Novel Transportation-Based Geometries, Gradient Flows, and Applications to Data Science

该项目将开发数学工具来研究数据科学和信号处理任务。虽然微分方程和变分方法为数据科学任务提供了有用的模型，但由于计算挑战和缺乏统计可靠性，许多现有模型的有效性在高维中降低。该项目将提供机器学习任务的方法，这些方法利用数据的几何形状，并且可以在高维中准确地近似。该项目将导致新的和更准确的方式来采样和表示数据分布。该项目将为培养新一代数学家提供机会，他们将获得应用分析的现代技术知识，并意识到数据科学中出现的重要问题。该项目将研究数据科学中的问题，图形上的偏微分方程（PDE）和信号分析中的任务。一个主要的努力将致力于研究采样的集合方法，如斯坦变分梯度下降和相关的模型。该模型将提供一个确定性的粒子为基础的方法，一般潜力的吉布斯分布采样。该项目将研究Stein几何和相关模型中的几何和梯度流。更广泛地说，基于集成的方法提供了一个有前途的途径来解决具有挑战性的采样问题（多模态，高度各向异性的能量景观）。它们通过平均场极限与偏微分方程的连接也允许分析研究和建模。研究人员和合作者将探索几种模型，并开发理论理解和计算方法。该项目还将研究基于非局部连续性方程和由此产生的非局部Wasserstein度量的概率测度空间中的路径。关于信号分析，该项目将研究信号空间上的基于变形的几何形状，考虑到传输和基于强度的差异。该奖项反映了NSF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而被认为值得支持。

This project will develop mathematical tools to study data science and signal processing tasks. While differential equations and variational approaches provide useful models for data science tasks, the effectiveness of many of the present models diminishes in high dimensions due to computational challenges and a lack of statistical reliability. This project will provide approaches to machine learning tasks that take advantage of the geometry of the data and can be accurately approximated in high dimensions. The project will lead to new and more accurate ways to sample and represent data distributions. The project will provide opportunities for training a new generation of mathematicians who will gain knowledge of modern techniques of applied analysis and be aware of important questions arising in data science.Motivated by problems in data science, partial differential equations (PDE) on graphs, and tasks in signal analysis, the project will investigate several distinct settings. A major effort will be devoted to studying ensemble methods for samplings, such as the Stein Variational Gradient Descent and related models. The models will provide a deterministic particle-based method for sampling Gibbs distributions for general potentials. The project will investigate the geometry and gradient flows in Stein geometry and related models. More broadly, ensemble-based methods provide a promising avenue to address challenging sampling problems (multimodal, highly anisotropic energy landscapes). Their connection to PDE via mean-field limits also allows for analytical study and modeling. The investigator and collaborators will explore several models and develop both theoretical understanding and computational approaches. The project will also study paths in the spaces of probability measures based on the nonlocal continuity equation and the resulting nonlocal Wasserstein metrics. Regarding signal analysis, the project will study deformation-based geometries on the space of signals that allow for both transportation and intensity-based differences.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.