Probabilistic Convergence and Stability of Random Mapper Graphs

Probabilistic Con v ergence and Stabilit y of Random Mapp er Graphs Adam Bro wn ∗ IST Austria Omer Bobro wski † T ec hnion - Israel Institute of T echnology Elizab eth Munc h ‡ Mic higan State Universit y Bei W ang § Univ ersity of Utah Abstract W e study the probabilistic conv ergence betw een the mapp er graph and the Reeb graph of a top ological space X equipp ed with a contin uous function f : X → R . W e ﬁrst give a categoriﬁcation of the mapp er graph and the Reeb graph by in terpreting them in terms of cosheav es and stratiﬁed co vers of the real line R . W e then in tro duce a v arian t of the classic mapper graph of Singh et al. (2007), referred to as the enhanced mapper graph, and demonstrate that such a construction appro ximates the Reeb graph of ( X , f ) when it is applied to p oin ts randomly sampled from a probability density function concentrated on ( X , f ). Our tec hniques are based on the interlea ving distance of constructible cosheav es and top ological estimation via kernel density estimates. F ollowing Munch and W ang (2018), we ﬁrst sho w that the mapp er graph of ( X , f ), a constructible R -space (with a ﬁxed op en co ver), approximates the Reeb graph of the same space. W e then construct an isomorphism b et ween the mapp er of ( X , f ) to the mapp er of a sup er-lev el set of a probability density function concentrated on ( X , f ). Finally , building on the approac h of Bobrowski et al. (2017), we show that, with high probability , we can recov er the mapp er of the sup er-lev el set given a suﬃciently large sample. Our work is the ﬁrst to consider the mapper construction using the theory of cosheav es in a probabilistic setting. It is part of an ongoing eﬀort to com bine sheaf theory , probability , and statistics, to supp ort top ological data analysis with random data. 1 In tro duction In recent years, top ological data analysis has b een gaining momentum in aiding knowledge discov ery of large and complex data. A great deal of work has b een fo cused on data mo deled as scalar ﬁelds. F or instance, scien tiﬁc simulations and imaging to ols pro duce data in the form of p oin t cloud samples equipp ed with scalar v alues, such as temp erature, pressure and grayscale intensit y . One wa y to understand and c haracterize the structure of a scalar ﬁeld f : X → R is through v arious forms of top ological descriptors, which provide meaningful and compact abstraction of the data. P opular topological descriptors can b e classiﬁed in to v ector-based ones such as p ersistence diagrams [ 25 ] and barco des [ 34 , 10 ], graph-based ones such as Reeb graphs [ 42 ] and their v arian ts merge trees [ 5 ] and contour trees [ 13 ], and complex-based ones such as Morse complexes, Morse-Smale complexes [33, 27, 26], and the mapp er construction [43]. F or a topological space X equipp ed with a fu nction f : X → R , the R e eb gr aph , denoted as R ( X , f ), enco des the connected components of the lev el sets f − 1 ( a ) for a ranging ov er R . It summarizes the structure of the data, represen ted as a pair ( X , f ), b y capturing the evolution of the top ology of its level sets. Research surrounding Reeb graphs and their v ariants has b een very active in recent years, from theoretical, computational and applications asp ects, see [ 6 ] for a survey . In the multiv ariate setting, Reeb spaces [ 28 ] generalize Reeb graphs and serve as top ological descriptors of multiv ariate functions f : X → R d . The Reeb graph is then a sp ecial case of a Reeb space for d = 1. One issue with Reeb spaces are their limited applicability to p oin t cloud data. T o facilitate their practical usage, a closely related construction called mapp er [43] w as introduced to capture the top ological structure ∗ E-mail: adam.bro wn@ist.ac.at † E-mail: omer@ee.tec hnion.ac.il ‡ E-mail: m uncheli@msu.edu § E-mail: beiwang@sci.utah.edu 1 of a pair ( X , f ) (where f : X → R d ). Given a top ological space X equipp ed with a R d -v alued function f , for the classic mapp er construction, we w ork with a ﬁnite go od cov er U = { U α } α ∈ A of f ( X ) for some indexing set A , such that f ( X ) ⊆ S U α . Let f ∗ ( U ) denote the cov er of X obtained by considering the path-connected comp onen ts of f − 1 ( U α ) for each α . The mapp er construction of ( X , f ) is deﬁned to b e the nerve of f ∗ ( U ), denoted as N f ∗ ( U ) , see Figure 1(h) for an example. By deﬁnition, the mapp er is an abstract simplicial complex; and its 1-dimensional skeleton is referred to as the classic mapp er gr aph in this pap er. As a computable alternative to the Reeb space, the mapp er has enjoy ed tremendous success in data science, including cancer researc h [ 38 ] and sp orts analytics [ 1 ]; it is also a cornerstone of sev eral data analytics companies suc h as Ayasdi and Alpine Data Labs. Many v ariants hav e b een studied in recent years. The α -R e eb gr aph [ 16 ] redeﬁnes the equiv alence relation b et ween p oin ts using op en in terv als of length at most α . The multisc ale mapp er [ 23 ] studies a sequence of mapp er constructions by v arying the granularit y of the co ver. The multinerve mapp er [ 14 ] computes the m ultinerve [ 29 ] of the connected cov er. The Joint Contour Net (JCN) [ 11 , 12 ] introduces quan tizations to the cov er elemen ts by rounding the function v alues. The extende d R e eb gr aph [3] uses co ver elemen ts from a partition of the domain without ov erlaps. Although the mapp er construction has b een widely appreciated by the practitioners, our understanding of its theoretical prop erties remains fragmentary . Some questions imp ortan t in theory and in practice center around its structure and its relation to the Reeb graph. Q1. Information conten t: What information is enco ded by the mapp er? Ho w muc h information can we reco ver ab out the original data from the mapp er by solving an inv erse problem? Q2. Stability: What is the structural stability of the mapp er with resp ect to p erturbations of its function, domain and cov er? Q3. Conv ergence: What is an appropriate metric under which the mapp er conv erges to the Reeb graph as the num b er of sampled p oin ts go es to inﬁnity and the granularit y of the cov er go es to zero? T o the b est of our knowledge, our work is the ﬁrst to address con vergence in a probabilistic setting . Given a mapp er construction applied to p oin ts randomly sampled from a probability densit y function, we pro ve an asymptotic result: as the num ber of p oin ts n → ∞ , the mapp er graph construction approximates that of the Reeb graph up to the granularit y of the cov er with high probability . Information, stabilit y and conv ergence. W e discuss our work in the con text of related literature in top ological data analysis. As many top ological descriptors, the mapp er summarizes the information from the original data through a lossy pro cess. T o quan tify its information con tent, Dey et al. [ 24 ] studied the topological information enco ded by Reeb spaces, mapp ers and multi-scale mapp ers, where 1-dimensional homology of the mapp er was sho wn to b e no ric her than the domain X itself. Carri´ ere and Oudot [ 14 ] characterized the information enco ded in the mapp er using the extended p ersistence diagram of its corresp onding Reeb graph. Gasparo vic et. al. [ 32 ] provided full descriptions of p ersisten t homology information of a metric graph via its in trinsic ˇ Cec h complex, a sp ecial type of nerve complex. In this pap er, we study the information conten t of the mapp er via a (co)sheaf-theoretic approach; in particular, through the notion of display lo c ale , we in tro duce an intermediate ob ject called the enhanc e d mapp er gr aph , that is, a CW complex with weigh ted 0-cells. W e show that the enhanced mapp er graph reduces the information loss during summarization and ma y b e of indep endent interest. In terms of stability , Carri ´ ere and Oudot [ 14 ] derived stability for the mapp er graph using the stability of extended p ersistence diagrams equipped with the b ottlenec k distance under Hausdorﬀ or W asserstein p erturbations of the data [ 20 ]. Our work is similar to [ 14 ] in a sense that w e study the stability of the enhanced mapp er graph with resp ect to p erturbation of the data ( X , f ), where the lo cal stability dep ends on ho w the cov er U is p ositioned in relation to the critical v alues of f . Ho wev er, we formalize the structural stabilit y of the enhanced mapp er graph using a categoriﬁcation of the mapp er algorithm and the interlea ving distance of constructible cosheav es. When f is a scalar ﬁeld and the connected cov er of its domain R consists of a collection of op en interv als, the mapp er construction is conjectured to recov er the Reeb graph precisely as the granularit y of the cov er go es to zero [ 43 ]. Babu [ 2 ] studied the ab o ve conv ergence using lev elset zigzag p ersistence mo dules and sho wed that the mapper con verges to the Reeb graph in the b ottlenec k distance. Munc h and W ang [ 37 ] c haracterized the mapp er using constructible cosheav es and prov ed the conv ergence b et ween the (classic) 2 mapp er and the Reeb space (for d ≥ 1) in interlea ving distance. The enhanced mapp er graph deﬁned in this pap er is similar to the geometric mapp er graph introduced in [ 37 ]. The diﬀerences b et w een the enhanced mapp er graph and geometric mapp er consist of technical c hanges in the geometric realization of each space as a quotient of a disjoint union of closed interv als. Prop osition 2.11 implies that the enhanced mapp er graph is isomorphic to the display lo cale of the mapp er cosheaf, giving theoretic signiﬁcance to the geometrically realizable enhanced mapp er graph. [ 24 ] established a con vergence result b et ween the mapp er and the domain under a Gromov-Hausdorﬀ metric. Carri´ ere and Oudot [14] sho wed conv ergence b et ween the (multinerv e) mapp er and the Reeb graph using the functional distortion distance [ 4 ]. The enhanced mapp er graph we deﬁne plays a role roughly analogous to the multinerv e mapp er in [ 14 ], although with several imp ortant distinctions. Most signiﬁcan tly is the fact that the enhanced mapp er graph is an R -space, and as such is not a purely combinatorial ob ject, in contrast to the multinerv e mapp er, which is a simplicial p oset. Carri ´ ere et al. [ 15 ] prov ed con vergence and provided a conﬁdence set for the mapp er using a b ottlenec k distance on certain extended p ersistence diagrams. They show ed that the mapp er is an optimal estimator of the Reeb graph and provided a statistical metho d for automatic parameter tuning using the rate of conv ergence. Like [ 15 ], this pap er studies a notion of consistency (detailed b elo w) for the mapp er algorithm. In contrast to [ 15 ], the results provided here use the Reeb distance on constructible R -graphs (deﬁned in Section 2) rather than b ottlenec k distances on extended p ersistence diagrams, and are applicable to more general top ological spaces (i.e., we do not require X to b e a smo oth manifold). Probabilistic mapper inference. This work is part of an eﬀort to harness the theory of probability and statistics to supp ort and analyze the use of top ological metho ds with random data. T o date, most of this eﬀort has b een put into problems related to the homology and p ersisten t homology of random p oin t clouds. The problem of homolo gic al infer enc e relates to the ability to recov er the homology (or p ersisten t homology) of an unknown space or function given random observ ations. In a noiseless setup this problem was studied in [ 39 , 7 , 18 , 21 , 44 ]. The noisy setup w as studied in [ 40 , 9 , 19 , 30 ]. Brieﬂy , these works provide metho ds to reco ver the homology , together with assumptions that guarantee correct recov ery with high probability . In man y of these, the results are asymptotic, taking the num b er of p oin ts n → ∞ . The main reason for taking limits, is that the mathematics b ecome more tractable, and provide simpler and more intuitiv e statements. Suc h asymptotic results can b e considered as pro ofs of c onsistency for such homology estimation pro cedures. In Section 3, we apply results of [ 9 ] to study consistency of the enhanced mapp er construction introduced in Section 2. The statistical techniques we use are similar to those developed in [17]. F or further discussion of the diﬀerences b et ween the tec hniques used in Section 3 and the results of [17], see [9]. In a wa y , the work here uses similar ideas to p erform “mapp er inference”, a type of structur al infer enc e , and prov es consistency . Other probabilistic studies related to applied topology mainly include limiting theorems (laws of large num b ers, and central limit theorems), and extreme v alue analysis for the homology and p ersisten t homology of random data (see e.g. [ 46 , 35 , 41 , 8 , 36 ]). How ev er, these are muc h more detailed quan titative statements than what we are lo oking for when working with the mapp er construction. Con tributions. W e highlight four contributions of this pap er. • First, in Section 2.3, we in tro duce and construct an enhanc e d mapp er gr aph . This graph retains more geometric information ab out the underlying space than the combinatorially deﬁned classic mapp er graph, m ultinerve mapp er graph, and geometric mapp er graph (deﬁned in [ 37 ]). Moreo ver, we show that the enhanced mapp er graph construction provides a concrete realization of the display lo cale of a constructible cosheaf. • Second, in Section 2.5, we give a categorical interpretation of the mapp er construction. This cate- goriﬁcation allows us to view mapp er construction as a functor from the category of cosheav es to the category of constructible cosheav es. W e can recov er a geometric realization of the mapp er construction from the categorical realization by taking enhanced mapp er graphs, i.e., the display lo cales, of the corresp onding constructible cosheav es. • Third, we prov e con vergence (Theorem 2.27) and stability (Theorem 4.4) for the mapp er cosheaf in the in terleaving distance. 3 • Finally , w e obtain results on the approximation qualit y of random mapp er graphs obtained from noisy data on spaces which are not assumed to b e manifolds (Theorem 4.2). Moreo ver, using the results of [ 22 ], each of our theorems are reinterpreted in terms of the geometrically-deﬁned enhanced mapp er graph and Reeb distance on R -graphs. This reinterpretation allows us to state our main result b elo w without referring to the machinery of cosheaf theory . Theorem (Corollary 4.3) L et R ( X , f ) b e the R e eb gr aph of a c onstructible R -sp ac e ( X , f ) , ˆ D π n b e the enhanc e d mapp er gr aph asso ciate d to the c oshe af ˆ D π n deﬁne d in Se ction 4, and d R ( · , · ) b e the R e eb distanc e deﬁne d in Se ction 2. Using the notation deﬁne d in Se ction 3, if ther e exists ε < δ U such that p is ε -c onc entr ate d on X , then lim n →∞ P  d R  ˆ D π n , R ( X , f )  ≤ r es f U  = 1 . In tuitively sp eaking, the ab o ve theorem states that we can recov er (a v ariant of ) the mapp er graph using the theory of cosheav es in a probabilistic setting. In particular, with high probability , the distance b et w een an enhanced mapp er graph and the Reeb graph is upp er b ounded by the resolution of the cov er (denoted as res f U , see Deﬁnition 2.26) as the nu mber of samples go es to inﬁnit y . The pro of of the theorem relies on t wo preliminary results. First, in Theorem 2.27, w e construct an interlea ving b et ween the Reeb cosheaf and mapp er cosheaf. Proposition 3.10 is the second key ingredient of the pro of, giving a probabilistic recov ery of the mapp er cosheaf from random p oin ts. By interpreting the enhanced mapp er graph in terms of cosheaf theory , we are able to simplify many of the pro ofs for con vergence and stabilit y . Generally , this paper illustrates the utility of combining sheaf theory with statistics in order to study robust top ological and geometric prop erties of data. Pictorial o v erview. T o b etter illustrate our k ey constructions, we give an example of an enhanced mapp er graph. As illustrated in Figure 1, given a top ological space equipp ed with a height function ( X , f ), we are in terested in studying how well its classic mapp er graph (h) (with a ﬁxed cov er) approximates its Reeb graph (b). In order to study this problem, we construct a categoriﬁcation of the mapp er graph, through the theory of constructible cosheav es (d). The display lo cale functor is used to recov er a geometric ob ject from these category-theoretic constructible cosheav es. The geometric realization of the display lo cale of the mapp er cosheaf is referred to as the enhanced mapp er graph (g). W e outline an explicit geometric realization of the enhanced mapp er graph as a quotient of a disjoint union of closed interv als (f ). The main result of the pap er, Theorem 4.2, giv es (with high probabilit y) a b ound on the interlea ving distance b et ween the Reeb cosheaf and the enhanced mapp er cosheaf. In order to interpret this result in terms of probabilistic conv ergence (Corollary 4.3), we apply the display lo cale functor to obtain the Reeb graph and the enhanced mapp er graph from their cosheaf-theoretic analogues. This pro cedure results (with high probability) in a b ound on the Reeb distance b et ween an enhanced mapp er graph and the Reeb graph of a constructible R -space with random data. 2 Bac kground In this section, we review the results of [ 22 ] together with [ 37 ], showing that the interlea ving distance b etw een the mapp er of the constructible R -space ( X , f ) relative to the op en cov er U of R and the Reeb graph of ( X , f ) is b ounded by the resolution of the op en co ver. Motiv ated by the categoriﬁcation of Reeb graphs in [ 22 ], we in tro duce a categoriﬁed mapp er algorithm, and restate the main results of [37] in this framework. Categoriﬁcation, in this con text, means that w e are in terested in using the theory of constructible coshea v es to study Reeb graphs and mapp er graphs. W e can accomplish this by deﬁning a cosheaf (the Reeb cosheaf ) whose display lo cale is isomorphic to a given Reeb graph. One goal (completed in [ 22 ]) of this approach is to use cosheaf theory to deﬁne an extended metric on the category of Reeb graphs. A natural candidate from the p ersp ectiv e of cosheaf theory is the interlea ving distance. Suppose we w ant to use the interlea ving distance of cosheav es to determine if tw o Reeb graphs are homeomorphic. W e can ﬁrst think of each Reeb graph as the display lo cale of a cosheaf, F and G , resp ectiv ely . This allows us to rephrase our problem as that of determining if the cosheav es, F and G , are isomorphic. In general, in terleaving distances cannot answ er this question, since the interlea ving distance is an extended pseudo -metric on the category of all coshea ves. In other words, having interlea ving distance equal to 0 is not enough to guarantee that F and G are isomorphic as cosheav es. This seems to suggest that the interlea ving distance is insuﬃcient for the study 4 (a) (c) (b) (d) (e) (f) (g) (h) Figure 1: An example of an enhanced mapp er graph. (a) An R -space ( X , f ) given by a top ological space X (in blue) equipp ed with a height function f : X → R . (b) Reeb graph of ( X , f ). (c) Nice cov er of R with op en interv als. (d) Visualization of the mapp er cosheaf. (e) Stratiﬁcation of R . (f ) Disjoint union of closed in terv als ( e D , in the notation of Section 2.3), with quotient isomorphic to the enhanced mapp er graph. (g) Enhanced mapp er graph ( D , in the notation of Section 2.3). (h) Classic mapp er graph of ( X , f ). of Reeb graphs. Ho wev er (due to results of [ 22 ]), if we restrict our study to the category of constructible coshea ves (ov er R ), we can av oid this subtlety . The interlea ving distance is in fact an extended metric on the category of constructible cosheav es. If tw o constructible coshea ves ha ve interlea ving distance equal to 0, then they are isomorphic as coshea ves. Therefore, the display lo cales of constructible cosheav es (ov er R ) are homeomorphic if the interlea ving distance b et w een the cosheav es is equal to 0. In other words, if we wan t to know if tw o Reeb graphs are homeomorphic, it is suﬃcien t to consider the interlea ving distance b et ween constructible cosheav es F and G , provided that the displa y lo cales of the constructible cosheav es reco ver the Reeb graphs. Therefore, in the remainder of this section, we deﬁne a mapp er cosheaf, and sho w that the Reeb cosheaf of a constructible R -space is a constructible cosheaf, and that the mapp er cosheav es are constructible. This allo ws us to use the commutativit y of diagrams and the interlea ving distance to prov e con vergence of the corresp onding display lo cales, that is, the Reeb graphs and the enhanced mapp er graphs. W e use the example in Figure 1 as a reference for v arious notions. 2.1 Constructible R -spaces W e b egin by deﬁning constructible R -spaces, which we consider to b e the underlying spaces for estimating the Reeb graphs, see Figure 1. Constructible R -spaces can b e considered as a class of top ological spaces whic h provide a natural setting for generalizing asp ects of classical Morse theory to the study of singular spaces. Like smo oth manifolds equipp ed with a Morse function, constructible R -spaces are top ological spaces equipp ed with a real v alued function f , whose ﬁb ers, f − 1 ( x ), satisfy certain regularity conditions. Sp eciﬁcally , the top ological structure of the ﬁb ers of the real v alued function are required to only change at a ﬁnite set of function v alues. The function v alues which mark changes in the top ological structure of ﬁb ers are referred to as critical v alues. Deﬁnition 2.1 ([ 22 ]) . An R -space is a p air ( X , f ) , wher e X is a top olo gic al sp ac e and f : X → R is a c ontinuous map. A constructible R -space is an R -sp ac e ( X , f ) satisfying the fol lowing c onditions: 1. Ther e exists a ﬁnite incr e asing se quenc e of p oints S = { a 0 , · · · , a n } ⊂ R , two ﬁnite sets of lo c al ly p ath- c onne cte d sp ac es { V 0 , · · · , V n } and { E 0 , · · · , E n − 1 } , and two sets of c ontinuous maps {  i : E i → V i } 5 and { r i : E i → V i +1 } , such that X is the quotient sp ac e of the disjoint union n a i =0 V i × { a i } t n − 1 a i =0 E i × [ a i , a i +1 ] by the r elations (  i ( x ) , a i ) ∼ ( x, a i ) and ( r i ( x ) , a i +1 ) ∼ ( x, a i +1 ) for al l i and x ∈ E i . 2. The c ontinuous function f : X → R is given by pr oje ction onto the se c ond factor of X . These ar e the obje cts of c ate gories R - space and R - space c , c onsisting of R -sp ac es and c onstructible R -sp ac es, r esp e ctively. Morphisms in these c ate gories ar e function-pr eserving maps; that is, ϕ : ( X , f ) → ( Y , g ) is given by a c ontinuous map ϕ : X → Y such that g ◦ ϕ ( x ) = f ( x ) . Example 2.2. A smo oth c omp act manifold X with a Morse function f c onstitutes a c onstructible R -sp ac e. F or instanc e, Figur e 1(a) il lustr ates a top olo gic al sp ac e X e quipp e d with a height function f ; the p air ( X , f ) is an R -sp ac e. Similarly, a height function f on a torus X gives rise to an R -sp ac e ( X , f ) in Figur e 6(a). In fact, X is not required to b e a manifold for ( X , f ) to b e an R -space. Throughout the remainder of this pap er, we assume that ( X , f ) is a constructible R -space. Deﬁnition 2.3 ([ 22 ]) . An R -graph is a c onstructible R -sp ac e such that the sets V i and E i ar e ﬁnite sets (with the discr ete top olo gy) for al l i . Example 2.4. The R e eb gr aph of a c onstructible R -sp ac e is an R -gr aph. F or instanc e, the R e eb gr aph of ( X , f ) in Figur e 1(b) is an R -gr aph. Similarly, the R e eb gr aph of a Morse function on a torus is an R -gr aph, se e Figur e 6(b). 2.2 Constructible cosheav es Shea ves and coshea ves are category-theoretic structures, called functors, whic h provide a framew ork for asso ciating data to open sets in a top ological space. These associations are required to preserve certain prop erties inherent to the top ology of the space. In this wa y , one can study the top ological structure of the space by studying the data asso ciated to each op en set by a given sheaf or cosheaf. In the following sections, w e will use cosheav es to enco de information ab out a constructible R -space by asso ciating op en interv als in the real line to sets of (path-)connected comp onen ts of ﬁb ers of the real v alued function corresp onding to the constructible R -space. Let In t b e the category of connected op en sets in R with inclusions which we refer to as interv als, and Set the category of ab elian groups with group homomorphism maps. W e ﬁrst deﬁne a cosheaf ov er R , which w e prop ose to b e the natural ob jects for categorifying the mapp er algorithm. Deﬁnition 2.5. A pre-cosheaf F on R is a c ovariant functor F : In t → Set . The c ate gory of pr e c oshe aves on R is denote d Set In t with morphisms given by natur al tr ansformations. A pr e-c oshe af F is a cosheaf if lim − → V ∈V F ( V ) = F ( U ) for e ach op en interval U ∈ In t and e ach op en interval c over V ⊂ Int of U , which is close d under ﬁnite interse ctions. The ful l sub c ate gory of Set In t c onsisting of c oshe aves is denote d Csh . Remark 2.6. We note that usual ly, c oshe aves ar e deﬁne d over the c ate gory of arbitr ary op en sets r ather than the c ate gory of c onne cte d op en sets. However, the c ate gory of c oshe aves deﬁne d over c onne cte d op en sets is e quivalent to the c ate gory of c oshe aves deﬁne d over arbitr ary op en sets, by the c olimit pr op erty of c oshe aves. When we deﬁne smo othing op er ations on c oshe aves in Se ction 2.4, ther e ar e imp ortant distinctions that wil l make cle ar the ne e d for the deﬁnition with r esp e ct to In t , as set-thickening op er ations do not pr eserve the c oshe af pr op erty otherwise. 6 Since we are interested in working with cosheav es which can b e describ ed with a ﬁnite amount of data, we will restrict our attention to a w ell-b eha ved sub category of Csh , consisting of constructible coshea ves (deﬁned b elo w). Constructibility can b e thought of as a type of “tameness” assumption for sheav es and coshea ves. Deﬁnition 2.7. A c oshe af F is constructible if ther e exists a ﬁnite set S ⊂ R of critical v alues such that F [ U ⊂ V ] is an isomorphism whenever S ∩ U = S ∩ V . The ful l sub c ate gory of Csh c onsisting of c onstructible c oshe aves is denote d Csh c . 2.3 The Reeb cosheaf and displa y locale functors W e introduce the Reeb cosheaf and display lo cale functors. These functors relate the category of constructible coshea ves to the category of R -graphs, and provide a natural categoriﬁcation of the Reeb graph [ 22 ]. In other w ords, via b oth Reeb cosheaf functor and displa y lo cale functors, one could consider the translation b et ween the data and their corresp onding categorical in terpretations. Let R f b e the R e eb c oshe af of ( X , f ) on R , deﬁned by R f ( U ) = π 0 ( X U ) , where X U := f − 1 ( U ) and π 0 ( X U ) denotes the set of path comp onen ts of X U . Deﬁnition 2.8. The Reeb cosheaf functor C fr om the c ate gory of c onstructible R -sp ac es to the c ate gory of c onstructible c oshe aves R - space c Csh c C is deﬁne d by C (( X , f )) = R f . F or a function-pr eserving map ϕ : ( X , f ) → ( Y , g ) , the r esulting morphism C [ ϕ ] is given by C [ ϕ ] : R f ( U ) = π 0 ◦ f − 1 ( U ) → π 0 ◦ g − 1 ( U ) = R g ( U ) induc e d by ϕ ◦ f − 1 ( U ) ⊆ g − 1 ( U ) . Deﬁnition 2.9. The costalk of a (pr e-)c oshe af F at x ∈ R is F x = lim ← − I 3 x F ( I ) . F or e ach c ostalk F x , ther e is a natur al map F x → F ( I ) (given by the universal pr op erty of limits) for e ach op en interval I c ontaining x . In order to related the Reeb and mapp er cosheav es to geometric ob jects, we make use of the notion of display lo c ale , in tro duced in [31]. Deﬁnition 2.10. The display lo cale of a c oshe af F (as a set) is deﬁne d as D ( F ) = a x ∈ R F x . A top olo gy on D ( F ) is gener ate d by op en sets of the form U I ,a = { s ∈ F x : x ∈ I and s 7→ a ∈ F ( I ) } , for e ach op en interval I ∈ In t and e ach se ction a ∈ F ( I ) . The display lo cale gives a functor from the category of cosheav es to the category of R -graphs, Csh c R - graph . D W e pro ceed by giving an explicit geometric realization of the display lo cale of a constructible cosheaf. Let F b e a constructible cosheaf with set of critical v alues R 0 ⊂ R . Let R 1 = R \ R 0 b e the complement of R 0 , so that we form a stratiﬁcation R = R 0 t R 1 , 7 See Figure 1(e) for an example (black p oin ts are in R 0 , their complements are in R 1 ). Let S 1 b e the set of connected comp onen ts of R 1 , i.e., the 1-dimensional stratum pieces. F or x ∈ R 0 , let I x denote the largest op en interv al containing x suc h that I x ∩ R 0 = { x } . Let ˜ D ( F ) := a V ∈ S 1 V × F ( V ) t a x ∈ R 0 { x } × F ( I x ) , where V is the closure of V and the pro duct C × ∅ of a set C with the empt y set is understo od to b e empty . Geometrically , ˜ D ( F ) is a disjoint union of connected closed subsets of R ; if the supp ort of F is compact, then ˜ D ( F ) is a disjoint union of closed interv als and p oin ts. Let π denote the pro jection map π : ˜ D ( F ) → R ( x, a ) 7→ x. Supp ose ( x, a ) ∈ V × F ( V ) ⊂ ˜ D ( F ) and x ∈ R 0 . W e hav e that V ∩ R 0 = ∅ and I x ∩ V 6 = ∅ (b ecause x lies on the b oundary of V ). By maximality of I x , we hav e the inclusion V ⊂ I x . Let ϕ ( x,a ) b e the map ϕ ( x,a ) : F ( V ) → F ( I x ) induced by the inclusion V ⊂ I x . W e can extend this map to the ﬁb er of π o ver x , ψ x : π − 1 ( x ) → F ( I x ) , where ψ x (( x, a )) := ϕ ( x,a ) ( a ) if ( x, a ) ∈ V × F ( V ) and ψ x (( x, a )) := a if ( x, a ) ∈ { x } × F ( I x ). Finally , we deﬁne an equiv alence relation of p oin ts in ˜ D ( F ). Suppose ( x, a ) , ( y , b ) ∈ ˜ D ( F ). Then ( x, a ) ∼ ( y , b ) if 1. x = y ∈ R 0 , and 2. ψ x ( a ) = ψ x ( b ) ∈ F ( I x ). Finally , let D ( F ) := ˜ D ( F ) / ∼ b e the quotient of ˜ D ( F ) by the equiv alence relation. The pro jection π factors through the quotient, giving a map ¯ π : D ( F ) → R . Prop osition 2.11. If F is a c onstructible c oshe af with set of critic al values S , then D ( F ) is a 1-dimensional CW-c omplex which is isomorphic (as an R -sp ac e) to the display lo c ale, D ( F ) , of F . Pr o of. W e will construct a homeomorphism γ : D ( F ) → D ( F ) which preserves the natural quotient maps ¯ f : D ( F ) → R and ¯ π : D ( F ) → R . Giv en x ∈ R 1 , w e ha ve that ¯ π − 1 ( x ) = { x } × F ( V ), where V is the connected component of R 1 whic h con tains x . Since F is constructible with resp ect to the c hosen stratiﬁcation, we hav e that F ( V ) ∼ = F x . This gives a bijection from ¯ π − 1 ( x ) to ¯ f − 1 ( x ). F or x ∈ R 0 , the ﬁb er ¯ π − 1 ( x ) is b y construction in bijection with F ( I x ). Again, since F is constructible and I x ∩ R 0 = B ( x ) ∩ R 0 for each suﬃciently small neighborho o d B ( x ) of x , we hav e that F ( I x ) ∼ = F x . These bijections deﬁne a map γ : D ( F ) → D ( F ), which preserves the quotient maps b y construction. All that remains is to show that γ is con tinuous. Supp ose x ∈ R 1 , and let V b e the connected comp onen t of R 1 whic h contains x , and B ( x ) b e an op en neigh b orho od of x suc h that B ( x ) ⊂ V . Then F y ∼ = F ( V ) for each y ∈ B ( x ), and F ( B ( x )) ∼ = F ( V ). Recall the deﬁnition of the basic op en sets U I ,a in the deﬁnition of display lo cale (with notation adjusted to b etter align with the current pro of ), U I ,a = ( s ∈ F y ⊂ a x ∈ R F x : y ∈ I and s 7→ a ∈ F ( I ) ) . Using the ab o ve isomorphisms to simplify the deﬁnition according to the curren t set-up, we get U B ( x ) ,a ∼ =    a ∈ a y ∈ B ( x ) F ( V )    . 8 Therefore, γ − 1 ( U B ( x ) ,a ) = B ( x ) × { a } , which is op en in the quotient top ology on D ( F ). Supp ose x ∈ R 0 , and let B ( x ) b e a neighborho o d of x suc h that B ( x ) ⊂ I x . Let V 1 and V 2 denote the t wo connected comp onen ts of R 1 whic h are contained in I x . If y ∈ B ( x ), then F y is isomorphic to either F ( V 1 ), F ( V 2 ), or F ( I x ). Moreov er, since F is constructible, w e hav e that F ( B ( x )) ∼ = F ( I x ). Let a 0 ∈ F ( I x ) corresp ond to a ∈ F ( B ( x )) under the isomorphism F ( I x ) ∼ = F ( B ( x )). F ollowing the deﬁnitions, we hav e that π − 1  γ − 1 ( U B ( x ) ,a )  =  V 1 ∩ B ( x )  × F [ V 1 ⊂ I x ] − 1 ( a 0 ) t  V 2 ∩ B ( x )  × F [ V 2 ⊂ I x ] − 1 ( a 0 ) t { x } × { a 0 } , where F [ V i ⊂ I x ] − 1 ( a 0 ) is understoo d t o b e a (p ossibly empt y) subset of F ( V i ). It follo ws that γ − 1 ( U B ( x ) ,a ) is op en in the quotien t top ology on D ( F ). Therefore, γ − 1 maps op en sets to op en sets, and we hav e shown that γ is a homeomorphism which preserv es the quotient maps ¯ f and ¯ π , i.e., ¯ f ( γ (( x, a ))) = ¯ π (( x, a )) = x . It follows from the prop osition that D ( F ) is indep enden t (up to isomorphism) of choice of critical v alues R 0 . Additionally , we no w note that w e can freely use the notation D ( F ) or D ( F ) to refer to the display lo cale of a constructible cosheaf o ver R . W e will con tinue to use b oth symbols, reserving D for the display lo cale of an arbitrary cosheaf, and using D when we w ant to emphasize the ab o ve equiv alence for constructible coshea ves. In [ 22 ], it is shown that the Reeb graph R ( X , f ) of ( X , f ) is naturally isomorphic to the display lo cale of R f . Moreo ver, the display lo cale functor D and the Reeb functor C are inv erse functors and deﬁne an equiv alence of categories b et w een the category of Reeb graphs and the category of constructible cosheav es on R . This equiv alence is closely connected to the more general relationships b et ween constructible cosheav es and stratiﬁed cov erings studied in [ 45 ]. The result allows us to deﬁne a distance b et ween Reeb graphs by taking the interlea ving distance b et w een the asso ciated constructible cosheav es as shown in the following section. 2.4 In terlea vings W e start by deﬁning the interlea vings on the categorical ob jects. Interlea ving is a typical to ol in top ological data analysis for quantifying proximit y betw een ob jects such as persistence mo dules and cosheav es. F or U ⊆ R , let U 7→ U ε := { y ∈ R | k y − U k ≤ ε } . If U = ( a, b ) ∈ Int , then U ε = ( a − ε, b + ε ). Deﬁnition 2.12. L et F and G b e two c oshe aves on R . A n ε -in terleaving b etwe en F and G is given by two families of maps ϕ U : F ( U ) → G ( U ε ) , ψ U : G ( U ) → F ( U ε ) which ar e natur al with r esp e ct to the inclusion U ⊂ U ε , and such that ψ U ε ◦ ϕ U = F [ U ⊂ U 2 ε ] , ϕ U ε ◦ ψ U = G [ U ⊂ U 2 ε ] for al l op en intervals U ⊂ R . Equivalently, we r e quir e that the diagr am F ( U ) F ( U ε ) F ( U 2 ε ) G ( U ) G ( U ε ) G ( U 2 ε ) ϕ U ϕ U ε ψ U ψ U ε c ommutes, wher e the horizontal arr ows ar e induc e d by U ⊆ U ε ⊆ U 2 ε . The in terleaving distance b etwe en two c oshe aves F and G is given by d I ( F , G ) := inf { ε | ther e exists an ε -interle aving b etwe en F and G } . No w that we hav e an interlea ving for elements of Csh c along with an equiv alence of categories b et ween Csh c and R - graph , we can dev elop this into an interlea ving distance for the Reeb graphs themselves. The in terleaving distance for Reeb graphs will b e deﬁned using a smo othing functor, which we construct b elo w. 9 Deﬁnition 2.13. L et ( X , f ) b e a c onstructible R -sp ac e. F or ε ≥ 0 , deﬁne the thick ening functor T ε to b e T ε ( X , f ) = ( X × [ − ε, ε ] , f ε ) , wher e f ε ( x, t ) = f ( x ) + t . Given a morphism α : X → Y , T ε ( α ) : X × [ − ε, ε ] → Y × [ − ε, ε ] ( x, t ) 7→ ( α ( x ) , t ) . The zero section map is the morphism ( X , f ) → T ε ( X , f ) induc e d by X → X × [ − ε, ε ] x 7→ ( x, 0) . Prop osition 2.14 ([ 22 , Prop osition 4.23]) . The thickening functor T ε maps R -gr aphs to c onstructible R -sp ac es, i.e., if ( G , g ) ∈ R - graphs then T ε ( G , g ) ∈ R - spaces c . In general, the thick ening functor T ε will output a constructible R -space, and not an R -graph. In order to deﬁne a ‘smo othing’ functor for R -graphs (following [ 22 ]), we need to introduce a Reeb functor, which maps a constructible R -space to an R -graph. Deﬁnition 2.15. The Reeb graph functor R maps a c onstructible R -sp ac e ( X , f ) to an R -gr aph ( X f , ¯ f ) , wher e X f is the R e eb gr aph of ( X , f ) and ¯ f is the function induc e d by f on the quotient sp ac e X f . The Reeb quotien t m ap is the morphism ( X , f ) → R ( X , f ) induc e d by the quotient map X → X f . No w w e can deﬁne a smo othing functor on the category of R -graphs. Deﬁnition 2.16. L et ( G , f ) ∈ R - graph . The Reeb smo othing functor S ε : R - graph → R - graph is deﬁne d to b e the R e eb gr aph of an ε -thickene d R -gr aph S ε ( G , f ) = R ( T ε ( G , f )) . The Reeb smo othing functor S ε deﬁned ab o ve is used to deﬁne an interlea ving distance for Reeb graphs, called the Reeb interlea ving distance. The Reeb interlea ving distance, deﬁned b elo w, can b e thought of as a geometric analogue of the interlea ving distance of constructible cosheav es. Let ζ ε F b e the map from ( F , f ) to S ε ( F , f ) given by the comp osition of the zero section map ( F , f ) → T ε ( F , f ) with the Reeb quotient map T ε ( F , f ) → R ( T ε ( F , f )). T o ease notation, we will denote the composition of ζ ε F : ( F , f ) → S ε ( F , f ) with ζ S ε ( F ,f ) : S ε ( F , f ) → S ε ( S ε ( F , f )) by ζ ε F ( ζ ε F ( F , f )). Deﬁnition 2.17. L et ( F , f ) and ( G , g ) b e R -gr aphs. We say that ( F , f ) and ( G , g ) ar e ε -in terleav ed if ther e exists a p air of function-pr eserving maps α : ( F , f ) → S ε ( G , g ) and β : ( G , g ) → S ε ( F , f ) such that S ε ( β ) ( α ( F , f )) = ζ ε F ( ζ ε F ( F , f )) and S ε ( α ) ( β ( G , g )) = ζ ε G ( ζ ε G ( G , g )) . That is, the diagr am ( F , f ) ζ ε F ( F , f ) ζ ε F ( ζ ε F ( F , f )) ( G , g ) ζ ε G ( G , g ) ζ ε G ( ζ ε G ( G , g )) α S ε ( α ) β S ε ( β ) c ommutes. The Reeb interlea ving distance , d R (( F , f ) , ( G , g )) , is deﬁne d to b e the inﬁmum over al l ε such that ther e exists an ε -interle aving of ( F , f ) and ( G , g ) : d R (( F , f ) , ( G , g )) := inf { ε : ther e exists an ε -interle aving of ( F , f ) and ( G , g ) } . 10 X AAAB83icbVDLSsNAFL3xWeur6tLNYCu4Kkk3dllw47KCfUATymQ6bYdOJmHmRiihv+HGhSJu/Rl3/o2TNgttPTBwOOde7pkTJlIYdN1vZ2t7Z3dvv3RQPjw6PjmtnJ13TZxqxjsslrHuh9RwKRTvoEDJ+4nmNAol74Wzu9zvPXFtRKwecZ7wIKITJcaCUbSSX/MjitMwzPqL2rBSdevuEmSTeAWpQoH2sPLlj2KWRlwhk9SYgecmGGRUo2CSL8p+anhC2YxO+MBSRSNugmyZeUGurTIi41jbp5As1d8bGY2MmUehncwjmnUvF//zBimOm0EmVJIiV2x1aJxKgjHJCyAjoTlDObeEMi1sVsKmVFOGtqayLcFb//Im6Tbqnlv3HhrVVrOoowSXcAU34MEttOAe2tABBgk8wyu8Oanz4rw7H6vRLafYuYA/cD5/AH+ekUg= 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See Remark 2.20. Remark 2.18. We should r emark on a te chnic al asp e ct of the ab ove deﬁnition. The c omp osition ζ ε F ◦ ζ ε F ( F , f ) is natur al ly isomorphic to ζ 2 ε F ( F , f ) . However, sinc e the deﬁnition of the R e eb interle aving distanc e r e quir es c ertain diagr ams to c ommute, it is ne c essary to sp e cify an isomorphism b etwe en ζ ε F ◦ ζ ε F ( F , f ) and ζ 2 ε F ( F , f ) if one would like to r eplac e ζ ε F ◦ ζ ε F ( F , f ) with ζ 2 ε F ( F , f ) in the c ommutative diagr ams. Ther efor e, we cho ose to work exclusively with the c omp osition of zer o se ction maps, r ather than working with diagr ams which c ommute up to natur al isomorphism. The remaining proposition of this section giv es a geometric realization of the interlea ving distance of constructible cosheav es. Prop osition 2.19 ([ 22 ]) . D ( F ) and D ( G ) ar e ε -interle ave d as R -gr aphs if and only if F and G ar e ε -interle ave d as c onstructible c oshe aves. Remark 2.20. Coshe aves ar e usual ly deﬁne d as functors on the c ate gory of op en sets inste ad of functors on the c onne cte d op en sets. We cho ose to use In t inste ad of Op en ( R ) due to te chnic al issues that arise when we b e gin smo othing the functors. Basic al ly, smo othing the functor do es not pr o duc e a c oshe af when the intervals ar e r eplac e d by arbitr ary op en sets in R . Consider the example of Fig. 2, wher e X is a line with map f pr oje ction onto R . Say U ε is the thickening of a set, U ε = { x ∈ R | | x − U | < ε } . Then we c an pick an ε so that A ε is two disjoint intervals, and ( A ∪ B ) ε is one interval. L et F b e the functor U 7→ π 0 f − 1 ( U ) which is a c oshe af r epr esenting the R e eb gr aph. Then the functor F ◦ ( · ) ε is not a c oshe af sinc e by the diagr am, ∅ = F (( A ∩ B ) ε ) F ( A ε ) = {••} {•} = F ( B ε ) colim = {• • •} F ( A ∪ B ) ε = {•} is not the c olimit of F ( A ε ) and F ( B ε ) . 1 2.5 Categoriﬁed mapp er In this section, we interpret classic mapp er (for scalar functions), a top ological descriptor, as a category theoretic ob ject. This interpretation, in terms of cosheav es and category theory , simpliﬁes man y of the argumen ts used to prov e conv ergence results in Section 4. W e ﬁrst review the classic mapp er and then discuss the categoriﬁed mapp er. The main ingredient needed to deﬁne the mapp er construction is a c hoice of cov er. W e sa y a cov er of R is go o d if all intersections are contractible. A cov er U is lo c al ly ﬁnite if for every x ∈ R , U x = { V ∈ U : x ∈ V } is a ﬁnite set. In particular, lo cally ﬁniteness implies that the cov er restricted to a compact set is ﬁnite. F or the remainder of the pap er, we work with nic e c overs whic h are go od, lo cally ﬁnite, and consist only of connected interv als, see Figure 1(c) for an example. 1 W e thank Vin de Silva for this counterexample. 11 W e will now introduce a categoriﬁcation of mapp er. Let U b e a nice cov er of R . Let N U b e the nerv e of U , endow ed with the Alexandroﬀ top ology . Consider the contin uous map η : R → N U x 7→ \ V ∈U x V , where the intersection T V ∈U x V is viewed as an op en simplex of N U . The mapp er functor M U : Set In t → Set In t can b e deﬁned as M U ( C ) = η ∗ ( η ∗ ( C )) , where η ∗ and η ∗ are the (pre)-cosheaf-theoretic pull-back and push-forward op erations resp ectiv ely . Ho wev er, rather than deﬁning η ∗ and η ∗ in generality , we choose to work with an explicit description of M U ( C ) given b elo w. F or notational conv enience, deﬁne I U : In t → In t U 7→ η − 1 (St( η ( U ))) , where St ( η ( U )) denotes the minimal op en set in N U con taining η ( U ) := ∪ x ∈ U η ( x ) (the op en star of η ( U ) in N U ). It is often conv enient to identify I U ( U ) with a union of op en in terv als in R . Lemma 2.21. Using the notation deﬁne d ab ove, we have the e quality I U ( U ) = [ x ∈ U \ V ∈U x V , wher e T V ∈U x V is viewe d as a subset of R (not as a simplex of N U ). Pr o of. If y ∈ S x ∈ U T V ∈U x V , then there exists an x ∈ U suc h that y ∈ V for all V ∈ U x . In other words, U x ⊆ U y . Therefore, η ( y ) ≥ η ( x ) in the partial order of N U . Therefore, η ( y ) ∈ St ( η ( U )). This implies that S x ∈ U T V ∈U x V ⊆ I U ( U ). F or the rev erse inclusion, assume that u ∈ I U ( U ), i.e., η ( u ) ∈ St ( η ( U )). This implies that there exists v ∈ U suc h that η ( u ) ≥ η ( v ). In other words, U v ⊆ U u . Therefore u ∈ ∩ V ∈U v V , and u ∈ S v ∈ U T V ∈U v V . Under this identiﬁcation, it is clear that I U ( U ) is an open set in R (since the op en cov er U is lo cally ﬁnite), and if U ⊂ V then I U ( U ) ⊂ I U ( V ). Moreov er, since T V ∈U x V is an interv al op en neighborho od of x and U is an op en interv al, then I U ( U ) is an op en interv al. Therefore, I U can b e viewed as a functor from In t to In t . Finally , we can give M U ( C ) an explicit description in terms of the functor I U . Deﬁnition 2.22. The mapp er functor M U : Set In t → Set In t is deﬁne d by M U ( C )( U ) := C ( I U ( U )) , for e ach op en interval U ∈ In t . Since I U is a functor from In t to In t , it follows that M U is a functor from Set In t to Set In t . Hence, M U ( C ) is a functor from the category of pre-cosheav es to the category of pre-cosheav es. In the following prop osition, we show that if C is a cosheaf, then M U ( C ) is in fact a constructible cosheaf. Prop osition 2.23. L et U b e a ﬁnite nic e op en c over of R . The mapp er functor M U is a functor fr om the c ate gory of c oshe aves on R to the c ate gory of c onstructible c oshe aves on R : M U : CSh → CSh c . Mor e over, the set of critic al p oints of M U ( F ) is a subset of the set of b oundary p oints of op en sets in U . 12 Pr o of. W e w ill ﬁrst show that if C is a cosheaf on R , then M U ( C ) is a cosheaf on R . W e ha ve already shown that M U ( C ) is a pre-cosheaf. So all that remains is to prov e the colimit prop ert y of cosheav es. Let U ∈ Int and V ⊂ In t b e a co ver of U b y op en interv als which is closed under in tersections. By deﬁnition of M U ( C ), w e ha ve lim − → V ∈V M U ( C )( V ) = lim − → V ∈V C ( I U ( V )) . Notice that I U ( V ) := {I U ( V ) : V ∈ V } forms an op en cov er of I U ( U ). Ho wev er, in general this cov er is no longer closed under intersections. W e will pro ceed b y showing that passing from V to I U ( V ) 0 := { T i ∈ I W i : { W i } i ∈ I ⊂ I U ( V ) } do es not change the colimit lim − → V ∈V C ( I U ( V )) . Supp ose I 1 and I 2 are tw o op en interv als in V suc h that I 1 ∩ I 2 = ∅ and I U ( I 1 ) ∩ I U ( I 2 ) 6 = ∅ . Recall that U 0 is the union of U with all intersections of cov er elements in U , i.e., the closure of U under intersections. By the identiﬁcation I U ( I i ) = [ x ∈ I i \ V ∈U x V , there exists a subset { W j } j ∈ J ⊂ U 0 suc h that I U ( I 1 ) ∩ I U ( I 2 ) = [ j ∈ J W j . Supp ose there exist V 1 , V 2 ∈ U 0 suc h that V i ( V 1 ∪ V 2 (i.e., one set is not a subset of the other), and V 1 ∪ V 2 ⊂ I U ( I 1 ) ∩ I U ( I 2 ). In other words, suppose that the cardinality of J , for any suitable choice of indexing set, is strictly greater than 1. Then there exists x 1 , x 2 ∈ I 1 suc h that x 1 ∈ V 1 \ V 2 and x 2 ∈ V 2 \ V 1 . Let w either b e a p oin t contained in V 1 ∩ V 2 (if V 1 ∩ V 2 6 = ∅ ) or a p oin t which lies b et ween V 1 and V 2 . Since I 1 is connected, we ha ve that w ∈ I 1 . A similar argument shows that w ∈ I 2 , which implies the contradiction I 1 ∩ I 2 6 = ∅ . Therefore, I U ( I 1 ) ∩ I U ( I 2 ) = W , for some W ∈ U 0 . Supp ose W = T k ∈ K W k for some { W k } k ∈ K ⊂ U , and let I 1 = J 1 , J 2 , · · · , J n = I 2 b e a c hain of op en interv als in V , such that J j ∩ J j +1 6 = ∅ . W e hav e that I 1 ∪ [ k ∈ K W k ∪ I 2 is connected, b ecause I 1 , I 2 , and S k ∈ K W k are interv als with S k ∈ K W k ∩ I 1 and S k ∈ K W k ∩ I 2 nonempt y . Therefore, for each j , J j ∩ W k 6 = ∅ for some k , i.e., W ⊂ I U ( J j ). In conclusion, we ha ve sho wn that I U ( I 1 ) ∩ I U ( I 2 ) ⊆ I U ( J j ) for each j . F ollowing the arguments in the pro of of Prop osition 4.17 of [22], it can b e shown that lim − → V ∈V C ( I U ( V )) = lim − → U ∈I U ( V ) C ( U ) = lim − → U ∈I U ( V ) 0 C ( U ) . Since C is a cosheaf, we can use the colimit prop ert y of cosheav es to get lim − → V ∈V M U ( C )( V ) = C ( I U ( U )) . Therefore M U ( C ) is cosheaf. W e will pro ceed to show that M U ( C ) is constructible. Let S b e the set of b oundary p oin ts for op en sets in U . Since U is a ﬁnite, go o d cov er of R , S is a ﬁnite set. If U ⊂ V are t wo op en sets in R suc h that U ∩ S = V ∩ S , then I U ( U ) = I U ( V ). Therefore M U ( F )( U ) → M U ( F )( V ) is an isomorphism. 13 W e use the mapp er functor to relate Reeb graphs (the display lo cale of the Reeb cosheaf R f ) to the enhanced mapper graph (the display locale of M U ( R f )). In particular, the error is con trolled b y the resolution of the cov er, as deﬁned b elo w. Deﬁnition 2.24. L et U b e a nic e c over of R and F a c oshe af on R . The resolution of U r elative to F , denote d res F U , is deﬁne d to b e the maximum of the set of diameters of U F := { V ∈ U : F ( V ) 6 = ∅} : res F U := max { diam( V ) : V ∈ U F } . Here we understand the diameter of op en sets of the form ( a, + ∞ ) or ( −∞ , b ) to b e inﬁnite. Therefore, the resolution res F U can take v alues in the extended non-negative num b ers R ≥ 0 t { + ∞} . Remark 2.25. If R f is a R e eb c oshe af of a c onstructible R -sp ac e ( X , f ) , then R f ( V ) 6 = ∅ if and only if V ∩ f ( X ) 6 = ∅ . Deﬁnition 2.26. Deﬁne res f U by res f U := max { diam( V ) : V ∈ U f } , wher e U f := { V ∈ U : V ∩ f ( X ) 6 = ∅} . The following theorem is analogous to [ 37 , Theorem 1], adapted to the current setting. Sp eciﬁcally , our deﬁnition of the mapp er functor M U diﬀers from the functor P K of [ 37 ], and the conv ergence result of [ 37 ] is pro ved for multiparameter mapp er (whereas the following result is only prov ed for the one-dimensional case). Theorem 2.27 (cf. [37, Theorem 1]) . L et U b e a nic e c over of R , and F a c oshe af on R . Then d I ( F , M U ( F )) ≤ res F U . Pr o of. If res F U = + ∞ , then the inequality is automatically satisﬁed. Therefore, we will work with the assumption that res F U < + ∞ . Let δ U = res F U < + ∞ . W e will prov e the theorem by constructing a δ U -in terleaving of the sheav es F and M U ( F ). Supp ose I ∈ Int . F or each x ∈ I , let W x = T V ∈U x V . Recall that I U ( I ) = [ x ∈ I W x . Ideally , we would construct an interlea ving based on an inclusion of the form I U ( I ) ⊂ I δ U . How ev er, this inclusion will not alwa ys hold. F or example, if U is a ﬁnite cov er, then it is p ossible for I to b e a b ounded op en interv al, and for I U ( I ) to b e unbounded. W e will include a simple example to illustrate this b ehavior. Suppose U = { ( −∞ , − 1) , ( − 2 , 2) , (1 , + ∞ ) } and let F b e the constant cosheaf supp orted at 0, i.e. F ( U ) = ∅ if 0 / ∈ U and F ( V ) = {∗} if 0 ∈ V . Consider the in terv al I = (0 , 3). F or each x ∈ (0 , 1] ⊂ I , we hav e that W x = ( − 2 , 2). If x ∈ (1 , 2) ⊂ I , then W x = ( − 2 , 2) ∩ (1 , + ∞ ). Finally , if x ∈ [2 , 3) ⊂ I , then W x = (1 , + ∞ ). Therefore, I U ( I ) = ( − 2 , + ∞ ), whic h is unbounded. How ev er, we observ e that F (( −∞ , − 1)) = ∅ , F (( − 2 , 2)) = {∗} , and F ((1 , + ∞ )) = ∅ . Therefore, (in the notation of Deﬁnition 2.24) U F = { ( − 2 , 2) } , and res F U = diam(( − 2 , 2)) = 4. Although I U ( I ) may b e unbounded, we can construct an interv al I 0 whic h is con tained in I δ U and satisﬁes the equality F ( I U ( I )) = F ( I 0 ). The remainder of the pro of will be dedicated to constructing such an in terv al. Let W := { U : U = ∩ a ∈ A W a for some A ⊂ I } b e an open co v er of I U ( I ) whic h is closed under intersections and generated by the op en sets W x . Then the colimit prop ert y of cosheav es gives us the equality F ( I U ( I )) = lim − → U ∈W F ( U ) . Let E := { e ∈ I : F ( W e ) = ∅} . If U = ∩ a ∈ A W a and A ∩ E 6 = ∅ , then F ( U ) = ∅ . Let W I \ E = { U ∈ W : U = ∩ a ∈ A W a for some A ⊂ I \ E } . W e should remark on a small technical matter concerning I \ E . In general, this set is not necessarily connected. If that is the case, we should replace I \ E with the minimal interv al whic h co vers I \ E . Going forward, w e will assume that I \ E is connected. Altogether we ha ve M U ( F )( I ) = F ( I U ( I )) = lim − → U ∈W F ( U ) = lim − → U ∈W I \ E F ( U ) = F   [ x ∈ I \ E W x   . 14 If x ∈ I \ E , then W x ∩ I 6 = ∅ and F ( W x ) 6 = ∅ . Therefore, W x ⊆ I δ U , since diam( W x ) ≤ δ U . Moreov er, [ x ∈ I \ E W x ⊆ I δ U . The ab o ve inclusion induces the following map of sets ϕ I : M U ( F )( I ) → F ( I δ U ) , whic h giv es the ﬁrst family of maps of the δ U -in terleaving. The second family of maps ψ I : F ( I ) → M U ( F )( I δ U ) , follo ws from the more ob vious inclusion I ⊂ I U ( I δ U ). Since the interlea ving maps are deﬁned by inclusions of in terv als, it is clear that the comp osition formulae are satisﬁed: ψ I δ U ◦ ϕ I = F [ I ⊂ I 2 δ U ] , ϕ I δ U ◦ ψ I = M U ( F ) [ I ⊂ I 2 δ U ] . Remark 2.28. One might think that The or em 2.27 c an b e use d to obtain a c onver genc e r esult for the mapp er gr aph of a gener al R -sp ac e. However, we should emphasize that the interle aving distanc e is only an extende d pseudo -metric on the c ate gory of al l c oshe aves. Ther efor e, even if the interle aving distanc e b etwe en F and M U ( F ) go es to 0, this do es not imply that the c oshe aves ar e isomorphic. We only obtain a c onver genc e r esult when r estricting to the sub c ate gory of c onstructible c oshe aves, wher e the interle aving distanc e gives an extende d metric. The displa y locale D ( M U ( R f )) of the mapp er cosheaf is a 1-dimensional CW-complex obtained by gluing the b oundary p oin ts of a ﬁnite disjoin t union of closed in terv als, see Figure 1(h). W e will refer to this CW-complex as the enhanc e d mapp er gr aph of ( X , f ) relative to U , see Figure 1(g). There is a natural surjection from D ( M U ( R f )) to the nerve of the connected cov er pull-back of U , N f ∗ ( U ) , i.e., from the enhanced mapp er graph to the mapp er graph, when the cov er U con tains op en sets with empt y triple in tersections. Using the Reeb interlea ving distance and the enhanced mapp er graph, we obtain and reinterpret the main result of [37] in the following corollary . Corollary 2.29 (cf. [37, Corollary 6]) . L et U b e a nic e c over of R , and ( X , f ) ∈ R - space c . Then d R ( R ( X , f ) , D ( M U ( R f ))) ≤ res f U . Throughout this section we introduce several categories and functors which we will now summarize. Let R - graph b e the category of R -graphs (i.e., Reeb graphs), R - space c the category of constructible R -spaces, Csh c b e the category of constructible coshea ves on R , S ε and T ε the smo othing and thick ening functors, D the display lo cale functor, and M U the mapp er functor. Altogether, we hav e the following diagram of functors and categories, R - graph Csh c R - space c T ε S ε C R M U . D Enhanced mapp er graph algorithm. Finally , w e brieﬂy describ e an algorithm for constructing the enhanced mapp er graph, following the example in Figure 1. Let ( X , f ) b e a constructible R -space (see Section 2.1). F or simplicity , supp ose that the co ver U consists of op en interv als, and con tains no nonempty triple 15 in tersections ( U ∩ V ∩ W = ∅ for all U, V , W ∈ U ). Let R 0 b e the union of b oundary p oin ts of cov er elements in the op en cov er U . Let R 1 b e the complement of R 0 in R . The set R 0 is illustrated with gray dots in Figure 1(e). W e b egin b y forming the disjoint union of closed interv als, a I I × π 0 ( f − 1 ( U I )) , where the disjoint union is taken o ver all connected comp onen ts I of R 1 , I denotes the closure of the op en in terv al I , and U I denotes the smallest op en set in U ∪ { U ∩ V | U, V ∈ U } whic h contains I . In other w ords, U I is either the intersection of tw o co ver elemen ts in U or U I is equal to a cov er elemen t in U . The sets π 0 ( f − 1 ( U I )) are illustrated in Figure 1(d). Notice that there is a natural pro jection map from the disjoint union to R , giv en by pro jecting each p oint ( y , a ) in the disjoint union onto the ﬁrst factor, y ∈ R . The enhanced mapp er graph is a quotient of the ab ov e disjoin t union b y an equiv alence relation on endp oin ts of interv als. This equiv alence relation is deﬁned as follows. Let ( y , a ) ∈ I × π 0 ( f − 1 ( U I )) and ( z , b ) ∈ J × π 0 ( f − 1 ( U J )) b e tw o elements of the ab o ve disjoint union. If y ∈ R 0 , then y is contained in exactly one cov er element in U , denoted by U y . Moreov er,if y ∈ R 0 , then there is a map π 0 ( f − 1 ( U I )) → π 0 ( f − 1 ( U y )) induced b y the inclusion U I ⊆ U y . Denote this map by ψ ( y ,I ) . An analogous map can be constructed for ( z , b ), if z ∈ R 0 . W e sa y that ( y , a ) ∼ ( z , b ) if t wo conditions hold: y = z is contained in R 0 , and ψ ( y ,I ) ( a ) = ψ ( z ,J ) ( b ). The enhanc e d mapp er gr aph is the quotient of the disjoint union by the equiv alence relation describ ed ab ov e. F or example, as illustrated in Figure 1, seven cov er elements of U in (c) give rise to a stratiﬁcation of R in to a set of p oin ts R 0 and a set of interv als R 1 in (e). F or eac h in terv al I in R 1 , w e lo ok at the set of connected comp onents in f − 1 ( U I ). W e then construct disjoint unions of closed in terv als based on the cardinalit y of π 0 ( f − 1 ( U I )) for eac h I ∈ R 1 . F or adjacen t interv als I 1 and I 2 in R 1 , suppose that I 1 is con tained in the cov er element V and I 2 is equal to the intersection of co ver elements V and W in U . W e consider the mapping from π 0 ( f − 1 ( U I 2 )) to π 0 ( f − 1 ( U I 1 )) (d). Here, we hav e that U I 2 = V ∩ W and U I 1 = V . W e then glue these closed interv als follo wing the ab o v e mapping, which giv es rise to the enhanced mapp er graph (g). App endix A outlines these algorithmic details in the form of pseudo co de. 3 Mo del Let X b e a compact lo cally path connected subset of R d . As stated in the introduction, study related to top ological inference usually splits b etw een noiseless and noisy settings. In the former, we assume that a given sample is drawn from X directly , while in the latter we allo w random p erturbations that pro duce samples in R d that need not b e on X , but rather in its vicinity . In this pap er, we address the noisy setting directly , using the mac hinery for sup er-lev el sets estimation developed in [ 9 ]. The basic inputs are a con tinuous function f : R d → R , and a probability density function p : R d → R . Our R -space of interest will b e ( X , f | X ), and we will assume w e are provided samples of X via p . Then, given a nice cov er U , we can compare the Reeb graph of ( X , f | X ) to the mapp er graph computed from the samples. 3.1 Setup In this section, we giv e some basic assumptions on f , p , U , and their interactions. W e start with some notation for the v arious sets of interest. Let X δ = { y ∈ R d : inf x ∈ X d ( x, y ) ≤ δ } b e the δ -thic kening of X , and let D L = p − 1 ([ L, + ∞ )) b e a sup er-lev el set of p . Giv en an op en set V ⊂ R , deﬁne X V := X ∩ f − 1 ( V ). Let X V δ := X δ ∩ f − 1 ( V ) b e the elements of X δ whic h map to V , and D V L := D L ∩ f − 1 ( V ). See 3 for an example of this notation. It is imp ortan t to note that X V δ is not necessarily equal to the δ -thick ening of X V . With this notation, w e will assume that p is ε -concen trated on X as deﬁned next with an example giv en in 4. Deﬁnition 3.1. A pr ob ability density function p is ε -concen trated on X if ther e exists op en intervals I 1 , I 2 , and a r e al numb er δ > 0 such that X ⊂ D L 1 ⊂ X δ ⊂ D L 2 ⊂ X ε , for any L 1 ∈ I 1 and L 2 ∈ I 2 . 16 Figure 3: This ﬁgure illustrates the inv erse images X V (in purple) and X V δ (union of tan and purple) for an ann ulus with height function and op en interv al V . Notice that in this example the δ -thic kening of X V w ould include X V δ as a prop er subset, hence ( X V ) δ 6 = X V δ . Deﬁnition 3.2. A pr ob ability density function p is concentrated on X if p is ε -c onc entr ate d on X for al l ε > 0 . W e now turn our attention to U , a nice cov er of R . Deﬁnition 3.3. The lo cal H 0 -critical v alue ov er V is deﬁne d as δ V = sup { δ | H 0 ( X V ) ∼ = − → H 0 ( X V δ ) } . L et U 0 := { V ⊂ R : V = ∩ α ∈ A U α for some { U α } α ∈ A ⊂ U } . The global H 0 -critical v alue ov er U is deﬁne d as δ U := min V ∈U 0 δ V . Throughout the pap er, we assume that the global H 0 -critical v alue ov er U is p ositiv e, i.e. δ U = min V ∈U 0 δ V > 0 . The p ositivit y of lo cal H 0 -critical v alues is nontrivial and often fails for constructible R -spaces which hav e singularities which lie ov er the b oundary of one of the op en sets in the op en cov er U . In future work, it would b e interesting to relax this assumption, and study conv ergence when the diameter of the union of op en sets V for which δ V = 0, is small. 3.2 Appro ximation by sup er-lev el sets In this section, we study how sup er-lev el sets of probability density functions (PDFs) can mo del the top ology of constructible R -spaces. W e need some further control ov er the relationship b et ween the PDF p and the cov er elements via the follo wing deﬁnition. Deﬁnition 3.4. Given an op en set V , we say that L is H 0 -regular ov er V if ther e exists ν > 0 such that for al l ε 1 < ε 2 ∈ ( L − ν, L + ν ) , the inclusion D ε 2 ⊂ D ε 1 induc es an isomorphism H 0 ( D V ε 2 ) ∼ = − → H 0 ( D V ε 1 ) . Throughout the pap er w e will assume that the PDF p is tame , in the sense that the set of p oints which are H 0 -regular ov er V is dense in R , for any given op en set V . Assume the global H 0 -critical v alue δ U is p ositiv e, and p is δ 2 -concen trated on X for some δ 2 suc h that 0 < δ 2 < δ U . By deﬁnition, there exist L 1 , L 2 and δ 1 suc h that 17 Figure 4: An example of the concen trated deﬁnition. The left side of the ﬁgure illustrates a probabilit y densit y function (PDF) p whic h is ε -concen trated on an annulus X . The center image illustrates the thick ened space X δ , b ounded by the red curves, and the sup er-lev el set D L 1 . The righ t side of the ﬁgure illustrates the thick ened space X ε , b ounded by the blue curv es, and the sup er-level set D L 2 . T ogether, we see that X ⊂ D L 1 ⊂ X δ ⊂ D L 2 ⊂ X ε . 1. X ⊂ D L 1 ⊂ X δ 1 ⊂ D L 2 ⊂ X δ 2 2. 0 < δ 1 < δ 2 < δ U 3. L 1 and L 2 are H 0 -regular ov er V for each V ∈ U 0 . The set of p oin ts which are H 0 -regular ov er V for each V ∈ U 0 is dense in R . If L 1 is not H 0 -regular o ver V for some V ∈ U 0 , then L 1 can b e turned into a regular v alue with an arbitrarily small p erturbation. Moreo ver, b y the Deﬁnition 3.1, this perturbation can b e done without breaking the c hain of inclusions X ⊂ D L 1 ⊂ X δ 1 ⊂ D L 2 ⊂ X δ 2 . W e therefore contin ue under the assumption that L 1 is H 0 -regular ov er V for eac h V ∈ U 0 . Prop osition 3.5. Assume that p is ε -c onc entr ate d on X for some ε < δ U . L et D ( V ) := Im  H 0 ( D V L 1 ) → H 0 ( D V L 2 )  . Then for e ach V ∈ U 0 , we have H 0 ( X V ) ∼ = D ( V ) and further for e ach V ⊂ W ∈ U 0 the fol lowing diagr am c ommutes, H 0 ( X V ) D ( V ) H 0 ( X U ) D ( U ) . ∼ = ∼ = The pro of will require the following tec hnical lemma. Lemma 3.6. Supp ose we have the fol lowing c ommutative diagr am of ve ctor sp ac es A B D C E ∼ = ∼ = with C ∼ = D ∼ = E . Then Im( D → B ) = Im( A → B ) and the map D ∼ = − → Im( A → B ) is an isomorphism of ve ctor sp ac es. 18 Pr o of. The map D → B is injective since D → E is an isomorphism and the diagram commutes. Therefore, Im ( D → B ) ∼ = D . Moreov er, since the diagram commutes, Im ( A → B ) ⊂ Im ( D → B ). Supp ose b ∈ Im ( D → B ), i.e., there exists d ∈ D whic h maps to b . Since C → D is an isomorphism, there exists c ∈ C whic h maps to d ∈ D . Let a ∈ A b e the image of c ∈ C under the map C → A . Since the diagram commutes, we ha ve that a ∈ A maps to b ∈ B under the map A → B . Therefore, b ∈ Im ( A → B ). W e hav e therefore shown that Im( A → B ) = Im( D → B ) ∼ = D . Pr o of of 3.5. Cho ose δ 2 > 0 such that δ 2 < δ U and p is δ 2 -concen trated on X . Applying the deﬁnition of δ 2 -concen trated, w e ha ve X ⊂ D L 1 ⊂ X δ 1 ⊂ D L 2 ⊂ X δ 2 . F or V ⊂ W w e ha ve the follo wing commutativ e diagram of vector spaces H 0 ( D V L 1 ) H 0 ( D V L 2 ) H 0 ( X V δ 1 ) H 0 ( D W L 1 ) H 0 ( D W L 2 ) H 0 ( X W δ 1 ) H 0 ( X W ) H 0 ( X V ) H 0 ( X W δ 2 ) H 0 ( X V δ 2 ) Since δ 1 < δ 2 < δ U , by deﬁnition of global H 0 -critical v alue ov er U , all four horizontal maps H 0 ( X V ) − → H 0 ( X V δ 1 ) − → H 0 ( X V δ 2 ) and H 0 ( X W ) − → H 0 ( X W δ 1 ) − → H 0 ( X W δ 2 ) are isomorphisms. Applying 3.6, we can conclude that H 0 ( X V ) − → D ( V ) and H 0 ( X W ) − → D ( W ) are isomorphisms of vector spaces. Since the diagram commutes, the image of D ( V ) under the map H 0 ( D V L 2 ) → H 0 ( D W L 2 ) is contained in D ( W ). Therefore, H 0 ( D V L 2 ) → H 0 ( D W L 2 ) induces a map D ( V ) → D ( W ), whic h completes the commutativ e diagram of the theorem. 3.3 P oin t-cloud mapper algorithm Giv en data { X 1 , · · · , X n } i.i.d ∼ p , where p is a PDF, we can estimate p using a kernel density estimator (KDE) of the form, ˆ p ( x ) := 1 C K nr d n X i =1 K r ( x − X i ) , where K ( x ) is a given k ernel function, K r := K ( x/r ), and C K is a constant deﬁned below. The kernel function should satisfy the following: 1. supp( K ) ⊂ B 1 (0), and K ( x ) is smo oth in B 1 (0). 2. K ( x ) ∈ [0 , 1], and max x K ( x ) = K (0) = 1, 3. R R d K ( x ) dx = C K with C K ∈ (0 , 1). Using ˆ p we can estimate the sup er-lev el sets D L using ˆ D L ( n, r ) := [ i : ˆ p ( X i ) ≥ L B r ( X i ) , (1) and the sets D V L using ˆ D V L ( n, r ) := ˆ D L ( n, r ) ∩ f − 1 ( V ) . (2) Cho ose ε i suc h that L i + 2 ε i , L i − 2 ε i are within the H 0 -regularit y range of L i o ver V for eac h V ∈ U and L 1 − 2 ε 1 > L 2 + 2 ε 2 . In the following, we will use the term “with high probabilit y” (w.h.p.) to mean that the probability of an even t to o ccur con verges to 1 as n → ∞ . 19 Prop osition 3.7. Fix L and V , and set ˆ D V L := ˆ D V L ( n, r ) . Fixing ε > 0 , ther e exists a c onstant C ε > 0 (indep endent of L and V ) such that if nr d ≥ C ε log n , then the fol lowing diagr am of inclusion r elations holds w.h.p., ˆ D V L + ε ˆ D V L − ε D V L D V L +2 ε D V L − 2 ε Pr o of. The pro of app ears as part of the pro of of Theorem 3.3 in [9]. Next, deﬁne the random vector space ˆ D i ( V ) := Im  H 0 ( ˆ D V L i + ε i ) → H 0 ( ˆ D V L i − ε i )  . Corollary 3.8. If nr d ≥ C ε i log n , then w.h.p. the r andom map H 0 ( D V L i ) → ˆ D i ( V ) is an isomorphism. Pr o of. The corollary follows from applying 3.6 to 3.7. F rom here on, unless otherwise stated, we will assume that r is chosen so that nr d ≥ max ( C ε 1 , C ε 2 ) log n , so that 3.8 holds for b oth ε 1 , ε 2 . Prop osition 3.9. F or every V ⊂ W ∈ U 0 , we have the fol lowing c ommutative diagr am w.h.p.,  H 0 ( D V L i ) ˆ D i ( V ) ∼ = H 0 ( D W L i ) ˆ D i ( W ) . ∼ = Pr o of. The pro of follows the same arguments as the pro of of Prop osition 3.5, and using Corollary 3.8. Finally , we deﬁne the following random vector space, ˆ D ( V ) := Im  H 0 ( ˆ D V L 1 + ε 1 ) → H 0 ( ˆ D V L 2 − ε 2 )  . Prop osition 3.10. Assume that p is ε -c onc entr ate d on X for some ε < δ U . F or every V ⊂ W ∈ U 0 , we have the fol lowing c ommutative diagr am w.h.p.,  H 0 ( X V ) ˆ D ( V ) ∼ = H 0 ( X W ) ˆ D ( W ) , ∼ = wher e the c onstants L 1 and L 2 (deﬁning ˆ D ) ar e given by the deﬁnition of ε -c onc entr ate d, and the c onstants ε 1 and ε 2 ar e given by the H 0 -r e gularity of L 1 and L 2 , r esp e ctively. 20 Pr o of. W e will use the assumption that L 1 − 2 ε 1 > L 2 + 2 ε 2 rep eatedly for eac h of the sup er-lev el set inclusions in the pro of. The inclusion of spaces ˆ D V L 1 − ε 1 ⊂ ˆ D V L 2 − ε 2 induces a homomorphism H 0 ( ˆ D V L 1 − ε 1 ) → H 0 ( ˆ D V L 2 − ε 2 ). Restricting the domain of this map, we get a homomorphism ˆ D 1 ( V ) → H 0 ( ˆ D V L 2 − ε 2 ). Since L 1 − ε 1 > L 2 + ε 2 > L 2 − ε 2 , the map ˆ D 1 ( V ) → H 0 ( ˆ D V L 2 − ε 2 ) factors through H 0 ( ˆ D V L 2 + ε 2 ) → H 0 ( ˆ D V L 2 − ε 2 ), forming the commutativ e diagram  ˆ D 1 ( V ) H 0 ( ˆ D V L 2 − ε 2 ) H 0 ( ˆ D V L 2 + ε 2 ) This implies that Im ( ˆ D 1 ( V ) → H 0 ( ˆ D V L 2 − ε 2 )) ⊂ ˆ D 2 ( V ), and gives a map ˆ D 1 ( V ) → ˆ D 2 ( V ), whic h w.h.p. completes the following commutativ e diagram,  H 0 ( D V L 1 ) ˆ D 1 ( V ) ∼ = H 0 ( D V L 2 ) ˆ D 2 ( V ) ∼ = where the horizontal maps are given by Corollary 3.8. Therefore, applying Prop osition 3.5, we hav e H 0 ( X V ) ∼ = − → D ( V ) w.h.p. ∼ = Im  ˆ D 1 ( V ) → ˆ D 2 ( V )  . Since ˆ D V L 1 + ε 1 ⊂ ˆ D V L 1 − ε 1 ⊂ ˆ D V L 2 + ε 2 ⊂ ˆ D V L 2 − ε 2 , w e hav e that Im  ˆ D 1 ( V ) → ˆ D 2 ( V )  = ˆ D ( V ). The map ˆ D ( V ) → ˆ D ( W ) in the statemen t of the proposition (and the commutativit y of the resulting diagram) is induced by the inclusion ˆ D V L 2 − ε 2  → ˆ D W L 2 − ε 2 in the following commutativ e diagram. 21 ˆ D V L 1 + ε 1 ˆ D V L 1 − ε 1 D V L 1 ˆ D W L 1 + ε 1 ˆ D W L 1 − ε 1 D W L 1 D W L 1 +2 ε 1 D V L 1 +2 ε 1 D W L 1 − 2 ε 1 D V L 1 − 2 ε 1 ˆ D V L 2 + ε 2 ˆ D V L 2 − ε 2 D V L 2 ˆ D W L 2 + ε 2 ˆ D W L 2 − ε 2 D W L 2 D W L 2 +2 ε 2 D V L 2 +2 ε 2 D W L 2 − 2 ε 2 D V L 2 − 2 ε 2 X V X W X V δ 1 X W δ 1 X V δ 2 X W δ 2 4 Main results In this section, we prov e conv ergence of the random enhanced mapp er graph to the Reeb graph, as well as stabilit y of the enhanced mapp er graph under certain p erturbations of the corresp onding real v alued function. Using the mo del describ ed in Section 3, we generate random data, which is used to deﬁne a cosheaf which estimates the connected comp onen ts of ﬁb ers of the real v alued function asso ciated to a given constructible R -space. In the pro of of Theorem 4.2, we sho w that, with high probability , the cosheaf constructed using random data is isomorphic to the mapp er functor applied to the Reeb cosheaf deﬁned in Section 2. W e then use the results established in Section 2 to translate the cosheaf theoretic statement into a geometric statement (Corollary 4.3) for the corresp onding R -graphs. T o b egin, we identify a suﬃcient condition for determining when a morphism of constructible cosheav es is an isomorphism. A morphism F → G of cosheav es is a family of maps F ( V ) → G ( V ), for each op en set V , whic h form a commutativ e diagram 22  F ( V ) G ( V ) F ( W ) G ( W ) for eac h pair of op en sets V ⊂ W . The morphism F → G is an isomorphism if each of the maps F ( V ) → G ( V ) is an isomorphism. Our ﬁrst result shows that for cosheav es of the form M U ( F ), it is suﬃcien t to consider only the maps M U ( F )( V ) → M U ( G )( V ) for op en sets V ∈ U 0 . Prop osition 4.1. L et C and D b e c oshe aves on R . An isomorphism M U ( C ) → M U ( D ) of c oshe aves is uniquely determine d by a family of isomorphisms M U ( C )( V ) → M U ( D )( V ) for e ach V ∈ U 0 , which form a c ommutative diagr am  M U ( C )( V ) M U ( D )( V ) ∼ = M U ( C )( W ) M U ( D )( W ) ∼ = for e ach p air V ⊂ W ∈ U 0 . Pr o of. Prop osition 2.23 shows that M U ( C ) and M U ( D ) are constructible cosheav es ov er R . The pro of then follo ws from [22, Prop osition 3.10]. Recalling the notation of Section 2 and Section 3, for a sup er-lev el set D L of p , let R D L b e the Reeb cosheaf of ( D L , f ) on R , deﬁned by R D L ( U ) = π 0 ( D U L ) for each op en set U ⊂ R . Let R ˆ D L b e the Reeb cosheaf of ( ˆ D L , f ) on R , deﬁned by R ˆ D L ( U ) = π 0 ( ˆ D U L ) where ˆ D L , ˆ D U L are deﬁned in (1) , (2) , resp ectiv ely , and U ⊂ R is an open set. W e should note that ( D L , f ) and ( ˆ D L , f ) are not apriori constructible spaces, so the cosheav es R D L and R ˆ D L are not necessarily constructible. Ho wev er, in what follo ws we will work exclusively with M U ( R D L ) and M U ( R ˆ D L ), which are constructible coshea ves by Prop osition 2.23. Let ˆ D π n b e the cosheaf deﬁned by ˆ D π n := M U  Im  R ˆ D L 1 + ε 1 → R ˆ D L 2 − ε 2  , with constants n , L 1 , L 2 , ε 1 , and ε 2 c hosen in Section 3. More explicitly , ˆ D π n maps an op en interv al U to elemen ts of the set R ˆ D L 2 − ε 2 ( I U ( U )) which lie in the image of the set R ˆ D L 1 + ε 1 ( I U ( U )) under the map induced b y the inclusion ˆ D L 1 + ε 1 ⊆ ˆ D L 2 − ε 2 . By Prop osition 2.23, ˆ D π n is a constructible cosheaf. Theorem 4.2. Assume ther e exists ε < δ U such that p is ε -c onc entr ate d on X , then lim n →∞ P  d I ( ˆ D π n , R X ) ≤ r es f U  = 1 . Pr o of. An inclusion of op en sets Y ⊂ Z induces a map π 0 ( Y ) → π 0 ( Z ) , of the corresp onding sets of path-connected comp onen ts of Y and Z resp ectiv ely . Eac h set of path-connected comp onen ts forms a basis for the homology group in degree 0. Therefore, the map from π 0 ( Y ) to π 0 ( Z ) extends to a map b et w een homology groups, resulting in the following commutativ e diagram 23  π 0 ( Y ) H 0 ( Y ) π 0 ( Z ) H 0 ( Z ) . By combining the preceding commutativ e diagram with Prop osition 3.10, we see that for every V ⊂ W ∈ U 0 , the following diagram commutes w.h.p.  π 0 ( X V ) ˆ D π n ( V ) ∼ = π 0 ( X W ) ˆ D π n ( W ) . ∼ = Notice that if V ∈ U 0 , then I U ( V ) = V . By Prop osition 4.1 we hav e that ˆ D π n w.h.p. ∼ = M U ( R X ) . Therefore, w.h.p. d I ( ˆ D π n , M U ( R X )) = 0 . Theorem 2.27, combined with the triangle inequality , implies the theorem. Corollary 4.3. L et R ( X , f ) b e the R e eb gr aph of a c onstructible R -sp ac e ( X , f ) , and D ( ˆ D π n ) b e the display lo c ale of the mapp er c oshe af deﬁne d ab ove. If ther e exists ε < δ U such that p is ε -c onc entr ate d on X , then lim n →∞ P  d R  D ( ˆ D π n ) , R ( X , f )  ≤ r es f U  = 1 . If p is concentrated on X , then the ab o ve corollary will hold for nice op en cov ers with arbitrarily small resolution, as long as δ U remains p ositiv e. Therefore, Corollary 4.3 implies that we can use random p oin t samples from p to construct mapp er graphs that are (w.h.p.) arbitrarily close (in the Reeb distance) to the Reeb graph of X . T o conclude, w e will turn our atten tion to the stabilit y of mapper coshea ves corresp onding to a constructible space ( X , f ) under p erturbations of the function f . The following theorem uses the machinery of cosheaf theory to prov e that the mapp er cosheaf is stable as long as the singular p oin ts of the constructible R -space X are suﬃciently “far aw ay” from the set of b oundary p oin ts of our op en cov er U . Theorem 4.4. Supp ose F and G ar e c onstructible c oshe aves over R , with a c ommon set of critic al values S . L et U b e a nic e op en c over of R , with set of b oundary p oints B . Assume that d I ( F , G ) < min {| s − b | : s ∈ S, b ∈ B } . Then d I ( M U ( F ) , M U ( G )) < d I ( F , G ) . Mor e over, if F is the R e eb c oshe af of ( X , f ) and G is the R e eb c oshe af of ( X , g ) , then d I ( M U ( F ) , M U ( G )) < || f − g || ∞ . Pr o of. Supp ose ϕ U : F ( U ) → G ( U ε ) and ψ U : G ( U ) → F ( U ε ) give an ε -in terleaving of F and G . Recall that M U ( F )( U ) = F ( I U ( U )) . Then ϕ I U ( U ) : M U ( F )( U ) → G ( I U ( U ) ε ) . 24 In general, this do es not giv e us an ε -in terleaving of M U ( F ) and M U ( G ), because I U ( U ) ε 6 = I U ( U ε ). Ho wev er, we will pro ceed by sho wing that each of these sets contain the same set of critical v alues. F ollowing the deﬁnition of I U , we see that for eac h U ∈ Int , I U ( U ) is an op en in terv al in R , with b oundary p oin ts contained in B . Therefore I U ( U ) ∩ S ⊂ I U ( U ) ε ∩ S . If the inclusion is not an equality , then there m ust exist s ∈ S suc h that s ∈ I U ( U ) ε and s / ∈ I U ( U ). In other w ords, if I U ( U ) ∩ S ( I U ( U ) ε ∩ S , then there exists s ∈ S and b ∈ B such that | s − b | < ε . Deﬁne N U ,ε ( U ) := I U ( U ε ) ∩ I U ( U ) ε . By the arguments ab o v e, if ε < min {| s − b | : s ∈ S, b ∈ B } , then I U ( U ) ∩ S = I U ( U ) ε ∩ S . It follo ws that N U ,ε ( U ) ∩ S = I U ( U ) ∩ S = I U ( U ) ε ∩ S , b ecause I U ( U ) ⊂ I U ( U ε ). By the deﬁnition of constructibility , this implies that the natural extension map G [ N U ,ε ( U ) ⊂ I U ( U ) ε ] (denoted by e for notational brevity) G ( N U ,ε ( U )) e − − − − → G ( I U ( U ) ε ) is an isomorphism, and therefore is inv ertible. The comp osition M U ( F )( U ) ϕ − → G ( I U ( U ) ε ) e − 1 − − → G ( N U ,ε ( U )) → G ( I U ( U ε )) = M U ( G )( U ε ) giv es an ε -in terleaving of M U ( F ) and M U ( G ), b ecause each map in the comp osition is natural with resp ect to inclusions. Therefore d I ( M U ( F ) , M U ( G )) < d I ( F , G ) . When F is the Reeb cosheaf of ( X , f ) and G is the Reeb cosheaf of ( X , g ), the second statemen t of the theorem is a direct consequence of the ab o v e inequality and [22, Theorem 4.4]. 5 Discussion In this pap er, w e work with a categoriﬁcation of the Reeb graph [ 22 ] and introduce a categoriﬁed version of the mapp er construction. This categoriﬁcation provides the framework for using cosheaf theory and interlea ving distances to study conv ergence and stability for mapp er constructions applied to p oin t cloud data. In this setting, the Reeb graph of a constructible R -space is realized as the displa y lo cale of a constructible cosheaf (whic h we refer to as the Reeb cosheaf, following [ 22 ]). In Section 2.5, w e deﬁne a mapp er functor from the category of coshea ves to the category of constructible cosheav es, giving a category theoretic interpretation of the mapp er construction. W e then deﬁne the enhanc e d mapp er gr aph to b e the display lo cale of the mapp er functor applied to the Reeb cosheaf. W e give an explicit geometric realization of the display lo cale as the quotien t of a disjoint union of closed in terv als, as illustrated in Figure 5. In Section 3, we giv e a mo del for randomly sampling p oin ts from a probability density function concentrated on a constructible R -space. After applying k ernel density estimates, we consider an enhanced mapp er graph generated b y the random data. The main result of the pap er, Theorem 4.2, then gives (with high probability) a b ound on the Reeb distance b et w een the Reeb graph and the enhanced mapp er graph generated by a random sample of p oin ts. Reﬁnemen t to classic mapp er graph. The enhanced mapp er graph suggests a few reﬁnements to the classic mapp er construction. Firstly , rather than an op en cov er U of f ( X ) (the image of the constructible R -space X in R ), it is more natural from the enhanced mapp er p ersp ectiv e to start with a ﬁnite subset R 0 of R . F rom this ﬁnite subset, the enhanced mapp er graph can b e computed by ﬁrst pro ducing a ﬁnite disjoint union of closed interv als, with eac h interv al asso ciated to a connected comp onen t of the complement of R 0 . Then, b y prescribing attaching maps on b oundary p oin ts of the disjoint union of closed interv als, one can obtain a combinatorial description of the enhanced mapp er graph as a graph with vertices lab eled with real n umbers. The e nhanced mapp er graph then has the structure of a stratiﬁed cov er of f ( X ), the image of the constructible R -space X in R . As suc h, the enhanced mapp er graph contains more information than the classic mapp er graph. Sp eciﬁcally , edges of the enhanced mapp er graph hav e a naturally deﬁned length which captures geometric information ab out the underlying constructible R -space. Therefore, the enhanced mapp er graph is naturally geometric, meaning that it comes equipp ed with a map to R . V ariations of mapper graphs. W e return to an in-depth discussion among v ariations of classic mapp er graphs. 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AAAB8nicbVBNT8JAEJ3iF+IX6tHLRjDxRFouciTx4hETAZPSkO2yhQ3bbbM7NSENP8OLB43x6q/x5r9xgR4UfMkkL+/NZGZemEph0HW/ndLW9s7uXnm/cnB4dHxSPT3rmSTTjHdZIhP9GFLDpVC8iwIlf0w1p3EoeT+c3i78/hPXRiTqAWcpD2I6ViISjKKV/PogpDqP5sO0PqzW3Ia7BNkkXkFqUKAzrH4NRgnLYq6QSWqM77kpBjnVKJjk88ogMzylbErH3LdU0ZibIF+ePCdXVhmRKNG2FJKl+nsip7Exszi0nTHFiVn3FuJ/np9h1ApyodIMuWKrRVEmCSZk8T8ZCc0ZypkllGlhbyVsQjVlaFOq2BC89Zc3Sa/Z8NyGd9+stVtFHGW4gEu4Bg9uoA130IEuMEjgGV7hzUHnxXl3PlatJaeYOYc/cD5/AMslkOI= Figure 5: An example of the enhanced mapp er graph D ( M U ( R f )) and the Reeb graph R ( T , f ) of the height function f on the torus T , with an op en cov er U of f ( T ) consisting of t wo op en in terv als. The maps q and f q are the natural quotient factorization of f obtained from the deﬁnition of the Reeb graph. Similarly , p and f p are the quotient map and factorization of f obtained from the deﬁnition of the enhanced mapp er graph. the enhanced mapp er graphs (g), geometric mapp er graphs (i) studied by [ 37 ], and multinerv e mapp er graphs (j), ha ve all b een shown to be interlea ved with Reeb graphs (b) [ 37 , 14 ]. T o further illustrate the subtle diﬀerences among the enhanced, geometric, mutinerv e and classic mapp er graphs, w e give additional examples in Figure 7 and Figure 8. In certain scenarios, some of these constructions app ear to b e identical or very similar to eac h other. W e would like to understand the information conten t asso ciated with the ab o v e v ariants of mapp er graphs, all of which are used as approximations of the Reeb graph of a constructible R -space. As illustrated in Figure 6, given an enhanced mapp er graph (g) and an op en cov er (c), one can recov er the the m ultinerve mapp er graph (j), the geometric mapp er graph (i), and the c lassic mapp er graph (k). In future w ork, it would b e interesting to quan tify precisely the reconstruction ordering of these v ariants with and without any knowledge of the op en co ver. In order to study conv ergence and stability of eac h v ariation of the mapp er graph, it is necessary to assign function v alues to vertices of the graph. F or the classic mapp er graph or multinerv e mapp er graph, each v ertex can b e assigned, for instance, the v alue of the midp oin t of a corresp onding interv al in R . How ev er, the displa y lo cale of a cosheaf o ver R admits a natural pro jection on to the real line, making a c hoice of function v alues unnecessary for the enhanced mapp er graph. F or this reason, we view the enhanced mapp er graph as a natural v ariation of the mapp er graph, well-suited for studying stabilit y and conv ergence, with a natural in terpretation in terms of cosheaf theory . Multidimensional setting and parameter tuning. It is natural to extend the enhanced mapp er graph (and more generally the categoriﬁcation of mapper graphs) to multidimensional Reeb spaces and m ulti- parameter mapp er through studying constructible coshea ves and stratiﬁed cov ers of R N , for N > 1. W e w ould also lik e to study the b eha vior of the parameter δ U for v arious constructible spaces and op en cov ers. In general, this parameter can v anish for “bad” choices of op en co ver U . It would b e worth while to extend the results of this pap er to obtain b ounds on the interlea ving distance when δ U v anishes. In conclusion, we hop e for the results of this pap er to promote the utility of combining metho ds from statistics and sheaf theory for the purp ose of analyzing algorithms in computational top ology . Ac knowledgemen ts. AB was supp orte d in p art by the Eur op e an Union ’s Horizon 2020 r ese ar ch and innovation pr o gr amme under the Marie Sklo dowska-Curie Gr ant A gr e ement No. 754411 and NSF IIS-1513616. OB was supp orte d in p art by the Isr ael Scienc e F oundation, Gr ant 1965/19. BW was supp orte d in p art by NSF IIS-1513616 and DBI-1661375. EM was supp orte d in p art by NSF CMMI-1800466, DMS-1800446, and CCF-1907591. We would like to thank the Institute for Mathematics and its Applic ations for hosting a workshop title d Bridging Statistics and Sheav es in May 2018, wher e this work was c onc eive d. 26 (a) (b) (c) (d) (e) (f) (g) (h) (j) (k) (i) Enhanced Geo- metric Multinerve Classic Figure 6: V ariations of mapp er graphs for the height function on a torus. (a) T orus with a height function. (b) Reeb graph. (c) Nice cov er. (d) Visualization of the mapp er cosheaf. (e) Stratiﬁcation of R . (f ) Disjoint union of closed interv als, e D ( M U ( R f )), with quotient isomorphic to the enhanced mapp er graph. (g) Enhanced mapp er graph, D ( M U ( R f )). (h) Disjoin t union of closed in terv als used to construct geometric mapp er graph [37]. (i) Geometric mapp er graph. (j) Multinerve mapp er graph. (k) Classic mapp er graph. Multinerve & classic (a) (c) (b) (d) (e) (f) (g) (h) (i) (j) Enhanced Geometric Figure 7: A return to the example illustrated in Figure 1. V ariations of mapp er graphs of a height function on a topological space. (a) A top ological space with a height function. (b) Reeb graph. (c) Nice co ver. (d) Visualization of the mapp er cosheaf. (e) Stratiﬁcation of R . (f ) Disjoin t union of closed interv als with quotien t isomorphic to the enhanced mapp er graph. (g) Enhanced mapp er graph. (h) Disjoint union of closed in terv als used to construct geometric mapp er graph [ 37 ]. (i) Geometric mapp er graph. (j) Multinerve and classic mapp er graph. Conﬂict of interest The authors declare that they hav e no conﬂict of interest. A Pseudo co de for the Enhanced Mapp er Graph Algorithm The following pseudo code (Algorithm 1) outlines an algorithm for computing the enhanced mapp er graph, whic h is stored as a graph G = ( F , E ) with a v ertex set F and an edge set E , together with a real-v alued function f : F → R . 27 (a) (b) (c) (d) (e) (f) (g) Enhanced & geometric (h) Multinerve & classic Figure 8: V ariations of mapp er graphs of a height function on a top ological space consisting of tw o line segmen ts. (a) A top ological space consisting of tw o line segmen ts. (b) Reeb graph. (c) Nice cov er. (d) Visualization of the mapp er cosheaf. (e) Stratiﬁcation of R . (f ) Disjoin t union of closed interv als with quotient isomorphic to the enhanced mapp er graph. (g) Enhanced and geometric mapp er graph. (i) Multinerv e and classic mapp er graph. The algorithm assumes that we are given sets π 0 ( f − 1 ( U )) (denoted by Σ in the pseudo co de) and set maps π 0 ( f − 1 ( U )) → π 0 ( f − 1 ( V )) (denoted by ρ in the pseudo co de) for v arious U ⊂ V ⊂ R . In other words, the algorithm assumes that there is an oracle (referred to as a set or acle ) that takes as input an inv erse mapping of an interv al and returns its corresp onding set of path-connected comp onen ts. It also assumes that there is a set-map or acle that keeps tracks of set maps b et w een a pair of path-connected comp onents (each comp onen t is denoted by s in the pseudo co de). In Section 3, we give a statistical approach for computing suc h sets and set maps through kernel densit y estimates. In Algorithm 1, let U = { U i } i ∈ A denote a ﬁnite set of pairwise intersecting op en interv als. F or simplicity , supp ose the index set A ⊂ Z con tains consecutiv e integers. That is, for each interv al U i := ( u − i , u + i ) (for some i ∈ A ), w e hav e u − i < u + i − 1 < u − i +1 < u + i < u + i +1 (assuming i − 1 , i + 1 ∈ A ). F or each interv al U i , Σ i := π 0 ( f − 1 ( U i )) denotes the set of path-connected comp onen ts. F or each path-connected comp onen t s ∈ Σ i , the pairs ( s, +) ∈ Σ i × { + , −} and ( s, − ) ∈ Σ i × { + , −} represen t the tw o vertices associated to the edge in the enhanced mapp er graph whic h corresp onds to s . Similarly , for each path-connected comp onen t t ∈ Σ ( i,i +1) , the pairs ( ρ − i ( t ) , +) and ( ρ + i ( t ) , − ) represent the tw o vertices asso ciated to the edge in the enhanced mapp er graph which corresp onds to t . F or clarit y , Figure 9 illustrates notations used in the pseudo code of Algorithm 1. It is based on a zo omed view of Figure 1(c)-(f ). The maps ρ − i and ρ + i deﬁne how the red vertices and blue vertices (as end p oin ts of in terv als) are glued together to form an enhanced mapp er graph. 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Input: • A ﬁnite set of pairwise intersecting op en in terv als: { U i := ( u − i , u + i ) } i ∈ A • F or each interv al, a set returned by a set oracle: – F or each U i , a set Σ i := π 0 ( f − 1 ( U i )) – F or each ( U i , U i +1 ), a set Σ ( i,i +1) := π 0 ( f − 1 ( U i ∩ U i +1 )) • F or each pair of interv als, a set map returned by a set-map oracle: F or each ( U i , U i +1 ), set maps ρ − i : Σ ( i,i +1) → Σ i , ρ + i : Σ ( i,i +1) → Σ i +1 . Output: • A graph G = ( F, E ) with a vertex set F and an edge set E ⊆ F × F • A function f : F → R Initialize F = ∅ and E = ∅ for i ∈ A do Set Σ + i := Σ i × { + } Set Σ − i := Σ i × {−} F ← F t Σ − i t Σ + i for s ∈ Σ i do E ← E t (( s, − ) , ( s, +)) f (( s, +)) := ( u − i +1 if i + 1 ∈ A u + i if i + 1 / ∈ A, f (( s, − )) := ( u + i − 1 if i − 1 ∈ A u − i if i − 1 / ∈ A, end end for ( i, i + 1) ∈ A × A do for t ∈ Σ ( i,i +1) do E ← E t (( ρ − i ( t ) , +) , ( ρ + i ( t ) , − )) end end 30 References [1] M. Alagappan. 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Probabilistic Convergence and Stability of Random Mapper Graphs

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