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

大量的高维数据在许多科学和工程学科中无处不在，如生物信息学、计算机视觉、神经成像和信号处理。这一建议是由新兴的工具来分析这些学科的数据，如非光滑的、基于流形的高维和多维数据学习。基于统计学、机器学习和优化之间的协同作用，本研究将侧重于开发新的非光滑流形优化算法和理论。该项目还将以现有的优化优势为基础，开发统计学和机器学习方面的新方法和理论。将开发软件包，使研究成果随时可供其他研究人员和实践者使用。此外，该项目将通过培养研究生来增强未来的技术劳动力。众所周知，高维数据的统计建模可能包括目标函数中的非光滑正则化，有些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。非光滑正则化与非凸流形约束的结合为设计具有收敛性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。本课题的研究成果将为基于非光滑流形的学习提供新的强大的分析工具和理论保障。将开发软件包，使研究成果随时可供其他研究人员和实践者使用。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。

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.