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

基于新型交通的几何形状、梯度流及其在数据科学中的应用

• 批准号: 2206069
• 负责人: Dejan Slepcev
• 金额: $ 37.82万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206069&HistoricalAwards=false
• 关键词: Novel; Transportation; Based; Geometries; Gradient

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.

该项目将开发数学工具来研究数据科学和信号处理任务。尽管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，许多当前模型的有效性降低了高维度。该项目将提供机器学习任务的方法，以利用数据的几何形状，并且可以在高维度中准确近似。该项目将导致新的，更准确的方法来采样和表示数据分布。该项目将为培训新一代数学家提供机会，他们将了解现代应用分析的现代技术，并意识到数据科学中引起的重要问题。数据科学中的问题，偏微分方程（PDE）在图形上以及信号分析中的任务，该项目将调查几个不同的设置。一项重大努力将致力于研究采样的集合方法，例如Stein变异梯度下降和相关模型。这些模型将提供基于确定的粒子方法，用于对吉布斯分布进行一般电势。该项目将研究Stein几何形状和相关模型中的几何形状和梯度流。更广泛的是，基于合奏的方法为解决挑战性采样问题（多模式，高度各向异性的能源景观）提供了有希望的途径。它们通过平均场限制与PDE的联系还可以进行分析研究和建模。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。该项目还将基于非局部连续性方程和产生的非本地瓦斯汀指标的概率度量空间中的路径研究。关于信号分析，该项目将在信号空间上研究基于变形的几何形状，这些几何形状允许运输和基于强度的差异。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响来审查标准的评估值得支持的。