An Introduction to Topological Data Analysis for Physicists: From LGM to FRBs
arXiv:1904.11044
Abstract
Topological Data Analysis (TDA) is a novel, and relatively new approach to analysing high-dimensional data sets. It does this by focussing on global properties like the shape and connectivity of the data giving it a significant advantage over more conventional tools based on cluster analysis, a localised property of the data. However, some of its mathematical foundations, like algebraic topology and discrete Morse theory, are perceived as an intimidatingly steep upramp into the subject. Consequently, it has enjoyed much less popularity as a data-analysis tool than less abstract methods. This article aims to change this. By focusing on a small set of simple examples, chosen primarily for their pedagogical value, we introduce and explain TDA's two principle branches; persistent homology and the Mapper algorithm. We then illustrate the universality of the method by discussing its application to the intriguing data set of fast radio burst (FRB) observations. We close the article with a discussion of the resilience of topological data analysis to noise and some statistical and computational challenges faced by the method.
arXiv comment47 pages, nearly the same number of figures
References in corpus (8)
- Catalogue of Topological Electronic Materials
- A Living Theory Catalogue for Fast Radio Bursts
- Fast radio bursts as synchrotron maser emission from decelerating relativistic blast waves
- Statistical Analysis and Parameter Selection for Mapper
- Persistent Homology and Non-Gaussianity
- Are fast radio bursts generated by cosmic string cusps?
- Topological Data Analysis for the String Landscape
- Practical Introduction to Clustering Data
Cited by in corpus (6)
- giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration
- Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology
- Topological data analysis and machine learning
- Photonic band structure design using persistent homology
- Dark soliton detection using persistent homology
- Noise-driven Topological Changes in Chaotic Dynamics