Hyperbolic deep learning has become a growing research direction in computer vision for the unique properties afforded by the alternate embedding space. The negative curvature and exponentially growing distance metric provide a natural framework for capturing hierarchical relationships between datapoints and allowing for finer separability between their embeddings. However, these methods are still computationally expensive and prone to instability, especially when attempting to learn the negative curvature that best suits the task and the data. Current Riemannian optimizers do not account for changes in the manifold which greatly harms performance and forces lower learning rates to minimize projection errors. Our paper focuses on curvature learning by introducing an improved schema for popular learning algorithms and providing a novel normalization approach to constrain embeddings within the variable representative radius of the manifold. Additionally, we introduce a novel formulation for Riemannian AdamW, and alternative hybrid encoder techniques and foundational formulations for current convolutional hyperbolic operations, greatly reducing the computational penalty of the hyperbolic embedding space. Our approach demonstrates consistent performance improvements across both direct classification and hierarchical metric learning tasks while allowing for larger hyperbolic models.

翻译：双曲深度学习凭借其替代嵌入空间所提供的独特性质，已成为计算机视觉领域日益增长的研究方向。负曲率与指数增长的距离度量为捕捉数据点间的层次关系提供了自然框架，并使其嵌入之间能够实现更精细的分离。然而，这些方法仍存在计算成本高昂且易失稳的问题，尤其是在尝试学习最适配任务与数据的负曲率时。当前的黎曼优化器未能考虑流形的变化，这严重损害了性能，并迫使采用较低的学习率以最小化投影误差。本文聚焦于曲率学习，通过为流行学习算法引入改进方案，并提出一种新颖的归一化方法以将嵌入约束在流形的可变表示半径内。此外，我们提出了黎曼AdamW的新颖公式、替代混合编码器技术，并为当前卷积双曲运算提供了基础公式，从而显著降低了双曲嵌入空间的计算代价。我们的方法在直接分类与层次度量学习任务中均展现出持续的性能提升，同时支持构建更大规模的双曲模型。