Persistent Homology in Topological Data Analysis

In this paper, we build up the theory and machinery required to understand persistent homology, and give an introductory overview to its tools and applications. At the highest level, persistent homology is an extension of homology, which is itself an extremely broad algebraic and topological field of study. Briefly, homology theory aims to concretely classify and rigorously define the notion of holes, boundaries, and volumes. These concepts are intuitively topological; for example, a sphere has a missing volume whilst a ball does not, and a torus has another hole in the middle that a sphere lacks. It turns out that these topological differences can be attacked algebraically, by essentially breaking the spaces down into closed ‘cycles’ of various dimensions, which may or may not be the boundaries of regions. The novelty of persistent homology is in applying homology theory to ‘uncertain’ spaces. The precise meaning is this: with homology, we are given a space, and then tasked with finding its holes. Persistent homology (and more generally the area of topological data analysis) aims to recover a topological space (characterized by its holes), given some set of incomplete information. The most usual case, which persistent homology deals with, would be sets of point cloud data, which we can consider to be finite subsets of arbitrary metric spaces. Obviously, we can give the data the discrete metric, but this might as well be topologically trivial. Instead, we want to image the data at various spatial resolutions; more precisely, we want to fatten up the point cloud space by looking at the covering space generated by taking larger and larger balls around each point. One can imagine, for example, a set of points sampled from a torus. As we increase the size of the balls around the points, the covering space gets fatter and fatter until we have something resembling a ‘bumpy’ torus. This is topologically a torus nonetheless, and we have recovered (in some loose sense, with some uncertainty) the original topological structure of the data. Unlike traditional methods of data analysis, which can also perform these kinds of classification tasks, persistent homology as constructed is dimensionless. We need not constrain ourselves to lower dimensional datasets, and can instead construct or