Klein Hyperbolic Metric Learning

Hyperbolic metric learning is highly effective in embedding hierarchical data structures. However, past work has predominantly focused on the conformal Poincaré model, leaving other geometries like the Klein model largely under-explored. In addition, the curved geodesics of the Poincaré model present a fundamental geometric misalignment with the linear projections dominating the feature transformation steps in the modern neural network backbones. In this paper, we investigate the Klein model, a projective model of hyperbolic geometry whose straight-line geodesics offer a structurally aligned alternative in modern encoders, for hyperbolic metric learning. By formalizing a framework based on Einstein gyrovector operations, we derive a numerically stable metric learning approach that mitigates the inherent optimization challenges of the Klein model. Extensive experiments on multiple image datasets for fine-grained image classification task show that the Klein model not only serves as a viable alternative to the Poincaré model but also achieves highly competitive performance by leveraging its unique geometric properties, without increasing parameter complexity. Our empirical findings establish the Klein model as an efficient geometric prior for hyperbolic metric learning.

Hyperbolic spaces have proven highly effective when organizing complex tree-like data. Currently, the standard approach often uses the Poincaré model, which connects data points with curved lines. This creates a structural mismatch with modern neural networks. To solve this, we introduce the Klein model to deep metric learning. Although it is a curved space, its shortest paths are strictly straight lines, perfectly matching how modern architectures work. We also developed a mathematically stable framework using Einstein gyrovector operations to make training smooth. Our image classification experiments prove that this straight-line hyperbolic approach avoids distortion and achieves highly competitive accuracy, all without making the model larger or more complex.