Doctor of Philosophy
Christensen, Daniel J.
This thesis presents a uniform treatment of different distances used in the applied topology literature. We introduce the notion of a locally persistent category, which is a category with a notion of approximate morphism that lets one define an interleaving distance on its collection of objects. The framework is based on a combination of enriched category theory and homotopy theory, and encompasses many well-known examples of interleaving distances, as well as weaker notions of distance, such as the homotopy interleaving distance and the Gromov–Hausdorff distance.
We show that the approach is not only an organizational tool, but a useful theoretical tool that allows one to formulate simple conditions under which a certain construction is stable, or under which an interleaving distance is, e.g., complete and geodesic. Being based on the well-developed theory of enriched categories, constructions in the theory of interleavings can be conveniently cast as enriched universal constructions.
We give several applications. We generalize Blumberg and Lesnick's homotopy interleaving distance to categories of persistent objects of a model category and prove that this distance is intrinsic and complete. We identify a universal property for the Gromov–Hausdorff distance that gives simple conditions under which an invariant of metric spaces is stable. We define a distance for persistent metric spaces, a generalization of filtered metric spaces, that specializes to known distances on filtered metric spaces and dynamic metric spaces, and use it to lift stability results for invariants of metric spaces to invariants of persistent metric spaces. We present a new stable invariant of metric measure spaces, the kernel density filtration, that encodes the information of a kernel density estimate for all choices of bandwidth. We study the interleaving distance in the category of persistent sets and show that, when restricted to a well-behaved subcategory that in particular contains all dendrograms and merge trees, one gets a complete and geodesic distance.
We relate our approach to previous categorical approaches by showing that categories of generalized persistence modules and categories with a flow give rise to locally persistent categories in a way that preserves both metric and categorical structure.
Summary for Lay Audience
Algorithms in data science often require an input as well as a choice of parameters. In order to avoid arbitrary choices, one can study the evolution of the output of the algorithms as the parameters range over all possible choices.
In the context of applied topology, many algorithms first construct a representation of a topological space and then compute an invariant of this space. For example, many clustering algorithms work by computing the connected components of a graph that encodes some of the topology of the data set. When letting the parameters range over all possible choices, instead of constructing a single topological space, the algorithm constructs a persistent topological space, that is, a topological space parametrized by the poset of real numbers, and then computes an invariant of this persistent space, yielding a parametrized invariant. For example, the connected components of a topological space give a clustering of the space, while the connected components of a persistent topological space give a hierarchical clustering. Parametrized invariants are often stable, meaning that they are robust to small perturbations of the input dataset, making them a convenient practical tool. Parametrized invariants are studied by Topological Persistence.
It was observed in the work of Chazal, Cohen-Steiner, Glisse, Guibas, Oudot, Bubenik, Scott, Lesnick, and others that category theory can be used to organize and strengthen stability and consistency results about topological persistence methods. Categories are used to group mathematical objects with comparable structures together, such as the collection of all topological spaces.
This thesis studies a notion of category whose objects can be treated as persistent or parametrized objects. We show the benefits of this approach by recovering and generalizing previous results in the persistence literature in a uniform way, as well as giving new applications.
Scoccola, Luis N., "Locally Persistent Categories And Metric Properties Of Interleaving Distances" (2020). Electronic Thesis and Dissertation Repository. 7119.
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