Gromov-Hausdorff Stable Signatures for Shapes Using Persistence

Eurographics Symposium on Geometry Processing 2009 Volume 28 (2009), Number 5 Marc Alexa and Michael Kazhdan (Guest Editors)

Gromov-Hausdorff Stable Signatures for Shapes using Persistence

Frédéric Chazal1 and David Cohen-Steiner2 and Leonidas J. Guibas3 and Facundo Mémoli4 and Steve Y. Oudot1

1INRIA Saclay 2INRIA Sophia 3Stanford University, Dept. Computer Science 4Stanford University, Dept. Mathematics

Abstract We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on the persistence diagrams of suitable filtrations built on top of these spaces. We prove the stability of our signatures under Gromov-Hausdorff perturbations of the spaces. We also extend these results to metric spaces equipped with measures. Our signatures are well-suited for the study of unstructured point cloud data, which we illustrate through an application in shape classification.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling—

1. Introduction integral invariants of [MCH∗06], the eccentricity functions of [HK03], the shape distributions of [OFCD02], the canon- In recent years, great advances have been made in the fields ical forms of [EK03], and the shape DNA and global point of data acquisition and shape modelling, and huge collec- signatures based spectral methods in [RWP05] and [Rus07], tions of digital models have been obtained. With the goal of respectively. The common underlying idea revolves around organizing these collections, it is very important to be able the computation and comparison of certain metric invariants, to define and compute meaningful notions of similarity be- or signatures, so as to ascertain whether two given data sets tween objects that exhibit invariance to different deforma- represent in fact the same object, up to a certain notion of tions or poses of the objects represented by the data. Prob- invariance. lems of this nature arise in areas such as molecular biology, metagenomics, face recognition, matching of articulated ob- The question of proving that a given family of signatures jects, graph matching, and pattern recognition in general. is indeed able to signal proximity or similarity of shapes in a reasonable way has been hardly addressed. In particular, In many practical applications, data are endowed with a the degree to which two shapes with similar signatures are notion of distance between their points, which turns them forced to be similar is in general not well understood. Con- into metric spaces. The information contained in such data versely, one can ask the more basic question of whether the sets typically takes the form of metric invariants, which it similarity between two shapes forces their signatures to be is of interest to capture and characterize. For instance, when similar. These questions cannot be completely well formu- the data have been obtained experimentally, the analysis of lated until one agrees on: (1) a notion of equality between these invariants can provide insights about the nature of the shapes, and (2) a notion of (dis)similarity between shapes. underlying phenomenological science. By regarding sampled shapes as finite metric spaces, one Relevant work. In the context of shape analysis, one typ- can use the Gromov-Hausdorff distance [Gro99] as a mea- ically wishes to be able to discriminate shapes under dif- sure of dissimilarity between shapes [MS04, MS05]. By en- ferent notions of invariance. Many approaches have been dowing the point clouds with different metrics, one obtains a proposed for the problem of (pose invariant) shape classi- great deal of flexibility in the different degrees of invariance fication and recognition, including the size theory of Frosini that one can encode in the measure of dissimilarity. For in- and collaborators [Fro90, CFL06a, CFL06b], the work of stance, using the Euclidean metric in Rd makes the Gromov- Hilaga et al. [HSKK01], the shape contexts [BMP02], the Hausdorff distance invariant under ambient rigid isome-

c 2009 The Author(s) Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. F. Chazal, D. Cohen-Steiner, L. Guibas, F. Mémoli, S. Oudot / Gromov-Hausdorff Stable Signatures for Shapes using Persistence tries [Mém08]. In contrast, using intrinsic metrics within the • Second, we extend the approach to the case of met- shapes makes the Gromov-Hausdorff distance blind to in- ric spaces endowed with real-valued functions. This latter trinsic isometries, such as when a same animal is represented step enables the design of new signatures for finite met- in different poses (Figure 1). Bronstein et al. [BBK09] sug- ric spaces by considering canonically-defined functions on gested to combine intrinsic and extrinsic metrics to increase these spaces, such as eccentricities (Section 4). As a result, the discriminating power of shape signatures. We illustrate more shapes can be discriminated using our approach. For this idea in Figure 6, where the first two shapes (circle and instance, in Figure 6, the sphere and ellipsoid cannot be helicoidal curve on a torus) have same length and are there- discriminated using standard Rips filtrations (see Definition fore isometric embeddings of a same object into R3. As a 2.7) because their diagrams are the same up to rescaling and result, signatures obtained from geodesic distances are iden- small noise (rows 2 and 3). In contrast, using a fraction of tical whereas signatures based on Euclidean distances differ. the eccentricity as function modifies the shapes of the 1- and 2-dimensional diagrams of the ellipsoid significantly, while the ones for the sphere hardly change due to the fact that the eccentricity is constant (row 4). Our signatures are also adapted to deal with measure met- 1 ric spaces [Gro99, Chapter 3. 2 ], where additional information is provided in the form of a collection of weights, one per point of the shape. In this case, stability is guaranteed under small perturbations in the Gromov-Wasserstein distance (Section 5). Since the computation of the Gromov-Hausdorff distance leads to hard combinatorial problems akin to the quadratic assignment problem (which is NP hard), the construction Figure 1: The database of shapes used in our experiments. of easily computable Gromov-Hausdorff stable invariants is very important. We show in Section 6 that the computa- bottleneck assignment Once the finite metric space representation of shapes has tion of our invariants yields simple problems been adopted, the question becomes whether a given fam- , which can be solved in polynomial time. In con- ily of signatures is stable under perturbations of the shape trast with other approaches that generally involve some del- in the Gromov-Hausdorff distance. A somewhat different icate numerical algorithms and/or require the availability of version of the problem is when the spaces are endowed a mesh of the underlying object [MS04, BBK06], the algo- with measures, a case in which perturbations are under- rithm for computing and comparing our signatures is entirely stood in the sense of the (l∞) Gromov-Wasserstein dis- combinatorial and can be applied on the sole input of a distance [Mém07]. These questions about stability of invari- tance matrix. ants are addressed in [Mém07] for a limited set of signa- Finally, our invariants prove to be useful for discriminat- tures [HK03, MCH∗06, BMP02, OFCD02]. ing between different classes of shapes in a classification task performed on the database of Figure 1. The results are Our contributions. Using tools from persistent homology presented in Section 7. [ELZ00, ZC05], we construct a novel class of Gromov- Hausdorff stable metric invariants. Our approach is in the 2. Mathematical background spirit of [CZCG05] and [dFL06,dFL05] in that we try to devise certain topologically motivated signatures that naturally 2.1. Metric spaces and Gromov-Hausdorff distance(s) combine the classificatory power of topology with the flexi- Recall that a metric space is a pair (X,d ) where X is a set bility of metrics. X and dX : X × X → R is a non-negative map such that for We enrich the pool of shape signatures available in the any x,y,z ∈ X, dX (x,y)= 0 if and only if x = y, dX (x,y)= literature in two ways (see Section 3): dX (y,x) and dX (x,z) ≤ dX (x,y)+ dX (y,z). • First, we extend zero-dimensional connectivity-based sig- In the sequel X denotes the collection of all compact met- natures such as the dengrograms of [CM09] or the invari- ric spaces. Two compact spaces (X,dX ) and (Y,dY ) are said ants computed by size theory [dFL06, dFL05] to higher- to be isometric if there exists a bijection φ : X → Y that ′ ′ dimensional homology classes, and we prove their stability preserves distances, namely: ∀x,x ∈ X, dY (φ(x),φ(x )) = ′ under small Gromov-Hausdorff perturbations. This greatly dX (x,x ). Such a map is called an isometry. The set of isom- increases the discriminative power of the approach, as illus- etry classes of compact metric spaces can be endowed with trated in Figure 6, where all shapes have been ε-sampled for a distance called the Gromov-Hausdorff distance [BBI01, a same value of ε and therefore all zero-dimensional dia- Thm. 7.3.30], whose definition uses the following notion of grams are the same (rows 2 and 3). correspondence between sets [BBI01, Def. 7.3.17]:

1 Definition 2.1 A correspondence between two sets X and we have dC(x,y) ≤ 2 kΓX,Y kl∞(C×C) and | fX (x) − fY (y)| ≤ Y is a subset C ⊂ X × Y such that: ∀ x ∈ X, ∃y ∈ Y s.t. k fX − fY kl∞(C) whenever (x,y) ∈ C. (x,y) ∈ C, and ∀ y ∈ Y, ∃x ∈ X s.t. (x,y) ∈ C. The set of 1 all correspondences between X and Y is denoted C(X,Y). Theorem 2.5 dGH defines a metric on the set of isomorphism classes of X1. Definition 2.2 The Gromov-Hausdorff distance between The proof of this result is a direct adaptation of the proof compact metric spaces (X,dX ), (Y,dY ) is: [BBI01, Thm. 7.3.30]. 1 dGH((X,dX ),(Y,dY )) = inf kΓX,Y kl∞(C×C), (1) 2 C∈C(X,Y) 2.2. Filtrations, persistence diagrams and stability + where ΓX,Y : X × Y × X × Y → R is defined by ′ ′ ′ ′ (x,y,x ,y ) 7→ |dX (x,x ) − dY (y,y )| and the notation Throughout the paper we consider simplicial homology with ′ ′ coefficients in a field, and we make a large use of topolog- kΓX,Y kl∞(C×C) stands for sup(x,y),(x′,y′)∈C ΓX,Y (x,y,x ,y ). ical persistence theory [ELZ00, ZC05]. Thorough introduc- Another way of defining the Gromov-Hausdorff distance tions to these subjects can be found in [Mun84,EH,CCS07]. is via common isometric embeddings of the two spaces. Recall that a simplicial complex K is a finite collection of Specifically, given two compact subsets X,Y of a same met- simplices such that every face of a simplex of K is also in K ric space (Z,dZ), the Hausdorff distance between X and Y is and the intersection of any two simplices is either empty or the quantity: a common face of each of them. A filtration K of a sim- Z plicial complex K is a nested sequence of subcomplexes dH(X,Y)= max{maxmindZ(x,y), maxmindZ(x,y)}. x∈X y∈Y y∈Y x∈X ∅ = Kα0 ⊆ Kα1 ⊆···⊆ Kαm = K, where α0 < α1 < ··· < αm The Gromov-Hausdorff distance is then defined as follows: is an ordered sequence of real numbers. The inclusion maps between the subcomplexes induce a directed system of vec- Definition 2.3 The Gromov-Hausdorff distance between tor spaces, called a persistence module, involving their k- compact metric spaces (X,dX ), (Y,dY ) is: dimensional homology groups: Z d ((X,d ),(Y,d )) = inf d (γ (X),γ (Y)), (2) 1 2 m GH X Y H X Y φ φ φm−1 Z,γX ,γY 0 1 Hk(Kα0 ) −→ Hk(Kα1 ) −→··· −→ Hk(Kαm ). (4) where γ γ range over all the isometric embeddings of X Y X , Y , The structure of this persistence module can be encoded as a into some same metric space (Z,dZ). multi-set of points DkK, called the k-th persistence diagram Definitions 2.2 and 2.3 are known to be equivalent [BBI01, of K (see Figure 2). Intuitively, each point p ∈ DkK encodes Thm. 7.3.25]. In particular, when the spaces are compact, the the lifespan of some k-dimensional homological feature ap- proof of equivalence shows that the infima in Eqs. (1) and (2) pearing at time px and dying at time py in the filtration K. are in fact minima. More formally, DkK is a multi-subset of the extended plane R2 R R We now turn our focus to compact metric spaces endowed , where = ∪ {−∞,+∞}, contained in the union of the extended diagonal ∆ = {(x,x) : x ∈ R} and of the grid with continuous real-valued functions. Let X1 be the collection of all such spaces: {α0,··· ,αm} × {α0,··· ,αm,α∞ =+∞}. The multiplicity of the points of ∆ is set to +∞, while the multiplicities of the R X1 = {(X,dX , fX ) | (X,dX ) ∈X , fX : X → continuous}. (αi,α j), 0 ≤ i < j ≤ +∞, are defined in terms of the ranks of j j i+1 ∗ We deem two spaces X,Y ∈X1 isomorphic whenever there the homomorphisms φi = φ j−1 ◦ ··· ◦ φi . See [CCSG 09] exists an isometry φ : X → Y s.t. fX = fY ◦φ. We then extend for a precise definition. the Gromov-Hausdorff distance as follows: The concept of persistence extends to the case of a fil- 1 tration defined by the sublevel-sets of a continuous func- Definition 2.4 Given X,Y ∈X1, dGH(X,Y)= tion f : X → R, under some tameness condition [CSEH05] 1 −1 ∞ ∞ stating that the family of inclusions f ((−∞,α]) ⊆ inf max kΓX,Y kl (C×C), k fX − fY kl (C) (3) C∈C(X,Y) 2 f −1((−∞,β]) induces at k-dimensional homology level a persistence module of the same (finite) type as in Eq. (4). where k fX − fY kl∞(C) stands for sup(x,y)∈C | fX (x) − fY (y)|. The definition of k-th persistence diagram of such a function Note that if fX and fY are equal constant functions, then one carries over. The use of persistence to devise signatures is recovers dGH. motivated by the following stability property: As in the case of the standard Gromov-Hausdorff dis- Theorem 2.6 (Stability [CSEH05]) Let X be a triangulable tance, the infimum in Eq. (3) is a minimum when the spaces space and f ,g : X → R two tame continuous functions. Then, are compact. Moreover, the proof of [BBI01, Thm. 7.3.25] the bottleneck distance between the persistence diagrams of shows that, for any correspondence C ∈C(X,Y), the disjoint 2 f and g in the extended plane R is at most k f − gk∞. union X ⊔Y can be endowed with a metric dC that coincides ∞ with dX over X × X and with dY over Y ×Y, and such that Recall that the bottleneck distance dB (A,B) between two

endowed with a real valued function f , we deﬁne a ﬁltra- tion R(X,dX , f )= {Rα(Xα)}α>0 where Xα denotes the set f −1((−∞,α]) ⊆ X.

It has been shown in [CO08] that for well-chosen values of α, the Rips complex Rα(X,dX ) of a finite metric space (X,dX ) can be used to recover the homology of an unknown shape that is sufficiently densely sampled by X. In this pa- ∞ ∞ per, we adopt a more general point of view by studying the 0.75 stability properties of the entire Rips filtration R(X,dX ).

0.5

2.3. Technical results

0.25 The following results will be used in the proofs but are not 0 0 0 necessary to understand the general approach. We begin with a standard result (see e.g. example 3.5.3 and exercise 3.5.4 Figure 2: Top row, left: a set X of 100 points sampled uni- formly at random in i.i.d. fashion from an offset of the unit of [BBI01]) that will be key in the next section: circle in the plane. The point cloud is endowed with the am- Lemma 2.8 Any finite metric space of cardinality n can be bient Euclidean metric d . Top row, right: the 2-skeleton E isometrically embedded into (Rn,l∞). of the Rips filtration R(X,dE ), containing 4,950 edges and 161,700 triangles. The simplices are colored accord- In view of this lemma, Rips filtrations of finite subsets X ing to their time of appearance in the filtration, from light of Rn endowed with the l∞-distance are of particular inter- blue to orange. Bottom row: the 0-dimensional (left) and 1- est for the rest of the paper. In particular they are closely dimensional (right) persistence diagrams of R(X,dE ). related to the so-called Cechˇ filtration of X within (Rn,l∞), which is also defined as a nested family of abstract simplicial complexes of vertex set X. Specifically, for all α > 0, 2 ∞ Rn ∞ multi-sets in (R ,l ) is the quantity minγ maxp A kp − the simplices of Cα(X, ,l ) correspond to the non-empty ∈ ∞ γ(p)k∞, where γ ranges over all bijections from A to B. subsets of X that are centers of open l -balls of same radius α with a non-empty common intersection in Rn. Theorem 2.6 has been recently extended to wider classes of functions and spaces using a new notion of prox- Lemma 2.9 ( [GM05]) For any finite set X ⊂ Rn and any n ∞ ∞ imity between persistence modules called ε-interleaving α > 0, one has Cα(X,R ,l )= Rα(X,l ). [CCSG∗09]. This extended stability result will be instru- n ∞ mental in the proof of Theorem 3.2. Note that the Cechˇ complex Cα(X,R ,l ) is nothing but ∞ From a practical point of view, the persistence diagrams the nerve of the collection of open l -balls of same radius α of a given filtration can be efficiently computed using the so- about the points of X. The topological structures of unions of Rn ∞ called persistence algorithm [ELZ00, ZC05]. In this paper open balls in ( ,l ) are related to the ones of their nerves we use a particular family of complexes, called Rips com- through the following extension of the Nerve Lemma which, plexes, to build filtrations on top of finite metric spaces (pos- more generally, relates good open covers to their nerves. sibly endowed with functions), and we use their persistence Given a topological space S and a family U = {Ux}x∈X of diagrams as signatures for the spaces, as shown in Figure 2. open subsets covering S, the family defines a good cover if for every finite subset Y of X the common intersection Definition 2.7 Given a finite metric space (X,d ) and a X y∈Y Uy is either empty or contractible. The nerve NU is parameter α > 0, the Rips complex Rα(X,dX ) is the ab- theT abstract simplicial complex of vertex set X such that stract simplicial complex of vertex set X, whose simplices x0,··· ,xk form a simplex if and only if Ux0 ∩···∩Uxk 6= ∅. correspond to the non-empty subsets of X of diameter less † ′ than 2α in the metric dX . The Rips filtration of (X,dX ), Lemma 2.10 (Persistent Nerve [CO08]) Let S ⊆ S be ′ noted R(X,dX ), is the nested family of Rips complexes ob- two paracompact spaces, and let U = {Ux}x∈X and U = ′ ′ tained by varying parameter α from 0 to +∞. When X is {Ux′ }x′∈X ′ be good open covers of S and S respectively, ′ ′ based on finite parameter sets X ⊆ X such that Ux ⊆ Ux x X S † for all ∈ . Then, the homotopy equivalences NU → Note that we depart from the usual definition, where the Rips and NU′ → S′ provided by the Nerve Theorem [Hat01, simplices correspond to subsets of X of diameter less than α. The 4G] commute with the canonical inclusions S ֒→ S′ and§ reason for this choice is purely formal: it boils down to eliminating ′ .factors of 2 in the indices. NU ֒→NU at homology level

3. Persistence based lower bounds for the persistence diagram of δX . The same is true for Y, there- Gromov-Hausdorff distance fore the persistence diagrams of R(X,dX ) and R(Y,dY ) lie at bottleneck distance at most ε of each other. Computing the Gromov-Hausdorff distance between two finite metric spaces appears to be practically intractable in Theorem 3.1 extends to finite metric spaces endowed with most cases. To distinguish such spaces from a metric point of real valued functions as follows: view, it is thus important to exhibit reasonable lower bounds whose computation is tractable. The results of this section Theorem 3.2 Let (X,dX ) and (Y,dY ) be two finite metric show that the bottleneck distances between persistence dia- spaces endowed respectively with f : X → R and g : Y → grams of Rips filtrations of finite metric spaces are reason- R. Then, for any k ∈ N, the bottleneck distance between able lower bounds. the persistence diagrams of the filtrations R(X,dX , f ) and 1 R(Y,dY ,g) is at most dGH((X,dX , f ), (Y,dY ,g)). Theorem 3.1 For any finite metric spaces (X,dX ) and (Y,dY ), for any k ∈ N, In the limit case where f = g ≡ 0, R(X,dX , f ) and ∞ R(Y,dY ,g) become respectively R(X,dX ) and R(Y,dY ), dB DkR(X,dX ),R(Y,dY ) ≤ dGH((X,dX ),(Y,dY )). 1 and we have dGH = dGH. Thus, we obtain the statement of Note that this lower bound on the Gromov-Hausdorff dis- Theorem 3.1 as a special case of the above result. tance is tight. For instance, take for X a set of two points at 1 distance 2 and for Y a set of two points at distance 2 + 2ε. Proof of Theorem 3.2. Let ε = dGH((X,dX , f ), (Y,dY ,g)). We follow the same strategy as in the proof of Theorem Then, (X,dX ) and (Y,dY ) can be isometrically embedded −1 3.1. For all α ∈ R, we let Xα = f ((−∞,α]) ⊆ X and into the real line, with X mapped to {0,2} and Y mapped −1 to {−ε,2 + ε}, which shows that their Gromov-Hausdorff Yα = g ((−∞,α]) ⊆ Y. As mentionned after definition distance is at most ε. Now, the 0-dimensional persistence di- 2.4, the infimum in Eq. (3) is in fact a minimum, realized by some correspondence C ∈C(X,Y), and the disjoint union agram of the Rips filtration of (X,dX ) is made of two points, namely (0,+∞) and (0,1), while the 0-dimensional persis- Z = X ⊔Y can be endowed with a suitable metric dZ such that the canonical inclusions γX : X ֒→ Z and γY : Y ֒→ Z tence diagram of the Rips filtration of (Y,dY ) is made of are isometries onto their images, and such that for every pair (0,+∞) and (0,1 + ε), hence their bottleneck distance is ε. 1 (x,y) ∈ C we have dZ(γX (x),γY (y)) ≤ 2 kΓX,Y kl∞(C×C) ≤ ε Proof of Theorem 3.1. Let ε = dGH((X,dX ),(Y,dY )). As and | f (x) − g(y)| ≤ k f − gkl∞(C) ≤ ε. By Lemma 2.8, we mentioned after Definition 2.3, the infimum in Eq. (2) is in can then embed the finite subspace (γX (X) ∪ γY (Y),dZ) into fact a minimum, which means that there is a metric space (Rn,l∞), where n = #X + #Y, through some isometric em- (Z,dZ) and two isometric embeddings γX : X → Z and γY : bedding γ. We thus get two isometric embeddings γ ◦ γX and Z Rn ∞ Y → Z such that the Hausdorff distance dH(γX (X),γY (Y)) is γ ◦ γY of X and Y respectively into ( ,l ), such that for ε. Consider now the subspace γX (X) ∪ γY (Y) ⊆ Z, endowed every pair (x,y) ∈ C we have kγ ◦ γX (x) − γ ◦ γY (y)k∞ ≤ ε with the induced metric. Since this subspace is finite, it can and | f (x) − g(y)| ≤ ε. It follows that the offsets filtrations n ∞ α α be isometrically embedded into (R ,l ), where n = #X + {γ ◦ γX (Xα) }α>0 and {γ ◦ γY (Yα) }α>0 are ε-interleaved, #Y, by Lemma 2.8. Let γ be the isometric embedding. We in the following sense: ∀α > 0, ∞ Z then have dH (γ◦γX (X),γ◦γY (Y)) = dH(γX (X),γY (Y)) = ε. α α+ε α+2ε n ∞ γ ◦ γX (Xα) ⊆ γ ◦ γY (Y(α+ε)) ⊆ γ ◦ γX (X(α+2ε)) . Hence, inside (R ,l ), the distance function δX to γ◦γX (X) α and the distance function δY to γ ◦ γY (Y) are ε-close in the Indeed, for any point p ∈ γ◦γX (Xα) , there is an x ∈ X such max norm. Moreover, since δX and δY are lower envelopes that kp−γ◦γX (x)k∞ ≤ α. Letting y ∈Y be such that (x,y) ∈ of finitely many piecewise-linear functions, they are them- C, our embedding ensures that kγ ◦ γX (x) − γ ◦ γY (y)k∞ ≤ ε selves piecewise-linear and therefore tame and continuous. and g(y) ≤ f (x)+ε ≤ α+ε. Hence, y belongs to Y(α+ε), and As a consequence, their persistence diagrams are ε-close in by the triangle inequality we have kp−γ◦γY (y)k∞ ≤ α+ε. α+ε the bottleneck distance, by Theorem 2.6. It follows that p ∈ γ ◦ γY (Yα+ε) . The second inclusion in the above equation follows by symmetry. Recall now that for all α ∈ R, the open α-sublevel-set of ∞ δX is the α-offset of γ ◦ γX (X), i.e. the union of the open l - Since the two offsets filtrations are ε-interleaved, ∗ balls of same radius α about the points of γ ◦ γX (X), noted the extended stability Theorem [CCSG 09, Thm α ∞ γ◦γX (X) . Since the l -balls are hypercubes, they are con- 4.4] guarantees that their persistence diagrams are vex and therefore their intersections are either empty or con- ε-close to each other in the bottleneck distance. By tractible. As a result, the Persistent Nerve Lemma 2.10 en- Lemma 2.10, so are the persistence diagrams of the n ∞ n ∞ sures that δX and the Cechˇ filtration C(γ ◦ γX (X),R ,l ) Cechˇ filtrations {Cα(γ ◦ γX (Xα),R ,l )}α>0 and n ∞ have identical persistence diagrams. Furthermore, Lemma {Cα(γ ◦ γY (Yα),R ,l )}α>0. By Lemma 2.9, so n ∞ 2.9 guarantees that C(γ ◦ γX (X),R ,l ) coincides with the are the persistence diagrams of the Rips filtrations ∞ ∞ ∞ Rips filtration R(γ ◦ γX (X),l ), which in return is equal to {Rα(γ ◦ γX (Xα),l )}α>0 and {Rα(γ ◦ γY (Yα),l )}α>0. R(X,dX ) since γ◦γX is an isometric embedding. As a conse- Finally, so are the diagrams of R(X,dX , fX ) and quence, the persistence diagram of R(X,dX ) coincides with R(Y,dY , fY ), since these filtrations coincide with

∞ ∞ X {Rα(γ ◦ γX (Xα),l )}α>0 and {Rα(γ ◦ γY (Yα),l )}α>0 An alternative to em. Instead of the eccentricies, one may respectively. consider using the following inductive family of functions: X X ′ X ′ f0 ≡ 0, and fm+1(x) = maxx′∈X dX (x,x )+ fm (x ) . It can be shown that for each m ∈ N the map hm : (X,dX ) 7→ 4. Persistence-based signatures for finite metric spaces X (X,dX , fm ) is in Hm. The complexity associated with com- 2 The results of section 3 suggest that the persistence di- puting hm is O(m · N ), where N is the cardinality of the agrams of Rips filtrations built on top of finite metric metric space. spaces endowed with well-chosen functions (see Definition 4.1) provide stable signatures to distinguish between differ- Remark 1 The above examples give rise to a large variety ent shapes. of maps in HL. Indeed, given h : (X,dX ) 7→ (X,dX , fX ) and ′ ′ ′ h : (X,dX ) 7→ (X,dX , fX ) in HL, observe that max(h,h ) and Definition 4.1 Let HL be the class of maps h : X →X1 that preserve the metric and satisfy for some L > 0 sup(h) : (X,dX ) 7→ (X,dX ,max( fX )) are both in HL. More- ′ R over, h + h ∈H2L, and for any λ ∈ , λh ∈H λ L. It is im- 1 | | k f − f k ∞ ≤ L · kΓ k ∞ (5) portant to note that the persistence diagrams of R( fX ) and X Y l (C) 2 X,Y l (C×C) R(λ fX ) are not related by a simple transformation: different for all (X,dX ),(Y,dY ) ∈ X and all C ∈ C(X,Y), where choices of λ give different signatures. h(X,dX ) = (X,dX , fX ) and h(Y,dY ) = (Y,dY , fY ). From Definitions 2.4 and 4.1 and Theorem 3.2 we obtain: 5. Introducing measures

Corollary 4.2 Let X,Y ∈X1 be finite. Then, for any h ∈HL, For the sake of generality, it is important to make the class d∞(D R( f ),D R( f )) of admissible functions as large as possible. Unfortunately, B k X k Y ≤ d (X,Y) max(1,L) GH it seems that certain canonical functions such as the average distance to a point are not directly controllable by the where h(X,dX ) = (X,dX , fX ), h(Y,dY ) = (Y,dY , fY ) and GH distance. This is intuitively clear since in averaging the R( fX )= R(X,dX , fX ), R( fY )= R(Y,dY , fY ). ′ ′ values ∑x′6=x dX (x,x )α(x ) there is a choice to be made for ′ ′ ′ 1 When the context is clear, we will abuse notations and omit the weights {α(x )}x ∈X , the choice α(x )= |X| being one some of the arguments in the expression R(X,dX , fX ). Be- of many possible. In order to incorporate this extra structure low we provide examples of maps that are in HL: (weights) into our formulas, we define a new metric on the class of spaces that arise in [Mém07, Mém08]. This metric The diameter map. Our easiest example is the map hdiam : belongs to the class of Gromov-Wasserstein metrics on met- (X,dX ) 7→ (X,dX , fX ) where fX : X → R is the constant ric spaces enriched with weights. function fX ≡ diam(X). It is easily seen that hdiam ∈H2. A measure metric space (mm space, for short) is a triple X (X,dX ,µX ) where (X,dX ) ∈X and µX is a Borel probability The eccentricity map. The map he1 : (X,dX ) 7→ (X,dX ,e1 ) X R X ′ ′ measure such that supp[µX ]= X. Here, supp[µX ] denotes the where e1 : X → is given by e1 (x)= maxx ∈X dX (x,x ) is support of X, i.e. the minimal closed set Z s.t. µ (X\Z)= 0. in H2. Indeed, assume X,Y ∈X and C ∈C(X,Y), and let X ′ ′ Let X w denote the collection of all mm spaces, and X w η = kΓX,Y kl∞(C×C). Pick (x,y),(x ,y ) ∈ C and notice that 1 the collection of all mm spaces endowed with continuous ′ ′ ′ dX (x,x ) ≤ dY (y,y )+ η ≤ maxdY (y,y )+ η. real-valued functions. Two mm spaces (X,dX ,µX , fX ) and y′∈Y (Y,dY ,µY , fY ) are said to be isomorphic if there exists an X Y −1 Hence, e1 (x) ≤ e1 (y)+ η for all (x,y) ∈ C. By symmetry, isometry φ : X →Y s.t. fX = fY ◦φ and µX φ (Z) = µY (Z) X Y |e1 (x) − e1 (y)| ≤ η for all (x,y) ∈ C. for all measurable sets Z ⊂Y. It turns out that we can metrize w the set of isomorphism classes of X1 by replacing corre- Higher-order eccentricities. A more general construction spondences between X and Y in X w by couplings, that is, N is as follows: given m ∈ , let hem : X →X1 be the map probability measures on X ×Y whose marginals on the first X X (X,dX ) 7→ (X,dX ,em), where em : X → R is defined by and second factors are µX and µY respectively. Since X,Y X are compact and µX ,µY have full support, the support of a ∀x0 ∈ X, em(x0)= max min dX (xi,x j), x1,··· ,xm 0≤i< j≤m coupling is a correspondence [Mém07, Lemma 1]. w 1 where x1,...,xm range over X. It turns out that hem is in H2. Given X,Y ∈X1 we then define dGW,∞(X,Y) to be the X In addition, observe that for m > 1, em provides more in- infimal η > 0 such that there exists a coupling µ for which X n 1 formation than just e1 : consider for instance S endowed both 2 kΓX,Y kl∞(C(µ)×C(µ)) ≤ η and k fX − fY kl∞(C(µ)) ≤ η, Sn N with the usual intrinsic metric. Then, e1 ≡ π for all n ∈ , where C(µ) denotes the correspondence supp[µ]. 1 2 S S w whereas e4 ≡ π/4 is different from e4 ≡ 2arcsin( 2/3) > For any X,Y ∈ X , we define the Gromov-Wasserstein X 1 π/4. Note however that the complexity of computingp em on distance dGW,∞(X,Y) to be equal to dGW,∞(X,Y), where m+1 w a finite metric space with N points is roughly O(N ). we regard X,Y as elements in X1 by endowing them

c 2009 The Author(s) Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd. F. Chazal, D. Cohen-Steiner, L. Guibas, F. Mémoli, S. Oudot / Gromov-Hausdorff Stable Signatures for Shapes using Persistence with the zero functions. One can prove that dGH(X,Y) ≤ 6.1. Thresholding ﬁltrations d X Y for all X Y w [Mém07, Proposition 6]. GW,∞( , ) , ∈X To deal with problem (1) above, we restrict the computa-

The approach of Sections 3 and 4 generalizes almost ver- tion of our filtrations as follows: given a filtration ∅ = Kα0 ⊆ R batim to this measured setting. In particular, we have the ···⊆ Kαn = K and a threshold t ∈ , we let m be such that w following analogous of Corollary 4.2. Let HL be the class αm−1 < t ≤ αm and we consider only the t-thresholded fil- w w of maps h : X →X1 that preserve the metric and the mea- tration ∅ = Kα0 ⊆···⊆ Kαm−1 ⊆ K. The reason why we re- sure and satisfy, for some L > 0, place Km by K is to preserve bottleneck distances, as will be shown below. The drawback is that in principle we should 1 k f − f k ∞ ≤ L · kΓ k ∞ (6) still have to compute the full complex K. Nevertheless, we X Y l (C(µ)) 2 X,Y l (C(µ)×C(µ)) will see that this is not necessary in practice. The threshold- w for all (X,dX ),(Y,dY ) ∈ X and all µ ∈ M(µX ,µY ). ing operation affects persistence diagrams in the following Here, h(X,dX ,µX ) = (X,dX ,µX , fX ), h(Y,dY ,µY ) = way (see also Figure 3): (Y,dY ,µY , fY ), and M(µX ,µY ) denotes the set of all Lemma 6.1 Given a filtration K and its t-thresholded variant couplings between (X,µX ) and (Y,µY ). t t K , there is a multi-bijection γ : DkK→ DkK such that: w w Corollary 5.1 Let X,Y ∈X be finite. Then for any h ∈HL , • the restriction of γ to the lower-left quadrant (−∞,t] × ∞ (−∞,t] is the identity; dB DkR(h(X)),DkR(h(Y)) ≤ dGW,∞(X,Y). the restriction of γ to the upper-left quadrant t max(1,L) • (−∞, ] × (t,+∞) is the vertical projection onto the line y = t; w The class HL is actually larger than HL. For instance, given • the restriction of γ to the line y =+∞ is the horizontal w (X,dX ,µX ) ∈ X and p ≥ 1 we define the p-eccentricity projection onto the extended half-plane (−∞,t] × R; function as • finally, the restriction of γ to the half-plane (t,+∞) × R 1/p is the projection onto the diagonal point (t,t). ′ p ′ sX,p(x) := dX (x,x ) µX (dx ) , ZX The proof of this rather intuitive result is omitted. Lemma ′ 6.1 implies that for any filtrations F,G and any threshold t: and for p = ∞, sX,∞(x) := maxx′∈X dX (x,x ). ∞ t t ∞ w w dB (DkF ,DkG ) ≤ dB (DkF,DkG). (7) Lemma 5.2 The p-eccentricity map hsp : X →X1 defined w by (X,dX ,µX ) 7→ (X,dX ,µX ,sX,p) is in H2 . This means that the results of Sections 3 through 5 still hold when the filtrations are replaced by their t-thresholded ver- w Proof . let X,Y ∈X and µ ∈M(µX ,µY ). Then, for all x ∈ X sions. In addition, Lemma 6.1 provides a way of computing p t and y ∈ Y, the triangle inequality in the l (µ)-norm implies: DkR (X,dX , fX ) without building R∞(X,dX , fX ) explicitly. Indeed, we know that R∞(X,dX , fX ) is the full simplex on kdX (x,·)k p − kdY (y,·)k p ≤ kΓX Y (x,y,·,·)k p . the vertex set X, therefore the filtration R(X,d , f ) has l (µ) l (µ) , l (µ) X X only one essential homology class [c], easily identified as the

Since µ ∈M(µX ,µY ), it follows easily that kdX (x,·)kl p(µ) = connected component created by the point of X with mini- t sX,p(x) and kdY (y,·)kl p(µ) = sY,p(y). Hence, mum fX -value. It follows that DkR (X,dX , fX ) can be obtained ﬁrst by computing the diagram of the truncated ﬁl- sX,p(x) − sY,p(y) ≤ kΓX,Y (x,y,·,·)kl p(µ) tration {Rα(Xα)} , and then by projecting the resulting 0<α≤t ≤ kΓX,Y (x,y,·,·)kl∞(C(µ)). multi-set (save the point corresponding to [c] when k = 0) t t In particular, for all (x,y) ∈ C(µ), vertically onto the lower-left quadrant (−∞, ] × (−∞, ]. ∞ sX,p(x) − sY,p(y) ≤ kΓX,Y kl∞(C(µ)×C(µ)), hence

ksX,p − sY,pkl∞(C(µ)) ≤ kΓX,Y kl∞(C(µ)×C(µ)), t w which means that hsp is in H2 (cf. Eq. (6)).

6. Algorithmic aspects In implementing the lower bounds given by Corollaries 4.2 and 5.1, one must take into account the facts that (1) the t ∞ computation of the full persistence diagrams may be too ex- pensive and (2) the bottleneck matching of the diagrams may Figure 3: Effect of thresholding on a persistence diagram. be time consuming. This section addresses these two issues.

‡ 6.2. Simplifying persistence diagrams stra’s algorithm on the 1-skeleton graph G(Xi) of the mesh . We then select a uniform subset Xr of 4K vertices using To address problem (2) above, we simplify each consid- i the Euclidean farthest point sampling procedure. Briefly, we ered persistence diagram D F as follows: given a parameter k first randomly choose a point from the vertex set. Then, we ε ≥ 0, we snap all points of D F lying within l∞-distance ε k choose the second point as the one at maximal distance from of the diagonal ∆ onto ∆. The resulting simplified diagram is the first one. Subsequent points are chosen always to maxi- called DεF. The triangle inequality implies that for any per- k mize the distance to the points already chosen. Let Xr denote sistence diagrams D F,D G and any parameters ε,ε′ ≥ 0, i k k this reduced model with 4K vertices. We further subsample Xr using the farthest point procedure, this time in the dis- ′ i ∞ ε ε ∞ ′ dB DkF,Dk G − dB DkF,DkG ≤ max{ε, ε }. (8) tance di computed using G(Xi), and retain only 300 points. rr We denote the resulting coarse set by Xi , which we endow In our experiments, we specify a maximal number N of off- with the distance di, properly normalized so that the diame- rr rr diagonal points to be kept per diagram. Specifically, we keep ter of Xi is 1. We also equip Xi with a probability measure ∞ the points with largest l -distance to ∆. For every diagram µi based on intrinsic Voronoi partitions, where the measure rr r DkF considered, we perform this simplification and record at point x ∈ Xi equals the proportion of points of Xi that are rr the incurred projection error ε ≥ 0, which remains small if closer to x than to any other point of Xi . From each model rr N is chosen sufficiently large. Xi we thus obtain a discrete mm-space Xi ,di,µi . Let S denote the collection of all 62 such mm-spaces. + w 6.3. Computing the bottleneck distance Let P := {1,2,3,∞}, H = {hsp , p ∈ P}⊂H2 and − w ′ H = {sup(hs )−hs , p ∈ P}⊂H (recall Remark 1). Let ε ε p p 4 Given two simplified persistence diagrams DkF and Dk G, H denote the union of these two sets of functions, and let each containing at most N off-diagonal points, we proceed as follows. Firstly, we discard the diagonal and consider only Λ := {0,0.01,0.1,0.2,...,0.9,1,2,3,4,5,10,30,50,70,90}. ′ ε ε DkF\ ∆ and Dk G\ ∆, whose total multiplicities are finite, For each pair of mm-spaces X and Y in S, for a choice of bounded from above by N. Secondly, we add arbitrary di- λ ∈ Λ and a function h ∈H we estimate the lower bound ′ agonal points to DεF\ ∆ and Dε G\ ∆, so that the resulting given by Corollary 5.1 as follows: k k t multi-sets { fi}i and {g j} j contain exactly N points each. Fi- 1. We first construct the thresholded filtrations R(λ · h(X)) t nally, we solve the following Bottleneck Assignment Prob- and R(λ · h(Y)) up to their 3-dimensional skeleta, as dis- ∗ d lem: compute c = minπ maxi fi,gπ(i) , where π ranges cussed in Section 6.1. The thresholding parameter t is over all permutations of the set {1,...,N}, and where chosen so as to guarantee that the 1-skeleta of the under- t t 1 lying complexes of R(λ · h(X)) and R(λ · h(Y)) con- d p,q = min kp − qk∞, max{|py − px|,|qy − qx|} . 2 tain no more than a fraction s of the (quadratic) number of edges of the complete graph. The value of s chosen in In plain words, d(p,q) measures the cost of matching p with our experiments is 7%, which allows a reasonable execu- q against the cost of matching both points with the diagonal ′ tion time of approximately 1.5 seconds per pair of mm- ∗ ∞ ε ε ∆. It can then be easily checked that c = dB (DkF,Dk G). spaces, on average. In addition, we have that d(p,q)= d(p,q′) whenever q,q′ ∈ 2. We use the software jplex [JPl] and the procedure of ∆, which makes the choice of the locations of the diago- Section 6.1 to compute the kth persistence diagrams of nal points inserted during the second step of our procedure each thresholded filtration, for k = 0,1,2. unimportant. In practice, we compute c∗ using the thresh- 3. We apply the simplification step described in Section olding algorithm of [BC99, Section 5], whose running time 6.2 with N = 30. For each k = 0,1,2, let Ek(X) = 2.5 is O N logN . εk(X) t εk(Y) t Dk R(λ · h(X)) and Ek(Y) = Dk R(λ · h(Y)) be the simplified diagrams, where εk(X) and εk(Y) are the 7. Application to shape classification projection errors. 4. We use the procedure of Section 6.3 to compute the quan- We exemplify the use of our framework on a shape clas- tity c(h,λ;X,Y)= sification problem under invariances. We used the publicly ∞ available database of (triangulated) shapes [SP04]. This max max d E (X),E (Y) − max{εX (k),εY (k)},0 k B k k database comprises 62 shapes from six different classes: , camel, cat, elephant, face, head and horse. Each max(1,λ · L(h)) class contains several different poses of the same shape. These poses are richer than just rigid isometries, as can be ‡ In this particular application we take advantage of the fact that seen from Figure 1. The number of vertices in the models a mesh is available for each of the shapes. In full generality, one range from 7K to 30K. We equip each model Xi with an in- can replace G(Xi) by some neighborhood graph in the ambient Eu- trinsic distance di between pairs of points in Xi using Dijk- clidean metric, with similar guarantees.

+ − where L(h) is 2 if h ∈ H and 4 if h ∈ H . By the probability Pe(M) of mis-classiﬁcation. We repeat this Corollary 5.1, Lemma 6.1 and Eq. (8), we know that procedure for 2K random choices of the training set. Using dGW,∞(X,Y) ≥ c(h,λ;X,Y). the same randomized procedure we obtain an estimate of the confusion matrix for this problem, whose entry (i, j) is the For any pair X,Y ∈ S, we can now bound d (X,Y) from GW,∞ probability that the classiﬁer will assign class j to a shape below by c(X,Y) := max c(h,λ;X,Y). Let M de- h∈H,λ∈Λ when the actual class is i — see Figure 4 (right). We ob- note the matrix with elements c(X,Y), X,Y ∈ S. Figure 4 tained P (M)= 4%. As Figure 5 shows, there is a shape of (left) shows the result of these computations on our database, e class camel that lies very close to the cluster formed by while Figure 5 shows the result of applying the metric multi- the class horse. Removal of this shape from the database dimensional scaling procedure implemented by the Matlab reduced the probability of error to 2%. command mdscale to M.

C 1

0.04 camel 8. Perspectives 0.9

10 0.035 0.8 cat In this paper we only used Euclidean and geodesic distances 0.7 20 0.03

0.6 elephant to illustrate our approach, however many other metrics may 0.025 30 ∗ 0.5 be considered, such as the diffusion distance [BBK ]. 0.02 face 0.4

40 0.015 0.3 The generality of our approach makes it applicable in a head 0.01 50 0.2 large variety of contexts beyond 3d shape classification. Our 0.005 0.1 horse 60 signatures are indeed well-defined, stable and computable 0 camel cat elephant face head horse 10 20 30 40 50 60 for any arbitrary finite metric space that is given in the form Figure 4: Left: estimate of the Gromov-Wasserstein distance of a distance matrix or equivalent representation. This makes computed on the database S. Right: estimated confusion ma- our approach well-suited for classification problems in gen- trix for the 1-nearest neighbor classification problem de- eral, even in cases where the data are high-dimensional or scribed in the text. The overall error rate is 4%. lying in some non-Euclidean space without any information on the embedding beside the inter-point distances.

MDSCALE with M 0.04 camel At a more theoretical level, let us point out that Theo- cat elephant rem 3.1 enables us to deﬁne limit Rips persistence diagrams 0.03 face head for compact metric spaces. Indeed, given a compact met- horse ric space (Z,dZ), consider a sequence of ﬁnite ε-samples of 0.02 (Z,dZ) such that ε converges to zero. The k-th Rips persis-

0.01 tence diagram of (Z,dZ) can then be deﬁned as the limit of the k-th persistence diagrams of the Rips ﬁltrations of

0 the sequence. This limit is independent of the choice of the sequence of ε-samples, since any two finite ε-samples of -0.01 (Z,dZ) lie at Gromov-Hausdorff distance 2ε of each other, which by Theorem 3.1 implies that the persistence diagrams -0.02 of their Rips filtrations are 2ε-close in the bottleneck distance. Thus, the limit Rips persistence diagram of (Z,dZ) is -0.03 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 well-defined, and by Theorem 3.1 it is stable under small Figure 5: MDS plot of the matrix M with labels corre- perturbations of the space in the Gromov-Hausdorff dis- R sponding to each class. The overall error is estimated to be tance. Similarly, for any real-valued function f : X → , 4%. Notice how there is excellent clustering of the different Theorem 3.2 enables to define limit Rips persistence dia- classes, save for a single shape of the class camel that lies grams for (X,dX , f ) and to guarantee their stability. This very close to the cluster formed by the class horse. Re- concept of limit persistence diagram of a compact metric moval of this shape reduces the classification error to 2%. space (X,dX ) is reminiscent of the approach to topological persistence developed in [CCSG∗09]. However, it may not be equivalent to the persistence diagram of the limit filtra- In order to evaluate the discriminative power contained in tion R(X,dX ) or R(X,dX , f ) in general. Finding the exact the matrix M, we consider a classification task as follows: relationships that exist between these diagrams is still open. We randomly select one shape from each class, form a training set T and use it for performing 1-nearest neighbor clas- Acknowledgements. This work was carried out within the sification (where nearest is with respect to the metric de- Associate Team TGDA: Topological and Geometric Data fined by M) of the remaining shapes. By simple comparison Analysis involving the Geometrica group at INRIA and the between the class predicted by the classifier and the actual Geometric Computing group at Stanford University. The au- class to which the shape belongs we obtain an estimate of thors wish to thank the referees for their suggestions to make

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circle helicoid sphere ellipsoid

geodesic / no function geodesic / no function geodesic / no function geodesic / no function ∞ ∞ ∞ ∞

1 1 1 1

0 0 0 0 0 0 0 0 euclidean / no function euclidean / no function euclidean / no function euclidean / no function ∞ ∞ ∞ ∞

1 1 1 1

0 0 0 0 0 0 0 0 euclidean / eccentricity euclidean / eccentricity euclidean / eccentricity euclidean / eccentricity ∞ ∞ ∞ ∞

1 1 1 1

0 0 0 0 0 0 0 0

Figure 6: Some toy examples, from left to right: the unit circle, a helical curve of same length 2π drawn on a torus, the unit sphere, an ellipsoid whose smallest equatorial ellipse has same length 2π as the equator of the sphere. On each shape, a uniform 0.0125-sample has been generated, on top of which various Rips ﬁltrations have been constructed using different metrics and functions: geodesic distance and no function (second row), Euclidean distance and no function (third row), Euclidean distance and 0.2 times the eccentricity (fourth row). The corresponding persistence diagrams are presented in rows 2 through 4: each picture shows the 0-dimensional (blue), 1-dimensional (red), and 2-dimensional (green) diagrams super-imposed.

c 2009 The Author(s) Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd.