CAREER: Variational and Geometric Methods for Data Analysis

University Of Utah, Salt Lake City UT

Investigators

Abstract

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.