\section{Introduction}
Segmentation of images, typically defined as a per-pixel or per-region classification task, plays a crucial role in the medical domain, with the U-Net architecture as a ubiquitous neural network~\cite{ronneberger2015u}. Since its inception, U-Net has become a staple in both academia and industry, due to its empirical effectiveness, modular design, and broad applicability to many data structures and tasks~\cite{azad2024medical}. Powered by continued advances in the design of network layers~\cite{alom2019recurrent, oktay2018attention, zhou2019unet++} and optimization~\cite{sudre2017generalised, isensee2021nnu}, U-Net has furthermore been extended to tasks such as image synthesis and reconstruction~\cite{laxman2021efficient, wang2022semantic, sun2023double}.

A core remaining issue for image segmentation problems in the medical domain is dealing with noise. The data acquisition process of medical images is likely to induce artifacts and random noise~\cite{nazir2024recent}. Consider, for example, speckle noise, electronic noise, backscatter noise, radiation interference, scanner-specific artifacts, or simple variations between patients being captured~\cite{kumar2025denoising}. The presence of such noise has a strong impact on the performance of U-Net~\cite{cheng2024continuous}. As a consequence, constant re-training for every new scanner~\cite{guo2024impact}, prior denoising steps~\cite{manakov2019noise}, or additional algorithmic modules~\cite{sheikh2022unsupervised} are needed to deal with noise. This limits the real-world integration of segmentation models and hampers scaling to larger data collections, which will invariably have multiple different noise properties.

In this work, we investigate the role of geometry for robustness in U-Net. Although many extensions and improvements have been introduced to U-Net, all share the same geometric foundations, namely Euclidean geometry. Recently, a wide range of works have shown that building deep learning in non-Euclidean geometries opens new doors for the field~\cite{dhingra2018embedding, liu2019hyperbolic, khrulkov2020hyperbolic, ayubcha2024improved}. Specifically, deep learning in hyperbolic space has shown strong potential for robustness, with hyperbolic neural networks being more resilient to adversarial attacks~\cite{van2024adversarial}, improving out-of-distribution detection~\cite{gonzalez2024hyperbolic}, and naturally capturing uncertainty~\cite{atigh2022hyperbolic}, without the need for additional data or optimization. Based on these success cases, this work studies the potential of hyperbolic deep learning for noise robustness in medical image segmentation.

This work introduces Hyperbolic U-Net, the first U-Net fully developed in the Poincar\'e ball of hyperbolic space. In our approach, input images are first mapped to hyperbolic space, after which all layers of the encoder-decoder model are performed on a hyperbolic manifold. We propose hyperbolic 2D transposed convolution and hyperbolic bilinear upsampling layers, since fully hyperbolic literature has so far focused mainly on encoders only~\cite{bdeir2024fully, van2023poincare}. We furthermore introduce a Newton approximation-scaled weight initialization, which ensures norm-preservation of all layers at the start of training. We hypothesize that hyperbolic geometry can improve robustness to noise in medical image segmentation. Specifically, the exponential volume growth of hyperbolic space may increase relative separation between features corresponding to different semantic classes, which could reduce the likelihood that noise-induced perturbations cross class boundaries. By embedding features in this space, Hyperbolic U-Net is expected to maintain segmentation accuracy under a variety of noise conditions. To test this hypothesis, we perform extensive experiments across multiple datasets and noise types, comparing Hyperbolic U-Net to standard Euclidean U-Nets and other competitive baselines. Results show that the Hyperbolic U-Net is highly robust to many variants of random noise, with minimal impact in performance even at high noise ratios, whereas a standard U-Net and nnU-Net setups consistently struggle with noise. The results highlight the power of hyperbolic geometry for medical image segmentation.