Optimal rates of convergence for persistence diagrams in Topological Data Analysis

Optimal rates of convergence for persistence diagrams in Topological Data Analysis

Frédéric Chazal, Marc Glisse , Catherine Labruère , Bertrand Michel frederic.chazal@inria.frmarc.glisse@inria.frclabruer@u-bourgogne.frbertrand.michel@upmc.fr
July 6, 2019
Abstract

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results.

1 Introduction

Motivations.

During the last decades, the wide availability of measurement devices and simulation tools has led to an explosion in the amount of available data in almost all domains of Science, industry, economy and even everyday life. Often these data come as point clouds sampled in possibly high (or infinite) dimensional spaces. They are usually not uniformly distributed in the embedding space but carry some geometric structure (manifold or more general stratified space) which reflects important properties of the “systems” from which they have been generated. Moreover, in many cases data are not embedded in Euclidean spaces and come as (finite) sets of points with pairwise distance information. This often happens, for example, with social network or sensor network data where each observation comes with a measure of its distance to the other observations: e.g., in a sensor network distributed in some domain, each sensor may not know its own position, but thanks to the strength of the signal received from the other sensors it may evaluate its distance from them. In such cases data are just given as matrices of pairwise distances between the observations, i.e. as (discrete) metric spaces. Again, although they come as abstract spaces, these data often carry specific topological and geometric structures.

A large amount of research has been done on dimensionality reduction, manifold learning and geometric inference for data embedded in, possibly high dimensional, Euclidean spaces and assumed to be concentrated around low dimensional manifolds; see for instance [42, 44, 31] and the references therein for recent results in this direction. However, the assumption of data lying on a manifold may fail in many applications. In addition, the strategy of representing data by points in Euclidean spaces may introduce large metric distortions as the data may lie in highly curved spaces, instead of in flat Euclidean spaces, raising many difficulties in the analysis of metric data. With the emergence of new geometric inference and algebraic topology tools, computational topology [28] has recently known an important development toward data analysis, giving birth to the field of Topological Data Analysis (TDA) [9] whose aim is to infer multiscale qualitative and quantitative relevant topological structures directly from the data. Topological persistence, more precisely persistent homology appears as a fundamental tool for TDA. Roughly, homology (with coefficient in a field such as, e.g., ) associates to any topological space , a family of vector spaces (the so-called homology groups) , , each of them encoding -dimensional features of . The Betti number of , denoted , is the dimension of and measures the number of -dimensional features of : for example, is the number of connected components of , the number of independent cycles or “tunnels”, the number of “voids”, etc…; see [32] for a formal introduction to homology. Persistent homology provides a framework [27, 46, 15] and efficient algorithms to encode the evolution of the homology of families of nested topological spaces indexed by a set of real numbers that can often be seen as scale parameters, such as, e.g., the sublevel sets of a function, union of growing balls, etc… The obtained multiscale topological information is then represented in a simple way as a barcode or persistence diagram; see Figure 3 and Section 2.2.
In TDA, persistent homology has found applications in many fields, including neuroscience [41], bioinformatics [33], shape classification [12], clustering [16] and sensor networks [22], to cite just a few. It is usually computed for a filtered simplicial complex built on top of the available data, i.e. a nested family of simplicial complexes whose vertex set is the data set (see Section 2.2). The obtained persistence diagrams are then used as “topological signatures” to exhibit the topological structure underlying the data; see Figure 1. The space of persistence diagrams is endowed with a metric, the so-called bottleneck distance, that allows to compare the obtained signatures and thus to compare the topological structure underlying different data sets. The relevance of this approach relies on stability results ensuring that close data sets, with respect to the so-called Gromov or Gromov-Hausdorff distance, have close persistence diagrams [17, 11, 15, 13]. However these results mainly remain deterministic and thus often restrict to heuristic or exploratory uses in data analysis.

The goal of this paper is to show that, thanks to recent results [15, 13], the use of persistent homology in TDA can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties.

Figure 1: A usual use of persistence in TDA.
Contribution.

In this paper we assume that the available data is the realization of a probability distribution supported on an unknown compact metric space. We consider the persistent homology of different filtered simplicial complexes built on top of the data and we study the rate of convergence of the associated persistence diagrams to some well-defined persistence diagram associated to the support of the probability distribution, with a minimax approach.

More precisely, we assume that we observe a set of points in a metric space , drawn i.i.d. from some unknown measure whose support is a compact set denoted . We then consider the persistent homology of the filtered simplicial complexes and built on top of and respectively and we establish convergence rates of the bottleneck distance between their persistence diagrams, ; see Section 3 for explicit results. It is important to notice that encodes topological properties of the support but not of the measure itself. As a consequence, to obtain explicit convergence rates we assume that satisfies the so-called -standard assumption for some constants : for any and any , . The following theorem illustrates the kind of results we obtain under such assumption.

Theorem (3.6 in Section 3): Let , and as above. Then for any measure satisfying the -standard assumption

where the constant only depends on and (not on ). Assume moreover that there exists a non isolated point in and consider any sequence such that . Then for any estimator of :

where is an absolute constant.

Our approach relies on the general theory of persistence modules and our results follow from two recently proven properties of persistence diagrams [13, 11, 15].
First, as can be any compact metric space (possibly infinite), the filtered complex is usually not finite or even countable and the existence of its persistence diagram cannot be established from the “classical” persistence theory [46, 27]. In our setting, the existence of follows from the general persistence framework introduced in [11, 15]. Notice that although this framework is rather abstract and theoretical it does not have any practical drawback as only persistence diagrams of complexes built on top of finite data are computed.
Second, a fundamental property of the persistence diagrams we are considering is their stability proven in [13]: the bottleneck distance between and is upper bounded by twice the Gromov-Hausdorff distance between and . This result establishes a strong connection between our persistence estimation problem and support estimation problems. Upper bounds on the rate of convergence of persistence diagrams are then easily obtained using the same arguments as the ones usually used to obtain convergence results for support estimation with respect to the Hausdorff metric. We take advantage of this general remark to find rates of convergence of persistence diagrams in general metric spaces (Section 3) and also in the more classical case where the measure is supported in (Section 4). Using Le Cam’s lemma, we also compute the corresponding lower bounds to check that the rates of convergence are optimal in the minimax sense.

Related works.

Although it is attracting more and more interest, the use of persistent homology in data analysis remains widely heuristic. There are relatively few papers establishing connections between persistence and statistics and, despite a few promising results, the statistical analysis of homology, persistent homology and more general topological and geometric features of data is still in its infancy.

One of the first statistical results about persistent homology has been given in a parametric setting, by Bubenik and Kim in [6]. They show for instance that for data sampled on an hypersphere according to a von-Mise Fisher distribution (among other distributions), the Betti numbers can be estimated with the parametric rate . However assuming that both the support and the parametric family of the distribution are known are strong assumptions which are hardly met in practice.

Closely related to our approach, statistical analysis of homology and of persistent homology has also been proposed very recently in [3] and [2] in the specific context of manifolds, i.e. when the geometric structure underlying the data is assumed to be a smooth submanifold of a Euclidean space. In the first paper, the authors exhibit minimax rates of convergence for the estimation of the Betti numbers of the underlying manifold under different models of noise. This approach is also strongly connected to manifold estimation results obtained in [31]. Our results are in the same spirit as [3] but extend to persistent homology and allow to deal with general compact metric spaces. In the second paper, the authors develop several methods to find confidence sets for persistence diagrams using subsampling methods and kernel estimators among other approaches. Although they tackle a different problem, it has some connections with the problem considered in the present paper that we briefly mention in Section 3.4.

Both [2] and our work start from the observation that persistence diagram inference is strongly connected to the better known problem of support estimation. As far as we know, only few results about support estimation in general metric spaces have been given in the past. An interesting framework is proposed in [23]: in this paper the support estimation problem is tackled using kernel methods. On the other hand, a large amount of literature is available for measure support estimation in ; see for instance the review in [19] for more details. Note that many results on this topic are given with respect to the volume of symmetric set difference - see for instance [4] and references therein - while in our topological estimation setting we need convergence results for support estimation in Hausdorff metric.

The estimator and the Devroye and Wise estimator [24], , where denotes the closed ball centered at with radius , are both natural estimators of the support. The use of is particularly relevant when the convergence of the measure of the symmetric set difference is considered but does not provide better results than in our Hausdorff distance setting. The convergence rate of to the support of the measure with respect to the Hausdorff distance is given in [21] in . Support estimation in has also been studied under various additional assumptions such as, e.g., convexity assumptions [25, 10, 18] or through boundary fragments estimation [35, 34] just to name a few. Another classical assumption is that the measure has a density with respect to the Lebesgue measure. In this context, plug-in methods based on non parametric estimators of the density have been proposed by [20] and [43]. We consider persistence diagram estimation in the density framework of [40] in Section 4 and show in this particular context that allows us to define a persistence diagram estimator that reaches optimal rates of convergence in the minimax sense.

A few other different methods have also been proposed for topology estimation in non deterministic frameworks such as the ones based upon deconvolution approaches [8, 39]. Several recent attempts have also been made, with completely different approaches, to study statistical persistence diagrams from a statistical point of view, such as [37] that studies probability measures on the space of persistence diagrams or [5] that introduces a functional representation of persistence diagrams, the so-called persistence landscapes, allowing to define means and variance of persistence diagrams. Notice that our results should easily extend to persistence landscapes.


The paper is organized as follows. Background notions and results on metric spaces, filtered simplicial complexes, and persistent homology that are necessary to follow the paper are presented in Section 2. The rates of convergence for the estimation of persistence diagrams in general metric spaces are established in Section 3. We also study these convergence rates in for a few classical problems in Section 4. Some numerical experiments illustrating our results are given in Section 5. All the technical proofs are given in Appendix.

2 Background

2.1 Measured metric spaces

Recall that a metric space is a pair where is a set and is a nonnegative map such that for any , if and only if , and . We denote by the set of all the compact subsets of . For a point and a subset , the distance of to is the minimum over all of . The Hausdorff distance between two subsets is the maximum over all points in of their distance to and over all points in of their distance to :

Note that is a metric space and can be endowed with its Borel -algebra.

Two compact metric spaces and are isometric if there exists a bijection that preserves distances, namely: , . Such a map is called an isometry. One way to compare two metric spaces is to measure how far these two metric spaces are from being isometric. The corresponding distance is called the Gromov-Hausdorff distance (see for instance [7]). Intuitively, it is the infimum of their Hausdorff distance over all possible isometric embeddings of these two spaces into a common metric space.

Definition 2.1.

Let and be two compact metric spaces. The Gromov-Hausdorff distance is the infimum of the real numbers such that there exist a metric space and subspaces and in which are isometric to and respectively and such that . The Gromov-Hausdorff distance defines a metric on the space of isometry classes of compact metric spaces (see Theorem 7.3.30 in [7]).

Notice that when and are subspaces of a same metric space then .

Measure.

Let be a probability measure on equipped with its Borel algebra. Let denote the support of the measure , namely the smallest closed set with probability one. In the following of the paper, we will assume that is compact and thus . Also note that .

The main assumption we will need in the following of the paper provides a lower bound on the measure . We say that satisfies the standard assumption if there exist , and such that

(2.2)

where denotes the open ball of center and radius in . This assumption is popular in the literature about set estimation (see for instance [19]) but it has generally been considered with in . Since is compact, reducing the constant to a smaller constant if necessary, we easily check that assumption (2.2) is equivalent to

(2.3)

where denotes the minimum between and . We then say that satisfies the -standard assumption.

2.2 Simplicial complexes on metric spaces

Figure 2: From left to right: the sublevelset of the distance function to a point set in , the -complex, and . The last two include a tetrahedron.
Geometric complexes.

The geometric complexes we consider in this paper are built on top of metric spaces and come as nested families depending on a real parameter. Topological persistence is used to infer and encode the evolution of the topology of theses families as the parameter grows. For a complete definition of these geometric filtered complexes built on top of metric spaces and their use in TDA, we refer to [13], Section 4.2. We only give here a brief reminder and refer to Figure 2 for illustrations. A simplicial complex is a set of simplexes (points, segments, triangles, etc) such that any face from a simplex in is also in and the intersection of any two simplices of is a (possibly empty) face of these simplices. Notice that we do not assume such simplicial complexes to be finite. The complexes we consider in this paper can be seen as a generalization of neighborhood graphs in dimension larger than .

Given a metric space which will also serve as the vertex set, the Vietoris-Rips complex is the set of simplices such that for all . The Čech complex is similarly defined as the set of simplices such that the closed balls have a non-empty intersection. Note that these two complexes are related by . Note also that these two families of complexes only depend on the pairwise distances between the points of .

When is embedded in some larger metric space , we can extend the definition of the Čech complex to the set of simplices such that the closed balls have a non-empty intersection in (not just in ). We can also define the alpha-complex or -complex as the set of simplices such that, for some that depends on the simplex, the closed balls and the complement of all the other balls for have a non-empty intersection in . In the particular case where , those two complexes have the same homotopy type (they are equivalent for our purposes) as the union of the balls for , as in Figure 2, and the -complex only contains simplices of dimension at most . Note that the union of the balls is also the -sublevel set of the distance to function , and as a consequence, those filtrations thus provide a convenient way to study the evolution of the topology of union of growing balls or sublevel sets of (see Figure 2 and Section 5 for more examples).

There are several other families that we could also have considered, most notably witness complexes [13]. Extending our results to them is straightforward and yields very similar results, so we will restrict to the families defined above in the rest of the paper.

All these families of complexes have the fundamental property that they are non-decreasing with ; for any , there is an inclusion of in , and similarly for the Čech, and Alpha complexes. They are thus called filtrations. In the following, the notation denotes one of the filtrations defined above.

Persistence diagrams.

An extensive presentation of persistence diagrams is available in [15]. We recall a few definitions and results that are needed in this paper.

Figure 3: A torus filtered by its -coordinate: , its persistence barcode, and its persistence diagram.
Figure 4: An -complex filtration, the sublevelset filtration of the distance function, and their common persistence barcode (they are homotopy equivalent).

We first give the intuition behind persistence. Given a filtration as above, the topology of changes as increases: new connected components can appear, existing connected components can merge, cycles and cavities can appear and can be filled, etc. Persistent homology is a tool that tracks these changes, identifies features and associates a lifetime to them. For instance, a connected component is a feature that is born at the smallest such that the component is present in , and dies when it merges with an older connected component. Intuitively, the longer a feature persists, the more relevant it is.

We now formalize the presentation a bit. Given a filtration as above, we can apply the -homology functor 111The notion of (simplicial) homology is a classical concept in algebraic topology that provides powerful tools to formalize and handle the notion of topological features of a simplicial complex in an algebraic way. For example the -dimensional homology group represents the -dimensional features, i.e. the connected components of the complex, represents the -dimensional features (cycles), represents the -dimensional features (cavities),… See, e.g. [32] for an introduction to simplicial homology. and get a sequence of vector spaces , where the inclusions induce linear maps . In many cases, this sequence can be decomposed as a direct sum of intervals, where an interval is a sequence of the form

(the linear maps are all the identity). These intervals can be interpreted as features of the (filtered) complex, such as a connected component or a loop, that appear at parameter in the filtration and disappear at parameter . An interval is determined uniquely by these two parameters. It can be represented as a segment whose extremities have abscissae and ; the set of these segments is called the barcode of . An interval can also be represented as a point in the plane, where the -coordinate indicates the birth time and the -coordinate the death time. The set of points (with multiplicity) representing the intervals is called the persistence diagram . Note that the diagram is entirely contained in the half-plane above the diagonal defined by , since death always occurs after birth. [15] shows that this diagram is still well defined even in cases where the sequence might not be decomposable as a finite sum of intervals, and in particular is well defined for any compact metric space [13]. Note that for technical reasons, the points of the diagonal are considered as part of every persistence diagram, with infinite multiplicity. The most persistent features (supposedly the most important) are those represented by the longest bars in the barcode, i.e. the points furthest from the diagonal in the diagram, whereas points close to the diagonal can be interpreted as noise.

Figure 5: Two diagrams at bottleneck distance .

The space of persistence diagrams is endowed with a metric called the bottleneck distance . Given two persistence diagrams, it is defined as the infimum, over all perfect matchings of their points, of the largest -distance between two matched points, see Figure 5. The presence of the diagonal in all diagrams means we can consider partial matchings of the off-diagonal points, and the remaining points are matched to the diagonal. With more details, given two diagrams and , we can define a matching as a subset of such that every point of and appears exactly once in . The bottleneck distance is then:

Note that points close to the diagonal are easily matched to the diagonal, which fits with their interpretation as irrelevant noise.

A fundamental property of persistence diagrams, proved in [15], is their stability. If and are two compact metric spaces then one has

(2.4)

Moreover, if and are embedded in the same metric space then one has

(2.5)

Notice that these properties are only metric properties: they do not involve here any probability measure on and .

3 Persistence diagrams estimation in metric spaces

Let be a metric space. Assume that we observe points in drawn i.i.d. from some unknown measure whose support is a compact set denoted .

3.1 From support estimation to persistence diagram estimation

The Gromov-Hausdorff distance allows to compare with compact metric spaces not necessarily embedded in . We thus consider as an element of (rather than an element of ). In the following, an estimator of is thus a function of which takes its values in and which is measurable for the Borel algebra induced by .

Let and be two filtrations defined on and . The statistical analysis of persistence diagrams proposed in the sequel starts from the following key fact: according to (2.4), for any :

(3.1)

where the probability corresponds to the product measure . Our strategy then consists in finding an estimator of the support which is close for the distance. Note that this general strategy of estimating in is not only of theoretical interest. Indeed as mentioned in the introduction, in some cases the space is unknown and the observations are just known through their matrix of pairwise distances , . The use of the Gromov-Hausdorff distance then allows to consider this set of observations as an abstract metric space of cardinality without taking care of the way it is embedded in .

This general framework embraces the more standard approach consisting in estimating the support by restraining the values of to . According to (2.5), in this case, for any :

(3.2)

Thanks to equations (3.1) and (3.2) the problem of persistence diagrams estimation boils down to the better known problem of estimating the support of a measure.

Let be a set of independent observations sampled according to endowed with the restriction of the distance . This finite metric space is a natural estimator of the support . In several contexts discussed in the following, shows optimal rates of convergence for the estimation of with respect to the Hausdorff and Gromov-Hausdorff distance. From (3.2) we will then obtain upper bounds on the rate of convergence of , and we will need to find the corresponding lower bounds to prove the optimality for topological inference issue.

In the next subsection, we tackle persistence diagram estimation in the general framework of abstract metric spaces. We will consider more particular contexts later in the paper.

3.2 Convergence of persistence diagrams

Cuevas and Rodríguez-Casal give in [21] the rate of convergence in Hausdorff distance of for some probability measure satisfying an -standard assumption on . In this section, we consider the more general context where is a probability measure satisfying an -standard assumption on a metric space , with . We give below the rate of convergence of in this context. The proof follows the lines of the proof of Theorem 3 in [21].

Theorem 3.3.

Assume that a probability measure on satisfies the -standard assumption. Then, for any :

Moreover, there exist two constants and only depending on and such that

and

Since the above theorem also holds when the Gromov distance is replaced by the Gromov-Hausdorff distance. In practice this allows to consider as an abstract metric space without taking care of the way it is embedded in the, possibly unknown, metric space .

Using (3.1) and (2.5), we then derive from the previous result the following corollary for the convergence rate of the persistence diagram toward .

Corollary 3.4.

Assume that the probability measure on satisfies the -standard assumption, then for any :

(3.5)

Moreover,

and

where and are the same constants as in Theorem 3.3.

3.3 Optimal rate of convergence

Let be the set of all the probability measures on the metric space satisfying the -standard assumption on .

The next theorem gives upper and lower bounds for the rate of convergence of persistence diagrams. The upper bound comes as a consequence of Corollary 3.4, while the lower bound is established using the so-called Le Cam’s lemma (see Lemma A.1 in Appendix).

Theorem 3.6.

Let be a metric space and let and . Then:

(3.7)

where the constant only depends on and (not on ). Assume moreover that there exists a non isolated point in and consider any sequence such that . Then for any estimator of :

where is an absolute constant.

Consequently, the estimator is minimax optimal on the space up to a logarithmic term as soon as we can find a non-isolated point in and a sequence in such that . This is obviously the case for the Euclidean space .

3.4 Confidence sets for persistence diagrams

Corollary 3.4 can also be used to find confidence sets for persistence diagrams. Assume that and are known and let . Then for ,

is a confidence region for of level . Nevertheless, in practice the coefficients and can be unknown. In , the coefficient can be taken equal to the ambient dimension in many situations. Finding lower bounds on the coefficient is a tricky problem that is out of the scope of the paper. Alternative solutions have been proposed recently in [2] and we refer the reader to this paper for more details.

4 Persistence diagram estimation in

In this section, we study the convergence rates of persistence diagram estimators for data embedded in . In particular we study two situations of interest proposed respectively in [40] and [31] in the context of measure support estimation. In the first situation the measure has a density with respect to the Lebesgue measure on whose behavior is controlled near the boundary of its support. In the second case, the measure is supported on a manifold. These two frameworks are complementary and provide realistic frameworks for topological inference in .

4.1 Optimal persistence diagram estimation for nonsingular measures on

Paper [40] is a significant breakthrough for level set estimation through density estimation. It presents a fully data-driven procedure, in the spirit of Lepski’s method, that is adaptive to unknown local density regularity and achieves a Hausdorff error control that is minimax optimal for a class of level sets with very general shapes. In particular, the assumptions of [40] describe the smoothness of the density near the boundary of the support.

In this section, we propose to study persistence diagram inference in the framework of [40] since this framework is very intuitive and natural. Nevertheless, we do not use the estimator of [40] for this task since we only consider here the support estimation problem (and not the more general level set issue as in [40]). Indeed, we will see that the estimator has the optimal rate of convergence for estimating the support according to , as well as for estimating the persistence diagram. We now recall the framework of [40, Section 4.3] corresponding to support set estimation.

Let be i.i.d. observations drawn from an unknown probability measure having density with respect to the Lebesgue measure and defined on a compact set . Let denote the support of , and let . The boundary of a set is denoted and for any , is the -inner of . The two main assumptions of [40] are the following:

: the density is upper bounded by and there exist constants , , such that for all with ,

: there exist constants and such that for all , and for all .

We denote by the set composed of all the densities on satisfying assumptions and , for a fixed set of positive constants , , , , , and .

Assumption describes how fast the density increases in the neighborhood of the boundary of the support: the smaller , the easier the support may be possible to detect. Assumption prevents the boundary from having arbitrarily small features (as for cusps). We refer to [40] for more details and discussions about these two assumptions and their connections with assumptions in other works.

For persistence diagram estimation, we are interested in estimating the support whereas the assumptions and involve the set . However, as stated in Lemma B.4 (given in Appendix B.3), these two sets are here almost identical in the sense that . Moreover, it can be proved that under assumptions and , the measure also satisfies the standard assumption with (see Lemma B.4). According to Proposition 3.6, the estimator thus converges in expectation towards with a rate upper bounded by . We also show that this rate is minimax over the sets by adapting the ideas of the proof given in [40] for the Hausdorff lower bound.

Proposition 4.1.
  1. For all ,

    where is a constant depending only on , , , , , and .

  2. There exists such that

    for large enough. The infimum is taken over all possible estimators of based on observations.

Remark.

Paper [40] is more generally about adaptive level set estimation. For this problem, Singh et al. define an histogram based estimator. Let denote the collection of cells, in a regular partition of into hypercubes of dyadic side length . Their estimator is the histogram , where . For estimating the level set , they consider the estimator

It is proved in [40] that achieves optimal rates of convergence for estimating the level sets, with chosen in a data driven way. Concerning support estimation, they also show that achieves optimal rates of convergence for estimating . We have seen that in this context it is also the case for the estimator . Since no knowledge of is required for this last estimator, we thus prefer to use this simpler estimator in this context.

4.2 Optimal rates of convergence of persistence diagram estimation for singular measures in

In this subsection, we consider the estimation of the support of a singular measure embedded in . A classical assumption in this context is to suppose that the support of the singular measure is a Riemannian manifold. As far as we know, rates of convergence for manifold estimation, namely for the estimation of the support of a singular probability measure supported on a Riemannian manifold of , have only been studied recently in [31] and [30]. These papers assume several noise models, which all could be considered in this context of persistence diagram estimation. However, for the sake of simplicity, we only study here the problem where no additional noise is observed, which is referred as the noiseless model in the first of these two papers. As before, upper bounds given in [31] on the rates of convergence for the support estimation in Hausdorff distance directly provide upper bounds on the rates of convergence of the persistence diagram of the support. Before giving the rates of convergence we first recall and discuss the assumptions of [31].

For any and any set , let . Let be the largest such that each point in has a unique projection onto , this quantity has been introduced by Federer in [29], it is called reach or condition number in the literature.

For a fixed positive integer , for some fixed positive constants , , and for a fixed compact domain in , [31] defines the set of probability measures on satisfying the two following assumptions:

  • The support of the measure is a compact Riemannian manifold (included in ) of dimension whose reach satisfies

    (4.2)
  • The measure is assumed to have a density with respect to -dimensional volume measure on , such that

    (4.3)

These two assumptions can be easily connected to the standard assumption. Indeed, according to [38] and using , for all there exists some constant such that for any , we have

and the same holds for according to . Thus, if we take for estimating the support in this context, we then obtain the rate of convergence according to Theorem 3.3. Nevertheless, this rate is not minimax optimal on the spaces as shown by Theorem 2 in [31]. Indeed the correct rate is . For proving this result, [31] proposes some “theoretical” estimator that can not be computed in practice. As far as we know, no usable and optimal estimator has been proposed in the literature for this issue. In consequence, the situation is the same for the estimation of persistence diagrams in this context. The following proposition shows that the optimal rates of convergence for support estimation are the same as for the persistence diagram estimation in this context.

Proposition 4.4.

Assume that we observe an -sample under the previous assumptions, then there exist two constants and depending only on such that

(4.5)

where the infimum is taken over all the estimators of the persistence diagram.

Of course this result is only of theoretical interest since it is not based on estimators which are usable in practice.

5 Experiments

A series of experiments were conducted in order to illustrate the behavior of the persistence diagrams under sampling of metric spaces endowed with a probability measure and to compare the convergence performance obtained in practice with the theoretical results obtained in the previous sections.

Spaces and data.

We consider four different metric spaces, denoted , , and hereafter, that are described below.

  • (Lissajous curve in ): the planar curve with the parametric equations , , (see Figure 6, left). Its metric is the restriction of the Euclidean metric in and it is endowed with the push forward by the parametrization of the uniform measure on the interval .

  • (sphere in ): the unit sphere in (see Figure 6, center). Its metric is the restriction of the Euclidean metric in and it is endowed with the uniform area measure on the sphere.

  • (torus in ): the torus of revolution in with the parametric equations , and , (see Figure 6, right). Its metric is the restriction of the Euclidean metric in and it is endowed with the push forward by the parametrization of the uniform measure on the square .

  • (rotating shape space): for this space we used a 3D character from the SCAPE database [1] and considered all the images of this character from a view rotating around it. We converted these images in gray color and resized these images to pixels (see Figure 7). Each is then identified with a point in where the coordinate is the level of gray of the pixel. Moreover, we normalized these images by projecting them on the unit sphere in . The metric space is the obtained subset of the unit sphere with the restriction of the Euclidean metric in . As it is parametrized by a circular set of views, it is endowed with the push forward of the uniform measure on the circle.

Figure 6: The spaces , and .
Figure 7: Images sampled from the space .
The experiments.

From each of the measured metric spaces , , and we sampled sets of points for different values of from which we computed persistence diagrams for different geometric complexes (see Table 1). For , and we have computed the persistence diagrams for the or -dimensional homology of the -complex built on top of the sampled sets. As -complexes have the same homotopy type as the corresponding union of balls, these persistence diagrams are the ones of the distance function to the sampled point set [26]. So, for each we computed the average bottleneck distance between the obtained diagrams and the persistence diagram of the distance to the metric space from which the points were sampled. For , as it is embedded in a very high dimensional space, computing the -complex is practically out of reach. So we have computed the persistence diagrams for the -dimensional homology of the Vietoris-Rips complex built on top of the sampled sets. The obtained results are described and discussed below.

  • Results for : we approximated the -dimensional homology persistence diagram of the distance function to the Lissajous curve by sampling with points and computing the persistence diagram of the corresponding -complex. As the Hausdorff distance between our sample and was of order we obtained a sufficiently precise approximation of for our purpose. The diagram is represented in blue on the left of Figure 8. For each , the average bottleneck distance between and the persistence diagrams obtained for the randomly sampled sets of size has been used as an estimate of where denotes the -complex filtration. is plotted as a function of on Figure 8, right. As expected, since the Lissajous curve is -dimensional, the points are close to a line of slope .

  • Results for and : the persistence diagrams and of the distance functions to and are known exactly and are represented in blue on Figures 9 and 10, left, respectively. Notice that we considered the -dimensional homology for and -dimensional homology for . For and for each , the average bottleneck distance between and the persistence diagrams obtained for the randomly sampled sets of size has been used as an estimate of where denotes the -complex filtration. is plotted as a function of on Figures 9 and 10, right. As expected, since the sphere and the torus are -dimensional, the points are close to a line of slope .

  • Results for : As in that case we do not know the persistence diagram of the Vietoris-Rips filtration built on top of , we only computed the -dimensional homology persistence diagrams of the Vietoris-Rips filtrations built on top of sets of points each, randomly sampled on . All these diagrams have been plotted on the same Figure 11, left. The right of Figure 11 represents a 2D embedding of one of the points sampled data set using the Multidimensional Scaling algorithm (MDS). Since is a set of images taken according a rotating point of view, it carries a cycle structure. This structure is reflected in the persistence diagrams that all have one point which is clearly off the diagonal. Notice also a second point off the diagonal which is much closer to it and that probably corresponds to the pinching in visible at the bottom left of the MDS projection.

Space (sampled sets for each ) range Geometric complex
300 -complex
100 -complex
100 -complex
20 250 Vietoris-Rips complex
Table 1: Sampling parameters and geometric complexes where denotes the set of integers .
Figure 8: Convergence rate for the persistence diagram of the -filtration built on top of points sampled on . Left: in blue the persistence diagram of the distance to (-dimensional homology); in red a persistence diagram of the -filtration built on top of points randomly sampled on . Right: the -axis is where is the number of points sampled on . The -axis is the of the estimated expectation of the bottleneck distance between the diagram obtained from an -filtration built on top of points sampled on and .
Figure 9: Convergence rate for the persistence diagram of the -filtration built on top of points sampled on . Left: in blue the persistence diagram of the distance to (-dimensional homology); in red a persistence diagram of the -filtration built on top of points randomly sampled on . Right: the -axis is where is the number of points sampled on . The -axis is the of the estimated expectation of the bottleneck distance between the diagram obtained from an -filtration built on top of points sampled on and .
Figure 10: Convergence rate for the persistence diagram of the -filtration built on top of points sampled on . Left: in blue the persistence diagram of the distance to (-dimensional homology); in red a persistence diagram of the -filtration built on top of points randomly sampled on . Right: the -axis is where is the number of points sampled on . The -axis is the of the estimated expectation of the bottleneck distance between the diagram obtain from -filtration built on top of points sampled on and .
Figure 11: Left: on the same figure the -dimensional homology persistence diagrams of the Vietoris-Rips filtration of sets of points sampled on . Right: the plot of the embedding of in using MDS.

6 Discussion and future works

In previous works, the use of persistent homology in TDA has been mainly considered with a deterministic approach. As a consequence persistence diagrams were usually used as exploratory tools to analyze the topological structure of data. In this paper, we propose a rigorous framework to study the statistical properties of persistent homology and more precisely we give a general approach to study the rates of convergence for the estimation of persistence diagrams. The results we obtain open the door to a rigorous use of persistence diagrams in statistical framework. Our approach, consisting in reducing persistence diagram estimation to another more classical estimation problem (here support estimation) is based upon recently proven stability results in persistence theory that are very general. As a consequence, our approach can be adapted to other frameworks. For example, the estimation of persistence diagrams of functions (e.g. densities) is strongly connected to the problem of the approximation of such functions with respect to the sup norm. In particular, building on ideas developed in [14] and [8], we intend to extend our results to persistence diagram estimation of distance-to-measure functions and topological inference from data corrupted by different kind of noise.

In another direction, an interesting representation of persistence diagrams as elements of a Hilbert space has recently been proposed in [5]. Our results easily extend to this representation of persistence diagrams called persistence landscapes. Following this promising point of view, we also intend to adapt classical kernel-based methods with kernels carrying topological information.

Appendix A Lecam’s Lemma

The version of Lecam’s Lemma given below is from [45] (see also [30]). Recall that the total variation distance between two distributions and on a measured space is defined by

Moreover, if and have densities and for the same measure on , then

Lemma A.1.

Let be a set of distributions. For , let take values in a metric space . Let and in be any pair of distributions. Let be drawn i.i.d. from some . Let be any estimator of , then