Persistent Homology via Quotient Spaces

Abstract

A point cloud can be endowed with a topological structure by constructing a simplicial complex using the points as vertices. Instead of assigning a single simplicial complex, Topological Data Analysis (TDA) employs multiple simplicial complexes, each representing the point cloud at a different resolution. These combine to form a filtration: a nested sequence of simplicial complexes which gives rise to persistent homology, a useful tool able to extract topological information from the point cloud. The Vietoris-Rips filtration is a popular choice in TDA, mainly for its simplicity and easy implementation for high-dimensional point clouds. Unfortunately, this filtration is often too large to construct fully. We introduce in this thesis a way of reducing a simplicial complex by identifying its vertices. Applying this technique to each simplicial complex in the Vietoris-Rips filtration results in a smaller filtration that can be shown to approximate the Vietoris-Rips filtration in terms of persistent homology.