\section{Conclusions}
In this paper, we propose Hyperbolic U-Net, the first U-Net architecture fully in hyperbolic space. We introduce hyperbolic formulations of three core Euclidean U-Net components. First, we formulate a geometric equivalent of transposed convolutions in hyperbolic space. Second, we generalize bilinear upsampling through geodesic interpolation. Third, we introduce a new weight initialization method that approximately preserves hyperbolic norms across layers, irrespective of input and output dimensions. Across extensive experiments spanning seven datasets, six noise types, three architectures, and five evaluation metrics, we find that Hyperbolic U-Net consistently demonstrates improved robustness to noise, supporting our hypothesis that hyperbolic geometry can provide more noise-tolerant embeddings. Analysis of the learned feature space suggests that the geometric properties of hyperbolic space, resulting in increased separation between classes, likely contribute to this robustness. Our results highlight the strong potential of hyperbolic geometry for medical image segmentation. A promising future direction is the design of hybrid Euclidean–hyperbolic architectures where early Euclidean layers can extract local features efficiently, while deeper hyperbolic layers enhance robustness to noise.
