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

CDS

• 批准号: 2324353
• 负责人: Peter Bubenik
• 金额: $ 22万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-10-01 至 2026-09-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2324353&HistoricalAwards=false
• 关键词: CDS& MSS; Topological; Learning; Multiparameter

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.

实验数据可能具有潜在的形状，其确定对于科学进步至关重要。在许多情况下，数据非常复杂，以至于最先进的方法无法对其形状提供充分的总结。拓扑数据分析，利用理论数学的思想来解决这一挑战。拓扑数据分析在形状随单个参数变化而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需要进一步研究。项目是开发数据形状的数学摘要可以轻松地使用数据科学中的其他工具进行多参数设置 总之，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并开发对科学家和工程师广泛使用的工具。佛罗里达大学将组织一个拓扑数据分析的主要工具，即持久同源性，该项目的主题涉及一种称为多参数持久同源性的更高级变体。该项目旨在开发新工具，以便轻松组合统计和机器学习的多参数持久同源性。多参数持久同源性的几种计算方法产生的摘要可以被视为指向度量空间上的带符号的形式和。研究人员已经证明，这些摘要（称为广义持久性图）具有 Wasserstein。距离，并且可以被视为指向度量空间上的自由 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，并开发相应的泛函分析和传输理论。景观是持久性图的非线性泛函。该项目将持久性景观扩展到广义持久性图。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了解有关他们的数据的更多信息。开发方法来选择负责检测到的拓扑信号反映的子群。它还将使用深度神经网络软件来生成负责学习分类的数据部分的稳定可视化。该奖项是 NSF 的法定使命，并被认为是值得的。通过评估提供支持基金会的智力价值和更广泛的影响审查标准。