The emergence of Deep Convolutional Neural Networks (DCNNs) has been a pervasive tool for accomplishing widespread applications in computer vision. Despite its potential capability to capture intricate patterns inside the data, the underlying embedding space remains Euclidean and primarily pursues contractive convolution. Several instances can serve as a precedent for the exacerbating performance of DCNNs. The recent advancement of neural networks in the hyperbolic spaces gained traction, incentivizing the development of convolutional deep neural networks in the hyperbolic space. In this work, we propose Hyperbolic DCNN based on the Poincar\'{e} Disc. The work predominantly revolves around analyzing the nature of expansive convolution in the context of the non-Euclidean domain. We further offer extensive theoretical insights pertaining to the universal consistency of the expansive convolution in the hyperbolic space. Several simulations were performed not only on the synthetic datasets but also on some real-world datasets. The experimental results reveal that the hyperbolic convolutional architecture outperforms the Euclidean ones by a commendable margin.

翻译：深度卷积神经网络（DCNN）的出现已成为计算机视觉领域实现广泛应用的一个普遍工具。尽管其具备捕捉数据内部复杂模式的潜在能力，但底层的嵌入空间仍然是欧几里得的，并且主要追求收缩性卷积。已有若干实例可作为DCNN性能受限的先例。近年来，双曲空间中的神经网络研究取得了进展，推动了在双曲空间中开发卷积深度神经网络的发展。本文中，我们提出了一种基于庞加莱圆盘的Hyperbolic DCNN。该工作主要围绕分析非欧几里得域中扩张性卷积的性质展开。我们进一步提供了关于双曲空间中扩张性卷积普适一致性的广泛理论见解。研究不仅在合成数据集上，也在一些真实世界数据集上进行了多次模拟实验。实验结果表明，双曲卷积架构以显著优势超越了欧几里得架构。