Hyperbolic deep learning has become a growing research direction in computer vision due to 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, current hyperbolic learning approaches are still prone to overfitting, computationally expensive, and prone to instability, especially when attempting to learn the manifold curvature to adapt to tasks and different datasets. To address these issues, our paper presents a derivation for Riemannian AdamW that helps increase hyperbolic generalization ability. For improved stability, we introduce a novel fine-tunable hyperbolic scaling approach to constrain hyperbolic embeddings and reduce approximation errors. Using this along with our curvature-aware learning schema for Riemannian Optimizers enables the combination of curvature and non-trivialized hyperbolic parameter learning. Our approach demonstrates consistent performance improvements across Computer Vision, EEG classification, and hierarchical metric learning tasks while greatly reducing runtime.

翻译：双曲深度学习因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获数据点间的层次关系提供了自然框架，并使其嵌入具有更精细的可分离性。然而，当前的双曲学习方法仍存在过拟合倾向、计算成本高昂且易失稳的问题，尤其在尝试学习流形曲率以适应不同任务和数据集时更为突出。为解决这些问题，本文推导了黎曼AdamW算法，以提升双曲泛化能力。为增强稳定性，我们提出了一种新颖的可微调双曲缩放方法，用于约束双曲嵌入并减少近似误差。将此方法与针对黎曼优化器的曲率感知学习方案相结合，能够实现曲率与非平凡化双曲参数学习的协同优化。我们的方法在计算机视觉、脑电图分类和层次度量学习任务中均展现出持续的性能提升，同时大幅降低了运行时间。