ATD: Algorithms for Data Analysis on Abstract Manifolds

ATD：抽象流形数据分析算法

• 批准号: 1830676
• 负责人: Michael Kirby
• 金额: $ 25万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-15 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1830676&HistoricalAwards=false
• 关键词: ATD; Algorithms; Data; Analysis; Abstract

This research program concerns the development of new mathematical algorithms for identifying critical information in data sets that has the potential to reveal biological or chemical threats to humans. The starting point for this work is the observation that modeling a threat over variations in its appearance serves as the foundation of robust detection algorithms. Mathematical tools from geometry and topology enable the design of algorithms for extracting information from sets of data that enhance traditional processing methods, leading to smarter sensors. The graduate students trained in this program will earn doctorates in mathematics while the undergraduates will have the opportunity to be mentored on topics of national interest early in their careers. This research addresses the development of mathematical modeling algorithms for data on abstract manifolds. In contrast to data clouds captured by sensors generating points that exist as a configuration in Euclidean space, this project encodes data as points on matrix manifolds. The mathematical framework requires only three features: the geometric context, the notion of distance between two points, and the ability to compute the perturbation of one point in the direction of another. Additionally, one can exploit information related to the statistics of points on matrix manifolds. This research program aims to explore the geometry of these spaces for the purposes of anomaly detection and characterization.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该研究计划涉及开发新的数学算法，用于识别数据集中的关键信息，这些信息有可能揭示对人类的生物或化学威胁。 这项工作的出发点是观察到，对威胁的外观变化进行建模是鲁棒检测算法的基础。 几何学和拓扑学的数学工具可以设计从数据集中提取信息的算法，从而增强传统的处理方法，从而产生更智能的传感器。 在这个程序中培养的研究生将获得数学博士学位，而本科生将有机会在他们的职业生涯早期就国家利益的主题进行指导。本研究针对抽象流形上数据的数学建模算法的发展。 与传感器捕获的数据云不同，传感器生成的点作为欧几里得空间中的配置存在，该项目将数据编码为矩阵流形上的点。 数学框架只需要三个特征：几何背景，两点之间距离的概念，以及计算一个点在另一个方向上的扰动的能力。 此外，可以利用与矩阵流形上的点的统计相关的信息。 该研究计划旨在探索这些空间的几何形状，以进行异常检测和表征。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Manifold curvature learning from hypersurface integral invariants

• DOI: 10.1016/j.laa.2020.05.020
• 发表时间: 2020-10
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: Javier Álvarez-Vizoso;M. Kirby;C. Peterson

Supervised Dimensionality Reduction and Visualization using Centroid-encoder

• 发表时间: 2020-02
• 期刊: ArXiv
• 作者: T. Ghosh;M. Kirby

Distributions of distances and volumes of balls in homogeneous lens spaces

均匀透镜空间中球的距离和体积的分布

• DOI: 10.1016/j.difgeo.2020.101712
• 发表时间: 2021
• 期刊: Differential Geometry and its Applications
• 影响因子: 0.5
• 作者: Balch, Brenden;Peterson, Chris;Shonkwiler, Clayton
• 通讯作者: Shonkwiler, Clayton

Self-organizing mappings on the flag manifold with applications to hyper-spectral image data analysis

• DOI: 10.1007/s00521-020-05579-y
• 发表时间: 2021-03-27
• 期刊: NEURAL COMPUTING & APPLICATIONS
• 影响因子: 6
• 作者: Ma, Xiaofeng;Kirby, Michael;Peterson, Chris
• 通讯作者: Peterson, Chris

Local eigenvalue decomposition for embedded Riemannian manifolds

• DOI: 10.1016/j.laa.2020.06.006
• 发表时间: 2020-11
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: Javier Álvarez-Vizoso;M. Kirby;C. Peterson