  This paper proposes a novel general framework of Riemannian conjugate
gradient methods, that is, conjugate gradient methods on Riemannian manifolds.
The conjugate gradient methods are important first-order optimization
algorithms both in Euclidean spaces and on Riemannian manifolds. While various
types of conjugate gradient methods are studied in Euclidean spaces, there have
been fewer studies on those on Riemannian manifolds. In each iteration of the
Riemannian conjugate gradient methods, the previous search direction must be
transported to the current tangent space so that it can be added to the
negative gradient of the objective function at the current point. There are
several approaches to transport a tangent vector to another tangent space.
Therefore, there are more variants of the Riemannian conjugate gradient methods
than the Euclidean case. In order to investigate them in more detail, the
proposed framework unifies the existing Riemannian conjugate gradient methods
such as ones utilizing a vector transport or inverse retraction and also
develops other methods that have not been covered in previous studies.
Furthermore, sufficient conditions for the convergence of a class of algorithms
in the proposed framework are clarified. Moreover, the global convergence
properties of several specific types of algorithms are extensively analyzed.
The analyses provide the theoretical results for some algorithms in a more
general setting than the existing studies and completely new developments for
the other algorithms. Numerical experiments are performed to confirm the
validity of the theoretical results. The results also compare the performances
of several specific algorithms in the proposed framework.
