MULTIPARAMETER PERSISTENT HOMOLOGY

A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
MIDDLE EAST TECHNICAL UNIVERSITY

BY

UĞUR BEKTAŞ CANTÜRK

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR
THE DEGREE OF MASTER OF SCIENCE
IN
MATHEMATICS

JUNE 2023
 Approval of the thesis:

MULTIPARAMETER PERSISTENT HOMOLOGY

submitted by UĞUR BEKTAŞ CANTÜRK in partial fulfillment of the requirements

for the degree of Master of Science in Mathematics Department, Middle East
Technical University by,

Prof. Dr. Halil Kalıpçılar

Dean, Graduate School of Natural and Applied Sciences

Prof. Dr. Yıldıray Ozan

Head of Department, Mathematics

Assoc. Prof. Dr. Mehmetcik Pamuk

Supervisor, Mathematics, METU

Examining Committee Members:

Assoc. Prof. Dr. Mustafa Gökhan Benli

Mathematics, METU

Assoc. Prof. Dr. Mehmetcik Pamuk

Mathematics, METU

Assoc. Prof. Dr. Tane Vergili

Mathematics, Karadeniz Technical University

Date: 09.06.2023
I hereby declare that all information in this document has been obtained and
presented in accordance with academic rules and ethical conduct. I also declare
that, as required by these rules and conduct, I have fully cited and referenced all
material and results that are not original to this work.

Name, Surname: Uğur Bektaş Cantürk

Signature :

iv
 ABSTRACT

MULTIPARAMETER PERSISTENT HOMOLOGY

Cantürk, Uğur Bektaş

M.S., Department of Mathematics
Supervisor: Assoc. Prof. Dr. Mehmetcik Pamuk

June 2023, 77 pages

The amount of the existing data is increasing rapidly and any way to understand
the shape of the given data is welcomed. In this thesis, we examine a method of
Topological Data Analysis which is called Persistent Homology. After a detailed
overview of 1-parameter case, we extend the structures to the multiparameter case
arising as a natural necessity. We present the similarities and differences between 1-
parameter and multiparameter persistent homology in an expository way. At the last
stage, we work on the interleaving, bottleneck, matching distances and their stability
properties. We also analyze the relations between these pseudometrics, and consider
some examples that help us to calculate them.

Keywords: persistent homology, distance, stability, multiparameter

v
 ÖZ

ÇOK DEĞİŞKENLİ KALICI HOMOLOJİ

Cantürk, Uğur Bektaş

Yüksek Lisans, Matematik Bölümü
Tez Yöneticisi: Doç. Dr. Mehmetcik Pamuk

Haziran 2023 , 77 sayfa

Var olan veri miktarı hızla artıyor ve verilen verinin şeklini anlamanın her yolu mem-
nuniyetle karşılanıyor. Bu tezde, Kalıcı Homoloji olarak adlandırılan Topolojik Veri
Analizi’nin bir metodunu inceliyoruz. 1-değişkenli durumun ayrıntılı bir incelemesin-
den sonra, yapıları, doğal bir gereksinim olarak ortaya çıkan çok değişkenli duruma
genişletiyoruz. 1-değişkenli ve çok değişkenli kalıcı homoloji arasındaki benzerlik-
leri ve farklılıkları açıklayıcı bir şekilde sunuyoruz. Son aşamada, serpiştirme, dar-
boğaz, eşleştirme mesafeleri ve kararlılık özellikleri üzerinde çalışıyoruz. Ayrıca, bu
psödometrikler arasındaki ilişkileri de inceliyoruz, ve onları hesaplamamıza yardımcı
olacak bazı örnekleri ele alıyoruz.

Anahtar Kelimeler: kalıcı homoloji, mesafe, istikrar, çok değişkenli

vi
To my family

vii
 ACKNOWLEDGMENTS

As a first thing, I would like to thank my supervisor Assoc. Prof. Dr. Mehmetcik
Pamuk for guiding me along my journey in master study. He introduced me to this
field, and I have learned so many things from him. With his help and encouraging me
about how to write my thesis, the work that started in his office turned into a master’s
thesis.

I also want to extend my thanks to everyone in the Department of Mathematics,

METU and Scientific and Technological Research Council of Türkiye (TÜBİTAK)
for their support via the project "Multidimensional Persistence and Toric Topology."

Lastly, thanks to my friends for their endless support, and I dedicate my thesis to my
parents: Sakine and Muharrem for everything they have done for me.

viii
 TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

CHAPTERS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 ONE-PARAMETER PERSISTENT HOMOLOGY . . . . . . . . . . . . . 5

2.1 Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Persistent Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Algebraic Background . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Persistence Module . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 The Bottleneck Distance in 1-Parameter . . . . . . . . . . . . . . . . 22

2.6 The Interleaving Distance in 1-Parameter . . . . . . . . . . . . . . . 24

3 MULTIPARAMETER PERSISTENT HOMOLOGY . . . . . . . . . . . . 31

3.1 Multiparameter Filtration and Its Examples . . . . . . . . . . . . . . 33

3.2 Multiparameter Persistence Module . . . . . . . . . . . . . . . . . . 36

4 METRICS AND STABILITY . . . . . . . . . . . . . . . . . . . . . . . . . 43

ix
 4.1 The Interleaving Distance . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 The Bottleneck Distance . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 The Matching Distance . . . . . . . . . . . . . . . . . . . . . . . . . 60

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

x
 LIST OF FIGURES

FIGURES

Figure 2.1 A filtration consisting of Vietoris-Rips complexes for some val-

ues of ϵ between ϵ = 0 and ϵ = 2.1 where we start with discrete set of
points, but for ϵ ≥ 2.1, the complex turns into a solid disk, [25]. . . . . 8

Figure 2.2 The barcode and the 0th and 1st persistence diagrams of the topo-
logical space M obtained by the height function, [13]. . . . . . . . . . . 20

Figure 3.1 (a) Set of points in Rn colorized by a local density estimate.

(b) Adding outliers with low density. (c) The barcode of (a) in 1-
dimensional holes. (d) The barcode of (b) in 1-dimensional holes, [8]. . 32

Figure 3.2 Subsets of R2 given in the plane with specified points and the
line L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 3.3 Degree-Rips bifiltration where degree parameter is 4, 3, 1 from

left to right, [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 3.4 Multi-cover bifiltration for the fixed values of k = 2, 3, [23]. . . 36

Figure 3.5 Bifiltration indexed by P = {0, 1, 2} × {0, 1, 2}, [8]. . . . . . . 40

Figure 4.1 Intervals where I1 and I2 are outer light purple squares, I3 is the
inner purple square and J is the red one, [4]. . . . . . . . . . . . . . . . 53

Figure 4.2 Decomposition of M and N with lines in R2 , [26]. . . . . . . . . 65

Figure 4.3 2-parameter persistence modules M and N with lines in R2 , [2]. 71

xi
xii
 CHAPTER 1

INTRODUCTION

The amount of data (discrete finite set of points in a metric space) we produce is
increasing rapidly and any way to understand the data at hand is welcomed. The
data can come from many different areas such as an engineering testbed, a biological
experiment or results of a questionnaire etc. A relatively new branch of mathematics,
called Topological Data Analysis (TDA), proposes to use topology to study properties
of data. It is an interdisciplinary field which has intersections with algebraic topology,
computational geometry, data analysis, computer science and statistics. The main
idea in TDA is to exploit topology to find out the features of the shape of data set and
present these features through an easily implementable summaries. In TDA, one also
defines (pseudo)metrics to distinguish data sets from each other again by examining
their shapes.

Persistent Homology (PH) is one of the most widely used methods in TDA. PH uses
homology which is a tool in algebraic topology to detect holes in topological spaces.
If we only consider a finite set of points, applying the homology functor directly to
this discrete set does not yield any interesting result. In this case, the only non-trivial
homology is the 0-th homology group which only counts the number of points. For
such a construction, one of the methods that PH suggests is instead of looking at the
data set directly, we consider the union of closed balls where centers are the data
points, and the radius is ϵ ≥ 0 (This is a very naive way to study data sets and we do
not mention this method in our thesis, but for motivational purposes, we want to point
this out here in the introduction section). However, we should not look for a single ϵ
since working with a fixed ϵ is unstable with respect to perturbations of data. Also,
such an approach cannot distinguish big holes from small ones. In fact, there exists

1
no canonical ϵ [23]. For these reasons, we examine all topological spaces that change
with respect to the proximity parameter ϵ. With this way, we obtain a nested sequence
of topological spaces called a filtration. Then, we apply the homology functor to these
topological spaces, or to simplicial complexes obtained from these data sets.

Now, we are ready to track changes in topological features via homology such as
where a homology class emerges and what happens to it along the filtration. After
using homology, we obtain persistence modules and by passing through their decom-
positions, we obtain a complete invariant called the barcode as a list of indecompos-
ables giving necessary information about lifespans of connected components, loops
and higher dimensional holes in 1-parameter case. Hence, by starting from low di-
mensional data, we can deduce higher dimensional results. In addition to a set of
discrete points, PH also works on data obtained by digital images, real valued func-
tions defined on a chosen topological space and so on.

We also compare persistence modules and measure "how far from being isomorphic"
by using stable metrics. We introduce the bottleneck and the interleaving distances,
and see that they are equal to each other in 1-parameter setting.

Although 1-parameter persistent homology has many beneficial sides, working on a

single parameter gives missing or unstable information for some type of data. For
instance, if we have the data with different density, then our ordinary parameter can-
not detect the differences caused by density. Therefore, adding a few new data points
behaving different than others, called as outliers in the literature, can make signifi-
cant changes in our results. That means ordinary persistence is very weak to outliers
in the data. To overcome this problem, the multiparameter persistent homology is
considered by Carlsson and Zomorodian [9] in 2009.

Shifting focus to the multiparameter setting brings new problems to us. The algebraic
structure of multiparameter persistence modules is much more complex, and there
exists no complete invariant for them, recall that for the 1-parameter case, barcode
is a complete invariant. This increases the importance of distances. We extend the
bottleneck and the interleaving distances to the multiparameter case. Moreover, by the
example of Bjerkevik [4], we see that they are no more equal to each other, and unlike
the interleaving distance, the bottleneck distance is not stable. Since computation of

2
the interleaving distance is so hard, this leads us to the matching distance as a stable
surrogate of the previously mentioned distances in the multiparameter setting [20].

This thesis is organized as follows:

In Chapter 2, we provide the necessary background for 1-parameter persistent homol-

ogy. We start Chapter 3 with the motivation of multiparameter persistent homology
and construct filtrations and persistence modules. Chapter 4 is all about pseudomet-
rics and their stability results in multiparameter setting. We give some examples to
understand how we can use them in practice. In this chapter, we also show that the
matching distance on different parametrizations of a line is always equal. Thus, re-
stricting the set of admissible lines to the suitable set having unique parametrization
for each line does not change dmatch .

3
4
 CHAPTER 2

ONE-PARAMETER PERSISTENT HOMOLOGY

2.1 Filtration

Definition 1. [27] A filtration indexed by {0, 1, . . . , n} is a nested sequence of

topological spaces:
K : K0 ⊆ K1 ⊆ · · · ⊆ Kn

where the number n is not necessarily finite.

Example 2.1.1. Let K be a compact topological space lying in some Euclidean space,
and h : K → R be a real valued function. For ai ∈ R, define the sublevel set Kai as

Kai := h−1 (−∞, ai ] = {x ∈ K | h(x) ≤ ai }

where i ∈ N. Clearly, Kai is a subspace of K for all ai ∈ R, and Kai ⊆ Kaj for
all ai < aj . Moreover, K = Kan for some an ∈ R since K is compact. Hence,
we construct a filtration of K indexed by {0, 1, . . . , n} by setting Ki := Kai for
i = 0, 1, . . . , n. In addition, it is called a sublevel set filtration.

Note here that a0 < a1 < . . . < an are the function values, and the filtering function h
is not necessarily continuous. One example of a filtering function, a height function,
is given in Figure 2.2.

Sometimes, to study complex geometric objects, it is more convenient to consider

them as discrete objects. We want to see them as the union of simple building blocks
glued together in a well defined set of rules. Our simple building blocks are called
simplices.

Definition 2. [5] Let x0 , . . . , xk ∈ Rn . An affine combination of these points is a

5
finite linear combination
k
X k
X
λi xi such that λi = 1,
i=0 i=0

where λ0 , . . . , λk ∈ R. We say that k + 1 points x0 , . . . , xk are affinely independent

if
k
X k
X
n
λi xi = 0 ∈ R and λi = 0,
i=0 i=0

then λi = 0 for all i ∈ {0, 1, . . . , k}. An affine combination is called as a convex

combination if λi is non-negative for all i ∈ {0, 1, . . . , k}. The set of all con-
vex combinations is said to be the convex hull of x0 , . . . , xk , and it is denoted by
conv{x0 , x1 , . . . , xk }.

A k-simplex σ is the convex hull of k + 1 affinely independent points. Note that

0-simplex is a vertex, 1-simplex is an edge, 2-simplex is a triangle and 3-simplex is a
tetrahedron. A face τ of a simplex σ is the convex hull of a non-empty subset of the
vertex set of σ, and it is denoted by τ ≤ σ.

Now, we put simplices together to form more complex shapes, called simplicial com-
plexes which are suitable for combinatorial purposes.

Definition 3. [5] A simplicial complex K in Rn is a finite collection of simplices

such that σ ∈ K and τ ≤ σ implies that τ ∈ K, and if σ0 , σ1 ∈ K, then σ0 ∩ σ1 is
either empty or a face of both σ0 and σ1 . A subcollection L of the simplicial complex
K is a subcomplex of K if it is a simplicial complex itself.

The above construction for the simplicial complex is geometric. In combinatorics, we

have a similar definition which is more abstract:

Definition 4. [27] An abstract simplicial complex A is a finite collection of sets

such that α ∈ A and β ⊆ α implies that β ∈ A. The sets in A are called simplices
and a subcomplex B of A is an abstract simplicial complex such that B ⊆ A.

Example 2.1.2. Let us see an example of a filtration obtained from a simplicial com-
plex K. Let h : K → R be a non-decreasing function, that is, for τ ≤ σ, we have
h(τ ) ≤ h(σ).

6
Consider the sublevel set Kai := h−1 (−∞, ai ]. Since h is non-decreasing, Kai is a
subcomplex of K for any ai ∈ R. Then, we have a filtration of K that is a nested
sequence of subcomplexes:

K0 ⊆ K1 ⊆ · · · ⊆ Kn = K

where Ki := Kai for some ai ∈ R.

Until now, we have seen examples in which a filtration ends up with a topological
space whose properties we want to understand. On the other hand, we may start with
a set of discrete points sitting in Rn . In this case, instead of attaining it as an ultimate
space, it becomes the initial subspace in the filtration. We can consider points in the
set as vertices, and by a proximity parameter which we will define, we can obtain a
filtration of simplicial complexes. There are many ways to obtain such a filtration,
and we introduce the Vietoris-Rips complex .

Definition 5. [17] Let X be a finite collection of points in Rn . The Vietoris-Rips

complex, denoted by Rips(X)ϵ , is an abstract simplicial complex whose k-simplices
are made of unordered (k + 1) distinct points such that all pairwise distances between
these points are less than or equal to ϵ.

From now on throughout this thesis, X denotes a finite set of points in some Euclidean
space unless otherwise stated.

Here, ϵ is called the proximity parameter. When ϵ = 0, we just have discrete set
of points, X = {xi }ni=1 . When ϵ > 0, we need to specify pairwise distances be-
tween points. If two points have a distance less than or equal to ϵ, then we add an
edge between them. After adding all possible edges, we obtain its 1-skeleton. Since
Vietoris-Rips complex is a flag complex, which means that it is maximal among all
abstract simplicial complexes with its 1-skeleton, it is enough to have 1-skeleton to
construct Rips(X)ϵ .

Clearly, as ϵ is increasing, the simplicial complex is getting bigger, so we can con-

struct a filtration by considering the corresponding Vietoris-Rips complexes for dif-
ferent values of the proximity parameter ϵ. Moreover, since we start with finitely
many points, for ϵ large enough, all the vertices in X get connected to each other. As

7
Figure 2.1: A filtration consisting of Vietoris-Rips complexes for some values of ϵ
between ϵ = 0 and ϵ = 2.1 where we start with discrete set of points, but for ϵ ≥ 2.1,
the complex turns into a solid disk, [25].

a result, the corresponding Vietoris-Rips complex becomes a solid disk eventually.

An example of a filtration using these complexes is given in Figure 2.1.

2.2 Persistent Homology

Thanks to the following filtration of topological spaces:

K : K0 ⊆ K1 ⊆ · · · ⊆ Kn ,

we can consider the corresponding sequence of homology groups to gain more infor-
mation. For i ≤ j, there exists an inclusion map Ki ,→ Kj which induces a homo-
morphism between homology groups: fpi,j : Hp (Ki ) → Hp (Kj ) for each dimension
p. Therefore, any filtration induces a sequence of homology groups connected by
homomorphisms induced by inclusions

Hp (K) : Hp (K0 ) → Hp (K1 ) → · · · → Hp (Kn ).

Remark 2.2.1. When we pass to induced maps, we actually apply homology functor
to inclusion maps with coefficients in Z. Moreover, all of Hp (Ki ) can also be seen
as Z-modules with scalar multiplication. In fact, if we take homology with field co-
efficients, then all the homology groups become vector spaces, and homomorphisms
between them are linear maps. We revisit these points when we define what a persis-
tence module is.

Now, let us give the formal definition of persistent homology groups:

8
Definition 6. [16] The p-th persistent homology groups Hpi,j (K) of K are the im-
ages of the homomorphisms induced by the inclusions Ki ,→ Kj ,

Hpi,j (K) := im{fpi,j : Hp (Ki ) → Hp (Kj )}

for 0 ≤ i ≤ j ≤ n. Note that Hpi,j (K) is not defined for i > j. Furthermore, the
p-th persistent Betti numbers are defined as the rank of the corresponding persistent
homology groups:
βpi,j := rank(Hpi,j (K)).

In fact, Hpi,j (K) measures the homology classes which persist or disappear through
the filtration levels. More specifically, we count the homology classes of Ki which
are still alive in Kj . Thus, Hpi,j (K) can also be defined as

Hpi,j (K) = Zp (Ki )/(Bp (Kj ) ∩ Zp (Ki )),

where Zp (Ki ) shows p-cycles in Ki , and Bp (Kj ) are p-boundaries in Kj .

/ Hpi−1,i (K).
Definition 7. [16] Let α ∈ Hp (Ki ). We say that α is born at Ki , if α ∈
Moreover, α dies at Kj , if fpi,j−1 (α) is a nontrivial element in Hp (Kj−1 ) while fpi,j (α)
is a trivial element in Hp (Kj ). Alternatively, we say that α dies at Kj if it merges with
an older class (a class that is born earlier) when we pass through from Kj−1 to Kj . In
other words, fpi,j−1 (α) ∈
/ Hpi−1,j−1 (K) but fpi,j (α) ∈ Hpi−1,j (K).

Whenever we say that two homology classes merge together at a particular level, we
mean that they become equal as homology group elements, and turns into a single
class. In such cases, we keep the class which is born earlier to keep better track of the
persistence of homology classes. If these two classes are born at the same time, then
we make a choice by keeping one of them. This rule is known as Elder Rule.

Definition 8. [16] Let α be born at Ki and die at Kj . If K is a sublevel set filtration,

then the difference in the corresponding function values, aj − ai , is called the persis-
tence of α, and it is denoted by pers(α) = aj −ai . In the case of a filtration consisting
of Vietoris-Rips complexes, it is taken to be ϵj − ϵi . We call the difference in index,
j − i, as the index persistence of the class. Furthermore, if α is born at Ki but never
dies, then α is called an essential class and its persistence and index persistence are
infinite. Even if we have a filtration consisting of finitely many spaces, we call the
classes that live in the homology group of the very last space as an essential class.

9
Definition 9. [16] Let µi,j
p be the number of p-dimensional classes that are born at

Ki and die at Kj . This number is called the multiplicity at (i, j).

Observe that we can express µi,j

p as a linear combination of the p-th persistent Betti

numbers:
µi,j i,j−1
p = (βp − βpi,j ) − (βpi−1,j−1 − βpi−1,j )

for all i < j and for all p. We can consider the meaning of βpi,j as the number of p-
dimensional classes born before or at Ki and die after Kj . So, in the equality above,
the left parenthesis gives the number of classes born before or at Ki and die exactly
at Kj while the right parenthesis gives the number of classes born before or at Ki−1
and die exactly at Kj .

2.3 Algebraic Background

We can compute persistent homology groups at each filtration level, and by studying
the lifespans of homology classes, we may draw some conclusions. Our aim is to
have a meaningful summary showing us the information coming from each filtration
level in a gathered form. However, as a first thing, we need to recall basic terminology
of the category theory. Let us define necessary parts:

Definition 10. [23] A partially ordered set (poset for short) is a set P with a binary
relation ≤ (the partial order) satisfying

i) x ≤ x (Reflexivity),

ii) x ≤ y and y ≤ x implies x = y (Anti-symmetry),

iii) x ≤ y and y ≤ z implies x ≤ z (Transivity),

for all x, y, z ∈ P . If x ≤ y or y ≤ x, then we say x and y are comparable. A

partially ordered set is called totally ordered if each pair of elements from the set
is comparable with each other. Furthermore, P op is the opposite poset of P , that is,
x ≤ y in P op if and only if y ≤ x in P .

10
Definition 11. [18, 23] A category C consists of:

i) A collection ObC of objects,

ii) A set Mor(X, Y ) of morphisms for each pair X, Y ∈ ObC and a distinguished
identity morphism idX ∈ Mor(X, X) for each X ∈ ObC,

iii) Compositions of morphisms, that is, for each f ∈ Mor(X, Y ) and g ∈ Mor(Y, Z),
we have g ◦ f ∈ Mor(X, Z), satisfying

f ◦ idX = f, idY ◦ f = f and (h ◦ g) ◦ f = h ◦ (g ◦ f )

for h ∈ Mor(Z, W ) where X, Y, Z, W ∈ ObC.

Remark 2.3.1. A poset (P, ≤) can be considered as a category with objects i, j ∈ P

and morphisms i ≤ j. Throughout this thesis, when we write the poset in bold, P , it
means the poset category of P .

Definition 12. [18, 23] Let C and D be categories. A functor F : C → D is a

mapping consisting of:

i) A choice of object F (X) ∈ ObD for each X ∈ ObC,

ii) A choice of morphism F (f ) ∈ Mor(F (X), F (Y )) for each f ∈ Mor(X, Y ),

satisfying F (g ◦ f ) = F (g) ◦ F (f ) and F (idX ) = idF (X) .

Remark 2.3.2. For notational purposes, later in the thesis, we might opt to denote
F (X) by FX .

Definition 13. [23] A natural transformation N : F → G between the two functors

F, G : C → D is a choice of morphism NX : FX → GX for each X ∈ ObC such
that the following diagram commutes:

F (f )
FX FY
NX NY

GX G(f )
GY

where f ∈ M or(X, Y ). N is called a natural isomorphism if each morphism NX is

an isomorphism.

11
We also mention the equivalence of two categories C and D:

Definition 14. [23] A functor F : C → D is said to be an equivalence between C

and D if there exists a functor G : D → C satisfying

G◦F ∼
= idC and F ◦ G ∼
= idD .

Next, we recall some algebraic definitions, but first let us fix some notation. Through-
out the rest of this chapter, R always denotes a commutative ring with unity, and
(T, ≤) denotes a totally ordered set with T ⊆ R.

Definition 15. [21] A ring ⟨R, +, ×⟩ is said to be a T -graded ring if it is written as

a direct sum,
R∼
M
= Ri ,
i∈T

such that the ring multiplication in R maps Ri × Rj to Ri+j . In particular, R0 is a

subring of R, and each Ri is a R0 -module for all i ∈ T . A graded ring R is non-
negatively graded if Ri = 0 for all i < 0.

Definition 16. [21] A T -graded module M over a graded ring R is a module which
is written as a direct sum,
M∼
M
= Mi ,
i∈T

so that the ring action is defined by Ri × Mj → Mi+j . Moreover, elements of Mi

are said to be homogeneous of degree i. This degree i is called grade. In particular,
each Mi is a R0 -module for all i ∈ T . A graded module M is non-negatively graded
if Mi = 0 for all i < 0.

Definition 17. [15, 21] An R-module M is said to be finitely generated if it has a

finite number of generators, that is, there exists a subset {m1 , . . . , mn } ⊆ M such
that for any m ∈ M ,
n
X
m= ri mi where ri ∈ R for all i.
i=1

The set {m1 , . . . , mn } is called a generating set of M , and it is minimal if its cardi-
X n
nality is minimal among its generating sets. The R-linear combinations ri mi is
i=1
n
X
called a relation if ri mi = 0.
i=1

12
Definition 18. [21] A free module M is an R-module which admits a basis, or the
zero module. Since a basis is linearly independent, M does not have any relation, and
if β is its basis, then
M∼
M
= R.
b∈β

Definition 19. [21] A sequence of R-module homomorphisms

fi−1 fi
... Mi−1 Mi Mi+1 ...

is called exact if ker(fi ) = im(fi−1 ) for all i.

Definition 20. [21] A presentation of an R-module M is an exact sequence

f g
F1 F0 M 0

where F0 and F1 are free R-modules. M is said to be finitely presented if F0 and F1

have finite bases.

Remark 2.3.3. Here, F0 is the free R-module generated by generators of M . Since

ker(g) = im(f ), and ker(g) corresponds to relations of M , im(f ) is the submodule
of relations. Furthermore, observe that coker(f ) = F0 / im(f ) ∼
= M.

Remark 2.3.4. Note that in the above definitions we can replace T which is a totally
ordered set with a poset.

2.4 Persistence Module

Definition 21. [28] A persistence module M indexed by T is a collection of R-

modules {Mi }i∈T together with homomorphisms, defined whenever i ≤ j,

φM
i,j : Mi → Mj

satisfying the conditions:

(i) φM
i,i = idMi

(ii) φM M M
i,k = φj,k ◦ φi,j for all i ≤ j ≤ k.

13
Observe that a persistence module M can be seen as a functor from T (the category
of T ) to the category of R-modules. Moreover, persistence modules form a cate-
gory whose morphisms are natural transformations between persistence modules. We
denote the category of T -indexed 1-parameter persistence modules by 1-mod.

Let us choose T as N for the time being and define a special type of persistence
modules:

Definition 22. [28] A persistence module M = {Mi , φM

i,j }i,j∈N is of finite type

if each Mi is a finitely generated R-module, and each φM

i,j is an isomorphism for

j ≥ i ≥ m for some m ∈ N.

Although we see the word module in the definition of persistence module, we have
not given a module structure yet. Let us put that structure on M = {Mi , φM
i,j }i,j∈N .

Consider the map φM

i,j as the shifting map denoted by

xj−i : Mi → Mj

for each i ≤ j ∈ N. Basically, xi corresponds to an i-shift upward in grading.

Moreover, we take R[x] with the standard grading, and we see a persistence module
as a graded module over R[x] thanks to the maps xi , i ∈ N. Formally, we define a
map α taking a persistence module, and sending it to a graded module over R[x]:
M
α(M ) = Mi ,
i∈N

such that R acts on Mi ’s individually, and the action of x is given by

x · (m0 , m1 , m2 , . . .) = (0, x(m0 ), x(m1 ), . . .)

= (0, φM M
0,1 (m0 ), φ1,2 (m1 ), . . .)

where mi ∈ Mi for all i ∈ N.

We define a category of graded modules by taking morphisms as the module homo-

morphisms f : M → N such that f (Mi ) ⊆ Ni for all i ∈ N. In fact, the map α gives
the following equivalence when we restrict persistence modules to be of finite type:

Theorem 2.4.1. [28] There exists an equivalence between the category of persistence
modules indexed by N of finite type over R and the category of finitely generated non-
negatively graded modules over R[x].

14
Thanks to Theorem 2.4.1, we can consider every persistence module indexed by N
of finite type over R with a module structure coming from a finitely generated non-
negatively graded module over the polynomial ring R[x]. In addition, a persistence
module is finitely generated if and only if it is of finite type.

Now, there is a question to answer. How can we decompose a finitely generated

graded module in a simple form? Here, the crucial point is the choice of R. For
instance, if we take R = Z, then we have a graded module over Z[x]. Finding a
decomposition of a Z[x]-module (or a graded module over Z[x]) is extremely hard.
Fortunately, we have a structure theorem coming from algebra which points out how
to take R:

Theorem 2.4.2. [28] Let D be a principal ideal domain. Then, every finitely gener-
ated D-module decomposes uniquely into cyclic D-modules in the following form:
M m 
β
D ⊕ D/di D ,
i=1

where di ∈ D such that di /di+1 and β ∈ Z. In a similar fashion, every finitely gen-
erated graded module M over a graded ring D which is a PID decomposes uniquely
into the following form:
n
M  m
M 
αi
x D ⊕ xγj D/dj D
i=1 j=1

where dj ∈ D are homogeneous such that dj /dj+1 and αi , γj ∈ Z. Moreover, xα

denotes an α-shift upward in grading. In both cases, the left and right parts of the
middle direct sum correspond to free portion and torsional portion, respectively.

Consider the set of polynomials given by k[x] where k is a field. It is an N-graded ring
with the usual polynomial addition and multiplication where Ri = {kxi | k ∈ k}. It
is also a principal ideal domain (PID) since k is a field. Therefore, if we take a finitely
generated graded module M over k[x], then by Theorem 2.4.2, M decomposes into
the form:
n
M m
 M 
M∼
= αi
x k[x] ⊕ xγj k[x]/(xβj ) . (2.1)
i=1 j=1

where βj ≥ 1. In other words, the reason behind working over polynomial ring
with field coefficients is to obtain a PID, and further a simple decomposition for such
persistence modules.

15
In Remark 2.2.1, we emphasized that if we apply homology functor to a filtration
with field coefficients, then homology groups Hp (Ki ) := Mi , i ∈ N become vector
spaces, and the maps fpi,j : Mi → Mj are linear for all i, j ∈ N. That is to say each
homology group becomes a k-module, and each linear map between them is a module
homomorphism. Therefore, we obtain a persistence module.

From now on, we always consider homology with field coefficients. Thus, a persis-
tence module means a collection of vector spaces and linear maps satisfying condi-
tions given in Definition 21. This in turn implies that a persistence module can be
seen as a functor from T to the category of vector spaces, Vect, over a fixed field k.
Furthermore, in the rest of the thesis, for ease of computation, the field k is chosen as
Z2 .

Let M1 and M2 be two persistence modules indexed by the same totally ordered set
T . We can define a new persistence module by considering their direct sum given as:

1 ⊕M2
(M1 ⊕ M2 )i = ((M1 )i , (M2 )i ) and φM
i,j = (φM M2
i,j , φi,j ).
1

We can further generalize this for a family of persistence modules {Mp | p ∈ J} with
the same index set T by considering their direct sums:
M M
M= Mp where Mi = (Mp )i .
p∈J p∈J

If the cardinality of J is infinite, then an element of Mi is a finite linear combination

of elements of (Mp )i , that is,
M
(Mp )i = {c1 m1 + . . . + cn mn | each cj ∈ k, mj ∈ (Mp )i for some i ∈ J}.
p∈J

We define the zero persistence module O as the persistence module such that Oi is
the zero vector space for all i ∈ T .

A persistence module M is said to be indecomposable if M ∼

= N1 ⊕ N2 implies
that N1 or N2 is the zero persistence module. M is called decomposable if it is not
indecomposable.

Definition 23. [8] An interval I in T is a non-empty subset of T satisfying the

condition: If i, j ∈ I and i ≤ k ≤ j, then k ∈ I.

16
Definition 24. [8] An interval module kI is a persistence module {Mi }i∈T satisfying
the conditions:

i) Mk = k for k ∈ I and Mk = 0 else.

ii) For k ≤ l, φk,l = idk where k, l ∈ I and φk,l is the zero map otherwise.

Theorem 2.4.3. [14] Let kI be an interval module. Then, it is indecomposable.

Proof. The set of morphisms, {ϕs }s∈T , from kI to itself is the endomorphism ring of
kI . Let us consider ϕs for some s ∈ I. It is a linear map from the field k to itself. So, it
sends x ∈ k to λx for some λ ∈ k. Since all diagrams are commutative, when we fix
ϕs as a multiplication by λ ∈ k, it implies each ϕt is also a multiplication by the same
λ for all t ∈ I. Thus, End(kI ) is isomorphic to k. Assume that kI is decomposable
such that kI ∼= N1 ⊕ N2 . Let π1 be the projection of kI to the first summand of its
decomposition. Clearly, π1 ∈ End(kI ) and π1 ◦ π1 = π1 . Thus, π1 is idempotent in k.
Since idempotents are just 0 and 1 in a field, π1 is the zero or identity map. If π1 = 0,
then it implies N1 = 0, and this gives a contradiction. If π1 = idkI , then it implies
N2 = 0 and this gives a contradiction. Therefore, kI is indecomposable.

Our first aim, while working with 1-parameter persistence modules, is to decompose
them as the sum of interval modules provided by Equation 2.1.

Theorem 2.4.4. [10] A persistence module indexed by N of finite type over k is

isomorphic to the direct sum of interval modules in the form:
n
M∼
M
= kIs
s=1

where Is = [is , js ) for is ∈ N, js ∈ N ∪ {∞}.

The index set in Theorem 2.4.4 can actually be extended to R with suitable finiteness
conditions (for more details, see [10, Proposition 3]).

Definition 25. [8, 11] Let M be a persistence module.

i) M is pointwise finite dimensional (p.f.d. for short) if dim(Mi ) < ∞ for all
i ∈ N.

17
 ii) M is said to be tame if for all i ∈ N, dim(Mi ) < ∞.

iii) If each linear map, φM

i,j for i < j, in the collection of M has finite rank, then M

is called q-tame.

Clearly, for a persistence module, being p.f.d. is equivalent to being tame. Further-
more, every p.f.d. persistence module is q-tame, but the converse is not true.
Remark 2.4.1. The relation between a p.f.d. persistence module and a persistence
module of finite type is the following: Every persistence module of finite type is
p.f.d. but the converse is not true.

We stated that finite type persistence modules over k indexed by N decompose into
interval modules. The following result of Botnan and Lesnick [8] generalizes the
result by extending the index set from N to T and persistence modules from finite type
to p.f.d. Thus, [8, Theorem 3.4] shows that 1-parameter p.f.d. persistence modules
are also interval decomposable:

Theorem 2.4.5. [8] Let M be a p.f.d. persistence module indexed by T . Then, M

decomposes into interval modules in the following way: There exists a unique multiset
B(M ) consisting of intervals such that

M∼
M
= kI .
I∈B(M )

The multiset B(M ) is one of our meaningful summaries which is called the barcode
of M .

We can observe that if we have a p.f.d. indecomposable persistence module, then it is

an interval module. Hence, being indecomposable and being an interval module are
equivalent for a p.f.d. persistence module. In addition, the following theorem reveals
that the multiset B(M ) is unique up to order.

Theorem 2.4.6. [12] (Krull–Remak–Schmidt–Azumaya) Suppose that a persistence

module M indexed by T can be decomposed into interval modules in two different
ways:
M∼ kIl ∼
M M
= = kIk .
Il ∈L Ik ∈K

Then, there is a bijection σ : L → K such that Il = Iσ(l) for all l.

18
The result of Theorem 2.4.5 is also obtained in the work of Dey and Wang [15]. They
stated that the result comes from a more general concept called quiver theory and
gives the result in the following form:

Proposition 2.4.7. [15] Any p.f.d. persistence module M indexed by T decomposes

uniquely into interval modules:

M∼
M
= k⟨bj ,dj ⟩
j∈J

where ⟨bj , dj ⟩ stands for one of the intervals: [bj , dj ], [bj , dj ), (bj , dj ], (bj , dj ).

Although, a q-tame persistence module is not necessarily p.f.d, we have a similar

result for q-tame persistence modules.

Proposition 2.4.8. [15] Any q-tame persistence module indexed by T decomposes

uniquely into interval modules.

Let us put together what we did so far. We start with a filtration indexed by N obtained
from the initial data. Then, we apply homology functor with field coefficients to this
filtration. So, we get a persistence module. If it is pointwise finite dimensional, then
we see that it is decomposed into interval modules. Since our indices comes from
N, all interval modules in the decomposition are of the form [i, j) which means a
homology class is born at i-th level, and dies at j-th level. We usually work on this
case, but the theory can be extended to the index set T ⊆ R.

From the beginning, understanding the shape of data is our main aim. This also
covers how we can make a distinction among them, and the answer is that we can
examine the corresponding persistence modules. Another question arises from here:
Is there any complete invariant for 1-parameter setting? First of all, an invariant
means a function from a set of structures, such as graded modules, to some set sending
isomorphic structures to the same value. Moreover, we say an invariant is complete
if it always assigns different values to non-isomorphic structures.

In our case, the structure set is the set of p.f.d. persistence modules and we send
them to their barcodes. With this construction, the barcode is a complete invariant.
Thus, if barcodes of two persistence modules are not equal to each other, then we can

19
Figure 2.2: The barcode and the 0th and 1st persistence diagrams of the topological
space M obtained by the height function, [13].

conclude that they are not isomorphic. Moreover, the computation of the barcode is
independent from the underlying field of a persistence module.

In Theorem 2.4.5, we defined the barcode of a persistence module. The other well-
known meaningful summary is the persistence diagram. Actually, a barcode and a
persistence diagram are very useful to visualize the persistent homology. If we work
on a p.f.d. persistence module indexed by N, then we can define the persistence
diagram of it as follows:

Definition 26. [16] The p-th persistence diagram is a multiset of points in the
extended real plane, R̄2 , and it is denoted as Dgmp (M ) for a persistence module M .
We draw a point at (i, j) if µi,j
p > 0. When the multiplicity increases, we make those

points slightly bigger. If a class is born at Ki and never dies, then we draw a point at
(i, ∞) where the second coordinate can be putted thanks to the extension of R2 . The
index persistence of a class is the vertical distance of the point to the diagonal.

Example 2.4.1. [13] Consider the surface M in R3 given by Figure 2.2. The filtration
function f : M → R is defined by a projection onto the value of z-coordinate. This
is an example of a height function, and we can construct the filtration by choosing
Ma = f −1 (−∞, a] for all a ∈ R.

The vertical red bars correspond to the lifespans of 0-dimensional homology classes.
The first homology class appears when Ma is non-empty for the first time for some

20
a ∈ R, and it never vanishes. On the other hand, after the second homology class
emerges, it merges with the former when the subspace becomes connected, and stops
at that level because of the Elder Rule.

We can also understand the behaviour of 1-dimensional homology classes. When a is

near enough to a1 , Fa is homeomorphic to two disjoint solid disks. Thus, we do not
have any 1-dimensional class. For a = ai where i ∈ {1, 2, 3}, there exists an 1-cycle,
σi , creating 1-dimensional homology class. Observe that σ3 is filled when a = a4 ,
so the homology class born at a3 dies at a4 . The other two cycles are never filled;
indeed, they span H1 (M ). This process is shown by the vertical blue bars.

After finding the barcode of M for 0 and 1-dimensional persistent homology, we

know birth and death places of each classes. Thus, we can obtain the corresponding
persistence diagrams, and show in a single diagram as in Figure 2.2 where the red
and blue points represent the 0th and 1st dimensional homology classes, respectively.
Moreover, the level of a for which M is fully obtained in the filtration gives the points
having infinite lifespan.

We can find βpi,j from persistence diagram by considering the upper, left quadrant
of the point (i, j). The quadrant is closed along its vertical right side since we can
consider classes die at Kk such that j < k, while the horizontal lower side is not
included since if some class dies at Kj , then it cannot be counted for βpi,j . The number
of points with their multiplicities is equal to βpi,j . Let us see this situation in the
following lemma:

Lemma 2.4.9. [16] Let K0 ⊆ K1 ⊆ · · · ⊆ Kn be a filtration. For 0 ≤ i ≤ j ≤ n

and every dimension p,
XX
βpi,j = µk,l
p .
k≤i j≤l

Remark 2.4.2. If we work on a p.f.d. persistence module indexed by arbitrary subset

T of R, then its decomposition can consist of the intervals of the form [bj , dj ], [bj , dj ),
(bj , dj ], (bj , dj ). When we pass to the persistence diagram, showing all of these four
different intervals with the same point (bj , dj ) ∈ R̄2 is one of the ways. However, this
implies that two non-isomorphic barcodes can have the same persistence diagram.
If we want to get rid of this situation, then we insert these intervals into persistence

21
diagrams by attaching different tick directions depending on the form of the interval.
For further details, see [12].

If we work on our classical way (i.e. p.f.d. persistence modules indexed by N), then
barcodes and persistence diagrams are equivalent to each other, hence the persistence
diagram is also a complete invariant on them.

We have seen how to obtain a persistence module by starting from a filtration. Then,
we have seen two meaningful summaries to visualize persistent homology: a barcode
and a persistence diagram. Now, what are the other ways to compare two persistence
modules? One of the answers is that we can define a distance between two persistence
modules, and try to draw some conclusions about whether they are isomorphic or not.
Furthermore, we can also classify non-isomorphic ones with respect to the distance.
In this part of the thesis, we discuss two well-known distances: the bottleneck and the
interleaving distances.

2.5 The Bottleneck Distance in 1-Parameter

We give the definition of the bottleneck distance by following [6]: Let us start with
two p.f.d. persistence modules M and N . Then, according to Theorem 2.4.5, we can
write both modules as sums of interval modules constructing the barcodes C and D
of M and N , respectively. Thanks to the relation between a barcode and a persis-
tence diagram, we can also see elements of the multisets C and D in their persistence
diagrams. Then, we define a matching X between C and D which is a bijection
′ ′ ′ ′
σ : C → D between C ⊆ C and D ⊆ D. Moreover, if we do not see an element
of C or D in the bijection σ, then the element is called unmatched, and we match this
element with its nearest point on the diagonal, {(x, x) | x ∈ R}, of the persistence
diagrams.
′ ′
Define the cost c(I, J) of a matching between I = ⟨a, b⟩ ∈ C and J = ⟨c, d⟩ ∈ D as

c(I, J) = max(|c − a|, |d − b|).

If I = ⟨a, b⟩ is unmatched, then define the cost of I as c(I) = (b − a)/2. We define

22
the cost of a matching X , c(X ), in the following way:
 
c(X ) := max sup c(I, J), sup c(I) .
(I,J)∈X unmatched I∈C∪D

We say that X is an ϵ-matching if c(X ) ≤ ϵ.

Definition 27. [8] The bottleneck distance between C and D is defined as

dB (C, D) := inf{c(X ) | X is a matching between C and D}.

Equivalently, we can define dB (C, D) as:

dB (C, D) = inf{ϵ ∈ [0, ∞] | there exists an ϵ-matching between C and D}.

We give the definition as the bottleneck distance between barcodes of M and N . It is

also equivalent to saying the bottleneck distance between M and N , i.e.,

dB (M, N ) := dB (C, D).

Remark 2.5.1. To find the bottleneck distance between two persistence modules, their
barcodes must exist, and we satisfy this condition by working on p.f.d. persistence
modules. However, this does not mean that we can find the bottleneck distance only
for p.f.d. ones. For instance, the definition can be extended to q-tame persistence
modules because of Proposition 2.4.8.

The importance of the bottleneck distance comes from the fact that it is stable under
small perturbations of the filtration function or the starting set of points. That is, the
perturbations cause smaller difference in terms of the bottleneck distance. Let us see
the stability of dB in detail:

Let X be a topological space with two filtration functions f, g : X → R. Then, define

the corresponding persistence modules M f and M g as

Mjf := Hi (f −1 (−∞, j]) Mjg := Hi (g −1 (−∞, j]).

where j ∈ R and i ∈ N is fixed. Assume that both of the persistence modules are
p.f.d. so that they decompose into interval modules. Therefore, we can pass through
the barcodes of them, namely B(M f ) and B(M g ) and see the stability feature:

23
Theorem 2.5.1. [6] The bottleneck distance satisfies the following:

dB (B(M f ), B(M g )) ≤ ||f − g||∞

where ||f − g||∞ := sup |f (x) − g(x)|.

x∈X

Let P and Q be two finite set of points in Rn having the same cardinality. Then,
we construct Vietoris-Rips complexes: Rips(P )ϵ , Rips(Q)ϵ with the same ϵ ∈ R
values. After applying homology functor, we obtain two persistence modules. With
the assumption making them pointwise finite dimensional, we have the barcodes:
B(Hi (Rips(P ))), B(Hi (Rips(Q))), where i ∈ N is fixed, and give the stability of
the bottleneck distance between them:

Theorem 2.5.2. [6] For any bijection σ : P → Q such that ||p − σ(p)|| ≤ ϵ for all
p ∈ P , we have

dB (B(Hi (Rips(P ))), B(Hi (Rips(Q)))) ≤ ϵ.

We find the bottleneck distance between two barcodes (or equivalently persistence
diagrams) of persistence modules. Instead, we can directly work on persistence mod-
ules, and look for a different distance structure.

2.6 The Interleaving Distance in 1-Parameter

We first introduce an ϵ-interleaving between two R-indexed persistence modules M

and N from a categorical point of view for ϵ ∈ [0, ∞).

Define a shift functor (−)(ϵ) : 1-mod → 1-mod by

M (ϵ)
M (ϵ)i = Mi+ϵ and φi,j = φM
i+ϵ,j+ϵ .

For a morphism α : M → N , we define α(ϵ) : M (ϵ) → N (ϵ) in the following way:

α(ϵ)i = αi+ϵ . The internal morphisms φM
i,i+ϵ : Mi → Mi+ϵ form a morphism from M

to M (ϵ).

Definition 28. [8, 23] We say that M and N are ϵ-interleaved if there exists mor-
phisms
ϕ : M → N (ϵ) ψ : N → M (ϵ)

24
such that the following diagrams commute for all i ≤ j:

φM
i,i+2ϵ
Mi Mi+2ϵ Mi+ϵ
ϕi ψi+ϵ ψi ϕi+ϵ

φN
i,i+2ϵ
Ni+ϵ Ni Ni+2ϵ

Alternatively, it can also be stated as follows: There exists an ϵ-interleaving between

M and N if there are two collections of linear maps

{ϕi : Mi → Ni+ϵ }i∈R , {ψi : Ni → Mi+ϵ }i∈R

making the triangle diagrams given above and the following parallelogram diagrams
commutative:

φM
i,j φM
i+ϵ,j+ϵ
Mi Mj Mi+ϵ Mj+ϵ
ϕi ϕj ψi ψj

φN
i+ϵ,j+ϵ φN
i,j
Ni+ϵ Nj+ϵ Ni Nj

Indeed, ϵ-interleaved relation measures how far two persistence modules from being
isomorphic. The next proposition is well-known in PH and easy to see.

Proposition 2.6.1. M and N are isomorphic persistence modules if and only if they
are 0-interleaved.

The following proposition is also well-known for which we provide our proof.

Proposition 2.6.2. If two persistence modules M and N are ϵ-interleaved, then they
are also δ-interleaved for all δ > ϵ.

′ ′
Proof. Define two linear maps ϕi : Mi → Ni+δ , ψi : Ni → Mi+δ as the following
compositions:

φM
i+ϵ,i+δ
Mi Mi+ϵ Mi+δ
′
ϕi ψi
ϕi ′
ψi
Ni+ϵ Ni+δ Ni
φN
i+ϵ,i+δ

25
With varying i ∈ R, we have two morphisms ϕ′ : M → N (δ), ψ ′ : N → M (δ).
We need to show triangle diagrams in the Definition 28 commutes. Observe that the
following diagram commutes:

Mi Mi+2ϵ Mi+ϵ+δ Mi+2δ

ϕi ψi+ϵ ψi+δ

Ni+ϵ Ni+δ

This implies that

Mi Mi+2δ
′ ′
ϕi ψi+δ

Ni+δ

commutes. In a similar way,

Mi+ϵ Mi+δ
ψi ϕi+ϵ ϕi+δ

Ni Ni+2ϵ Ni+ϵ+δ Ni+2δ

commutes. Thus,

Mi+δ
′ ′
ψi ϕi+δ

Ni Ni+2δ

also commutes. Thus, M and N are δ-interleaved.

Definition 29. [15] Let M and N be two R-indexed persistence modules. We define
their interleaving distance dI (M, N ) as

dI (M, N ) := inf{ϵ | M and N are ϵ-interleaved}.

If no such ϵ exists, then define dI (M, N ) = ∞. In this case, we also say they are
∞-interleaved.

We need to emphasize that being 0-interleaved and having interleaving distance as 0

are not the same thing. If the interleaving distance of two persistence modules, M

26
and N , is 0, then we cannot guarantee that they are 0-interleaved. So, dI (M, N ) = 0
does not imply M and N are isomorphic in all cases.
Example 2.6.1. Let M and N be two persistence modules such that M0 = Z2 , Mi = 0
for all i ∈ R \ {0} and Ni = 0 for all i ∈ R. Then, they are not isomorphic, but
dI (M, N ) = 0. As a second example, consider interval modules k(3,5) and k[3,5) .
Although the interleaving distance between them is 0, they are not isomorphic.

Definition 30. [22] Metrics and Pseudometrics

An extended pseudometric on some set X is a function d : X ×X → [0, ∞] satisfying

the followings:

i) d(x, x) = 0 for all x ∈ X.

ii) d(x, y) = d(y, x) for all x, y ∈ X.

iii) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X where d(x, y), d(y, z) < ∞.

An extended metric is an extended pseudometric with an additional property: If

d(x, y) = 0, then x = y. Moreover, the word "extended" is added since distance
can be ∞.

Our last example implies that the interleaving distance is an extended pseudometric on
persistence modules indexed by R. On the other hand, if we work on such modules
indexed by N, and exclude classes born and died at the same time, then dI is an
extended metric.

Throughout this thesis, the word "distance" always means an extended pseudometric.

The following proposition shows how the interleaving distance changes when we take
direct sums of persistence modules.

Proposition 2.6.3. [12] Let M1 , M2 , N1 and N2 be persistence modules. Then,

dI (M1 ⊕ M2 , N1 ⊕ N2 ) ≤ max(dI (M1 , N1 ), dI (M2 , N2 )).

Moreover, for two families of persistence modules {Mp | p ∈ J} and {Np | p ∈ J}

where each persistence module is indexed by R, let us take
M M
M= Mp and N = Np ,
p∈J p∈J

27
then we have
dI (M, N ) ≤ sup{dI (Mp , Np ) | p ∈ J}.

The interleaving distance and the bottleneck distance have different constructions,
and we want to find a relation between them: Are they equal to each other? If not, can
we say that one of them is always bigger than the other? These are natural questions,
and to answer these questions, we need more.

Proposition 2.6.4. [12] Let I = ⟨a, b⟩, J = ⟨c, d⟩ be two intervals and kI , kJ be the
corresponding interval modules. Then, we have

dI (kI , kJ ) ≤ c(I, J) = max(|c − a|, |d − b|)

where the right hand side of the inequality comes from a matching between I and J
in the process to find the bottleneck distance. We can also consider I is unmatched as
another case, i.e., it is matched with its nearest point on the diagonal. Thus, we have
the following analogous equality:
1
dI (kI , O) = c(I) = (b − a)
2
where O is the zero persistence module.

Now, we are ready to give the main result of 1-parameter persistent homology.

Theorem 2.6.5. (The Isometry Theorem) [22, Theorem 3.4] Let M and N be two
p.f.d. persistence modules. Then,

dI (M, N ) = dB (B(M ), B(N )).

In the literature, the inequality dB (B(M ), B(N )) ≤ dI (M, N ) is called the algebraic
stability theorem, and the converse inequality dI (M, N ) ≤ dB (B(M ), B(N )) is said
to be the converse algebraic stability theorem.

Proof. [12] Let us show that dI (M, N ) ≤ dB (B(M ), B(N )). Since M and N are
pointwise finite dimensional,
M M
M= kIi and N = kJj ,
i∈L1 j∈L2

where Ii ∈ B(M ) and Jj ∈ B(N ). Let X be a matching between the barcodes B(M )
and B(N ) where its cost equals to ϵ. We have three cases:

28
 i) If Ii and Jj are matched, then from Theorem 2.6.4, dI (kIi , kJj ) ≤ c(Ii , Jj ) ≤ ϵ.

ii) If Ii is unmatched, then from Theorem 2.6.4, dI (kIi , O) = c(Ii ) ≤ ϵ. Moreover,

for each unmatched interval module, add the zero persistence module O to the
direct sum decomposition of N by adding a new index to L2 .

iii) If Jj is unmatched, then from Theorem 2.6.4, dI (O, kJj ) = c(Jj ) ≤ ϵ. More-
over, for each unmatched interval module, add O to the direct sum decomposi-
tion of M by adding a new index to L1 .

Therefore, index sets in the direct sums become same in terms of cardinality, call it
L. So, after reindexing interval modules with respect to matched ones, we have
M M
M= kIi and N = kJi
i∈L i∈L

where dI (kIi , kJi ) ≤ ϵ for all i ∈ L. Thus, by Theorem 2.6.3, dI (M, N ) ≤ ϵ. Since
the bottleneck distance is the infimum of ϵ values, dI (M, N ) ≤ dB (B(M ), B(N )).
Another proof for the converse algebraic stability theorem is done by Lesnick [22].
The proof of the algebraic stability theorem is given by Chazal, Cohen-Steiner, Glisse,
Guibas and Oudot [12] (also see the work of Bauer and Lesnick [3]).

We have seen that the bottleneck distance is stable, and the interleaving distance is an
extended pseudometric. Since these distances are equal to each other for p.f.d. per-
sistence modules, we conclude that they are both stable and extended pseudometrics.

29
30
 CHAPTER 3

MULTIPARAMETER PERSISTENT HOMOLOGY

There is an obvious question to answer: Why do we need to construct multiparameter

persistent homology? Why 1-parameter construction is not enough for us? One of the
main reasons is that 1-parameter persistence is not robust to outliers. Consider finite
set of points in Rn . If we add some points as noise, then the persistent homology of
the new set of points can be significantly different from the persistent homology of
the former.

Let us see this situation in an example, see Figure 3.1, given by Botnan and Lesnick [8].
As demonstrated, by adding finitely many outliers, the barcode changes a lot. Before
adding outliers, we can easily detect the circular shape of the data from the long bar
by applying the first homology functor. However, after adding them, we cannot see
such a long bar in the barcode, so we cannot determine the circular shape of the data
although it still has that shape. To handle outliers in our data, we need to pass through
multiparameter persistent homology by introducing new parameters such as the den-
sity parameter, that is, a function taking greater value in dense regions of the data than
sparse regions of it. With the help of new parameters, we understand the shape of the
data with higher density, and we can get rid of the complexity coming from outliers.

In many applications of persistent homology, one parameter is not enough to make

full analysis. To extend definitons of filtration and persistence module, let us give the
necessary background:

Let P1 , P2 , . . . , Pn be posets. Define the product poset P1 × P2 × . . . × Pn as

(x1 , x2 , . . . , xn ) ⪯ (y1 , y2 , . . . , yn ) if and only if xi ≤ yi for all i ∈ {1, 2, . . . , n}.

Definition 31. [8] An interval I in a poset P is a non-empty subset of P satisfying

31
Figure 3.1: (a) Set of points in Rn colorized by a local density estimate. (b) Adding
outliers with low density. (c) The barcode of (a) in 1-dimensional holes. (d) The
barcode of (b) in 1-dimensional holes, [8].

the followings:

i) If i, j ∈ I and i ≤ k ≤ j, then k ∈ I.

ii) If i, j ∈ I, then there are i = k0 , k1 , . . . , kn = j ∈ I such that ks and ks+1 are

comparable for all s ∈ {0, 1, . . . , n}.

Example 3.0.1. Consider three subsets shown in Figure 3.2:

L d
b
c
a e

I1 I2 I3

Figure 3.2: Subsets of R2 given in the plane with specified points and the line L.

32
 i) I1 is not an interval in R2 since a, b ∈ I1 and a ≤ c ≤ b, but c ∈
/ I1 . We also see
this situation by the restriction of L to I1 . It has two disconnected components
implying our result.
ii) I2 is an interval.
iii) I3 is not an interval because although d, e ∈ I3 , there exists no elements {ki }ni=0
of I3 making second condition satisfied.

As in 1-parameter, intervals are one of our main interests. However, intervals are more
complicated in multiparameter case and to obtain particular results, we sometimes
consider their subsets such as rectangles:

Definition 32. [8] A rectangle in T1 ×. . .×Tn is an interval of the form I1 ×. . .×In ,

where Ii is an interval in totally ordered set Ti for all i ∈ {1, . . . , n}.

3.1 Multiparameter Filtration and Its Examples

Definition 33. [8] Let Top be the category of topological spaces and P be a poset.
Then, a P -indexed filtration is a functor F : P → Top such that Fx ⊆ Fy for all
x ≤ y ∈ P where Fx := F (x). If we take P = T1 × T2 × . . . × Tn where each Ti
is a totally ordered set for all i ∈ {0, 1, . . . , n}, then we call F a multiparameter or
n-parameter filtration.

In fact, Definition 33 is a generalization of a 1-parameter filtration (Definition 1).

Example 3.1.1. For a topological space K, we consider the sublevel set filtration
S(f ) : P → Top via a not necessarily continuous function f : K → P defined as
S(f )p := {k ∈ K | f (k) ≤ p}. For instance, the diagram of the 2-parameter sublevel
set filtration S(f ) : N × N → Top can be seen as follows:

.. .. ..
. . .

F0,2 F1,2 F2,2 ...

F0,1 F1,1 F2,1 ...

F0,0 F1,0 F2,0 ...

33
Figure 3.3: Degree-Rips bifiltration where degree parameter is 4, 3, 1 from left to
right, [8].

It’s time to see a few constructions for bifiltrations where number of parameters is 2,
and (X, dX ) is a finite metric space.

Example 3.1.2. [8] Let us give the degree-Rips bilfiltration, DRips(X). We have
two parameters: ϵ is a scale paramater, and d is a degree parameter. DRips(X)d,ϵ is a
maximal subcomplex of Rips(X)ϵ only including vertices have degree at least d − 1
in the 1-skeleton of Rips(X)ϵ . For an illustration, see Figure 3.3.

Example 3.1.3. [8,23] Let f : X → R be a function. The sublevel-Rips bilfiltration

obtained from f is the function Ripssub (f ) : R × [0, ∞) → Simp defined as

Ripssub (f )a,ϵ := Rips(f −1 (−∞, a])ϵ ,

where a ∈ R is the parameter changing the data set of points with the inverse image
of f , and ϵ ∈ [0, ∞) is the scale parameter of the Vietoris-Rips complex. In addition,
Simp shows the category of simplicial complexes.

Similarly, the superlevel-Rips bifiltration obtained from f is the function Ripssuper (f ) :

Rop × [0, ∞) → Simp defined as

Ripssuper (f )a,ϵ := Rips(f −1 [a, ∞))ϵ .

34
The crucial part of the above filtrations is the choice of f . For instance, we can choose
f as fr : X → R defined by

fr (x) = |{y ∈ X | dX (x, y) ≤ r}|,

where r > 0 is fixed. Moreover, fr is a density function, and in general, the superlevel-
Rips bifiltration obtained from a density function is called a density-Rips bifiltration.
Remark 3.1.1. Choosing a different r > 0 in the construction of fr can change func-
tion values. This dependency on r can lead to deficiencies in the usage of the density-
Rips bifiltration. On the other hand, the degree-Rips bifiltration does not have such
a dependency. In addition, we can define the degree-Rips bifiltration by using fr ,
but this time r is not fixed, and it equals to the scale parameter, ϵ, of Vietoris-Rips
complex.

For another example, define f : X → R+ ∪ {0} by

1 X
f (x) = dX (x, y),
|X| y∈X

where |X| is the number of the elements of X. Thus, f gives the average distance
of x to the points of X. It is said to be an eccentricity function, and we call the
superlevel-Rips bifiltration obtained from f as the eccentricity-Rips bifiltration.

Instead of using sublevel or superlevel filtrations, we can take the inverse image of
the function on finite intervals. By this way, we can define a trifiltration. To do that,
start with the set I = {(a, b) | a ≤ b} where a, b ∈ R. So, I can be seen as a poset
with the poset relation induced by Rop × R.
Example 3.1.4. [8] Let f : X → R be a function. The interval-Rips trifiltration
Rips(f )(a,b),ϵ : I × [0, ∞) → Simp is defined as

Rips(f )(a,b),ϵ := Rips(f −1 [a, b])ϵ .

Example 3.1.5. [23] We have another bifiltration method called the multi-cover bi-
filtration defined as follows: Let X ⊆ Rn be finite, and Cov(X) : Nop × R → Top
be

Cov(X)k,ϵ = {y ∈ Rn | ∃k points {xi }ki=1 ∈ X such that ||y − xi || ≤ ϵ, ∀i ∈ T },

where T = {1, . . . , k}.

35
 Figure 3.4: Multi-cover bifiltration for the fixed values of k = 2, 3, [23].

Before our last example for a multiparameter filtration, let us define the barycentric
subdivision of a simplicial complex K. Then its barycentric subdivision, denoted
by bary(K) is an abstract simplicial complex whose (k − 1)-simplices are of the
form {σ1 , σ2 , . . . , σk } where each σi is a simplex of K such that σi ⊂ σi+1 for all
i ∈ {1, 2, . . . , k − 1}. Observe that we are taking simplices of K as vertices of
bary(K).

Example 3.1.6. [8] For m ∈ N+ , define S(K)m as the maximal subcomplex of

bary(K) such that vertices of S(K)m are only the simplices of K with dimension at
least m − 1. With varying m, corresponding subcomplexes give a (N+ )op filtration in
1-variable.

To construct the subdivison-Rips bifiltration, start with X. Consider the Vietoris-Rips

complexes with scale parameter ϵ. For each Rips(X)ϵ , pass through its barycen-
tric subdivision. Then, include the second parameter m assigning subcomplexes of
bary(Rips(X)ϵ ). So, we obtain the subdivision-Rips bifiltration given as

bary(X)ϵ,m : [0, ∞) × (N+ )op → Simp.

3.2 Multiparameter Persistence Module

Definition 34. [8] Let Vect be the category of k-vector spaces where morphisms
correspond to linear maps. For (P, ≤) a poset, a P -indexed persistence module is a
functor M : P → Vect. Define Mi := M (i). For i ≤ j, φM
i,j denotes the linear map

Mi → Mj . If we take P = T1 × T2 × . . . × Tn where each Ti is a totally ordered

36
set for all i ∈ {0, 1, . . . , n}, then we call M a multiparameter or an n-parameter
persistence module. Furthermore, they form a category shown by n-mod where
morphisms are taken as natural transformations.

In 1-parameter case, we defined persistence module as a collection of R-modules

where R is a ring. Then, we took R = k to obtain results, and we had k-vector spaces
in the collection of a persistence module like in this situation. In fact, if we take n = 1
in Definition 34, then it becomes the same definition in 1-parameter case in which we
choose R = k.

Example 3.2.1. If we apply homology functor to the filtration obtained in the Example
3.1.1, then we obtain the persistence module M : N × N → Vect with the following
commutative diagram:

.. .. ..
. . .

M0,2 M1,2 M2,2 ...

M0,1 M1,1 M2,1 ...

M0,0 M1,0 M2,0 ...

So, the pipeline of multiparameter persistent homology is the same as 1-parameter

case. After starting with a multiparameter filtration, we apply homology functor and
we obtain corresponding multiparameter persistence module.

We directly extend definitions of graded rings and modules introduced in 15 and 16

to n-parameter setting by taking T = T1 × T2 × . . . × Tn where each Ti is a totally
ordered set.

For the present, we take T = Nn or Zn . Linear maps of a persistence module M ,

namely φM
i,j , i ≤ j ∈ T , can be seen as a map

xj−i = xj11 −i1 xj22 −i2 . . . xjnn −in : Mi → Mj .

37
These maps give the structure of a graded module:

M∼
M
= Mi over k[x1 , . . . , xn ].
i∈T

Conversely, starting with a graded module over k[x1 , . . . , xn ] gives us an n-parameter

persistence module. In fact, we have equivalence between their categories, and re-
stricting persistence modules to finite type (with its obvious extension to the multipa-
rameter case) yields the following:

Theorem 3.2.1. [9] The category of persistence modules over k of finite type indexed
by T and the category of finitely generated T -graded modules over k[x1 , . . . , xn ] are
equivalent categories.

In 1-parameter persistent homology, we have seen that if we take a finitely generated

graded module over k[x], then it decomposes uniquely by the Structure Theorem
given in 2.4.2. What allows us to use it is that k[x] is a PID. However, in n-parameter
case, we take such a module over k[x1 , . . . , xn ], and since k[x1 , . . . , xn ], n ≥ 2, is no
more a PID, we cannot use the Structure Theorem. To see why k[x1 , . . . , xn ], n ≥ 2,
is not a PID, consider the ideal generated by x1 and x2 : we cannot write it by a single
generator.

Let T = Rn , and P n be a ring analogue of k[x1 , . . . , xn ] such that powers of inde-

terminates come from the set [0, ∞). For i = (i1 , . . . , in ) ∈ [0, ∞)n , xi denotes
xi11 . . . xinn ∈ P n . Then, P n is an Rn -graded ring, and there exists an equivalence be-
tween categories of n-parameter persistence modules indexed by Rn and Rn -graded
modules over P n , [22].

The next step is to decide how to decompose our persistence module. We cannot
use the Structure Theorem due to the reason we have mentioned above. On the other
hand, we can still decompose p.f.d. persistence modules:

Theorem 3.2.2. [8, Theorem 4.2] For a poset P and a p.f.d. n-parameter persistence
module M : P → Vect,

(i) M is written as a direct sum of indecomposables {M λ }λ∈Λ so that

M∼
M
= M λ.
λ∈Λ

38
 (ii) Suppose that M is separated into indecomposables in two different ways:

M∼ Mλ ∼
M M
= = Mγ.
λ∈Λ γ∈Γ

Then, there is a bijection σ : Λ → Γ such that M λ ∼

= M σ(λ) for all λ ∈ Λ.

Remark 3.2.1. The proof of Theorem 3.2.2 has huge background, but Botnan and
Crawley-Boevey proved the first part of the theorem in their article [7] in the recent
past. The second part is coming from the Azumaya–Krull–Remak–Schmidt Theorem,
[1].

In n-parameter case where n ≥ 2, the main problem is that we do not know the
exact list of isomorphism classes of indecomposables, and this makes everything very
complex. In 1-parameter case, those indecomposables are just interval modules and
the set of intervals is called the barcode.

Moreover, the related question is: If we define the barcode on multiparameter p.f.d.
persistence modules as a set of indecomposable components of them obtained by
Theorem 3.2.2 (i), does it fulfill all the properties satisfying by the barcode of a 1-
parameter persistence module?

Definition 35. [8] Let B be a multiset of subsets of P . Then, we say B is a good

barcode of the persistence module M if

rank(Mi → Mj ) = |{S ∈ B | i, j ∈ S}|

for all i, j ∈ P .

Observe that every barcode coming from p.f.d. 1-parameter persistence modules is
automatically a good barcode. However, if we start with a p.f.d. n-parameter persis-
tence module M , then there is not always a good barcode of M . Let us see this by
the following example.

Example 3.2.2. [8] Start with the bifiltration shown in Figure 3.5. If we apply
H1 (−; Z2 ), then the corresponding persistence module M can be seen by the follow-
ing diagram:

39
 Figure 3.5: Bifiltration indexed by P = {0, 1, 2} × {0, 1, 2}, [8].

id
k k 0

id (1 0)

(1 0)T (1 1)
k k2 k

(0 1)T id

id
0 k k

Clearly, this persistence module is p.f.d. and let us show that it cannot have a good
barcode.

Say B is a good barcode of it. Observe the followings:

i) rank (M(0,1) → M(2,1) ) = 1, so there is I ∈ B such that (0, 1), (2, 1) ∈ I.

ii) rank (M(0,1) → M(1,2) ) = 1, so there is J ∈ B such that (0, 1), (1, 2) ∈ J.

iii) rank (M(1,0) → M(2,1) ) = 1, so there is K ∈ B such that (1, 0), (2, 1) ∈ K.

Since dim(M(0,1) ) = 1, we have I = J, and since dim(M(2,1) ) = 1, this makes

I = K. Thus, I = J = K. This implies that (1, 0), (1, 2) ∈ I. However,

rank(M(1,0) → M(1,2) ) = 0.

So, this gives a contradiction. Therefore, M does not have a good barcode.

Hence, extending the barcode to p.f.d. multiparameter persistence modules by keep-

ing all of its properties is not the case. Furthermore, as we mentioned earlier, working

40
on indecomposables without knowing what they are exactly does not allow us to go
further. Thus, unlike 1-parameter case, the barcode is not a complete invariant among
multiparameter p.f.d. persistence modules, for details see [9].

On the other hand, we can restrict ourselves to interval-decomposable persistence

modules to know the list of indecomposables.

In multiparameter setting, the definition of an interval module is same as in 1-parameter

case given in Definition 24. Moreover, interval modules are indecomposable in mul-
tiparameter persistence, and a proof of this is the direct extension of the proof given
for Theorem 2.4.3.

Definition 36. [8] A persistence module M indexed by P is said to be interval

decomposable if there exists a multiset B(M ) of intervals in P such that

M∼
M
= kI .
I∈B(M )

Here, we define B(M ) as the barcode of M .

Definition 37. [4] Let M be an interval decomposable persistence module indexed

by T = T1 × . . . × Tn where each Ti is totally ordered set for all i ∈ {1, . . . , n}. If
all I ∈ B(M ) are rectangles in T , then we call M rectangle decomposable.

By definitons, rectangle decomposable modules forms a subset of interval decompos-

able modules. Sometimes, we restrict persistence modules into one of these two types
of modules.

Now, let us see an important kind of persistence modules. For P be a poset and
a ∈ P , Qa is a persistence module given by

k if a ≤ x,

a
Qx =
0 otherwise.



idk if a ≤ x,

Qax,y =
0

otherwise.

Example 3.2.3. Consider Q1,0 where all maps between k’s are identity and P = N2 .

41
 .. .. ..
. . .

0 k k ...

0 k k ...

0 k k ...

Definition 38. [22] A persistence module F is called free if there exists a multiset A
of elements of P such that
F ∼
M
= Qa .
a∈A

Definition 39. [23] A presentation of a persistence module M is a morphism f :

F1 → F0 of free persistence modules such that F0 / im(f ) ∼
= M . If F1 and F0 are
finitely generated, then M is called finitely presented.

From [23, Proposition 6.40], we can see that F0 and F1 are generated by the set of
generators, and the set of elements generating all relations of M , respectively. Thus,
it is enough to show these sets have finite cardinality to call a persistence module
finitely presented.

42
 CHAPTER 4

METRICS AND STABILITY

We have seen that the barcode does not have an extension to p.f.d. multiparameter
persistence modules. Instead, Carlsson and Zomorodian proposed the rank invariant
for multiparameter setting [9].

Definition 40. [9] The rank invariant of M denoted as ρM is defined by

ρM (i, j) := rank (φM

i,j )

for every i ≤ j ∈ Rn .

In the literature, the rank invariant is also called persistent Betti numbers, which are
already defined for the 1-paramater case in the Definition 6.

The rank invariant draws the attention because it is easy to compute and equivalent
to the barcode in 1-parameter case implying that the rank invariant is complete in
1-parameter setting. However, its extension to multiparameter case is no more a com-
plete invariant, and this can be seen by the following example:

Example 4.0.1. [24] Consider the persistence modules:

.. .. .. ..
. . . .
id id2 id id2
(1 0)T id 2 (1 0)T id2
k k2 ... k k2 ...
(0 1)T (1 0)T
id id
0 k ... 0 k ...
M N

43
While M is the free module with two generators at (1, 0) and (0, 1), that is,

M∼
= Q1,0 ⊕ Q0,1 ,

N has another generator at (1, 1), and the homology classes of N at (1, 0) and (0, 1)
generates a single class at (1, 1) as a result of the relation ⟨(1, 0) − (0, 1)⟩.

Observe that ranks of the following maps

M(1,0) ⊕ M(0,1) → M(1,1) N(1,0) ⊕ N(0,1) → N(1,1)

(x, y) → (x, y) (x, y) → (x, 0)

are 2 and 1, respectively. Therefore, M and N are not isomorphic to each other. Alter-
natively, if we write M and N as a direct sum of indecomposables, then we can see
that there is no bijection giving isomorphisms between corresponding components.
Hence, they are not isomorphic by Theorem 3.2.2. On the other hand, ranks of all
linear maps, φM N
i,j and φi,j for i ≤ j, are equal to each other. This implies that although

M and N are not isomorphic, they have the same rank invariant. Therefore, the rank
invariant is not complete in multiparameter persistent homology.

All in all, prominent candidates, namely the barcode and the rank invariant, are not
complete. Furthermore, there is no known complete invariant for multiparameter
persistent homology. Therefore, the duty of understanding whether two persistence
modules are isomorphic or not belongs to distances. We extend the familiar ones to
n-parameter case, and define another one: the matching distance.

4.1 The Interleaving Distance

In this subsection, we extend the definition of the interleaving distance to the multipa-
rameter persistent homology. However, we first give the definition of an ϵ-interleaving
n
between two functors F, G : Rn → C where C is an arbitrary category. Let C R de-
note the category of functors from Rn to C.

For ϵ ∈ [0, ∞), we define the shift functor as

n n
(−)(⃗ϵ) : C R → C R

44
by
F (⃗ϵ)
F (⃗ϵ)i = Fi+⃗ϵ and φi,j = φFi+⃗ϵ,j+⃗ϵ

where ⃗ϵ = (ϵ, . . . , ϵ) ∈ Rn , and φFi,j : Fi → Fj denotes the corresponding morphism

of i ≤ j in C for the functor F : Rn → C.

For a natural transformation N : F → G, we define N (⃗ϵ) : F (⃗ϵ) → G(⃗ϵ) in the

following way: N (⃗ϵ)i = Ni+⃗ϵ. Moreover, the internal morphisms φFi,i+⃗ϵ : Fi → Fi+⃗ϵ
form a natural transformation from F to F (⃗ϵ).

Definition 41. [8] For ϵ ∈ [0, ∞), two functors F, G : Rn → C are said to be
ϵ-interleaved if there exist morphisms

ϕ : F → G(⃗ϵ) ψ : F → G(⃗ϵ)

satisfying

φF
i,i+2⃗
ϵ
Fi Fi+2⃗ϵ Fi+⃗ϵ
ϕi ψi+⃗ϵ ψi ϕi+⃗ϵ

φG
i,i+2⃗
ϵ
Gi+⃗ϵ Gi Gi+2⃗ϵ
Definition 42. [8] Let F and G be two functors from Rn to C. We define their
interleaving distance dI (F, G) as

dI (F, G) := inf{ϵ | F and G are ϵ-interleaved}.

If no such ϵ exists, then we set dI (F, G) = ∞.

In the above construction, if we specify C = Top, then functors become Rn -indexed

filtrations, and we can obtain dI between such filtrations. Similarly, taking C as Vect
causes functors to be Rn -indexed persistence modules and enables us to extend dI to
multiparameter setting.

Like in 1-parameter case, two multiparameter persistence modules are said to be iso-
morphic persistence modules if and only if they are 0-interleaved. Moreover, if two
n-parameter persistence modules are ϵ-interleaved, then they are also δ-interleaved
for all δ > ϵ. The proof of it is just the extension of the proof of Proposition 2.6.2.

Recall that, for 1-parameter case, we state that both the bottleneck distance and the
interleaving distance are stable.

45
For multiparameter case, let us first give a general definition of what we mean by a
stable distance: Let X be any topological space with two functions f, g : X → Rn
defined on it. We can obtain persistence modules M f and M g via maps f and g as
homology of sublevel set filtrations S(f ) and S(g) respectively, that is,

Mjf := Hi (S(f )j ) Mjg := Hi (S(g)j )

where j ∈ Rn and i ∈ N is fixed. In addition, consider Rn with the max norm given
as
||r||∞ = max |ri |
i

where r = (r1 , . . . , rn ) ∈ Rn . After these constructions, we are ready to define the

notion of a stable distance:

Definition 43. [22] Let d be a distance (i.e., extended pseudometric) on multipa-

rameter persistence modules. We say that d is i-stable if it satisfies the following
inequality:
d(M f , M g ) ≤ sup∥f (x) − g(x)∥∞ ,
x∈X
f g
where M and M are defined as above. Moreover, d is called stable if it is i-stable
for all i ≥ 0.

Theorem 4.1.1. [22] The interleaving distance dI is stable.

Proof. Let sup∥f (x) − g(x)∥∞ = ϵ. Then, we have S(f )j ⊆ S(g)j+⃗ϵ and S(g)j ⊆
x∈X
S(f )j+⃗ϵ. These inclusion maps give an ϵ-interleaving between S(f ) and S(g). Ap-
plying homology functor to these filtrations brings out an ϵ-interleaving between
Hi (S(f )) and Hi (S(g)) which follows from the naturality of the homology functor.
Therefore, M f and M g are ϵ-interleaved. Hence,

dI (M f , M g ) ≤ ϵ = sup∥f (x) − g(x)∥∞ .

x∈X

Thus, small perturbations of functions lead to small changes in the corresponding

persistence modules in terms of the interleaving distance. In fact, we also look for a
stable pseudometric to be as sensitive as perturbations of functions. This quest reveals
the universality property:

46
Definition 44. [22] For i ≥ 0, a pseudometric d on n-mod is called i-universal if
it is i-stable and for any i-stable pseudometric d′ on n-mod, we have d′ (M, N ) ≤
d(M, N ) for all n-parameter persistence modules M and N obtained from i-th ho-
mology of sublevel set filtrations.

Moreover, d is called universal if it is stable and for any stable pseudometric d′ on

n-mod, we have d′ (M, N ) ≤ d(M, N ) for all n-parameter persistence modules M
and N obtained from i-th homology of sublevel set filtrations for some i ≥ 0.

Now, suppose we have two n-parameter persistence modules M and N . Recall that to
define the interleaving distance between M and N , these modules are not necessarily
coming from two filtering functions defined on a single topological space. So, if we
want to define stability for a more general setup, that is, for arbitrary persistence mod-
ules, we need a result guaranteeing the existence of such functions and a topological
space. The following theorem solves this problem when the modules are over a prime
field, that is, Q or Zp where p is prime:

Proposition 4.1.2. [22, Proposition 5.8] Let k be a prime field, and M and N be
ϵ-interleaved n-parameter persistence modules whose underlying field is k. Then, for
any i ≥ 1, there exist a CW-complex X and continuous functions f, g : X → Rn
such that

M∼
= Hi (S(f )), N∼
= Hi (S(g)), sup∥f (x) − g(x)∥∞ = ϵ.
x∈X

Therefore, a persistence module over a prime field can be obtained from the i-th
homology of a sublevel set filtration up to isomorphism for all i ≥ 1. This implies
that we can write the statement of Definition 44 for arbitrary persistence modules over
a prime field except for the definiton of 0-universal.

Theorem 4.1.3. [22, Theorem 5.5] For a prime field k and i ≥ 1, dI is i-universal.

Proof. Let M and N be n-parameter persistence modules such that dI (M, N ) = ϵ.

Then, for any δ > 0, there is an (ϵ+δ)-interleaving between them. So, by Proposition
4.1.2, there exists a CW complex X and continuous functions f, g : X → Rn such
that M ∼ = Hi (S(f )), N ∼
= Hi (S(g)), sup ∥f (x) − g(x)∥∞ = ϵ + δ. Let us take
x∈X

47
any i-stable pseudometric d on n-mod where i ≥ 1, then by Definition 43,

d(M, N ) ≤ sup∥f (x) − g(x)∥∞ = ϵ + δ for all δ > 0.

x∈X

Thus, we have

d(M, N ) ≤ ϵ = dI (M, N ).

In addition, we need to consider the case with dI (M, N ) = ∞. Then, trivially

d(M, N ) ≤ ∞ = dI (M, N ).

Corollary 4.1.4. [22, Corollary 5.6] For a prime field k, dI is universal.

Proof. We have already showed dI is stable. Let d be a stable pseudometric on n-

parameter persistence modules. So, it is also 1-stable. Since dI is 1-universal, we
have d(M, N ) ≤ dI (M, N ) for all n-parameter persistence modules.

However, universality of dI over a non-prime field have not showed yet. Proving or
disproving it is an open problem.

Being ϵ-interleaved and having the interleaving distance as ϵ are different. We can-
not guarantee that two persistence modules are ϵ-interleaved when the interleaving
distance between them is ϵ. In fact, for finitely presented ones, they are equivalent:

Theorem 4.1.5. (The Closure Theorem) [22, Theorem 6.1] If M and N are two
finitely presented Rn -indexed persistence modules and dI (M, N ) = ϵ, then M and
N are ϵ-interleaved.

The interleaving distance is an extended pseudometric in the multiparameter persis-

tent homology. However, if we work on finitely-presented Rn -indexed ones, then
having dI of two such modules as 0 implies that they are 0-interleaved by the Closure
Theorem. Hence, they are isomorphic. So, we have the following corollary:

Corollary 4.1.6. [22] dI is an extended metric on finitely presented Rn -indexed per-

sistence modules.

48
4.2 The Bottleneck Distance

When we define the bottleneck distance for multiparameter persistence modules,

we need to examine all matchings between indecomposable summands as in the 1-
parameter case. However, while working with objects of n-mod, decomposing a
given module into known indecomposable summands is a big problem even the mod-
ule is known to be p.f.d.

Hence, we restrict our attention to interval decomposable persistence modules, and

define the bottleneck distance on them.

Let M and N be two n-parameter interval decomposable persistence modules indexed

by Rn , and let C and D be the barcodes of M and N , respectively, that is

M∼ kI and N ∼
M M
= = kJ .
I∈C J∈D

′ ′ ′
We define a matching between C and D as a bijection σ : C → D where C ⊆ C and
′
D ⊆ D. Moreover, if we do not see an element of C or D in the bijection σ, then the
element is unmatched.

Definition 45. [4] With the above construction, we say a matching is an ϵ-matching
if it satisfies the following conditions:

′
i) for all I ∈ C , kI and kσ(I) are ϵ-interleaved.

ii) for all unmatched I ∈ C, kI is ϵ-interleaved with the zero module.

iii) for all unmatched J ∈ D, kJ is ϵ-interleaved with the zero module.

Observe that if we have an ϵ-matching, then all pairs in the above construction are
ϵ-interleaved with each other. So, they are all δ-interleaved for all δ > ϵ. This implies
that we have a δ-matching as well.

Definition 46. [4] The bottleneck distance between C and D is defined as

dB (C, D) := inf{ϵ | there is an ϵ-matching between C and D}.

If no such ϵ exists, then dB (C, D) = ∞.

49
Note that this definition of the bottleneck distance for multiparameter persistence
modules is consistent with the 1-parameter case. To see this, let us first suppose that
elements of the barcodes that we consider are of the form [a, b) for some a, b ∈ R.
For an arbitrary persistence module M , we define M ′ so that

if M ∼ k⟨ai ,bi ⟩ , then M ′ ∼

M M
= = k[ai ,bi ) .
⟨ai ,bi ⟩∈C [ai ,bi )∈C ′

Observe that dB (M, M ′ ) = 0. Similarly, define N ′ for another persistence module

N . Then, by transitivity of the bottleneck distance, we have

dB (M, N ) = dB (M ′ , N ′ ).

This implies that the consistency argument can be proved by working on the barcodes
we have supposed, that is, it does not matter whether we include the endpoints or not.

It is enough to show that the cost of a matching between C and D is smaller than or
equal to ϵ if and only if there exists an ϵ-matching between C and D as in Definition
45. First, start with a matching X between C and D such that c(X ) ≤ ϵ. So, there
′ ′ ′ ′ ′
exists a bijection σ : C → D where C ⊆ C and D ⊆ D. Let I ∈ C and σ(I) = J.
Thus, c(I, J) ≤ ϵ. From the Proposition 2.6.4, we have

i) dI (kI , kJ ) ≤ c(I, J) ≤ ϵ, so kI and kJ are ϵ-interleaved since both interval

modules are finitely presented.
′
ii) dI (kL1 , O) = c(L1 ) ≤ ϵ where L1 ∈ C \ C . So, kL1 is ϵ-interleaved with the
zero module.
′
iii) dI (kL2 , O) = c(L2 ) ≤ ϵ where L2 ∈ D \ D . So, kL2 is ϵ-interleaved with the
zero module.

Hence, the matching X is an ϵ-matching.

Conversely, let us start with an ϵ-matching. Say I = [a, b) and J = [c, d) are matched
with each other. Consider them as two interval modules kI and kJ . Then, they are
ϵ-interleaved. To create the desired matching, we examine some cases. As a first one,
if both intervals does not include any subinterval of the form [t, t + 2ϵ] for an arbitrary
t ∈ R, then we can classify them as unmatched, and obtain a matching with cost at
most ϵ.

50
In any other case, one of the intervals include such a subinterval. We claim that when
they match with each other, we obtain c(I, J) ≤ ϵ. Assume that

c(I, J) = max{|c − a|, |d − b|} > ϵ.

Our aim is to get a contradiction by using the ϵ-interleaving between kI and kJ . Let
us suppose that c(I, J) = |c − a| > ϵ and without loss of generality, c > a. So,
c > a + ϵ. Let [t, t + 2ϵ] ⊆ [a, b) for some t ∈ R. Then, we have the following
non-commutative diagram

(kI )a = k (kI )a+2ϵ = k

(kJ )a+ϵ = 0

and this contradicts with the ϵ-interleaving between modules. Thus, c(I, J) ≤ ϵ.

Let [t, t + 2ϵ] ̸⊆ [a, b) for any t ∈ R and [s, s + 2ϵ] ⊆ [c, d) for some s ∈ R. Then, the
following diagram does not commute

(kI )c+ϵ = 0

(kI )c = k (kI )c+2ϵ = k

and this also contradicts with the ϵ-interleaving. Hence, c(I, J) ≤ ϵ.

Similar arguments show that if c(I, J) = |d − b| > ϵ where at least one of I or J

include a closed subinterval of length 2ϵ, then we get a contradiction as before. Thus,
c(I, J) ≤ ϵ, and we obtain a matching with cost ≤ ϵ in each case.

In addition, say L = [a, b) is matched with the zero module. Then, kL is ϵ-interleaved
b−a
with the zero persistence module. So, 2
≤ ϵ, and this implies that c(L) ≤ ϵ. If
we combine all of this, and write in a single matching X , then c(X ) ≤ ϵ. So, both
constructions for the bottleneck distance in 1-parameter are equivalent to each other.

Moreover, if we have an ϵ-matching between the barcodes C and D, then we can con-
struct an ϵ-interleaving between the corresponding interval decomposable persistence
modules. Therefore, we always have the following statement:

dI (M, N ) ≤ dB (C, D).

51
To see this, start with an ϵ-matching. Let

M∼ kI and N ∼
M M
= = kJ .
I∈C J∈D

If kI and kJ are matched with each other, then they are ϵ-interleaved with morphisms
ϕI : kI → kJ (ϵ) and ψI : kJ → kI (ϵ). Furthermore, if kL is matched with the zero
module for L ∈ C \C ′ (resp. D\D′ ), then there exist two morphisms ϕL : kL → O and
ψL : O → kL (ϵ) (resp. ϕL : O → kL (ϵ) and ψL : kL → O) giving an ϵ-interleaving.
In this case, we just add the zero module to the direct sum of N (resp. M ). So, this
makes no change in persistence modules up to isomorphism. Also, corresponding C
and D become the same in cardinality after adding new elements with empty support
coming from the zero modules. Then, we can write a bijection β : C → D so that kI
and kJ := kβ(I) are ϵ-interleaved.

Observe that for M ∼

M
= kI , we have
I∈C
M 
M (ϵ) ∼
M
= kI (ϵ) = kI (ϵ)
I∈C I∈C
M 
since M (ϵ)i = Mi+⃗ϵ ∼
M
= kI = (kI )i+⃗ϵ where i ∈ Rn . Similarly, we have
i+⃗ϵ
I∈C I∈C
M 
N (ϵ) ∼
M
= kJ (ϵ) = kJ (ϵ).
J∈D J∈D

Define two maps ϕ : M → N (ϵ) ∼ kJ (ϵ) and ψ : N → M (ϵ) ∼

M M
= = kI (ϵ) by
J∈D I∈C
M  M   M  M  
ϕ (kI )i := ϕI (kI )i and ψ (kJ )i := ψI (kβ(I) )i
I∈C I∈C J∈D I∈C

where i ∈ Rn . Since each ϕI and ψI are morphisms making all diagrams in Definition
41 commute, ϕ and ψ give an ϵ-interleaving between M and N .

We have seen that in the 1-parameter case, dI = dB via Theorem 2.6.5. So, we
can directly ask that are they also equal to each other in the multiparameter setting?
Actually, we have already showed dI (M, N ) ≤ dB (C, D) by the explanation given in
the previous paragraph. So, it is all about whether the converse is true or not. The
answer is no, and it can be seen by the following counterexample given by Bjerkevik,
[4, Example 5.1]:

52
 3
I1
2
J
1
I3
1 2 3

I2

Figure 4.1: Intervals where I1 and I2 are outer light purple squares, I3 is the inner
purple square and J is the red one, [4].

Let M = kI1 ⊕ kI2 ⊕ kI3 and N = kJ where

I1 = (−3, 1) × (−1, 3), I2 = (−1, 3) × (−3, 1),

I3 = (−1, 1) × (−1, 1), J = (−2, 2) × (−2, 2),

and intervals can be seen in Figure 4.1.

So, C = B(M ) = {I1 , I2 , I3 } and D = B(N ) = {J}. Consider all possible match-
ings between barcodes.

Matching 1: I1 is matched with J, I2 and I3 are unmatched.

Matching 2: I2 is matched with J, I1 and I3 are unmatched.

Matching 3: I3 is matched with J, I1 and I2 are unmatched.

Matching 4: I1 , I2 , I3 and J are unmatched.

In any case, at least one of I1 or I2 is unmatched. So, they are matched with the zero
module. Without loss of generality, consider the interleavings between kI1 and O.
The morphisms
ϕ : kI1 → O(⃗ϵ) ψ : O → kI1 (⃗ϵ)

are the zero maps where ⃗ϵ = (ϵ, ϵ). Then, we just need to examine triangle diagrams.
Take ϵ = 2, then the internal morphisms of kI1 :
kI1
φp,p+(4,4) : k(I1 )p → k(I1 )p+(4,4)

53
is just the zero map since at least one of k(I1 )p and k(I1 )p+(4,4) is the zero vector space
for I1 = (−3, 1) × (−1, 3). So, we have one of the following cases:

φp,p+(4,4) =0 φp,p+(4,4) =0 φp,p+(4,4) =0

k 0 0 k 0 0

ϕp =0 ψp+(2,2) =0 ϕp =0 ψp+(2,2) =0 ϕp =0 ψp+(2,2) =0

0 0 0

In any case, our diagram commutes. Since we are taking ϕp = 0 = ψp for all p ∈ R2 ,
the other triangle diagram also commutes. Thus, kI1 and O are 2-interleaved.

Claim 1: kI1 and O are not (2 − δ)-interleaved for all δ ∈ R such that 0 < δ ≤ 2.

Let p = (−3 + δ, −1 + δ). So, p ∈ I1 ⇒ (kI1 )p = k. Moreover,

p + 2⃗ϵ = (−3 + δ, −1 + δ) + 2(2 − δ, 2 − δ)

= (1 − δ, 3 − δ).

Thus, p + 2⃗ϵ ∈ I1 ⇒ (kI1 )p+2⃗ϵ = k. So, we have the following triangle diagram:

φp,p+2⃗ϵ=id
k k

ϕp =0 ψp+⃗ϵ=0
0

This diagram does not commute since the horizontal map is the identity map, but
composition of other two maps is just the zero map. This shows that our claim is true.
Thus, kI1 and O are ϵ-interleaved only for all ϵ ≥ 2. The same is true for kI2 . Since
all of four matchings include I1 or I2 as unmatched, they cannot be an ϵ-matching for
any ϵ < 2.

Furthermore, with a similar argument, we can see that kI3 is 1-interleaved with the
zero persistence module.

Now, let us try to find interleavings between kIi and kJ where i ∈ {1, 2, 3}.

Claim 2: kIi and kJ are 2-interleaved for all i ∈ {1, 2, 3}.

In any case, if we take ϕ : kIi → kJ (2) and ψ : kJ → kIi (2) as zero morphisms, then
they are trivial natural transformations. For triangle diagrams, since composition of

54
above morphisms is zero, we just need to show horizontal maps are always zero. This
is true since
φp,p+2(2,2) = φp,p+(4,4) : kKp → kKp+(4,4) = 0

for all K ∈ {I1 , I2 , I3 , J}. Therefore, all triangle diagrams also commutes.

Hence, kIi and kJ are 2-interleaved for all i ∈ {1, 2, 3}. So, all of 4 matchings be-
tween barcodes are 2-matching. Since we showed that they cannot be an ϵ-matching
for ϵ < 2, dB (C, D) = 2.

In the next step, we try to find dI (M, N ) by investigating following interleavings:

Claim 3: None of kIi for i ∈ {1, 2, 3} is not (1 − δ)-interleaved with kJ for any
0 < δ ≤ 1.

Let us work on kI1 . Consider p = (−3 + δ, 1 + δ) with the following diagram:

(kI1 )p (kI1 )p+2(−−→

1−δ)

(kJ )p+(−−−
→
1−δ)

Then, p = (−3 + δ, 1 + δ) ∈ I1 ⇒ (kI1 )p = k. Moreover,

−−−→
p + (1 − δ) = (−3 + δ, 1 + δ) + (1 − δ, 1 − δ) = (−2, 2) ∈
/ J ⇒ (kJ )(−2,2) = 0.
−−−→
p+2(1 − δ) = (−3+δ, 1+δ)+(2−2δ, 2−2δ) = (−1−δ, 3−δ) ∈ I1 ⇒ (kI1 )(−1−δ,3−δ) = k.

So, the diagram becomes:

id
k k

0 0
0

and it does not commute. Hence, kI1 is not (1 − δ)-interleaved with kJ for any
0 < δ ≤ 1. The same situation for kI2 and kI3 can be seen in a similar way.

Claim 4: kI1 and kJ are 1-interleaved.

Define morphisms ϕI1 : kI1 → kJ (1) and ψ I1 : kJ → kI1 (1) as follows:

55
If i ∈ I1 and i + (1, 1) ∈ J, then let ϕIi 1 = id where i ∈ R2 . If at least one of i or
i + (1, 1) is not an element of the corresponding interval, then let ϕIi 1 be the zero map.
Similarly, if j ∈ J and j + (1, 1) ∈ I1 , then let ψjI1 = id where j ∈ R2 . If at least one
of j or j + (1, 1) is not an element of the corresponding interval, then let ψjI1 be the
zero map.

Then, let us show triangle diagrams commute:

(kI1 )i (kI1 )i+(2,2)

I I
ϕi 1 1
ψi+(1,1)
(kJ )i+(1,1)

As a first case, let i, i + (2, 2) ∈ I1 , i + (1, 1) ∈ J. So, our diagonal maps are in the
form: k → k defined as the identity map. So, the corresponding diagram:

id
k k

id id
k

commutes.

The only case where we do not have commutativity happens if i, i + (2, 2) ∈ I1 ,

i + (1, 1) ∈
/ J. We have

id
k k

0 0
0

and it is clearly non-commutative. Observe that if both i and i + (2, 2) is an element

of I1 , then i ∈ (−3, −1) × (−1, 1). This implies that i + (1, 1) ∈ J. However, we
have already assumed that i + (1, 1) ∈
/ J. So, this case cannot happen. Thus, we
have commutative diagrams in all cases. For other triangle diagrams, the situation is
similar. So, kI1 and kJ are 1-interleaved. Since we know they are not ϵ-interleaved
for all ϵ < 1, dI (kI1 , kJ ) = 1.

In addition, we can define morphisms ϕI2 , ψ I2 for kI2 and ϕI3 , ψ I3 for kI3 in a similar

56
way. It can be shown
dI (kI2 , kJ ) = 1 = dI (kI3 , kJ )

with the same idea.

In the next step, we try to find two morphisms

ϕ : M → N (1) and ψ : N → M (1)

showing M and N are 1-interleaved. As a first idea, we can try to define ϕ by using
the following diagram:

kI1 ⊕ kI2 ⊕ kI3

ϕ
π I1

kI1 kJ (1)
ϕI1

where π I1 is the projection of M onto the first component. Let ϕ = ϕI1 ◦ π I1 , so

ϕp = ϕIp1 ◦ πpI1 . Take p = (−0.9, −2.9) ∈ I2 , then p + (1, 1) ∈ J and p + (2, 2) ∈ I2 .
Indeed, the diagram

Mp Mp+(2,2) 0⊕id ⊕0
0⊕k⊕0 0⊕k⊕0
=⇒
ϕp ψp+(1,1)
Np+(1,1) k

However, π I1 = 0 since p ∈
/ I1 ⇒ (kI1 )p = 0. So, ϕ = ϕI1 ◦ π I1 = ϕI1 ◦ 0 = 0. Thus,
the diagram cannot commute. Hence, this way of defining ϕ does not make M and N
1-interleaved.

Now, try another way. In any situation, we have one of the followings:

i) p ∈ I1 ∩ I2 :

Observe that I1 ∩ I2 = I3 . Thus, p ∈ I1 ∩ I2 ⇒ Mp = k ⊕ k ⊕ k. Also,

p + (1, 1) ∈ J ⇒ Np (1) = k. Define ϕp : k ⊕ k ⊕ k → k by ϕp = ϕIp1 ◦ πpI1 .

ii) p ∈ I1 \ I2 :

Define ϕp : k ⊕ 0 ⊕ 0 → Np (1) by ϕp = ϕIp1 ◦ πpI1 .

So, the function ϕ on the cases i) and ii) is defined in the same way.

57
 iii) p ∈ I2 \ I1 :

Define ϕp : 0 ⊕ k ⊕ 0 → Np (1) by ϕp = ϕIp2 ◦ πpI2 .

iv) p ∈
/ I1 ∪ I2 :

Define ϕp = 0.

For the reverse function, we define ψ : N → M (1) by

ψ : N = kJ → M (1) = kI1 (1) ⊕ kI2 (1) ⊕ kI3 (1)

(kJ )p → (ψ I1 ((kJ )p ), ψ I2 ((kJ )p ), ψ I3 ((kJ )p )) where p ∈ R2 .

With the above construction of ϕ : M → N (1) and ψ : N → M (1), all diagrams are
commutative. Thus, M and N are 1-interleaved.

Claim 5: M and N are not ϵ-interleaved for ϵ < 1.

Let us assume there exists such an interleaving. So, we have following morphisms
ϕ : M → N (ϵ) and ψ : N → M (ϵ). Moreover, we can define ϕIi and ψ Ii as functions
commuting following diagrams:

ψ
kIi kJ (kI1 ⊕ kI2 ⊕ kI3 )(ϵ)
ϕIi
f Ii π Ii
ψ Ii
kI1 ⊕ kI2 ⊕ kI3 ϕ
kJ (ϵ) kIi (ϵ)

where f Ii is the inclusion map onto the i-th component of M .

However, ϕIi and ψ Ii show that kIi and kJ are ϵ-interleaved for some ϵ < 1 by the
following arguments:

ϕIi and ψ Ii are well-defined morphisms. This implies that parallelogram diagrams
are commutative. Moreover,

I Ii
fp i φp,p+2⃗ϵ πp+2⃗
ϵ
(kIi )p Mp Mp+2⃗ϵ (kIi )p+2⃗ϵ
ϕp ψp+⃗ϵ
I I
ϕpi i
ψp+⃗ϵ
(kJ )p+⃗ϵ

58
and
φp−⃗ϵ,p+⃗ϵ φp+⃗ϵ,p+3⃗ϵ
(kIi )p−⃗ϵ (kIi )p+⃗ϵ (kIi )p+3⃗ϵ

ϕp−⃗ϵ ψp ϕp+⃗ϵ ψp+2⃗ϵ

(kJ )p φp,p+2⃗ϵ (kJ )p+2⃗ϵ

commutes, and this contradicts with Claim 3 whose correctness is already shown.
So, we get a contradiction. Thus, M and N are not ϵ-interleaved for ϵ < 1. Thus,
dI (M, N ) = 1. Hence, we obtained

dB (C, D) = 2 ̸= 1 = dI (M, N ).

So, the bottleneck distance and the interleaving distance are not equal to each other
in the multiparameter persistent homology.

Moreover, let us choose k as a prime field. We know dI is universal over such a

field. Let us assume that dB is stable, then by definition of universality, we have
dB (B(L1 ), B(L2 )) ≤ dI (L1 , L2 ) for all persistence modules L1 and L2 . However,
we have obtained that dI (M, N ) = 1 < 2 = dB (C, D), and this contradicts with our
assumption. Therefore, the bottleneck distance is not stable in multiparameter setting.

Since dI ≤ dB , the bottleneck distance cannot be smaller than the interleaving dis-
tance. So, we can ask that can we bound the bottleneck distance from above by using
the interleaving distance? The answer comes from the following theorem for the case
of rectangle decomposable persistence modules:

Theorem 4.2.1. [4, Theorem 4.3] Let M and N be rectangle decomposable per-
sistence modules indexed by Rn . If M and N are ϵ-interleaved, then there exists a
(2n − 1)ϵ-matching between their barcodes, B(M ) and B(N ). Thus,

dB (B(M ), B(N )) ≤ (2n − 1)dI (M, N ).

3
M
In our example, M = kIi and N = kJ are rectangle decomposable persistence
i=1
modules indexed by R2 , and they satisfy Theorem 4.2.1, but with respect to this ex-
ample, upper bound is not optimal. However, Bjerkevik show this bound is optimal
for n = 2, and give another example satisfying the bound: for details, see [4, Exam-
ple 5.2].

59
Lastly, both of these distances have negative sides in practice. The bottleneck distance
is not stable, so it is unsuitable to use, and as our previous example shows, even if
we work on interval decomposable bipersistence modules, finding the interleaving
distance between them is so hard. Therefore, we are searching for a stable distance
which is easier to compute compared to the interleaving distance.

4.3 The Matching Distance

Computing distances in multiparameter are more difficult than the case of 1-parameter.
As an idea, we can restrict n-parameter persistence modules to 1-parameter ones.
Therefore, we can use our knowledge and methods in 1-parameter such as taking ad-
vantage of easy computability of the bottleneck distance. Informally, we make this
restriction by intersecting n-parameter persistence modules with lines which we will
define in the next definition.

Again, we take Rn with the max norm:

||r||∞ = max |ri |

i

where r = (r1 , . . . , rn ) ∈ Rn .

Definition 47. [20] Let L be a line in Rn with a direction vector m

⃗ = (m1 , . . . , mn )
whose coordinates take value in R. So, the equation of the line is L : u = tm+b where
t ∈ R, and b = (b1 , . . . , bn ) ∈ Rn is an arbitrary point on the line. If m
⃗ ∈ (0, ∞)n ,
then the line L : u = tm + b is called an admissible line.

Let M be an n-parameter persistence module. We can obtain a 1-parameter persis-

tence module, denoted by ML , by restricting M to an admissible line L so that

(ML )i := Mu

where u = im + b, i ∈ R. Moreover, for i ≤ j ∈ R, φM M

i,j := φu,v where u = im + b
L

and v = jm + b.

Remark 4.3.1. We do not count lines with direction vector having a negative coor-
dinate as admissible lines. The reason is that if we restrict n-parameter persistence
module to one of such a line, points of the intersection of the persistence module and

60
 ML
the line are not comparable with each other. Thus, we cannot define linear maps, φi,j
where i ≤ j, of the persistence module.

Definition 48. [20] The matching distance dmatch on rank invariants of q-tame per-
sistence modules M and N is defined as follows:

dmatch (M, N ) := dmatch (ρM , ρN ) := sup mL .dB (B(ML ), B(NL ))

L

⃗ = (m1 , . . . , mn ) ∈ (0, ∞)n , and L

where mL = min {mi } for a direction vector m
i=1,...,n
varies in the set of admissible lines.

Note that ML and NL are restrictions of q-tame persistence modules M and N .

Hence, ML and NL are 1-parameter q-tame persistence modules. So, by Proposition
2.4.8, we can guarantee that barcodes of ML and NL exist. Thus, we can compute the
matching distance between M and N by the above argument.

Recall that if a direction vector of a line L has at least one coordinate as 0, then we
don’t call this line as admissible because mL = 0 for this line. So, when we multi-
ply the bottleneck distance with mL in the process of finding the matching distance,
numbers coming from there are just zero. Thus, they do not have any effect on the
computation.

Moreover, the size of the set of admissible lines causes problems in the computation
of the matching distance. So, we look for a smaller set where the result of dmatch
remains same. We observe that each line in the set of admissible lines is represented
by infinitely many different parametrizations. Hence, it is meaningful to try to pa-
rameterize the set of admissible lines uniquely in terms of the parameters m
⃗ and b,
provided that the computation gives the same number for different parametrizations
of the same line. To achieve this, we first restrict the set of admissible lines by taking
direction vectors m
⃗ as unit vectors with respect to the max norm. That is, we require
⃗ ∈ (0, 1]n because for different direction vectors, we can obtain the same line. As
m
an example, consider the following three lines with direction vectors

m
⃗ 1 = (1/2, . . . , 1/2), m
⃗ 2 = (1, . . . , 1), m
⃗ 3 = (2, . . . , 2)

and take b = 0 in each one. Then, they all give the same line.

61
This situation also happens for b ∈ Rn . For instance, let n = 2 and take three lines
with the same direction vector m
⃗ = (1, 1) and b = (0, 0), (1, 1), (2, 2), respectively.
Then, we have the same line for all of them. In addition, if we take

L1 : t(1, 1) + (1/2, −1/2) and L2 : t(1, 1) + (1/4, −3/4),

we observe that L1 = L2 . Hence, we need to restrict values of b under a condition,

and letting b = (b1 , . . . , bn ) ∈ Rn so that ni=1 bi = 0 is promising.
P

⃗ ∈ (0, 1]n is
After these restrictions, let Λ be the set of lines L : u = tm + b where m
a unit vector in Rn with respect to the max norm, and ni=1 bi = 0.
P

Proposition 4.3.1. Every admissible line has a representative in the set Λ.

Pn
/ (0, 1]n and
Proof. Let L : u = tm + b be an admissible line where m ∈ i=1 bi ̸= 0.
Define
m
m′ :=
||m||∞
Observe that m′ ∈ (0, 1]n , ||m′ ||∞ = 1 and the line u = sm′ + b is same as L since
 
′ s m s
sm + b = .||m||∞ . +b= m + b.
||m||∞ ||m||∞ ||m||∞

So, we obtain direction vector of the line as we want. Now, consider the hyperplane
x1 + . . . + xn = 0 in Rn . Normal vector of this hyperplane is ⃗n = (1, . . . , 1) ∈ Rn
⃗ ′ ̸= 0. Thus, m
and ⃗n · m ⃗ ′ is not parallel to the hyperplane. In addition, dimensions
of the hyperplane (n − 1) and the line (1) are complement of n. This implies that the
hyperplane and the line u = sm′ + b intersect with each other. Therefore, there exists
b′ = (b′1 , . . . , b′n ) ∈ L such that ni=1 b′i = 0. Thus, we rewrite L as u = sm′ + b′ .
P

Proposition 4.3.2. Every line in the set Λ is uniquely determined by m

⃗ and b.

Proof. Let L1 : tm + b and L2 : th + d be equal lines in Λ. Then, their direction

⃗ and ⃗h are unit vectors,
vectors are parallel. So, let h = cm for some c ∈ R. Since m
there exists i, j ∈ {1, . . . , n} such that mi = 1 and hj = 1. So, hi = c.1 = c and
we also have 1 = hj = cmj . Moreover, we know that c = hi ∈ (0, 1]. Suppose
that c ̸= 1, then 1 = cmj implies that mj > 1, but this contradicts with the fact
mj ∈ (0, 1]. Thus, c = 1, and h = m.

62
Therefore, L1 : tm + b and L2 : tm + d. Then, let t1 m + b = t2 m + d for some
t1 , t2 ∈ R. So, (t1 − t2 )m = d − b. If we consider each coordinate, we have

(t1 − t2 )mi = di − bi
n
X n
X
⇒ (t1 − t2 )mi = d i − bi
i=1 i=1
n
X n
X n
X
⇒ (t1 − t2 ) mi = di − bi
i=1 i=1 i=1
n
X
⇒ (t1 − t2 ) mi = 0.
i=1

Since mi > 0 for all i ∈ {1, . . . , n}, t1 − t2 = 0. Thus, t1 = t2 and t1 m + b = t1 m + d

implies that b = d. So, by choosing m
⃗ and b, we decide lines in the set Λ uniquely.

Since we uniquely obtain lines in Λ, we take a line L only once in the computa-
tion of the bottleneck distance. However, we can ask whether the computation of
mL .dB (ML , NL ) is always same for different parametrizations of an admissible line
L. If we change constant b, it creates the same shifting in 1-parameter persistence
modules ML and NL , so the bottleneck distance does not change. Let us see this
situation in practice: Say

L1 : u = t(m1 , m2 , . . .) + (x1 , x2 , . . .) and L2 : u = t(m1 , m2 , . . .) + (y1 , y2 , . . .)

are two different parametrizations of the same line L. Assume that when we com-
pute the bottleneck distance between barcodes of ML1 and NL1 , the interval modules
[a1 , c1 ) and [b1 , d1 ) are matched with each other. Consider the corresponding interval
modules of [a1 , c1 ) and [b1 , d1 ) in the decompositions of ML2 and NL2 , respectively:
[a2 , c2 ) and [b2 , d2 ). So, we have

a1 (m1 , m2 , . . .) + (x1 , x2 , . . .) = a2 (m1 , m2 , . . .) + (y1 , y2 , . . .)

b1 (m1 , m2 , . . .) + (x1 , x2 , . . .) = b2 (m1 , m2 , . . .) + (y1 , y2 , . . .)
⇒ (x1 − y1 , x2 − y2 , . . .) = (a2 − a1 )(m1 , m2 , . . .)
= (b2 − b1 )(m1 , m2 , . . .)
⇒ a2 − a1 = b 2 − b 1
⇒ b 1 − a1 = b 2 − a2 .

63
Similarly,
d 1 − c1 = d 2 − c2 .

Costs coming from the matchings of pairs are equal to each other:

max{|b1 − a1 |, |d1 − c1 |} = max{|b2 − a2 |, |d2 − c2 |}.

In a similar way, if [a1 , c1 ) is matched with the zero module, then we obtain the
following relation with the corresponding interval module [a2 , c2 ):
  c −a c 2 − a2  
1 1
c1 − a1 = c2 − a2 ⇒ c k[a1 ,c1 ) = = = c k[a2 ,c2 ) .
2 2
Hence, costs are again equal to each other. This implies that dB gives the same num-
ber although we change the parametrization of the line with respect to b.

Moreover, let us investigate what happens to dB if we take a k-multiple of the direc-

tion vector where k > 0: Say

L1 : u = t(m1 , m2 , . . .) + (x1 , x2 , . . .) and L2 : u = t(km1 , km2 , . . .) + (x1 , x2 , . . .)

are two different parametrizations of the same line L. If [a, c) and [b, d) are matched
in some matching between ML1 and NL1 , then
ha c  h b d
, and ,
k k k k
are matched with each other in the corresponding matching between ML2 and NL2 .
Thus, we have the following relation between costs:
 
c k[a,c) , k[b,d) = max{|b − a|, |d − c|}
n b−a d−c o
= k. max ,
 k  k
= k.c k[ ka , kc ) , k[ b , d )
k k

Similarly, if an interval module is matched with the zero module, its cost is divided
by k in the case of L2 . Therefore, the corresponding bottleneck distance is divided by
k, but mL is multiplied by k, so we have the same result for mL .dB . Thus, matching
distance does not be affected from different parametrizations of the same admissible
line.
Remark 4.3.2. As a result of above parts, the matching distance between M and N
can be found by only considering lines from Λ. From now on, L always varies in Λ
when we compute dmatch .

64
 L
5

∼
= ⊕
1

1 5
M M1 M2

L
5

∼
= ⊕
1

1 5
N N1 N2

Figure 4.2: Decomposition of M and N with lines in R2 , [26].

Furthermore, the matching distance is an extended pseudometric. To see why it is not

an extended metric, we examine the following example:

Example 4.3.1. [26] In Figure 4.2, we have two 2-parameter interval decomposable
persistence modules

M∼
= M1 ⊕ M2 and N ∼
= N1 ⊕ N2 .

We claim first that they are not isomorphic. To see this, assume otherwise. Then, by
Theorem 3.2.2 (ii), there is a bijection between index sets {1, 2} giving isomorphisms
between the corresponding modules. Thus, N1 is isomorphic to either M1 or M2 .

To see that neither is possible, consider the linear maps

(N1 )(0,1) → (N1 )(1,1) and (N1 )(1,0) → (N1 )(1,1) .

These are by definition just the identity maps from the field k to itself. However,

(M1 )(0,1) = 0 → (M2 )(1,1) = k and (M2 )(1,0) = 0 → (M2 )(1,1) = k are zero maps.

Hence, none of the modules M1 and M2 have both maps as identity. So, N1 is not
isomorphic to any of M1 and M2 . This gives a contradiction. Thus, M and N are not
isomorphic.

On the other hand, if we restrict M and N to lines L ∈ Λ, then it can be seen from
Figure 4.2 that corresponding 1-parameter persistence modules ML and NL are equal
to each other. Thus, we have

dB (B(ML ), B(NL )) = 0 for all L ∈ Λ.

65
So, we have

dmatch (M, N ) = 0.

This implies that dmatch is not an extended metric.

Moreover, we can also make an inference about the interleaving distance. To achieve
this, let us first observe that both of M and N are finitely presented. Let 1(i,j) show
the multiplicative identity of k at the grade (i, j) ∈ R2 .

M1 is generated by 1(1,0) , and the relations are generated by 1(1,5) and 1(5,0) . Hence,
taking F1 = Q(1,5) ⊕ Q(5,0) , and F0 = Q(1,0) in Definition 39 makes M1 finitely
presented. In the other ones, we just state the generators of F0 and F1 :

i) For M2 , 1(0,1) generates F0 , and 1(0,5) , 1(5,1) generate F1 .

ii) For N1 , 1(1,0) , 1(0,1) generate F0 , and 1(0,5) , 1(5,0) , 1(1,1) generate F1 .

iii) For N2 , 1(1,1) generates F0 , and 1(1,5) , 1(5,1) generate F1 .

So, all of them are finitely presented. This implies that direct sums M and N are
also finitely presented. Now, let us assume that dI (M, N ) = 0. Then, since they are
also finitely presented, there exists a 0-interleaving between M and N by the Closure
Theorem. However, being 0-interleaved means that these modules are isomorphic to
each other, and we have observed that they are not. Therefore, our assumption is
wrong. Thus, dI (M, N ) > 0 = dmatch (M, N ).

The matching distance shows up as a computable surrogate of other distances. Re-

cently, the exact computation of the matching distance gets attention in the theory
of multiparameter persistence modules, [19] and [2]. Researchers mainly work on
2-parameter persistence modules because visualizing them is rather easy and results
in 2-parameter are more likely to be generalized. Kerber et. al. showed that the com-
putation of the matching distance can be done in polynomial time [19]. Later, Brooks
et. al. restricted the set of lines Λ to some set which is enough to consider to find the
matching distance [2]. Moreover, they have provided both geometric interpretation
and easy implementation.

66
In the next step, we want to understand whether the matching distance is stable or
not. We examine this obscurity with the help of the interleaving distance.
 
ϵ
Lemma 4.3.3. [20] If M and N are ϵ-interleaved, then ML and NL are mL
-
interleaved.

Proof. In the proof, we use the following notation:

M (u) := Mu
ϕ(u + ⃗ϵ) := ϕu+⃗ϵ,
φM (u, u + 2⃗ϵ) := φM
u,u+2⃗ϵ.

Since M and N are ϵ-interleaved, there exist two morphisms ϕ : M → N (⃗ϵ) and
ψ : N → M (⃗ϵ) such that

ψ(u + ⃗ϵ) ◦ ϕ(u) = φM (u, u + 2⃗ϵ),

and
ϕ(u + ⃗ϵ) ◦ ψ(u) = φN (u, u + 2⃗ϵ)

where u ∈ Rn , so that triangle diagrams commute. Our aim is to show the existence
   
of two morphisms ϕL : ML → NL mϵL and ψL : NL → ML mϵL such that
   
ϵ ML ϵ
ψL s + ◦ ϕL (s) = φ s, s + 2
mL mL
and    
ϵ NL ϵ
ϕL s + ◦ ψL (s) = φ s, s + 2
mL mL
where s ∈ R.
 
Consider an arbitrary s ∈ R with points u(s) = sm + b, u′ (s) = s + ϵ
mL
m + b and
 
u′′ (s) = s + 2 mϵL m + b on L. Consider each coordinate:

u(s) = (sm1 + b1 , . . . , smn + bn )

    
′ ϵ ϵ
u (s) = s+ m1 + b1 , . . . , s + mn + bn
mL mL
ϵ
= u(s) + (m1 , . . . , mn )
mL
≥ u(s) + ⃗ϵ

67
since mL = min {mi }.
i=1,...,n
   
ϵ ϵ
Let us define ϕL : ML → NL mL
and ψL : NL → ML mL
by

ϕL (s) = ϕ(u′ − ⃗ϵ) ◦ φM (u, u′ − ⃗ϵ) and ψL (s) = ψ(u′ − ⃗ϵ) ◦ φN (u, u′ − ⃗ϵ)

where ML (s) = M (u) and NL (s) = N (u).

Since u(s) ≤ u′ (s)−⃗ϵ for all s ∈ R, u ≤ u′ −⃗ϵ. Hence, linear maps φM (u, u′ −⃗ϵ) and
φN (u, u′ − ⃗ϵ) are well-defined. Therefore, ϕL and ψL are well-defined morphisms.
 
Notice that ψL (s) is defined from NL (s) = N (u) to ML s + mL = M (u′ ). Also,
ϵ

we have φM (u + ⃗ϵ, u′ ) ◦ ψ(u) from N (u) to M (u′ ) and this gives the following
equality:
ψL (s) = φM (u + ⃗ϵ, u′ ) ◦ ψ(u)

by the following commutative diagram coming from definition of morphisms:

M (u + ⃗ϵ) M (u′ )

N (u) N (u′ − ⃗ϵ)

Consider
 
ϵ
ψL s + ◦ ϕL (s) = φM (u′ + ⃗ϵ, u′′ ) ◦ ψ(u′ ) ◦ ϕ(u′ − ⃗ϵ) ◦ φM (u, u′ − ⃗ϵ).
mL
Since φM (u′ − ⃗ϵ, u′ + ϵ) = ψ(u′ ) ◦ ϕ(u′ − ⃗ϵ),
 
ϵ
ψL s + ◦ ϕL (s) = φM (u′ + ⃗ϵ, u′′ ) ◦ φM (u′ − ⃗ϵ, u′ + ϵ) ◦ φM (u, u′ − ⃗ϵ)
mL
= φM (u, u′′ )
 
ML ϵ
=φ s, s + 2 .
mL
 
Similarly, ϕL (s) is defined from ML (s) = M (u) to NL s + ϵ
mL
= N (u′ ). So, we
can write ϕL (s) as
ϕL (s) = φN (u + ⃗ϵ, u′ ) ◦ ϕ(u).

Consider
 
ϵ
ϕL s + ◦ ψL (s) = φN (u′ + ⃗ϵ, u′′ ) ◦ ϕ(u′ ) ◦ ψ(u′ − ⃗ϵ) ◦ φN (u, u′ − ⃗ϵ).
mL

68
Since φN (u′ − ⃗ϵ, u′ + ϵ) = ϕ(u′ ) ◦ ψ(u′ − ⃗ϵ),
 
ϵ
ϕL s + ◦ ψL (s) = φN (u′ + ⃗ϵ, u′′ ) ◦ φN (u′ − ⃗ϵ, u′ + ϵ) ◦ φN (u, u′ − ⃗ϵ)
mL
= φN (u, u′′ )
 
ML ϵ
=φ s, s + 2 .
mL
 
ϵ
Thus, triangle diagrams commute. Therefore, ML and NL are mL
-interleaved.

Theorem 4.3.4. [20] For arbitrary p.f.d. persistence modules M and N , we have

dmatch (M, N ) ≤ dI (M, N ).

Proof. Let L ∈ Λ. By Lemma 4.3.3, if M and N are ϵ-interleaved, then ML and NL

 
are mϵL -interleaved. Thus,

mL .dI (ML , NL ) ≤ dI (M, N ).

Since M and N are pointwise finite dimensional, so are ML and NL . Thus, from the
Isometry Theorem in 1-parameter, dI (ML , NL ) = dB (B(ML ), B(NL )). So,

dmatch (M, N ) = sup mL .dB (B(ML ), B(NL )) ≤ dI (M, N ).

L

So, the matching distance between p.f.d. persistence modules is a lower bound for
a stable distance, namely the interleaving distance. Thus, it satisfies the comparison
in Definition 43 where M f and M g need to be p.f.d. Hence, the matching distance
defined between Rn indexed p.f.d. persistence modules is also stable.

We want to point out that computing the matching distance is not straightforward
since we need to take infinitely many lines into account. On the other hand, if we
know the interleaving distance between two persistence modules, finding the match-
ing distance can be easier in some cases.

Example 4.3.2. Recall Figure 4.1, where we have two persistence modules

M = kI1 ⊕ kI2 ⊕ kI3 and N = kJ .

69
We have already seen that dI (M, N ) = 1. To calculate the matching distance, we
consider the line L1 : u = t(1, 1) where t ∈ R. Then, mL1 = 1 and

ML1 = k(−1,1) ⊕ k(−1,1) ⊕ k(−1,1) and NL1 = k(−2,2) .

Therefore, we have the following induced barcodes

B(ML1 ) = (−1, 1) ⊕ (−1, 1) ⊕ (−1, 1) and B(NL1 ) = (−2, 2).

We find the bottleneck distance by following the construction given in Subsection

2.5. We have 4 matchings between B(ML1 ) and B(NL1 ). Three of them are exactly
the same: one copy of (−1, 1) is matched with (−2, 2) and the other copies are left
unmatched. Hence, we find their cost as

max{c( (−1, 1), (−2, 2) ), c( (−1, 1) )} = max{1, 1} = 1.

In the last matching, all of the intervals are unmatched. Thus, its cost is

max{c( (−1, 1) ), c( (−2, 2) )} = max{1, 2} = 2.

Hence, the bottleneck distance is equal to

dB (B(ML1 ), B(NL1 )) = min{1, 2} = 1.

Observe that, this implies

1 = mL1 .dB (B(ML1 ), B(NL1 )) ≤ sup mL .dB (B(ML ), B(NL ))

L

= dmatch(M,N )
≤ dI (M, N ) = 1.

Thus, we have dmatch(M,N ) = 1.

In the above example, note first that the interleaving distance is an upper bound for
the matching distance since we start with p.f.d. persistence modules. Moreover, the
computation of the matching distance on the line L1 realizes this upper bound. Since
the matching distance is calculated as the supremum through all admissible lines,
we can immediately conclude that the matching distance between M and N is 1.
However, this method is impractical because calculating dI is much more difficult.

We may expect to catch the matching distance by considering only lines with slope
1, but the next example shows that, unlike the previous one, it is not sufficient to find
the matching distance.

70
 Figure 4.3: 2-parameter persistence modules M and N with lines in R2 , [2].

Example 4.3.3. [2] Let M and N be 2-parameter rectangle decomposable persistence

modules given as

M = M1 ⊕ M2 = kR1 ⊕ kR2 and N = N1 ⊕ N2 = kR3 ⊕ kR4

where

R1 = [0, 7) × [0, 7), R2 = [0, 7) × [4, 11),

R3 = [0, 7) × [0, 11), R4 = [0, 7) × [4, 7).

According to Figure 4.3, observe that for the lines L : y = x + p where p ≤ 0 or

p ≥ 4, restrictions of M and N to such lines are equal, that is, ML = NL . Hence,
dB (B(ML ), B(NL )) = 0. So, the matching distance coming from such lines is just 0.

On the other hand, if we consider the lines L : y = x + p where 0 < p < 4, they are
uniquely parametrized as
 −p p 
L : t(1, 1) + , , t∈R
2 2
so that L ∈ Λ. Let us consider barcodes of restrictions of M and N to such lines.
hp p h p p
B(ML ) = B((M1 )L ) ⊕ B((M2 )L ) = ⊕ 4 − ,7 +
,7 − ,
2 2 2 2
hp p h p p
B(NL ) = B((N1 )L ) ⊕ B((N2 )L ) = , 7 + ⊕ 4 − ,7 − .
2 2 2 2
In all of these lines, mL = 1. So, to analyze effects of these lines on the matching
distance, we only consider the corresponding bottleneck distances. Now, consider all
the matchings:

71
σ1 : (Mi )L is matched with (Ni )L for i = 1, 2. Then, c(σ1 ) = p.

σ2 : (M1 )L is matched with (N2 )L and (M2 )L is matched with (N1 )L . Then,
c(σ2 ) = 4 − p.

σ3 : (M1 )L is matched with (N1 )L , (M2 )L and (N2 )L is unmatched. Then,


 3+p if p ≤ 3,

2
c(σ3 ) =
p

if p > 3.

σ4 : (M1 )L is matched with (N2 )L , (M2 )L and (N1 )L is unmatched. Then,


4 − p if p ≤ 1 ,

2
c(σ4 ) =
7

if p > 12 .
2

σ5 : (M2 )L is matched with (N1 )L , (M1 )L and (N2 )L is unmatched. Then,


4 − p if p ≤ 1,

c(σ5 ) =
 7−p

if p > 1.
2

σ6 : (M2 )L is matched with (N2 )L , (M1 )L and (N1 )L is unmatched. Then,


 7 if p ≤ 7 ,

2 2
c(σ6 ) =
p if p > 7 .

2

σ7 : They are all unmatched. Then, c(σ7 ) = 72 .

Remember that we take infimum of all matchings to find the bottleneck distance. If
p < 2, then minimum of costs of all the matchings is equal to p happening in σ1 . This
implies that dB is smaller than 2 for the case p < 2. If p > 2, then 4 − p coming
from σ2 is the minimum of costs of all matchings. Therefore, dB is also smaller than
2 for the case p > 2. Lastly, we consider all matchings when p = 2 and see that the
corresponding bottleneck distance is 2 which is the biggest one among them.

On the other hand, consider the line L′ passing through (0, 0) and (7, 11). Its equation
is y = 11
7
x. So, the parametrization of it making L′ an element of Λ is
7 
L′ : t , 1 , t ∈ R.
11

72
Thus, we have

B(ML′ ) = B((M1 )L′ ) ⊕ B((M2 )L′ ) = [0, 7) ⊕ [4, 11),

B(NL′ ) = B((N1 )L′ ) ⊕ B((N2 )L′ ) = [0, 11) ⊕ [4, 7).

7
By analyzing all matchings, we see that dB (B(ML′ ), B(NL′ )) = 4. Since mL′ = 11
,
we have
7
mL′ .dB (B(ML′ ), B(NL′ )) = .4
11
> 1.2 = mL .dB (B(ML ), B(NL ))

where L : y = x + 2. Since we take supremum in the computation of the matching

distance, it cannot happen by a restriction of a line with slope 1 in this example. We
28
also want to remark that at this point all we can say is that dmatch (M, N ) ≥ 11
.

When the lines are in R2 , we work on the representations coming from Λ instead of
the equations of the form: y = mx + c where m, c ∈ R. The reason is that the first
way can be easily extended to n-parameter case where n ≥ 3.

73
74
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