Keywords:

Contractibility of a persistence map preimage

Jacek Cyranka and Konstantin Mischaikow

Department of Mathematics, Rutgers, The State University of New Jersey,

110 Frelinghusen Rd, Piscataway, NJ 08854-8019, USA

Department of Computer Science and Engineering, University California, San Diego,

9500 Gilman Drive, La Jolla, CA 92093-0404, USA

jcyranka@gmail.com, mischaik@math.rutgers.edu

July 29, 2019

Abstract. This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of solutions snapshots, what conclusions can be drawn about solutions of the original dynamical system? In this paper we provide a definition of a persistence diagram for a point in modeled on piecewise monotone functions. We then provide conditions under which time series of persistence diagrams can be used to guarantee the existence of a fixed point of the flow on that generates the time series. To obtain this result requires an understanding of the preimage of the persistence map. The main theorem of this paper gives conditions under which these preimages are contractible simplicial complexes.

#### Keywords:

Topological data analysis, persistent homology, dynamical systems, fixed point theorem.

## 1 Introduction

Topological data analysis (TDA), especially in the form of persistent homology, is rapidly developing into a widely used tool for the analysis of high dimensional data associated with nonlinear structures. That topological tools can play a role in this subject should not be unexpected given the central role of nonlinear functional analysis in the study of geometry, analysis, and differential equations. What is perhaps surprising is that, to the best of our knowledge, there has been no systematic attempt to make use of nonlinear functional analysis as a tool to process information obtained via persistent homology.

Persistent homology is often used as a means of data reduction. A typical example takes the form of a complicated scalar function defined over a fixed domain, where the geometry of the sub-(super)-level sets is encoded via homology. Of particular interest to us are settings in which the scalar function arises as a solution to a partial differential equation, we are interested in tracking the evolution of the function, but experimental data only provides information on the level of digital images of the process. Furthermore, for many problems capturing the dynamics requires a long time series of rather large digital images. Thus, rather than storing the full images, one can hope to work with a time series of persistence diagrams.

The simplest mathematical analogy is that of attempting to describe the evolution of a partial differential equation via observations of the evolution of finite dimensional vectors obtained through a Galerkin projection. Obviously, since this is a central problem in numerical analysis this problem has been addressed in a variety of ways, including functional analysis. However, we are interested in going a step further. One can imagine that a piecewise linear function is associated with each finite dimensional vector, and to continue the analogy, the sublevel set persistence diagram for this function is computed. We would like to draw conclusions about the dynamics of the original partial differential equation from the time series of the persistence diagrams. This is an extremely ambitious goal and far beyond our capabilities at the moment. A much simpler question is the following: if there is an attracting region in the space of persistence diagrams, under what conditions can we conclude that there is a fixed point for the partial differential equation.

This paper represents a first step towards answering the simpler question. Theorem 9.1 shows that given a flow with a global compact attractor and a neighborhood of a particular form in the space of persistence diagrams, if under the dynamics the neighborhood is mapped into itself, then there exists a fixed point for the flow. Returning to the partial differential equation analogy, one could consider a system defined over an interval where the dynamics on the global attractor is obtained via a finite difference approximation.

The challenge, even in the simplified finite dimensional setting, is that to obtain results one must understand the topology of pre-images of sets under persistent homology, a topic for which there are only limited results. That the structure of pre-images is complicated follows directly from the fact that persistent homology can provide tremendous data reduction, but in a highly nonlinear fashion. We emphasize that simple examples show that the preimage set of a straight line path in a persistence diagram is nonconvex, see Figure 1.

With this in mind the primary goal of this paper is to show that for a reasonable class of bounded piecewise monotone continuous functions on a bounded interval the pre-image of a persistence diagram is composed of contractible, simplicial sets. The importance of this result is that it opens the possibility of applying standard algebraic topological tools, e.g. Lefschetz fixed point theorem, Conley index, to dynamics that is observed through the lens of persistent homology.

To state our goal precisely requires the introduction of notation. Throughout this paper denotes a simplicial complex composed out of vertices and edges. More specifically, where denotes the -skeleton of , i.e. the set of vertices

 S(0)N={ | i=1,…,N},

and denotes the set of edges

 S(1)N={ | i=1,…,N−1}.

Observe that can be viewed as a simplicial decomposition of a bounded interval . We are interested in sublevel set filtrations associated with piecewise monotone continuous functions on , and thus we are interested in the following filtration on .

###### Definition 1.1.

Let . For , define

 SN(z,θ):={σ∈SN:f(z,σ)≤θ}

where , is given by

 f(z,σ):={zjif σ=,max{zj,zj+1}if σ=.
###### Definition 1.2.

Given , we can reorder the coordinates of such that

 zj1≤zj2≤⋯≤zjN.

The sublevel-set filtration of at is given by

 SN(z,zj1)⊆SN(z,zj2)⊆⋯⊆SN(z,zjN).

and denoted by .

Since is a finite filtration of simplicial complexes we can use the classical results from persistent homology [11, 6] to compute the persistent homology of . Since for fixed , is completely determined by , we treat this as a map

 Dgm:RN→Per

where denotes the space of all persistence diagrams. Furthermore, there are a variety of topologies that can be put on such that becomes a continuous map [3, 4].

###### Remark 1.3.

If we replace by in the definition of and set
, then the appropriate modification of Definition 1.2 leads to a superlevel-set filtration of . The results of this paper are equally applicable in this setting, but the proofs must be modified appropriately.

Since is not a linear space, is far from a linear function. However, some restrictions of its image can be obtained from the values of the coordinates of . For this reason it is useful to think of as a real valued function defined over its coordinate indices, i.e.  where , .

###### Definition 1.4.

Let . A local minimum of is a sequence of indices such that , where , and it holds that

• if , then ,

• if , then .

A local maximum of is a sequence of indices such that , where , and it holds that

• if , then ,

• if , then .

A local extremum of is either a local minimum of or a local maximum of . A boundary extremum is a local extremum where or . A local extremum that is not a boundary extremum is an interior extremum.

We say that is monotone increasing (monotone decreasing) over the set of indices if (.

We make the following assumption for the remainder of this paper

A

The values of all local extrema of are distinct.

The important consequence of assumption A is the following proposition that is proven in Section 2.

###### Proposition 1.5.

If satisfies assumption A, then each persistence point in is unique.

As indicated above, given we are interested in the topology of . For the moment we focus on the pre-image of a single persistence diagram for which it is useful to catalog the order of the local extrema.

###### Definition 1.6.

The critical value set is given by

 E+:={h=(h1,…,hn)∈Rn:n % is odd, and h2k±1

Set . We will also use the simpler notation .

###### Remark 1.7.

To obtain results concerning superlevel set persistence one requires the use of

 E−:={h=(h1,…,hn)∈Rn:n % is odd, and h2k±1>h2k for all k} (2)
###### Definition 1.8.

We define a map as follows. Assume has local extrema that are either interior extrema or boundary minima. Observe that this implies that is odd. Let be an index associated with the -th extremum ordered such that . Observe that

 cv(z):=(zi1,…,zin)∈En+⊂E+.
###### Example 1.9.

Let . The interior extrema are and . The boundary extrema are and . Thus and or . Finally, .

As is discussed in Section 2, if , then . Given , set

 C(h):={z∈RN:cv(z)=h}. (3)

Then, is constant on .

Using results of [5] in Section 2 we prove the following result.

###### Proposition 1.10.

Let . Then the preimage of is composed of a finite number of mutually disjoint components

 Dgm−1(diag)=I(diag)⋃i=1C(hi)

where and all the , are related by permutations of the coordinates.

The main result of this paper is as follows.

###### Theorem 1.11.

Assume satisfies assumption A. Let

 Dgm−1(Dgm(z))=I(Dgm(z))⋃i=1C(cv(z)i).

Then for each , is an union of a finite contractible set of polytopes.

Moreover, let for some , and satisfies assumption A, then is a compact simplicial set.

As is described below the proof of Theorem 1.11 occupies the majority of this paper (Sections 2 through 7). In Section 8 we prove Theorem 8.5. This is an extension of Theorem 1.11 to regions in of a particular form and is obtained by providing an explicit homotopy contracting the preimage of the region to the preimage of a single diagram that lies in the region. Theorem 9.1, described above, follows almost directly from Theorem 8.5, and is presented in Section 9.

The proof of Theorem 1.11 is extremely technical, involving an induction proof based on the number of persistence points or equivalently the number of local extrema pairs and requires substantial bookkeeping of subsets of .

We begin in Section 2 with the necessary background information on sub-level set filtrations and persistence diagrams. We also recall the Nerve Theorem, which is the main tool that we use for our eventual proof of contractbility of the preimage set. The new content is a basic form of bookkeeping provided by convex polytopes covering the preimage set. By doing so we provide an easy proof that the preimage set is a finite union of polytopes (Proposition 2.6).

Unfortunately, we are unable to prove contractibility using the representation presented in Section 2. Thus, in Section 3 we introduce more complicated class of coverings of . Fortunately, this class of coverings has the structure of a meet semi-lattice that we use extensively as a new form of bookkeeping.

Section 4 describes the preimage set in terms of star-shaped sets, generated by semi-lattice morphisms on the set of multi-indices. However, analyzing the topology of the nerve complex defined using those star-shaped sets directly turns out to be hard, and its intersection structure is unclear. Instead, we introduce a covering using more coarse sets.

Section 5 provides an argument for the initial step of the induction argument. To provide the reader with some intuition we illustrate the results with an example for a particular case.

Section 6 provides a motivating example for the contractibility inductive argument. We carry out formally the inductive argument for contractibility of the coarse covering in Section 7.

### 1.1 Acknowledgements

The work of JC and KM was partially supported by grants NSF-DMS-1125174, 1248071, 1521771 and a DARPA contracts HR0011-16-2-0033.

In addition KM was partially supported by DARPA contract FA8750-17-C-0054 and NIH grant R01 GM126555-01.

## 2 Preliminaries

#### Notation

Throughout this paper and denote natural numbers where .

### 2.1 Sublevel-set filtrations and persistence diagrams

Before we introduce finite dimensional dynamical systems we study in this paper, let us first introduce the crucial tool that we use to describe graphs determined by a finite number of points. We compute persistent homology of sublevel-set filtrations of one dimensional simplicial complexes. The vertices of the simplicial complex have associated values, determined by a graph. We define the persistence map as the a map assigning to each sublevel-set filtration its persistence diagram or (equivalently) barcode. Existing literature about persistent homology is very rich, see e.g. [3, 6, 11] and references provided there. Here we briefly recall persistent homology theory to the extent it is required by our theory. We also prove some preliminary results about the preimage (fiber) of the persistence map.

Let be fixed. We define the persistence map as the map associating the filtered complex its persistence diagram, or sometimes referred to as the barcode [7, 11].

Observe that is an one dimensional filtered cell complex. The persistent homology of cell complexes is computed using by now standard persistent homology algorithms provided in [6].

By persistence diagram we mean a set of points of the extended plane

 E=R×R∪{∞}.

We denote the space of persistence diagrams by

 Per⊂E,

For any diagram , and for each point , it holds that . Moreover, in the one-dimensional setting there is necessarily one point having the coordinates .

Here we do not study in detail the structure and decomposition of the persistence map. We define the persistence map, as the map which takes a function and a vector to the persistence diagram encoding -th order persistence homology of . Also, note that in our setting we deal only with the persistence map of degree , associated with connected components.

It is known that the persistence map is -Lipschitz, after equipping the category of persistent vector spaces with the interleaving distance [3].

There exists a vast literature on algorithms computing the persistence diagram for general simplicial inputs, see e.g. [6, 11] and references cited there. But, in the literature regarding computational homology the space of admissible functions is chosen in a way admitting a notion of discrete Morse functions, which in turn provides the standard notion of critical points. As a consequence, the cells where components of the sub-level set filtration are born/die can be indicated uniquely by algorithms computing the persistent homology. However, the algorithm can be easily altered to make it work in the setting of plateaus.

In our setting motivated by dynamical systems we do not make any assumption that would provide the standard notion of critical points. A function can include ’plateaus’, i.e. pieces of constant value. As a consequence the local extrema, where the components of the sublevel-set filtration are born or die are not necessary unique.

Let

 diag∈Per

be a persistence diagram composed out of a finite number of points .

Where the first point corresponds to the component of infinite persistence, we have

 diag={p1=(p1,∞), p2=(pb2,pd2), …, pn=(pbn,pdn)}, (4)

where it holds that , and

 pbi≠pbj for i≠j, and pdj>pdj+1.

Due to the Elder Rule [5, 6] property of the persistence map it must hold that

 p11. (5)

The diagram has its associated barcode, which we denote by , and which is composed out of the following intervals

 I1=[p1,∞),I2=[pb2,pd2), …,In=[pbn,pdn). (6)

Observe that from (5) the following inclusion on barcodes must hold

 Ij⊂I1.

We say that a function realizes the barcode (6) if it holds that . It was shown in [5] that there is at least one piecewise linear function realizing the barcode, and this argument can be transferred directly to our setting.

To distinguish functions realizing the barcode , we group them into equivalence classes. We remark that to characterize equivalence classes we can use the notion of chiral merge tree as defined in [5].

It was shown in [Corollary 5.5, [5]] that the number of chiral merge trees realizing a barcode is equal to , where denotes the number of intervals in containing .

Now we recall the known result saying that is composed out of a finite number of components. For the moment we use an abstract notion , which we will make precise later. Each component is represented by a chiral merge tree, which was proved in [5] in the case of Morse functions. The functions we consider in our setting are not necessarily Morse, below we provide a suitable lemma fitted to our setting.

###### Lemma 2.1.

Let . Then the preimage is composed out of a finite number of components

 Dgm−1(diag)=I(Dgm(z))⋃i=1C(cv(z)i),

where . is such that for it holds that

• , are not path connected, i.e. that for any and , there is no continuous path connecting with in .

• each is characterized by the equivalence class of the associated chiral merge tree.

###### Sketch of a proof.

The proof follows naturally from the results in [5]. If two vectors in are such that the chiral merge trees associated with filtration functions , are different. It follows that have the same number of extrema, whose heights are determined by the persistence diagram , but appearing in different permutations. Hance any continuous path modifying into must change the heights of the local minima, and hence, the persistence diagram along this path must not be constant. Therefore, there is no continuous path in , which connects with . ∎

The case of super-level set filtrations is completely analogous, and we leave redoing the super-level case as an exercise for the reader.

### 2.2 Topology of C(h)

Let us fix example . In this subsection we show that defined in (3) is the union of a finite set of polytopes.

The set of strictly ordered multi-indices is given by

 LNm:={(y1,…,ym)∈{1,2,…,N}m:yj
###### Definition 2.2.

Let and . A point is -monotone if the following conditions are satisfied.

1. for .

2. is monotone increasing on and monotone decreasing on .

3. If is odd, then is monotone increasing on .

4. If is even, is monotone decreasing on .

Observe that if is -monotone, then .

###### Example 2.3.

If is as in Example 1.9, then is -monotone for or .

Given define by

 Sh(y)=Sh,N,n(y):={z∈RN:z is (y,h)-% monotone}. (7)

Let

 ¯a,¯b>0

denote global bounds imposed on the first and the last components respectively and denote the subset of having boundary elements bounded by

 B=B(¯a,¯b)={z∈RN:z1∈[−¯a,¯a], zN∈[−¯b,¯b]}. (8)

Let , define the compact version of by

 Sh(y)=Sh,N,n(y):={z∈B:z is (y,h)-% monotone}. (9)

Because most of the arguments are precisely the same regardless if we consider a compact or noncompact version of (the application in Section 9 is the sole exception), we do not provide distinct symbols for these two objects.

###### Example 2.4.

Let , , , and . The possible coordinate values of are shown in Figure 2. Observe that is an unbounded convex set.

Let us relate a preimage component with the definition of -monotonicity. First, observe that

 C(h) ={z∈RN:z is (y,h)-monotone for% any y∈LNn}, and (10a) C(h) =⋃y∈LNnSh(y). (10b) We define analogously the compact version of C(h):LNn→2B, using compact Sh’s.
###### Proposition 2.5.

Given , and . The set defined by (7) is a closed convex polytope. defined as (9) is additionally compact.

###### Proof.

From the definition is the intersection of a finite number of half-spaces , etc.. Therefore is a closed convex polytope. In case it is additionally compact, as the boundary points have restricted values. ∎

###### Proposition 2.6.

Given . The set is a finite union of polytopes. is a compact simplicial set.

###### Proof.

Follows directly from Prop 2.5 showing that for each is a closed convex polytope (and in case , is compact), and the identity (10b). ∎

###### Remark 2.7.

defined by (7) is also a unbounded convex set, where only the boundary points take the unbounded values, see Figure 2. However, defined as is obviously a compact convex set. Analogously, in the unbounded case is a finite union of polytopes, and is additionally compact, hence a simplicial set.

All elements of the proof of Theorem 1.11 are the same regardless if we consider the polytope sets being unbounded (over ) or compact (over ).

### 2.3 Nerve Theorem

The fundamental tool in our study of the topology of the persistence map preimage is the Nerve Theorem, below we recall it as Theorem 2.8. It is elementary to observe that any preimage set of interest () is a finite union of polytopes, see Proposition 2.6. However, the polytopes constructing can have arbitrary topology. By finding an appropriate family of sets that covering (denoted in the sequel by ’s), we show that the nerve of that covering is the full simplex, and hence, is homotopy equivalent to a point (contractible). See also Example 6.1. Let denote the support of a simplicial complex, let . Let be a family of sets. The nerve records the “intersection pattern” of . It is the simplicial complex denoted by , with vertex set and with simplices given by

 N(A)={J⊂[n]:⋂i∈JAi≠∅}.
###### Theorem 2.8 (Nerve Theorem [1, 2], version provided in [10]).

Let be subcomplexes of a finite simplicial complex that together cover (each simplex of is in at least one ), and let . Suppose that the intersection is empty or contractible for each nonempty . Then

 ∥N({A1,A2,…,An})∥≃∥K∥;

i.e., the nerve is homotopy equivalent to .

### 2.4 Meet semi-lattices

###### Definition 2.9.

A meet-semilattice is a set with a binary operation satisfying

1. (associativity) ,

2. (symmetry) , and

3. (idempotency) .

A classical example of a meet-semilattice comes from set theory. Given a set , let denote its power set. Then, is a meet-semilattice.

## 3 Indexing sets

As indicated in the Introduction the proof of Theorem 1.11 makes use of a complicated class of coverings. The remainder of this section introduces the notation used to index these coverings.

###### Definition 3.1.

For , we denote the set of multi-indices by and define it to consist of the vectors that satisfy the following four conditions for all .

1. .

The order relation in is summarized in Figure 3. Condition (ii) implies that the assumption that is necessary. Observe that if , then consists of a single element.

###### Example 3.2.

The simplest nontrivial example of a set of multi-indices is

 L34 ={(w1,w2,w3,w4):w1≤w2,w3≤w4,w1

Define by where

 (w∧w′)k={min{wk,w′k}if k odd,max{wk,w′k}if k even.
###### Theorem 3.3.

is a meet-semilattice.

###### Proof.

We begin by showing that if , then . We need to show that satisfies conditions (i)-(iv) of Definition 3.1. Since and

 w′′2k−1:=min{w2k−1,w′2k−1}≤max{w2k,w′2k}=w′′2k

and thus (i) is satisfied. To see that (ii) is satisfied we note that and implies that

 w′′2k−1:=min{w2k−1,w′2k−1}

Similar arguments apply to (iii) and (iv).

Checking that is associative, symmetric, and idempotent is straightforward. ∎

We denote the set of ordered multi-indices by

 LN2n:={(x1,…,x2n)∈{1,…,N}2n:x2k−1≤x2k for 1≤k≤n and x2k

Observe that . However, is not a sub meet-semilattice of . To see this assume that and that . Let . Then , which implies that .

###### Example 3.4.

The simplest nontrivial example of a set of ordered multi-indices is

 L34 ={(x1,x2,x3,x4):x1≤x2

Recall the strictly ordered multi-indices defined in Section 2.2 and observe that

 LN2n⊂LN2n⊂LN2n.

Again, is not a sub meet-semilattice of .

We now define functions that takes arbitrary multi-indices and produces sets of pairs of strictly ordered multi-indices, i.e., maps , . Set

 (12)

For , define by

 (13)

And

 φn−2(w)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩{(y,y)∈LNn×LNn:w2(n−2)
###### Example 3.5.

If and , then . Furthermore, if , then

 φ1(w)={((1,3,7),(1,5,7)),((2,3,7),(2,5,7))}.
###### Example 3.6.

If and , then , for . Furthermore, if , then

 φ1(w) ={((1,2,4,5),(1,3,4,5)),((1,2,4,6),(1,3,4,6)),((1,2,5,6),(1,3,5,6))} φ2(w) ={((1,2,3,5),(1,2,4,5)),((1,2,3,6),(1,2,4,6))}

The action of all the restricted to ordered multi-indices is relatively easy to describe. Define by

 ρ(x)={y∈LNn:y1≤x1, x2(k−1)
###### Example 3.7.

Let , . Then ,

 ρ(112244)={y∈L64:y1≤1, 1
###### Lemma 3.8.

Let . If , then and

 {(y,y):y∈ρ(x)}=n−2⋂k=1φk(x).
###### Proof.

If , then and hence

 φk(x)={(y,y):yk≤x2k−1, x2k

Therefore,

 n−2⋂k=1φk(x)={(y,y):y1≤x1, x2(k−1)

which by definition of is equal to the set . ∎

For , set

 PN2n:={(x1,…,x2n)∈{1,…,N}2n:x2k−1=x2k

Observe that , but it is not a sub-lattice of .

###### Example 3.9.

The simplest nontrivial example is

 P34 ={(x1,x2,x3,x4):x1=x2

We make use of ordered multi-indices (pairs) on the restricted domain, i.e.

 Ln−3,N−32:={(x1,x2)∈{n−3,…,N−3}2:x1≤x2}.

in conjunction with a map given by

 θ((p1,p2)):={(y1,…,y2(n−1)): y2(n−3)≤p1≤p2
###### Example 3.10.

Let and . Then,

 Ln−3,N−32=L2,32={(x1,x2)∈{2,3}2:x1≤x2}={(22),(23),(33)}

Then

 θ:L2,32→2P58,

and hence we have

 θ(22) ={11223344,11223355,11224455}, θ(23) ={11224455}, θ(33) ={11224455,11334455,22334455}.
###### Proposition 3.11.

It holds that

 ⋃p∈Ln−3,N−32p=(p