Since optimization on Riemannian manifolds relies on the chosen metric, it is appealing to know that how the performance of a Riemannian optimization method varies with different metrics and how to exquisitely construct a metric such that a method can be accelerated. To this end, we propose a general framework for optimization problems on product manifolds where the search space is endowed with a preconditioned metric, and we develop the Riemannian gradient descent and Riemannian conjugate gradient methods under this metric. Specifically, the metric is constructed by an operator that aims to approximate the diagonal blocks of the Riemannian Hessian of the cost function, which has a preconditioning effect. We explain the relationship between the proposed methods and the variable metric methods, and show that various existing methods, e.g., the Riemannian Gauss--Newton method, can be interpreted by the proposed framework with specific metrics. In addition, we tailor new preconditioned metrics and adapt the proposed Riemannian methods to the canonical correlation analysis and the truncated singular value decomposition problems, and we propose the Gauss--Newton method to solve the tensor ring completion problem. Numerical results among these applications verify that a delicate metric does accelerate the Riemannian optimization methods.

翻译：由于黎曼流形上的优化依赖于所选度量，了解不同度量如何影响黎曼优化方法的性能，以及如何精巧地构造度量以加速方法，具有重要意义。为此，我们提出一个基于预条件度量的乘积流形优化问题通用框架，并在此度量下发展了黎曼梯度下降法和黎曼共轭梯度法。具体而言，该度量由旨在近似代价函数黎曼海森矩阵对角块的算子构造而成，具有预条件效果。我们阐释了所提方法与变度量方法之间的关系，并表明多种现有方法（如黎曼高斯-牛顿法）可通过具有特定度量的所提框架进行解释。此外，我们针对典型相关分析和截断奇异值分解问题，定制了新的预条件度量并调整了所提黎曼方法，同时提出高斯-牛顿法求解张量环完备问题。这些应用中的数值结果验证了精心设计的度量确实能够加速黎曼优化方法。