.10
12

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Published in Transactions on Machine Learning Research (04/2023)
00 88 (2(t)
(=² 12)0 dt
0
r [u-
8 2 [(1 + 2p 2p-8) 8)
pt 1-3
2F3
2F3
gi
22p+1-B
T T(++1= 1-)1-1) T 1 p+1 -)
1
p - # 1--1--197 9
p+ 1
1
p- y (p- r (u+
2u+8-2p
1 1+ 1 2 + 4
2F3
24 + 12?
23
(H 1)2
p- + p #
and holds for 3 + 2/ > O, and 2p 1. Sctting 3 = 2, a: p m $ 24) yields
d-1 and p + d41 (where
2
2
1.J2 d-1(t) (t)
(ar2 + 12) 14 1+1 dt
7(m D)T (2v + d)
v+
1F2 1+
220+d P (v 1+1
2
['m + + d)
m + v + d' a²
T(v m)
m + d m
F2
2
(2a)2m 2d [ 1 I (m d+1)
-----1 2m + d a²
We remark that the 2Fs hypergcometric functions from the formula collapse to F2 hypergoometric functions in the application above, this is due to repeated parameter appearing in both case; v + d+1 in the first instance and m d+1 in the second. With this integral computed we can conclude that
2
(m)) = 2T 111
I(2v + d)
l'm v)lind [(m +d) FF2
V 2 + 1 -
(2x)) T (u+ 2
+ d
2x 1
T(v -y-m) m) Kendd 1F2
m7 + d m7 1 v
+
I(v) (m + =+1) (2a)
)
2m + +d' a²
Applying 23
we can write this as
28T (v + 2) I(m - 2)mmddd T(v)a2" T(m +d)
D v d
(m. (2m))
v
m + d a²
2T d+3
(-1) Kmndd
m + d
1
+
Fg
l(v) I'm + w)ll' (m [ (2a)
+ 1 2m
d
a²
as required.
Proposition 5.2 Statements 2 and 3.
Proof. The following result provides the large parameter asymptotic behaviour of a 1122 of the same style the first term in 30 it is adapted from Luke (1699] 7.3(11)
(a),z)
F2
= 1 + Σ
+ +0
(41)
+ m
(b m) + m
m2n+2
where m-b 0 The next result adapted from Luke 1969 7.3(8) and provides the large parameter asymptotic behaviour of F2 of the same style as the second term in 30
13

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Published in Transactions on Machine Learning Research (04/2023)
F2
B + m 2m + X 12 =1 =1+
m
(a + m)
+ +0
(42)
(B + mm)m(2mm + X 1),j!
m
Applying Stirling's formula 11 can deduce that the constant Kmn,d
27
grows asymptotically
m d- 2d 17 T d
Bom,d
(43)
In addition, Stirling's formula also gives the following asymptotics for the Gamma functions involving appearing in (30
T(m - v) T'm + v + d mn244d
and
e
2m
(44)
[(m 1 - 1/I (m + d+1 2
2T m
d-1
m7
Using these asymptotic components in 30 we can deduce that, for large m, we have
2 I' ( 2² T (1) T ( m=+
(-1)"
-
(m)
1+0
2d-1
2am
m
Clearly the second component of the above asymptolic decays al an exponentially fast rate and so, to leading order, we have
2 r (v 5) 22 T ( m++2
b (A)
Lemma 3.1 shows that the native space Noi possesses the stated reproducing kernel properties. The norm
equivalence of Nem to the Sobolev space of order + y follows from Remark 3.1 and the established asymptotic decay rate of (A).
B Results Associated to the F Family
Here we present the proof of the statements of Proposition 5.4 associated to the family.
Proof of Proposition 5.4 Statement
Proof. For brevity we shall write bm,d for bm, in this proof. Applying (2 we have
V 2 (2) b B(a,u+T)
I Im + d - 1)
(2)
B(a,v)
mll (m= + d-1) (a + v + 7 m2 (m + d+1
The following identitics are taken from Prudnikov et al. 1981a) Appendix 1.6
(x)2) 225
1
+
and (z) = (z) (x + m)2g.
(45)
2
Applying these we can show that
B(a,u+T) (T) (2)+42
22j
(attm)
[++++1)
I+m
[+m+1
bm (T, ar
B(o,v) (a + u mm
a+t++tm
(atut+mmt
and
b
a, [m + d - 1)
8
ootm
2
atm+t
2
T+m
2
(xtm+t)
2m+d-2 1' m + d-1
0+1+7+m
aut+ +m++
(m + d+
j!
2
(m, at v) I'm +
V
a+m atmil I+m rim
d+1 ;1 2
2m+d-2 T (m + d-1
r
atutrim
a
2
14

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Published in Transactions on Machine Learning Research (04/2023)
where, in the final line, recognise the infinite series the 1F3 hypergeometric function.
Proposition 5.4 Statements 2 and 3.
Proof. We begin by examining the asymptotic decay of the multiple of the 4F3 hypergeometrie function from 33 i.e. we consider
Vi T(m dd-11 1) v(T, (a)m(a)
Cm,llT,a,v)=
r (3) 2m r (m + )
B(a,v) (a+ +-+++Tp) mmm!!
11m+d-1) + + (v)((T ( 2m T (m +
aa)+((ttt)) VT
) T(m+a +
In the case where m is large apply Stirling's asymptotic formula
to deduce that
Cm,llT,,,u F(v + (a)((++))) VT
(46)
((a)(())((+) r (2)) 1+
We now move on the asymptotic decay of the 1F3. hypergeometric function from formula is taken from Prudnikov et al. (1983) 7.2.3(9)
33
The following
F+1 9+1
B Op :2 3oo Pg
T(3+o) ((3)T
[
t0-2(1 1011--t---- -1)--1 FFg
Cpp
zt
dt.
Pq
Applying this to the a F3 hypergeometric function from 33 with 2 =1, 3= atm+1 that B+0 o a +
and o m+d-a (such
2
d+1) we have that
2
a-m+1
atm Tomm
1+m++1
F3
d+1 atu+r+m atutr-m+13
(47)
I (m + d+1)
m+a-1
m 2a-aa--
m
t)
F
dt
m+a+1
I m-d-o
atvir a+t+r+1
The following identity is taken from
Luke 1969) 7.3(3)
Ft
Opt1 +r by +r
it
t)8
1
dit
n
d2
+ 0
(111))
pm+1 )
2r
pk
k=2
p+1
where g =
by Σ
-T,
di = rr) + Σ b
(48)
1=1 j=1
j=1
j=1
dy = Σ B
and
|arg(1 (1 - t)l < n.
The quantitics 8R.s above depend only on the parameters apt1 and by. For p we can use 48 to write the FF hypergeometric function appearing in the integral 47 follows
a
1 +
T-1 + m a+v+r+1
dit
82,1 + 22
3 F2
2
/
= (1 1) - +o
0
atutr
m
m
2
m3
We can the above to write
a+m+1 a+m r+m s+m+1
F3
+ d+1 atvtrtm atut +m :1
(49)
I' (m= + d+1) n2 m
/d/ 431,2 m²
432,2
1
2
Imdua (0)
I (1)
Im (2)
+0
m m+d
m²
m3
15

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Published in Transactions on Machine Learning Research (04/2023)
where
Im,dvv;a (=)= =
1= +5
+V
m
a
a+1
d
T 2 + a + +) r (
+
= B
+ +
( af + j = 0,
2
2
In the case where m is large apply Stirling's asymptotic formula
11 to deduce that
I,.dvvaaa)) ( +
(m
2
d-1 2
0,,,, and
n Vm.
+v
m+++++
2x
These two results allow to deduce that, when is large, we have the following asymptotic formula.
a+m+l
atm
r+m
++m+11
r [ + v) 2m+d+v
(1+ d1 + (52,1t + )e,2t²)
4B3
2
atvtr-m atvtr-m+ ;1
2 2 2
(50)
m + d+1
2/T m +
m²
2
Bringing
50 and
46
together, we can conclude that
T(p+ + ((a)I(v)I(T)
2 +v) m ml
( 1
bm.d
T [
Lemma 3.1 shows that the native space Noy possesses the stated reproducing kernel properties. The norm equivalence of NNs the Sobolev space of order v+ follows from Remark 3.1 and the established asymptotic decay rate of (A).
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18


Reviews:
[{"review_id": "IHwCssD15I", "reviewer": "Reviewer_GKnX", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "In particular, the authors focus on \"Sobolev-type kernels\", a concept that to the best of this Reviewer's knowledge, is not widely known in the ML community (not even in the kernel community I would say).\n\nThough I have worked with kernels, and understand the RKHS formulation and its role in kernel methods, I had a tough time understanding the objective, the contribution and the novelty of this work. I list a few comments below to support my recommendation\n\n- Reading through the paper, and even though the authors list their contributions in the Introduction, it is hard to identify which parts of the manuscript are still background material and which are novel definitions or results. Perhaps the contributions are Prop 4.1, Lemma 5.1, Prop 5.3-5.4? It seems so but I couldn't confirm it.\n\n- In general, I understand the the authors proved properties for some kernels/spaces, and even connected with existing kernels. I would say that the source of my confusion is that the value of these properties is not explained. Why is it important, for instance, to identify the Sobolev space associated to a given kernel? The authors state that a consequence of this property is its use on kernel cubature, however, there is not evidence shown by the authors of a practical advantage of their proposal\n\n- I would say that the theoretical value of the paper is not clear for the  wider readership of TMLR. If the paper had applications perhaps it would reach a more general audience. \n\n- I would like to clarify that I am not saying that every paper has to have experiments, but in this case the theoretical contribution has not been clearly presented.\n\n- Another piece of evidence that this paper might appeal to a more theoretical audience and it is out of scope in TMLS is that it contains limited references of ML venues. Aside from a couple of IEEE and ICML references, most of them are from applied maths journals. Again, this is not a disadvantage but reveals that the readership of TMLR might not identify the contributions of this work.\n\nMinor: typo on $\\mathbb{S}^\\infty$ in second line of Section 2", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The article is of theoretical nature and it studies RKHSs produced by geodesically isotropic kernels defined over spheres. ", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Because of my appreciation of the paper presented above, it is difficult to suggest specific changes. \n\nI would say that the main point to resolve is to thoroughly explain the impact of the authors' work in the ML community, hopefully through discussion, illustrations and experiments. ", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "VtEheZP6K5", "reviewer": "Reviewer_c7Q3", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Overall the article is well-written and tackles an interesting question in the field of kernel-based cubature. Indeed, an RKHS is defined typically through the associated kernel, and it is not immediately clear how to characterize the smoothness of the functions belonging to such an RKHS based solely on the expression defining the kernel. In practice, one would compare the RKHS to classical functional spaces such as Sobolev spaces for which the smoothness is well-established. Characterizing the smoothness of RKHSs containing functions defined on the hypercube is a classic topic. Extending these results to hyperspheres was achieved for some dot product kernels [1]. In particular, in these works, the Fourier expansion through spherical harmonics is used instead of the classical Fourier expansion. This article extends these results to kernels defined through the geodesic distance. However, I identify two weaknesses in this work:\n- The equivalence between some RKHSs and Sobolev spaces on the hyperspheres is already known in the literature [1]. In particular, cubature rules defined by two 'equivalent' kernels will achieve the same rate of convergence. I believe that the article does not emphasize enough on the benefits we may get from using geodesically isotropic kernels in this context and settles for a brief mention in the introduction ''... The statement is correct, albeit quite unnatural: constructive criticism in Porcu et al. (2018) and Porcu et al. (2021) show that direct constructions on the sphere would be preferable...''. It is preferable to give more insight on the relevance of this class of kernels.\n- The arguments were illustrated for three families of kernels that are equivalent to the same class of functional spaces (Sobolev spaces), altoutgh they seem to be valid for a larger class of kernels. In particular, the study proposed in this work does not cover RKHSs of infinitely smooth functions; see [2] for an example. It would be interesting to harness the potential of Proposition 4.1 by investigating the necessary and sufficient conditions on the Shoenberg sequence so that (12) is satisfied. Such a result would be applicable to a larger class of kernels. \n\n\n\n\n[1] Brauchart, Johann, E. Saff, I. Sloan, and R. Womersley. \"QMC designs: optimal order quasi Monte Carlo integration schemes on the sphere.\" Mathematics of computation 83, no. 290 (2014)\n\n[2] Minh, Ha Quang, Partha Niyogi, and Yuan Yao. \"Mercer’s theorem, feature maps, and smoothing.\" International Conference on Computational Learning Theory. Springer, Berlin, Heidelberg, (2006)\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This article studies the smoothness of reproducing kernel Hilbert spaces associated with geodesically isotropic kernels defined on the sphere. This study is relevant in the field of numerical integration. For instance, the rate of convergence of the worst-case error of integration of some kernel-based Cubature formulas depends on the smoothness of RKHS. The authors showed that the so-called ‘Fourier–Schoenberg sequence’ characterizes a geodesically isotropic kernel. This sequence is usually available for kernels defined on the infinite-dimensional sphere. The authors showed how to ‘project’ this sequence to finite-dimensional hyperspheres. Finally, they illustrated these results in the case of three families of kernels.\n\n", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I would recommend to give more insight on the relevance of geodesically isotropic kernels in the field of numerical integration. \n\nMinor comments:\n\n- After equation 3, it seems that it is enough to assume that $b_{m,d}$ is a summable sequence. Why do we need the assumption that it is a probability mass sequence? \n- In equation 19: the notation is a bit confusing. The operator is defined on the set of sequences, not on the set of scalars.", "broader_impact_concerns": "None ", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "uRwMxvJpdH", "reviewer": "Reviewer_RopF", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\nThe paper is well organized.\nThe main contribution is to close the gap between thoery and practice of kernel cubature and kernel discrepancy on spheres by carefully deriving the mathematical expressions of Fourier-Schoenberg sequences for several  parametric classes of members $\\psi$ .\n\n\nWeaknesses:\nI'm not familiar with the Riemann Stein's method. Perhaps the application of kernel methods on sphere in machine learning is limited, so some discussions on the real application will be helpful to broader readers.\n\nTypos:\npage 2: \"The theory of kernel cubature on is well-developed\", on what.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies the properties of Hilbert spaces defined by certain kernels on sphere. Specifically, the conditions on equivalence between Hilbert spaces and Sobolev spaces are presented. By giving the explicit Fourier-Schoenberg sequences, this works closes the gap of kernels spaces equivalence assumptions in Barp etal. (2022).  Thus, it extends the studies of kernel methods from Euclidean spaces to non-Euclidean space.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "No", "audience_explanation": "", "requested_changes": "More discussions on the applications of kernel methods on sphere will be welcome.", "broader_impact_concerns": "No", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper derives theoretical properties of reproducing kernel Hilbert spaces (RKHS) associated with geodesically isotropic kernels on $\mathbb{S}^{d}$. In particular, it gives a characterization of the smoothness of such RKHS. More precisely, it gives some conditions which implies that a RKHS with geodesically isotropic kernel belongs to the Sobolev space of $\mathbb{S}^d$ of order $\gamma>0$. These results are illustrated through three classes of isotropic kernels. 
Although the paper does not really much detail on potential applications to ML, in my opinion, this may be of interest for further studies and may find natural applications for data defined on hyperspheres.

Before final acceptance, the authors should proofread their paper. I found a few typos. In particular, 

- Eq (16) and other places, $S^{d-1}$
- p3 $ \sigma^2 \psi(\theta(\xi,\eta)$

Decision: Accept (poster)