CAREER: Geometry-inspired approaches to information theory and learning

信息论和几何学紧密地交织在一起；一个领域的理论发展刺激了另一个领域的重大突破。在几何学中，大量文献致力于研究几何物体的低维投影的大小和形状。然而，在信息论中很少有相似之处。另一方面，信息论有丰富的工具来分析几何中没有合适的类比的极值问题。该项目旨在结合这两个领域的各自优势，以扩展可用于几何和信息理论研究的数学工具集。该项目的一个重要目标是通过这种研究获得的结果和见解的实际应用。受现代数据分析应用的启发，如随机投影、层析成像重建和基于神经网络的算法，该项目将探索现代机器学习中受几何启发的方法。计划中的研究是高度跨学科的，并将工程、数学和计算机科学的不同社区聚集在一起。该项目将通过教学和研究指导，促进本科生和研究生的智力发展。此外，研究者将通过威斯康星数学、科学和工程人才搜索定期与K-12学生接触，并在当地的初中和高中就几何和信息理论相关的主题发表演讲。该项目由三个不同的技术重点组成。在第一篇文章中，将发展熵的新概念，以表征随机变量的低维投影的随机性，解决概率几何化文献中的明显差距。在第二个重点，一个新的框架，解决几何极值问题，利用信息不等式将发展。新的信息不等式将被研究并应用于x射线和氡变换的层析重建和分析。最后，第三个重点将通过使用源自几何、信息理论和最优运输的工具，考虑机器学习理论和应用的新方向。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.