Collaborative Research: Machine Learning on Stratified Matrix Manifolds Under Group Actions -- Foundations, Algorithms and Applications

Advances in machine learning and artificial intelligence (AI) are fueling an unprecedented revolution in science, technology and productivity. However, the analysis of complex datasets from domains such as neuroscience and quantum physics remains a challenge and accurately learning from such large and complex datasets requires novel tools. This project will help advance national AI competitiveness by developing new theory and algorithms for biological and physical sciences that require data reduction and generation. The intellectual contributions of the project will be amplified through curriculum innovations, workforce development, open science, and interdisciplinary applications. Educationally, the investigators will co-develop cross-institution graduate seminars on machine learning and research experiences for undergraduates, training students in data-driven computation. This project will develop novel methodologies for machine learning on stratified matrix manifolds under group actions, integrating topology, geometry, and machine learning to advance data-enabled discovery across scientific domains. The work is organized into three thrusts. (1) Mathematical and Statistical Foundations: defining stratified matrix manifolds and metrics across ranks, developing noise-robust O(k)-equivariant dimensionality reduction, and approximating stratified vector bundles. (2) Machine Learning Algorithms: designing equivariant dimensionality reduction, optimal transport, and flow-matching algorithms that operate seamlessly across strata while preserving manifold structure. (3) Scientific Applications: demonstrating the resulting methods on problems in computational neuroscience (e.g., detecting reconfiguration of neural stimulus spaces) and quantum science (e.g., symmetry-aware dimensionality reduction for Kohn–Sham density functional theory). By exploiting the intrinsic structure of stratified manifolds, this project will enable novel analysis of symmetry-constrained, rank-varying data in domains where current methods are prohibitively expensive or inaccurate. 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. NSF Award ID: 2603388 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Harlin Lee | Institution: University of North Carolina at Chapel Hill, CHAPEL HILL, NC | Award Amount: $380,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2603388 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2603388.html