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Multidimensional Persistence and Noise
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0001-6007-9273
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(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-168011OAI: oai:DiVA.org:kth-168011DiVA: diva2:813850
Note

QS 2015

Available from: 2015-05-25 Created: 2015-05-25 Last updated: 2016-05-16Bibliographically approved
In thesis
1. Invariants for Multidimensional Persistence
Open this publication in new window or tab >>Invariants for Multidimensional Persistence
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The amount of data that our digital society collects is unprecedented. This represents a valuable opportunity to improve our quality of life by gaining insights about complex problems related to neuroscience, medicine and biology among others. Topological methods, in combination with classical statistical ones, have proven to be a precious resource in understanding and visualizing data. Multidimensional persistence is a method in topological data analysis which allows a multi-parameter analysis of a dataset through an algebraic object called multidimensional persistence module. Multidimensional persistence modules are complicated and contain a lot of information about the input data. This thesis deals with the problem of algorithmically describing multidimensional persistence modules and extracting information that can be used in applications. The information we extract, through invariants, should not only be efficiently computable and informative but also robust to noise.

In Paper A we describe in an explicit and algorithmic way multidimensional persistence modules. This is achieved by studying the multifiltration of simplicial complexes defining multidimensional persistence modules. In particular we identify the special structure underlying the modules of n-chains of such multifiltration and exploit it to write multidimensional persistence modules as the homology of a chain complex of free modules. Both the free modules and the homogeneous matrices in such chain complex can be directly read off the multifiltration of simplicial complexes.

Paper B deals with identifying stable invariants for multidimensional persistence. We introduce an algebraic notion of noise and use it to compare multidimensional persistence modules. Such definition allows not only to specify the properties of a dataset we want to study but also what should be neglected. By disregarding noise the, so called, persistent features are identified. We also propose a stable discrete invariant which collects properties of persistent features in a multidimensional persistence module.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2015. vii, 31 p.
Series
TRITA-MAT-A, 2015:07
Keyword
computational topology
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-167644 (URN)978-91-7595-613-8 (ISBN)
Public defence
2015-06-11, F3, Lindstedtsvagen 26, KTH, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20150525

Available from: 2015-05-25 Created: 2015-05-22 Last updated: 2015-05-25Bibliographically approved
2. Homotopy Theory and TDA with a View Towards Category Theory
Open this publication in new window or tab >>Homotopy Theory and TDA with a View Towards Category Theory
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains three papers. Paper A and Paper B deal with homotopy theory and Paper C deals with Topological Data Analysis. All three papers are written from a categorical point of view.

In Paper A we construct categories of short hammocks and show that their weak homotopy type is that of mapping spaces. While doing this we tackle the problem of applying the nerve to large categories without the use of multiple universes. The main tool in showing the connection between hammocks and mapping spaces is the use of homotopy groupoids, homotopy groupoid actions and the homotopy fiber of their corresponding Borel constructions.

In Paper B we investigate the notion of homotopy commutativity. We show that the fundamental category of a simplicial set is the localization of a subset of the face maps in the corresponding simplex category. This is used to define ∞-homotopy commutative diagrams as functors that send these face maps to weak equivalences. We show that if the simplicial set is the nerve of a small category then such functors are weakly equivalent to functors sending the face maps to isomorphisms. Lastly we show a connection between ∞-homotopy commutative diagrams and mapping spaces of model categories via hammock localization.

In Paper C we study multidimensional persistence modules via tame functors. By defining noise systems in the category of tame functors we get a pseudo-metric topology on these functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

Abstract [sv]

Denna avhandling innehåller tre artiklar. Artikel A och Artikel B handlar om homotopiteori och Artikel C handlar om topologisk dataanalys. Alla tre artiklar är skrivna från en kategorisk synvinkel.

I Artikel A konstruerar vi kategorier av korta hängmattor och visar att dess svaga homotopityper är ekvivalenta med avbildningsrum. Samtidigt som vi gör detta så tacklar vi även problemet med att applicera nerv-funktorn till stora kategorier utan att använda sig av multipla universum. Huvudverktyget för att visa kopplingen mellan hängmattor och avbildningsrum är användandet av homotopigruppoider, deras verkan samt den homotopiska fibern av deras respektive Borel-konstruktioner.

I Artikel B undersöker vi konceptet homotopisk kommutativitet. Vi visar att fundamentalkategorin hos en simpliciell mängd är lokaliseringen av en delmängd av sido-avbildningarna i den korresponderande simpliciella kategorin. Detta används för att definiera ∞-homotopiskt kommuterande diagram som funktorer som skickar dessa sido-avbildningar till svaga ekvivalenser. Vi visar att om den simpliciella mängden är nerven av en liten kategori så är sådana funktorer svagt ekivalenta till funktorer som skickar sido-avbildningarna till isomorfier. Slutligen så visar vi på en koppling mellan ∞-homotopiskt kommuterande diagram och avbildningsrum hos modellkategorier via hängmatte-lokalisering.

I Artikel C studerar multidimensionella persistensmoduler via tama funktorer. Genom att definiera brussystem i kategorin av tama funktorer så får vi en pseudo-metrisk topologi på dessa funktorer. Vi visar hur denna pseduo-metrik kan användas för att identifiera persistenta egenskaper hos kompakta multidimensionella persistensmoduler. För att räkna antalet sådana persistenta egenskaper så introducerar vi karakteristik-räknings-invarianten och visar att tilldelandet av denna variant till kompakta tama funktorer är en 1-Lipschitz operation. För endimensionell persistens så förklarar vi hur, genom att välja lämpigt brussystem, karakteristik-räknings-invarianten identifierar samma persistenta egenskaper som den streckkods-konstruktionen.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. 23 p.
Series
TRITA-MAT-A, 2016:05
Keyword
Homotopy theory, Topological Data Analysis, Category theory, Mapping spaces, Homotopy commutative diagrams
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-186189 (URN)978-91-7729-003-2 (ISBN)
Public defence
2016-06-07, Kollegiesalen, Brinellvägen 8, Stockholm, 15:00 (English)
Opponent
Supervisors
Note

QC 20160516

Available from: 2016-05-16 Created: 2016-05-04 Last updated: 2016-05-16Bibliographically approved

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