EXPLORING PERSISTENT LOCAL HOMOLOGY IN TOPOLOGICAL DATA ANALYSIS

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1 EXPLORING PERSISTENT LOCAL HOMOLOGY IN TOPOLOGICAL DATA ANALYSIS Brittany Terese Fasy Montana State University Comuter Science Deartment Bozeman, MT Bei Wang University of Utah Scientific Comuting and Imaging Institute Salt Lae City, UT ABSTRACT Toological data analysis (TDA) has raidly grown in oularity in recent years. One of the emerging tools is ersistent local homology, which can be used to extract local structure from a dataset. In this aer, we rovide a survey that exlores this new tool, emhasizing its use in data analysis. Index Terms ersistent homology, comutational toology, stratified learning, data analysis 1. INTRODUCTION Toological data analysis (TDA) bridges research from comutational geometry and toology, algebraic toology, machine learning, and statistics in studying datacentric roblems. Recently, the field has roduced a collection of innovative techniues in studying the shae of data. Two fundamental tass in the field of TDA are reconstruction (how to assemble discrete samles into global structures) and inference (how to infer structure). In this aer, we resent ersistent local homology (PLH) as a recent develoment in TDA that is a combination of the concets of ersistent homology and local homology. Roughly seaing, the homology of a toological sace measures its toological features such as connected comonents, tunnels, and voids. Local homology studies the homology within a local neighborhood relative to its boundary. PLH turns the algebraic concet of local homology into a multi-scale notion by constructing extended series of homology grous [1, 2]. It is, in our oinion, the tool within TDA that studies the local structure of data. We exlore PLH from theoretical, algorithmic and alication ersectives, with Partially suorted by NSF and NIH award This research is suorted by NSF-IIS an emhasis on its alication in data analysis. Furthermore, this aer unifies results develoed by various researchers over the ast few years. Section 2 gives a brief theoretical and algorithmic exosition of PLH. Section 3 surveys alications of PLH in several ey areas, including road networ analysis, clustering and stratification learning. Section 4 concludes by contemlating ossible future directions. 2. PERSISTENT LOCAL HOMOLOGY We give theoretical bacground for defining PLH, namely, homology, ersistent homology and local homology. We cover the relevant notions from their smooth/continuous setting to the corresonding discrete/simlicial setting, where the former is simle for theory and the latter is aroriate for algorithms. For more details, see [3, 4] for a gentle introduction to TDA and [5] for algorithmic foundations. Homology and ersistent homology. Homology deals with toological features of a sace. Given a toological sace X, the zero-, one- and two-dimensional homology grous, denoted as H 0 (X), H 1 (X) and H 2 (X) resectively, corresond to comonents, tunnels and voids of X. Formally, the construction of homology grous begins with a chain comlex C(X) that encodes information about X, which is a seuence of abelian grous C 0 (X), C 1 (X),... connected by homomorhisms nown as the boundary oerators : C (X) C 1 (X). The -th homology grou H (X) = er ( )/im ( +1 ). We wor rimarily with relative homology grous H (X, A) (for A X) which is defined by the same formula, but using boundary mas on the uotient saces C (X)/C (A) C 1 (X)/C 1 (A). A more nuanced way to describe the shae of X is using ersistent homology, which is a multi-scale notion

2 of homology. Its most common form alies to a seuence of toological saces connected by inclusions, called a filtration. That is, we may consider the finite seuence = X 0 X 1 X n = X. Alying homology to this seuence, the homology grous are connected from left to right by homomorhisms induced by inclusion X i X j (for i j), denoted as f i,j : H (X i ) H (X j ). The ran of the homology grous changes as the index increases. When the ran increases (euivalently, the ma f i 1,i is not surjective), we call this a birth event at X i, and when the ran decreases (the ma f j 1,j is not injective), we call it a death event at X j. Persistent homology airs the birth events and the death events as a multi-set of oints in the lane, called the ersistence diagram; see [5]. Local homology. Homology studies the structure of the entire toological sace; however, we are often interested in studying the local structures in the data as well. A standard tool to use for studying local structure is local homology. The -th local homology grou of X at a oint x 0 X is the relative homology H (X, X x 0 ). Alternatively, it can be defined as the limit of the homology of X relative to everything excet a shrining neighborhood around x, lim r 0 H (X, X\U r ), where U r is a neighborhood of x 0 with radius r. Persistent local homology. In PLH, we adat a multiscale notion of local homology based on ersistence. For a fixed distance α, we consider the thicened version of X, that is, we let X α denote the subset of R d that is at most distance α from X. We fix a neighborhood V R d of x in the ambient sace R d, and let U α = V X α. We then comute the homology of U α relative to its boundary U α. We construct an α-filtration by allowing α to range from zero to the diameter of V. For all α < α, the inclusion U α U α induces a homomorhism on homology: H(U α, U α ) H(U α, U α ). In other words, we are interested in the ersistent (local) homology defined by the relative homology grous H(U α, U α ), where α is the arameter that defines the filtration. We denote the corresonding ersistence diagram by Dgm(X; V ). In addition, we could consider another notion of PLH, referred to as an r-filtration, constructed by fixing the thicening arameter α and varying the radius r of V, see [6] for details. Distances between ersistence diagrams. Given two ersistence diagrams, D 1 and D 2, we may want to comare the diagrams. And, in fact, a well-defined distance between ersistence diagrams is the bottlenec distance: W (D 1, D 2 ) := inf f : D 1 D 2 su x f(x), (1) x D 1 where f : D 1 D 2 is bijection between the diagrams D 1 and D 2. This distance between diagrams is stable both for general filtrations [7, 8] as well as for local homology filtrations [9, 10]. Persistent local homology comutation. Comuting PLH from otentially noisy oint cloud samles is more challenging than comuting ersistent homology in general (see [2]), due to the combinatorial comlexity of comuting U α, the boundary of local neighborhoods. Delaunay comlexes and their variants have tyically been emloyed to guarantee theoretical correctness [1, 11]; although, their construction does not scale well with dimension. On the other hand, Vietoris- Ris comlexes have been used more commonly in TDA due to their algorithmic simlicity and robust comutation in ractice (e.g. [12, 13]). It has been shown that α- and r-filtrations of PLH could be aroximated by constructing families of Vietoris-Ris comlexes [6]. Some rogress has been made towards comuting PLH using even more comact combinatorial structures such as Grah Induced Comlexes [14]. In other contexts, when the objects are convex shaes (e.g., straight road segments as studied in [9]), a family of Čech filtrations can rovide uic comutations, rovided that the radius of the neighborhoods is small enough. 3. APPLICATIONS OF PLH The original alication of PLH is in stratification learning and clustering artitioning an object of interest into ieces of uniform dimension. The role of PLH has exanded to include alications to road networ analysis and beyond. We briefly exlore these alications next Stratification Learning and Clustering A classic roblem in learning focuses on inferring the structure of data from oint cloud samles. In manifold learning, we assume the samles are drawn from a manifold; more generally, in stratification learning, we assume the oints are samled from a stratified sace (i.e., a mixture of ossibly intersecting manifolds). Previous wor in ure mathematics has focused on the study of

3 stratified saces under smooth and continuous settings without comutational considerations of noisy and discrete data [15, 16]. Statistical aroaches that rely on inferences of mixture models or local dimension estimation reuire either strict geometric assumtions (e.g. linearity) or may not handle comlex singularities [17, 18] (as illustrated in Fig 1). Recently, toological aroaches have led to new theoretical and algorithmic results by addressing these issues. The main objective is to cluster the oint cloud samles, according to the structure of their underlying manifold ieces and how the ieces interact with one another, at multile scales. In articular, the local structure of a sufficiently samled stratified sace could be studied based on PLH [1]; and oint cloud data (PCD) could be clustered by the manifold ieces they belong to based on how the PLH of nearby samled oints ma into one another [11]. For examle, consider the sace X in R 2 in Fig. 2. For each air (, ), let f = φ X (,, r) and g = φ X (,, r). Then the oints and are considered to have the same local structure if f and g are both isomorhisms, that is, their local structures ma into each other bijectively via their intersection; euivalently, if er f = co f = 0 and if er g = co g = 0. For a fixed r, the local homology classes (in this case, holes or tunnels within the local neighborhoods) are labeled in their corresonding locations. In (a), X contains four ieces of onedimensional manifolds (colored in red, in, urle and blue). In (b), X itself is a one-dimensional manifold. Both and belong to different clusters in (a) and the same cluster in (b). However, the notion of local be- (a) α 2 α 1 f γ g 1 β 1 α 3 x y (b) α 1 f g β 1 γ 1 Fig. 1. Left: A stratified sace is constructed by attaching a dis to the tunnel of a inched torus. Structures surrounding comlex singularities are highlighted at oints x and y. It is difficult to cluster the samled oints based on local dimension estimation alone, esecially surrounding these comlex singularities. Middle and right: samled oints are clustered at two scales. Stratification learning in the clustering setting relies on ingredients of PLH and intersection homology [19, 20, 21]. A crucial comonent is to study a ersistent version of local homology intersection ma. Intuitively, two nearby oints belong to the same cluster, if they loo the same locally and their local neighborhoods are glued together in a nice way. Assume we are given a stratified sace X embedded in R d, define B r (x) to be a ball of radius r in R d centered at x that corresonds to a neighborhood of x X. For a fixed radius r, and for every air of oints, R d whose neighborhoods intersect, we define the following relative homology ma φ X (,, r): H(X B r (), X B r ()) (2) H(X B r () B r (), X (B r () B r ())). Fig. 2. (a) and do not have the same local structure at radius r since er f 0, that is, extra local structure exists in the neighborhood of ; (b) and have the same local structure at radius r since both f and g are isomorhisms, that is, local structures of and ma to the local structure of the intersection bijectively. comes unclear in the context of the uncertainty induced from samling, therefore a ersistence version of the above local homology intersection ma is constructed to define same local structure across multile scales Road Networ Comarison Road networs are changing every day, due to evolving city architecture caused by new road construction, temorary (or ermanent) road closures, etc. Understanding where and by how much the road networ has changed is a new roblem in transortation research. Traditional algorithms for comaring road networs have been heuristic in nature, failing to rovide theoretical guarantees. The first non-heuristic distance measure between road networs was resented in [9]. This algorithm, exlained next, uses PLH.

4 Suose X and Y are two grahs embedded in D, a comact subsace of R d, and let ε > 0 be a locality arameter given a riori. The grahs X and Y reresent the two road networs that we wish to comare. For each x D, we loo at Vx ε = B ε (x), the ball centered for X at x of radius ε. The local signature at x (for locality arameter ε) is the ersistence diagram S X (x, ε) := Dgm(X; Vx ε ). Liewise, we define the local signature for Y as S Y (x, ε) := Dgm(Y; Vx ε ). A local signature at x D is a descritor (in this case, a ersistence diagram) that is used to describe the local structure as witnessed by x. The arameter ε is used to determine how far x can see. (a) Local Structures (b) Signature restricted to X Fig. 3. PLH comares structures resent in local neighborhoods; see (a). Choosing neighborhoods centered on each oint of X, we can visualize the PLH distance between grahs embedded in the same domain; see (b). Instead of comaring the grahs X and Y directly, we comare the local signatures S X (x, ε) and S Y (x, ε). We define the local distance signature ψ ε : D R by ψ ε (x) = W (S X (x, ε), S Y (x, ε)), where W (, ) is the Bottlenec distance as defined in the revious section; see Figure 3, which illustrates the distance between two grahs generated from GPS trajectories in Athens. 1 While the local signatures are very insightful, sometimes it is necessary to uantify the distance between two grahs with a single number. To this end, we integrate the local distance over D as well as over a range of values for ε in order to obtain a distance between our grahs X and Y: Definition 3.1. The PLH distance metric is: r1 d LH (X, Y) = ω(r) η(x)ψ ε (x) dx dε, 0 where η : D R and ω : [0, r 1 ] R are non-negative weight functions that integrate to unity. We note here that as long as both weight functions 1 Data is available at D are everywhere ositive, d LH is a metric; see [9]. The PLH Metric is a owerful tool in road networ analysis, as it rovides an actual metric between road networs (reresented as grahs embedded in the lane), as well as rovides a visualization of the differences between the networs. In [22], this aroach is comared against other distances between embedded networs Other Alications PLH can be used in (local) dimension estimation [10, 23]. Its variant, ersistent local cohomology, could be used to analyze branching structures in high-dimensional oint cloud data for scientific visualization [24]. Stratification learning based on PLH could also be formulated in the context of grah reconstruction from noisy oint cloud, to classify oints as either vertices or edges, and to determine the degree of the vertices [25, 26]. Thining of PLH as a local feature, it can be used as an inut to a machine learning algorithm. Bendich et al. [10] describe a method, called multi-scale local shae analysis (MLSA), for extracting local geometric and toological structures in data sets. In articular, they use PLH as features in a machine learning algorithm to classify LIDAR data sets, demonstrating that MLSA outerforms PLH and PCA alone, when used to train an SVM classifier. 4. DISCUSSION In this aer, we defined ersistent local homology (PLH), illustrating the ower of this tool in several alications, including stratified learning and clustering, road networ analysis, and machine learning. We have discussed single arameter filtrations, but we acnowledge that the study of PLH can also benefit from advances in multi-arameter ersistence, allowing us to vary multile arameters (e.g., both α and r) simultaneously. Moreover, using PLH in new alication domains will resent new and interesting challenges. For examle, PLH can be alied in scientific visualization and medical image analysis, where one studies structures such as vascular networs.

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