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 methods 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 (MLR). Experimentation on standard vision tasks demonstrates the effectiveness of our HCNN framework and the Lorentz model in both hybrid and fully hyperbolic settings. Overall, we aim to pave the way for future research in hyperbolic computer vision by offering a new paradigm for interpreting and analyzing visual data. Our code is publicly available at https://github.com/kschwethelm/HyperbolicCV.

翻译：真实世界中的视觉数据展现出内在的层级结构，这些结构可以在双曲空间中得到有效表示。双曲神经网络（HNNs）是在这类空间中学习特征表示的一种有前景的方法。然而，当前计算机视觉领域的方法仍依赖欧几里得主干网络，仅在任务头部将特征投影到双曲空间，这限制了它们充分发掘双曲几何优势的能力。为解决这一问题，我们提出了HCNN，这是首个专为计算机视觉任务设计的全双曲卷积神经网络（CNN）。基于洛伦兹模型，我们推广了CNN的基本组件，并提出了卷积层、批归一化及多项逻辑回归（MLR）的新颖公式。在标准视觉任务上的实验表明，我们的HCNN框架和洛伦兹模型在混合与全双曲设置中均具有有效性。总体而言，我们旨在通过提供一种解释与分析视觉数据的新范式，为双曲计算机视觉的未来研究铺平道路。我们的代码已公开于https://github.com/kschwethelm/HyperbolicCV。