An invitation to statistics in Wasserstein space

157 Pages·2020·4.876 MB·English

by  Panaretos V.M., Zemel Y

SPRINGER BRIEFS IN PROBABILITY
AND MATHEMATICAL STATISTICS
Victor M. Panaretos
Yoav Zemel
An Invitation
to Statistics in
Wasserstein
Space
SpringerBriefs in Probability and Mathematical
Statistics
Editor-in-Chief
GesineReinert,UniversityofOxford,Oxford,UK
MarkPodolskij,UniversityofAarhus,AarhusC,Denmark
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NinaGantert,TechnischeUniversita¨tMu¨nchen,Mu¨nich,Nordrhein-Westfalen,
Germany
TailenHsing,UniversityofMichigan,AnnArbor,MI,USA
RichardNickl,UniversityofCambridge,Cambridge,UK
SandrinePe´che´,Universite´ ParisDiderot,Paris,France
YosefRinott,HebrewUniversityofJerusalem,Jerusalem,Israel
AlmutE.D.Veraart,ImperialCollegeLondon,London,UK
MathieuRosenbaum,Universite´ PierreetMarieCurie,Paris,France
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Victor M. Panaretos • Yoav Zemel
An Invitation to Statistics
in Wasserstein Space
VictorM.Panaretos YoavZemel
InstituteofMathematics StatisticalLaboratory
EPFL UniversityofCambridge
Lausanne,Switzerland Cambridge,UK
ISSN2365-4333 ISSN2365-4341 (electronic)
SpringerBriefsinProbabilityandMathematicalStatistics
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Preface
A Wasserstein distance is a metric between probability distributions μand νon a
groundspaceX,inducedbytheproblemofoptimalmasstransportationorsimply
optimaltransport.Itreflectstheminimaleffortthatisrequiredinordertoreconfig-
ure the mass of μto produce the mass distribution of ν. The ‘effort’ corresponds
to the total work needed to achieve this reconfiguration, where work equals the
amountofmassattheorigintimesthedistancetotheprescribeddestinationofthis
mass. The distance between origin and destination can be raised to some power
otherthan1whendefiningthenotionofwork,givingrisetocorrespondinglydiffer-
entWassersteindistances.WhenviewingthespaceofprobabilitymeasuresonX
as a metric space endowed with a Wasserstein distance, we speak of a Wassertein
Space.
Mass transportation and the associated Wasserstein metrics/spaces are ubiqui-
tous in mathematics, with a long history that has seen them catalyse core devel-
opments in analysis, optimisation, and probability. Beyond their intrinsic mathe-
matical richness, they possess attractive features that make them a versatile tool
for the statistician. They frequently appear in the development of statistical the-
ory and inferential methodology, sometimes as a technical tool in asymptotic the-
ory, due to the useful topology they induce and their easy majorisation; and other
timesasamethodologicaltool,forexample,instructuralmodellingandgoodness-
of-fit testing. A more recent trend in statistics is to consider Wasserstein spaces
themselves as a sample and/or parameter space and treat inference problems in
such spaces. It is this more recent trend that is the topic of this book and is
coming to be known as ‘statistics in Wasserstein spaces’ or ‘statistical optimal
transport’.
Fromthetheoreticalpointofview,statisticsinWassersteinspacesrepresentsan
emerging topic in mathematical statistics, situated at the interface between func-
tional data analysis (where the data are functions, seen as random elements of an
infinite-dimensional Hilbert space) and non-Euclidean statistics (where the data
satisfy non-linear constraints, thus lying on non-Euclidean manifolds). Wasser-
vii
viii Preface
stein spaces provide the natural mathematical formalism to describe data collec-
tions that are best modelled as random measures on Rd (e.g. images and point
processes). Such random measures carry the infinite-dimensional traits of func-
tional data, but are intrinsically non-linear due to positivity and integrability re-
strictions.Indeed,contrarilytofunctionaldata,theirdominatingstatisticalvariation
arisesthroughrandom(non-linear)deformationsofanunderlyingtemplate,rather
than the additive (linear) perturbation of an underlying template. This shows op-
timal transport to be a canonical framework for dealing with problems involving
the so-called phase variation (also known as registration, multi-reference align-
ment, or synchronisation problems). This connection is pursued in detail in this
book and linked with the so-called problem of optimal multitransport (or optimal
multicoupling).
Inwritingourmonograph,wehadtwoaimsinmind:
1. To present the key aspects of optimal transportation and Wasserstein spaces
(Chaps.1 and 2) relevant to statistical inference, tailored to the interests and
backgroundofthe(mathematical)statistician.Thereare,ofcourse,classictexts
comprehensivelycoveringthisbackground.1Buttheirchoiceoftopicsandstyle
ofexpositionareusuallyadaptedtotheanalystand/orprobabilist,withaspects
mostrelevantforstatisticiansscatteredamong(much)othermaterial.
2. To make use of the ‘Wasserstein background’ to present some of the funda-
mentalsofstatisticalestimationinWassersteinspaces,anditsconnectiontothe
problem of phase variation (registration) and optimal multicoupling. In doing
so, we highlight connections with classical topics in statistical shape theory,
suchasProcrustesanalysis.Onthesetopics,nobook/monographappearstoyet
exist.
The book focusses on the theory of statistics in Wasserstein spaces. It does
not cover the associated computational/numerical aspects. This is partially due
to space restrictions, but also due to the fact that a reference entirely dedicated
to such issues can be found in the very recent monograph of Peyre´ and Cu-
turi [103]. Moreover, since this book is meant to be a rapid introduction for
non-specialists, we have made no attempt to give a complete bibliography. We
have added some bibliographic remarks at the end of each chapter, but these are
in no way meant to be exhaustive. For those seeking reference works, Rachev
[106] is an excellent overview of optimal transport up to 1985. Other recent re-
viewsareBogachevandKolesnikov[26]andPanaretosandZemel[101].Thelat-
ter review can be thought of as complementary to the present book and surveys
some of the applications of optimal transport methods to statistics and probability
theory.
1E.g.byRachevandRu¨schendorf[107],Villani[124,125],AmbrosioandGigli[10],Ambrosio
etal.[12],andmorerecentlybySantambrogio[119].
Preface ix
StructureoftheBook
Thematerialisorganisedintofivechapters.
• Chapter1presentsthenecessarybackgroundinoptimaltransportation.Starting
withMonge’soriginalformulation,itpresentsKantorovich’sprobabilisticrelax-
ation and the associated duality theory. It then focusses on quadratic cost func-
tions(squarednormedcost)andgivesamoredetailedtreatmentofcertainimpor-
tantspecialcases.Topicsofstatisticalconcernsuchastheregularityoftransport
maps andtheir stabilityunder weak convergence of theorigin/destination mea-
suresarealsopresented.Thechapterconcludeswithaconsiderationofmoregen-
eralcostfunctionsandthecharacterisationofoptimaltransportplansviacyclical
monotonicity.
• Chapter 2 presents the salient features of ((cid:2) -)Wasserstein space starting with
2
topologicalpropertiesofstatisticalimportance,aswellasmetricpropertiessuch
ascoveringnumbers.Itcontinueswithgeometricalfeaturesofthespace,review-
ing the tangent bundle structure of the space, the characterisation of geodesics,
andthelogandexponentialmapsasrelatedtotransportmaps.Finally,itreviews
the relationship between the curvature and the so-called compatibility of trans-
portmaps,roughlyspeakingwhencanoneexpectoptimaltransportmapstoform
agroup.
• Chapter 3 starts to shift attention to issues more statistical and treats the prob-
lem of existence, uniqueness, characterisation, and regularity of Fre´chet means
(barycenters) for collections of measures in Wasserstein space. This is done by
means of the so-called multimarginal transport problem (a.k.a. optimal multi-
transportoroptimalmulticouplingproblem).Thetreatmentstartswithfinitecol-
lectionsofmeasures,andthenconsidersFre´chetmeansfor(potentiallyuncount-
ably supported) probability distributions on Wasserstein space and associated
measurabilityconcerns.
• Chapter 4 considers the problem of estimation of the Fre´chet mean of a prob-
ability distribution in Wasserstein space, on the basis of a finite collection of
i.i.d. elements from this law observed with ‘sampling noise’. It is shown that
thisproblemisinextricablylinkedtotheproblemofseparationofamplitudeand
phasevariation(a.k.a.registration)ofrandompointpatters,wherethefocusison
estimating the maps yielding the optimal multicoupling rather than the Fre´chet
meanitself.Nonparametricmethodologyforsolvingeitherproblemisreviewed,
coupledwithassociatedasymptotictheoryandseveralillustrativeexamples.
• Chapter 5 focusses on the problem of actually constructing the Fre´chet mean
and/or optimal multicoupling of a collection of measures, which is a necessary
step when using the methods of Chap.4 in practice. It presents the steepest de-
scentalgorithmbasedonthegeometricalfeaturesreviewedinChap.2andacon-
vergence analysis thereof. Interestingly, it is seen that the algorithm is closely
relatedtoProcrustesalgorithmsinshapetheory,andthisconnectionisdiscussed
indepth.Severalspecialcasesarereviewedinmoredetail.
x Preface
Eachchaptercomeswithsomebibliographicnotesattheend,givingsomeback-
groundandsuggestingfurtherreading.Thefirsttwochapterscanbeusedindepen-
dently as a crash course in optimal transport for statisticians at the MSc or PhD
level depending on the audience’s background. Proofs that were omitted from the
main text due to space limitations have been organised into an online supplement
accessibleatwww.somewhere.com
Acknowledgements
Wewishtothankthreeanonymousreviewersfortheirthoughtfulfeedback.Weare
especiallyindebtedtooneofthem,whoseanalyticalinsightswereparticularlyuse-
ful.Anyerrorsoromissionsare,ofcourse,ourownresponsibility.VictorM.Panare-
tos gratefully acknowledges support from a European Research Council Starting
Grant. Yoav Zemel was supported by Swiss National Science Foundation Grant #
178220. Finally, we wish to thank Mark Podolskij and Donna Chernyk for their
patienceandencouragement.
Lausanne,Switzerland VictorM.Panaretos
Cambridge,UK YoavZemel