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  1. Hyperbolic neural networks have been popular in the re- cent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in develop- ing these networks lies in the nonlinearity of the embed- ding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group which is a semi-Riemannian manifold, i.e. a mani- fold equipped with an indefinite metric. Most existing meth- ods (with some exceptions) use local linearization to de- fine a variety of operations paralleling those used in tra- ditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) fol- lowed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower- dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transforma- tions, which are the natural isometric transformations in hyperbolic spaces. This projection is computationally effi- cient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it al- lows for weight sharing. The nested hyperbolic space rep- resentation is the core component of our network and there- fore, we first compare this representation – independent of the network – with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projec- tion. Finally, we present experiments demonstrating com- parative performance of our network on several publicly available data sets. 
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