标题： 凸优化的内在黎曼近似梯度方法 显示英文标题

标题： The Intrinsic Riemannian Proximal Gradient Method for Convex Optimization

摘要： 我们考虑了一类在Hadamard流形上的（可能强）测地凸优化问题，其中目标函数分解为光滑函数和一个可能非光滑函数的和。 我们引入了一种内在的凸黎曼近似梯度（CRPG）方法，该方法在非光滑步骤中使用流形近似映射，而无需在嵌入空间或切空间中进行操作。 建立了凸问题的次线性收敛速率和强凸问题的线性收敛速率，并推导出了推广欧几里得情况的基本近似梯度不等式。 我们在双曲空间和对称正定矩阵流形上的数值实验表明，相较于现有方法具有显著的计算优势。 显示英文摘要

摘要： We consider a class of (possibly strongly) geodesically convex optimization problems on Hadamard manifolds, where the objective function splits into the sum of a smooth and a possibly nonsmooth function. We introduce an intrinsic convex Riemannian proximal gradient (CRPG) method that employs the manifold proximal map for the nonsmooth step, without operating in the embedding or tangent space. A sublinear convergence rate for convex problems and a linear convergence rate for strongly convex problems is established, and we derive fundamental proximal gradient inequalities that generalize the Euclidean case. Our numerical experiments on hyperbolic spaces and manifolds of symmetric positive definite matrices demonstrate substantial computational advantages over existing methods.