[Paper Review] An Introduction to Topological Data Analysis for Physicists: From LGM to FRBs

Jeff Murugan, Duncan Robertson|arXiv (Cornell University)|Apr 24, 2019

Topological and Geometric Data AnalysisComputer Science19 references17 citations

TL;DR

This paper introduces Topological Data Analysis (TDA) to physicists, focusing on persistent homology and the Mapper algorithm as tools to extract topological structure from noisy, high-dimensional data. It demonstrates their application to fast radio burst (FRB) data, revealing spatial clustering patterns, and highlights TDA’s robustness to noise and potential for statistical inference in astrophysics and cosmology.

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.

Motivation & Objective

• To bridge the gap between topological data analysis (TDA) and physics by making TDA accessible to physicists.
• To address the limitations of traditional clustering and dimensionality reduction in high-dimensional, noisy data common in astrophysics and particle physics.
• To demonstrate the utility of TDA—specifically persistent homology and the Mapper algorithm—for identifying topological patterns in real astrophysical datasets like FRBs.
• To provide a pedagogical foundation for TDA techniques, emphasizing their robustness to noise and geometric invariance.
• To advocate for TDA as a powerful tool for uncovering hidden structures in complex data, with applications in cosmology, quantum materials, and high-energy physics.

Proposed method

• Construct a simplicial complex from data points using a filtration based on distance thresholds (e.g., ε-neighborhoods), enabling topological analysis across multiple scales.
• Apply persistent homology by computing homology groups and Betti numbers across a filtration, tracking topological features (connected components, loops, voids) as the scale parameter ε varies.
• Use the Euler-Poincaré formula to relate Betti numbers and the alternating sum of simplices in the complex, enabling efficient computation via linear algebra.
• Implement the Mapper algorithm by applying a filter function to data, clustering points within overlapping intervals, and constructing a graph that captures the data’s topological structure.
• Visualize results using persistence diagrams and barcodes to represent the birth and death of topological features, and Mapper graphs to reveal clustering and connectivity patterns.
• Validate robustness by applying TDA to noisy synthetic data and real FRB data, showing that topological features persist despite perturbations.

Experimental results

Research questions

• RQ1 How can topological data analysis (TDA) be used to extract meaningful, robust topological features from high-dimensional, noisy astrophysical data such as FRB observations?
• RQ2 To what extent do persistent homology and the Mapper algorithm preserve the underlying topological structure of data when noise is introduced?
• RQ3 Can TDA reveal spatial clustering patterns in FRB data that are not apparent through standard statistical or clustering methods?
• RQ4 What are the computational and statistical challenges in applying TDA to real-world datasets, and how can they be mitigated?
• RQ5 How can TDA be extended to infer cosmological parameters or characterize the topology of large-scale structures in the universe?

Key findings

• Persistent homology and the Mapper algorithm successfully recover the underlying topological structure of data, even when the data is corrupted by mild noise.
• The Mapper algorithm is computationally efficient and capable of processing datasets of 2000 points in seconds on a mid-range laptop, making it practical for real-world applications.
• Application of TDA to FRB data reveals spatial clustering patterns that suggest non-uniform distribution, indicating potential physical or astrophysical structure.
• Persistence diagrams applied to CMB anisotropy data can constrain cosmological parameters such as local non-Gaussianity to Δf_NL = 35.8 at 68% confidence level, demonstrating proof-of-concept for cosmological inference.
• Statistical inference tools for TDA outputs—such as confidence regions for persistence diagrams and Mapper graphs—have been developed, enabling rigorous topological inference.
• Despite computational costs, especially for persistent homology on high-dimensional data, TDA offers a robust, coordinate- and metric-independent approach to data analysis that complements traditional statistical methods.