Collaborative Research: New Methods, Theory and Applications for Nonsmooth Manifold-Based Learning

协作研究：非平滑流形学习的新方法、理论和应用

• 批准号: 2243650
• 负责人: Shiqian Ma
• 金额: $ 15万
• 依托单位: William Marsh Rice 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: 2022-10-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2243650&HistoricalAwards=false
• 关键词: Collaborative; Research; New; Methods; Theory

Massive high-dimensional data are ubiquitous in many scientific and engineering disciplines, such as bioinformatics, computer vision, neuroimaging, and signal processing. This proposal is motivated by emerging tools for analyzing data from these disciplines, such as nonsmooth, manifold-based learning with high-dimensional and multidimensional data. Building on the synergy among statistics, machine learning, and optimization, this research will focus on the development of new optimization algorithms and theory for nonsmooth manifold optimization. The project will also build on existing optimization strengths to develop new methods and theory in statistics and machine learning. Software packages will be developed to make the research outcomes readily available to other researchers and practitioners. In addition, the project will enhance the future technical workforce through the training of graduate students. It is known that statistical modeling of high-dimensional data may include the non-smooth regularization in the objective function, and some may even involve non-convex manifold constraints such as orthogonality constraints. The manifold-based learning offers a powerful framework for dimension reduction and signal processing. The combination of non-smooth regularization and non-convex manifold constraints brings new opportunities and challenges for designing optimization algorithms with convergence guarantees and also for developing new statistical methods and theory. The research outcomes of this project will provide new powerful analytic tools in nonsmooth manifold-based learning with theoretical guarantees. Software packages will be developed to make the research outcomes readily available to other researchers and practitioners.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.

海量高维数据在许多科学和工程学科中无处不在，例如生物信息学、计算机视觉、神经成像和信号处理。这一提议的动机是新兴的工具，用于分析这些学科的数据，如非光滑，基于流形的学习与高维和多维数据。基于统计学、机器学习和优化之间的协同作用，本研究将专注于非光滑流形优化的新优化算法和理论的发展。该项目还将在现有优化优势的基础上开发统计和机器学习的新方法和理论。将开发软件包，使其他研究人员和从业人员能够随时获得研究成果。此外，该项目还将通过培训研究生来加强未来的技术劳动力。众所周知，高维数据的统计建模在目标函数中可能包含非光滑正则化，有的甚至可能涉及非凸流形约束，如正交性约束。基于流形的学习为降维和信号处理提供了一个强大的框架。非光滑正则化和非凸流形约束的结合为设计具有收敛保证的优化算法以及发展新的统计方法和理论带来了新的机遇和挑战。本项目的研究成果将为非光滑流形学习提供新的强有力的分析工具和理论保证。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。