Hyperbolic Convolution via Kernel Point Aggregation

TL;DR

This paper introduces HKConv, a novel hyperbolic convolution method that effectively captures local features in hyperbolic space, improving the performance of hyperbolic neural networks on various tasks.

Contribution

HKConv is a new trainable hyperbolic convolution that correlates local features with fixed kernel points and respects hyperbolic geometry properties.

Findings

HKConv achieves state-of-the-art results in multiple tasks.

HKConv is equivariant to permutation and invariant to parallel transport.

The method effectively learns local hyperbolic features.

Abstract

Learning representations according to the underlying geometry is of vital importance for non-Euclidean data. Studies have revealed that the hyperbolic space can effectively embed hierarchical or tree-like data. In particular, the few past years have witnessed a rapid development of hyperbolic neural networks. However, it is challenging to learn good hyperbolic representations since common Euclidean neural operations, such as convolution, do not extend to the hyperbolic space. Most hyperbolic neural networks do not embrace the convolution operation and ignore local patterns. Others either only use non-hyperbolic convolution, or miss essential properties such as equivariance to permutation. We propose HKConv, a novel trainable hyperbolic convolution which first correlates trainable local hyperbolic features with fixed kernel points placed in the hyperbolic space, then aggregates the output features within a local neighborhood. HKConv not only expressively learns local features according to the hyperbolic geometry, but also enjoys equivariance to permutation of hyperbolic points and invariance to parallel transport of a local neighborhood. We show that neural networks with HKConv layers advance state-of-the-art in various tasks.