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中进行推理的一种重要方式。在许多现代科学和工程领域中，产生了大规模的高维数据集，每个对象内具有丰富的结构信息，允许人们在单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，重要的是要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现有文献中基本上未探索的高效学习和推理策略提供了新的线索。PI旨在利用数据中低维结构的两个来源：（i）数据的流形，以及（ii）组变换之间的代数一致性，以设计高度准确和计算效率高的统计方法，用于从社会，生物医学和比较生物科学中提取大量复杂数据集的模式。该项目将涉及教育和培训下一批学生，并为他们提供在数据科学领域工作所需的工具。通过举办研讨会，传播研究成果，建立不同领域之间的联系，也是该项目的重要内容。该项目的目标是开发新的几何调和分析方法，以提取信息，并对配备了群变换的大规模数据集进行推理。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）跨频率信道的角同步，（ii）公共主丛的多个相关联的向量丛上的扩展向量扩散映射，以及（iii）通过群的多个不可约表示在分子的构象空间和生物解剖表面的形状空间中进行群落检测，有价值的两两互动。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力领域应用，包括（1）cryo-EM和cryo-electron tomography（ET）图像降噪，（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各学科，其中观测是嘈杂的，不完整的，并可能通过一个未知的群体元素的行动潜在的转换修改。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.