CAREER: Optimal Transport Beyond Probability Measures for Robust Geometric Representation Learning

机器学习是现代人类社会的关键技术支柱，它使算法的发展能够自动发现有价值的数据模式。它的深远影响遍及各个工业领域，包括医疗保健、汽车、能源和娱乐。此外，它已成为科学研究中不可或缺的工具，使科学家能够分析数据，构建和验证假设，并做出预测，从而加快科学探索和发现的进程。尽管机器学习取得了成功，但人们对机器学习的许多基础问题和理论方面仍然知之甚少，这带来了与此类技术相关的不必要的后果。这些问题中的关键是确定如何准确量化机器学习模型的不确定性，并辨别何时它们的预测是值得信赖的。同样重要的是提高这些方法的效率和健壮性，特别是当它们需要从有限的数据或演示中学习模式时。为了解决其中的一些问题，研究者将研究机器学习的数学基础，使用最优运输、积分几何和测量理论的工具。本项目开发的基础工具预计将导致下一代机器学习方法，以其效率、不确定性意识、可解释性和鲁棒性而闻名，在医疗保健、交通运输和国防领域具有潜在的优势。这项研究将与综合教育和推广活动相结合，以鼓励从高中到研究生学习的各个学术水平的研究参与。将特别强调弱势群体的参与，包括田纳西州中部的少数民族和农村服务机构，并积极促进STEM学科的多样性和包容性，特别是在人工智能和机器学习教育方面。这个项目的动机是测量高维数学对象之间有意义的距离是现代机器学习的核心。它旨在探索新的几何距离对机器学习方法的效率和鲁棒性的影响。该项目的研究议程包括三个时间顺序阶段：1)利用研究者之前在（不平衡）最优传输、传输Lp和平坦和弯曲空间中的广义切片距离方面的工作，同时考虑其统计、几何和拓扑性质，开发可扩展的、超越概率测量的基于传输的最佳度量；2)为提议的度量创建欧几里德嵌入，以促进它们与传统机器学习过程的集成，从而实现高效的分类和聚类；3)将基于传输的嵌入与几何深度表征学习模型相结合，进行高维研究，评估其对几何深度学习方法性能和鲁棒性的影响。该项目的第一阶段为扩展的度量类（包括正矢量和带符号矢量度量）开发计算效率高的距离，并探索它们的度量结构、拓扑、测地线和稳定性。第二阶段集中于第一阶段引入的基于传输的度量的有效嵌入技术，研究它们的规律性、稳定性和计算方面。第三阶段研究第二阶段开发的基于传输的嵌入在不同对称群中的不变性和等变性，并将这些嵌入整合到几何深度神经结构中。最后，该项目旨在通过结合积分几何、测量理论、最优运输、数理统计和机器学习的见解来促进跨学科合作，从而鼓励在这些高度相关领域的知识交流。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Machine learning is a pivotal technological pillar in modern human society, enabling the development of algorithms that automatically uncover valuable data patterns. Its profound influence extends across various industrial sectors, encompassing healthcare, automotive, energy, and entertainment. Furthermore, it has become an indispensable tool in scientific research, empowering scientists to analyze data, construct and validate hypotheses, and make predictions, thereby expediting the progress of scientific exploration and discovery. Despite its success, many foundational questions and theoretical aspects of machine learning remain poorly understood, posing unwanted ramifications associated with such technologies. Critical among these issues is determining how to accurately quantify the uncertainty of machine learning models and discerning when their predictions are trustworthy. Equally important is enhancing the efficiency and robustness of these methods, especially when they are required to learn patterns from limited data or demonstrations. To address some of these issues, the investigator will study the mathematical foundations of machine learning, using tools from optimal transport, integral geometry, and measure theory. The foundational tools developed in this project are anticipated to lead to the next generation of machine learning methods, notable for their efficiency, uncertainty awareness, interpretability, and robustness, with potential benefits in healthcare, transportation, and national defense. This research will be integrated with comprehensive education and outreach initiatives to encourage research involvement across academic levels, from high school to graduate studies. Particular emphasis will be placed on engaging disadvantaged groups, including minority and rural serving institutions in Middle Tennessee, and actively promoting diversity and inclusivity in STEM disciplines, particularly in artificial intelligence and machine learning education.This project is motivated by the fact that measuring meaningful distances between high-dimensional mathematical objects is central to modern machine learning. It aims to explore the impact of novel geometric distances on the efficiency and robustness of machine learning methods. The project's research agenda comprises three chronological phases: 1) developing scalable optimal transport-based metrics extending beyond probability measures, leveraging the investigator's prior work in (unbalanced) optimal transport, transport Lp, and generalized sliced distances in both flat and curved spaces, while considering their statistical, geometric, and topological properties; 2) creating Euclidean embeddings for the proposed metrics to facilitate their integration with traditional machine learning processes for efficient classification and clustering; and 3) combining transport-based embeddings with geometric deep representation learning models, and conducting high-dimensional studies to assess their impact on the performance and robustness of geometric deep learning methods. Phase 1 of the project develops computationally efficient distances for extended classes of measures, including both positive and signed vector measures, and explores their metric structure, topology, geodesics, and stability. Phase 2 concentrates on efficient embedding techniques for the transport-based metrics introduced in Phase 1, investigating their regularity, stability, and computational aspects. Phase 3 examines the invariance and equivariance of the transport-based embeddings developed in Phase 2 in relation to different symmetry groups and integrates these embeddings into geometric deep neural architectures. Lastly, the project aims to foster interdisciplinary collaboration by combining insights from integral geometry, measure theory, optimal transport, mathematical statistics, and machine learning, thereby encouraging knowledge exchange in these highly relevant fieldsThis 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.