where
denotes a Riemannian manifold with the Riemannian metric , is a
nonempty, compact, geodesically convex set, and is
geodesically convex and geodesically -smooth. Here -convex function may be non-convex in
the usual Euclidean space but convex along the manifold, and thus can be
solved by a global optimization solver.
Riemannian Manifold
A Riemannian Manifold is a real smooth
manifold equipped with
a Riemannian metric . Let
denote the inner product of ; and the norm of is defined as , where the metric induces an inner product
structure in each tangent space associated with every .
A geodesic is a constant speed curve
that is locally distance minimizing. Let and , then an exponential map maps to
on , such that there is a
geodesic with and . If there is a unique
geodesic between any two points in , the
exponential map has inverse , i.e., , and the geodesic is the unique shortest
path with , where is the geodesic distance between
.
Parallel transport maps a vector to , and preserves inner products and norm,
that is, , where .
For any and
any geodesic with and for such that ,
then is geodesically
convex (G-convex), and an equivalent definition is formulated
as follows;
where is
the Riemannian gradient of at
. A function is
called geodesically -strongly convex (-strongly G-convex) if for
any , the
following inequality holds
A differential function is
geodesically -smooth
(G-L-smooth) if its gradient is -Lipschitz, i.e.,
Global Convergence Rate
Analysis
provided the global complexity analysis of first-order algorithms for
-convex optimization, and designed
the following Riemannian gradient descent rule:
where is the iterate index,
is an exponential
map at , is a step-size or learning rate, and
is the
Riemannian gradient of at .
References
[1]@inproceedings{liu2017accelerated,
title={Accelerated First-order Methods for Geodesically Convex
Optimization on Riemannian Manifolds.}, author={Liu, Yuanyuan and Shang,
Fanhua and Cheng, James and Cheng, Hong and Jiao, Licheng} }