Computation and Visualization of Multi-Parameter Topological Invariants of Data

拓扑数据分析（Topological Data Analysis，TDA）是统计学的一个较新的分支，其目标是应用拓扑学来开发研究数据的粗尺度、全局、非线性、几何特征的工具。持续同源性是TDA研究最广泛的工具，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。持久同源性产生数据的不变量，称为条形码，通过将数据与称为过滤的嵌套拓扑空间序列相关联，然后应用标准拓扑和代数构造。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性的点云数据，单个滤波不足以丰富到对数据中的感兴趣结构进行编码。这促使考虑多维持久同源性，关联到数据的拓扑空间，同时配备了两个或多个过滤。多维持久同源性产生的代数不变量的数据远比在1-D设置复杂。因此，需要新的方法来处理这些不变量在实践中。本项目的目标是在2-D环境中引入这种方法，具体地说，本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性方法扩展到2-D环境。所提出的工具提供了一个交互式可视化的条形码的限制的2-D持久性模块仿射1-D线。计算方法的核心是一种新的数据结构，基于平面线排列，可以对这些条形码进行快速查询。该工具还提供了一个可视化的多级贝蒂数的2-D持久性模块。有人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，其方式与过去10到15年来普通的持续同源性被应用于数据研究的方式大致相同。该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.