Computation and Visualization of Multi-Parameter Topological Invariants of Data

拓扑数据分析（TDA）是统计学中一个相对较新的分支，其目标是应用拓扑来开发研究数据的粗尺度、全局、非线性和几何特征的工具。持续同源性是研究最广泛的TDA工具，已应用于许多科学和工程领域，包括图像处理、生物系统的时间序列数据和传感器网络。通过将数据与称为过滤的嵌套拓扑空间序列关联，然后应用标准拓扑和代数结构，持久同构产生数据的不变量，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，单个过滤不够丰富，无法编码数据中感兴趣的结构。这激发了对多维持久同调的考虑，它将数据关联到同时配备两个或多个过滤的拓扑空间。多维持久同调产生的数据的代数不变量远比一维设置复杂得多。因此，在实践中处理这些不变量需要新的方法。这个项目的目标是在二维环境中引入这样的方法。具体来说，该项目是开发算法和设计实用的软件工具，将探索性数据分析的常用持久同源性方法扩展到二维环境。所提出的工具提供了二维持久性模块对仿射一维线的条形码限制的交互式可视化。计算方法的核心是一种基于平面线排列的新颖数据结构，在这种结构上可以对这些条形码执行快速查询。这些工具还提供了二维持久化模块的多级Betti数的可视化。建议将这些工具应用于科学数据的研究——特别是来自生物系统的数据——在很大程度上与过去10到15年将普通持久同源性应用于数据研究的方式相同。这个项目将打算为相应的方法建立统计基础。

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.