Poincar\'e ResNet

TL;DR

This paper presents Poincaré ResNet, a novel hyperbolic residual network operating directly on pixel data in hyperbolic space, addressing key training challenges with innovative initialization, normalization, and computation graph solutions.

Contribution

It introduces the first end-to-end hyperbolic residual network for visual data, overcoming initialization, normalization, and computational challenges in hyperbolic deep learning.

Findings

Successfully trained deep hyperbolic ResNets on visual data.

Proposed identity-based initialization preserves norms across layers.

Introduced Poincaré midpoint batch normalization as an efficient alternative.

Abstract

This paper introduces an end-to-end residual network that operates entirely on the Poincar\'e ball model of hyperbolic space. Hyperbolic learning has recently shown great potential for visual understanding, but is currently only performed in the penultimate layer(s) of deep networks. All visual representations are still learned through standard Euclidean networks. In this paper we investigate how to learn hyperbolic representations of visual data directly from the pixel-level. We propose Poincar\'e ResNet, a hyperbolic counterpart of the celebrated residual network, starting from Poincar\'e 2D convolutions up to Poincar\'e residual connections. We identify three roadblocks for training convolutional networks entirely in hyperbolic space and propose a solution for each: (i) Current hyperbolic network initializations collapse to the origin, limiting their applicability in deeper networks. We provide an identity-based initialization that preserves norms over many layers. (ii) Residual networks rely heavily on batch normalization, which comes with expensive Fr\'echet mean calculations in hyperbolic space. We introduce Poincar\'e midpoint batch normalization as a faster and equally effective alternative. (iii) Due to the many intermediate operations in Poincar\'e layers, we lastly find that the computation graphs of deep learning libraries blow up, limiting our ability to train on deep hyperbolic networks. We provide manual backward derivations of core hyperbolic operations to maintain manageable computation graphs.