Hyperbolic Convolutional Neural Networks
Andrii Skliar Maurice Weiler
Deep Learning is mostly responsible for the surge of interest in Artificial
Intelligence in the last decade. So far, deep learning researchers have been
particularly successful in the domain of image processing, where Convolutional
Neural Networks are used. Although excelling at image classification,
Convolutional Neural Networks are quite naive in that no inductive bias is set
on the embedding space for images. Similar flaws are also exhibited by another
type of Convolutional Networks - Graph Convolutional Neural Networks. However,
using non-Euclidean space for embedding data might result in more robust and
explainable models. One example of such a non-Euclidean space is hyperbolic
space. Hyperbolic spaces are particularly useful due to their ability to fit
more data in a low-dimensional space and tree-likeliness properties. These
attractive properties have been previously used in multiple papers which
indicated that they are beneficial for building hierarchical embeddings using
shallow models and, recently, using MLPs and RNNs.
  However, no papers have yet suggested a general approach to using Hyperbolic
Convolutional Neural Networks for structured data processing, although these
are the most common examples of data used. Therefore, the goal of this work is
to devise a general recipe for building Hyperbolic Convolutional Neural
Networks. We hypothesize that ability of hyperbolic space to capture hierarchy
in the data would lead to better performance. This ability should be
particularly useful in cases where data has a tree-like structure. Since this
is the case for many existing datasets \citep{wordnet, imagenet, fb15k}, we
argue that such a model would be advantageous both in terms of applications and
future research prospects.