Simplicial Complexes and Persistent Homology

Abstract

Topological data analysis is a novel field of mathematics in which topology is applied to problems in data analysis. In this paper we will study a particular tool from topological data analysis called persistent homology. First, we develop the theory of simplicial complexes and their topological counterparts polyhedra. We then define a particular type of homology theory called simplicial homology which, intuitively, captures n-dimensional `holes' of polyhedra. As a motivating example for the definition of persistence homology, we will consider the Cech complex of a point cloud X in euclidean space. The Cech complex is constructed as a parametrized simplicial complex, that for each choice of a parameter gives a simplicial homology. By looking at how this simplicial homology changes as we go from one choice of the parameter to another, we then define the persistence homology of a filtration. The persistence diagram of a filtration turns out to be a particularly useful representation of persistence homology. Furthermore the space of all persistence diagrams admits a metric called the bottleneck distance with which we can compare the persistence diagrams of different filtrations. Lastly, we show an important result called The Bottleneck Stability which ensures that the bottleneck distance is stable with respect to small perturbations of the filtration.