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Understanding the Topology and the Geometry of the Persistence Diagram Space Via Optimal Partial Transport Vincent Divol, Théo Lacombe

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Understanding the Topology and the Geometry of the Persistence Diagram Space Via Optimal Partial Transport Vincent Divol, Théo Lacombe Understanding the Topology and the Geometry of the Persistence Diagram Space via Optimal Partial Transport Vincent Divol, Théo Lacombe To cite this version: Vincent Divol, Théo Lacombe. Understanding the Topology and the Geometry of the Persistence Diagram Space via Optimal Partial Transport. 2019. hal-01970466v1 HAL Id: hal-01970466 https://hal.inria.fr/hal-01970466v1 Preprint submitted on 9 Jan 2019 (v1), last revised 17 Jul 2020 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Understanding the Topology and the Geometry of the Persistence Diagram Space via Optimal Partial Transport Vincent Divol and Th´eoLacombe [email protected] Datashape, Inria Saclay Abstract We consider a generalization of persistence diagrams, namely Radon measures supported on the upper half plane for which we define natural extensions of Wasserstein and bottleneck distances between persis- tence diagrams. Such measures naturally appear in topological data analysis when considering continuous representations of persistence diagrams (e.g. persistence surfaces) but also as limits for laws of large numbers on persistence diagrams or as expectations of probability distributions on the persistence diagrams space. Introducing a formalism originating from the theory of optimal partial transport, we build a convenient framework to prove topological properties of this new space, which will also hold for the closed subspace of persistence diagrams. New results include a characterization of convergence with respect to Wasserstein metrics, and the existence of barycenters (Fr´echet means) for any distribution of diagrams. We also showcase the strength of this framework by providing several statistical results made meaningful thanks to this new formalism. 1 Introduction Topological Data Analysis (TDA) is an emerging field in data analysis that has found applications in computer vision [30], material science [21, 25], shape analysis [10, 37], to name a few. The aim of TDA is to provide interpretable descriptors of the underlying topology of a given object. One of the most used (and theoretically studied) descriptors in TDA is the persistence diagram. This descriptor consists in a locally finite multiset of 2 points in the upper half plane Ω := f(t1; t2) 2 R ; t2 > t1g, each point in the diagram corresponding informally to a topological feature (connected component, loop, hole) of a given object. The space of persistence diagrams D is usually equipped with partial matching metrics dp, sometimes called Wasserstein distances [17, Chapter VIII.2]: for p 2 [1; +1) and a; b in D, define 1 ! p X p dp(a; b) := inf kx − π(x)k ; (1) π2Γ(a;b) x2a[@Ω where Γ(a; b) is the set of partial matchings between a and b, i.e. bijections between a [ @Ω and b [ @Ω, @Ω := f(t; t); t 2 Rg being the boundary of Ω, namely the diagonal (see Figure 1). When p ! 1, we recover the so-called bottleneck distance: d1(a; b) := inf max kx − π(x)k: (2) π2Γ(a;b) x2a[@Ω An equivalent viewpoint, developed in [11, Chapter 3], is to consider a persistence diagram as a measure of P the form a = x2X nxδx, where X ⊂ Ω is locally finite and nx 2 N for all x 2 X, so that a is a locally finite measure supported on Ω with integer mass on each point of its support. Considering persistence diagrams with such a perspective suggests to study more general Radon measures1 supported on the upper half-plane Ω. Such objects appear naturally in several applications, e.g. when taking representations of persistence diagrams such as persistent surfaces [1], studying laws of large numbers for persistence diagrams [15, 16, 19], linear expectations of random persistence diagrams [12], or when estimating barycenters of persistence diagrams [28]. In Section 3, we study the space M of Radon measures supported on Ω. For finite p ≥ 1 (the case p = 1 is studied in Section 3.3), we define the persistence of µ 2 M as Z p Persp(µ) := d(x; @Ω) dµ(x); (3) Ω 1A short reminder about Radon measure theory is provided in Appendix A. 1 Ω @Ω Figure 1: An example of optimal partial matching between two diagrams. The bottleneck distance between these two diagrams is|by definition—the length of the longest edge in this matching, while their Wasserstein distance dp is the p-th root of the sum of all edge lengths to the power p. where d(x; @Ω) is the distance from a point x 2 Ω to the diagonal @Ω, and we define p M := fµ 2 M; Persp(µ) < 1g: (4) p p p Let D := D\M be the space of persistence diagrams with finite persistence p. We equip M with metrics Dp (see Definition 2.1), originally introduced in a work of Figalli and Gigli [18]. These metrics appear as extensions p p of metrics dp, as they both coincide on D (Proposition 3.2). Elements of the metric space (M ;Dp) are referred to as persistence measures in the following. As Dp is closed in Mp (Corollary 3.1), most properties of Mp hold for Dp too (e.g. being Polish, Proposition 3.3). v A sequence of Radon measures (µn)n is said to converge vaguely to a measure µ, denoted by µn −! µ, if for any continuous compactly supported function f :Ω ! R, µn(f) ! µ(f). We prove the following equivalence between convergence for the metric Dp and the vague convergence: p Theorem 3.4. Let 1 ≤ p < 1. Let µ, µ1; µ2;::: be measures in M . Then, ( v µn −! µ, Dp(µn; µ) ! 0 , (5) Persp(µn) ! Persp(µ): This characterization in particular holds for persistence diagrams in Dp. This result is analog to the characterization of convergence of probability measures in the Wasserstein space (see [39, Theorem 6.9]). A proof for Radon measures supported on a bounded set can be found in [18, Proposition 2.7]. Our work consists in extending this result to non-bounded sets, in particular to the upper half plane Ω. As a corollary of this theorem, we obtain the following general result on the continuity of linear representations of persistence measures (and diagrams). Corollary 3.2. Let p 2 [1; +1), d ≥ 1, and f :Ω ! Rd be a continuous bounded function. The feature map p d R p Φ: M ! R defined by Φ(µ) = Ω d(x; @Ω) f(x)dµ(x) is continuous with respect to Dp. This new result can be compared to the recent works [26, Proposition 3.2] and [15, Theorem 3], which show that linear feature maps can have more regularity (e.g. Lipschitz or Hlder) under additional assumptions. We also show in Section 3.2, that the problem of computing the Dp metric between two measures of finite mass can be turned into the known problem of computing a Wasserstein distance (see Section 2) between two measures with the same masses, a result having practical implications for the computation of distances dp between diagrams. Section 4 studies Fr´echet means (i.e. barycenters, see Definition 4.1) for probability distributions of persistence measures. In the specific case of persistence diagrams, the study of Fr´echet means was initiated in [31, 36], where authors prove the existence of barycenters for certain types of distributions [31, Theorem 28]. We show that this existence result is actually true for any distributions of persistence measures, and adapt them to persistence diagrams. Namely, we prove the following results: Theorem 4.3. For any probability distribution P supported on Mp with finite p-th moment, the set of barycenters of P is not empty. Theorem 4.4. If P is supported on Dp and has a finite p-th moment, then P admits a barycenter in Dp. 2 Theorem 4.3 follows the work of [29, Theorem 2] (itself following the seminal paper of Agueh and Carlier [2]), where authors prove the existence of Fr´echet mean of probability measures endowed with the Wasserstein metric. We adapt this result to the space of Radon measures endowed with the Dp metrics. Section 5 presents two applications of Theorem 3.4: Proposition 5.1 gives a law of large numbers in terms of the metric Dp, while Proposition 5.3 states a stability result between (Cech)ˇ diagrams and input point cloud in a random setting. 2 Elements of optimal transport In this section, (Ω; d) denotes a Polish space. Optimal transport is a widely developed theory providing tools to study and compare probability measures supported on Ω [38, 39, 33]. 2.1 Wasserstein distances Given two probability measures µ, ν supported on (Ω; d), the Wasserstein-p distance (p ≥ 1) induced by the metric d between these probability measures is defined as 1 ZZ p p Wp;d(µ, ν) := inf d(x; y) dπ(x; y) ; (6) π2Π(µ,ν) Ω×Ω where Π(µ, ν) denotes the set of transport plans between µ and ν, that is the set of measures on Ω × Ω which have respective marginals π(1) = µ, π(2) = ν. When there is no ambiguity on the distance d used, we simply write Wp instead of Wp;d. In order to have Wp finite, µ and ν are required to have a finite p-th moment, that R p is there exists x0 2 Ω such that Ω d(x; x0) dµ(x) (resp.
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