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

如今，大量数据的可用性不断增加，主要是由于技术的不断进步。因此，大量的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。高维和多维数据的非光滑流形学习由于其固有的非凸性和非光滑性而变得非常复杂。这个项目将解决基于非光滑流形学习的统计和计算问题，并探索其新的应用。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。

Nowadays, the availability of massive data is continuously increasing, primarily due to the continued advancement of technology. As a consequence, massive high-dimensional data are ubiquitous in many scientific and engineering disciplines, such as bioinformatics, computer vision, neuroimaging, and signal processing. The nonsmooth manifold-based learning with high-dimensional and multidimensional data is in general complicated due to its intrinsic non-convexity and non-smoothness. This project will address both statistical and computational issues of nonsmooth manifold-based learning and explore its new applications. 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.