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: 2308597
• 负责人: Shiqian Ma
• 金额: $ 31.68万
• 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 2024-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2308597&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，机器学习以及信息科学和工程等多样化的领域中无处不在，从而创造了算法有效解决此类问题的需求。该项目将研究非凸优化的重要但较不发达的领域，即非平滑和非平滑的利马尼亚优化。该项目的结果将提供有关非凸优化问题的重要类别的见解，并将导致开发解决方案的新工具。关于非凸优化问题的新教学材料将用于教育在这一重要类别中的下一代学生。这种探索的社会影响将是使基因表达，自主驾驶和癌症研究等领域的新应用有益。虽然现有的理论和算法的利曼优化通常要求目标函数可区分，但相比之下，该项目侧重于非平滑和非平滑和非lipschitz riemannian优化。特别是，该项目将研究几种非平滑优化的算法，这些算法在Riemannian环境中较少发达，包括乘数的歧管交替方向方法，惯性歧管近端近端方法，随机歧管近端点算法和歧管副标测 - 歧管 - 歧管 - 线性Algorithm。为了使用非lipschitz物镜进行利曼式优化，研究人员将得出相应的最佳条件，然后设计基于平滑技术的两种算法，即Riemannian平滑梯度下降方法和Riemannian平滑信任区域方法。提出的算法将被实施以解决现实世界中的应用程序，例如单细胞RNA测序数据的聚类，以及在自主驾驶中的3D对象检测和3D跟踪。这一奖项反映了NSF的法定任务，并通过评估该基金会的知识绩效和广泛的影响来评估NSF的法定任务，并被视为值得进行的支持。