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，但由于计算方面的挑战和缺乏统计可靠性，许多现有模型的有效性在高维情况下会降低。该项目将提供机器学习任务的方法，利用数据的几何形状，可以在高维上精确地近似。该项目将带来新的、更准确的采样和表示数据分布的方法。该项目将为培养新一代数学家提供机会，他们将获得现代应用分析技术的知识，并了解数据科学中出现的重要问题。受数据科学问题、图上的偏微分方程（PDE）和信号分析任务的启发，该项目将研究几种不同的设置。主要的工作将致力于研究采样的集合方法，如斯坦因变分梯度下降和相关模型。该模型将为一般势的吉布斯分布的采样提供一种确定性的基于粒子的方法。该项目将研究斯坦因几何和相关模型中的几何和梯度流。更广泛地说，基于集成的方法为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）提供了一条有希望的途径。它们通过平均场限制与PDE的联系也允许分析研究和建模。研究者和合作者将探索几个模型，并发展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。在信号分析方面，该项目将研究基于变形的信号空间几何形状，以考虑交通和强度的差异。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。

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.