TY - GEN

T1 - Differentiating through the fŕechet mean

AU - Lou, Aaron

AU - Katsman, Isay

AU - Jiang, Qingxuan

AU - Belongie, Serge

AU - Lim, Ser Nam

AU - Sa, Christopher De

N1 - Publisher Copyright: © 2020 37th International Conference on Machine Learning, ICML 2020. All rights reserved.

PY - 2020

Y1 - 2020

N2 - 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 Fŕechet 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 Fŕechet mean for arbitrary Riemannian manifolds. Then, focusing on hyperbolic space, we derive explicit gradient expressions and a fast, accurate, and hyperparameter-free Fŕechet mean solver. This fully integrates the Fŕechet mean into the hyperbolic neural network pipeline. To demonstrate this integration, we present two case studies. First, we apply our Fŕechet 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 Fŕechet 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.

AB - 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 Fŕechet 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 Fŕechet mean for arbitrary Riemannian manifolds. Then, focusing on hyperbolic space, we derive explicit gradient expressions and a fast, accurate, and hyperparameter-free Fŕechet mean solver. This fully integrates the Fŕechet mean into the hyperbolic neural network pipeline. To demonstrate this integration, we present two case studies. First, we apply our Fŕechet 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 Fŕechet 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.

UR - http://www.scopus.com/inward/record.url?scp=85105139516&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:85105139516

SP - 6349

EP - 6359

JO - 37th International Conference on Machine Learning, ICML 2020

JF - 37th International Conference on Machine Learning, ICML 2020

T2 - 37th International Conference on Machine Learning, ICML 2020

Y2 - 13 July 2020 through 18 July 2020

ER -