  Hyperbolic spaces, which have the capacity to embed tree structures without
distortion owing to their exponential volume growth, have recently been applied
to machine learning to better capture the hierarchical nature of data. In this
study, we generalize the fundamental components of neural networks in a single
hyperbolic geometry model, namely, the Poincar\'e ball model. This novel
methodology constructs a multinomial logistic regression, fully-connected
layers, convolutional layers, and attention mechanisms under a unified
mathematical interpretation, without increasing the parameters. Experiments
show the superior parameter efficiency of our methods compared to conventional
hyperbolic components, and stability and outperformance over their Euclidean
counterparts.
