Persistent Homology Techniques for Big Data and Machine Intelligence: A Survey

Abstract

Persistent homology (PH) is a powerful burgeoning technique from Topological data analysis (TDA) that leverages machinery drawn from algebraic topology. PH records the appearance and disappearance of essential topological features of an object that persist across various scales or resolutions, and it is immune to noise. PH is independent of parameters, dimension and coordinates. In recent years, PH is catching an eye of the machine learning community due to the challenges offered by nature of data available today. But due to unapproachable introductory literature on PH, beginners who have the mathematical aptitude but no background in algebraic topology find it difficult to understand the concepts and techniques involved. On the other side, researchers are working effortlessly to bring together machine learning and TDA, as both combined can do wonders. This paper is an attempt to introduce the theory of PH step by step for the beginner and illustrate the concepts involved by using toy data sets. The main purpose of this research work is to introduce extensions of PH that helps researchers to apply tools from statistics and machine learning.