Collaborative Research: AF: Medium: A Unified Framework for Geometric and Topological Signature-Based Shape Comparison

数据分析的一个基本方面是比较数据集的能力，以便测量（不）相似性并量化数据中存在的模式。 然而，数据往往过于庞大和复杂，无法对其进行整体分析，因此，需要使用不同的技术来汇总数据，以便处理更小、更易于管理的数据表示。本项目通过数学的透镜来研究数据比较问题，使用几何和拓扑签名来简洁地表示这些形状。这个项目将考虑各种不同类型的形状数据，这些数据存在于一些更大的几何或拓扑空间中（例如，GIS轨迹、点集、网格、3D扫描或图形），并考虑可用于简洁地表示这些形状的代数、几何和图形签名的类别。 该项目主要利用拓扑数据分析这一新兴但发展迅速的领域，将同源或同伦等拓扑工具与几何测量相结合，以创建用于分析数据形状的强大分析工具。研究生和本科生将紧密结合到项目中，并将特别努力让代表性不足的群体的学生参与进来。研究团队的其他努力包括计划一个专注于该领域女性的研讨会，以及在她们自己的大学中扩大多样性和包容性的努力。该项目专注于在签名之上具有一些共同的底层注释框架的形状，这些注释框架通常是来自原始嵌入的额外结构或几何信息。这项研究包括两个主要部分。 首先，研究人员开始对算法和方法进行原则性研究，以开发一个统一的框架，该框架利用多个签名进行形状比较。这一阶段的目标是提供理论结果以及对各种数据集和特征的经验评估。该项目的第二个主要组成部分是研究逆问题，其目的是从签名组合中重建形状。 众所周知，这些问题对于几何或拓扑签名来说是非常困难的，因为它们必然是有损的，并且会删除某些类型的信息。 在该项目的过程中，研究人员还开发了一个形状签名工具包，可以计算一系列签名和距离，在软件中加入现有的距离概念和在项目过程中开发的新概念。该项目由数学基础核心计划和刺激竞争研究的既定计划（EPSCoR）共同资助该奖项反映了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。

A fundamental aspect for data analysis is the ability to compare data sets, in order to measure (dis)similarity and quantify patterns present in the data. However, data is often too large and complex to analyze in its entirety, and therefore different techniques are used to summarize the data in order to work with smaller, more manageable representations of it. This project studies the data-comparison problem through the lens of mathematics, using geometric and topological signatures to represent these shapes concisely. This project will consider a variety of different kinds of shape data which live in some larger geometric or topological space (e.g., GIS trajectories, point sets, meshes, 3d scans, or graphs), and consider classes of algebraic, geometric, and graphical signatures which can be used to represent these shapes concisely. The project draws primarily upon the nascent yet rapidly developing area of topological data analysis, where tools from topology like homology or homotopy are combined with geometric measures to create robust analysis tools for analyzing the shape of data. Graduate and undergraduate students will be tightly integrated into the project, and special efforts will be made to involve students from underrepresented groups. Additional efforts by the research team include planning a workshop focused on women in this field, as well as broadening diversity and inclusion efforts in their own universities.The project focuses on shapes that have some common underlying annotation framework on top of the signature, which is usually additional structural or geometric information from the original embedding. The research consists of two major components. In the first, the investigators are initiating a principled study of algorithms and approaches to develop a unified framework which leverages multiple signatures for shape comparison. The goal of this phase is to provide theoretical results as well as empirical evaluations on a variety of data sets and signatures. The second major component of the project studies inverse problems, which aim to reconstruct shapes from a combination of signatures. Such problems are notoriously difficult for geometric or topological signatures, as they are necessarily lossy and remove certain types of information. During the course of the project, the investigators are also developing a shape signatures toolkit that enables computation of a range of signatures and distances, adding to the software both existing notions of distance and new ones developed over the course of the project.This project is jointly funded by the Algorithmic Foundations Core Program and by the Established Program to Stimulate Competitive Research (EPSCoR).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.