Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Frédéric Chazal
Vin de Silva
Marc Glisse
Steve Oudot
The Structure
and Stability
of Persistence
Modules
123
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é é
Fr d ric Chazal Vin de Silva
(cid:129)
Marc Glisse Steve Oudot
(cid:129)
The Structure and Stability
of Persistence Modules
123
Frédéric Chazal MarcGlisse
Inria Saclay– Île-de-France Inria Saclay– Île-de-France
Palaiseau Palaiseau
France France
VindeSilva SteveOudot
PomonaCollege Inria Saclay– Île-de-France
Claremont, CA Palaiseau
USA France
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-319-42543-6 ISBN978-3-319-42545-0 (eBook)
DOI 10.1007/978-3-319-42545-0
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Preface
Ourintention,atthebeginning,wastowriteashortpaperresolvingsometechnical
issuesinthetheoryoftopologicalpersistence.Specifically,wewishedtopresenta
clean easy-to-use frameworkfor continuous-parameter persistence,building onthe
well-studied case of discrete-parameter persistence. Over a number of years, we
gradually abandoned the idea of a short paper. It apparently takes us about a
hundred pages to explain things in a simple way. We take the time to develop the
concepts that make everything flow: decorated reals, quiver calculus, r-measures,
q-tameness. Our main concern has been to make the mathematics lucid. We hope
the reader will find the ideas useful and the presentation clarifying.
Palaiseau, France Frédéric Chazal
Claremont, CA, USA Vin de Silva
Palaiseau, France Marc Glisse
Palaiseau, France Steve Oudot
May 2016
v
Acknowledgments
Many important ideas in this monograph have their origins in earlier work with
David Cohen-Steiner and Leo Guibas [15] and Gunnar Carlsson [8]. We deeply
appreciate their intellectual influence.
We aregrateful toseveralpeoplewho directly helped improve this monograph.
William Crawley-Boevey has been an invaluable source of expertise on module
decomposition. The current form of Theorem 2.8, proved by him, is particularly
convenientforourexposition.MichaelLesnickprovideduswithanearlyversionof
his work [42], drew our attention to the paper of Webb [48], and corrected our
original statement of Theorem 2.8. Peter Landweber read the manuscript closely
and gave us many detailed suggestions and corrections. Finally, we thank the
anonymous referees whose suggestions have greatly improved this monograph.
This work has been supported by several research grants:
(cid:129) Digiteo Foundation. C3TTA and a Digiteo Chair (held by VdS).
(cid:129) European Commission. Computational Geometric Learning (255827).
(cid:129) Agence Nationale de Recherche. GIGA (ANR-09-BLAN-0331-01).
(cid:129) DARPA. Sensor Topology and Minimal Planning (HR0011-07-1-0002).
(cid:129) Simons Foundation. Simons Fellowship (grant #267571 to VdS).
We gratefully acknowledge the support of these funding agencies. We also thank
the Institute for Mathematics and its Applications, for hosting and partially sup-
porting a sabbatical (VdS), with funds provided by the National Science
Foundation.
Lastbutnotleast,wethankourhomeinstitutionsfortheirongoingsupport:Inria
Saclay – Île-de-France (FC, MG, SO) and Pomona College (VdS).
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 A Brief History of Persistence . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Application: Stable Descriptors for Metric Spaces . . . . . . . . . . . 5
1.4 Application: Stable Clustering Using Persistence . . . . . . . . . . . 7
1.5 Recommended Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Persistence Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Persistence Modules Over a Real Parameter . . . . . . . . . . . . . . . 15
2.2 Index Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Module Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Interval Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Interval Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 The Decomposition Persistence Diagram . . . . . . . . . . . . . . . . . 24
2.7 Quiver Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Rectangle Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 The Persistence Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 The Persistence Measure (Continued) . . . . . . . . . . . . . . . . . . . 33
3.3 Abstract r-Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Equivalence of Measures and Diagrams . . . . . . . . . . . . . . . . . . 38
3.5 Non-finite Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 Measures and Diagrams in the Extended Plane . . . . . . . . . . . . . 44
3.7 The Measure Persistence Diagram . . . . . . . . . . . . . . . . . . . . . 47
3.8 Tameness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9 Tameness (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.10 Vanishing Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.11 Vanishing Lemmas (Continued) . . . . . . . . . . . . . . . . . . . . . . . 58
3.12 Finite Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
ix
x Contents
4 Interleaving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1 Shifted Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Interleaving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Interleaving (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 The Interpolation Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 The Interpolation Lemma (Continued) . . . . . . . . . . . . . . . . . . . 78
5 The Isometry Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1 The Interleaving Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 The Bottleneck Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 The Bottleneck Distance (Continued) . . . . . . . . . . . . . . . . . . . 89
5.4 The Isometry Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 The Converse Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . 92
5.6 The Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.7 The Measure Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . 99
5.8 The Measure Stability Theorem (Continued) . . . . . . . . . . . . . . 103
6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1 Partial Interleavings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Extended Persistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Chapter 1
Introduction
We intend this monograph to be a self-contained presentation of the theory of
persistence modules over the real line. We give the best proofs we know of the
mostimportantresults.Eachtheoremislocatedatanappropriatelevelofabstraction
(webelieve).
• Newcomerswillfindthistobeanaccessibleintroductiontothealgebraicfounda-
tionsofpersistence.Theywilllearnwhatpersistencemodulesare,howtoaccess
their internal structure, their different degrees of ‘tameness’, how to construct
theirpersistencediagramsmathematically,andhowtoshowthatthosediagrams
arestable.
• Expertswillfindthatourmethodsaddprecisionandpowertowhattheyalready
know. We construct persistence modules in great generality and show that they
can be manipulated just as easily as the traditional examples. We give evidence
thatthisgreatergeneralityoccursnaturally‘inthewild’.
Persistencemodulesarethemathematicalobjectattheheartoftheyoung,rapidly
growingfieldoftopologicaldataanalysis.Thisfield—ablendofcomputerscience,
algebraictopologyandstatistics—isfoundedontheassumptionthatscientificdata
sets carry information in their internal structure and that sometimes this internal
structure is topological. Persistence modules were designed to carry topological
informationaboutadatasetatmanydifferentscalessimultaneously.Thisinformation
canbeextractedintheformofaninvariant—thepersistencediagramorbarcode—
thatcanbecomputedeffectivelyandisstatisticallyrobust.
New researchers in the field have to come to terms with multiple aspects of
persistence.Theymustlearnhowdatacanbeturnedintogeometricobjects(usually
anestedfamilyofsimplicialcomplexes).Theymustlearntoworkwiththealgorithms
thatturnthesegeometricobjectsintobarcodesorpersistencediagrams.Theyneed
algebraictopologytoknowhowtointerpretthesebarcodes.And,increasinglyoften
nowadays,theymustknowenoughstatisticaltheorytodrawvalidinferencesfrom
theresultsofthesecalculations.Thereisbynowagreatdealofliteraturecovering
©TheAuthor(s)2016 1
F.Chazaletal.,TheStructureandStabilityofPersistenceModules,
SpringerBriefsinMathematics,DOI10.1007/978-3-319-42545-0_1
2 1 Introduction
thesedifferentfacets,writtenbyandintendedforresearchersacrossalargerangeof
disciplines.
Ourfocusisnarrower.Inthismonograph,weareconcernedalmostexclusively
with the mathematical properties of persistence modules. We have several reasons
fordoingthis:
• The existing literature is largely built around particular instances of topological
persistence,suchasthesublevelsetpersistenthomologyofaMorsefunctionona
compactmanifold.Whilethiscanbeusefulfordevelopingintuition,itdoescreate
biasinhowthesubjectisunderstood.Wewishtocorrectthisbias.
• Several components of the theory are algebraic in nature. The fact that most of
theusualexamplescomefromtopologysometimescreatesadifferentimpression.
Herewegiveapurelyalgebraicpresentationofthemainingredients:thepersis-
tencediagram,tameness,stability.Topologicalargumentsareusedwhenstudying
thepropertiesofpersistencemodulesobtainedfromtopologicaldata,butarenot
neededotherwise.
• Variationssuchasimageandkernelpersistencecanbetreatedequally.
More specifically, we are interested in persistence modules indexed by a single
realparameter.Muchofourworkgoestoshowthatacontinuousparametercanbe
handled just as effectively as a discrete parameter; this fills a gap in the literature.
Ontheotherhand,wemakenoattempttodiscussmultiparameterpersistence.Itis
acomplicatedsubjectandatsomepointwillbereadyforitsownbook.
Withinitsscope,themonographisalmostentirelyself-contained.Wedoappeal
to two off-the-shelf technical results from module theory (Theorems 2.7 and 2.8).
When discussing persistent homology, we assume that the reader is familiar with
thestandardpropertiesofsimplicialcomplexesandhomologyfunctors.Sinceitis
importanttoremainconnectedtothelargerworldoftopologicaldataanalysis,we
briefly present two applications in Sects.1.3 and 1.4, and a themed reading list in
Sect.1.5.
1.1 ABriefHistoryofPersistence
Theearlyhistoryofpersistenceisconcernedwiththequantity
rs =rank(H(X )→H(X ))
t s t
for an object X represented at two different scales s,t, and where H is homology.
Thisappearedintheearly1990sintheworkofFrosini[34],withdifferentnotation
and under the name ‘size function’. Independently, a few years later, Robins [46]
introduced the term ‘persistent Betti numbers’ for quantities of the form rε , and
ε+ρ
notedtheirstabilitywithrespecttoHausdorffdistance.
