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We explore the evolution of daily returns of four major US stock market indices during the technology crash of 2000, and the financial crisis of 2007-2009. Our methodology is based on topological data analysis (TDA). We use persistence…

Mathematical Finance · Quantitative Finance 2017-11-22 Marian Gidea , Yuri Katz

Topological Data Analysis (TDA) is an emergent field that aims to discover topological information hidden in a dataset. TDA tools have been commonly used to create filters and topological descriptors to improve Machine Learning (ML)…

Machine Learning · Computer Science 2021-02-09 Rolando Kindelan , José Frías , Mauricio Cerda , Nancy Hitschfeld

Portfolio management is essential for any investment decision. Yet, traditional methods in the literature are ill-suited for the characteristics and dynamics of cryptocurrencies. This work presents a method to build an investment portfolio…

Portfolio Management · Quantitative Finance 2020-09-09 Rodrigo Rivera-Castro , Polina Pilyugina , Evgeny Burnaev

Interacting, self-propelled particles such as epithelial cells can dynamically self-organize into complex multicellular patterns, which are challenging to classify without a priori information. Classically, different phases and phase…

Quantitative Methods · Quantitative Biology 2021-01-19 Dhananjay Bhaskar , William Y. Zhang , Ian Y. Wong

Topological data analysis (TDA) has become an attractive area for the application of quantum computing. Recent advances have uncovered many interesting connections between the two fields. On one hand, complexity theoretic results show that…

Quantum Physics · Physics 2025-11-06 Nhat A. Nghiem

Topological Data Analysis (TDA) is a recent approach to analyze data sets from the perspective of their topological structure. Its use for time series data has been limited to the field of financial time series primarily and as a method for…

Machine Learning · Computer Science 2019-06-18 Rodrigo Rivera-Castro , Polina Pilyugina , Alexander Pletnev , Ivan Maksimov , Wanyi Wyz , Evgeny Burnaev

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such…

Computational Geometry · Computer Science 2023-06-21 David Loiseaux , Mathieu Carrière , Andrew J. Blumberg

Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that…

Methodology · Statistics 2020-03-17 Yu-Min Chung , William Cruse , Austin Lawson

Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We show that a computational problem closely related to a core task in TDA --…

Quantum Physics · Physics 2024-10-29 Casper Gyurik , Alexander Schmidhuber , Robbie King , Vedran Dunjko , Ryu Hayakawa

The study of topology is strictly speaking, a topic in pure mathematics. However in only a few years, Topological Data Analysis (TDA), which refers to methods of utilizing topological features in data (such as connected components, tunnels,…

Applications · Statistics 2019-09-25 Nalini Ravishanker , Renjie Chen

Topological Data Analysis (TDA) offers a suite of computational tools that provide quantified shape features in high dimensional data that can be used by modern statistical and predictive machine learning (ML) models. In particular,…

Cryptography and Security · Computer Science 2023-07-06 Dominic Gold , Koray Karabina , Francis C. Motta

Topological Data Analysis (TDA) has been praised by researchers for its ability to capture intricate shapes and structures within data. TDA is considered robust in handling noisy and high-dimensional datasets, and its interpretability is…

Machine Learning · Computer Science 2024-01-10 Funmilola Mary Taiwo , Umar Islambekov , Cuneyt Gurcan Akcora

Over the last two decades, topological data analysis (TDA) has emerged as a very powerful data analytic approach which can deal with various data modalities of varying complexities. One of the most commonly used tools in TDA is persistent…

Methodology · Statistics 2025-12-08 Anass El Yaagoubi Bourakna , Moo K. Chung , Hernando Ombao

In the present article we describe and discuss a framework for applying different topological data analysis (TDA) techniques to a music fragment given as a score in traditional Western notation. We first consider different sets of points in…

Algebraic Topology · Mathematics 2022-12-13 Alberto Alcalá-Alvarez , Pablo Padilla-Longoria

Topological clustering, and its main algorithm ToMATo, is a clustering method from Topological Data Analysis (TDA) which has been applied successfully in several applications during the last few years. This is due to its high versatility,…

Machine Learning · Computer Science 2026-05-15 Ludo Andrianirina , Mathieu Carrière

Topological data analysis (TDA) is a rising branch in modern applied mathematics. It extracts topological structures as features of a given space and uses these features to analyze digital data. Persistent homology, one of the central tools…

Algebraic Topology · Mathematics 2025-05-26 Chuan-Shen Hu

Traditional risk measures in finance, predominantly based on the second moment of return distributions or tail risk heuristics (VaR/CVaR), fail to account for the intrinsic geometric structure of market dynamics. This paper introduces a…

General Topology · Mathematics 2026-04-16 Gabriel Santana , Jemirson Ramirez

Topological Data Analysis (TDA) gives practioners the ability to analyse the global structure of cybersecurity data. We use TDA for anomaly detection in host-based logs collected with the open-source Logging Made Easy (LME) project. We…

Machine Learning · Computer Science 2022-04-28 Thomas Davies

TDA (topological data analysis) is a relatively new area of research related to importing classical ideas from topology into the realm of data analysis. Under the umbrella term TDA, there falls, in particular, the notion of persistent…

Algebraic Topology · Mathematics 2019-06-03 Facundo Memoli , Kritika Singhal

Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description…

Algebraic Topology · Mathematics 2025-11-04 Vincent P. Grande , Michael T. Schaub