CAREER: Geometry and Learning for Manifold-Structured Data in 3D and Beyond

随着现代技术和计算能力的进步，三维及更高维度的数据处理和分析成为医学成像、计算化学、计算生物学、社会网络等各个领域普遍存在的任务。在实际的许多问题中，数据通常与一定的相干和非线性结构相关联。在数学上，这允许我们将数据集建模为在通常嵌入高维环境空间的低维流形上采样的点。与使用成熟的工具处理平面域上的函数的图像和信号处理不同，流形结构数据由于其复杂的几何结构而更具挑战性。例如，由于嵌入、转换或表示的多样性，相同的几何对象可以采用非常不同的坐标表示（想象相同的人体形状可以有不同的姿态作为其近等距嵌入歧义）。这些模糊性形成了一个无限维的等维群，使流形结构数据分析和学习中的高级任务更具挑战性。为了克服这些挑战，从理论和计算的角度开发新的工具来处理流形结构化数据变得越来越重要。本课题旨在探讨几何偏微分方程（PDEs）和学习理论与内在数据分析之间的桥梁连接，以研究流形结构数据的分析和学习。该项目的主要目标包括三个组成部分。第一部分是研究一种基于几何偏微分方程的方法框架，用于表示为不完全点间距离的流形的数据结构。第二部分是克服使用内在描述符处理非近等距流形的性能差的挑战。第三部分提出了一种定义流形上几何卷积的新方法。这为构造所提出的几何卷积神经网络提供了一个构建块，用于对流形结构数据进行深度学习。通过与生物医学工程师的合作，人类大脑映射等应用也将得到探索。本次工作所产生的新方法和研究成果将为解决流形结构数据分析中的问题提供新的方法，并将以适当的方式整合到我未来的教学和课程项目中。教育计划是提供独特的机会，培养对探索几何和学习流形结构数据感兴趣的本科生和研究生，并向公众开放。该奖项反映了美国国家科学基金会的法定使命，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。

With the advance of modern technology and computing power, processing and analyzing of data in three and higher dimensions becomes a ubiquitous task in diverse fields such as medical imaging, computational chemistry, computational biology, social networks and many others. For many problems in practice, data is commonly associated with certain coherent and nonlinear structure. Mathematically, this allows us to model data sets as points sampled on manifolds usually of low dimensions embedded in a high dimensional ambient space. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data is far more challenging due to their complicated geometry. For example, the same geometric object can take very different coordinate representations due to the variety of embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). These ambiguities form an infinite dimensional isometric group and make higher-level tasks in manifold-structured data analysis and learning even more challenging. To overcome these challenges, it becomes increasingly important to develop new tools in both theoretical and computational point of views for processing manifold structured data. This project proposes to investigate analyzing and learning of manifold-structured data by bridging connection from geometric partial differential equations (PDEs) and learning theory to intrinsic data analysis. The major objectives of this project contain three components. The first part is to investigate a framework of geometric-PDEs-based methods to a data structure for manifolds represented as incomplete inter-point distance. The second part is to overcome the challenge of poor performance using intrinsic descriptors to handling not nearly isometric manifolds. In the third part, a new method of defining geometric convolution on manifolds is considered. This provides a building block of constructing the proposed geometric convolutional neural network for conducting deep learning on manifold-structured data. By collaborating with biomedical engineers, applications such as human brain mappings will also be explored. The new methodologies and research findings resulting from the proposed work will lead to new ways of tackling problems in manifold-structured data analysis and will be integrated into my future teaching and course projects in appropriate ways. The education plan is to provide unique opportunities to train undergraduate and graduate students interested in exploring geometry and learning on manifold-structured data, and to reach out the general public.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.