CAREER: Geometry-inspired approaches to information theory and learning

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Information theory and geometry are closely intertwined; theoretical developments in one domain have stimulated significant breakthroughs in the other. In geometry, a large body of literature is devoted to studying the sizes and shapes of low-dimensional projections of geometric objects. However, few parallels exist in information theory. On the other hand, information theory has a rich set of tools to analyze extremal problems, which have no suitable analogs in geometry. This project aims to combine the individual strengths of both areas to expand the set of mathematical tools that can be brought to bear upon the study of geometry and information theory. An important goal of the project is the practical applications of results and insights gained through such a study. Motivated by applications in modern data analysis, such as random projections, tomographic reconstruction, and neural network-based algorithms, this project will explore geometry-inspired approaches in modern machine learning. The planned research is highly interdisciplinary and serves to bring together different communities in engineering, mathematics, and computer science. The project will contribute to the intellectual development of undergraduate and graduate students through teaching and research mentorship. Besides, the investigator will engage regularly with K-12 students via the Wisconsin Math, Science, and Engineering Talent Search, and deliver talks at local middle and high schools on topics related to geometry and information theory.The project consists of three distinct technical thrusts. In the first thrust, novel notions of entropy will be developed to characterize the randomness of lower-dimensional projections of a random variable, addressing a conspicuous gap in the geometrization of probability literature. In the second thrust, a new framework for solving extremal problems in geometry using information inequalities will be developed. Novel information inequalities will be investigated and applied to tomographic reconstruction and analysis of X-Ray and Radon transforms. Finally, the third thrust will consider new directions in the theory and applications of machine learning by using tools derived from geometry, information theory, and optimal transport.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.