CAREER: Variational and Geometric Methods for Data Analysis

该项目将使用应用数学的工具来开发和分析用于大规模数据分析的可扩展方法，特别是聚类，几何处理，图像分析和分析数据中的高阶相互作用等问题。一个具体的最终目标是使用N方向场来创建四边形网格，这是可以证明的高质量，并在工程有限元模拟中有直接的应用。PI计划通过研究研讨会和学生项目吸引、参与和教育犹他州大学的本科生和研究生。学生将受益于接触这个跨学科领域，并成为研究的一个组成部分，参加定期小组会议和讨论，并有机会在会议上展示他们的研究成果。拟议的教育计划包括开发两门课程，优化导论和数据科学导论，这两门课程将为整个犹他州大学科学与工程学院的学生提供服务。作为犹他州大学ACCESS项目的一部分，一个针对即将入学的女本科生的为期七周的暑期强化项目，一个关于数据分析的新课程将促进这一代表性不足的群体对STEM学科的研究。该项目将开发和分析基于几何、变分原理和偏微分方程的新计算方法，用于数据分析。由于相关的变分特征和随机过程，这些方法是几何和/或物理解释，并具有可证明的属性，补充和扩展传统的数据分析方法从统计和计算机科学。拟议的研究计划有三个主要目标。第一个目标是研究与图划分问题的Cheeger公式相关的基础问题，以及与图曲率和Merriman-Bence-Osher（MBO）扩散生成运动的连接。特别是，一个新的概率解释的MBO方法将导致有效的算法，系统地平衡分区组件。第二个目标是使用向量场的泛化，当N = 4时称为N方向场或交叉场，用于几何处理和图像分析中的各种任务。这项工作是很好的动机，最近的进展PI和他的研究生对生成的边界对齐的四边形网格的基础上金兹伯格-朗道理论。在某种程度上，这将涉及到MBO算法和相关的李雅普诺夫函数的扩展，以近似调和映射的图像在广义集。第三个目标是开发分析单纯复形的有效方法，推广和扩展分析图的方法。为了克服单纯复形的非局部和多尺度性质所带来的固有计算成本，PI将在保留相关广义拉普拉斯算子频谱的基础上开发高效的稀疏化算法。该奖项反映了NSF的法定使命，并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，被认为值得支持。

This project will use tools from applied mathematics to develop and analyze scalable methods for large-scale data analysis, especially for the problems of clustering, geometry processing, image analysis, and analyzing high-order interactions in data. One specific end-goal is to use N-direction fields to create quad meshes, which are provably high-quality, and have immediate applications in finite element simulations for engineering. The PI plans to attract, involve, and educate students at both the undergraduate and graduate levels at the University of Utah through research seminars and student programs. Students will benefit from exposure to this interdisciplinary field and be an integral part of the research, participating at regular group meetings and discussions, and given the opportunity to present their research findings at conferences. The proposed education plan includes the development of two courses, Introduction to Optimization and Introduction to Data Science, which will serve students throughout the University of Utah's Schools of Science and Engineering. A new course on Data analysis as part of the University of Utah's ACCESS program, a seven-week intensive summer program for incoming female undergraduates, will foster study of STEM disciplines for this underrepresented group.This project will develop and analyze new computational methods based on geometry, variational principles, and partial differential equations for data analysis. Due to associated variational characterizations and stochastic processes, these methods are geometrically and/or physically interpretable and have provable properties, complementing and extending traditional data analytic methods from statistics and computer science. The proposed research plan has three primary goals. The first goal is to study foundational questions related to the Cheeger formulation of the graph partitioning problem and connections to graph curvature and Merriman-Bence-Osher (MBO) diffusion generated motion. In particular, a new probabilistic interpretation of the MBO method will lead to efficient algorithms that systematically balance partition components. The second goal is to use a generalization of vector fields, called N-direction fields or cross fields when N=4, for a variety of tasks in geometry processing and image analysis. This work is well-motivated by recent progress of the PI and his graduate student on the generation of boundary-aligned quadrilateral meshes based on the Ginzburg-Landau theory. In part, this will involve the extension of the MBO algorithm and associated Lyapunov function to approximate harmonic maps with image in generalized sets. The third goal is to develop efficient methods for analyzing simplicial complexes, generalizing and extending methods for analyzing graphs. To overcome the inherent computational costs due to the non-local and multi-scale nature of simplicial complexes, the PI will develop efficient sparsification algorithms based on preserving the spectrum of associated generalized Laplacian operators.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.