CDS&E-MSS: Topological Learning with Multiparameter Persistent Homology

实验数据可能具有一种潜在的形式，其确定对科学进步至关重要。在许多情况下，数据非常复杂，最先进的方法无法提供对其形状的适当总结。这个项目的主题，拓扑数据分析，使用理论数学的思想来解决这个挑战。拓扑数据分析在单个参数变化的情况下已经取得了成功，但在多个参数变化的情况下还需要进一步的研究。该项目的目标是开发多参数设置中数据形状的数学摘要，可以很容易地与数据科学中的其他工具相结合。总而言之，该项目将利用数学、计算机科学和统计学的思想来推进理论和开发对科学家和工程师广泛使用的工具。将在佛罗里达大学组织一次会议，以促进年轻STEM研究人员的职业发展。作为拓扑数据分析的主要工具，持久同源性已经得到了广泛的应用。这个项目的主题涉及一个更高级的变体，称为多参数持久同源。该项目的目标是开发新的工具，以便轻松地将多参数持久同源性与统计学和机器学习相结合。多参数持久同调的几种计算方法产生的总结可以看作是点度量空间上的有符号形式和。研究者已经证明了这些总结，称为广义持久性图，具有Wasserstein距离，并且可以被视为点度量空间上的自由Banach空间的元素。该项目将为广义持久性图生成连续线性泛函，并发展相应的泛函分析和最优传输理论。研究者的持久性景观是持久性图的非线性函数。这个项目将持久性领域扩展到一般化的持久性图。当检测到拓扑信号或构建拓扑分类时，研究人员希望使用它来更多地了解他们的数据。该项目将开发选择负责检测到的拓扑信号的子种群的方法。它还将使用深度神经网络软件来生成负责学习分类的部分数据的稳定可视化。该奖项反映了美国国家科学基金会的法定使命，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。

Experimental data may have an underlying shape whose determination can be crucial for scientific advances. In many cases, the data is sufficiently complex that state-of-the-art methods do not provide an adequate summary of its shape. The subject of this project, topological data analysis, uses ideas from theoretical mathematics to address this challenge. Topological data analysis has been successful in settings where the shape varies as a single parameter changes, but is in need of further research in settings where multiple parameters vary. The goal of this project is to develop mathematical summaries of the shape of data in the multiple parameter setting that may be easily combined with other tools in data science. In summary, this project will use ideas in mathematics, computer science, and statistics to advance theory and develop tools that are of broad use to scientists and engineers. A conference at the University of Florida will be organized in order to advance careers of young STEM researchers. A main tool of topological data analysis, persistent homology is now well established. The subject of this project concerns a more advanced variant called multiparameter persistent homology. The goal of this project is to develop new tools for easily combining multiparameter persistent homology with statistics and machine learning. Several of the computational approaches to multiparameter persistent homology produce summaries which may be viewed as signed formal sums on a pointed metric space. The investigator has shown that these summaries, called generalized persistence diagrams, have a Wasserstein distance, and may be viewed as elements of the free Banach space on a pointed metric space. This project will produce continuous linear functionals for generalized persistence diagrams and develop a corresponding theory of functional analysis and optimal transport. The investigator’s persistence landscape is a nonlinear functional for persistence diagrams. This project will extend the persistence landscape to generalized persistence diagrams. When a topological signal is detected or a topological classification is constructed, researchers would like to use this to learn more about their data. This project will develop methods for selecting a sub-population responsible for a detected topological signal. It will also use deep neural network software to produce stable visualizations of the parts of the data responsible for a learned classification.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.