Approximation in kernel-based spaces, optimal subspaces and approximation of eigenfunctions

110 Pages·2016·1.7 MB·English

Sede Amministrativa: Università degli Studi di Padova
Dipartimento di Matematica
___________________________________________________________________
SCUOLA DI DOTTORATO DI RICERCA IN SCIENZE MATEMATICHE
INDIRIZZO MATEMATICA COMPUTAZIONALE
CICLO XXVIII
Approximation in kernel-based spaces, optimal subspaces and
approximation of eigenfunctions
Direttore della Scuola : Ch.mo Prof. Pierpaolo Soravia
Coordinatore d’indirizzo: Ch.mo Prof. Michela Redivo Zaglia
Supervisore :Ch.mo Prof. Stefano De Marchi
Dottorando: Gabriele Santin

ABSTRACT
Kernel-basedapproximationmethodsprovideoptimalrecoveryproce-
dures in the native Hilbert spaces in which they are reproducing. Among
other, kernels in the notable class of continuous and strictly positive def-
inite kernels on compact sets possess a series decomposition in L - or-
2
thonormaleigenfunctionsofaparticularintegraloperator.
The interest for this decomposition is twofold. On one hand, the sub-
spaces generated by eigenfunctions, or eigenbasis elements, are L -optimal
2
trialspacesinthesenseofwidths. Ontheotherhand,suchexpansionisthe
fundamentaltoolofsomeofthestateoftheartalgorithmsinkernelapprox-
imation. Despite these reasons motivate a great interest in the eigenbasis,
foragivenkernelthisdecompositionisgenerallycompletelyunknown.
In this view, this thesis faces the problem of approximating the eigen-
basisofgeneralcontinuousandstrictlypositivedefinitekernelsongeneral
compactsetsoftheEuclideanspace,foranyspacedimension.
We will at first define a new kind of optimality that is based on a error
measurement closer to the one of standard kernel interpolation. This new
width is then analyzed, and we will determine its value and characterize
its optimal subspaces, which are spanned by the eigenbasis. Moreover,
thisoptimalityresultissuitabletobescaledtosomeparticularsubspaceof
thenativespace, andthisrestrictionallowsustoprovenewresultsonthe
construction of computable optimal trial spaces. This situation covers the
standardcaseofpoint-basedinterpolation,andwillprovidealgorithmsto
approximate the eigenbasis by means of standard kernel techniques. On
thebasisofthenewtheoreticalresults,asymptoticestimatesontheconver-
genceofthemethodwillbeproven. Thesecomputationswillbetranslated
intoeffectivealgorithms,andwewilltesttheirbehaviorintheapproxima-
tionoftheeigenspaces. Moreover, twoapplicationsofkernel-basedmeth-
odswillbeanalyzed.
iii

RIASSUNTO
Imetodikernelfornisconoproceduredimigliorapprossimazionenegli
spazi di Hilbert nativi, ovvero gli spazi in cui tali kernel sono reproducing
kernel. Nel caso notevole di kernel continui e strettamente definiti positivi
su insiemi compatti, e` nota l’esistenza di una decomposizione in una se-
rie data dalle autofunzioni (ortonormali in L ) di un particolare operatore
2
integrale.
L’interesse per questa espansione e` motivata da due ragioni. Da un
lato, i sottospazi generati dalle autofunzioni, o elementi della eigenbasis,
sono i trial space L -ottimali nel senso delle widhts. D’altro canto, tale es-
2
pansionee` lostrumentofondamentaleallabaseinalcunideglialgoritmidi
riferimento utilizzati nell’approssimazione con kernel. Nonostante queste
ragioni motivino decisamente l’interesse per le eigenbasis, la suddetta de-
composizionee` generalmentesconosciuta.
Alla luce di queste motivazioni, la tesi affronta il problema dell’ap-
prossimazione delle eigenbasis per generici kernel continui e strettamente
definitipositivisugenericiinsiemicompattidellospazioeuclideo,perogni
dimensione.
Inizieremo col definire un nuovo tipo di ottimalita` basata sulla misura
dell’errore tipica dell’interpolazione kernel standard. Il nuovo concetto di
widthsara` analizzato, nesara` calcolatoilvaloreecaratterizzatiirispettivi
sottospazi ottimali, che saranno generati dalla eigenbasis. Inoltre, questo
risultato di ottimalita` risultera` essere adatto ad essere ristretto ad alcuni
particolari sottospazi dello spazio nativo. Questa restrizione ci permet-
tera´ di dimostrare nuovi risultati sulla costruzione di trial space ottimali
che siano effettivamente calcolabili. Questa situazione include anche il
casodell’interpolazionekernelbasatasuvalutazionipuntuali,efornira` al-
goritmi per approssimare le autofunzioni tramite metodi kernel standard.
Forniremo inoltre stime asintotiche di convergenza del metodo basate sui
nuovi risultati teorici. I metodi presentati saranno implementati in algo-
ritminumerici,enetesteremoilcomportamentonell’approssimazionedegli
autospazi. Infine analizzeremo l’applicazioni dei metodi kernel a due di-
versiproblemidiapprossimazione.
v

ACKNOWLEDGMENTS
First and foremost, I wish to thank my Advisor Prof. Stefano De Marchi,
who introduced me to this fascinating topic and pushed me toward inter-
estingquestionsandproblems. Hehasbeenamentorinabroadsense,and
hegavememanygreatopportunities.
My gratitude goes also to all the members of the research group Con-
structive Approximation and Applications at the University of Padova and
Verona. Since the writing of my Bachelor thesis I found there an exciting
and open environment, always committed to motivating and promoting
youngstudents.
Thecentralpartofthisthesishasbeenwrittenthankstothesupervision
of Prof. Robert Schaback. Working with him has been a pleasure and an
honor.
I would like to thank Prof. Gregory Fasshauer, who accepted to be the
Referee of this thesis, and whose comments and suggestions on the first
draftcontributedtoimprovethiswork.
ManythanksalsotothemembersofmynewresearchgroupNumerical
Mathematics at the University of Stuttgart, and especially to Prof. Bernard
Haasdonk,whogavemeallthetimeIneededtofinalizethisthesis.
Not mentioning everything my family did and do for me, my parents
Maria Antonia and Angelo teached and always encouraged me to follow
myaspirations,whileputtingthemintherightperspective. Specialthanks
to them, and to my brother and sisters Caterina, Francesco e Cecilia, with
Laura,TobiaandRiccardo.
IwishtothankalsoEmilia,LuciaandFulvio,whoembracedmeasina
secondfamily.
TheyearsinPadovawouldnothavehadthesamemeaningwithoutmy
friendsandcolleaguesAlessio,Anna,Andrea,Cristina,Daniele,Daria,Da-
vide,Davide,Federico,Francesco,Giovanni,Laura,Luigi,Luisa,Mariano,
Marta,Matteo,MicheleandValentina,withwhomIsharedcountlessbeau-
tifulmoments. Manythankstoeachofthem,andinparticulartoLaurafor
her priceless friendship, and to Federico, who shared with me his passion
forMathematics,andwhooftenbelievedinmemuchmorethanhowIdid
bymyself.
Finallyandaboveall,thankstoMaria.
vii

Contents
Abstract iv
Riassunto vi
Acknowledgments viii
Introduction 1
1 Preliminaries 5
1.1 Kernelsandkernel-basedspaces . . . . . . . . . . . . . . . . 5
1.2 Kernel-basedinterpolationandapproximation . . . . . . . . 7
1.3 Errorbounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 EmbeddinginL (Ω)andtheeigenbasis . . . . . . . . . . . . 10
2
1.5 Generalizedinterpolation . . . . . . . . . . . . . . . . . . . . 11
2 Optimalsubspacesandapproximationofeigenfunctions 15
2.1 Generalsubspacesofthenativespace . . . . . . . . . . . . . 16
2.1.1 Kernel-basedapproximation . . . . . . . . . . . . . . 16
2.1.2 Otherapproximationprocesses . . . . . . . . . . . . . 18
2.2 OptimalsubspaceswithrespecttothePowerFunction . . . 19
2.2.1 Reviewoftheknownresults . . . . . . . . . . . . . . 19
2.2.2 OptimalitywithrespecttothePowerFunction . . . . 22
2.3 Computableoptimalsubspaces . . . . . . . . . . . . . . . . . 23
2.3.1 Restrictiontoaclosedsubspace . . . . . . . . . . . . . 24
2.3.2 Thecaseofp . . . . . . . . . . . . . . . . . . . . . . . 26
n
2.3.3 Thecaseofκ . . . . . . . . . . . . . . . . . . . . . . . 27
n
2.3.4 Approximationofthetrueeigenspaces . . . . . . . . 28
2.4 Convergenceofthediscreteeigenvalues . . . . . . . . . . . . 29
2.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 Intermediatespaces . . . . . . . . . . . . . . . . . . . 30
2.5.2 Recoveryoflinearfunctionals . . . . . . . . . . . . . . 32
ix
3 Algorithms 35
3.1 Backgroundfactsonchangeofbasis . . . . . . . . . . . . . . 36
3.2 Orthogonalbasiswithrespecttotwogeneralinnerproducts 37
3.2.1 Directconstructionofthediscreteeigenbasis . . . . . 39
3.3 WeightedSingularValueDecompositionbasis . . . . . . . . 40
3.3.1 Motivationanddefinition . . . . . . . . . . . . . . . . 41
3.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.3 Weighteddiscreteleast-squaresapproximation . . . . 45
3.4 Connectionsbetweenthetwoalgorithms . . . . . . . . . . . 48
4 Computationalaspectsandnumericalexperiments 51
4.1 Greedyconstructionofthediscreteeigenbasis . . . . . . . . 52
4.1.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 FastcomputationoftheWSVDbasis . . . . . . . . . . . . . . 59
4.2.1 TheLanczosmethodandtheapproximationoftheSVD 60
4.2.2 Constructionofthebasis . . . . . . . . . . . . . . . . . 60
4.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.4 Approximation . . . . . . . . . . . . . . . . . . . . . . 63
4.2.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Applications 71
5.1 Partitionofunitywithlocalstabilization . . . . . . . . . . . . 71
5.1.1 BackgroundfactsonthePUM . . . . . . . . . . . . . 72
5.1.2 Thealgorithm . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 ReconstructionofmedicalimagesfromCTscans . . . . . . . 75
5.2.1 BackgroundfactsontheRadontransform . . . . . . . 76
5.2.2 Kernel-basedreconstruction . . . . . . . . . . . . . . . 77
5.2.3 Symmetricformulation . . . . . . . . . . . . . . . . . 78
5.2.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 83
Appendices
A Constructionofnativespacesfromsequences 89
A.1 Nativespacefromsequences . . . . . . . . . . . . . . . . . . 89
A.2 Relationbetweendifferentnativespaces . . . . . . . . . . . . 90
A.3 Projectorsandsubspaces . . . . . . . . . . . . . . . . . . . . . 91
Bibliography 93
x