  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, the first 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.
Experimentation on standard vision tasks demonstrates the superiority of our
HCNN framework and the Lorentz model 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.

Computer Vision and Pattern Recognition;Machine Learning