Collaborative Research: CIF: Small: New Theory and Applications of Non-smooth and Non-Lipschitz Riemannian Optimization

合作研究：CIF：小：非光滑和非Lipschitz黎曼优化的新理论和应用

• 批准号: 2007823
• 负责人: Lingzhou Xue
• 金额: $ 21.48万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-10-01 至 2024-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2007823&HistoricalAwards=false
• 关键词: Collaborative; Research; CIF; Small; New

Non-convex optimization problems are ubiquitous in fields as diverse as data science, machine learning, and information science and engineering, thereby creating a need for algorithms to efficiently solve such problems. This project will study an important but less developed area in non-convex optimization, namely non-smooth and non-Lipschitz Riemannian optimization. The outcomes of this project will provide insights into important classes of non-convex optimization problems, and will lead to the development of new tools for solving them. New teaching material on non-convex optimization problems will be produced for educating the next generation students in this important class of applications. The societal impact of this exploration will be to benefit new applications in areas such as gene expression, autonomous driving and cancer studies. While existing theory and algorithms for Riemannian optimization usually require the objective function to be differentiable, in contrast this project focuses on non-smooth and non-Lipschitz Riemannian optimization. In particular, the project will study several algorithms for non-smooth optimization that are less developed in the Riemannian setting, including the manifold alternating direction method of multipliers, the inertial manifold proximal gradient method, the stochastic manifold proximal point algorithm, and the manifold prox-linear algorithm. For Riemannian optimization with a non-Lipschitz objective, the investigators will derive the corresponding optimality conditions and then design two algorithms that are based on a smoothing technique, namely the Riemannian smoothing gradient descent method and the Riemannian smoothing trust region method. The proposed algorithms will be implemented to solve real-world applications such as the clustering of single-cell RNA sequencing data, and 3D object detection and 3D tracking in autonomous driving.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.

非凸优化问题在数据科学、机器学习、信息科学与工程等各个领域普遍存在，因此需要算法来有效地解决此类问题。该项目将研究非凸优化中一个重要但欠发达的领域，即非光滑和非李普希茨黎曼优化。该项目的成果将提供对重要类别的非凸优化问题的见解，并将导致解决这些问题的新工具的开发。我们将制作关于非凸优化问题的新教材，用于教育下一代学生这一重要的应用类别。这项探索的社会影响将有利于基因表达、自动驾驶和癌症研究等领域的新应用。虽然黎曼优化的现有理论和算法通常要求目标函数可微，但本项目侧重于非平滑和非 Lipschitz 黎曼优化。特别是，该项目将研究黎曼环境中较少开发的几种非光滑优化算法，包括乘子流形交替方向法、惯性流形近端梯度法、随机流形近点算法和流形近似值法。 -线性算法。对于非Lipschitz目标的黎曼优化，研究人员将推导相应的最优性条件，然后设计两种基于平滑技术的算法，即黎曼平滑梯度下降法和黎曼平滑置信域法。所提出的算法将用于解决现实世界的应用，例如单细胞RNA测序数据的聚类，以及自动驾驶中的3D物体检测和3D跟踪。该奖项反映了NSF的法定使命，并通过评估认为值得支持利用基金会的智力优势和更广泛的影响审查标准。