[Paper Review] Topological data analysis and machine learning

Daniel Leykam, Dimitris G. Angelakis|arXiv (Cornell University)|Jun 30, 2022

Topological and Geometric Data Analysis4 citations

TL;DR

This paper reviews topological data analysis (TDA) applications in physics and machine learning, emphasizing persistent homology for detecting phase transitions and extracting topological features from complex datasets. It demonstrates how TDA compresses high-dimensional data into interpretable topological invariants, enabling efficient machine learning pipelines and offering a robust alternative to traditional methods in condensed matter and quantum systems.

ABSTRACT

Topological data analysis refers to approaches for systematically and reliably computing abstract ``shapes'' of complex data sets. There are various applications of topological data analysis in life and data sciences, with growing interest among physicists. We present a concise yet (we hope) comprehensive review of applications of topological data analysis to physics and machine learning problems in physics including the detection of phase transitions. We finish with a preview of anticipated directions for future research.

Motivation & Objective

• To bridge the gap between topological data analysis (TDA) and physics by making TDA techniques accessible to physicists unfamiliar with the field.
• To demonstrate how TDA, particularly persistent homology, can extract meaningful topological features from high-dimensional, noisy physical datasets.
• To show that TDA features can serve as effective, low-dimensional inputs for machine learning models, reducing reliance on computationally expensive neural networks.
• To explore the potential of quantum TDA algorithms for future exponential speedups in topological feature computation.
• To advocate for TDA as a standard, reliable tool for analyzing complex physical systems, from classical to quantum materials.

Proposed method

• Uses persistent homology to compute Betti numbers across filtration parameters, capturing topological features like loops and voids in point clouds.
• Applies the Mapper algorithm and persistence diagrams to visualize and quantify shape differences in physical systems, such as spin configurations or particle distributions.
• Integrates TDA-derived features into machine learning pipelines as input representations, replacing raw data or high-dimensional embeddings.
• Proposes quantum algorithms for computing combinatorial Laplacians and persistent Betti numbers, enabling potential exponential speedups on fault-tolerant quantum computers.
• Employs quantum phase estimation and shallow quantum circuits to implement boundary operators and detect cycles in simplicial complexes.
• Uses representative cycles and visual insets in persistence diagrams to improve interpretability and link abstract topological features to physical structures.

Experimental results

Research questions

• RQ1 How can topological data analysis be used to detect phase transitions in physical systems without prior knowledge of order parameters?
• RQ2 What are the most effective TDA techniques for extracting interpretable, robust features from high-dimensional, noisy physical data?
• RQ3 In what ways can TDA features improve the performance and interpretability of machine learning models in physics applications?
• RQ4 What are the prospects and limitations of quantum algorithms for accelerating topological data analysis in physical systems?
• RQ5 How can TDA be made accessible and trustworthy for non-specialist physicists in condensed matter and quantum materials research?

Key findings

• Persistent homology successfully identifies topological features such as loops and clusters in spin systems, enabling unsupervised detection of phase transitions.
• TDA-derived features serve as effective, low-dimensional inputs for machine learning, outperforming raw data in some cases and reducing computational cost.
• Quantum TDA algorithms, such as those using quantum phase estimation, can efficiently compute combinatorial Laplacians and detect zero modes corresponding to persistent cycles.
• Recent quantum algorithms show potential for exponential speedups in computing persistent Betti numbers and Wasserstein distances, though practical realization remains challenging.
• Visualizing representative cycles alongside persistence diagrams enhances interpretability and links abstract topological features to physical structures.
• TDA has already demonstrated superior performance in complex problems such as predicting biomolecule binding affinities, suggesting strong potential in physics applications.