Abstract
In this paper, we generalize the Stochastic Gradient Descent (SGD) and RMSProp algorithms to the setting of Riemannian optimization. SGD is a popular method for large
scale optimization. In particular, it is widely used to train
the weights of Deep Neural Networks. However, gradients computed using standard SGD can have large variance, which is detrimental for the convergence rate of the
algorithm. Other methods such as RMSProp and ADAM
address this issue. Nevertheless, these methods cannot be
directly applied to constrained optimization problems. In
this paper, we extend some popular optimization algorithm
to the Riemannian (constrained) setting. We substantiate
our proposed extensions with a range of relevant problems
in machine learning such as incremental Principal Component Analysis, computating the Riemannian centroids of
SPD matrices, and Deep Metric Learning. We achieve competitive results against the state of the art for fine-grained
object recognition datasets