  Real-world visual data exhibit intrinsic hierarchical structures that can be
represented effectively in hyperbolic spaces. Hyperbolic neural networks (HNNs)
are a promising approach for learning feature representations in such spaces.
However, current HNNs in computer vision rely on Euclidean backbones and only
project features to the hyperbolic space in the task heads, limiting their
ability to fully leverage the benefits of hyperbolic geometry. To address this,
we present HCNN, a fully hyperbolic convolutional neural network (CNN) designed
for computer vision tasks. Based on the Lorentz model, we generalize
fundamental components of CNNs and propose novel formulations of the
convolutional layer, batch normalization, and multinomial logistic regression.
{Experiments on standard vision tasks demonstrate the promising performance of
our HCNN framework in both hybrid and fully hyperbolic settings.} Overall, we
believe our contributions provide a foundation for developing more powerful
HNNs that can better represent complex structures found in image data. Our code
is publicly available at https://github.com/kschwethelm/HyperbolicCV.
