Convergence Analysis of Gradient Algorithms on Riemannian Manifolds without Curvature Constraints and Application to Riemannian Mass
Convergence Analysis of Gradient Algorithms on Riemannian Manifolds without Curvature Constraints and Application to Riemannian Mass
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DOI：
10.1137/19m1289285
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
Jinhua Wang;Xiangmei Wang;Chong Li;J. Yao
中科院分区：
文献类型：
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作者：
Jinhua Wang;Xiangmei Wang;Chong Li;J. Yao
我们研究了一般黎曼流形（无曲率限制）上优化问题的梯度算法（采用一般步长）的收敛性问题。在局部凸性/拟凸性（分别对应弱尖锐极小值）的假设下，建立了局部/全局收敛（分别对应线性收敛）结果。作为应用，分别探讨了采用固定步长和Armijo步长的梯度算法在寻找黎曼 $L^p$（$p\in[1,+\infty)$）质心时的线性收敛性质，这尤其扩展和/或改进了文献[Afari2013]中的相应结果。
We study the convergence issue for the gradient algorithm (employing general step sizes) for optimization problems on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity (resp. weak sharp minima), local/global convergence (resp. linear convergence) results are established. As an application, the linear convergence properties of the gradient algorithm employing the constant step sizes and the Armijo step sizes for finding the Riemannian $L^p$ ($p\in[1,+\infty)$) centers of mass are explored, respectively, which in particular extend and/or improve the corresponding results in \cite{Afsari2013}.