PERSISTENCE DIAGRAM BUNDLES A MULTIDIMENSIONAL GENERALIZATION OF VINEYARDS ABIGAIL HICKOK

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PERSISTENCE DIAGRAM BUNDLES:

A MULTIDIMENSIONAL GENERALIZATION OF VINEYARDS

ABIGAIL HICKOK

Abstract. I introduce the concept of a persistence diagram (PD) bundle, which is the

space of PDs for a ﬁbered ﬁltration function (a set {fp:Kp→R}p∈Bof ﬁltrations that

is parameterized by a topological space B). Special cases include vineyards, the persistent

homology transform, and ﬁbered barcodes for multiparameter persistence modules. I prove

that if Bis a smooth compact manifold, then for a generic ﬁbered ﬁltration function, Bis

stratiﬁed such that within each stratum Y⊆B, there is a single PD “template” (a list of

“birth” and “death” simplices) that can be used to obtain the PD for the ﬁltration fpfor

any p∈Y. If Bis compact, then there are ﬁnitely many strata, so the PD bundle for a

generic ﬁbered ﬁltration on Bis determined by the persistent homology at ﬁnitely many

points in B. I also show that not every local section can be extended to a global section

(a continuous map sfrom Bto the total space Eof PDs such that s(p)∈PD(fp) for all

p∈B). Consequently, a PD bundle is not necessarily the union of “vines” γ:B→E; this

is unlike a vineyard. When there is a stratiﬁcation as described above, I construct a cellular

sheaf that stores suﬃcient data to construct sections and determine whether a given local

section can be extended to a global section.

1. Introduction

In topological data analysis (TDA), our aim is to understand the global shape of a data

set. Often, the data set takes the form of a collection of points in Rn, called a point cloud,

and we hope to analyze the topology of a lower-dimensional space that the points lie on.

TDA has found applications in a variety of ﬁelds, such as biology [29], neuroscience [12], and

chemistry [27].

We use persistent homology (PH), a tool from algebraic topology [21]. The ﬁrst step of

persistent homology is to construct a ﬁltered complex from our data; a ﬁltered complex is a

nested sequence

(1) Kr0⊆ Kr1⊆ ··· ⊆ Krn⊆ ···

of simplicial complexes. For example, one of the standard ways to build a ﬁltered complex

from point cloud data is to construct the Vietoris–Rips ﬁltered complex. At ﬁltration-

parameter value r, the Vietoris–Rips complex Krincludes a simplex for every subset of

points within rof each other. In persistent homology, one studies how the topology of Kr

changes as the ﬁltration parameter-value rincreases. As rgrows, new homology classes

(which represent “holes” in the data) are born and old homology classes die. One way of

summarizing this information is a persistence diagram: a multiset of points in the extended

plane R2. If there is a homology class that is born at ﬁltration-parameter value band dies

at ﬁltration-parameter value d, the persistence diagram contains the point (b, d).

Date: August 15, 2023.

arXiv:2210.05124v2 [math.AT] 11 Aug 2023

2 ABIGAIL HICKOK

Figure 1. An example of a vineyard. There is a persistence diagram for each

time t. Each curve is a vine in the vineyard. (This ﬁgure is a slightly modiﬁed

version of a ﬁgure that appeared originally in [26].)

Developing new methods for analyzing how the topology of a data set changes as multiple

parameters vary is a very active area of research [7]. For example, if a point cloud evolves

over time (i.e., it is a dynamic metric space), then one maybe interested in using time as

a second parameter, in addition to the ﬁltration parameter r. Common examples of time-

evolving point clouds include swarming or ﬂocking animals whose positions and/or velocities

are represented by points ( [15, 25, 36]). In such cases, one can obtain a ﬁltered complex

r0⊆ Kt

r1⊆ ··· ⊆ Kt

rnat every time tby constructing, e.g., the Vietoris–Rips ﬁltered

complex for the point cloud at time t. It is also common to use the density of the point

cloud as a parameter ( [6, 10, 30]). Many other parameters can also vary in the topological

analysis of point clouds or other types of data sets.

One can use a vineyard [14] to study a 1-parameter family of ﬁltrations {Kt

r0⊆ Kt

r1⊆

···Kt

rn}t∈Rsuch as that obtained from a time-varying point cloud. At each t∈R, one can

compute the PH of the ﬁltration Kt

r0⊆ Kt

r1⊆ ··· ⊆ Kt

rnand obtain a persistence diagram

PD(t). A vineyard is visualized as the continuously-varying “stack of PDs” {PD(t)}t∈R.

See Figure 1 for an illustration. As t∈Rvaries, the points in the PDs trace out curves

(“vines”) in R3. Each vine corresponds to a homology class (i.e., one of the holes in the

data), and shows how the persistence of that homology class changes with time (or, more

generally, as some other parameter varies). However, one cannot use a vineyard for a set of

ﬁltrations that is parameterized by a space that is not a subset of R. For example, suppose

that we have a time-varying point cloud whose dynamics depend on some system-parameter

values µ1, . . . , µm∈R. Many such systems exist. For example, the D’Orsogna model is

a multi-agent dynamical system that models attractive and repulsive interactions between

particles [11]. Each particle is represented by a point in a point cloud. In certain parameter

regimes, there are interesting topological features, such as mills or double mills [31]. For

each time t∈Rand for each µ1, . . . , µm∈R, one can obtain a ﬁltered complex

(2) Kt,µ1,...,µm

r0⊆ Kt,µ1,...,µm

r1⊆ ··· ⊆ Kt,µ1,...,µm

PERSISTENCE DIAGRAM BUNDLES 3

by constructing, e.g., the Vietoris–Rips ﬁltered complex for the point cloud at time tat

system-parameter values µ1, . . . , µm. A parameterized set of ﬁltered complexes like the one

in (2) cannot be studied using a vineyard for the simple reason that there are too many

parameters.

Such a parameterized set of ﬁltered complexes cannot be studied using multiparameter

PH [8], either. A multiﬁltration is a set {Ku}u∈Rnof simplicial complexes such that Ku⊆ Kv

whenever u≤v.Multiparameter PH is the F[x1, . . . , xn] module obtained by applying

homology (over a ﬁeld F) to a multiﬁltration. (For more details, see reference [8].) The

parameterized set of ﬁltered complexes in (2) is not typically a multiﬁltration because it is

not necessarily the case that Kt,µ1,...,µm

ri̸⊆ Kt′,µ′

1,...,µ′

rifor all values of t,t′,{µi}, and {µ′

i}.

Not only is there not necessarily an inclusion Kt,µ1,...,µm

ri−→ Kt′,µ′

1,...,µ′

ri, but there is not any

given simplicial map Kt,µ1,...,µm

ri→ Kt′,µ′

1,...,µ′

ri. Therefore, we cannot use multiparameter PH.

In what follows, we will work with a slightly diﬀerent notion of ﬁltered complex than that

of (1). A ﬁltration function is a function f:K → R, where Kis a simplicial complex, such

that every sublevel set Kr:= {σ∈ K | f(σ)≤r}is a simplicial complex (i.e., f(τ)≤f(σ)

if τis a face of σ). For every r≤s, we have that Kr⊆ Ks. A simplex σ∈ K appears

in the ﬁltration at r=f(σ). By setting {ri}= Im(f), where ri< ri+1, we obtain a

nested sequence as in (1). Conversely, given a nested sequence of simplicial complexes, the

associated ﬁltration function is f(σ) = min{ri|σ∈ Kri}, with K=SiKri.

1.1. Contributions. I introduce the concept of a persistence diagram (PD) bundle, in which

PH varies over an arbitrary “base space” B. A PD bundle gives a way of studying a ﬁbered

ﬁltration function, which is a set {fp:Kp→F}p∈Bof functions such that fpis a ﬁltration of

a simplicial complex Kp. At each p∈B, the sublevel sets of fpform a ﬁltered complex. For

example, in (2), we have B=Rn+1 and we obtain a ﬁbered ﬁltration function {ft,µ1,...,µm:

K → R}(t,µ1,...,µm)∈Rm+1 by deﬁning ft,µ1,...,µmto be the ﬁltration function associated with

the ﬁltered complex in (2). The associated PD bundle is the space of persistence diagrams

PD(fp) as they vary with p∈B(see Deﬁnition 3.2). In the special case in which Bis an

interval in R, a PD bundle is equivalent to a vineyard.

I prove that for “generic” ﬁbered ﬁltration functions (see Section 4.2), the base space Bcan

be stratiﬁed in a way that makes PD bundles tractable to compute and analyze. Theorem

4.15 says that for a “generic” ﬁbered ﬁltration function on a smooth compact manifold B,

the base space Bis stratiﬁed such that within each stratum, there is a single PD “template”

that can be used to obtain P D(fp) at any point pin the stratum. Proposition 4.5 shows

that all “piecewise-linear” PD bundles (see Deﬁnition 3.4) have such a stratiﬁcation. The

template is a list of (birth, death) simplex pairs, and the diagram PD(fp) is obtained by

evaluating fpon each simplex. In particular, when Bis a smooth compact manifold, the

number of strata is ﬁnite, so the PD bundle is determined by the PH at a ﬁnite number of

points in the base space.

I show that unlike vineyards, PD bundles do not necessarily decompose into a union of

“vines”. More precisely, there may not exist continuous maps γ1, . . . , γm:B→Esuch that

(3) PD(fp) =

[

i=1

γi(p)

for all p∈B. This is a consequence of Proposition 5.3, in which it is shown that nontrivial

global sections are not guaranteed to exist. That is, given a point z0∈P D(fp0) for some

4 ABIGAIL HICKOK

p0∈B, it may not be possible to extend p07→ z0to a continuous map s:B→E:= {(p, z)|

z∈P D(fp)}such that s(p)∈PD(fp) for all p. This behavior is a feature that gives PD

bundles a richer mathematical structure than vineyards.

For any ﬁbered ﬁltration with a stratiﬁcation as described above (see Theorem 4.15 and

Proposition 4.5), I construct a “compatible cellular sheaf” (see Section 6.2) that stores the

data in the PD bundle. Rather than analyzing the entire PD bundle, which consists of

continuously varying PDs over the base space B, we can analyze the cellular sheaf, which is

discrete. For example, in Proposition 6.4, I prove that an extension of p07→ z0to a global

section exists if a certain associated global section of the cellular sheaf exists. A compatible

cellular sheaf stores suﬃcient data to reconstruct the associated PD bundle and analyze its

sections.

Though not the focus of this paper, I also give a simple example of vineyard instability in

Appendix A.1. It is often quoted in the research literature that “vineyards are unstable”;

however, this “well-known fact” has been shared only in private correspondence and, to the

best of my knowledge, has never been published. The example of vineyard instability is

furnished from an example in Proposition 5.3.

1.2. Related work. PD bundles are a generalization of vineyards, which were introduced

in [14]. Two other important special cases of PD bundles are the ﬁbered barcode of a

multiparameter persistence module [4] and the persistent homology transform (B=Sn)

from shape analysis [2,32]. I discuss the special case of ﬁbered barcodes in detail in Section

3.2.2; the base space Bis a subset of the space of lines in Rn. The persistent homology

transform (PHT) is deﬁned for a constructible set M⊆Rn+1. For any unit vector v∈Sn,

one deﬁnes the ﬁltration Mv

r={x∈M|x·v≤r}(i.e., the sublevel ﬁltration of the

height function with respect to the direction v). PHT is the map that sends v∈Snto

the persistence diagram for the ﬁltration {Mv

r}r∈R. The signiﬁcance of PHT is that it

is a suﬃcient statistic for shapes in R2and R3[32]. Applications of PHT are numerous

and include protein docking [34], barley-seed shape analysis [3], and heel-bone analysis in

primates [32].

For PHT, Curry et al. [18] proved that the base space Snis stratiﬁed such that the PHT

of a shape Mis determined by the PH of {Mv

r}r∈Rfor ﬁnitely many directions v∈Sn(one

direction vper stratum). This is related to the stratiﬁcation given by Theorem 4.15, in which

I show that a “generic” PD bundle whose base space Bis a compact smooth manifold (such

as Sn) is similarly stratiﬁed and thus determined by ﬁnitely many points in B(one p∈B

per stratum). The primary diﬀerence between the stratiﬁcations in [18] and Theorem 4.15

is that in [18], each stratum is a subset in which the order of the vertices of a triangulated

shape M(as ordered by the height function) is constant, whereas in Theorem 4.15, each

stratum is a subset in which the order of the simplices (as ordered by the ﬁltration function)

is constant. The stratiﬁcation result of the present paper (Theorem 4.15) applies to general

PD bundles, while [18] applies only to PHT.

The stratiﬁcation that we study in the present paper is used in [23] to develop an algorithm

for computing “piecewise-linear” PD bundles (see Deﬁnition 3.4). The algorithm relies on

the fact that for any piecewise-linear PD bundle on a compact triangulated base space B,

there are a ﬁnite number of strata, so the PD bundle is determined by the PH at a ﬁnite

number of points in B.

The existence (or nonexistence) of nontrivial global sections in PD bundles is related to

the study of “monodromy” in ﬁbered barcodes of multiparameter persistence modules [9].

PERSISTENCE DIAGRAM BUNDLES 5

(a)

(b)

(c)

(d)

(e)

Figure 2. An example of a ﬁltration. The simplicial complex Kihas the

associated ﬁltration-parameter value i. (This ﬁgure appeared originally in

[24].)

Cerri et al. [9] constructed an example in which there is a path through the ﬁbered barcode

that loops around a “singularity” (a PD in the ﬁbered barcode for which there is a point

in the PD with multiplicity greater than one) and ﬁnishes in a diﬀerent place than where it

starts.

1.3. Organization. This paper proceeds as follows. I review background on persistent

homology in Section 2. In Section 3, I give the deﬁnition of a PD bundle, with some

examples, and I compare PD bundles to multiparameter PH. In Section 4, I show how to

stratify the base space Binto strata in which the (birth, death) simplex pairs are constant

(see Theorem 4.15 and Proposition 4.5). I discuss sections of PD bundles and the existence

of monodromy in Section 5. I construct a compatible cellular sheaf in Section 6.2. I conclude

and discuss possible directions for future research in Section 7. In Appendix A.1, I use the

example of monodromy from Section 5 to construct an example of vineyard instability. In

Appendix A.2, I provide technical details that are needed to prove Theorem 4.15.

2. Background

We begin by reviewing persistent homology (PH) and cellular sheaves. For a more thor-

ough treatment of PH, see [19, 28], and for more on cellular sheaves, see [16, 22].

2.1. Filtrations. Consider a simplicial complex K. A ﬁltration function on Kis a real-

valued function f:K → Rsuch that if τ∈ K is a face of σ∈K, then f(τ)≤f(σ). The

ﬁltration value of a simplex σ∈ K is f(σ). The r-sublevel sets Kr:= {σ∈ K | f(σ)≤r}

form a ﬁltered complex. The condition that f(τ)≤f(σ) if τ⊆σguarantees that Kris a

simplicial complex for all r. For all s≤r, we have Ks⊆ Kr. The parameter ris the ﬁltration

parameter. For an example, see Figure 2.

For example, suppose that X={xi}M

i=1 is a point cloud. Let Kbe the simplicial complex

that has a simplex σwith vertices {xj}j∈Jfor all J⊆ {1, . . . , M}. The Vietoris–Rips

ﬁltration function is f(σ) = 1

2maxj,k∈J{∥xj−xk∥}, where {xj}j∈Jare the vertices of σ.

2.2. Persistent homology. Let f:K → Rbe a ﬁltration function on a ﬁnite simplicial

complex K, and let {Kr}r∈Rbe the associated ﬁltered complex. Let r1<··· < rNbe the

ﬁltration values of the simplices of K. These are the critical points at which simplices are

added to the ﬁltration. For all s∈[ri, ri+1), we have Ks=Kri.

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PERSISTENCEDIAGRAMBUNDLES:AMULTIDIMENSIONALGENERALIZATIONOFVINEYARDSABIGAILHICKOKAbstract.Iintroducetheconceptofapersistencediagram(PD)bundle,whichisthespaceofPDsforafiberedfiltrationfunction(aset{fp:Kp→R}p∈BoffiltrationsthatisparameterizedbyatopologicalspaceB).Specialcasesincludevineyards,thepersis...

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PERSISTENCE DIAGRAM BUNDLES A MULTIDIMENSIONAL GENERALIZATION OF VINEYARDS ABIGAIL HICKOK

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