thesis on topological data analysis

New mathematically well-founded theories have given birth to the field of. Topological Data Analysis (TDA). TDA is a field of mathematics that analyzes. data from a fundamentally di ̇erent perspective. TDA is mainly motivated by the. idea of utilizing powerful methods and approaches in geometry and topology to.

Topological data analysis is an expanding field that attempts to obtain qualitative information from. a data set using topological ideas. There are two common methods of topological data analysis: persistent homology and the Mapper algorithm; the focus of this thesis is on the latter.

Topological Data Analysis (TDA) uses ideas from topology to study the "shape" of data. It provides a set of tools to extract features, such as holes, voids, and connected components, from complex high-dimensional data. This thesis presents an introductory exposition of the mathematics underlying the two main

The recent application of algebraic and computational topology to data science has led to the development of a new field known as Topological Data Analysis (TDA) (Carlsson, 2009).TDA techniques are based on the observation that data (e.g., a set of points in a Euclidean space) can be interpreted as elements of a geometric object.

dimensional and complex data context it is challenging to differentiate between significant information and noise, let alone to visualize the data. Mathematical methods enable us to strategically analyze the data and look for structures, rela-tionshipsandhiddenfeatures. Topological Data Analysis (TDA) presents one approach to tackle this chal ...

This thesis aims to serve as an introduction to Topological Data Analysis (TDA), a collec-tion of methods that seek to quantify the topological and geometric features of data using algebraic topology. The theory behind persistent homology, a stable multi-scale approach for characterizing the structure of data, is presented here.

An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists Frédéric Chazal and Bertrand Michel February 26, 2021 Abstract Topological Data Analysis (tda) is a recent and fast growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data.

1 Introduction and Motivation. Topological data analysis (tda) is a recent field that emerged from various works in applied (algebraic) topology and computational geometry during the first decade of the century.Although one can trace back geometric approaches to data analysis quite far into the past, tda really started as a field with the pioneering works of Edelsbrunner et al. (2002) and ...

The rapidly developing field of topological data analysis represents data via graphs rather than as solutions to equations or as decompositions into clusters. This Review discusses the methods and ...

topology, and he has spent the last 20 years on the development of topological data analysis. He is also passionate about the transfer of scientiÞc Þndings to real-world applications, leading him to found the topological data analysis-based company Ayasdi in 2008. Mikael Vejdemo-Johansson is Assistant Professor in the Department of Mathematics

Stanford University. Yusu primarily works in topological and geometric data analysis, developing effective and theoretically justiÞed algorithms for data analysis using geometric and topological ideas, as well as in applying them to practical domains. She received the DOE Early Career Principal Investigator Award in 2006 and NSF Career Award ...

Topological data analysis (TDA) is an emerging field whose goal is to provide mathematical and algorithmic tools to understand the topological and geometric structure of data. ... This thesis will ...

Topological data analysis refers to approaches for system-atically 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 review of applica-tions of topological data analysis to physics ...

USING TOPOLOGICAL DATA ANALYSIS. A Thesis . by . JONG WOON CHOI . Submitted to the Office of Graduate and Professional Studies of . Texas A&M University . in partial fulfillment of the requirements for the degree of . MASTER OF SCIENCE . Chair of Committee, John E. Killough. Committee Member, Eduardo Gildin . Vivek Sarin . Head of Department, A ...

This book gathers the proceedings of the 2018 Abel Symposium, which was held in Geiranger, Norway, on June 4-8, 2018. The symposium offered an overview of the emerging field of "Topological Data Analysis". This volume presents papers on various research directions, notably including applications in neuroscience, materials science, cancer ...

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 believed to promote an intuitive understanding of model behavior. However, claims regarding the power and usefulness of TDA have only been partially tested in ...

3.8 Euler Characteristics in Topological Data Analysis 228 3.9 Exploratory Data Analysis with Mapper 231 3.10 Summary 233 3.11 Suggestions for Further Reading 234 4 Dimensionality Reduction, Manifold Learning, and Metric Geometry 235 4.1 A Quick Refresher on Eigenvectors and Eigenvalues 238 4.2 Background on PCA and MDS 239 4.3 Manifold ...

Topological Data Analysis denotes the set of algorithms and methods to define and retrieve the underlying structure of the shapes in the data. Utilizing topological inference in data mining and generally data science is recent, while computational geometry and computational topology have been examined in the area of applied mathematics for many ...

1 Introduction and Motivation. Topological data analysis (tda) is a recent field that emerged from various works in applied (algebraic) topology and computational geometry during the first decade of the century.Although one can trace back geometric approaches to data analysis quite far into the past, tda really started as a field with the pioneering works of Edelsbrunner et al. (2002) and ...

Topological Data Analysis A Thesis submitted to Indian Institute of Science Education and Research Pune in partial ful llment of the requirements for the BS-MS Dual Degree Programme by Rajdeep Haldar Indian Institute of Science Education and Research Pune Dr. Homi Bhabha Road, Pashan, Pune 411008, INDIA. April, 2020 Supervisor: Prof. Sourish Das

The topological data analysis algorithm uses graphs to find high-dimensional holes in a data set. For the implementation of this algorithm on a quantum computer one can use Hamiltonian simulation based on Trotterization (which will be explained in Sections 2.2.4 and 2.2.5) or based on sparse access (Section 3).

This thesis is a mathematical exposition of the theory behind Topological Data Analysis (TDA) complemented by two applications in medicine and financial realm. We start by establishing the foundation of homology theory, then study the reconstruction of the underlying manifold from point cloud data. Followed by the theory of persistent homology ...

Topological data analysis Material database and statistics. By applying the above high-throughput screening method, we have successfully identified the band representations and band topologies for 9991 MPID entries in PhononDB@kyoto-u and 1516 MPID entries in the Materials Project. As the ab initio calculations were performed on a finite-size ...

This thesis aims to serve as an introduction to Topological Data Analysis (TDA), a collection of methods that seek to quantify the topological and geometric features of data using algebraic topology. The theory behind persistent homology, a stable multi-scale approach for characterizing the structure of data, is presented here.

Abstract. Childhood maltreatment may adversely affect brain development and consequently influence behavioral, emotional, and psychological patterns during adulthood. In this study, we propose an analytical pipeline for modeling the altered topological structure of brain white matter in maltreated and typically developing children. We perform topological data analysis (TDA) to assess the ...

Wang Y Ombao H Chung MK Topological data analysis of single-trial electroencephalographic signals Ann Appl Stat 2018 12 3 1506 3852686 Google Scholar; 23. Yan Y Wu X Li C He Y Zhang Z Li H Li A Wang L Topological eeg nonlinear dynamics analysis for emotion recognition IEEE Trans Cognit Develop Syst 2023 15 2 625 638 Google Scholar Cross Ref; 24.

High-purity polycrystalline silicon, as a core raw material in the photovoltaic industry, has a trade structure whose robustness affects the supply security of the entire photovoltaic industry. Using social network analysis methods and dependence indicators, this study constructs a Polycrystalline Silicon Trade Dependency Network (PSTDN) from 1995 to 2019, and performs descriptive statistical ...

As a new generation of porous materials, metal-organic frameworks (MOFs) are the promising candidates for NH 3 uptake. However, to obtain the stable MOFs for a good NH 3 adsorption capacity and reliable recyclability under mild conditions remains a big challenge. Herein, we report a novel topological 3D porous indium-organic framework (InOF), [In 8 (μ 2-OH) 6 (μ 2-H 2 O) 3 L 6 Cl 6]∙5DMF ...