CRII: AF: Enriched Topological Summaries for Inverse Problems

CRII：AF：反问题的丰富拓扑摘要

• 批准号: 1850052
• 负责人: Justin Curry
• 金额: $ 17.41万
• 依托单位: SUNY at Albany
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1850052&HistoricalAwards=false
• 关键词: CRII; AF; Enriched; Topological; Summaries

With Big Data comes the need for succinct data summaries that are easy to visualize and understand. Topological Data Analysis (TDA) offers a particular suite of computational tools for summarizing and visualizing shape in high-dimensional data sets. These tools have been used in recent years to guide new discoveries in cancer research, neuroscience, materials science, image analysis, and many areas where shape, broadly construed, is of importance. However, by summarizing data in a succinct way certain important differences between data sets can often go undetected. This project provides a new mathematical framework for precisely measuring how lossy the most popular methods in TDA are. By developing new ways for quantifying how large-scale differences go undetected, the project offers enrichments of current topological methods to obtain novel data science tools with greater distinguishing power. The awarded funds will go primarily to fund a graduate student to aid the PI in basic research and development of these enriched topological summaries. Computational experiments on data sets of public interest, e.g. time-varying socio-economic indicators and weather data, will be carried out to test performance of these tools against current state of the art methods. Additionally, the PI will incorporate these novel methods into the data science curriculum at the PI's host institution and recruit students from historically under-represented groups to be trained as the nation's next generation of data scientists.This project is an ambitious extension of earlier work undertaken by the PI and provides a targeted attack on the inverse problem for the main objects of study in topological data analysis: the merge tree, which tracks how connected components of the sub-level set of a function evolves; Reeb graphs, which tracks connected components of the fiber of a function; and the barcode/persistence diagram, which is a collection of intervals/points in the plane that represent an algorithmic pairing of critical points of a function on a space. The PI showed in earlier work how merge trees determine the associated barcode and provided a precise enumeration of how many distinct merge trees have the same barcode. By identifying functions on the real line that are related by an orientation preserving coordinate transformation, the PI showed how a novel enrichment of the merge tree---the chiral merge tree---faithfully captures these equivalence classes of functions and offers exponential distinguishing power over the barcode for time-series analysis. This project aims to carry out similar analysis for functions on surfaces, with an eye toward improving the classification performance of persistent homology in image analysis, and for Reeb graphs, to better integrate with level-set persistent homology. By carrying out a careful study of inverse problems in each of these settings, novel enrichments analogous to the chiral merge tree will be developed. Additionally, to make these enriched topological summaries useful for data classification tasks, novel metrics will be defined for comparing each of these summaries and algorithms will be developed for the efficient computation of these metrics.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.

随着大数据的出现，需要简洁的数据摘要，易于可视化和理解。拓扑数据分析（TDA）提供了一套特殊的计算工具，用于总结和可视化高维数据集的形状。近年来，这些工具被用于指导癌症研究、神经科学、材料科学、图像分析和许多广泛理解形状的重要领域的新发现。然而，通过以一种简洁的方式总结数据，往往无法发现数据集之间的某些重要差异。该项目提供了一个新的数学框架，用于精确测量TDA中最流行的方法的损耗。通过开发量化大规模差异未被发现的新方法，该项目提供了当前拓扑方法的丰富内容，以获得具有更大区分能力的新型数据科学工具。获得的资金将主要用于资助一名研究生，以帮助PI进行基础研究和开发这些丰富的拓扑摘要。我们会就公众感兴趣的数据集（例如时变社会经济指标和天气数据）进行计算实验，以测试这些工具与当前最先进方法的表现。此外，PI将把这些新方法纳入PI所在机构的数据科学课程，并从历史上代表性不足的群体中招募学生，培训他们成为美国下一代的数据科学家。该项目是PI早期工作的雄心勃勃的扩展，并为拓扑数据分析中的主要研究对象提供了针对逆问题的有针对性的攻击：合并树，它跟踪函数的子层次集的连接组件如何演变；Reeb图，它跟踪一个函数纤维的连接组件；还有条形码/持久性图，它是平面上的区间/点的集合，代表了空间上一个函数的临界点的算法配对。PI在早期的工作中展示了合并树如何确定相关的条形码，并提供了有多少不同的合并树具有相同条形码的精确枚举。通过识别实数线上与方向保持坐标变换相关的函数，PI显示了合并树的一种新的丰富——手性合并树——如何忠实地捕获这些等价的函数类，并为时间序列分析提供了条形码上的指数区分能力。本项目旨在对曲面上的函数进行类似的分析，着眼于提高图像分析中持久同调的分类性能，对于Reeb图，更好地与水平集持久同调相结合。通过在这些设置中对逆问题进行仔细的研究，将开发出类似于手性合并树的新颖富集。此外，为了使这些丰富的拓扑摘要对数据分类任务有用，将定义用于比较这些摘要的新指标，并将开发用于有效计算这些指标的算法。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。

项目成果

Classification of Constructible Cosheaves

可施工的陪轮分类

• 发表时间: 2020
• 期刊: Theory and applications of categories
• 影响因子: 0.5
• 作者: Curry, J;Patel, A
• 通讯作者: Patel, A

Moduli spaces of morse functions for persistence

持久性莫尔斯函数的模空间

• DOI: 10.1007/s41468-020-00055-x
• 发表时间: 2020
• 期刊: Journal of Applied and Computational Topology
• 作者: Catanzaro, Michael J.;Curry, Justin M.;Fasy, Brittany Terese;Lazovskis, Jānis;Malen, Greg;Riess, Hans;Wang, Bei;Zabka, Matthew
• 通讯作者: Zabka, Matthew