Statistical and Computational Aspects of Geometry- and Topology-Based Machine Learning

由于复杂的高维数据在各种科学领域中大规模生成，探索性数据分析对于更好地理解数据生成过程至关重要。事实上，数据分析管道的主要步骤可以说是使用无监督机器学习方法，帮助数据分析师有效地可视化和理解正在分析的数据。本项目将着重于加深对这些方法的理解，以便能够更好地解释这些程序的产出。在这个项目中开发的新方法将传播到应用领域的基础上现有的合作的PI。将通过开放源码软件包向广大公众提供所制定方法的执行情况。该项目还将培训研究生和本科生（来自社会经济弱势背景）在统计数据科学方面取得成功。更具体地说，该项目的主要目标是开发统计和计算方法，以提取高维数据集中可用的低维几何和拓扑结构。因此，该项目的贡献将位于统计机器学习以及几何和拓扑数据分析的交叉点。PI将通过几何透镜对现有方法进行更深入的理解，并提出基于拓扑透镜的无监督机器学习的新方法。在第一部分中，PI将研究在构造高维数据的低维嵌入时出现某种几何正交锥结构的原因，使用非线性降维技术如核主成分分析，扩散映射，非局部方法如ISOMAP和局部线性嵌入（LLE），以及拓扑方法如均匀流形近似和投影（UMAP）。在第二部分中，PI将开发和分析新的降维技术，保留高维数据中可用的拓扑信息。最后，PI将从理论和方法的角度研究拓扑正则化技术在回归和分类中的应用。该奖项反映了NSF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。

As complex high-dimensional data is generated at a large-scale across a wide variety of scientific fields, exploratory data analysis is crucial to gain a better understanding about the data generating process. Indeed, the primary step in the data analysis pipeline arguably is to use unsupervised machine learning methods that help the data analyst to effectively visualize and understand the data being analyzed. This project will concentrate on developing a deeper understanding of such methods so as to enable interpreting the outputs of such procedures better. The novel methodology developed in this project will be disseminated to the applied fields based on existing collaborations of the PIs. Implementations of the developed methodologies will be made available for use by the wider public via open-source packages. This project will also train graduate students and undergraduate students (from socio-economically disadvantaged backgrounds) for a successful career in statistical data science. More specifically, the main goal of this project is to develop statistical and computational methods to extract low-dimensional geometric and topological structure available in high-dimensional datasets. The contributions of this project will hence lie at the intersection of statistical machine learning, and geometric and topological data analysis. The PIs will work both on developing a deeper understanding of existing methodology via a geometric lens, and on proposing novel methodology for unsupervised machine learning based on a topological lens. In the first part, the PIs will study the reason for the emergence of a certain geometric orthogonal cone structures when constructing low-dimension embeddings of high-dimensional data, with non-linear dimension reduction techniques like kernel principal component analysis, diffusion maps, non-local method like ISOMAP and Local Linear Embedding (LLE), and topological methods like Uniform Manifold Approximation and Projection (UMAP). In the second part, the PIs will develop and analyze novel dimension reduction techniques that preserve the topological information available in high-dimensional data. Finally, the PIs will examine the use of topological regularization techniques for regression and classification, from a theoretical and methodological perspective.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.