TRIPODS: Topology, Geometry, and Data Analysis (TGDA@OSU):Discovering Structure, Shape, and Dynamics in Data

TRIPODS：拓扑、几何和数据分析 (TGDA@OSU)：发现数据中的结构、形状和动力学

• 批准号: 1740761
• 负责人: Facundo Memoli
• 金额: $ 150万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-10-01 至 2023-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1740761&HistoricalAwards=false
• 关键词: TRIPODS; Topology; Geometry; Data; Analysis

This project will advance the methodological and theoretical foundations of data analytics by considering the geometric and topological aspects of complex data from mathematical, statistical and algorithmic perspectives, thus enhancing the synergy between the Computer Science, Mathematics, and Statistics communities. Furthermore, this project will benefit a range of impactful scientific areas including medicine, neuronanatomy, machine learning, geographic information systems, mechanical engineering designs, and political science. The research products will be implemented and disseminated through software packages and tutorials, allowing widespread application by industrial and academic practitioners. Through this project, the PIs will develop curricula for cross-disciplinary, undergraduate and graduate education. There is already extant data science curriculum offered jointly between Statistics and Computer Science and Engineering at The Ohio State University (OSU), including the recent Data Analytics undergraduate major, providing a platform to develop new courses and an opportunity to engage future industry leaders in basic research. Additionally, this project aims to develop partnerships with the Translational Data Analytics and the Mathematical Biosciences Institutes at OSU, as well as other internal and external research and education centers. Plans for workshops and summer schools are included for outreach and training purposes.In the past few decades, a large number of models, methods, and algorithmic frameworks have been developed for data science. However, as data become increasingly more complex, the field faces new challenges. In particular, the non-Euclidean nature, the higher order connectivity, the hidden global cues, and the dynamics regulating the data pose further challenges to existing methods. This project will explore and leverage the geometric and topological structures inherent in the data to tackle some of these problems. The main aims are to discover, model and reveal information in the form of (i) structures in data, (ii) shapes from data, and (iii) dynamics underlying data. This project leverages concepts from mathematical areas of differential and algebraic topology and geometry, applied statistics and combinatorics, and computational areas of algorithms, graph theory, and statistical/machine learning. Research in geometric and topological data analysis has brought forth the need to recast and reinvestigate classical concepts in statistics and mathematics in the context of finite data, approximations, and noise. This project investigates explicit or hidden structures behind data, such as cluster trees, which are the basis for understanding and efficient processing of data. Additionally, the PIs aim to model the precise shape behind data globally or locally, which are essential for providing a platform where various statistical analyses can be carried out. Particular examples include the shape space of surface models and the tree space of phylogenetic trees. Finally, this project will consider dynamics in the data, where the interplay between temporal and topological/geometric features can lead to deeper insights. All of these areas will inevitably be enriched by new applications.

该项目将通过从数学，统计和算法的角度考虑复杂数据的几何和拓扑方面来推进数据分析的方法和理论基础，从而增强计算机科学，数学和统计社区之间的协同作用。此外，该项目将使一系列有影响力的科学领域受益，包括医学，神经解剖学，机器学习，地理信息系统，机械工程设计和政治学。 研究成果将通过软件包和教程加以实施和传播，使工业界和学术界的从业人员能够广泛应用。 通过这个项目，PI将开发跨学科，本科和研究生教育的课程。 俄亥俄州州立大学（OSU）的统计学和计算机科学与工程之间已经联合提供了现有的数据科学课程，包括最近的数据分析本科专业，提供了一个开发新课程的平台，并有机会让未来的行业领导者参与基础研究。 此外，该项目旨在与俄勒冈州立大学的转化数据分析和数学生物科学研究所以及其他内部和外部研究和教育中心建立伙伴关系。 在过去的几十年里，数据科学已经开发了大量的模型、方法和算法框架。 然而，随着数据变得越来越复杂，该领域面临着新的挑战。 特别是，非欧几里德性质，高阶连通性，隐藏的全局线索，以及调节数据的动态性对现有方法提出了进一步的挑战。 这个项目将探索和利用数据中固有的几何和拓扑结构来解决其中的一些问题。 其主要目的是发现，建模和揭示信息的形式（i）数据中的结构，（ii）数据的形状，以及（iii）数据的动态。 该项目利用了微分和代数拓扑和几何，应用统计和组合数学，算法，图论和统计/机器学习的计算领域的数学领域的概念。 在几何和拓扑数据分析的研究提出了需要重铸和重新调查统计和数学的背景下，有限的数据，近似和噪音的经典概念。 该项目研究数据背后的显式或隐藏结构，例如聚类树，这是理解和有效处理数据的基础。 此外，PI旨在对全球或本地数据背后的精确形状进行建模，这对于提供一个可以进行各种统计分析的平台至关重要。 具体的例子包括表面模型的形状空间和系统发育树的树空间。 最后，该项目将考虑数据中的动态，其中时间和拓扑/几何特征之间的相互作用可以导致更深入的见解。 所有这些领域都将不可避免地因新的应用而得到丰富。

项目成果

Polluted bootstrap percolation in three dimensions

• DOI: 10.1214/20-aap1588
• 发表时间: 2017-06
• 期刊: The Annals of Applied Probability
• 作者: Janko Gravner;A. Holroyd;David J Sivakoff

A Topological Regularizer for Classifiers via Persistent Homology

• 发表时间: 2018-06
• 作者: Chao Chen;Xiuyan Ni;Qinxun Bai;Yusu Wang

Analysis of shape data: From landmarks to elastic curves

形状数据分析：从地标到弹性曲线

• DOI: 10.1002/wics.1495
• 发表时间: 2020
• 期刊: WIREs Computational Statistics
• 作者: Bharath, Karthik;Kurtek, Sebastian
• 通讯作者: Kurtek, Sebastian

Estimation of Spatial Deformation for Nonstationary Processes via Variogram Alignment

通过变差函数对齐估计非平稳过程的空间变形

• DOI: 10.1080/00401706.2021.1883481
• 发表时间: 2021
• 期刊: Technometrics
• 影响因子: 2.5
• 作者: Qadir, Ghulam A.;Sun, Ying;Kurtek, Sebastian
• 通讯作者: Kurtek, Sebastian

Locally-Weighted Elastic Comparison of Planar Shapes

平面形状的局部加权弹性比较

• 发表时间: 2018
• 期刊: IEEE Workshop on Differential Geometry in Computer Vision and Machine Learning
• 作者: Strait, Justin;Kurtek, Sebastian;MacEachern, Steven N.
• 通讯作者: MacEachern, Steven N.