Tame representations in Topological Data Analysis: decompositions, invariants and metrics
Time: Thu 2023-06-15 14.00
Location: F3, Lindstedtsvägen 26 & 28, Stockholm
Subject area: Mathematics
Doctoral student: Francesca Tombari , Matematik (Inst.)
Opponent: Peter Bubenik, University of Florida
Supervisor: Professor Wojciech Chachólski, Matematik (Avd.); Professor Patrizio Frosini, University of Bologna
This thesis is a compilation of results that can be framed within the field of applied topology. The starting point of our study is objects presenting a possibly complex intrinsic geometry. The main goal is then to simplify, without trivializing, the geometric information characterising these objects by choosing an appropriate representation. Thus, besides being simple and compact, the chosen representation should maintain the wealth of features of the initial object.In Topological Data Analysis (TDA), this simplification process can be done by assigning to each geometric object a functor indexed by a suitable poset.The most important fact about these functors is that, under appropriate hypotheses on the geometric object, they are discretisable. Being discretisable in this context means that they can be finitely encoded by a finite poset mapping to the original indexing poset. It is then possible to make one step further by computing invariants for the representations obtained. Desirable features for such invariants are to be effectively computable and suitable to describe metrics on. Comparing them gives, in fact, a good approximation of the comparison of the underlying geometric objects which are our primary interest.
Paper A studies decompositions of simplicial complexes that are induced by coverings of their vertices. These decompositions are inspired by data analysis where commonly the data is given by a distance space, to which a filtered simplicial complex can be associated. We study how the homotopy type of a decomposed complex differs from the initial one, both for generic and for metric simplicial complexes.
Another model to perform data analysis from a topological perspective is given by the theory of group equivariant nonexpansive operators.In Paper B, we show that such operators form a complementary tool to persistent homology in the context of TDA. We propose a categorical structure incorporating both models and then we study the functoriality of persistence.
In Paper C we investigate suitable indexing posets for tame functors. The attention is focused on upper semilattices, which are particularly well suited for this purpose. Another class of posets that have similar properties to upper semilattices is the one of realisations, which we introduce here. Their similarities are both combinatorial, in particular concerning a notion of dimension that we introduce, and related to homological algebra for the tame functors indexed by them. In Paper C we also propose a method based on Koszul complexes to compute homological invariants for tame functors indexed either by upper semilattices or realisations. This question is then expanded in Paper D, where we study homological invariants relative to a chosen class of projectives, possibly different to the standard ones. We propose a framework to translate from the relative to the standard setting, where Koszul complexes are available to perform the computations. We also identify an obstruction for such translation to be possible and characterise it for several examples of relative projectives.
In Paper E we study the geometrical properties of a well-established metric in 2-parameter persistent homology, called the matching distance. Motivated by the need for effectiveness in the computation of such metric, we study its geometric properties.In particular, we show how to take advantage of the differential geometric structure of the underlying objects to understand the properties of the metric.
In Paper F we study the category of discretisable functors with values in non-negative chain complexes. In this category, we are particularly interested in cofibrant indecomposables, which require a model structure to be defined. Thus, we first identify a new class of posets indexing the functors for which a projective model structure exists and give a characterisation of cofibrant indecomposables there. In the case, the indexing poset is not of this type, we outline a technique to construct arbitrarily complicated cofibrant indecomposables.