# Convergence of the reach for a sequence of Gaussian-embedded manifolds

## Abstract

Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifold *M* into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the fundamental smoothness parameters of manifold learning theorems is the reach, or critical radius, of *M*. Roughly speaking, the reach is a measure of a manifold’s departure from convexity, which incorporates both local curvature and global topology. This paper develops limit theory for the reach of a family of random, Gaussian-embedded, manifolds, establishing both almost sure convergence for the global reach, and a fluctuation theory for both it and its local version. The global reach converges to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory.

## Keywords

Gaussian process Manifold Random embedding Critical radius Reach Curvature Asymptotics Fluctuation theory## Mathematics Subject Classification

Primary 60G15 57N35 Secondary 60D05 60G60## 1 Introduction

This paper has two themes to it. One lies in the general area of the geometry of Gaussian processes, or random fields, over general spaces, and is about random embeddings. The second is more topological, and can be seen as putting probability measures on spaces of manifolds, and then studying the behavior of their reach. Both are motivated from recent results in manifold learning.

### 1.1 Gaussian embeddings

*m*-dimensional, compact, smooth manifolds, without boundary, and which will be denoted by

*M*. On

*M*, we define a centered, unit variance, smooth, Gaussian process \(f:\,M\rightarrow {\mathbb R}\), the distribution of which is characterized by its covariance function \(\mathbb {C}:\,M\times M\rightarrow {\mathbb R}\). Taking \(k\ge 1\), we also define a \({\mathbb R}^k\)-valued process

*k*processes in an infinite sequence of i.i.d. copies of

*f*. It is not hard to check (under the mild side requirements that will be made formal later) that (1.1) defines, with probability one, an embedding (i.e. an injective homeomorphism) \(f^k(M)\) of

*M*into \({\mathbb R}^k\) for all \(k\ge 2m+1\), akin to what one would expect from the Whitney embedding theorem. We call this a Gaussian embedding of

*M*.

*self-normalised, Gaussian, embedding*of

*M*.

However, although all of *M*, *f* and the ambient spheres are smooth, it is not so clear how smooth these embeddings are going to be as \(k\rightarrow \infty \). On the one hand, the self-normalisation in (1.2) ensures that \(h^k(M)\) lies in a fixed radius sphere. On the other hand, high-dimensional spheres are strange objects, with surface areas tending to zero as the dimension grows. Thus, given the increasing independence added into the mapping with each new *f* component, it is not at all a priori clear whether the embeddings eventually become rough, and perhaps fractal, or whether there is some sort of strong law behavior that leads to deterministic behavior in the limit. If the latter case is correct (which it is) then an associated fluctuation theory is called for.

The main results of this paper resolve these issues, at least in the framework of the reach of the self-normalised Gaussian embeddings \(h^k(M)\), as \(k\rightarrow \infty \).

### 1.2 Reach

The modern notion of reach seems to have appeared first in the classic paper [9] of Federer, in which he introduced the notion of sets with positive reach and their associated curvatures and curvature measures. Working in an Euclidean setting, Federer was able to include, in a single framework, Steiner’s tube formula for convex sets and Weyl’s tube formula for \(C^2\) smooth submanifolds of \({\mathbb R}^n\). The importance of this framework extended, however, far beyond tube formulae, as it became clear that much of the theory surrounding convex sets could be extended to sets that were, in some sense, locally convex, and that the reach of a set was precisely the way to quantify this property.

To be just a little more precise, but going beyond the Euclidean setting, we start with a smooth manifold *N* embedded in an ambient manifold \(\widehat{N}\). Then the local reach at a point \(x\in N\) is the furthest distance one can travel, along any vector based at *x* but normal to *N* in \(\widehat{N}\), without meeting a similar vector originating at another point in *N*. The (global) reach of *N* is then the infimum of all local reaches. The reach is related to local properties of *N* through its second fundamental form, but also to global structure, since points on *N* that are far apart in a geodesic sense might be quite close in the metric of the ambient space \(\widehat{N}\). The reach of a manifold is also known as its ‘critical radius’ for a good geometrical reason described below. (See the paragraph following (2.3).) However, to avoid any possible confusion, we shall use only the term ‘reach’ throughout this paper.

We shall give precise definitions in the following section, noting for now that beyond its importance in tube formulae and other classical areas of Differential Geometry and Topology, the notion of positive reach has recently begun to play an important role in the literature of Topological Data Analysis (TDA) in general, and manifold learning via algebraic techniques in particular. We shall discuss this briefly at the end of Sect. 2.

### 1.3 Main results and structure of the paper

*N*at the point \(x\in N\), while

*N*. We, however, are interested in the reach of \(h^k(M)\), and the main result of this paper is Theorem 4.3, which states that there is a deterministic function \(\sigma ^2_c(f,x), x\in M\), such that, with probability one, and uniformly in \(x\in M\),

The remainder of the paper is organised as follows: In the following section we have collected some general results about positive reach that were a large part of the motivation for our study. The reader uninterested in motivation can skip all but the definition of reach in Sect. 2.1. The reader interested in knowing more about the history and applications of positive reach is referred to the excellent survey by Thale [25], or Chapter 7 of [6], which discusses reach in the context of TDA.

After the brief Sect. 3 devoted to notations, Sect. 4 defines Gaussian processes on manifolds and associated notions such as the induced metric. It also introduces the constant \(\sigma ^2_c(f)\). Much of this section is a quick summary of the material in [1] needed for this paper, and once this is done we have everything defined well enough to state the main result of the paper.

The real work starts in Sect. 5, in which we develop specific representations for the reach of a \(S^{k-1}\) embedded manifold which form the basis of all that follows. Some of the results here already exist in the literature, and the proofs of these are relegated to an “Appendix”. Some are new and full proofs are given in situ. Sect. 6 lists four lemmas, from which, together with the representation of Sect. 5, the proof of the a.s. convergence in the main theorem follows easily. Following this, Sects. 7–10, which is where the hardest work lies, then prove these lemmas, one at a time. In Sect. 11 we turn to the fluctuation result of (1.5), both proving it and describing the limit process. Two technical appendices complete the paper.

## 2 Critical radius and positive reach

### 2.1 The definition

Throughout the paper our underlying manifold *M* will satisfy the following assumptions:

### Assumption 2.1

*M* is an *m*-dimensional manifold, compact, boundaryless, oriented, \(C^3\), and connected.

Sometimes we shall assume that *M* is associated with a Riemannian metric *g*, and sometimes that it is embedded in a smooth Riemannian manifold \((\widehat{M},{{\widehat{g}}})\). The main example that we shall need for this paper for an embedding space is the unit sphere \({{\widehat{M}}}=S^{k-1}\), but we shall also meet the simple Euclidean case \({{\widehat{M}}}={\mathbb R}^k\) when discussing tube formulae below. In the first example, geodesics are along great circles, and the associated Riemannian distance is measured via angular distance. In the second, the geometry is the standard Euclidean one.

As an aside, we note that all our results could be extended to the case of manifolds with boundary, and even stratified manifolds satisfying the kind of side conditions endemic to [1]. However, then we would also have to suffer through all the heavy notation endemic to [1], which seemed unnecessary, given that our primary motivation was to establish a general principle rather than the most general result possible.

For the main result of the paper, all of the conditions in Assumption 2.1 are required. This is not true for some of the lemmas along the way, but for ease of exposition we shall generally adopt all the conditions throughout the paper. For the fluctuation result, we shall even need that *M* is \(C^6\), and we will add that assumption when needed. Of course, if the majority of the authors were topologists rather than probabilists, we would probably just have assumed that *M* is ‘smooth’ (i.e. \(C^\infty \)) and then not have been concerned with optimal levels of differentiability.

*x*in \(\widehat{M}\), is given by the local diffeomorphism

*x*in the direction \(\eta _X\mathop {=}\limits ^{\Delta }X/\Vert X\Vert \in \ S(T_x\widehat{M})\), the (sphere of) unit tangent vectors at

*x*. The notion of reach is closely related to the radius of the largest ball around the origins in \(T_x\widehat{M}\), \(x\in M\), for which all the exponential maps are, in fact, diffeomorphisms.

*M*in \(\widehat{M}\) at

*x*, in a direction \(\eta \in S(T_x\widehat{M})\), is defined by

*M*which is closer to \(\exp ^{\widehat{M}}_{x}(p\,\eta )\) than

*x*is. The local reach of

*M*in \(\widehat{M}\) at the point

*x*is defined as

*x*, of

*M*in \(\widehat{M}\). Taking an infimum over the entire manifold finally gives the global reach of

*M*in \(\widehat{M}\):

*r*and dimension

*n*over the manifold

*M*, but in such a way that the ball only touches

*M*at a single point. The largest choice of radius that allows this is the critical radius/reach.

The simplest Euclidean example is provided when *M* is a convex set, in which case its reach will be infinite. In fact, infinite reach characterizes these convex sets. If *M* is a sphere, then its reach is equal to its radius. If *M* is the disjoint union of two spheres, the reach is the minimum of the two radii and half of the closest distance between the spheres.

On the other hand, if \({{\widehat{M}}}\) is itself a sphere, and *M* a great circle, then the reach of *M* (in angular coordinates, and as a subset of the ambient sphere) will be \(\pi /2\). In general, the reach of a closed submanifold of a sphere will be no more than \(\pi /2\).

This is all you need to know about reach to skip to Sect. 4 and read the rest of the paper. The rest of this section is motivational.

### 2.2 Medial axis

*M*and its local feature size, notions which have been developed in the Computational Geometry community [3]. Given

*M*embedded in \({{\widehat{M}}}\), define the set

*G*is called the medial axis, and for any \(x\in M\) the local feature size

*s*(

*x*) is the distance of

*x*to the medial axis. It is easy to check that

### 2.3 On tube formulae

As mentioned earlier, the birthplace of the notion of reach is Weyl’s volume of tubes formula, a classical result in Differential Geometry, and an extension of the much earlier Steiner’s tube formula for convex sets in \({\mathbb R}^n\). Interestingly, Weyl’s original paper [27] was motivated by a question raised by Hotelling [11] related to the derivation of confidence regions around regression curves. Both of these papers still make for fascinating (but not easy) reading today, and both generated enormous literatures, one mathematical (e.g. [10]) and one statistical (e.g. [12] and the literature referenced there). For its importance to Probability see, for example, [1] and the references therein.

*M*in \({\widehat{M}} ={\mathbb R}^k\) to be

*k*-dimensional Lebesgue volume, \(\omega _{n}\) denotes the volume of a unit

*n*-dimensional ball, and the \({\mathcal {L}}_j{(M)}\) are the Lipschitz–Killing curvatures of

*M*. These are also known as quermassintegrales, Minkowski, Dehn and Steiner functionals, and intrinsic volumes, although in many of these cases the indexing and normalisations differ. It is worth noting, as Weyl established in what he considered the part of [27] that required more than “what could have been accomplished by any student in a course of calculus”, that these functionals are intrinsic. That is, they are independent of the embedding of

*M*into \({\mathbb R}^k\). (See for example, Lemma 10.5.1 in [1], where this fact is given a probabilistic proof in the notation we use here.)

It is hard to overstate the importance of (2.4), along with its variants for more general ambient spaces. The fact that the formula ceases to hold for \(\rho \) larger than the reach means that all the applications of tube formulae also fail at some point, and it is knowing where this point is that makes the reach such an important parameter of a manifold.

### 2.4 Condition numbers, manifold learning and learning homology

Standard manifold learning scenarios usually start with a ‘cloud’ of points \({\mathcal {X}}=\{x_1,\ldots ,x_n\}\) in some high dimensional space, which are believed to be sampled from an underlying manifold *M* of much lower dimension *m*, with or without additional noise. (Additional noise will mean that the points need not lie on *M* itself, but rather are sampled from some region near *M*.) A classical problem is to construct a set which approximates *M* in a useful fashion. This is a well known problem with a vast literature, and ‘useful’ here is usually taken to be mean physical closeness in some norm.

More recently, a new literature has appeared, motivated by ideas from Algebraic Topology, in which the aim of physical closeness is replaced with the aim of correctly recovering the topology of *M*. Two of the earliest papers in this area are [18, 19] (but see also [8]) and it is these papers that were in fact the original motivators of the current one.

*M*, and the recovery method—or at least the theorems describing its properties—relies on knowing the reach \(\tau \) of

*M*. In this case, choosing an \(\varepsilon \in (0, \tau /2)\), the simple union of \(\varepsilon \)-balls centered at the points of \({\mathcal {X}}\) is chosen as the estimator of

*M*. That is

*M*, then

*M*is a deformation retract of \(M_{estimate}\), and so both sets have the same homology. For the second stage, it shows that if a large enough sample is taken then one can bound, from below, the probability of the sample being dense enough. The final result is that, for all small \(\delta \), if

*M*with probability at least \(1-\delta \). A corresponding result for the case of sampling with noise is given in [18].

We have brought the above equations to show, explicitly, how the reach appears in the complexity of this estimation problem. The smaller the reach, is, the smaller one is forced to take \(\varepsilon \), and the larger the sample size *n* needs to be for a given estimation accuracy.

Of course, for a given problem, one does not know what *M* is, and so, a fortiori, little is known about its reach. Consequently, in the spirit of Smale’s two step procedure, we need to enrich the second stage by also averaging over possible *M*. The current paper is a step in this direction, by formulating a class of random Gaussian manifolds and beginning a study of their reach.

Moreover, the main result of the paper has an immediate application in the manifold learning situation. Although Theorem 4.3 relates only to a very specific random embedding of *M* into a high dimensional sphere, a liberal interpretation of the theorem implies that the part of the complexity of the estimation problem depending on the reach is more or less independent of any embedding of *M* into a higher dimensional space. The import of this is that there is no ‘curse of dimensionality’, related to reach, that involves the dimension of the ambient space.

Of course, we can only make these claims for the Gaussian-embedding that we study, but the fact that they are proven in the Gaussian case will alleviate concerns among practitioners, in general, that ambient dimension has an effect on reach. This was not known until now, even for a special case.

A second practical implication of this paper is the introduction, albeit implicit, of a new class of smooth random manifolds that are both reasonable and mathematically tractable. Recalling the two stage paradigm of Smale above, it would be interesting, and probably useful, to introduce into the TDA setting the notion of Bayesian optimization. In terms of the above homology learning example, by this we mean minimizing not the probability of correctly identifying the homology for a fixed (but unknown) *M*, but rather minimizing the expectation of some cost function of this probability, averaged over a (random) family of possible *M*. The calculations of the current paper, along with those of [15] which address issues of the asymptotic isometry of Gaussian-embedded Riemannian manifolds, show that the model introduced here allows for tractable mathematical manipulation.

## 3 Some (standard) notation

Many of our proofs, and even some of our definitions, freely use standard notation from Differential Geometry. Since we expect that not all readers will be familiar with this, we include here a brief notational guide. There are many standard texts to which one can turn for details. Lee’s book [17] is our favourite, but the quick and dirty treatment in Chapter 7 of [1] also suffices.

We are working with a Riemannian manifold (*M*, *g*), for which the Riemannian metric is, for each \(x\in M\), an inner product \(g_x\) on the tangent space \(T_xM\) to *M* at *x*. If \(\{(U_\alpha ,\phi _\alpha )\}_\alpha \) is an atlas for *M*, then for each chart \((U_\alpha ,\phi _\alpha )\) we shall often need a (local) orthonormal frame field \(X^{\alpha }=\{X_1^\alpha ,\dots ,X^\alpha _m\}\) for the tangent bundle \(T^\alpha M\mathop {=}\limits ^{\Delta }\{T_xM,\ x\in U_\alpha \}\), where orthonormality is in the metric *g*. We shall refer to this later as “choosing an orthonormal frame field”, without specific reference to charts or the index \(\alpha \). Since all our later definitions and calculations are local (i.e. can be carried out in terms of local charts) this is not a problem (and global issues such as parallelizability do not arise.)

*F*in direction \(X_x\) at

*x*. At various times we will make use of all of these possible notations, so as to make individual formulae either clear and/or compact.

*F*is the unique continuous vector field on

*M*such that

*X*. If

*F*is a function of more than one parameter, say

*F*(

*x*,

*y*), then we will denote the gradient with respect to

*x*as \(\nabla _xF(x,y)\), etc.

*M*,

*g*).

It is standard that \(\nabla ^2F\) could also have been defined to be \(\nabla (\nabla F)\), which is from where the notation comes. Recall that in the simple Euclidean case the Hessian is typically considered to be the \(N\times N\) matrix \(H_F=(\partial ^2 F/\partial x_i\partial x_j)_{i,j=1}^N\). In the more general setting above, \(H_F\) defines the two-form by setting \(\nabla ^2f(X,Y)= XH_FY'\). In this case (3.2) follows from simple calculus.

We shall need the obvious, but important, fact that if *x* is a critical point of *F* (i.e. \(\nabla F(x)=0\)) then \(XF(x)=0\) for all \(X\in T_xM\) and so by (3.2) it follows that \( \nabla ^2F(X,Y)(x)= XYF(x)\). Consequently, at these points, the Hessian is independent of the metric *g*.

This concludes our brief excursion into notation. We can now turn to some details on Gaussian fields on manifolds before stating our main results.

## 4 Gaussian processes on manifolds, and the main theorem

As mentioned earlier, our basic reference for Gaussian processes is [1]. Here we shall only give the very minimum in definitions and notation needed for this paper.

### 4.1 Gaussian processes on Riemannian manifolds

We start, as usual, with a \(C^3\) compact manifold *M*, with or without an associated Riemannian metric *g*. (For the novice, Sect. 3 explains these terms and some of the following notation.)

*T*(

*M*) of

*M*defined by

*X*,

*Y*are vector fields with values \(X_x,Y_x\) in the tangent space \(T_x M\). We shall assume throughout that \(\mathbb {C}\) is positive definite on \(M\times M\), from which it follows that

*g*is a well defined Riemannian metric, which we call the

*metric induced by f*.

From now on, we shall make one of two—essentially complementary—assumptions:

### Assumption 4.1

If, in the above setting, we are given a manifold *M* as in Assumption 2.1 and a Gaussian process \(f:\,M\rightarrow {\mathbb R}\), but no metric on *M*, we shall assume that *M* is endowed with the metric induced by *f*.

If, on the other hand, we start with a Riemannian manifold (*M*, *g*), then we shall choose a Gaussian process in such a way that the metric induced by (4.1) is precisely *g*.

The fact that, given a metric *g*, there exists a Gaussian process inducing this metric, is a consequence of the Nash embedding theorem (cf. proof of Theorem 12.6.1 in [1]). This assumption is crucial to all that follows, and there is no known general topological or geometric theory for Gaussian processes when the metric on *M* is not the one induced by the process.

The only additional assumptions that we require relate to smoothness and non-degeneracy for *f*, but for this we need some notation. Thus we write, from now on, \(\nabla \) for the Levi-Civita connection of (*M*, *g*), and \(\nabla ^2\) for the corresponding covariant Hessian. Fix an orthonormal (with respect to *g*) frame field \(E=(E_1,\dots ,E_m)\) in *T*(*M*). The specific choice of *E* is not important.

### Assumption 4.2

*M*, and, for each \(x\in M\), the joint distributions of the \((1+m+m(m+1)/2)\)-dimensional random vector

We shall also assume that \(\mathbb {E}\{f^2(x)\}\), the variance of *f*, is constant, and for convenience, we take the constant to be one. No other homogeneity assumptions are required.

*f*guaranteeing the differentiability requirements of the assumption are implicit in Corollary 11.3.2 of [1]. If

*M*is a Euclidean domain with smooth boundary, then these conditions require that the various sixth-order partial derivatives of the covariance function satisfy

*K*and \(\alpha >0\), and where \(C^{(6)}\) is a generic sixth-order partial derivative of the covariance function. For a general manifold corresponding conditions in terms of charts and atlases suffice. See Chapter 12 of [1] for details.

The degeneracy conditions (4.2) are close to trivial, but important. Together with smoothness, they ensure that the sample paths of *f* are a.s. Morse over *M*.

As an aside regarding Assumption 4.2, we note that the requirement that \(f\in C^3(M)\) can probably be done away with. It arises as a side issue in a tightness argument in Sect. 9.3, which requires a uniform bound on increments of fourth order derivatives of \(\mathbb {C}\). A (much) more complicated argument would probably require only that \(f\in C^{2+\epsilon }\) for some \(\epsilon >0\), but rather than lose sight of the forest for the trees we are happy to live with the extra smoothness. In fact, in order to prove the fluctuation result (1.5), we shall even have to assume that \(f\in C^6(M)\). We shall explain how the need for these high levels of smoothness arise below, when we have the requisite notation.

### 4.2 The parameter \(\sigma ^2_c(f)\)

*f*, then the limit behaves well. For example, just to be certain that \(\lim _{y\rightarrow x} f^x(y)\) is well defined requires that \(f\in C^2\).

(In fact, ratios of the 0 / 0 nature appear throughout the proofs, with denominators such as \(1-\mathbb {C}(x,y)\) (as above), \(1-\mathbb {C}^2 (x,y)\), and even \((1-\mathbb {C}^2 (x,y))^2\), all of which are problematic as \(y\rightarrow x\). For the a.s. convergence of (1.4), this leads to the requirement that \(f\in C^3(M)\). For the fluctuation result (1.5) we will even need to assume that \(f\in C^6(M)\). While these conditions seem, at first, rather severe, they seem to be necessary and not just a consequence of our method of proof. For example, as far as the \(C^3\) requirement is concerned, it comes from the fact that local reach involves curvature, and so second order derivatives. However, the definition of \(f^x\) itself involves \(\nabla f\), leading to the requirement for three derivatives for *f* (or at least slightly more than \(C^2\)). For details, see Sect. 9.4 and, in particular, the proof of Lemma 12.2).

Associated with the Gaussian process *f* are a reproducing kernel Hilbert space, *H*, and an \(L_2\) space, \(\mathcal H\), which is the completion of the span of *f* over *M*. Writing *S*(*H*) and \(S(\mathcal H)\) for the unit spheres of these spaces, there is an isometry, \(\Psi \) between *M*, when given the metric *g* induced by *f*, and the embedded submanifold \(\Psi (M)\subset S(\mathcal H)\), determined by \(\Psi (x)=f(x)\), for all \(x\in M\). There are also isometries between \(S(\mathcal H)\) and *S*(*H*), and so between *M* and a corresponding subset of *S*(*H*), the details of which can be found, for example, in Chapter 3 of [1], but which date back to the earliest history of Gaussian processes.

It turns out that \(\sigma ^2_c(f,x)\) is precisely the cotangent squared of the local reach of \(\Psi (M)\) at the point *f*(*x*), when \(S(\mathcal H)\) is considered as a submanifold of \(\mathcal H\). It follows immediately that \(\sigma ^2_c(f)\) is the cotangent squared of the corresponding global reach. Similar statements can be made about the isometric embedding of *M* into *S*(*H*), but would take longer to explain. The bottom line, however, is that both \(\sigma ^2_c(f,\cdot )\) and \(\sigma ^2_c(f)\) are basic quantities inherently connected to (*M*, *g*) when it is viewed via isometric embeddings into larger spaces, and that there is a lot of Hilbert sphere geometry lying behind the asymptotics of this paper.

These observations are relatively recent. In our current notation, they can be found in Section 14.4.3 of [1], but see also [22] and the references therein.

*f*, defined by

*u*, the difference

*M*, as well as some minor side conditions on both

*M*and

*f*. The definition of \(\sigma ^2_c(f)\) is also correspondingly changed. See, for example, [23] for a discussion of why local convexity is required. In fact, what is required is close to positive reach, and the reason that (4.7) fails for zero reach is much the same reason that tube formulae fail. But that is another story.) In our setting, it is now known that

### 4.3 Main result

### Theorem 4.3

*M*be a manifold satisfying Assumption 2.1, and let \(f:\,M\rightarrow {\mathbb R}\) be a Gaussian process satisfying Assumptions 4.1 and 4.2. Assume that \(\sigma ^2_c(f)\), as defined by (4.6), is finite. Consider the embedding (1.2) of

*M*into the unit sphere in \(\mathbb {R}^k\), and let \(\theta _k\) be the global reach of the random manifold \(h^k(M)\). Then, with probability one,

*M*is \(C^6\), and the sample paths of

*f*are a.s. \(C^6\) on

*M*, then there exists a sequence \(\bar{\gamma }_k\) of random processes from \(M\rightarrow {\mathbb R}\), such that, for all \(x\in M\),

*M*with supremum norm, and

We defer all further discussion of the fluctuation result of (4.11) and (4.12) until Sect. 11, where it will be restated as Theorem 11.1, and the (rather involved) definition of the process \(\gamma \) will appear. Until then we shall concentrate on the a.s. convergence of (4.10).

As an aside, note that a variation of some of the easier arguments in the following sections show that the sequence of mappings \(h^k\) tends, with probability one, to an isometry, in the sense that the associated pullbacks to *M* of the usual metric on \(S^{k-1}\) tend to the induced metric (4.1) on *M*. We provide a rigorous treatment of this result in [15], albeit with the self-normalisation of (1.2) replaced by a \(\sqrt{k}\) normalisation. We also prove there that this gives rise to the a.s. convergence of a class of intrinsic functionals of \(h^k(M)\) to the corresponding functionals on (*M*, *g*). We refer you to [15] for details.

## 5 Computation of the reach

This section contains two purely geometric lemmas from which follow the probabilistic computations that make up most of the paper. The first gives a characterisation of the reach of general submanifolds of spheres, and the second does the same for the specific submanifolds \(h^k(M)\) in terms of the functions \(f^k\). To start, recall that geodesic distance on the sphere is measured in terms of angles, \(r\in [0,\pi )\). Let *M* be a submanifold of \(S^{k-1}\), and \(\eta _x\) a unit normal vector at \(x\in M\).

We can now state the following characterisation, which implicitly assumes, as we shall from now on, that *M* has dimension at least one. As stated it is identical to Lemma 2.1 of [22], restricted to our setting. ([22] treats the more general setting of stratified manifolds.). Furthermore, as pointed out there, the proof is essentially the same as the proof given in [12] for the one-dimensional case. Nevertheless, because of its importance to this paper, and (only) for the sake of completeness, we give the proof in “Appendix 1”.

An important point to note for the statement of this lemma is that since the manifold *M* is embedded in \(\mathbb {R}^k\), we can, and do, treat all tangent spaces \(T_xM\) as affine subspaces of \(\mathbb {R}^k\), with origin at *x*.

### Lemma 5.1

*M*be a submanifold of \(S^{k-1}\), satisfying the conditions of Assumption 2.1. Let \(T^{\perp }_x M \subset T_x S^{k-1} \subset T_x\mathbb {R}^k\) be the normal space of

*M*at

*x*as it sits in \(S^{k-1}\), viz. the affine subspace of \(\mathbb {R}^k\) which is the orthogonal complement of \(\text {span}(T_x M\oplus \{x\})\subset T_x\mathbb {R}^k\) in \(T_x\mathbb {R}^k\). Then the reach, \(\theta (x)\), at

*x*is given by

We are now in a position to derive the global reach of our random manifold \(h^k(M)\). The result is given in the next lemma. However, before stating and proving the lemma, we need some preparatory definitions.

*M*, and define the \(k\times (m+1)\) matrix

*k*is large enough (\(k\ge 2m+1\)). By the independence of the \(f_j\) and the non-degeneracy of Assumption 4.2, the rows of \(L_x\) are a.s. linearly independent, and so \(L_x^ {T}L_x\) is a.s. invertible. The matrix \(P_x\) orthogonally projects vectors in \({\mathbb R}^k\) onto

*k*samples of the correlation coefficient between

*f*(

*x*) and

*f*(

*y*); viz.

*k*i.i.d. realizations of it at

*y*by

### Lemma 5.2

*M*be a manifold satisfying the conditions of Assumption 2.1, embedded into \(S^{k-1}\) via the embedding map defined in (1.2). Assume that

*f*satisfies Assumptions 4.1 and 4.2. Then, with probability one, the reach of \(h^k(M)\) is given by

### Proof

*f*is centered Gaussian, its derivatives are also centered Gaussians. Furthermore, setting

## 6 Four key lemmas and the proof of the main theorem

The proof of Theorem 4.3 follows from the four lemmas stated below and is given at the end of this section. Throughout this section we shall assume, without further comment, that *M* satisfies the conditions of Assumption 2.1. The conditions on *f* vary, since not all the lemmas require the same level of smoothness. All the conditions, however, are implied by Assumptions 4.1 and 4.2.

We start by showing that the first two terms in (5.5) converge uniformly, with probability one, to 1.

### Lemma 6.1

*M*, with i.i.d. components, each a centered, unit variance Gaussian process over

*M*, with a.s. continuous sample paths. Then, with probability one,

### Lemma 6.2

The third lemma (after some trivial calculations) will—see below—give us that the remaining term in (5.5) converges to the parameter \(\sigma ^2_c(f)\).

### Lemma 6.3

It will follow from the proof of this lemma that \(f^x(y)\) is bounded even when we are arbitrarily close to diag(\(M\times M\)). This is needed to ensure that all the terms defined in (5.5) are, a priori, well defined.

The final step we need is the following.

### Lemma 6.4

We now show how to prove the main result as a straightforward consequence of the previous four lemmas.

### Proof of Theorem 4.3

## 7 Proof of Lemma 6.1

### Theorem 7.1

*X*be a Borel random variable with values in a separable Banach space

*B*with norm \(\Vert \cdot \Vert _{\text {B}}\). Let \(S_n\) be the partial sum of

*n*i.i.d. realizations of

*X*. Then,

*B*is

*C*(

*M*) (continuous functions over

*M*), equipped with the sup norm. The mean zero condition is trivial. To check the moment condition on the norm of \((f)^2-1\), note that

## 8 Proof of Lemma 6.3

*f*.)

### 8.1 The limit (8.1) is well defined

The proof is basically an application of L’Hôpital’s rule. To start, we take an orthonormal frame field \(X=\{X_1,\dots ,X_m\}\) for the tangent bundle of *M*, where orthonormality is in the induced metric *g* of (4.1) and with the conventions described in Sect. 3.

*c*be a \(C^2\) curve in

*M*such that

*f*. Thus, to find the true limit, another application of L’Hôpital’s rule is necessary, and so we have

*y*to

*x*, is also well defined. As a consequence, we also have that, for each finite

*k*,

### 8.2 Completing the proof

We now turn to the proof of the lemma, establishing (6.1).

This, however, follows exactly along the lines of the proof of Lemma 6.1, again applying Theorem 7.1. We need only take as our Banach space \(C_b(M^{*})\), the bounded, continuous functions on \(M^{*}\) with supremum norm, and as our basic random variable \(X=(f^{x}(y))^2-\text {Var}\left( f^x(y)\right) \).

The previous subsection establishes the a.s. boundedness of *X* needed to make the argument work.

## 9 Proof of Lemma 6.2

*k*, there is no easy way to find a uniform bound on the ratio, since both numerator and denominator tend to zero.

### 9.1 Outline of the proof

*x*and

*y*, refrain from writing out explicitly the summation indices and their range in some situations where they are obvious, and also introduce

### 9.2 Fi-di convergence of \({\widetilde{\zeta }}_k\), and characterising the limit

The main result of this section is the following.

### Lemma 9.1

The proof will rely on the following result of Anderson.

### Theorem 9.2

([4], Theorem 4.2.3) Let \(\{U(k)\}\) be a sequence of *d*-component random vectors and *b* a fixed vector such that \(\sqrt{k}(U(k)-b)\) has the limiting distribution \(\mathcal {N}(0,T)\) as \(k\rightarrow \infty \). Let *g*(*u*) be a vector-valued function of *u* such that each component \(g_j(u)\) has a nonzero differential at \(u=b\), and let \(\psi _b\) be the matrix with (*i*, *j*)-th component \(({\partial g_j(u)}/{\partial u_i})|_{u=b}\). Then \(\sqrt{k}(g(U(k))-g(b))\) has the limiting distribution \(\mathcal {N}(0,\psi _b^\prime T\psi _b)\).

Proof of Lemma 9.1 As one might guess from the complicated form of (9.9) the calculations involved are somewhat tedious, and so we shall concentrate on making the main steps clear.

*U*are the maximum likelihood estimators of the corresponding elements of

*b*. It then follows from standard estimation theory (e.g. [4], Theorem 3.4.4) that \(\sqrt{k}(U(k)-b)\) has a limiting normal distribution with mean 0 and some covariance matrix

*T*, the specific structure of which does not concern us at the moment.

*U*above it is easy to relate the \(\widehat{\mathbb {C}}\)s to the \(\bar{C}\)s, and if we now define a function \(g:\,\mathbb {R}^{3n}\rightarrow \mathbb {R}^n\) by

The first is the condition on the differential required by Theorem 9.2, but this is trivial. The second is to derive the exact form (9.9) of the limiting covariances, which, while not intrinsically hard, is a long and tedious calculation. The calculation starts by writing out the covariance function for \(\widehat{\mathbb {C}}\) and computing moments, all of which involve Gaussian variables. Fortunately, most of the detailed calculations that we need were carried out long ago in the statistical literature and, can be found, for example, in [13] [e.g. Chapter 41, Example 41.6]. What remains is to send \(k\rightarrow \infty \) in these expressions, and deduce (9.9). We shall not go through the tedious details here. \(\square \)

### 9.3 Tightness of \( {\widetilde{\zeta }}_k\)

*m*and \(4m^2\) components of the Riemannian differential and Hessian, respectively.

#### 9.3.1 \(\alpha _k\) converges weakly in \({B^{(2)}}\)

To this end, note that the summands in (9.13) are i.i.d. copies of the random function \(f\otimes f:\,M\times M\rightarrow {\mathbb R}\) defined by \((f\otimes f) (x,y) = f(x)f(y)\). If we endow \(M\times M\) with the topology induced by the Riemannian distance \(d_{M\times M}\) (this is the metric we use in place of the semi-metric in the theorem from [26]), then \(M\times M\) is compact in this topology. Since the convergence of the finite dimensional distributions of (9.13) follows from Theorem 9.2, all that is left to check for weak convergence of (9.13) is tightness.

In order to show tightness, we first need to set up some notation, in particular for Taylor expansions on \(M\times M\), in terms of Riemannian normal coordinates.

*M*, and take \(U_1\times U_2\) around \((x_0,y_0)\) in \(M\times M\). Then, \((x^i:y^i)\) give us the following definition of coordinates in the product space \(U_1\times U_2\):

*v*in the product tangent space splits uniquely as the sum of \(v_1 \in T_{x_0}U_1\) and \(v_2 \in T_{y_0}U_2\). This further gives us the following definition for the exponential map in \(M\times M\):

*v*in this space can be written as

*v*is given in Riemannian normal coordinates by \(t(v^1\cdots v^{2m})\), geodesics are locally minimizing, and so along with the previous fact, we have \(\Vert v\Vert ^2=d^2_{M\times M}((x,y),(x_0,y_0))\). Also, importantly, since the Christoffel symbols vanish at the centers of the normal charts, covariant derivatives at the centers reduce to usual partial derivatives. Therefore, working with normal coordinates is useful in local calculations.

*K*. The expectation here is bounded above by

*f*has unit variance, and, for a differential operator

*D*of any order, writing \(D\mathbb {C}^{x_0y_0}\) for \(D\mathbb {C}^{xy}|_{(x_0,y_0)}\), we have that it is equal to

*O*(1) and \(O(\Vert v\Vert )\) cancel. We check this thoroughly below. The second order terms can be trivially bounded using the facts that \(f\in C^3\) and \(|v_i|\le \Vert v\Vert .\) This technique of bounding gives the required constant

*K*independent of the points, but does not offer too much insight. Consequently, we illustrate how this can be done for one case only.

*M*:

*M*, and with probability one, to 1, we have (e.g. [5] [Theorem 4.4]) the joint weak convergence of the pair

#### 9.3.2 \(\Delta _k\) converges weakly in \({B^{(2)}}\)

*k*, bound \(\Gamma _k\) from above and below by two sequences of processes, which converge to the same limit. These bounds [cf. (9.21) below] involve a common term, the weak convergence of which is known, and a smaller term, which converges a.s. and uniformly to zero.

The bound depends on the following algebraic inequality, due to Cartwright and Field [7].

### Theorem 9.3

*a*,

*b*] \((0<a<b)\), whose arithmetic and geometric means are denoted by \(AM_w\) and \(GM_w\), respectively. Then,

This completes the proof of the weak convergence of \(\Delta _k\).

### 9.4 The continuous mapping argument

Recall that \({\widetilde{\zeta }}\) is at least \(C^2\) over \(M\times M\) because of weak convergence in \({B^{(2)}}\). The same (and more) is true for the covariance function \(\mathbb {C}\), so the issue of continuity of *H* is trivial if we restrict \(\gamma \) to a region away from the diagonal of \(M\times M\).

*H*, and thus complete the proof of Lemma 6.2.

*c*in

*M*such that

*y*approaches

*x*along \(X_x\),

However, all terms here are well defined, finite, and non-zero with probability one, so we are done.

## 10 Proof of Lemma 6.4

*k*i.i.d. copies of this process can, asymptotically, be a.s. bounded by \(C\sqrt{\log k}\) for some finite

*C*.

*k*,

## 11 Fluctuation theory for Local Reaches

The main result of this section is Theorem 11.1, which contains what is needed to complete the statement of Theorem 4.3, in that the limit process for (11.1) is now described in (formidable) detail.

To make that detail appear a little more natural, we shall do a little algebra before stating the theorem.

### 11.1 Some algebra and rearrangements

*x*and

*y*as \(k\rightarrow \infty \). However, looking back over the proof, in particular the final inequality (10.2), we see that the same is true for \(\sqrt{k}E^{x,k}(y)\), from which it follows that we can ignore the error term in (11.2). In addition, we also know from Theorem 4.3 that

*M*and \(M^{*}\), respectively,

*M*, while the last term will converge to \(V^{xy}\) times a Gaussian process on \(M^{*}\). Unfortunately, all the limit processes will be correlated, which is what makes the precise description of this a little long-winded, as we now see.

### 11.2 The fluctuation result

### Theorem 11.1

*f*and

*M*satisfy the assumptions of Theorem 4.3, including the conditions that

*M*is \(C^6\) and that, with probability one, \(f\in C^6(M)\). Then there exists a sequence \(\bar{\gamma }_k\) of random processes from \(M\rightarrow {\mathbb R}\), such that, for all \(x\in M\),

*M*with supremum norm, and

- 1.\(\eta (y)\) is a centered Gaussian process over
*M*with correlation function$$\begin{aligned} \mathbb {E}\{\eta (y_1)\eta (y_2)\}=2(\mathbb {C}(y_1,y_2))^2. \end{aligned}$$ - 2.\(\beta (x,y)\) is a centered Gaussian process over \(M^{*}\) with correlation function$$\begin{aligned} \mathbb {E}\{\beta (x_1,y_1)\beta (x_2,y_2)\}=2\left( \mathbb {E}\{f^{x_1}(y_1)f^{x_2}(y_2)\}\right) ^2. \end{aligned}$$
- 3.\(\zeta (x,y)\) is a centered Gaussian process over \(M^{*}\) with correlation function$$\begin{aligned}&\mathbb {E}\{ \zeta (x_1,y_1) \zeta (x_2,y_2)\} \\&\quad =\, \frac{1}{(1-(\mathbb {C}^{x_1y_1})^{2})(1-(\mathbb {C}^{x_2y_2})^{2})}\\&\qquad \times \bigg \{\frac{1}{2}\mathbb {C}^{x_1 y_1}\mathbb {C}^{x_2y_2} \left[ (\mathbb {C}^{y_1 x_2})^2+(\mathbb {C}^{y_1 y_2})^2+(\mathbb {C}^{x_1 x_2})^2+(\mathbb {C}^{x_1 y_2})^2\right] \\&\qquad +\,\mathbb {C}^{x_2y_1}\left[ \mathbb {C}^{x_1 y_2}-\mathbb {C}^{x_1 x_2}\mathbb {C}^{x_2y_2}\right] +\mathbb {C}^{y_1 y_2}\left[ \mathbb {C}^{x_1 x_2}-\mathbb {C}^{x_1 y_2}\mathbb {C}^{x_2y_2}\right] \\&\qquad -\,\mathbb {C}^{x_1 y_1}\left[ \mathbb {C}^{x_1 x_2}\mathbb {C}^{x_1 y_2}+\mathbb {C}^{y_1 y_2}\mathbb {C}^{x_2y_1}\right] \bigg \}. \end{aligned}$$

*M*. None of the cross-covariances are dependent on the particular choice of vector field.

Although Theorem 11.1 takes a lot of space to state, its main implication is simple: The limiting fluctuations of the local reach of \(h^k(M)\) are bounded by a functional of a Gaussian process on \(M^{*}\). The detailed structure of this Gaussian process is complicated, and depends, in terms of properties such as differentiability, on the underlying covariance of *f*. For example, while the limit is a.s. continuous, it will not typically be differentiable, and fine sample path properties such as Hölder continuity will depend on the behaviour of the underlying covariance \(\mathbb {C}\) in ways that are not at all obvious.

## 12 Proof of Theorem 11.1

We start with two lemmas, and then use these to complete the proof in the Sect. 12.2.

### 12.1 Two lemmas

### Lemma 12.1

### Proof

Most of the pieces that make up the proof of Theorem 11.1 are actually already in hand. For a start, by Lemmas 6.1 and 6.3 we know \(\Sigma ^{(2)}_k(x,y)\) and \(\Sigma ^{(1)}_k(y)\) converge to the deterministic limits \(V^{xy}\) and 1, respectively, where the convergence is almost surely uniform in \((x,y)\in M^{*}\) and \(y\in M\). The corresponding weak convergence is, obviously, implied by this. Secondly, in Sect. 9.3.1 we established the weak convergence of \(N_k\) in \(C_b(M)\) [cf. (9.20) and the last paragraph of Sect. 9.3.2].

Note that to this point we have relied on results that arose in earlier parts of the paper, and these required only that \(f\in C^3(M)\). The additional level of differentiability required by the lemma, and so also by Theorem 11.1, comes from the following lemma, which completes the collection by establishing the weak convergence of \(B_k\).

In view of the fact that all the limit processes are either deterministic or Gaussian, applications of Slutsky’s theorem and the Cramér–Wold device then complete the proof, modulo calculating all the limit variances and covariances, for which we do not intend to write out the details. \(\square \)

### Lemma 12.2

Under the conditions of Lemma 12.1, and with the notation of the previous section, \(B_k\Rightarrow \beta \) in \(C_b(M^{*})\).

### Proof

The proof follows along the same lines as the proof of the weak convergence of \(\alpha _k\) described in Sect. 9.3.1.

*M*, with the conventions of Sect. 3. Write the corresponding Riemannian normal basis vectors as \(\{{\partial }/{\partial x^i}\}\), and \(\left\{ {\partial }/{\partial x^i}:{\partial }/{\partial y^i}\right\} \) as the corresponding basis for the tangent spaces on \(M\times M\). In this basis, we have

*H*say, between functions on \(M^{*}\) via

*H*is continuous, with probability one, for the probability measure supported on the paths of \(\Lambda \). (This is not straightforward, since, as we shall soon see, we once again run into 0 / 0 issues for \((H(\Lambda ))(x,y)\) as \(x\rightarrow y\).) As a first step in checking this continuity, we need to know something about \(\Lambda \), and the function space on which the convergence of the numerator in (12.5) to \(\Lambda \) occurs.

*M*is one-dimensional. While notationally much simpler than the general case (although we shall see in a moment that it is hardly ‘simple’) it is indicative of the general situation. In the general case the limit in (12.8) will be taken along a specific path of

*y*’s, for which the final direction of approach to

*x*will be what plays the role of the single dimension in the following calculation.

*g*-norm of \(\frac{\partial }{\partial x}\) is one that

*H*of (12.6), we need to have convergence not only of the sum \(k^{-1/2} \sum \Lambda _\ell \), but also at least four of its derivatives. That is, we need weak convergence in the Banach space \(B^{(4)}\) of four times continuously differentiable functions on \(M\times M\), equipped with the norm

Now that we know what to do, the rest is, at least in principle, straightforward, and the proof follows along the same lines of the proof of the weak convergence of \(\alpha _k\) we treated in Sect. 9.3.1. Convergence of finite dimensional distributions follows from Theorem 9.2, while tightness requires the computation of moments of increments of the \(\Lambda _\ell \) and their first four derivatives. Note, however, that \(\Lambda _\ell (x,y)\) involves \(f_\ell ^x(y)\). Since we have seen that \(f_\ell ^x(\cdot )\), as a function on *M*, basically possesses one less level of differentiability that *f* itself, requiring four derivatives for \(\Lambda _\ell \) ultimately leads to requiring \(f\in C^5(M)\). In addition, since the arguments is Sect. 9.3.1 relied on a Taylor expansion, one further derivative is required, which is why the lemma, and so Theorem 11.1, require \(f\in C^6(M)\).

We leave the details to the reader. While they are long and involved, the fact that all random variables are either Gaussian or squares of Gaussians means that there is no more involved than Wick’s formula and accounting. \(\square \)

### 12.2 Proof of Theorem 11.1

## Notes

### Acknowledgements

We would like to thank Takashi Owada for useful discussions, and two referees for helpful comments.

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