Differentiating through the Fr\'echet Mean

TL;DR

This paper introduces a method to differentiate through the Fréchet mean on Riemannian manifolds, enabling its integration into hyperbolic neural networks for improved geometric learning.

Contribution

It provides a novel way to differentiate through the Fréchet mean on arbitrary Riemannian manifolds and develops explicit gradients and a fast solver for hyperbolic space.

Findings

Achieved state-of-the-art results on hyperbolic graph datasets.

Developed a hyperbolic batch normalization method.

Integrated Fréchet mean into hyperbolic neural networks successfully.

Abstract

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.