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

Vanderbilt University, Nashville TN

Investigators

Abstract

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. 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 fields 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.