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

海量高维数据普遍存在于生物信息学、计算机视觉、神经成像、信号处理等众多科学和工程学科中。这一建议是由用于分析这些学科的数据的新兴工具推动的，例如使用高维和多维数据的非平滑、基于流形的学习。基于统计学、机器学习和优化之间的协同作用，本研究将致力于非光滑流形优化的新的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方面的新方法和新理论。将开发软件包，以便其他研究人员和从业人员随时可以获得研究成果。此外，该项目还将通过培养研究生来加强未来的技术队伍。众所周知，高维数据的统计建模可能包含目标函数中的非光滑正则化，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带来了新的机遇和挑战，也为发展新的统计方法和理论带来了新的机遇和挑战。本课题的研究成果将为基于非光滑流形的学习提供新的强有力的分析工具和理论保障。该奖项反映了NSF的法定使命，通过使用基金会的学术价值和更广泛的影响审查标准进行评估，被认为是值得支持的。

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.