Applied Harmonic Analysis Methods for G-Invariant Representations and Matrix Analysis Problems

About This Grant

This project aims to develop and analyze tools from applied harmonic analysis with the goal of understanding and capitalizing on the formal concept of redundancy in mathematics and applications. The mathematical challenges involved in this project are formulated to directly contribute to more interpretable and explainable machine learning algorithms as well as address complex optimization problems that are currently intractable with existing computational resources. In addition to advancing foundational research, the project will play a key role in training graduate students, equipping them with cutting-edge skills in mathematics and computation. This will help cultivate a globally competitive STEM workforce capable of solving real-world challenges. The project will pursue two main research thrusts. The first thrust involves the development of both theory and algorithms for invariant coorbit representations. This includes designing stable Euclidean embeddings of the metric quotient space and analyzing their analytic and geometric properties. These embeddings will be applied to both optimization and machine learning tasks, with a focus on sorting-based neural architectures that yield easier interpretability and explainability. The second thrust involves the development and analyses of optimal factorizations of positive semi-definite self-adjoint operators and associated quadratic bounds. These results will be applied to a special form of the blind source separation problem. To achieve these goals, the research will integrate methods from optimization, harmonic analysis, computational invariant theory, and machine learning. This interdisciplinary approach aims to advance a mathematical framework for constructing invariant coorbit representations, deriving optimal factorizations of positive semi-definite operators, and computing quadratic bounds relevant to learning and optimization tasks. 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.