  Recent advances in deep representation learning on Riemannian manifolds
extend classical deep learning operations to better capture the geometry of the
manifold. One possible extension is the Fr\'echet mean, the generalization of
the Euclidean mean; however, it has been difficult to apply because it lacks a
closed form with an easily computable derivative. In this paper, we show how to
differentiate through the Fr\'echet mean for arbitrary Riemannian manifolds.
Then, focusing on hyperbolic space, we derive explicit gradient expressions and
a fast, accurate, and hyperparameter-free Fr\'echet mean solver. This fully
integrates the Fr\'echet mean into the hyperbolic neural network pipeline. To
demonstrate this integration, we present two case studies. First, we apply our
Fr\'echet mean to the existing Hyperbolic Graph Convolutional Network,
replacing its projected aggregation to obtain state-of-the-art results on
datasets with high hyperbolicity. Second, to demonstrate the Fr\'echet mean's
capacity to generalize Euclidean neural network operations, we develop a
hyperbolic batch normalization method that gives an improvement parallel to the
one observed in the Euclidean setting.
