Strong topology on the set of persistence diagrams
aa r X i v : . [ m a t h . GN ] M a y STRONG TOPOLOGY ON THE SET OF PERSISTENCEDIAGRAMS
VOLODYMYR KIOSAK, ALEKSANDR SAVCHENKO, AND MYKHAILO ZARICHNYI
Abstract.
We endow the set of persistence diagrams with the strong topol-ogy (the topology of countable direct limit of increasing sequence of boundedsubsets considered in the bottleneck distance). The topology of the obtainedspace is described.Also, we prove that the space of persistence diagrams with the bottleneckmetric has infinite asymptotic dimension in the sense of Gromov. Introduction
Topological Data Analysis (TDA) is a field in applied mathematics concen-trated around investigation of big data by topological methods. Imposing metricstructures in the data set allows for applying techniques from algebraic topology.In this way, the notion of persistent homology was introduced [4].The persistent homology plays an important role in TDA. The persistencediagrams are used to characterize persistent homology and thus to describegeometric properties of data. The set of all persistence diagrams can be en-dowed with different metrics. The most known are the Wasserstein metric andbottleneck metric.The spaces of persistence diagrams are object of considerations in numerouspublications (see, e.g., [10, 5, 9, 11, 13]). In particular, in [11] a characterizationtheorem for compact subsets in the space of persistence diagrams is proved.It is proved in [2] that the space of persistent diagrams is of infinite asymptoticdimension in the sense of Gromov. This concerns the Wasserstein metric onthe set of persistence diagrams. Answering a question from [2] we prove ananalogous result for the bottleneck metric on this set.As we remark below, the set of all persistence diagrams is nothing but theinfinite symmetric power of the upper (positive) half-plane. In this note weconsider the strong (direct limit) topology on this set. One of our results is thatthe space of the persistence diagrams with this topology is homeomorphic tothe countable direct limit of the euclidean spaces.
Mathematics Subject Classification.
Key words and phrases. persistence diagram, bottleneck distance. Preliminaries
Persistence diagrams.
Let∆ = { ( x, y ) ∈ R | x = y } , ˆ X = { ( x, y ) ∈ R | x ≤ y } , and X = ˆ X \ ∆. For any n ∈ N , let ˆ X n = { ( x, y ) ∈ ˆ X | y ≤ n } , X n = ˆ X n \ ∆.A persistence diagram is a function µ : X → Z + such that µ ( a ) = 0 for allbut finitely many a ∈ ˆ X and µ ( a ) = 0 for all a ∈ ∆. The support of µ is the setsupp( µ ) = { a ∈ ˆ X | µ ( a ) > } .By D we denote the set of all persistence diagrams. Given n ∈ N , we denoteby D n the set of all µ ∈ D such that | supp( µ ) | ≤ n .2.2. Bottleneck distance.
Let µ ∈ D . A sequential representation of µ is afinite sequence ( a , . . . , a k ) such that the following are satisfied:(1) for every a ∈ supp( µ ), |{ i ≤ m | a = a i }| = µ ( a );(2) if a i / ∈ supp( µ ), then a i ∈ ∆.The number k is said to be the length of the representation ( a , . . . , a k ).By S k , the group of permutations of the set { , . . . , k } is denoted.Let µ, ν ∈ D . We define d ( µ, ν ) = inf { min { max { ρ ( a i , b σ ( i ) ) | ≤ i ≤ k } | σ ∈ S k }| ( a , . . . , a k ) , ( b , . . . , b k ) are sequential representations of µ and ν respectively , k ∈ N } . (The assignment a i b σ ( i ) , i = 1 , . . . , k , is said to be a matching (see, e.g.,[6] for details).The function d is known to be a metric on D (the bottleneck metric; see, e.g.,[6]).2.3. Space R ∞ . Recall that the direct limit of the increasing sequence of topo-logical spaces X ⊂ X ⊂ . . . (here X n is a subspace of X n +1 , for each n ) isthe set X = ∪ ∞ n =1 X n endowed with the strongest topology inducing the originaltopology on each X n . The obtained topological space is denoted by lim −→ X n .We identify every ( x , . . . , x n ) ∈ R n with ( x , . . . , x n , ∈ R n +1 . Thus, R n is regarded as a subspace in R n +1 . We denote by R ∞ the direct limit of thesequence R ⊂ R ⊂ R ⊂ . . . .A characterization theorem for the space R ∞ is proved by K. Sakai [12]. Theorem 2.1 (Characterization Theorem for R ∞ ) . Let X be a countable di-rect limit of finite-dimensional compact metrizable spaces. The following areequivalent. (1) X is homeomorphic to R ∞ ; TRONG TOPOLOGY ON THE SET OF PERSISTENCE DIAGRAMS 3 (2) for every finite-dimensional compact metrizable pair ( A, B ) and everyembedding f : B → X there exists an embedding ¯ f : A → X that extends f . Asymptotic dimension.
Let Y be a metric space. A family A of subsetsof X is said to be uniformly bounded if sup { diam( A ) | A ∈ A} < ∞ . Given D >
0, we say that a family A is D -disjoint if, for every distinct A, B ∈ A ,dist(
A, B ) ≥ D We say that the asymptotic dimension of Y does not exceed n , if, for any D >
0, there exists a uniformly bounded cover U of Y such that U = ∪ ni =0 U i ,where every family U i is D -disjoint, i = 0 , . . . , n . The notion of asymptoticdimension is defined by M. Gromov [8]. See, e.g., [1] for properties of theasymptotic dimension. 3. Main results
Strong topology on the space of persistence diagrams.
For every n ∈ N , let D n = { µ ∈ D | | supp( µ ) | ≤ n and supp( µ ) ⊂ X n } and D ∞ = lim −→ D n . Theorem 3.1.
The space D ∞ is homeomorphic to R ∞ .Proof. For any n ∈ N , define a map ξ n : ˆ X nn → D n as follows: ξ n ( a , . . . , a n )( a ) = |{ i | a = a i }| , for all a ∈ X n . Note that this map is clearly continuousand it admits a factorization ξ n = ξ ′ n ξ ′′ n , where ξ ′′ : ˆ X nn → ( ˆ X n / ( ˆ X n ∩ ∆) n ) n is the factorization map, where ∗ n stands for ˆ X n ∩ ∆ (actually, ξ ′′ n = q n , where q : ˆ X n → ˆ X n / ( ˆ X n ∩ ∆) n is the factorization map). Therefore, D n is the orbitspace of the action of the group S n on the space ( ˆ X n / ( ˆ X n ∩ ∆)) n by permuta-tion of coordinates. In other words, D n is homeomorphic to the n th symmetricpower SP n ( ˆ X n / ( ˆ X n ∩ ∆)). The orbit containing ( x , . . . , x n ) will be denotedby [ x , . . . , x n ].We denote by ∗ n the point q ( ˆ X n ∩ ∆) ∈ ˆ X n / ( ˆ X n ∩ ∆). Identifying ∗ n with ∗ n +1 ,one can consider ˆ X n / ( ˆ X n ∩ ∆) as a subset of ˆ X n +1 / ( ˆ X n +1 ∩ ∆). Then identifying[ x , . . . , x n ] ∈ SP n ( ˆ X n / ( ˆ X n ∩ ∆)) with [ x , . . . , x n , ∗ n +1 ] ∈ SP n +1 ( ˆ X n +1 / ( ˆ X n +1 ∩ ∆)) we finally obtain that D ∞ is homeomorphic to the spacelim −→ SP n ( ˆ X n / ( ˆ X n ∩ ∆)) = SP ∞ (lim −→ ˆ X n / ( ˆ X n ∩ ∆) . The latter space is known as the infinite symmetric power construction [7].For every n ∈ N , the space ˆ X n / ( ˆ X n ∩ ∆) is homeomorphic to the 2-dimensionaldisc. Therefore, the space SP n ( ˆ X n / ( ˆ X n ∩ ∆)) is a finite dimensional abso-lute retract (AR), see [14]. Similarly as in [15] one can prove that the space SP ∞ (lim −→ ˆ X n / ( ˆ X n ∩ ∆) (and also the space D ∞ ) is homeomorphic to R ∞ . VOLODYMYR KIOSAK, ALEKSANDR SAVCHENKO, AND MYKHAILO ZARICHNYI
For the sake of completeness, we provide here a proof based on Sakai’s Charac-terization Theorem 2.1. Let (
A, B ) be a finite-dimensional compact metrizablepair an let f : B → D ∞ be an embedding. Then there exists n ∈ N such that f ( B ) ⊂ D n . As we already remarked, D n is an absolute retract. Thus, thereexists an extension g : A → D n of f .We denote by α : A → A/B the quotient map. Since the quotient space
A/B is finite-dimensional, there exists an embedding i : A/B → [0 , m , for some m .Without loss of generality we assume that α ( A/B ) = 0, α ( A \ B ) ⊂ (0 , m ,and m > n . Write i ( a ) = ( i ( a ) , . . . , i m ( a )).Now, given x ∈ A , write g ( x ) as [ g ( x ) , . . . , g n ( x )]. Then define¯ f ( x ) =[ g ( x ) , . . . , g n ( x ) , (1 , i ( α ( x ))) , . . . , (2 n + 1 , n + 1 + i ( α ( x ))) , (2 n + 2 , n + 2 + i ( α ( x )) , . . . , (4 n + 3 , n + 3 + i ( α ( x ))) ,. . . (( m − n + 1) + 1 , ( m − n + 1) + 1 + i m ( α ( x ))) , . . . , ( m (2 n + 1) + 1 , m (2 n + 1) + 1 + i m ( α ( x )))] . First, note that ¯ f is well defined and continuous. If x ∈ B , then i k ( α ( x )) = 0, k = 1 , . . . , n (2 n + 1), and therefore¯ f ( x ) = [ g ( x ) , . . . , g n ( x ) , (1 , , . . . , ( n (2 n + 1) , n (2 n + 1)))] = f ( x ) , because of our identifications.Clearly, ¯ f is well-defined and continuous. Since A is compact, in order toprove that ¯ f is an embedding it is sufficient to prove that ¯ f is injective. Let x, y ∈ A , x = y . If x ∈ B and y ∈ A \ B , then | supp( ¯ f ( x )) | ≤ n and, since i k ( α ( y )) = 0, for some k ≤ m , we conclude that | supp( ¯ f ( y )) | ≥ n + 1. If x, y ∈ A \ B , then there exists k ≤ m , such that i k ( α ( x )) = i k ( α ( y )). Since | supp( g ( x )) | ≤ n , we see that there exists j ≤ n + 1 such that( j (2 n + 1) + 1 , j (2 n + 1) + 1 + i j ( α ( x )) ∈ supp( ¯ f ( x )) \ supp( ¯ f ( y ))and therefore ¯ f ( x ) = ¯ f ( y ). The other cases being treated similarly, this provesinjectivity of ¯ f . By Theorem 2.1, D ∞ is homeomorphic to R ∞ . (cid:3) Asymptotic dimension of the space of persistence diagrams.
Weconsider the set D of all persistence diagrams with the bottleneck metric. Given n ∈ N , consider the set K n = { [( n, n + ( n + 1) + t ) , (2 n + 1 , n + 1 + ( n + 1) + t ) , . . . , ( n − n + ( n − , ( n − n + ( n −
2) + ( n + 1) + t n − )] , ( n + ( n − , n + ( n −
1) + ( n + 1) + t n )] | ( t , . . . , t n ) ∈ [0 , n ] n } TRONG TOPOLOGY ON THE SET OF PERSISTENCE DIAGRAMS 5
Clearly, K n is isometric to the cube [0 , n ] n endowed with the l ∞ -metric. Thiseasily follows from the observation that every matching between two points from K n realizing the bottleneck distance consists of pairs of points from X so thateach pair lies on a vertical line.Since n can be chosen arbitrarily large, the known properties of the asymptoticdimension (see, e.g., [1]) imply the following. Theorem 3.2. asdim D = ∞ . Remarks
We conjecture that an analog of Theorem 3.1 can be proved for the space D ∞ = lim −→ D n in the case when every D n is endowed with the Wassersteinmetric (also Wasserstein p, q metric considered in [2]).Since many spaces of persistence diagrams are infinitely-dimensional, onecan expect that the methods of infinite-dimensional topology, in particular, thetheory of infinite-dimensional manifolds, will be useful in their investigations.In [5], the space D bN of bounded persistent diagrams with less than N points ismentioned. We consider the space ˜ D N of exactly N points (taking into accountthe multiplicities), N ∈ N . Having in mind the mentioned identification ofpersistence diagrams and symmetric powers one can derive from [14, Theorem4.5] that the space ˜ D N is homeomorphic to the euclidean space R N . This leadsto the question of description of topology of the subspace ˜ D ≤ N = ∪ i ≤ N ˜ D i of D .In this note we restricted ourselves with persistence diagrams of finite support.In some publications, persistence diagrams with countably many points are alsoconsidered. In particular, it is known that the latter spaces are complete in theWasserstein metric. In the subsequent publications we are going to considerthe geometry of the complete spaces of persistence diagrams and some of theirsubspaces. References [1] G.Bell, A.Dranishnikov, Asymptotic dimension, Topol. Appl. 155, 2008, 1265–1296.[2] G. Bell, A. Lawson, C. Neil Pritchard, D. Yasaki. The space of persistence diagrams hasinfinite asymptotic dimension. Preprint, arXiv:1902.02288[3] D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry, AMS GSM 33, 2001.[4] G. Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46:255–308, 2009[5] M. Carri`ere, M. Cuturi, S. Oudot, Sliced Wasserstein Kernel for Persistence Diagrams,Preprint, arXiv:1706.03358[6] F. Chazal, V. de Silva, M. Glisse, and S. Oudot, The structure and stability of persistencemodules, SpringerBriefs in Mathematics, Springer, [Cham], 2016.[7] A. Dold, R. Thom, Quasifaserungen und unendliche symmetrische Produkte, Annals ofMathematics, Second Series, 67(1958): 239–281.
VOLODYMYR KIOSAK, ALEKSANDR SAVCHENKO, AND MYKHAILO ZARICHNYI [8] M. Gromov, Asymptotic invariants of infinite groups, in: Geometric Group Theory, vol.2, Sussex, 1991, in: London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ.Press, Cambridge, 1993, pp. 1–295.[9] C. Li, M. Ovsjanikov, and F. Chazal, Persistence-based structural recognition. In Pro-ceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 1995–2002, 2014.[10] Y. Mileyko, S. Mukherjee, and J. Harer, Probability measures on the space of persistencediagrams. In: Inverse Problems 27.12 (2011), 124007 (22 p.)[11] J. A. Perea, E. Munch, F. A. Khasawneh, Approximating Continuous Functions onPersistence Diagrams Using Template Functions, Preprint, arXiv:1902.07190[12] K. Sakai, On R ∞ and Q ∞ -manifolds, Topol. Appl., Volume 18, Issue 1, September 1984,69–79.[13] K. Turner, G. Spreemann, Same but Different: distance correlations between topologicalsummaries, Preprint, arXiv:1903.01051[14] C. H. Wagner. Symmetric, cyclic, and permutation products of manifolds. Warszawa:Instytut Matematyczny Polskiej Akademi Nauk, 1980.[15] M. M. Zarichnyi, Free topological groups of absolute neighborhood retracts and infinite-dimensional manifolds, Dokl. Akad. Nauk SSSR 266:3 (1982), 541–544. (In Russian) Institute of Engineering Odessa State Academy of Civil Engineering andArchitecture Didrihson st., 4, Odessa, 65029, Ukraine
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