Applications of Topology to Data Analysis

Abstract

This thesis aims to serve as an introduction to Topological Data Analysis (TDA), a collection of methods that seek to quantify the topological and geometric features of data using algebraic topology. The theory behind persistent homology, a stable multi-scale approach for characterizing the structure of data, is presented here. Further, an algorithm to compute persistence diagrams, a standard representation of persistent homology, is also discussed. An overview of some stable vectorized representations of persistent homology that are better suited for statistical and machine learning tasks is also given. The remainder of the thesis addresses how these techniques can help analyze images and time series data. Subsequently, a topological pipeline for image classification is put forth. Application of TDA to biological images and financial time series data is also presented to motivate the broad scope of these techniques.