CAREER: Machine learning, Mapping Spaces, and Obstruction Theoretic Methods in Topological Data Analysis

数据分析可以被描述为从观察中提取信息和以原则性方式理解模式的双重过程。这一进程和以数据为中心的技术的部署最近在许多科学领域带来了前所未有的进步，并随着以知识为基础的经济和系统的出现而增加了全球繁荣。在更高的层面上，这场革命是由两个推动力推动的：允许收集复杂数据集的现代技术，以及我们用来理解它们的理论和算法。尽管如此，尽管有很多好处，但从数据中提取可操作的知识是困难的。在不受控制的环境中收集的观测数据通常是高维的、复杂的和嘈杂的；即使使用受控实验，它们背后的复杂系统--如气象学、化学、医学和生物学的系统--也可以产生具有非常重要的潜在拓扑结构的数据集。这指的是诸如不相连部分(即，簇)的数量、是否存在空洞或数据空间的可定向性之类的属性。通过这项职业奖资助的研究将利用代数拓扑学的想法来解决数据科学问题，如复杂数据集的可视化和表示，以及在设计用于预测和分类的学习系统时非平凡拓扑学带来的挑战。这项工作将通过创建一个在线TDA(拓扑数据分析)学院纳入国际和平研究所的教育计划，其双重目的是降低数据科学家和学者进入该领域的门槛，以及增加在计算数学领域服务不足的社区的代表性。这个项目为研究生提供了研究培训的机会。理解拓扑空间之间的映射集导致了丰富和复杂的数学，因为它包含了代数不变量，如同伦群和广义(余)同调理论。虽然一些数据科学问题是映射空间问题的离散版本-包括非线性降维和有监督学习-但目前缺乏相应的理论和算法处理。这一职业奖项将有助于纠正这种情况。这里阐述的研究计划试图启动一种新颖的研究计划，解决如何将数据集的底层拓扑用于数据建模(例如，降维)以及当学习复杂数据空间之间的映射时(例如，在监督学习中)的理论和算法。这项工作将产生通过对空间进行分类来计算数据的拓扑感知和稳健的多尺度协调的方法、对数据集之间地图稳健扩展的拓扑障碍的计算理论，以及引入现代深度学习范例以学习非欧几里德数据集之间的地图。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。

Data analysis can be described as the dual process of extracting information from observations, and of understanding patterns in a principled manner. This process and the deployment of data-centric technologies have recently brought unprecedented advances in many scientific fields, as well as increased global prosperity with the advent of knowledge-based economies and systems. At a high level, this revolution is driven by two thrusts: the modern technologies which allow for the collection of complex data sets, and the theories and algorithms we use to make sense of them. That said, and for all its benefits, extracting actionable knowledge from data is difficult. Observations gathered in uncontrolled environments are often high-dimensional, complex and noisy; and even when controlled experiments are used, the intricate systems that underlie them --- like those from meteorology, chemistry, medicine and biology --- can yield data sets with highly nontrivial underlying topology. This refers to properties such as the number of disconnected pieces (i.e., clusters), the existence of holes or the orientability of the data space. The research funded through this CAREER award will leverage ideas from algebraic topology to address data science questions like visualization and representation of complex data sets, as well as the challenges posed by nontrivial topology when designing learning systems for prediction and classification. This work will be integrated into the educational program of the PI through the creation of an online TDA (Topological Data Analysis) academy, with the dual purpose of lowering the barrier of entry into the field for data scientists and academics, as well as increasing the representation of underserved communities in the field of computational mathematics. The project provides research training opportunities for graduate students.Understanding the set of maps between topological spaces has led to rich and sophisticated mathematics, for it subsumes algebraic invariants like homotopy groups and generalized (co)homology theories. And while several data science questions are discrete versions of mapping space problems --- including nonlinear dimensionality reduction and supervised learning --- the corresponding theoretical and algorithm treatment is currently lacking. This CAREER award will contribute towards remedying this situation. The research program articulated here seeks to launch a novel research program addressing the theory and algorithms of how the underlying topology of a data set can be leveraged for data modeling (e.g., in dimensionality reduction) as well as when learning maps between complex data spaces (e.g., in supervised learning). This work will yield methodologies for the computation of topology-aware and robust multiscale coordinatizations for data via classifying spaces, a computational theory of topological obstructions to the robust extension of maps between data sets, as well as the introduction of modern deep learning paradigms in order to learn maps between non-Euclidean data sets.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.