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

Differentiating through the Fr\'echet Mean