Recently, there has been a raising surge of momentum for deep representation learning in hyperbolic spaces due to their high capacity of modeling data like knowledge graphs or synonym hierarchies, possessing hierarchical structure. We refer it as hyperbolic deep neural network in this paper. Such a hyperbolic neural architecture potentially leads to drastically compact models with much more physical interpretability than its counterpart in Euclidean space. To stimulate future research, this paper presents a coherent and comprehensive review of the literature around the neural components in the construction of hyperbolic deep neural networks, as well as the generalization of the leading deep approaches to the hyperbolic space. It also presents current applications around various machine learning tasks on several publicly available datasets, together with insightful observations and identifying open questions and promising future directions.
Spatial-temporal graph convolutional networks (ST-GCNs) have been successfully applied for dynamic graphs representation learning, such as modeling skeleton-based human actions. However, ST-GCNs embed these non-Euclidean graph structures into Euclidean space, which is not the natural space to represent such structures as embedding them in this space incurs a large distortion. In this work, we make use of hyperbolic non-Euclidean geometry and construct compact ST-GCNs in the hyperbolic space. It can be shown that hyperbolic ST-GCNs (HST-GCNs) outperform the corresponding Euclidean counterparts. Additionally, these compact hyperbolic models can be used to increase the performance of large complex Euclidean models. Moreover, we show that the same or even better performance of large Euclidean models can be achieved by fusing the scores of smaller Euclidean models and a compact hyperbolic model. This in turn leads to reducing the total number of model parameters and hence model size. To validate the performance of these hyperbolic networks, we conducted extensive experiments on NTU RGB+D, NTU RGB+D 120 and Kinectics-Skeleton datasets for human action recognition.
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