CRII: AF: Enriched Topological Summaries for Inverse Problems

伴随着大数据而来的是对易于可视化和理解的简明数据摘要的需求。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中的形状。近年来，这些工具被用来指导癌症研究、神经科学、材料科学、图像分析以及许多广义上认为形状很重要的领域的新发现。然而，通过以简洁的方式总结数据，数据集之间的某些重要差异通常不会被发现。该项目为精确测量TDA中最流行的方法的有损程度提供了一个新的数学框架。通过开发新的方法来量化大规模差异如何未被发现，该项目提供了对当前拓扑方法的丰富，以获得具有更大区分能力的新型数据科学工具。授予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总结。将对公共利益的数据集进行计算实验，例如时变的社会经济指标和天气数据，以测试这些工具相对于当前最先进方法的性能。此外，PI将把这些新方法纳入PI主办机构的数据科学课程，并从历史上代表性不足的群体中招收学生作为国家下一代数据科学家。这个项目是PI早期工作的雄心勃勃的延伸，并为拓扑数据分析中的主要研究对象提供了有针对性的反问题攻击：跟踪函数次水平集的连通分量如何演变的合并树；跟踪函数纤维的连通分量的Reeb图；以及条形码/持久性图，它是表示空间上函数的临界点的算法配对的平面中的间隔/点的集合。PI在前面的工作中展示了合并树如何确定关联的条形码，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。PI通过识别实线上由方向保持坐标变换关联的函数，展示了一种新的合并树的丰富-手性合并树-如何忠实地捕获这些等价的函数类，并提供比用于时间序列分析的条形码更指数的区分能力。该项目旨在对曲面上的函数进行相似分析，以期提高持久同调在图像分析中的分类性能，以及对于Reeb图，更好地与水平集持久同调相结合。通过在这些设置中的每一个中对反问题进行仔细的研究，将开发类似于手性合并树的新型富集物。此外，为了使这些丰富的拓扑摘要对数据分类任务有用，将定义新的指标来比较这些摘要，并将开发算法来有效计算这些指标。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。

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.