Conformal Field Theory, Cryo-Electron Microscopy, and Neural Networks

该项目将概率应用于数学物理，统计和机器学习三个独立的领域。前两个领域是共形场理论（CFT）和低温电子显微镜。前者是工具的理论高能物理，而对于后者最近的进展，在探测器技术和软件算法已经允许实现在近原子分辨率的生物分子结构的测定。 该项目结合了概率和表征理论的技术来研究它们。第三个领域是神经网络，它使用的方法在大数据实践中非常有效，但在理论上是神秘的。该项目应用概率工具从理论上巩固和改进这些方法。该奖项将支持教育，指导和推广活动，包括经验机器学习的本科生和硕士研究项目以及美国国家数学奥林匹克队的培训。更详细地说，该项目侧重于CFT中的共形引导，冷冻电子显微镜的非凸优化以及神经网络的数据增强。 项目的第一部分应用PI先前的工作，给出Liouville共形块的概率构造，以表征其性质，并完成Liouville理论的共形引导程序。 项目的第二部分应用PI先前的工作来表征和改进一阶优化算法，用于在实践中的冷冻EM中的最大似然估计。 该项目的第三部分应用了PI之前的工作，将数据增强和随机优化相结合，以创建理论上有原则的数据增强方法。该奖项反映了NSF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。

This project applies probability to three separate areas of mathematical physics, statistics, and machine learning. The first two areas are conformal field theory (CFT) and cryo-electron microscopy. The former is instrumental for the theoretical high-energy physics, while for the latter recent advances in detector technology and software algorithms have allowed to achieve the determination of biomolecular structures at near-atomic resolution. The project combines techniques from probability and representation theory to study them. The third area, neural networks, uses methods which are extremely effective in practice of Big Data, yet theoretically are mysterious. The project applies probabilistic tools to theoretically ground and improve these methods. The award will support the educational, mentoring, and outreach activities including undergraduate and masters research projects in empirical machine learning and training of the US national math olympiad team. In more detail, the project focuses on the conformal bootstrap in CFT, non-convex optimization for cryo-electron microscopy, and data augmentation for neural networks. The first part of the project applies prior work of the PI giving a probabilistic construction of Liouville conformal blocks to characterize their properties and complete the conformal bootstrap program for Liouville theory. The second part of the project applies prior work of the PI to characterize and improve first-order optimization algorithms for maximum-likelihood estimation in cryo-EM in practice. The third part of the project applies prior work of the PI connecting data augmentation and stochastic optimization to create theoretically principled data augmentation 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.