In the Euclidean setting the proximal gradient method and its accelerated variants are a class of efficient algorithms for optimization problems with decomposable objective. In this paper, we develop a Riemannian proximal gradient method (RPG) and its accelerated variant (ARPG) for similar problems but constrained on a manifold. The global convergence of RPG is established under mild assumptions, and the O(1/k) is also derived for RPG based on the notion of retraction convexity. If assuming the objective function obeys the Rimannian Kurdyka–Łojasiewicz (KL) property, it is further shown that the sequence generated by RPG converges to a single stationary point. As in the Euclidean setting, local convergence rate can be established if the objective function satisfies the Riemannian KL property with an exponent. Moreover, we show that the restriction of a semialgebraic function onto the Stiefel manifold satisfies the Riemannian KL property, which covers for example the well-known sparse PCA problem. Numerical experiments on random and synthetic data are conducted to test the performance of the proposed RPG and ARPG.

在欧几里得设置中，近端梯度法及其加速的变体是针对具有可分解目标的优化问题的一类有效算法。在本文中，我们针对相似的问题但受限于流形开发了一种黎曼近邻梯度方法（RPG）及其加速变体（ARPG）。RPG的全球收敛性是在温和的假设下建立的，并且O（1 / kRPG也是基于缩回凸度的概念而得出的。如果假设目标函数服从Rimannian Kurdyka–Łojasiewicz（KL）属性，则进一步表明RPG生成的序列收敛到单个固定点。与在欧几里得环境中一样，如果目标函数满足指数的黎曼KL性质，则可以确定局部收敛速度。此外，我们表明，将半代数函数限制到Stiefel流形上可以满足黎曼KL性质，例如，它涵盖了众所周知的稀疏PCA问题。进行了随机和合成数据的数值实验，以测试所提出的RPG和ARPG的性能。