1 Introduction↩︎

The connection between the graph Laplacian and the Laplace–Beltrami operator is a central theme in geometric data analysis and has attracted substantial interest from both applied and theoretical communities. On the one hand, from a theoretical perspective, this relationship offers a discretization framework for studying differential operators on manifolds via point clouds or sampled data [1]–[4]. On the other hand, from an applied viewpoint, this connection underpins a wide array of algorithms in machine learning and signal processing, including manifold learning techniques such as Laplacian eigenmaps [5]–[7] or Diffusion Maps [8], [9]. Understanding the convergence of the graph Laplacian to the Laplace–Beltrami operator — notably the role of scaling, sampling density, and boundary behavior — remains crucial in ensuring that discrete approximations faithfully capture the geometry and analysis of the underlying continuous space. This paper aims at tackling this question on a class of smooth Riemannian manifolds having possibly singular boundary.
To describe our framework and main results, let us consider independent and identically distributed (i.i.d.) random variables \(\{X_i\}_{i \ge 1} \sim X\) with common law \(\mathbb{P}_X\) on a metric measure space \((M,d,\mu)\). For any \(t>0\) and \(n \in \mathbb{N}\backslash \{0\}\), define \[L_{n,t}f(x):=\frac{1}{nt^{d/2+1}}\sum_{i=1}^{n}e^{-\frac{d(x,X_j)^2}{t}}(f(x)-f(X_j))\] for any \(x \in M\) and \(f\) in the space \(\mathcal{C}(M)\) of continous functions on \(M\). Here \(d\) is a positive integer. We call \(L_{n,t}\) the intrinsic Graph Laplacian with \(d\)-dimensional Gaussian kernel associated with the family \(\{X_1,\ldots,X_n\}\) and the parameter \(t\). In case \((M,d,\mu)\) is a subset of some ambient Euclidean space \(\mathbb{R}^D\), then the intrinsic distance \(d\) might be replaced by the extrinsic Euclidean norm \(\|\cdot\|\), in which case we call \(L_{n,t}\) the extrinsic Graph Laplacian.

We define the (intrinsic/extrinsic) \(d\)-dimensional Gaussian operator at time \(t\) as the expected value of \(L_{n,t}\) with respect to \(\mathbb{P}_X\), that is \[L_tf(x):=\mathbb{E}_X\left[\frac{1}{t^{d/2+1}}e^{-\frac{d(x,X)^2}{t}}(f(x)-f(X))\right]\] for any \(x \in X\) and \(f \in \mathcal{C}(M)\). It follows from the Strong Law of Large Numbers that \[L_{n,t}f(x) \to L_tf(x) \qquad \text{almost surely as n \to \infty}\] for any \(x,f,t\) as above. Note that if \(\mathbb{P}_X\) is absolutely continuous with respect to \(\mu\) with Radon–Nikodym derivative \(p \in L^1(M,\mu)\), then the Gaussian operator rewrites as \[L_t f(x) = \frac{1}{t^{d/2+1}} \int_M e^{-\frac{d(x,y)^2}{t}} (f(x) - f(y)) p(y) \, d\mu(y).\]

After the work of various authors, e.g. [6], [10], it is by now known that when \(M\) is a \(d\)-dimensional smooth submanifold of some Euclidean space \(\mathbb{R}^D\), for any sufficiently regular \(f\) and \(p\), the extrinsic Gaussian operator satisfies \[\label{eq:int} L_tf(x)\to c_d \, \Delta_{M,p}f(x) \qquad \text{ as t\to 0}\tag{1}\] at any interior point \(x \in M\), where \(c_d\) is a dimensional constant and \(\Delta_{M,p}\) is the weighted Laplace operator given by \[\Delta_{M,p}f(x) :=p(x)\Delta_M{f}(x) + \frac{1}{2}\nabla{p}(x) \cdot \nabla{f}(x),\] with \(\Delta_M\) the classical Laplace–Beltrami operator of \(M\). In [11] (see also [12]), this pointwise result for the extrinsic Gaussian operator was extended to three types of singular subsets of \(\mathbb{R}^D\), namely submanifolds with boundaries, intersections of them, and submanifolds with edge-type singularities. For instance, if \(x\) belongs to the non-empty boundary of some smooth submanifold \(M \subset \mathbb{R}^D\), then \[\label{eq:bound} L_tf(x) = \frac{c'_d}{\sqrt{t}}p(x)\partial_\nu f(x)+ o\left(\frac{1}{\sqrt{t}}\right) \qquad \text{ as t \to 0,}\tag{2}\] where \(\partial_\nu f(x)\) is the inner normal derivative of \(f\) at \(x\) and \(c_d'\) is another dimensional constant.

Our first main result extends both 1 and 2 in the setting of smooth Riemannian manifolds with kinks. These spaces which we introduce in Section 3 might be understood as manifolds with possibly non-empty and irregular boundary. To distinguish these possibly non-smooth boundaries with classical smooth ones, we use the terminology border : a smooth manifold with kinks \(M\) thus divides into an interior part \(int M\) and a border part \(\partial M\). Manifolds with kinks include notably manifolds without boundary, manifolds with smooth boundary, and manifolds with corners as considered e.g. in [13]–[18]. Other relevant spaces like cones, pyramids, even cusps, fall into the framework of manifolds with kinks while they are not manifolds with corners.

If \(M\) is a \(d\)-dimensional manifold with kinks, then any \(x \in M\) admits a tangent space \(T_xM \simeq \mathbb{R}^d\) defined as the usual space of smooth derivations at \(x\). A key concept in our analysis is the notion of inward tangent cone \[I_xM \subset T_xM\] which consists in all the directions emanating from \(x\) and pointing towards the interior of \(M\). In particular, \(I_xM\) coincides with \(T_xM\) if \(x\) is an interior point, with a half-subspace of \(T_xM\) if \(x\) is a classical boundary point, and with a binary fraction of \(T_xM\) if \(x\) is a corner point, see Lemma 9.

We also introduce on manifolds with kinks the notions of \(C^k\) functions for \(k \in \mathbb{N}\cup\{\infty\}\), and tangent, cotangent and tensor bundles, see Section 3.3 and 3.4. We then define vector fields as the sections of the tangent bundle, differential forms as alternating sections of the cotangent bundle, Riemannian metrics as the sections of the symmetric, positive definite covariant 2-tensor bundle. Like for manifolds without/with boundary, we define the differential of order \(k\) of a \(\mathcal{C}^k\) function as the suitable symmetric covariant \(k\)-tensor field which acts on contravariant \(k\)-tensor fields. Note that we shall use \(Z^{(k)}\) to denote the contravariant \(k\)-tensor field \(Z \otimes \ldots \otimes Z\) where \(Z\) is a vector field. Any Riemannian metric \(g\) on a manifold with kinks \(M\) allows to specify the gradient and hessian of a \(\mathcal{C}^2\) function \(f\), through the classical relations \[df(Z) = g(\nabla^g f,Z), \qquad d^{(2)}f(Z \otimes Z') = g([\mathrm{Hess}^g f] Z,Z').\] A Riemannian metric additionally identifies those tangent vectors at a point \(x\) that have norm one, whose set we denote by \(S^g_xM\). We also define \[S^gI_xM :=I_x M \cap S^g_xM\] which we endow with the \((d-1)\)-dimensional Hausdorff measure \(\sigma\) associated with the distance induced on \(T_xM\) by the scalar product \(g(x)\).

For the purpose of our analysis, we need minimal regularity on the boundary of the manifolds with kinks that we consider. The weakest assumption we can work with is Lipschitz and Continuously Directionally Differentiable regularity, LCDD for short, see Section 3 for the precise definition. For any positive integer \(\ell\) we set \(c_\ell :=\Gamma((\ell+1)/2)/2\) where \(\Gamma\) is the classical Gamma function. Then our first main result writes as follows.

The recent preprint [19] provides a result that goes in the same direction as Theorem 1, but with less details on the expression of the second order differential \(A_p\) depending on the singular behaviour at \(x\) of the space under consideration. Our approach allows for a neat identification of the operator in several cases, including the following ones.

• If \(x\) is an interior point, then \(I_xM\) coincides with \(T_xM\). Then the symmetry of \(T_xM\) with respect to the antipodal map \(z \mapsto -z\) implies that \(v(x)=0\), while \(A_p(x)\) reduces to the weighted Laplacian \(p(x)\Delta_g{f}(x) + \frac{1}{2}g_x(\nabla^g{p}(x), \nabla^g{f}(x))\) where \(\Delta_g\) is the Laplace–Beltrami operator associated with the metric \(g\). This matches up with 1 .

• In case \(x\) is a boundary point, we obtain \[I_xM = \{ v \in T_xM : g(\partial_\nu f(x),v)\ge 0\}\] where \(\partial_{\nu}^g f(x)\) is the inner normal derivative of \(f\) at \(x\), hence \(\partial_{v_g(x)} f(x)\) coincides with \(\partial_{\nu}^g f(x)\) and we recover 2 .

• Lastly, if \(x\) is an essential corner of depth \(k\), then we obtain \[\label{eq:Rdk} I_xM \simeq \bigcap_{1 \le i \le k} \{ v \in T_xM : g(\partial_{i} f(x),v)\ge 0\}\tag{3}\] where each \(\partial_{i} f(x)\) corresponds to differentiation along a half-line direction in the local model \[\mathbb{R}^n_k :=\mathbb{R}^k_+ \times \mathbb{R}^{d-k}.\] Then \(\mathop{}\!\mathrm{d}_x f (v_g(x)) = \sum_i \partial_{i} f (x)\).

Theorem 1 opens the door to a natural Neumann boundary condition on manifolds with kinks, that would be \(\partial_{v(x)} f(x)=0\) for any \(x \in \partial M\). A natural problem one could investigate after that is the spectral convergence of the Graph Laplacian generated from finitely many samples \(\{X_1,\ldots,X_n\}\) to the suitable Neumann Laplacian as sample size goes to infinity. Note that this question is studied in [20] with the so-called symmetrized AMV operator on manifolds with smooth boundary.
Our second main result tells us how to choose bandwidth \(t\) as a function of the sample size \(n\) so that the difference between the graph Laplace operator \(L_{n,t}\) and the operator \[\mathcal{L}_tf(x) :=-\frac{c_d}{\sqrt{t}}\left( p(x) \, \partial_{v(x)}f(x) + o(1)\right) - c_{d+1} \bigg( p(x) A_gf(x) + [p,f]_g(x) \bigg)\] identified in Theorem 1 converges to zero in probability or almost surely. We refer to Definition 29 for the notion of \(\alpha\)-subexponentiality, which encompasses boundedness, subexponentiality and subgaussianity for a real-valued random variable.

Note the difference in the sufficient condition between (1) and (2) : the former doesn’t depend on the tail decay rate of \(f(X),\) whereas the latter does.

We obtain Theorem 2 in Section 5 through concentration estimates. Such estimates — studied in e.g. [10]–[12] — quantify the error in approximating the Gauss operator by the graph Laplace operator. Our results generalize this line of work in two key directions:

1. We work in the setting of Riemannian manifolds with kinks, whose singularities can be significantly more severe than those of manifolds with boundary considered in [11].

2. Unlike most previous works, we do not assume that the function \(f : M \to \mathbb{R}\) is bounded. In particular, our results hold on noncompact manifolds provided they have finite total volume.

Regarding point (2), previous concentration bounds rely on Hoeffding’s or Bernstein’s inequalities for bounded random variables. In contrast, we employ more general forms of these inequalities that are applicable to subgaussian or subexponential random variables with appropriate tail decay.

Let us conclude by mentioning that Theorem 2 guided us to choose the bandwidth parameter \(t>0\) correctly in the numerical experiments presented in Section 6.

Acknowledgments.↩︎

The authors are funded by the Research Foundation – Flanders (FWO) via the Odysseus II programme no. G0DBZ23N. They both thank Dominic Joyce for helpful answers concerning manifolds with corners, and Iosif Pinelis who suggested the proof of ?? .

2 Preliminaries↩︎

Throughout the paper, we write \(0_d\) for the origin of \(\mathbb{R}^d\), we let \(\mathbb{B}_r^d(x)\) stand for the Euclidean ball of radius \(r\) centered at \(x \in \mathbb{R}^d\), and we write \(\mathbb{B}_r^d\) and \(\mathbb{B}^d\) instead of \(\mathbb{B}_r^d(0_d)\) and \(\mathbb{B}_1^d(0_d)\) respectively. We let \((e_1,\ldots,e_d)\) denote the canonical basis of \(\mathbb{R}^d\). For \(y \in \mathbb{R}^d\) we will often denote by \(y'\) the \(d-1\) first coordinates of \(y\) in the canonical basis of \(\mathbb{R}^d\), and we let \(y_d\) be the last one. We consider the open upper half space \(\mathbb{H}^d:=\{y \in \mathbb{R}^d : y_d > 0\}\) and the Euclidean unit sphere \(\mathbb{S}^{d-1}:=\{y \in \mathbb{R}^d : y_1^2 + \ldots + y_d^2 = 1\}\). We let \(\sigma\) be the usual surface measure on \(\mathbb{S}^{d-1}\) which coincides with the \((d-1)\)-dimensional Hausdorff measure of the Euclidean distance. We also write \(\mathcal{L}^d\) for the \(d\)-dimensional Lebesgue measure in \(\mathbb{R}^d\).

For a subset \(A\) of the Euclidean space \(\mathbb{R}^d\), we let \(\bar{A}\) be the closure of \(A\) and \(int(A)\) be its interior. We recall that a rigid motion of \(\mathbb{R}^d\) is a linear map \(R : y \mapsto Ay+b\) where \(A\in O(d)\) and \(b\in \mathbb{R}^d\). The epigraph and strict epigraph of \(\gamma : \mathbb{R}^d \to \mathbb{R}\) are respectively defined as \[epi(\gamma) :=\{ y \in \mathbb{R}^d : y_d \ge \gamma(y')\} \quad \text{and} \quad \mathring{epi}(\gamma) :=\{ y \in \mathbb{R}^d : y_d > \gamma(y')\}.\] When \(\gamma\) is continuous, we obviously have \[\label{eq:epi61} epi(\gamma) = \overline{\mathring{epi}(\gamma)}.\tag{4}\] The directional derivative of \(\gamma\) at \(x \in \mathbb{R}^d\) in the direction \(v \in \mathbb{R}^d\) is defined as \[\gamma'(x;v) :=\lim_{t \downarrow 0} \frac{\gamma(x+tv)-\gamma(x)}{t}\] whenever the limit exists.

2.1 Boundary points↩︎

For an open set \(\Omega \subset \mathbb{R}^d\) with a non-empty boundary \(\partial \Omega\), we will call boundary points of \(\Omega\) the elements in \(\partial \Omega\). Inspired by [21], we define the regularity of boundary points as follows.

As customary, we may write smooth instead of \(\mathcal{C}^\infty\), and Lipschitz instead of \(\mathcal{C}^{0,1}\). We will essentially work with boundary points that are \(\mathcal{C}^0\), \(\mathcal{C}^1\), Lipschitz, or CDD. Lipschit and CDD shall be abreviated to LCDD.

The next proposition shows that, from a topological point of view, there is no difference between a \(\mathcal{C}^0\) boundary point and a boundary point of the upper half space \(\mathbb{H}^d\).

2.2 Diffeomorphisms and inward sectors↩︎

For \(k\) a positive integer, we shall use the following notion of \(\mathcal{C}^k\) diffeomorphism between possibly non-open subsets of \(\mathbb{R}^d\), see e.g. [17] for details.

We will mostly use this definition in the case where \(A\) and \(B\) are the closures of open subsets of \(\mathbb{R}^d\) with a non-empty boundary. In this regard, let us introduce a useful definition.

It is easily seen that the inward tangent sector of a point belonging to an open set is \(\mathbb{R}^d\), and that at a \(\mathcal{C}^1\) boundary point it is a half space. Moreover, the inward tangent sector at the origin of \(\mathbb{R}^d_k\) is \(\mathbb{R}^d_k\) itself.

Let us provide relevant properties of diffeomorphisms between closures of open sets.

Before closing this subsection, let us explain how the inward tangent sector provides us with a convenient way to define cusps.

2.3 Boundary cones of open Euclidean subsets↩︎

In this section, we provide classical notions from convex analysis, see e.g. [22] for more details. We recall that \(C\subset \mathbb{R}^d\) is called a cone if it is invariant under multiplication by a positive number, i.e. if \(\lambda v \in C\) for any \(v\in C\) and \(\lambda > 0.\)

The Bouligand tangent cone is also known as tangent cone or contingent cone in the literature. It is a closed set. One can easily check that our definition is equivalent to the one given in [22]. Note also that \[\tilde{F}_a(A) \subset F_a(A) \subset T^B_aA\] with possibly strict inclusion : for instance, \(\tilde{F}_{0_d}(\bar{\mathbb{H}}^d) = \mathbb{H}^d \varsubsetneq \bar{\mathbb{H}}^d = F_{0_d}(\bar{\mathbb{H}}^d)\). Moreover, in our setting, the following holds.

For suitably regular boundary points, the Bouligand tangent cone satisfies the following additional properties.

2.4 Non-fluctuating boundary and local blow-ups↩︎

Consider an open subset \(\Omega\subset \mathbb{R}^d\) with a non-empty boundary, and a boundary point \(x\in \partial \Omega.\) For any \(t>0\), set \[\Omega_{x,t}:=\frac{\Omega-x}{{t}} \, \cdot\] We are interested in finding the limit in the sense of convergence of characteristic functions of the sets \(\Omega_{x,t}\) as \(t\to 0\). When this limit exists, we call it the blow up of \(\Omega\) at \(x.\) With this in mind, we introduce the following.

Roughly speaking, a domain with non-fluctuating boundary at a boundary point \(x\) is so that whenever there exist points on a line spanned by \(v\) arbitrary close to \(x\) that stay outside of \(\Omega,\) then there must be a segment of that line that also remains outside \(\Omega.\) On the co if \(\Omega\) has a fluctuating boundary in the direction \(v,\) then there is a line \(t\mapsto x+tv\) which intersects \(\partial \Omega\) infinitely many times without a segment \(\{x+tv: 0\le t \le t_0\}\) being properly contained in \(\partial \Omega.\) This explains the terms fluctuating and non-fluctuating.

We shall use the following lemma.

Now we focus on particular boundary points.

We are now in a position to state and prove the following important property.

Let us conclude with providing a large class of domains with non-fluctuating boundary at a boundary point. We recall that a set \(A \subset \mathbb{R}^d\) is convex if the segment \([x,y]:=\{(1-t)x+ty : t \in [0,1]\}\) belongs to \(A\) for any \(x,y \in A\).

Then the following holds.

3 Manifolds with kinks↩︎

In this section, we introduce the notion of manifolds with kinks.

3.1 Topological manifolds with kinks↩︎

Recall that a topological space is called paracompact if any open cover has a locally finite open refinement, and that a Hausdorff topological space is paracompact if and and only if it admits partitions of unity subordinate to any open cover. For two topological spaces \(X,Y\) and a subset \(A \subset X\), a map \(\varphi : A \to Y\) is a topological embedding if it is an homeomorphism onto its image.

Note that a border chart cannot be an interior chart, since for \((U,\phi)\) to be an interior chart, \(\phi(x)\) has to be an interior point of \(\phi(U),\) contradicting the fact that for a border chart, \(\phi(x)\) is a boundary point of \(\phi(U).\) What may not be immediate is that a point cannot be simultaneously an interior point with respect to one chart and a border point with respect to another chart. This is precisely the point of the next lemma.

We are now in a position to introduce our definition of topological manifold with kinks.

The next proposition shows that from a topological point of view, there is no difference between a manifold with kinks and a manifold with boundary, just like there is no topological difference between a manifold with corners and a manifold with boundary. The proof is an immediate consequence of Proposition 5 which implies that the codomain of border charts are all locally homeomorphic.

3.2 Differentiable manifolds with kinks↩︎

Let us now consider differentiable structures on topological manifolds with kinks. We let \(k\) be a positive integer that we keep fixed for the whole section.

The previous definition is classical for interior points and becomes relevant for border points only. In particular, it allows to define the regularity of border points thanks to the next key result.

Note that the previous claim fails if the transition maps are Hölder continuous. On the other hand, increasing the regularity of the border points independently of the border charts would also need us to increase the regularity of the transition maps.
We can now define differentiable manifolds with kinks.

The word “kink” suggests particular boundary points where the manifold presents some type of singularity. With this in mind, we classify border points of \(\mathcal{C}^1\) manifolds with kinks as follows.

Note that, according to our definition, any point on the boundary of a smooth manifold with boundary is a \(\mathcal{C}^1\) boundary point, and any corner on a manifold with corner is an essential corner. But the notion of essential kink captures wilder singularities, as illustrated below. Therefore, the category of manifolds with kinks is strictly larger than the ones of manifolds with corners, of manifolds with boundary, and of manifolds without boundary, and contains them all.

We conclude this section with some definitions regarding fluctuating boundaries (recall Definition 6 for the case of open Euclidean subsets).

3.3 Tangent space and inward sector↩︎

In Subsection 2.3, we recalled the concepts of Bouligand tangent cone, feasible direction cone and open feasible direction cone at a point of a subset of \(\mathbb{R}^d\), and we derived some results on these cones at suitably regular boundary points. The goal of the present subsection is to introduce the corresponding notions for smooth manifolds with kinks and to derive peculiar properties at essential kinks.

3.3.1 Tangent space↩︎

From now on, we work only with smooth manifolds with kinks, i.e. those manifolds with kinks for which the transition maps are all \(\mathcal{C}^{\infty}\). This is to make sure that the tangent space defined below as the space of smooth derivations (i.e. linear forms on the space of smooth functions) is finite dimensional. Indeed, for any \(\mathcal{C}^k\) manifold with \(k < +\infty\), the space of \(\mathcal{C}^k\) derivations is infinite dimensional : see e.g. [23].

In order to introduce smooth derivations on manifolds with kinks, we must first define what a smooth function is. This is what we do in the next definition, largely inspired by the context of manifolds with corners (see e.g. [18]).

We are now in a position to define smooth derivations and tangent spaces as follows.

We can now define the differential of a smooth function between two smooth manifolds with kinks. There is no change with the case of classical manifolds, see e.g. [27].

Let us now establish the following natural result.

3.3.2 Inward sector↩︎

Let us now introduce the notion of inward tangent sector for smooth manifolds with kinks. This is analogous to the inward tangent sector of Euclidean domains as discussed in Section 2.2. Our definition builds upon the classical characterization of the tangent space in terms of initial velocities of curves, which holds true for manifolds without/with boundary (see e.g. [27]). For manifolds with boundary, one can identify inward tangent vectors at boundary points by specifying the domain of the curves we choose. This is how we came up with the following natural definition.

It follows from 6 that any element \([\gamma] \in I_xM\) canonically defines a smooth derivation \(f \mapsto (f\circ \gamma)'(0)\) belonging to \(T_xM\). As such, \[\label{eq:inclusion} I_xM \subset T_xM.\tag{7}\] The converse is obvious when \(x \in int M\). We discuss the case \(x \in \partial M\) in Lemma 9 below.

We shall use the natural convention which denotes equivalent classes \([\gamma]\) as \(\gamma'(0)\), and think of these objects as initial velocities pointing towards the interior of \(M\).

We are now in a position to characterize border points in terms of their inward tangent sector.

3.3.3 Strictly inward sector↩︎

This section generalizes the Euclidean open feasible direction cone to the setting of manifolds with kinks. A natural attempt to do so might be to consider \[\{ \gamma'(0) : \gamma \colon I \to int(M) \text{ of class } \mathcal{C}^1 \text{ with } I = [0,\epsilon)\text{ for some \epsilon > 0 and } \gamma(0)=x \}/\sim\] where \(x\) is a border point of a smooth manifold with kinks, and \(\sim\) is like in 6 . A problem with this set is that even if a curve \(\gamma\) might entirely lie within \(int(M)\), the initial velocity vector \(\gamma'(0)\) may still be a boundary vector.

For instance, take \(M:=[0,\infty)\times [0,\infty) \subset \mathbb{R}^2, x:=0_2\), and consider \(\gamma(t):=(t+t^2,t^2)\) for \(t>0\) small. Then \(\gamma'(0)=(1,0)\) belongs to the previous set, but if we consider \(M\) as a subset of \(\mathbb{R}^2,\) its open feasible direction cone is \((0,+\infty)\times (0,+\infty)\) that does not contain the boundary vector \((1,0).\)

For this reason, we use a different definition. Before introducing it, let us provide a definition of cusp in this context.

We are now in a position to define the strictly inward tangent sector at a point of a smooth manifold with kinks.

3.4 Riemannian manifolds with kinks↩︎

In this section, we develop a suitable notion of Riemannian metric for manifolds with kinks. There is no particular difference compare to the case of manifolds without/with boundary, but we provide details for completeness.

3.4.1 Tangent bundle↩︎

Let us first define the tangent bundle on a smooth manifold with kinks. The definition is basically the same as for manifolds without/with boundary.

Let us check the following natural result.

3.4.2 Covariant \(k\)-tensor bundle↩︎

Let \(k\) be a positive integer. Recall that a covariant \(k\)-tensor on a vector space \(V\) is an element of the \(k\)-fold tensor product \(V^* \otimes \dots \otimes V^*\) or, equivalently, a \(k\)-linear map \(V \times V \times \dots \times V \to \mathbb{R}.\) We denote by \(T^k(V^*)\) the space of all covariant \(k\)-tensors on \(V\). Then we can define the covariant \(k\)-tensor bundle on a smooth manifold with kinks as follows.

Acting like in the previous subsection, one can easily show the following. We omit the proof for brevity.

3.4.3 Riemannian metrics↩︎

Let us define covariant tensor fields on manifold with kinks.

We are now in a position to define smooth Riemannian metrics on smooth manifolds with kinks.

3.4.4 Riemannian distance and volume measure↩︎

The Riemannian distance defined via a length-minimizing problem extends with no change to the context of manifolds with kinks. We recall the definition for completeness and refer to [27], for instance, for more details.

In the same way, the definition of the Riemannian volume measure carry over to manifolds with kinks and behaves like in the case of manifolds without boundary.

3.5 Extension of Riemannian manifolds with kinks↩︎

For our purposes, we need to extend beyond the border any smooth Riemannian metric on a smooth manifold with kinks \(M\). To this aim, we shall flow \(M\) into its interior using a suitable semiflow. We adapt an argument for manifolds with corners that goes back to [15] at least, see also [16].

3.5.1 Vector fields and semiflows↩︎

Let us begin by defining vector fields and their associated semiflows on smooth manifolds with kinks.

Let us ensure that any smooth manifold with kinks admits a stricly inward-pointing vector field.

Recall the definition of integral curve.

Then the following existence result holds.

3.5.2 Submanifolds with kinks↩︎

Let us now provide a definition of submanifold adapted to the context with kinks, inspired by [16] who introduced submanifolds with corners.

Our main result now is that any smooth manifold with kinks can be embedded into a manifold with boundary having same dimension. The proof actually embeds a smooth manifold with kinks into its interior.

3.5.3 Riemannian extension↩︎

We will next show that under this assumption, any \(\mathcal{C}^2\) Riemannian metric can be extended from \(M\) to the manifold \(\tilde{M}\) without boundary given by Lemma 27.

4 Asymptotic behavior of the intrinsic Gaussian Operator↩︎

In this section, we prove Theorem 1.

4.1 Localisation↩︎

Recall that for a metric measure space \((M,\mathop{}\!\mathrm{d},\mu)\), the intrinsic \(d\)-dimensional Gaussian operator at time \(t>0\) is defined by \[L_t f(x) :=\frac{1}{t^{d/2+1}}\int_{M} \exp\left(-\frac{\mathop{}\!\mathrm{d}^2(x,y)}{t}\right) (f(x)-f(y)) \mathop{}\!\mathrm{d}\mu(x)\] for any \(f \in \mathcal{C}(M)\) and \(x \in M\). We are interested in the short-time behavior of this operator. In this regard, the next simple lemma implies that all the information about this behavior is contained in small balls. The proof is immediate so we skip it.

As a consequence, we get \[L_tf(x) = \frac{1}{t^{d/2+1}} \int_{B_{t^\eta }(x)} \exp\left(-\frac{\mathop{}\!\mathrm{d}^2(x,y)}{t}\right) (f(x)-f(y)) d\mathrm{\mu}(y) + O(t^{-d/2-1}e^{-t^{2 \eta -1}})\] and so Theorem 1 follows from studying the asymptotic expansion of the first term in the previous right-hand side. Let us give a name to this quantity: \[\label{eq:localized} \bar{L}_{t,\eta } f (x) :=\frac{1}{t^{d/2+1}} \int_{B_{t^\eta }(x)} \exp\left(-\frac{\mathop{}\!\mathrm{d}^2(x,y)}{t}\right) (f(x)-f(y)) d\mathrm{\mu}(y)\tag{8}\]

4.2 Change of coordinates↩︎

Let us put ourselves in the context of Theorem 1.

We consider a smooth Riemannian manifold with kinks \((M^d,g)\), a density \(p \in \mathcal{C}_{\ge 0}^2(M)\), a function \(f \in \mathcal{C}^3(M)\), a point \(x \in M\) that is LCDD border (note that the case were \(x\) is interior is covered by 1 ), and we choose \(\eta \in (0,1/2)\). We let \((\tilde{M},\tilde{g})\) be the open Riemannian manifold of which \((M,g)\) is a submanifold, see Lemma 27 and Proposition 28. We denote by \(\tilde{B}_r(x)\) the \(\tilde{g}\)-ball of radius \(r\) centered at \(x\), by \(\tilde{\exp}_x\) the \(\tilde{g}\)-exponential map at \(x\), and by \(\tilde{\gamma}_{x,v}\) the unique maximal \(\tilde{g}\)-geodesic with initial point \(x\) and initial velocity \(v \in T_x \tilde{M}\). Since \(x\) is an interior point for \(\tilde{M}\), there exists \(R>0\) such that \(\tilde{\exp}_x\) is a diffeomorphism from \(\mathbb{B}_R^d \subset T_x \tilde{M}\) onto \(\tilde{B}_R(x)\). Moreover, since \((\tilde{M},\tilde{g})\) extends \((M,g)\), the set of \(\mathcal{C}^1\) curves joining \(x\) to \(y \in \tilde{B}_R(x)\) and lying entirely in \(M\) is a subset of those curves lying in \(\tilde{M}\), so that \[\label{eq:ineq95distance} \tilde{\mathop{}\!\mathrm{d}}(x,y) \le \mathop{}\!\mathrm{d}(x,y).\tag{9}\] As a consequence, for any \(r \in (0,R)\), \[B_r(x) \subset \tilde{B}_r(x).\] This implies, in particular, that \((B_R(x), \tilde{\exp}_x^{-1}|_{B_r(x)})\) is a border chart centered at \(x\) of \(M\).

As usual, we identify the Euclidean space \((T_x \tilde{M},g(x))\) with \((\mathbb{R}^d,\cdot)\) by choosing a \(g(x)\)-orthonormal basis \((w_1,\ldots,w_d)\) of \(T_x \tilde{M}\) and mapping each \(w_i\) to the \(i\)-th element of the canonical basis of \(\mathbb{R}^d\). We will freely use this identification in the sequel. Let us also point out that \((T_xM,g(x))\) and \((T_x\tilde{M},\tilde{g}(x))\) coincide as Euclidean spaces. We define \[\Omega :=\tilde{\exp}_x^{-1}(B_R(x)).\] This is an open subset of \(T_x\tilde{M} \simeq \mathbb{R}^d\) having \(0_d\) as a \(\mathcal{C}^0\) boundary point.

4.3 Euclidean calculation↩︎

The previous change of coordinates allows us to make a calculation in the Euclidean space \(\mathbb{R}^d\). This is what we do in this subsection. We provide a general result which might allow for considering union of intersecting manifolds with kinks in a future work.

Let us recall the notations. For any \(y \in \mathbb{R}^d\) we set \(y^{(k)} :=y\otimes \ldots \otimes y \in (\mathbb{R}^d)^{\otimes^k}\). If \(f \in \mathcal{C}^{\infty}(U)\) for some open subset \(U \subset \mathbb{R}^d\), we let \(\mathop{}\!\mathrm{d}^{(k)}_xf\) denote the differential of order \(k\) of \(f\) at \(x\), which we see here as a \(k\)-linear symmetric map from \((\mathbb{R}^d)^{\otimes^k}\) to \(\mathbb{R}\), and we set \[\|\mathop{}\!\mathrm{d}^{(k)}_xf\|_{op} :=\sup_{y \in \mathbb{R}^d\backslash \{0_d\}} \frac{\mathop{}\!\mathrm{d}^{(k)}_xf(y^{(k)})}{|y|^{k}} \, \cdot\] We shall also write \(\mathop{}\!\mathrm{d}_x^{(0)}f(y^{(0)})\) for \(f(x)\), in which case \(\|\mathop{}\!\mathrm{d}^{(0)}_xf\|_{op}= f(x)\). We consider a cone \(\mathcal{C} \subset \mathbb{R}^d\) and a scalar product \(g\) on \(\mathbb{R}^d\), and we set \(S^g \mathcal{C} :=\mathcal{C} \cap \{\xi \in \mathbb{R}^d : g(\xi,\xi)=1\}.\)

To prove this proposition, we need a simple preliminary lemma. We provide a proof for completeness.

We are now in a position to prove Proposition 30.

4.4 Conclusion↩︎

Let us now explain how to reach the conclusion of Theorem 1. From Lemma 13 and Proposition 29, we get \[L_tf(x) = \frac{1}{t} \int_{\mathbb{B}^d_{t^{\eta}}\cap \tilde{F}_{0_d}\Omega} e^{-\frac{\|\xi\|^2}{t}} (\tilde{f}(0_d) - \tilde{f}(\xi))q(\xi)\mathop{}\!\mathrm{d}\xi + o\left( \frac{1}{\sqrt{t}}\right) + O(t^{-d/2-1}e^{-t^{2 \eta -1}}) \qquad \text{as t\to 0.}\] From Proposition 8 and Proposition 9, we can replace \(\tilde{F}_{0_d}\Omega\) by \(I_{0_d}\bar{\Omega}\) in the previous domain of integration. Then we apply Proposition 30 to obtain that \[\begin{align} \frac{1}{t} \int_{\mathbb{B}^d_{t^{\eta}}\cap I_{0_d}\bar{\Omega}} e^{-\frac{\|\xi\|^2}{t}} (\tilde{f}(0_d) - \tilde{f}(\xi))q(\xi)\mathop{}\!\mathrm{d}\xi \end{align}\] equals \[- \frac{c_d}{\sqrt{t}} q(0_d) \partial_{v_{\mathcal{I}_{0_d}\bar{\Omega}}} \tilde{f}(0_d) - c_{d+1} \bigg(q(0_d) A_{I_{0_d}\bar{\Omega}} \tilde{f}(0_d) + [q,\tilde{f}]_{I_{0_d}\bar{\Omega}}(0_d)\bigg) + O(\sqrt{t})\] as \(t \downarrow 0\). But \[q(0_d) = p(\tilde{\exp}_x(0_x)) \sqrt{\det g(0_d)} = p(x),\] \[\begin{align} \partial_{v_{I_{0_d}\bar{\Omega}}} \tilde{f}(0_d) = d_{0_d}\tilde{f}(v_{I_{0_d}\bar{\Omega}}) & = d_xf \left( d_{0_d}\tilde{\exp}_x\left( \int_{S^g I_{0_d}\bar{\Omega}} \theta \mathop{}\!\mathrm{d}\sigma (\theta)\right)\right)\\ & = d_xf \left( \int_{S^g I_{0_d}\bar{\Omega}} d_{0_d}\tilde{\exp}_x(\theta) \mathop{}\!\mathrm{d}\sigma (\theta)\right)\\ & = d_xf \left( \int_{S^gI_xM} \theta \mathop{}\!\mathrm{d}\sigma(\theta)\right) = d_xf(v_g(x)) = \partial_{v_g(x)}f(x), \end{align}\] where we have used Lemma 8 to perform the change of variable which provides us with the last line. Likewise, we obtain \[A_\mathcal{C}\tilde{f}(0_d) = A_gf(x),\] \[\begin{align} [q,\tilde{f}]_{I_{0_d}\bar{\Omega}}(x) = [p,f]_g(x). \end{align}\] This concludes the proof of Theorem 1.

5 Concentration estimates↩︎

The goal of this section is to establish Theorem 2.

5.1 \(\alpha\)-Subexponential Random Variables↩︎

We begin by defining the class of \(\alpha\)-subexponential random variables introduced by Götze, Sambale and Sinulis in [29].

The following lemma is useful for our purposes. We provide a quick proof for completeness.

The next statement is taken from [29]. It is a concentration result for sample means of \(\alpha\)-subexponential random variables.

5.2 Convergence in probability↩︎

In this section, we fall back onto our original setup, i.e. we consider \(\{X_i\}_{\mathbb{N}} \sim X\), a sequence of iid random variables taking values in a Riemannian manifold with kinks \((M,g)\). Assume the law \(\mathbb{P}_X\) has a \(\mathcal{C}^2(M)\) density function \(p,\) and consider a function \(f \in \mathcal{C}^3(M)\) and a number \(t>0\).

We are firstly interested in finding out the concentration of \(L_{n,t}f(x)\) around its expected value \(L_tf(x)\), for \(x \in M\) being either an interior point or an LCDD border point. This is given by the following result.

Let us recall the operator \[\mathcal{L}_tf(x) :=-\frac{c_d}{\sqrt{t}}\left( p(x) \, \partial_{v(x)}f(x) + o(1)\right) - c_{d+1} \bigg( p(x) A_gf(x) + [p,f]_g(x) \bigg)\] identified in Theorem 1. Since the latter differ from \(L_tf(x)\) of a \(o(\sqrt{t})\) term, we immediately deduce the following from the previous proposition.

We are now in a position to prove (1) in Theorem 2. Consider \(t_n \to 0\) such that \(\sqrt{n}\, t_n^{\frac{d}{2} + 1} \to \infty\) as \(n \to \infty\). Then \[C_1 \exp \left( - C_2 \left( n\, t^{\frac{d}{2} + 1} \epsilon \right)^{\alpha} \right)\to 0 \qquad \text{and} \qquad C_1 \exp \left( - C_2 \left( \sqrt{n}\, t^{\frac{d}{2} + 1} \epsilon \right)^{\alpha} \right) \to 0 \qquad \text{as n \to +\infty}.\] This shows that \[\mathbb{P} \left( \left| L_{n,t_n} f(x) - \mathcal{L}_{t_n} f(x) \right| \ge \epsilon \right) \to 0 \qquad \text{as n \to +\infty}.\]

5.3 Almost sure convergence↩︎

Let us now prove (2) in Theorem 2. To this aim, recall that a sequence of real-valued random variables \(\{Z_n\}\) completely converges to another real-valued random variable \(Z\) if for any \(\epsilon>0\), \[\sum_{n} \mathbb{P}(|Z_n - Z| \ge \epsilon) < +\infty.\] It is easily seen from the first Borel–Cantelli lemma that complete convergence implies almost sure convergence. We shall need an elementary lemma.

We can now prove (2) in Theorem 2. Consider \(t_n \to 0\) such that \(\left(\sqrt{n}\, t_n^{\frac{d}{2} + 1} \right)^{\alpha}/\ln(n)\to \infty\) as \(n \to \infty\). For any \(\epsilon >0\), \[\begin{align} \sum_{n} \mathbb{P} \left( \left| L_{n,t_n} f(x) - \mathcal{L}_{t_n} f(x) \right| \ge \epsilon \right) & \le C_1 \sum_{n} \exp \left( - C_2 \left( n\, t^{\frac{d}{2} + 1} \epsilon \right)^{\alpha} \right)\\ & = C_1 \sum_{n} \exp \left( - b_n\epsilon^\alpha \right) \end{align}\] with \(b_n :=C_2 \left( n\, t^{\frac{d}{2} + 1} \right)^{\alpha}\). The assumption on \(\{t_n\}\) implies that \(b_n/\ln(n) \to \infty\) as \(n\to \infty\), hence Lemma 16 yields that \[\sum_{n} \mathbb{P} \left( \left| L_{n,t_n} f(x) - \mathcal{L}_{t_n} f(x) \right| \ge \epsilon \right) \le \sum_{n} \exp \left( - b_n\epsilon^\alpha \right) < +\infty.\] Thus \(|L_{n,t_n}f(x) - \mathcal{L}_{t_n}f(x)|\) completely converges to \(0\), thus it converges almost surely.

6 Numerical simulations↩︎

6.1 Experiments for a 3D ball↩︎

We test the value of \(L_{n,t}f\) at these points, when \(n:=10^8,f(x,y,z):=x+y+z,\) (whose Laplacian is zero) and the kernel bandwidths \(t:=t_n,\) varies logarithmically from \(0.05\) down to \(0.01\) (20 values). Note that Theorem 1.2, part (1) gives us a condition of the convergence in probability of \(L_{n,t_n}\) for \(n\to \infty, t_n\to 0, \sqrt{n}t_n^{\frac{d}{2}+1}\to \infty.\) In our experiments, the values of \(\sqrt{n}t_n^{\frac{d}{2}+1}\) vary between \(\sqrt{10^8}\times (0.01)^{\frac{3}{2}+1}=0.01\) and \(\sqrt{10^8}\times (0.05)^{\frac{3}{2}+1}=5.56.\) These values are not large, but one can already see that the behavior of \(\sqrt{t_n}L_{n,{t_n}}\) matches up with the asymptotics provided by our theory, and the behavior remains consistent as we decrease \(t:=t_n\) in our range.
The two tables below show us how the graph Laplace operator \(L_{n,t}f\) and its expectation \(L_tf\) behave asymptotically. Below ‘int’ corresponds to an interior point (the origin of the ball), and ‘bd’ corresponds to a boundary point (\((1,0)\)), and scaled Laplacians mean \(\sqrt{t}L_{n,t}\) for discrete and \(\sqrt{t}L_t\) for continuous graph Laplacians.

Below are the plots for the above tables:

6.2 Experiments for a 3D cube↩︎

We experiment with the asymptotic behavior of the graph Laplacian for a uniform random sample on the unit 3D hypercube \([0,1]^3\) (so \(d:=3\) here) \(abcdefgh,\) at the following points \(I,F,E,V\) below. We test the value of \(L_{n,t}f\) at these points, when \(n:=10^8,f(x,y,z):=x+y+z,\) (whose Laplacian is zero) and the kernel bandwidths \(t:=t_n,\) varies logarithmically from \(0.05\) down to \(0.01\) (20 values). Note that Theorem 1.2, part (1) gives us a condition of the convergence in probability of \(L_{n,t_n}\) for \(n\to \infty, t_n\to 0, \sqrt{n}t_n^{\frac{d}{2}+1}\to \infty.\) In our experiments, the values of \(\sqrt{n}t_n^{\frac{d}{2}+1}\) vary between \(\sqrt{10^8}\times (0.01)^{\frac{3}{2}+1}=0.01\) and \(\sqrt{10^8}\times (0.05)^{\frac{3}{2}+1}=5.56.\) These values are not large, but one can already see that the behavior of \(\sqrt{t_n}L_{n,{t_n}}\) matches up with the asymptotics provided by our theory, and the behavior remains consistent as we decrease \(t:=t_n\) in our range.

• Interior point : \[I \;=\;\frac{a+b+c+\dots+h}{8}.\]

• Face midpoint : \[F \;=\;\frac{a+v+d+e}{4}.\]

• Edge midpoint : \[E \;=\;\frac{a+b}{2}.\]

• Vertex : \[V = a.\]

Below are the tables and plots giving us the numerical results and the plots.

References↩︎