CAREER: Variational and Geometric Methods for Data Analysis

该项目将使用应用数学中的工具来开发和分析大规模数据分析的可伸缩方法，特别是针对聚类、几何处理、图像分析和分析数据中的高阶相互作用的问题。一个特定的最终目标是使用N方向场来创建四边形网格，这种网格可以被证明是高质量的，并在工程有限元模拟中立即应用。PI计划通过研究研讨会和学生项目来吸引、参与和教育犹他大学的本科生和研究生。学生将从接触这一跨学科领域中受益，并成为研究的组成部分，参加定期的小组会议和讨论，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，最优化导论和数据科学导论，将为犹他大学科学学院和工程学院的学生提供服务。作为犹他大学Access项目的一部分，一门新的数据分析课程将为这一代表性不足的群体培养STEM学科的学习。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。该项目为即将入学的女本科生提供了一个为期七周的强化暑期课程。由于相关的变分特征和随机过程，这些方法在几何和/或物理上是可解释的，并且具有可证明的性质，补充和扩展了来自统计学和计算机科学的传统数据分析方法。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公式有关的基本问题，以及与图曲率和Merriman-Bence-Osher(MBO)扩散产生的运动之间的联系。特别是，对MBO方法的新的概率解释将导致系统地平衡分区分量的高效算法。第二个目标是使用矢量场的泛化，称为N方向场或交叉场，当N=4时，用于几何处理和图像分析中的各种任务。这项工作是由PI和他的研究生在基于Ginzburg-Landau理论生成边界对齐的四边形网格方面取得的最新进展而进行的。在一定程度上，这将涉及MBO算法和相关的Lyapunov函数的扩展，以逼近广义集上的像的调和映射。第三个目标是发展分析单纯复形的有效方法，推广和推广分析图的方法。为了克服单纯复形的非局部和多尺度特性带来的固有计算成本，PI将基于保留相关广义拉普拉斯算子的谱来开发高效的稀疏算法。这一奖项反映了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 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.