CAREER: Reeb graph learning: Classification, Clustering, and Embedding of Graphical Signatures

可用数据的不断增加是现代生活的一个普遍特征。这些数据有多种形式，例如点、图像和网络。网络(在数学术语中相当于称为图)本身可以是大型数据集中的数据点。例如，人们可能希望比较许多不同物种的基因调控网络，或者比较不同分子的嵌入图以预测治疗特性。本文主要研究拓扑数据分析(TDA)领域中出现的具有附加结构的图，TDA是一种对数据中的形状和结构进行编码的现代数据分析方法。本项目的研究目标是建立必要的机器学习理论，以利用这类图形数据的丰富结构，并提供理论保证和实用算法。这一研究计划与全面整合的教育计划相结合，包括举办一系列讲习班，以促进TDA中的初级研究人员与领域科学家之间的跨学科合作，以及创建开放源码和教育材料，使更多跨学科研究人员可以使用新开发的方法。更详细地说，这个项目关注的是图形签名，这是配备了实值函数的拓扑图，进一步编码了所表示的结构的拓扑或几何的某些方面。这些结构包括利用率很高的结构，如Reeb图、映射器图、合并树和等高线树。最近的工作已经开发了用于比较这些对象的度量，允许将例如Reeb Grahs的空间视为度量空间。为该项目开发的机器学习界面将利用并尊重可用的指标。虽然已经有旨在将一般图与机器学习相接口的工作，并且分别针对其他拓扑签名，例如持久同调，但还没有工具用于考虑附加功能结构的图形签名。项目团队将进一步开发随机Reeb图理论来生成数据集，我们可以在这些数据集上测试方法，以及在植物形态领域的应用中测试结果。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The constant increase in available data is a ubiquitous feature of modern life. This data comes in many forms such as points, images, and networks. Networks (equivalently called graphs in mathematical terminology) can themselves be data points in a large data set. For instance, one may wish to compare gene regulatory networks across many different species, or compare embedded graphs from different molecules to predict therapeutic properties. This research focuses on graphs with additional structure, which arise in the field of Topological Data Analysis (TDA), a modern take on data analysis that encodes shape and structure in data. The research objective of this project is to build the machine-learning theory necessary to utilize the rich structure of this kind of graph data which comes with both theoretical guarantees and practical algorithms. This research plan is combined with a fully integrated education plan including a series of workshops to stimulate interdisciplinary collaborations between starting researchers in TDA with domain scientists, as well as creation of open source code and educational materials to make the newly developed methods available to more interdisciplinary researchers. In more detail, this project focuses on graphical signatures, which are topological graphs equipped with a real-valued function further encoding some aspect of the topology or geometry of the structure represented. These include highly utilized constructions such as Reeb graphs, mapper graphs, merge trees, and contour trees. Recent work has developed metrics for comparing these objects, allowing for treating the space of, e.g., Reeb grahs as a metric space. The machine-learning interfaces developed for this project will utilize and respect the metrics available. While there is work aimed interfacing general graphs with machine learning, and separately for other topological signatures such as persistent homology, no tools yet exist for graphical signatures which take the additional function structure into account. The project team will further develop a theory of random Reeb graphs to generate data sets on which we can test the methods, as well as test the results in applications arising from the field of plant morphology.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.