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.

3.

Scolamiero, Martina

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

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.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Chachólski, Wojciech

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Lundman, Anders

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Ramanujam, Ryan

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Öberg, Sebastian

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Multidimensional Persistence and Noise2017In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 17, no 6, p. 1367-1406Article in journal (Refereed)

Abstract [en]

In this paper, we study multidimensional persistence modules (Carlsson and Zomorodian in Discrete Comput Geom 42(1):71–93, 2009; Lesnick in Found Comput Math 15(3):613–650, 2015) via what we call tame functors and noise systems. A noise system leads to a pseudometric topology on the category of tame functors. We show how this pseudometric 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 one-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.