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Invariants for Multidimensional Persistence
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Algebra and Geometry)ORCID iD: 0000-0001-6007-9273
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 [en]
computational topology
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-167644ISBN: 978-91-7595-613-8 (print)OAI: oai:DiVA.org:kth-167644DiVA: diva2:813485
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
List of papers
1. Combinatorial presentation of multidimensional persistent homology
Open this publication in new window or tab >>Combinatorial presentation of multidimensional persistent homology
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A multifiltration is a functor indexed by Nr that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural Nr-graded R[x1,…,xr]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the Nr-graded R[x1,…,xr]-modules that can occur as R-spans of multifiltrations of sets are the direct sums of monomial ideals.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-168007 (URN)
Note

QS 2015

Available from: 2015-05-25 Created: 2015-05-25 Last updated: 2015-05-25Bibliographically approved
2. Multidimensional Persistence and Noise
Open this publication in new window or tab >>Multidimensional Persistence and Noise
Show others...
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-168011 (URN)
Note

QS 2015

Available from: 2015-05-25 Created: 2015-05-25 Last updated: 2016-05-16Bibliographically approved

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Scolamiero, Martina

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