Application driven Topological Data Analysis

应用程序驱动的拓扑数据分析

• 批准号: EP/R018472/1
• 负责人: U Tillmann
• 金额: $ 362.78万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR018472%2F1
• 关键词: Application; driven; Topological; Data; Analysis

Modern science and technology generates data at an unprecedented rate. A major challenge is that this data is often complex, high dimensional, may include temporal and/or spatial information. The "shape" of the data can be important but it is difficult to extract and quantify it using standard machine learning or statistical techniques. For example, an image of blood vessels near a tumor looks very different than an image of healthy bloodvessels; statistics alone cannot quantify this shape because it is the shape that matters. The focus of this proposal is to study the shape of data, through the development of new mathematics and algorithms, and build on existing data science techniques in order to obtain and interpret the shape of data. A theoretical field of mathematics that enables the study of shapes is topology. The ability to compute the shape (its topology) of complicated shapes is only possible with advanced mathematics and algorithms. The field known as topological data analysis (TDA), enables one to use topology to study the shape of data, such as loops in a blood vessel network. In particular, an algorithm within TDA known as persistent homology, provides a topological summary of the shape of the data (e.g., features such as holes) at multiple scales. A key success of persistent homology is the ability to provide robust results, even if the data are noisy. There are theoretical and computational challenges in the application of these algorithms to large scale, real-world data.The aim of this project is to build on current persistent homology tools, extending it theoretically, computationally, and adapting it for practical applications. Our core team is composed of experts in pure and applied mathematicians, computer scientists, and statisticians whose combined expertise covers cutting edge pure mathematics, mathematical modeling, algorithm design and data analysis. This core team will work closely with our collaborators in a range of scientific and industrial domains. Some of the application challenges we have set out include:Can we detect a tumor by looking at the shape of images of blood vessels? Can we design new materials by looking at the shape of molecules using topology? How can we design such molecules? Can we detect anomalies in security data? And importantly, how can we accelerate algorithms to obtain topological characteristics of data in real time?

现代科学技术以前所未有的速度生成数据。一个主要的挑战是，这些数据通常很复杂，高维，可能包括时间和/或空间信息。数据的“形状”可能很重要，但是很难使用标准的机器学习或统计技术提取和量化它。例如，肿瘤附近血管的图像看起来与健康的血液疗法的图像截然不同。仅统计数据就无法量化这种形状，因为它的形状很重要。该提案的重点是通过开发新数学和算法来研究数据的形状，并以现有数据科学技术为基础，以获取和解释数据形状。拓扑是一种使形状研究的数学领域是拓扑。只有通过高级数学和算法才能计算复杂形状的形状（拓扑）的能力。称为拓扑数据分析（TDA）的领域使人们能够使用拓扑来研究数据的形状，例如血管网络中的循环。特别是，TDA中称为持续同源性的算法提供了一个拓扑摘要，以多个尺度上的数据形状（例如，孔等特征）。持续同源性的关键成功是能够提供可靠的结果，即使数据很嘈杂。这些算法将这些算法应用于大规模的现实数据，存在理论和计算挑战。该项目的目的是建立在当前的持久同源工具的基础上，从理论上讲，计算并将其用于实际应用。我们的核心团队由纯粹和应用数学家，计算机科学家和统计学家的专家组成，他们的专业知识涵盖了尖端纯数学，数学建模，算法设计和数据分析。这个核心团队将与我们的合作者紧密合作，在一系列科学和工业领域。我们提出的一些应用挑战包括：我们可以通过查看血管形状来检测肿瘤吗？我们可以使用拓扑来查看分子的形状来设计新材料吗？我们如何设计这样的分子？我们可以检测安全数据中的异常情况吗？重要的是，我们如何加速算法以实时获得数据的拓扑特征？

项目成果

SOFSEM 2023: Theory and Practice of Computer Science - 48th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2023, Nový Smokovec, Slovakia, January 15-18, 2023, Proceedings

SOFSEM 2023：计算机科学的理论与实践 - 第 48 届计算机科学理论与实践当前趋势国际会议，SOFSEM 2023，斯洛伐克 Nová Smokovec，2023 年 1 月 15-18 日，论文集

• DOI: 10.1007/978-3-031-23101-8_8
• 发表时间: 2023
• 作者: Didimo W

Discrete Geometry and Mathematical Morphology - Second International Joint Conference, DGMM 2022, Strasbourg, France, October 24-27, 2022, Proceedings

离散几何与数学形态学 - 第二届国际联合会议，DGMM 2022，法国斯特拉斯堡，2022 年 10 月 24-27 日，论文集

• DOI: 10.1007/978-3-031-19897-7_31
• 发表时间: 2022
• 作者: Anosova O

Bounds for the asymptotic distribution of the likelihood ratio

似然比渐近分布的界限

• DOI: 10.1214/19-aap1510
• 发表时间: 2020
• 期刊: The Annals of Applied Probability
• 作者: Anastasiou A

Principal Components Along Quiver Representations

• DOI: 10.1007/s10208-022-09563-x
• 发表时间: 2021-04
• 期刊: Foundations of Computational Mathematics
• 影响因子: 3
• 作者: A. Seigal;H. Harrington;Vidit Nanda

A Survey of Vectorization Methods in Topological Data Analysis

• DOI: 10.1109/tpami.2023.3308391
• 发表时间: 2022-12
• 期刊: IEEE Transactions on Pattern Analysis and Machine Intelligence
• 影响因子: 23.6
• 作者: Dashti Ali;Aras T. Asaad;M. Jiménez;Vidit Nanda;Eduardo Paluzo-Hidalgo;M. Soriano-Trigueros