EAGER: The Exploration of Geometric and Non-Geometric Structure in Data

机器学习的目标是从数据中提取有用的信息。尽管研究人员可用于分析的数据量在不断增加，但许多数据都没有标签，这意味着这些数据没有标签表明它们与特定学习任务的关联。因此，理解无监督推理是机器学习的关键问题之一。此外，为某个任务注释的数据可能很难使用，即使对于仅有微小差异的任务也是如此。这在文献中被称为迁移学习问题。为了最大限度地利用可用信息，机器学习算法需要基于严格的数学模型来获取、分析和使用关于数据的现实结构假设。拟议的工作为从事这一项目的学生提供了一个接触广泛主题的机会，包括机器学习、统计学、几何和应用数学。学生将在机器学习和数据分析中学习理论和算法开发技能的结合。这项工作的成果将通过在期刊、会议、各种场所的演讲，包括教程和课程笔记上的出版物向广大科学界传播。与本项目相关的材料将纳入《少年派？S》和《少年派》的课程。该计划亦会为有兴趣的本科生提供暑期研究和实习机会，让他们参与与该项目有关的研究。在这个急切的项目中，将开始对数据的两种结构假设进行探索。将探索数据中的几何结构，如集群的层次结构和密度。将探索使用偏序处理非几何数据，基于概率模型进行偏序排序，解决零点学习和迁移学习等问题。通过在这些框架内探讨从数据进行推理的问题，该项目的输出将成为应对机器学习挑战和开发高效算法的垫脚石，以在理论和实践中推进最先进的技术。有人争辩说，这些模型和提出的数学/算法机制是服从理论分析的，并将提供对真实数据属性的洞察。拟议工作的结果将扩大机器学习方法的范围，以理论上有充分依据的方式分析更复杂的数据。

The goal of machine learning is to extract useful information from data. While the amount of data available to researchers for analysis is ever increasing, much of the data are unlabeled, meaning that the data come without labels indicating their associations with specific learning tasks. Thus understanding unsupervised inference is one of the key problems in machine learning. In addition, data annotated for a certain task may be difficult to use even for tasks only slightly different. This is known as the problem of transfer learning in the literature. To make the most of the available information, machine learning algorithms need to to obtain, analyze and use realistic structural assumptions about the data based on rigorous mathematical models. The proposed work offers students working on this project an opportunity to be exposed to a broad spectrum of topics including machine learning, statistics, geometry and applied mathematics. Students will learn a combination of theory and algorithm development skills in machine learning and data analysis. The results of this work will be disseminated to the broad scientific community through publications in journals, conferences, presentations in various venues, including tutorials and course notes. The material related to this project will be incorporated in PI?s and co-PI's courses. The PIs will also create summer research and practice opportunities for interested undergraduate students in research related to the project. In this EAGER project an exploration of two types of structural assumptions on the data will be started. Geometric structures in data will be explored, such as hierarchical structure of clusters and density. The use of partial orders for non-geometric data will be explored, based on probabilistic models for partial rankings an orders for problems such as zero-shot learning and transfer learning. By approaching the problem of inference from data within these frameworks, output of this project will be a stepping stone to the challenges of machine learning and to developing efficient algorithms to advance the state-of-the-art both in theory and practice. It is argued argue that these models and the proposed mathematical/algorithmic machinery are amenable to theoretical analysis and will provide insight into properties of real data. Results from the proposed work will broaden the scope of machine learning methods to analyze more complex data in a theoretically well-founded manner.