1 Introduction
The Nerve Theorem [1] is an important link between topological spaces and discrete geometric and topological algorithms. It is at the heart, either implicitly or explicitly, of many foundational algorithms in the rapidly growing field of topological data analysis. Example problems include surface reconstruction [2, 3, 4], function reconstruction [5], homology inference [6, 7, 8] , coordinate-free sensor network coverage [9, 10], shape analysis [11], data modeling [12, 13, 14], and clustering [15, 16].
A cover of a simplicial complex is a collection of subcomplexes such that . ^{1}^{1}1The notation is intended to help the reader remember that , (“double U”) is the union of the ’s. The nerve of is the abstract simplicial complex defined as follows
A cover is good if for every the simplicial complex is empty or contractible. The Nerve Theorem equates the homotopy type and thus homology of the covered simplicial complex to that of the nerve of a good cover . This theorem allows one to construct algorithms that compute topological properties of the nerve of a particular good cover and relate the output back to infer properties of the covered space.
In persistence theory, one often works with filtered topological spaces: , where for all . The Persistent Nerve Lemma of Chazal and Oudot [2] proves that the Nerve Theorem extends in the most natural way to the persistence setting. In particular, it implies that the union filtration and the nerve filtration have the same persistent homology assuming is a good cover of for all .
The requirement of good covers to ensure topological theoretical guarantees has significant algorithmic implications as it significantly reduces the spaces one can work in. Many algorithms in topological data analysis utilize a standard pipeline where one considers a nice sample of some space, builds a simplicial complex from this sample called the Čech complex, which is the nerve of metric balls, and uses the fact that this has the same topology of the metric balls. This means that algorithms depending on the Nerve Theorem can only be applied to spaces that admit covers by convex sets in Euclidean spaces or smooth manifolds of sufficiently large convexity radius [5] can be considered. Even one small hole in an intersection of cover elements can render nerve-based computational algorithms invalid as the theory rests upon some interpretation of this theorem. This also has implications for triangulations of covers of surfaces that have marginal measurement errors, because the errors can cause the cover elements to no longer be convex for example. Nerves are also in coverage testing for homological sensor networks [10, 9], however the idealized model of Euclidean balls as coverage regions differs significantly from the very jagged coverage regions measured in practice, particularly when taking into account the affect of physical obstacles on real-life sensors’ detection ranges.
In this paper we introduce a parametrization of the good cover condition for simplicial cover filtrations called an -good cover, which roughly says that the homology of the cover elements’ intersections only persists for some amount of time. Our main result is as follows. Given a simplicial cover filtration that is an -good cover filtration of the corresponding covered simplicial filtration, there exists a constructive -interleaving between the -dimensional persistence modules of the a finite simplicial filtration and its covers’ nerve filtration, which implies a tight bound on the bottleneck distance between each modules persistence diagram. We assume no structure on the simplicial cover filtration other than that the simplicial complexes are finite. This persistence module interleaving notably results from a “pseudo” interleaving between chain complexes where we only require the maps compositions are chain homotopic to the identity chain maps between scales. Due to the spaces in question being finite simplicial complexes, the chain maps used to construct the homological interleaving are computable.
A corollary of the Generalized Persistent Nerve Theorem is the algebraic Persistent Nerve Lemma for simplicial filtrations and covers. We also reduce the interleaving distance and thus the bottleneck distance bound by a factor of in Section 5 by considering a time-scale shifted nerve filtration assuming one knows an upper bound on the -goodness of the cover.
History of the Problem
In August , Govc and Skraba posted the solution to a very similar problem, among other results in their paper An Approximate Nerve Theorem to the arXiv. They originally assumed that the reduced homology of the -wise intersections of a simplicial cover filtrations’ cover elements are -interleaved with the module, a condition they call an -acyclic cover. Their paper arrived at an identical bound, up to differences in definitions. They restricted their hypotheses to a filtered simplicial complex that is induced by a cover on the complete complex. The cover filtration at each scale was thus defined by the simplicial filtration at that scale’s intersection with the cover. Unfortunately, this assumption is too restrictive to imply a simplicial version of the Persistent Nerve Lemma which was a primary goal of this research for us, as it does not account for covers which have no inherent relation to each other outside of inclusions.
In September , we submitted our results for presentation at the 26th Fall Workshop on Computational Geometry, which notably implied the Persistent Nerve Lemma as a corollary, for our space assumptions, as originally desired. Soon afterwards, Govc and Skraba updated their arXiv submission relaxing their cover filtration assumption, and their paper has recently been accepted to a journal [17].
Though our papers both prove a similar result, the approaches are very different and utilize and develop different tools. Govc and Skraba utilized a construction from homological algebra called a spectral sequence, as well as their novel right and left persistence module interleavings to prove their theorem, which is actually a direct result of the the module interleavings they compute between the pages of the spectral sequences and the persistence modules of the nerve and space filtrations.
In contrast, our proof technique focuses on constructing maps between the chain complexes of the nerve filtration, space filtration, and the barycentric decomposition of the so-called blow-up complex. We go from the homological -goodness condition to corresponding chain maps and chain homotopies between them among the chain complexes of the spaces of interest. This is more inline with the approach traditionally used to proof the Nerve Theorem and the homotopy version of the Persistent Nerve Lemma. Due to the fact that these chain maps are defined on chain complexes of simplicial complexes and regular CW-complexes, they are computable in practice. Our module interleaving and bottleneck distance bound results are a consequence of a chain-theoretic generalization of an interleaving only up to chain homotopy, rather than being purely one concerning persistent homology. There are also the novel contributions of the creation of chain map between the barycentric decomposition of the nerve of a cover and the space filtration at a further time scale, and the use a technique we call lifting to form a chain map into the barycentric decomposition of the blowup complex. The lack of existence of such a map is the reason the Nerve Theorem fails when the cover is not good, which gives credence to the notion that the maps we construct are “natural” choices.
Related Work
Apart from the Persistent Nerve Lemma and it’s original homotopy version, researchers have examined related problems concerning covers and their associated nerves with respect to persistence theory.
Botnan and Spreemann [18] proved that if three cover filtrations are -interleaved, where two of the filtrations are good and sandwich the third in the interleaving, then the bottleneck distance between the persistence module of the nerve of one of the good cover filtrations and the arbitrary one is upper bounded by .
Dey et al. [19] prove that for a cover whose elements are path-connected, the -dimensional homology of the map from the covered space to the nerve of the cover is surjective. Using this result, they also prove that if there exists a so-called cover map between two covers, then the -dimensional homology of the simplicial map resulting from the cover map between the covers’ nerves is surjective.
2 Background
This is an overview of the combinatorial, topological and algebraic structures used in the paper to prove the Generalized Persistent Nerve Theorem. See Hatcher’s Algebraic Topology [20] for further reference on chain complexes and homology, and Chazal et al.’s The Structure and Stability of Persistence Modules [21] for more on persistence theory.
2.1 Simplicial Complexes
A geometric simplex is the convex closure of a set of affinely independent points. A (geometric) simplicial complex is a collection of simplices such that for each simplex in , each of its subsimplices are in , and the intersection of two simplices is in or is empty. An abstract simplicial complex over a finite vertex set is a subset of the powerset closed under taking subsets. For a simplex , its dimension is . A simplex of dimension is called a -simplex of . The dimension of is the dimension of its largest simplex.
Each finite abstract simplicial complex has a corresponding geometric simplicial complex. Consider the function , where . For each maximal of an abstract simplicial complex , its geometric realization is , i.e. the convex closure of the image of its vertices under . The geometric realization of is which is equipped with the subspace topology inherited from the Euclidean topology. This construction allows for discussion of topological properties of abstract simplicial complexes and is functorial in the sense that given two abstract simplicial complexes with a simplicial map between them, the geometric realization carries the simplicial map to a continuous structure-preserving map between the complexes’ realizations.
CW-complexes are topogical structures that generalize the gluing procedures used to construct simplicial complexes. A CW-complex is defined inductively, starting with a collection of -cells, vertices, and then for natural , is the union of and some -cells whose boundaries are glued via continuous maps called attaching maps to the -cells of . A CW-complex is called finite if for some .
As an example, the -sphere can be viewed as a CW-complex with one -cell and one -cell where the -cell’s boundary is attached to the -cell via the constant map. A simplicial complex is CW-complex under the obvious gluing procedure. Given two finite CW-complexes and , is a finite CW-complex with cells of the form , where and , and .
A closed cover of a simplicial complex is a collection of simplicial complexes all defined over the same vertex set such that and is a subcomplex of of . A space is contractible if it has the homotopy type of a point. For example, all -simplices are contractible. A cover where every nonempty intersections of finitely many elements of is is called a good cover. Given a closed cover indexed over indexign set , the nerve of is the abstract simplicial complex
The following is a version of the Nerve Theorem which relates the homotopy type of the nerve of a cover to that of the covered space. See Corollary 4G.3 in [20] for the more general topological formulation.
Theorem 1 (Nerve Theorem).
If is a closed cover of a finite simplicial complex such that every non-empty intersection of finitely many of the elements in is contractible, then is homotopy equivalent to the geometric realization of the nerve, .
2.2 Chain Complexes and Homology
We restrict ourselves to chain complexes and homology groups over to simplify boundary computations, but the constructions hold for general finite fields as well. Given a simplicial complex , and a non-negative integer , a simplicial -chain over is a formal sum of the form , where each and each is a unique -simplex of . Collectively these simplicial
-chains form a vector space/abelian group over
called the -dimensional chain group, denoted by , which has a natural basis consisting of the -simplices of . Formally, . For each there exists a linear map , called the simplicial boundary map, defined on a -simplex by , where is the -th face of or alternatively the simplex spanned by the vertices of with removed. The boundary map extends linearly to arbitrary -chains and has the property that , or for short. We denote the sequence of simplicial chain groups with the appropriate boundary maps as and call it the simplicial chain complex of .Given a CW-complex , we can similarly define the cellular chain complex , where each is naturally isomorphic to the vector space over over with a basis being the -cells in . For each basis -cell in , there is a boundary map , where is computed by the cellular boundary formula (see page 140 in Hatcher[20] for the exact formula). For a so-called regular CW-complex, which simplicial complexes and their products are, all the coefficients .
Given two simplicial complexes and , a chain map is a collection of maps such that is a homomorphism and for all , i.e. the following diagram commutes.
A continuous map induced by a map between the vertices of and yields a simplicial chain map , and a cellular map yields a cellular chain map . Given two chain maps , a chain homotopy between them is a a sequence of maps , such that , or for short, . This is equivalent to Diagram 2.2 commuting. We will drop the subscripts when the dimension is clear. If a chain homotopy exists between and , then and are said to be chain homotopic, written .
Given a simplicial complex and its simplicial chain complex , the -dimensional simplicial homology group as . The rank of this groups is the number of linearly independent -dimensional holes in the space. Collectively, the homology groups of are denoted by . If two spaces have the same homotopy type, then their corresponding chain maps are chain homotopic, which then implies their homology groups are isomorphic in all dimensions. The reduced homology is the homology computed from the chain complex where one adjoins a copy of to . Functionally this means that so that the one-point space has trivial reduced homology groups over all dimensions.
Given two finite CW-complexes and , there is a natural isomorphism , where
is the tensor product. In particular,
, where the basis of is the collection of products , where is a -cell in and is a -cell in for . Each basis elements is identified under the isomorphism with . The product chain complex has the boundary map defined on any basis element by and it extends linear to all chains.The cellular homology of a CW-complex is defined in the same manner as simplicial homology except instead with cellular chain complex groups and boundaries. In fact, for a finite simplicial complex , , which is a result of the cellular and simplicial chain complexes being canonically isomorphic — the -simplices are the -cells when is viewed as a CW-complex.
2.3 Filtrations and Persistence
A filtration is a sequence of topological spaces such that if and only if . If a filtration consists of simplicial complexes, it is known as a simplicial filtration, and if each of the simplicial complexes is finite, it is known as a finite simplicial filtration. Filtrations often arise as the sublevel sets of a real-valued function on a topological space or simplicial complex.
Persistent homology is the changes in the homology of a filtration as it ranges over the interval . To be precise, it is the computation of the “birth” and “death” scales of homological features under the homology maps induced by inclusion, , for all such that . The persistent homology data of a filtration is contained in its persistence module, denoted by , which consists of the spaces over all scales, and the aforementioned maps for . The birth and deaths scales of -dimensional homological features in a filtration are represented in a filtration’s -dimensional persistence diagram, denoted by . This is a multiset with elements being points in the plane , where and are the birth and death scales respectively of features, and for all with infinite multiplicity. When discussing the persistence diagrams collectively for all dimensions we write . The following theorem provides a condition under which we can say that two filtrations have identical persistence diagrams, often called the Persistence Equivalence Theorem — see chapter 26 of [22].
Theorem 2 (Persistence Equivalence Theorem).
Consider two filtrations , with point-wise finite dimensional persistence modules. If their persistence modules and are isomorphic then the filtrations have identical persistent homology and .
Note that is isomorphic to if and only if there are natural isomorphisms for all .
A persistence diagram is finite if it has finitely many off-diagonal points. The standard metric on the space of persistence diagrams is the bottleneck distance , which is efficiently computable for finite diagrams. For two finite diagrams and it is defined as
where is the set of all bijections . Two finite persistent diagrams are equivalent if and only if the bottleneck distance between them is . A major result in topological data analysis is that the bottleneck distance stable with respect to perturbations of the function generating the diagram, which is known as the Stability Theorem [23].
Given two filtrations , , their persistence modules and are -interleaved if there exists collections of homomorphisms and , and , such that and , and these maps commute with all and for all . This is known as an additive interleaving. Persistence module interleavings and their persistence diagrams’ bottleneck distances are related by the Algebraic Stability Theorem (see [24, Thm 4.4]),
Theorem 3 (Algebraic Stability Theorem).
Given two filtrations and such that for all , , if and are -interleaved then for all .
Let be a collection of simplicial filtrations, where and for all and , is a finite subcomplex of an ambient simplicial complex. Let , the collection of simplicial complexes at scale from each filtration . For each non-empty , let yielding a simplicial filtration . Note that in this notation and . For a collection of filtrations there is an associated nerve filtration . For each scale the union of over the elements of is the simplicial complex defined as and the the union filtration with respect to is denoted by . For each , is a cover and thus we say that is a cover filtration of , or cover for short. We call a good cover filtration, or good cover for short, of if is a good cover of for all .
The previous definitions allow for the statement of the Persistent Nerve Lemma which the main theorem of this paper generalizes. The following lemma was originally formulated by Frédéric Chazal and Steve Oudot in [2] as a generalization of the Nerve Theorem to filtrations.
Lemma 4.
Let be two finite simplicial complexes with good covers, and respectively, such that for all . There exists homotopy equivalences and that commute with the topological inclusions and .
Viewing each from a good cover filtration as a good cover of , Theorem 2 can be applied to the construction of the homotopy equivalences in the proof of Lemma 4 to achieve the following fundamental persistent homology result.
Theorem 5 (Persistent Nerve Lemma).
Given a collection of finite simplicial filtrations where is a good cover filtration of , then .
3 The Generalized Persistent Nerve Theorem
In the Persistent Nerve Lemma a primary assumption is that is a good cover of for all , i.e. is a good cover filtration of . However, there are common situations where a simplicial cover may not be good as shown in the figures below.
In both cases these good cover violations are relatively small and intuitively ought to be able to be made insignificant for purposes of, for example, recovering the homotopy type of a triangulated space. Persistent homology is an ideal theory for quantifying what is meant by a small violation of the good cover condition. The following is our natural generalization of a good cover filtration.
Definition 6.
A cover filtration is an -good cover of a filtration if for all non-empty , and all , .
Though the definition of an -good covers is stated in terms of homology of the inclusions of cover intersections, it is in fact weaker than the assumption that is null-homotopic, which would align better with the traditional notion of a good cover. We choose to still use the term “good” despite this choice. The main goal of this paper is providing a theorem concerning persistent homology so nothing would be gained by working at the homotopy level.
There is the following relation between the definitions of an -good cover and a good cover. If is a good finite simplicial cover then for each each non-empty intersection of cover elements of , is homotopy equivalent to a point. This implies that , so is a -good cover. However, the converse does not hold — the -skeleton of the Poincaré -sphere has trivial reduced homology groups but is not contractible.
Recall the definition of the nerve filtration . The following theorem called the Generalized Persistent Nerve Theorem provides a tight bound of on the bottleneck distance between the -dimensional persistence diagrams of the nerve filtration and a simplicial cover filtration, given that is an -good cover filtration of .
Theorem 7 (Generalized Persistent Nerve Theorem).
Given a finite collection of finite simplicial filtrations , where and all are subcomplexes of a sufficiently large simplicial complex, if is an -good cover filtration of , then
As Theorem 7 is true for all dimensions it implies that for , as well as implying Theorem 5 (the Persistent Nerve Lemma) for the case of finite simplicial filtrations. See Appendix A for a construction that realizes the bottleneck distance bound over all dimensions.
The Generalized Persistent Nerve Theorem can be seen as an extension of the Persistent Nerve Lemma analogous to how the Algebraic Stability Theorem for persistence modules extends the Persistence Equivalence Theorem to interleaved modules, by viewing -good cover filtrations as perturbations or approximations of good cover filtrations, the ideal object.
This relationship is summarized in the following table.
Equivalence | Approximation | |
---|---|---|
Persistence Modules | Persistence Equivalence Theorem | Stability Theorem |
Nerves | Persistent Nerve Lemma | Gen. Persistent Nerve Theorem |
4 Proof Construction
Fix a cover filtration consisting of finite simplicial filtrations, where and all the are subcomplexes of some sufficiently large simplicial complex. Assume that is an -good cover filtration of the simplicial filtration . Fix a dimension for the remainder of the proof which will the be maximal dimension considered when discussing chain complexes, i.e. for any space and likewise for homology groups.
The procedure to prove Theorem 7 is as follows. First we construct a diagram of chain complexes and chain maps that yield a -interleaving between the filtered chain complexes and for all . This chain complex interleaving is analogous to that defined previously between persistence diagrams, except one we only require the appropriate compositions be chain homotopic to the identity chain maps rather than equivalent. Applying homology to said chain complex diagram, the chain maps that are chain homotopic to the identity become homologically equivalent to identity maps so there is a true -interleaving between the persistence modules and for each . The theorem is proved then by applying the Algebraic Stability Theorem (Theorem 3).
4.1 The Nerve Diagram and Blowup Complex
For each , define as the directed graph with vertex set , and edges of the form for any non-empty such . Note the vertices of this graph are in correspondence with the simplices of and thus form the -skeleton of the barycentric subdivision of , while the entire (undirected) graph is its -skeleton. The edges correspond to inclusions between intersections of cover elements of the form .
can be given the structure of an abstract simplicial complex, where the -simplices are sequences of vertices such that for and , , there exists an edge . Some call this complex built from an acyclic graph or poset a flag complex. From now on we will use the notation to refer to the simplicial representation. A fact that will be important later on is that its geometric realization is homeomorphic as a topological space to .
From and we define the finite CW-complex that glues together all the realizations of the simplices of paired with their corresponding cover elements’ intersection in .
Note that this is the barycentric decomposition of the so-called (Mayer–Vietoris) blow-up complex central to the proof of the Nerve Theorem, called the realization of a diagram of spaces in Hatcher [20], and other more recent persistent homology research, e.g. Zomorodian and Carlsson’s work on localized homology [25]. Other readers familiar with combinatorial topology and discrete geometry may recognize this as the homotopy colimit of the categorical diagram between and Top constructed from the cover elements’ interesections’ correspondence with the vertices and the inclusions with the edges of . One may refer to Kozlov [26], p.262, for the relevant definition or Welker et al.’s treatise on homotopy colimits and their applications [27]. This is expanded upon in Appendix B.
The associated filtration is denoted . By definition we have that for all . As the filtration organizes and combines the nerves and the covered simplicial complexes, it is easier to define maps from it rather than from .
Using the filtration we will now reduce the proof to constructing a -interleaving between and as follows. There are natural projection maps for each , where and when It is well-known fact that that is a homotopy equivalence (see 4G.2 in [20]) for covers of paracompact spaces, so as finite simplicial complexes are paracompact.
Moreover, two projections and commute with the inclusions and , yielding the following commutative homological diagram for all scales such that .
(1) |
Define the filtration . There are also natural projection maps for each , where and . When is a good cover of , and by extension is a good cover filtration of , the projection maps are homotopy equivalences that commute with the filtration inclusions. This is a central component to the proof the Nerve Theorem and Lemma 4.
However, under our assumptions, we do not have the homotopy equivalences resulting from the good cover condition, so instead we must find an interleaving between and to prove our theorem. As is homeomorphic to , this is sufficient. In the next section we will construct the chain maps mapping a basis chain in to a chain in which will result in an interleaving by symmetrizing.
4.2 The Chain Maps
For the remainder of the proof we will not use the geometric realization vertical bars when discussing basis cellular chains corresponding to tensor products of geometric simplices viewed as cells to avoid cumbersome presentation. For an abstract -simplex where , the following two shorthands will be used, and . These are the geometric realizations of the restriction of to the first -vertices and the -th through -th vertices respectively. For notational simplicity, given some and some vertex , , where denotes the removal of vertex .
For each non-empty , pick a vertex where , and note that for all . Consider the map which is the extension of the linear map sending each vertex of to to . This results in the chain map defined as follows.
The following lemma will be used to show the existence of a chain homotopy between the inclusion chain maps and the vertex chain maps.
Lemma 8.
Fix . Given non-empty , where , and is ’s corresponding vertex, consider the inclusion chain map and . There exists a chain homotopy from to .
Proof.
We construct the chain homotopy by induction on dimension to prove that for all that there exists such that . In the base case we can consider . Now for some , assume there exists such that .
Let be the -chain such that . Observe that for , for any -simplex ,
The second line follows by the inductive hypothesis as is a -chain. The third line follows as and and are chain maps so they commute with the boundary operators.
This proves that is a cycle so is a cycle, and as is an -good cover. There then must exist a boundary such that . Define . By the above calculations this choice of satisfies the inductive hypothesis so we are done. ∎
By Lemma 8, for a given and , there exists a chain homotopy, which we denote as , between the identity chain map and the constant chain map . By definition we have the equality .
Denote for short for the remainder of the paper when referencing the For , define the map for a cellular basis element , where is a -cell and is a -cell corresponding to the geometric realization of the abstract -simplex in , as
Note that this is well-defined despite ’s domain each being as for any basis cellular chain , is a simplex of for some by the definition of .
Recall the two projections from the barycentric decomposition of the blow-up complex , and . For a cellular chain -chain , the projection-induced chain maps and are defined as follows,
and
Define the chain map for a basis -simplex to be the following
The chain maps and are induced by topological maps so they are chain maps by construction, while it is not immediately apparent it is a chain map.
Lemma 9.
The map as defined above, where is a chain map for all dimensions less than or equal to .
Proof.
Denote for this proof to make it clear what dimension is being worked in. We will prove that is a chain map by induction on the basis of for arbitrary and . These are the simplices of resulting from abstract simplices in of the form .
In the base case, where , then for some vertex , so we have that .
Now assume that for some , the following holds for any given basis -chain , . Now consider a basis -chain . We have the following equalities, defining .
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