Distance in Topological Data Analysis

Data analysis has undergone a remarkable transformation in recent years, spurred by the explosion of data generated from diverse sources ranging from scientific experiments to social media interactions. Topological data analysis (TDA) is a recent field of applied mathematics that provides algebraic topology and computational geometry tools for inferring relevant information from possibly complex data. Persistent homology is a crucial concept in TDA because it identifies and tracks topological features across multiple resolutions, thereby providing a robust summary of shape and structure in the data. It turns out that the classical Hausdorff distance is unsuitable for comparing simplicial complexes, prompting the need for a revised notion of distance on the space of finite sets of simplices in R^d. This dissertation is centered around a simplicial complex metric. By focusing on individual abstract simplicial complexes before they are incorporated into filtered complexes, this work contributes to the preprocessing and representation of data before persistence is applied. Moreover, noting that F₂-homology cycle representatives may be described as simplicial subcomplexes, we also discuss the notion of optimal homology bases as a concise summary of homology present in a simplicial complex. By computing distances between individual simplicial complexes, this work offers a means to assess the similarity of topological features encoded by homology cycles.