Witnessed kDistance
Abstract
Distance functions to compact sets play a central role in several areas of computational geometry. Methods that rely on them are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The recently introduced distance to a measure offers a solution by extending the distance function framework to reasoning about the geometry of probability measures, while maintaining theoretical guarantees about the quality of the inferred information. A combinatorial explosion hinders working with distance to a measure as an ordinary power distance function. In this paper, we analyze an approximation scheme that keeps the representation linear in the size of the input, while maintaining the guarantees on the inference quality close to those for the exact but costly representation.
Keywords
Distance function Orderk Voronoi diagram Geometric inference Surface reconstruction Wasserstein noise1 Introduction
The problem of recovering the geometry and topology of compact sets from finite point samples has seen several important developments in the previous decade. Homeomorphic surface reconstruction algorithms have been proposed to deal with surfaces in R^{3} sampled without noise [1] and with moderate Hausdorff (local) noise [13]. In the case of submanifolds of a higher dimensional Euclidean space [20], or even for more general compact subsets [5], it is also possible, at least in principle, to compute the homotopy type from a Hausdorff sampling. If one is only interested in the homology of the underlying space, the theory of persistent homology [15] applied to Vietoris–Rips complexes provides an algorithmically tractable way to estimate the Betti numbers from a finite Hausdorff sampling [7].
All of these constructions share a common feature: they estimate the geometry of the underlying space by a union of balls of some radius r centered at the data points P. A different interpretation of this union is the rsublevel set of the distance function to P, \(\operatorname {d}_{P}: x\mapsto \min_{p\in P} \x  p\\). Distance functions capture the geometry of their defining sets, and they are stable to Hausdorff perturbations of those sets, making them wellsuited for reconstruction results. However, they are also extremely sensitive to the presence of outliers (i.e., data points that lie far from the underlying set); all reconstruction techniques that rely on them fail even in presence of a single outlier.
To counter this problem, Chazal, CohenSteiner, and Mérigot [6] developed a notion of distance function to a probability measure that retains the properties of the (usual) distance important for geometric inference. Instead of assuming an underlying compact set that is sampled by the points, they assume an underlying probability measure μ from which the point sample P is drawn. The distance function \(\mathrm {d}_{\mu,m_{0}}\) to the measure μ depends on a mass parameter m _{0}∈(0,1). This parameter acts as a smoothing term: a smaller m _{0} captures the geometry of the support better, while a larger m _{0} leads to better stability at the price of precision. Crucially, the function \(\mathrm {d}_{\mu,m_{0}}\) is stable to the perturbations of the measure μ under the Wasserstein distance. Defined in Sect. 2.2, this distance evaluates the cost of the optimal way to transport one measure onto another, where we pay for the squared distance a unit mass travels. Consequently, the Wasserstein distance between the underlying measure μ and the uniform probability measure on the point set P is small even if P contains some outliers, since individual points support only a small fraction of the mass. The stability result ensures that distance function \(\mathrm {d}_{\mathbf {1}_{P}, m_{0}}\) to the uniform probability measure 1 _{ P } on P retains the geometric information contained in the underlying measure μ and its support.
Computing with Distance Functions to Measures
The difficulty in applying these methods is that to get an equality in Eq. (1) the minimum number of barycenters to store is the same as the number of sites in the orderk Voronoi diagram of P, making this representation unusable even for modest input sizes [9]. Our solution is to construct an approximation of the distance function d_{ P,k }, defined by the same equation as (1), but with \(\bar{p}\) ranging over a smaller subset of barycenters. In this article, we study the quality of approximation given by a linearsized subset—the witnessed barycenters, defined as the barycenters of any k points in P whose orderk Voronoi cell contains at least one of the sample points. The algorithmic simplicity of the scheme is appealing: we only have to find the k−1 nearest neighbors for each input point. We denote by \(\mathrm{d}^{\mathrm{w}}_{P,k}\) and call witnessed kdistance the function defined by Eq. (1), where \(\bar{p}\) ranges over the witnessed barycenters.
Contributions
 (H)

We assume that the “ground truth” is an unknown probability measure μ supported on a compact set K whose dimension is bounded by ℓ. This means that there exists a positive constant α _{ μ } such that for every point x in K and every radius r<diam(K), one has μ(B(x,r))≥α _{ μ } r ^{ ℓ }.
Our first result asserts in a quantitative way that if the uniform measure to a point cloud P is a good Wasserstein approximation of μ, then the witnessed kdistance to P provides a good approximation of the distance to the underlying compact set K. We denote by ∥.∥_{∞} the norm of uniform convergence on ℝ^{ d }, that is, \(\f\_{\infty}:= \sup_{x \in \mathbb{R}^{d}} f(x)\).
Witness Bound
(Theorem 4.4)
The above bound is only a constant times worse than a similar bound for the exact kdistance which was proven in [6]. In other words, under the hypothesis of this theorem the quality of the inference is not significantly decreased when replacing the exact kdistance by the witnessed kdistance.
Outline
The relevant background appears in Sect. 2. We present our approximation scheme together with a general bound of its quality in Sect. 3. In Sect. 4, we give the proof of the Witnessed Bound. The convergence of the uniform measure on a point cloud sampled from a measure of low complexity appears in Sect. 5. We illustrate the utility of these two bounds with an example and a topological inference statement in our final Sect. 6.
2 Background
We begin by reviewing the relevant background on measure theory, Wasserstein distances, and distances to measures.
2.1 Measure
Let us briefly recap the few concepts of measure theory that we use. A nonnegative measure μ on the space ℝ^{ d } is a map from (Borel) subsets of ℝ^{ d } to nonnegative numbers, which is additive in the sense that \(\mu(\bigcup_{i\in \mathcal{N}} B_{i} ) = \sum_{i} \mu(B_{i})\) whenever (B _{ i }) is a countable family of disjoint (Bore)l subsets. The total mass of a measure μ is mass(μ):=μ(ℝ^{ d }). A measure μ with unit total mass is called a probability measure. The support of a measure μ, denoted by spt(μ), is the smallest closed set whose complement has zero measure. The expectation or mean of μ is the point \(\mathbb{E}(\mu) = \int_{\mathbb{R}^{d}} x\,\mathrm {d}\mu(x)\); the variance of μ is the number \(\sigma_{\mu}^{2} = \int_{\mathbb{R}^{d}} \x  \mathbb{E}(\mu)\^{2}\,\mathrm {d}\mu(x)\).
Although the results we present are often more general, the typical probability measures we have in mind are of two kinds: (i) the probability measure defined by rescaling the volume form of a lowerdimensional submanifold of the ambient space and (ii) discrete probability measures that are obtained through noisy sampling of probability measures of the previous kind. For any finite set P with N points, recall that 1 _{ P } is the uniform measure supported on P, i.e., the sum of Dirac masses centered at p∈P with weight 1/N.
2.2 Wasserstein Distance
A natural way to quantify the distance between two measures is the quadratic Wasserstein distance. This distance measures the L^{2}cost of transporting the mass of the first measure onto the second one. Note that it is possible to define Wasserstein distances with other exponents; for instance, the L^{1} Wasserstein distance is commonly called the earth mover’s distance [22]. A general study of this notion and its relation to the problem of optimal transport appear in [24]. We first give the general definition and then explain its interpretation when one of the two measures has finite support.
Discrete Target Measure
Wasserstein Noise

Consider a probability measure μ and f:ℝ^{ d }→ℝ, the density of a probability measure centered at the origin, with finite variance \(\sigma^{2} := \int_{\mathbb{R}^{d}} \x\^{2} f(x)\,\mathrm {d}x\). Denote by ν the result of the convolution of μ by f. Then, the quadratic Wasserstein distance between μ and ν is at most σ. This follows for instance from [24, Proposition 7.17].

Let P denote a set of N points drawn independently from the measure ν. Suppose also that the ν has small tails, e.g., ν(ℝ^{ d }∖B(0,r))≤exp(−cr ^{2}) for some constant c. Then, the Wasserstein distance W_{2}(ν,1 _{ P }) between ν and the uniform probability measure on P converges to zero as N grows to infinity. Examples of such asymptotic convergence results are called “the uniform law of large numbers” and are common in statistics (see for instance [4] and references therein).
2.3 DistancetoMeasure and kDistance
In [6], the authors introduce a distance to a probability measure as a way to infer the geometry and topology of this measure in the same way the geometry and topology of a set is inferred from its distance function. Given a probability measure μ and a mass parameter m _{0}∈(0,1), they define a distance function \(\operatorname {d}_{\mu,m_{0}}\) which captures the properties of the ordinary distance function to a compact set that are used for geometric inference.
Definition 2.1
For any point x in ℝ^{ d }, let δ _{ μ,m }(x) be the radius of the smallest ball centered at x that contains a mass at least m of the measure μ. The distance to the measure μ with parameter m _{0} is defined by \(\operatorname {d}_{\mu,m_{0}}(x) = m_{0}^{1/2} (\int_{m=0}^{m_{0}} \delta_{\mu,m}(x)^{2}\,\mathrm {d}m)^{1/2}\).
3 Witnessed kDistance
In this section we describe a simple scheme for approximating the distance to a uniform measure together with a general error bound. The main contribution of our work, presented in Sect. 4, is the analysis of the quality of this approximation when the input points come from a measure concentrated on a lowerdimensional subset of the Euclidean space.
3.1 kDistance as a Power Distance
Given a set of points U={u _{1},…,u _{ n }} in ℝ^{ d } with weights (w _{ u }), we call the power distance to U the function pow_{ U } obtained as the lower envelope of all the functions \(\operatorname {d}_{u}^{2}: x \mapsto \u  x\^{2}  w_{u}\), where u ranges over U. By Proposition 3.1 in [6], we can express the square of any distance to a measure as a power distance with nonpositive weights. The following proposition recalls this property in the case of kdistance; it is equivalent to the wellknown fact that the orderk Voronoi diagrams can be written as the power diagrams for a certain set of points and weights [3].
Proposition 3.1
Proof
From the proof of Proposition 3.1, we also see that the only barycenters that actually play a role in Eq. (3) are the barycenters of k points of P whose orderk Voronoi cell is not empty. However, the dependence on the number of nonempty orderk Voronoi cells makes computation intractable even for moderately sized point clouds in the Euclidean space [9]. One way to avoid this difficulty is to replace the kdistance to P by an approximate kdistance, defined as in Eq. (3), but where the minimum is taken over a smaller set of barycenters. Then, the question is: Given a point set P, can we replace the set of barycenters \(\mathrm{Bary}^{k}_{P}\) in the definition of kdistance by a small subset B while controlling the approximation error \(\\mathrm{pow}^{1/2}_{B}  \mathrm {d}_{P,k}\_{\infty}\)?
Replacing the kdistance with another power distance is especially attractive since many geometric and topological inference methods relying on distance functions continue to hold when one of the functions is replaced by a good approximation in the class of power distances. When this is the case, and some sampling conditions are met, it is possible, for instance, to recover the homotopy type of the underlying compact set (see the Reconstruction Theorem in [6]).
3.2 Approximating by Witnessed kDistance
We consider a subset of the barycenters suggested by the input data. The answer to our question is affirmative if we accept a multiplicative error.
Definition 3.2
Computing the set of all witnessed barycenters of a point set P requires only finding the k−1 nearest neighbors of every point in P. This search problem has a long history in computational geometry [2, 8, 17], and now has several practical implementations. Even a bruteforce approach with the running time O(dn ^{2}), where n is the number of points in P, is significantly better than the Ω(n ^{⌊d/2⌋} k ^{⌈d/2⌉}) lower bound on the number of cells in orderk Voronoi diagrams [9]. (Note that this lower bound holds as n/k→∞, which is not the case in our problem; finding similar lower bounds when n/k is constant is an open problem.)
General Error Bound
Because the distance functions we consider are defined by minima, and \(\mathrm{Bary}_{\mathrm{w}}^{k}(P)\) is a subset of Bary^{ k }(P), the witnessed kdistance is never less than the exact kdistance. In the lemma below, we give a general multiplicative upper bound. This lemma does not assume any special property for the input point set P. However, even such a coarse bound can be used to estimate Betti numbers of sublevel sets of d_{ P,k }, using arguments similar to those in [7].
Lemma 3.3
(General Bound^{1})
Proof
4 Approximation Quality
Let us briefly recall our hypotheses. There is an ideal, wellconditioned measure μ on ℝ^{ d } supported on an unknown compact set K. We also have a noisy version of μ, i.e., another measure ν with W_{2}(μ,ν)≤σ, and we suppose that our data set P consists of N points independently sampled from ν. In this section we give conditions under which the witnessed kdistance to P provides a good approximation of the distance to the underlying set K.
4.1 Dimension of a Measure
First, we make precise the main assumption (H) on the underlying measure μ, which we use to bound the approximation error made when replacing the exact by the witnessed kdistance. We require μ to be low dimensional in the following sense.
Definition 4.1
A measure μ on ℝ^{ d } is said to have dimension at most ℓ with constant α _{ μ }>0 if the amount of mass contained in the ball B(p,r) is at least α _{ μ } r ^{ ℓ }, for every point p in the support of μ and every radius r smaller than the diameter of this support. If μ is said to have dimension at most ℓ, this means that there exists a constant α _{ μ }.
The important assumption here is that the lower bound μ(B(p,r))≥αr ^{ ℓ } should be true for some positive constant α and for r smaller than a given constant R. The choice of \(R = \operatorname{diam}(\mathrm{spt}(\mu))\) provides a normalization of the constant α _{ μ } and slightly simplifies the statements of the results.
Let M be an ℓdimensional compact submanifold of ℝ^{ d }, and f:M→ℝ a positive weight function on M with values bounded away from zero and infinity. Then, the dimension of the volume measure on M weighted by the function f is at most ℓ. A quantitative statement can be obtained using the Bishop–Günther comparison theorem; the bound depends on the maximum absolute sectional curvature of the manifold M, as shown in Proposition 4.9 in [6]. Note that the positive lower bound on the density is really necessary. For instance, the dimension of the standard Gaussian distribution \(\mathcal{G}(0,1)\) on the real line is not bounded by 1, nor by any positive constant, because the density of this distribution decreases to zero faster than any function r↦1/r ^{ ℓ } as one moves away from the origin.
It is easy to see that if m measures μ _{1},…,μ _{ m } have dimension at most ℓ, then so does their sum. Consequently, if (M _{ j }) is a finite family of compact submanifolds of ℝ^{ d } with dimensions (d _{ j }), and μ _{ j } is the volume measure on M _{ j } weighted by a function bounded away from zero and infinity, the dimension of the measure \(\mu=\sum_{j=1}^{m} \mu_{j}\) is at most max_{ j } d _{ j }.
4.2 Bounds
In the remainder of this section, we bound the error between the witnessed kdistance \(\mathrm{d}^{\mathrm{w}}_{P,k}\) and the (ordinary) distance \(\operatorname {d}_{K}\) to the compact set K. We start from a proposition from [6] that bounds the error between the exact distance to measure and d_{ K }.
Theorem 4.2
Proof
To make this bound concrete, let us construct a simple example where the term corresponding to the Wasserstein noise and the term corresponding to the smoothing have the same order of magnitude.
Example
In the previous theorem, when ν=1 _{ P } is the uniform measure on a point cloud P and m _{0}=k/P, we get the exact bound on the kdistance.
Corollary 4.3
(Exact Bound)
In the main theorem of this section, the exact kdistance in Corollary 4.3 is replaced by the witnessed kdistance. Observe that the new error term is only a constant factor off from the old one.
Theorem 4.4
(Witnessed Bound)
Before proving the theorem, we start with an auxiliary lemma showing that a measure ν close to a measure μ satisfying an upper dimension bound (as in Definition 4.1) remains concentrated around the support of μ.
Lemma 4.5
(Concentration)
Let μ be a probability measure satisfying the dimension assumption, and let ν be another probability measure. Let m _{0} be a mass parameter. Then, for every point p in the support of μ, ν(B(p,η))≥m _{0}, where \(\eta = m_{0}^{1/2} \mathrm{W}_{2}(\mu, \nu) + 4 \alpha_{\mu}^{1/\ell}m_{0}^{1/2+1/\ell}\).
Proof
Proof of the Witnessed Bound Theorem
5 Convergence under Empirical Sampling
One term remains moot in the bounds in Corollary 4.3 and Theorem 4.4, namely the Wasserstein distance W_{2}(μ,1 _{ P }). In this section, we analyze its convergence. The rate depends on the complexity of the measure μ, defined below. The moral of this section is that if a measure can be well approximated with few points, then it is also well approximated by random sampling.
Definition 5.1
The complexity of a probability measure μ at a scale ε>0 is the minimum cardinality of a finitely supported probability measure ν that εapproximates μ in the Wasserstein sense, i.e., such that W_{2}(μ,ν)≤ε. We denote this number by \(\mathcal{N}_{\mu}(\varepsilon)\).
Observe that this notion is very close to the εcovering number of a compact set K, denoted by \(\mathcal{N}_{K}(\varepsilon)\), which counts the minimum number of balls of radius ε needed to cover K. It is worth noting that if measures μ and ν are close—as are the measure μ and its noisy approximation ν in the previous section—and μ has low complexity, then so does the measure ν. The following lemma shows that measures satisfying the dimension assumption have low complexity. Its proof follows from a classical covering argument that appears, for example, in Proposition 4.1 of [18].
Lemma 5.2
(Dimension–Complexity)
Let K be the support of a measure μ of dimension at most ℓ with constant α _{ μ } (as in Definition 4.1). Then, for every positive ε, \(\mathcal{N}_{\mu}(\varepsilon) \leq 5^{\ell}/(\alpha_{\mu}\varepsilon^{\ell})\).
Combining this lemma with the theorem below, we get a bound on how well a measure satisfying an upper bound on its dimension is approximated by empirical sampling.
Theorem 5.3
(Convergence)
Proof
Sampling from a Perturbation of the Measure
A result similar to the Convergence Theorem follows when the samples are drawn not from the original measure μ, but from a “noisy” approximation ν. When the measure ν is also supported on a compact set, this follows directly from the Convergence Theorem.
Corollary 5.4
(FixedDiameter Sampling)
Proof
Perturbations with Noncompact Support
In many cases the perturbed measure ν is not compactly supported, and the previous corollary does not apply. It is still possible to recover similar results under a stronger assumption than a simple bound on the Wasserstein distance between μ and ν.
To give a flavor of such a result, we consider the simple case where ν is a convolution of μ with an isotropic centered Gaussian distribution with variance σ ^{2}, that is: \(\nu = \mu * \mathcal{G}(0, (\sigma^{2}/d)\mathbf{I})\). We will make use of the following bound on the sum of squared norms of random Gaussian vectors.
Lemma 5.5
Proof
Corollary 5.6
(Gaussian Convolution Sampling)
Proof
Remark
We note that the result of Corollary 5.6 can be extended to more general models of noise. The crucial point is to be able to control the Wasserstein distance between 1 _{ P } and 1 _{ Q }, where P and Q are point sets obtained by sampling N points from μ and ν. Estimates of this kind can be obtained, for instance, if there exist random variables X and Y with distributions μ and ν such that the random variable Z=∥X−Y∥^{2} is subexponential, i.e., ℙ(Z≥t)≤exp(−ct). We refer the interested reader, for example, to Proposition 5.16 in [23].
6 Discussion
Complexes
Since we are working with the power distance to a weighted point set (the witnessed barycenters U), we can employ different simplicial complexes commonly used in the computational geometry literature [16]. Recall that an (abstract) simplex is a subset of some universal set, in our case the witnessed barycenters U; a simplicial complex is a collection of simplices, where each subset of every simplex belongs to the collection.
Note that the Vietoris–Rips complex VR_{ r }(U) does not, in general, have the homotopy type of \(\mathrm{pow}_{U}^{1}(\infty,r]\). It is, however, possible to prove inference results for homology given an interleaving property, i.e., there exists a constant α≥1 such that Č_{ r }(U)⊆VR_{ r }(U)⊆Č_{ αr }(U). The inclusion Č_{ r }(U)⊆VR_{ r }(U) always holds, simply by definition. However, the second inclusion does not necessarily hold if the weights are positive, as the following example demonstrates.
Example
Consider the weighted point set U made of the three vertices (u,v,w) of an equilateral triangle with unit side length and weights w _{ u }=w _{ v }=w _{ w }=1/4. Then, for any nonnegative r, the Vietoris–Rips complex VR_{ r }(U) contains the triangle, while the Čech complex Č_{ r }(U) contains this triangle only as soon as \((r^{2} + 1/4)^{1/2} \geq 1/\sqrt{3}\), i.e., \(r \geq 1/\sqrt{12}\). In this case, there is no α such that the inclusion VR_{ r }(U)⊆Č_{ αr }(U) holds for every positive r.
On the other hand, the following lemma shows that when the weights (w _{ u })_{ u∈U } are nonpositive, the inclusion VR_{ r }(U)⊆Č_{2r }(U) always holds. This property lets us extend the usual homology inference results from Vietoris–Rips complexes to the (weighted) Vietoris–Rips complexes associated with the witnessed kdistance.
Lemma 6.1
If U is a point cloud with nonpositive weights, VR_{ r }(U)⊆Č_{2r }(U).
Proof
Now, choose a simplex σ in the Vietoris–Rips complex VR_{ r }(U). Let v be its vertex with the smallest weight. By the previous paragraph, we know that v belongs to every ball B(u,(2r)^{1/2}+w _{ u }), for every u∈σ. Therefore, all these balls intersect, and, by definition, σ belongs to the Čech complex Č_{2r }(U). □
Inference
Suppose we are in the conditions of the hypothesis (H). Additionally, we assume that the support K of the original measure μ has a weak feature size larger than R. This means that the distance function d_{ K } has no critical value in the interval (0,R). A consequence of this hypothesis is that all the offsets \(K^{r} = \mathrm {d}_{K}^{1}[0,r]\) of K are homotopy equivalent for r∈(0,R). Suppose again that we have drawn a set P of N points from a nearby measure μ. The following theorem combines the results of Sects. 4 and 5.
Theorem 6.2
(Approximation)
 (D)
 If the diameter of the support of ν does not exceed D, thenwith probability at least$$\bigl\\mathrm{d}^{\mathrm{w}}_{P,k}  \mathrm {d}_K\bigr\_\infty \leq 15 m_0^{1/2} \sigma + 12m_0^{1/\ell}\alpha_\mu^{1/\ell} $$$$1  \bigl(2 \mathcal{N}_\mu(\sigma) + 1\bigr) \exp\biggl(\frac{32 N\sigma^4}{D^4 \mathcal{N}_\mu(\sigma)^2}\biggr). $$
 (G)
 If ν is a convolution of μ with a Gaussian, \(\nu = \mu * \mathcal{G}(0, (\sigma^{2}/d)\mathbf{I})\), thenwith probability at least$$\bigl\\mathrm{d}^{\mathrm{w}}_{P,k}  \mathrm {d}_K\bigr\_\infty \leq 12m_0^{1/2} \sigma + 12 m_0^{1/\ell}\alpha_\mu^{1/\ell} $$In both statements, \(\mathcal{N}_{\mu}(\sigma)\) is the complexity of measure μ, as in Definition 5.1, and α _{ μ } is the parameter in Definition 4.1.$$1  \exp(Nd/2)  \bigl(2\mathcal{N}_\mu(\sigma) + 1\bigr) \exp\biggl(\frac{2 N\sigma^4}{D^4\mathcal{N}_\mu(\sigma)^2} \biggr). $$
Choice of the Mass Parameter
Footnotes
 1.
The authors thank Daniel Chen for strengthening an earlier version of this bound.
Notes
Acknowledgements
The authors would like to thank the anonymous referees for their insightful feedback. This work has been partly supported by a grant from the French ANR, ANR09BLAN033101, NSF grants FODAVA 0808515, CCF 1011228, and NSF/NIH grant 0900700. Quentin Mérigot would also like to acknowledge the support of the Fields Institute during the revision of this article. Dmitriy Morozov would like to acknowledge the support at LBNL of the DOE Office of Science, Advanced Scientific Computing Research, under award number KJ0402KRD047, under contract number DEAC0205CH11231.
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