  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.
