Computation and Visualization of Multi-Parameter Topological Invariants of Data

数据多参数拓扑不变量的计算和可视化

• 批准号: 1521552
• 负责人: Matthew Wright
• 金额: $ 21.02万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1521552&HistoricalAwards=false
• 关键词: Computation; Visualization; Multi; Parameter; Topological

Topological data analysis (TDA) is a relatively new branch of statistics whose goal is to apply topology to develop tools for studying the coarse-scale, global, non-linear, geometric features of data. Persistent homology, the most widely studied tool for TDA, has been applied to many areas of science and engineering, including image processing, time series data in biological systems, and sensor networks. Persistent homology yields invariants of data, called barcodes, by associating to the data a sequence of nested topological spaces called a filtration, and then applying standard topological and algebraic constructions. However, for many data sets of interest, such as point cloud data with noise or non-uniformities in density, a single filtration is not rich enough to encode the structure of interest in the data. This motivates the consideration of multidimensional persistent homology, which associates to the data a topological space simultaneously equipped with two or more filtrations. Multi-D persistent homology yields algebraic invariants of data far more complex than in the 1-D setting. New methodology is thus required for working with these invariants in practice. The goal of this project is to introduce such methodology in the 2-D setting.Specifically, this project is to develop algorithms and design practical software tools that extend the usual persistent homology methodology for exploratory data analysis to the 2-D setting. The proposed tools provide an interactive visualization of the barcodes of the restriction of a 2-D persistence module to affine 1-D lines. At the heart of the computational approach is a novel data structure, based on planar line arrangements, on which one can perform fast queries for these barcodes. The tools also provide a visualization of the multi-graded Betti numbers of a 2-D persistence module. It is proposed to apply the tools to the study of scientific data - especially data arising from biological systems - in much the same way that ordinary persistent homology has been applied to the study of data over the last ten to fifteen years. This project will intend to establish statistical foundations for the corresponding methodology.

拓扑数据分析（TDA）是一个相对较新的统计分支，其目标是应用拓扑来开发研究数据的粗尺度，全球，非线性，几何特征。持续性同源性是TDA研究的最广泛研究的工具，已应用于许多科学和工程领域，包括图像处理，生物系统中的时间序列数据和传感器网络。持续的同源性通过与数据与一个称为过滤的嵌套拓扑空间相关联，从而产生称为条形码的数据，然后应用标准拓扑和代数结构。但是，对于许多感兴趣的数据集，例如具有噪声的点云数据或密度不均匀性，单个过滤不足以编码数据中感兴趣的结构。这激发了对多维持续同源性的考虑，该同源物与数据同时配备了两个或更多过滤的数据。多-D持续同源性产生的数据的代数不变性要比1-D设置更为复杂。因此，在实践中与这些不变的人合作需要新的方法。该项目的目的是在2D设置中引入这种方法。特别是，该项目是开发算法和设计实用的软件工具，以将通常的持久同源性方法扩展到探索性数据分析到2D设置。所提出的工具提供了限制2-D持续模块对仿射1-D线的限制条形码的交互式可视化。计算方法的核心是基于平面线布置的新型数据结构，可以在其上对这些条形码进行快速查询。这些工具还提供了2-D持续模块的多分级Betti数字的可视化。有人建议将工具应用于科学数据的研究，尤其是由生物系统引起的数据，与过去十到十五年中普通持续的同源性研究相同。该项目将打算为相应方法建立统计基础。