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.

翻译：现实世界的视觉数据表现出固有的层次结构，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）是在此类空间中学习特征表示的一种有前景的方法。然而，当前计算机视觉中的双曲神经网络依赖欧几里得主干网络，仅在任务头中将特征投影到双曲空间，这限制了其充分利用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Lorentz模型，我们推广了CNN的基本组件，并提出了卷积层、批归一化以及多项逻辑回归的新公式。在标准视觉任务上的实验证明了我们的HCNN框架及Lorentz模型在混合和全双曲设置中的优越性。总体而言，我们相信我们的贡献为开发更强大的HNNs奠定了基础，使其能更好地表示图像数据中的复杂结构。我们的代码公开于https://github.com/kschwethelm/HyperbolicCV。