Data analysis using discrete cubical homology

It is a highly intuitive and powerful tool, based on a simple observation from topology that data has a shape and understanding this shape is key to analyzing the data. The main tool of topological data analysis is persistence homology, a way of quantifying n -dimensional "holes" obtained from the data in question. The essential premise behind persistence homology is known as topological inference, which is the assumption that the data is a finite sample from some large (typically infinite) topological space. One goal, therefore, is to reproduce characteristics or features of that space from the finite sample. The process typically involves several steps, namely: building a filtered graph (often called the Rips graph) from a finite metric space, taking the associated filtration of flag complexes, and finally computing homology groups of these complexes. It is worth observing that in this pipeline, (filtered) graphs are merely an intermediate step in the construction, and not an object of independent interest. In contrast, in the present paper, we work with data that is most naturally presented in the form of a graph, without an assumption that it was sampled from a topological space. For example, consider a collection of time series of stock prices of all companies traded on a given exchange.