RI: Small: Learning discrete structure from continuous spaces

科学、医学、商业和工程越来越多地受到数据驱动。假设、诊断、决策和设计是通过收集和分析大量数据来寻找有意义的模式来做出的。机器学习领域的目标是开发出对数据中的模式进行推断和推理的方法。该奖项支持的研究解决了从未标记数据中学习的问题，这是数据科学中更令人烦恼的问题之一。PI分析和开发算法来解决诸如聚类之类的问题，即寻找相似对象的组，以及理解数据中不断变化的属性。通过将机器学习、建模和几何数据分析相结合，该工作为现代数据分析注入了新的思想和方法，有助于构建实用的无监督和无监督学习算法，并分析了它们的性质和适用范围。从事这个项目的学生有一个独特的机会接触到广泛的主题，包括机器学习、统计学、几何学和应用数学。在更技术性的层面上，拟议研究的统一观点是，许多这些无监督学习问题可以被视为通过离散数据的透镜来恢复底层连续空间的结构或不变量。这项工作从这个角度考虑了无监督学习的一些重要方面，包括密度模型中的层次聚类、数据量化、图形聚类和估计，以及从数据中学习度量结构。该项目还考虑将这些想法应用于监督学习，特别是在帮助将算法扩展到大数据方面。虽然这个项目的工作集中在理论分析上，但这些都是为了实用的算法、实现和应用而开发的。这个奖项反映了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。

Science, medicine, business, and engineering are increasingly data-driven. Hypotheses, diagnoses, decisions, and designs are made by gathering and analyzing a wealth of data in search of meaningful patterns. It is the goal of the field of machine learning to develop methods for inferring and reasoning about patterns in the data. The research supported by this award addresses the question of learning from unlabeled data, one of the more vexing problems of data science. The PI's analyze and develop algorithms for problems such as clustering, i.e., finding groups of similar objects, as well as understanding continuously changing attributes in data. By integrating machine learning, modeling and geometric data analysis, this work injects new ideas and methodologies to modern data analysis, helps build practical algorithms for unsupervised and unsupervised learning and analyze their properties and domains of applicability. Students working on this project have a unique opportunity to be exposed to a broad spectrum of topics including machine learning, statistics, geometry and applied mathematics.On a more technical level, the unifying perspective for the proposed research is that many of these unsupervised learning problems can be viewed as recovering structure or invariants of the underlying continuous space through the lens of the discrete data. This work takes that point of view to consider a number of important aspects of unsupervised learning including hierarchical clustering in the density model, data quantization, graphon clustering and estimation, as well as learning metric structure from data. The project also considers applications of these ideas to supervised learning, particularly in helping to scale algorithms to large data. While the work on this project concentrates on theoretical analyses, these are developed with a view toward practical algorithms, implementations and applications.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.