  Stochastic gradient descent is a simple approach to find the local minima of
a cost function whose evaluations are corrupted by noise. In this paper, we
develop a procedure extending stochastic gradient descent algorithms to the
case where the function is defined on a Riemannian manifold. We prove that, as
in the Euclidian case, the gradient descent algorithm converges to a critical
point of the cost function. The algorithm has numerous potential applications,
and is illustrated here by four examples. In particular a novel gossip
algorithm on the set of covariance matrices is derived and tested numerically.
