  This paper considers the problem of minimizing the summation of a
differentiable function and a nonsmooth function on a Riemannian manifold. In
recent years, proximal gradient method and its invariants have been generalized
to the Riemannian setting for solving such problems. Different approaches to
generalize the proximal mapping to the Riemannian setting lead versions of
Riemannian proximal gradient methods. However, their convergence analyses all
rely on solving their Riemannian proximal mapping exactly, which is either too
expensive or impracticable. In this paper, we study the convergence of an
inexact Riemannian proximal gradient method. It is proven that if the proximal
mapping is solved sufficiently accurately, then the global convergence and
local convergence rate based on the Riemannian Kurdyka-\L ojasiewicz property
can be guaranteed. Moreover, practical conditions on the accuracy for solving
the Riemannian proximal mapping are provided. As a byproduct, the proximal
gradient method on the Stiefel manifold proposed in~[CMSZ2020] can be viewed as
the inexact Riemannian proximal gradient method provided the proximal mapping
is solved to certain accuracy. Finally, numerical experiments on sparse
principal component analysis are conducted to test the proposed practical
conditions.
