Computation and Visualization of Multi-Parameter Topological Invariants of Data

拓扑数据分析(TDA)是一个相对较新的统计学分支，其目标是应用拓扑学开发工具来研究数据的粗尺度、全局、非线性、几何特征。持久同源是TDA中研究最广泛的工具，已被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。持久同调通过将称为过滤的嵌套拓扑空间序列与数据相关联，然后应用标准的拓扑和代数构造，来产生数据的不变量，称为条形码。然而，对于许多感兴趣的数据集，例如具有噪声或密度不均匀的点云数据，单个过滤器不足以编码数据中的感兴趣结构。这激发了多维持久同调的考虑，多维持久同调将同时配备两个或更多过滤的拓扑空间与数据相关联。多维持久同调产生比一维设置中复杂得多的数据的代数不变量。因此，在实践中使用这些不变量需要新的方法。这个项目的目标是将这种方法引入到二维环境中，具体地说，这个项目是开发算法和设计实用的软件工具，将通常用于探索性数据分析的持久同源方法扩展到二维环境中。所提出的工具提供了2-D持久性模块对仿射1-D线的限制的条形码的交互式可视化。计算方法的核心是一种基于平面线排列的新型数据结构，人们可以在该结构上执行对这些条形码的快速查询。这些工具还提供了2-D持久性模块的多等级Betti数的可视化。建议将这些工具应用于科学数据的研究--特别是生物系统产生的数据--就像过去10至15年来将普通的持久同源性应用于数据研究一样。该项目将为相应的方法奠定统计基础。

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.