Towards Geometry-Informed Machine Learning: A Comprehensive Framework for Recognizing and Leveraging Heterogeneous Geometric Structure in Data

This project supports the development of more efficient and sustainable machine learning methods using inherent structure in the data. Structured data arises in many scientific and industrial applications, including relational structure in complex social and biological systems, hierarchical structure in information and language systems, as well as symmetries in scientific data that derive from fundamental laws of physics.

The project aims to develop methods for identifying, characterizing, and leveraging such structure in machine learning and data science applications. Research findings will be incorporated into graduate courses and graduate and undergraduate students from potentially diverse backgrounds will be mentored as part of this project, contributing to the training of the next generation of applied mathematicians.

In addition, ideas and concepts with direct relation to the proposed research will be incorporated into STEM outreach activities with the goal of sharing the research findings with the broader community.

The project aims to develop a novel computational framework for leveraging geometric structure in data that is applicable to settings without pre-existing knowledge on data geometry. Geometric representation learning will be formalized as a model selection problem, where the respective geometric characteristics are learned from data. The project’s results will contribute to the field by providing a systematic analysis of the benefits of geometric machine learning methods compared to classical Euclidean approaches.

With that, the project aims to develop a deeper theoretical understanding of geometric machine learning and offer practical, empirically validated guidelines for the application of such methods.

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.