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We study the representation capacity of deep hyperbolic neural networks(HNNs) with a ReLU activation function. We establish the first proof that HNNscan $\varepsilon$-isometrically embed any finite weighted tree into ahyperbolic space of dimension $d$ at least equal to $2$ with prescribedsectional curvature $\kappa<0$, for any $\varepsilon> 1$ (where $\varepsilon=1$being optimal). We establish rigorous upper bounds for the network complexityon an HNN implementing the embedding. We find that the network complexity ofHNN implementing the graph representation is independent of the representationfidelity/distortion. We contrast this result against our lower bounds ondistortion which any ReLU multi-layer perceptron (MLP) must exert whenembedding a tree with $L>2^d$ leaves into a $d$-dimensional Euclidean space,which we show at least $\Omega(L^{1/d})$; independently of the depth, width,and (possibly discontinuous) activation function defining the MLP.

Capacity Bounds for Hyperbolic Neural Network Representations of Latent Tree Structures