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，大量的文献致力于研究几何对象的低维投影的大小和形状。然而，在信息论中几乎没有类似的东西。另一方面，信息论有丰富的工具来分析极值问题，这在几何中没有合适的类似物。这个项目的目的是联合收割机的个人优势，这两个领域扩大了一套数学工具，可以带来承担后，几何和信息理论的研究。该项目的一个重要目标是实际应用通过这一研究获得的结果和见解。受现代数据分析应用的启发，如随机投影，断层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的几何启发方法。计划中的研究是高度跨学科的，旨在汇集工程，数学和计算机科学的不同社区。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，研究员将通过威斯康星州数学，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论相关的主题进行演讲。该项目包括三个不同的技术方向。在第一个推力，熵的新概念将被开发来表征随机变量的低维投影的随机性，解决概率文献的几何化中的一个明显的差距。在第二个推力，一个新的框架解决极值问题的几何信息不等式将开发。新的信息不等式将被调查和应用到层析重建和分析的X射线和Radon变换。最后，第三个重点是通过使用几何学、信息论和最佳运输工具，考虑机器学习理论和应用的新方向。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.