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

这个项目将概率应用于数学物理、统计学和机器学习这三个不同的领域。前两个领域是共形场理论(CFT)和低温电子显微镜。前者是理论高能物理的工具，而后者是探测器技术和软件算法的最新进展，使其能够在近原子分辨率下确定生物分子结构。该项目结合了概率论和表征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的实践中非常有效，但在理论上却是神秘的。该项目应用概率工具从理论上为这些方法奠定基础，并对其进行改进。该奖项将支持教育、指导和推广活动，包括本科生和硕士在经验式机器学习和美国国家数学奥林匹克代表队培训方面的研究项目。更详细地说，该项目集中在CFT中的共形自举，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的前人工作，给出了Liouville共形块的概率构造，刻画了它们的性质，完成了Liouville理论的共形Bootstrap程序。项目的第二部分应用PI的前人工作，在实践中刻画和改进了低温EM中最大似然估计的一阶优化算法。该项目的第三部分应用PI将数据增强和随机优化相结合的先前工作，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智力优势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.