Collaborative Research: Geometric Harmonic Analysis in Learning and Inference: Theory and Applications

对象的两两比较是人类学习从海量数据集进行推理的重要方式。在许多现代科学和工程领域中，产生的大规模高维数据集包含了每个对象中丰富的结构信息，允许人们在各个对象之间进行详细的成对比较。为了保留数据的精细结构信息，重要的是同时考虑标量相似性度量和描述数据点之间关系的变换。当变换允许诸如群之类的代数结构时，额外的代数刚性约束为有效的学习和推理策略提供了新的线索，这些策略在现有文献中基本上没有被探索过。PI的目标是利用数据中低维结构的两个来源：(I)数据背后的流形，和(Ii)群变换之间的代数一致性，以设计出高度准确和计算高效的统计方法，从来自社会、生物医学和比较生物科学的海量复杂数据集中提取模式。该项目将包括教育和培训下一批学生，并为他们配备从事数据科学工作的必要工具。通过组织研讨会传播研究成果和建立不同领域之间的联系也是拟议工作的重要方面。该项目的目标是开发新的几何调和分析方法，以提取信息并在配备了分组变换的大规模数据集上进行推理。这将涉及以下三个相互关联的目标的基础理论工作和算法开发：(I)跨频率通道的角同步，(Ii)公共主丛的多个相关向量丛上的扩展向量扩散图，以及(Iii)通过群值两两相互作用的多重不可约表示在分子构象空间和生物解剖表面的形状空间中进行群落检测。在实践方面，PI建议将这些新开发的技术应用于生物医学和比较生物科学中的高影响领域，包括(1)冷冻EM和冷冻电子断层扫描(ET)图像去噪，(2)进化和比较生物学中的形状空间分析，以及(3)学习生物分子机器的构象空间和动力学结构。在项目期间开发的技术将广泛适用于跨学科，其中观察到的噪声、不完整，并可能通过未知群体元素的行动进行潜在转化而被修改。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。

Pairwise comparison of objects is an important way human beings learn to reason from massive data sets. In many modern science and engineering fields, large-scale high dimensional data sets are generated with abundant structural information within each object, allowing people to conduct detailed pairwise comparisons between individual objects. To preserve the fine structural information of the data, it is important to take into account both the scalar similarity measure and the transformations that describes the relation between the data points. When the transformations admit an algebraic structure such as a group, the additional algebraic rigidity constraints shed new lights upon efficient learning and inference strategies largely unexplored in existing literature. The PIs aim to utilize the two sources of low-dimensional structures in data: (i) the manifold underlying the data, and (ii) the algebraic consistency among the group transformations, to devise highly accurate and computationally efficient statistical methods for extracting patterns in massive complex data sets emerging from social, biomedical, and comparative biological sciences. This project will involve educating and training the next wave of students, and equipping them with the necessary tools to work in data science. Dissemination of research results and building connections among different fields through organizing workshops are also important aspects of the proposed work.The goal of the project is to develop novel geometric harmonic analysis methods to extract information and perform inference on large-scale datasets equipped with group transformations. This will involve foundational theoretical work and algorithm development in the following three interrelated objectives: (i) angular synchronization across frequency channels, (ii) extended vector diffusion maps on multiple associated vector bundles of a common principal bundle, and (iii) community detection in conformation spaces of molecules and shape spaces of biological anatomical surfaces through multiple irreducible representations of group-valued pairwise interactions. On the practical side, the PIs propose to apply these newly developed techniques to high impact domain applications in biomedical and comparative biological sciences, including (1) cryo-EM and cryo-electron tomography (ET) image denoising, (2) shape space analysis in evolutionary and comparative biology, and (3) learning conformation spaces and dynamical structures of biomolecular machines. The techniques developed during the project period will be broadly applicable across disciplines, where the observations are noisy, incomplete, and possibly modified by a latent transformation through the action of an unknown group element.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.