HypCBC: Domain-Invariant Hyperbolic Cross-Branch Consistency for Generalizable Medical Image Analysis
Authors: Francesco Di Salvo
Sebastian Doerrich
Jonas Alle
Christian Ledig
Venue: Accepted by TMLR ()

Abstract: Robust generalization beyond training distributions remains a critical challenge for deep neural networks. This is especially pronounced in medical image analysis, where data is often scarce and covariate shifts arise from different hardware devices, imaging protocols, and heterogeneous patient populations. These factors collectively hinder reliable performance and slow down clinical adoption. Despite recent progress, existing learning paradigms primarily rely on the Euclidean manifold, whose flat geometry fails to capture the complex, hierarchical structures present in clinical data. In this work, we exploit the advantages of hyperbolic manifolds to model complex data characteristics. We present the first comprehensive validation of hyperbolic representation learning for medical image analysis and demonstrate statistically significant gains across eleven in-distribution datasets and three ViT models. We further propose an unsupervised, domain-invariant hyperbolic cross-branch consistency constraint. Extensive experiments confirm that our proposed method promotes domain-invariant features and outperforms state-of-the-art Euclidean methods by an average of $+2.1\%$ AUC on three domain generalization benchmarks: Fitzpatrick17k, Camelyon17-WILDS, and a cross-dataset setup for retinal imaging. These datasets span different imaging modalities, data sizes, and label granularities, confirming generalization capabilities across substantially different conditions.

Published in Transactions on Machine Learning Research (02/2026)
HypCBC: Domain-Invarian Hyperbolic Cross-Branch Con- sistency for Generalizable Medical Image Analysis
Francesco Di Salvo
francesco.di-sadee di bww.ddaaeee.uuuuubnbbbee.r..... de
zAlLab Bambery University of Bamberg, Germany
Scbastian Docrrich
aAlLab Bambeer
University of Bamberg, Germany
Jonas Alle
jonas. manuss.llee@uui-bambbboe de
aAlLab Bambeer
University of Bamberg, Germany
Christian Ledig
christion. wewiieldididdnnnaaaaaaea
zAlLab Bamberg
Universily of Bamberg, Germany
Reviewed on OpenReview: https: optp//vieerre...mm net/ form? id= 1spGp
Abstract
Robust generalization beyond training distributions remains a critical challenge for deep neural networks. This especially pronounced in medical image analysis, where data often scarce and covariate shifts arise from different hardware devices, imaging pro- tocols, and heterogencous patient populations. These factors collectively hinder reliable performance and slow down clinical adoption. Despite recent progress, existing learning paradigms primarily rely on the Euclidean manifold, whose flat geometry fails to cap- lure the complex, hierarchical structures present in clinical data. In this work, we ex- ploit the advantages of hyperbolic manifolds to model complex data characteristics. We present the first comprehensive validation of hyperbolic representation learning for medical image analysis and demonstrate statistically significant gains across eleven in-distribution datasets and three VIT models. We further propose unsupervised, domain-invariant hyperbolic cross- branch consistency constraint. Extensive experiments confirm that our proposed method promotes domain-invariant features and outperforms state-of-tho-an Eu- clidean methods by an average of 2.1%% AUC on three domain generalization benchmarks: Fitzpatrick17k, Camelyon WILDS, and a cross-dataset setup for retinal imaging. These datasets span different imaging modalities, data sizes, and label granularities, confirming generalization capabilities across substantially different conditions. The code is available at httpa//mww.nnecooosesoo////yeeee
1 Introduction
Deep lcarning models have achieved remarkable success over the past decade, yiclding exceptional results in various computer vision lasks ranging from image classification to instance segmentation. However, ensuring that models generalize reliably across distribution shifts remains a fundamental challenge, especially in safety-critical applications. For instance, autonomous driving systems must generalize across varying scenery, lighting, and weather conditions at test time (Bijelic et al., 2020; Kumar & Muhammad, 2023). Similarly, medical imaging models face different domain generalization challenges, e-g-, shifts in patient populations,

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tissue-staining protocols, or scanner manufacturers (Ktena et al., 2024; Čevora et al. 2024). These may seem subtle compared to outdoor settings, yet. they still strongly affect performance. Robustness to such subtle but consequential domain changes is therefore essential for safe deployment of Al in hospitals and clinics. Most domain-shift remedics in medical imaging span data augmentations (Zhang et al., 2018; Di Salvo et al., 2024) and Euclidean representation learning (Arjovsky et al., 2019; Sagawa el al., 2020; Krueger el al., 2021).
While effective in many settings, Euclidean embeddings offer a uniform, flat geometry that may not align with the hicrarchical relationships often present in clinical data. On the other hand, the hyperbolic manifold has gained notable traction in recent years (Mettes et al., 2024). Indeed, it offers natural remedy: its constant negative curvature mirrors hicrarchical structures by allocating exponentially increasing space for finer-grained distinctions, and it has shown notable performance in many vision tasks. However, end-to-end hyperbolic networks can be unstable on large datasets (Ayubcha et al., 2024) and remain underutilized in medical imaging. To overcome these stability challenges, we project Euclidean embeddings from frozen foundation model into a lightweight hyperbolic manifold and demonstrate its clear advantages over Euclidean baselines across eleven medical datasets.
We then introduce HypCBC, hyperbolic two-branch training strategy with invarin-n cross-branch consistency (CBC) regularization, as illustrated in Figure 1. This approach learns both fine-grained and domain-agnostic representations, yielding consistent. and substantial domain generalization improvements on three datasets: Fitzpatrick17k (dermatology), CamelyonI7-WILDS (histopathology), and a cross-dataset retinal imaging benchmark.
To motivate and validate the learning dynamics of our two-branch strategy, also report two targeted ab- lation studies. First, we vary the latent dimensional- ity of a single-branch model to quantify how its capac- ity influences domain invariance versus label discrimina- tion. Second, we compare low-dimensional against high- dimensional regularization, demonstrating that the for- mer yields significant robustness gains that cannot be ex- plained by mere increase of parameter count. In sum- mary, our contributions are:
y euc
hyp
y cbc
cbc y 2
y
hyp Z 128D
hyp
Z 2D
euc h
We demonstrate that hyperbolic embeddings significantly outperform (p < 0.05) Euclidean ones in classification accuracy across eleven in-distribution (ID) medical imaging datasets. These span a diverse range of imaging modalities (9), sample sizes (102-109), and label granulari- ties (2-11).
128D
h
euc
f
LCE
LKL --D
expa O
We introduce novel hyperbolic two-branch training strategy with a domain-invariant consis- lency constraint. This approach significantly en- hances domain generalization (DG) performance, measured by the area under the receiver operat- ing curve (AUC), in dermatology, histopathology, and retinal imaging.
Figure 1: Given a frozen Euclidean feature ex- tractor that outputs f, ERM applies Fu- clidcan lincar probe over a 128D projection h128D- HypERM additionally uses a fixed ex- ponential map expf2s to classify over hyperbolic embeddings Z128D- Our method, HypCBC, in- troduces second projection hęp followed by expε to yield 32D- The logits of this low- dimensional branch are used as targets in the KL loss, promoting domain-agnostic information transfer into the high-dimensional branch.
Ablation studies on bottleneck dimension and manifold geometry reveal two key findings. First, a 2D low-dinensional branch effectively balances domain invariance with label discrimination. Sec- ond, hyperbolic regularization exhibits consistent robustness improvements, unlike its Euclidean counterpart.
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2 Related work
2.1 Hyperbolic manifold
Hyperbolic spaces have emerged a powerful tool for modeling hierarchical and tree-like data structures (Mettes et al., 2024). Recent work demonstrates their effectiveness across natural language processing (Dhin- gra et al., 2018), few-shot learning (Guo et al., 2022; Khrulkov et al., 2020), hierarchical classification (Dhall et al,, 2020), metric learning (Ermoloy et al., 2022: Bi et al., 2025), semantic segmentation (Atigh et al., 2022), out-of-distribution detection (Guo et. al., 2022), category discovery (Liu el al., 2025), and anomaly detection (Li et al., 2024; Gonzalez-Jimenez et al., 2025). Despite these successes, hyperbolic representations remain underexplored in medical contexts. Existing efforts include fino-grained classification (Yu ct al., 2022; Ramirez et al., 2025), multi-modal neuroimaging (Ayubcha et al., 2024), and anomaly detection (Gonzalez- Jimenez et al., 2025). However, end-to-end hyperbolic networks can exhibit numerical instability and sub- stantially reduced training efficiency, with convergence requiring significantly more epochs, an effect that becomes more pronounced on larger datasets (Ayubcha et al., 2024). To combine the strengths of hyperbolic embeddings with stable training, we follow the projection-based approach of Ermolov al. (2022), but un- like their end-to-end fine-tuning of the backbone, we freeze a pre-trained Euclidean backbonc and append weigt hyperbolic projection layers. We evaluate these representations for domain generalization, an- alyzing how embedding dimensionality influences the separation of domain-specifie versus domain-agnostic features.
2.2 Domain Generalization
The community has doveloped a varicty of domain generalization methods, primarily representation--carning techniques such as adversarial learning (Ganin et. al., 2016), invariant risk minimization (Arjovsky el. al., 2019), and meta-learning (Li et al., 2018a), often relying on domain labels. To boost robustness, image and latent augmentation strategies such as AugMix (Hendrycks et al., 2019) and MixStyle (Zhou et al., 2024) are also commonly utilized. Furthermore, recent works have shown that targeted augmentations offer greater gains than domain-agnostic ones (Gao et al., 2023; Di Salvo et al., 2024).
Our approach takes advantage of the hyperbolic manifold, whose constant negative curvature more accurately reflects the complexity of clinical data. Similar to Domain Adversarial Neural Networks (DANN) (Ganin et al., 2016), we introduce second branch to promote domain invariance throughout the network. However, instead of relying on domain labels and adversarial training strategy, our method achieves invariance in fully unsupervised manner. We accomplish this goal by employing low-dimensional manifold, which has been empirically demonstrated not to discriminate between domains (ef. Section 5.1).
Concurrently, Bi ot al. (2025) embed hyperbolic geometry directly into VMamba via end-to-end state- space modeling for fine-grained domain generalization. By contrast, our method is backbone-agnostic and introduces lightweight hyperbolic projections with cross- cross-branch consistency on frozen features, targeting unsupervised domain generalization with minimal architectural overhead.
2.3 Information transfer and consistency
Knowledge distillation, originally proposed to compress large "teacher" networks into smaller "students", has also been adopted for domain generalization as form of regularization that transfers domainn-nvariint information. Empirical studics suggest that carly network layers often encode domain-spocific information (Zhou et al., 2024). To this extent, prior works exploit this property by distilling final-layer predictions into randomly selected intermediate classifiers to encourage invariance throughout the network (Sultana et al., 2022). Subsequent refinements introduce logit softening to further stabilize the distillation process (Galappaththige et al., 2024).
In this work, we reinterpret such knowledge transfer as a cross-branch consistency regularization. Specifically, the low-dimensional (i.e., domain-invariant) hyperbolic branch provides a compact reference that constrains the high-dimensional branch via a consistency objective, encouraging out relying on classical teacher-student distillation.
representations with-
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h1
91
(x,3) E X x y
expi2e
Ces 11)
128D
Poincard ball and disk
eXP2
(cx(52,3)
b2
X
Figure 2: Given an input image-label pair (x, y) E & x V; we extract an image embodding f= ((x) E IR", where n depends on the chosen backbone. This is projected via two heads into Euclidean embeddings h R128 and h2 E RR2. Each mapped into its respective Poincaré ball D by expe, yielding hyperbolic embeddings 21 and Z2. Both branches incur cross-entropy losses on their Multiclass Logistic Regression (MLR) logits y1 and ý22 In addition, f22 also supervises ý1 via KL-based consistency term. The high- dimensional branch captures fine-grained, class-specific features for inference, and the diw-dimensional branch enforces an information bottleneck that promotes domain-invariant representations.
While Euclidean knowledge transfer (e.g., distillation) is well studied in prior works, to the best of our knowledge, its application in hyperbolic space remains limited. Recent work by Yang et al. (2025) introduce a hyperbolic knowledge distillation approach for cross-domain few-shot learning, Their method relies on multiple domain-specific teacher models, meta-learning, and access to target-domain data at test time, placing closer to domain adaptation than domain generalization. In contrast, our setting assumes no domain labels and no access to target-domain data at any stage. We thus focus improving robustness to entirely unseen domains under standard domain generalization protocol.
3 Method
3.1
Hyperbolic space
The n-dimensional hyperbolic space H" naturally offers a geometry suited for complex, hierarchical data structures, such as medical images. While Euclidean space has flat geometry, tee., with curvature = 0, hyperbolic space has a conslant negative cureature, which can effectively capture the inherent hierarchical feature relations of image data (Mettes et al., 2024).
Among the several isometric models of hyperbolic space, (I>.." to represent an n-dimensional hyperbolic space (Khrulkov et al., 2020; Atigh et al., 2022; Ermolov et al., 2022; Guo et al., 2022).
the widely adopted Poincaré ball model
Following the notation of Khrulkov et al. (2020), we define the manifold as D (x E TR" e||x|| < 1), where c > O is scaling factor controlling the magnitude of the negative curvature. The metric (i.e., rule for measuring distances) of this space is given by:
=
x= 1- 1
(1)
where gF = In is the standard Euclidean metric. In simple terms, the conformal factor A scales the usual Euclidean distances, adapting them the curved geometry of the Poincaré ball. To perform vector operations similar to addition in Euclidean space, the gyrovector formalism (Ungar, 2009) adopted, which defines the Mubius addition of two points x,y Da in Equation 2.
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x G. y = (1 20(x, y) clly
(2)
+ 2c(x, y) e 1
This operalion gemeralizes the familiar concept of vector addition to our curved space. Based on Möbius addition, the geodesic (t.e., (oortest-path) distance between two points x, y E Di is given by:
(x, = Vc
arctanh (vol - x cB yll)
Notably, as the curvature parameter e approaches 0, the hyperbolic distance converges to twice the Euclidean distance, i.e,, lim (x,) 2
To fill the gap between conventional feature extractors (which operate Euclidean space) and our hyperbolic representation, employ the exponential map. This bijective mapping projects Euclidean vector v E R" onto the hyperbolic manifold at chosen base point Xg (usually set to O). The exponential map is defined as:
expx (v) =0=xx xB te (tanh (eAEllvIl 2
Velvvl
This exponential mapping ensures that the features are faithfully transferred from the Euclidean to the hyperbolic manifold. Its inverse, i.e., the logarithmic map, allows points in the hyperbolic space to be projected back into the Euclidean space.
3.2 Multi-branch learning
To further exploit hyperbolic geomelry for domain generalization, draw inspiration from Domain- Adversarial Neural Networks (DANN) (Ganin et al., 2016) while eliminating the need for explicit domain labels and gradient reversal. As illustrated in Figure 2, given an image-label pair (x,y) E X x y and a frozen Euclidean backbone Φ, we first extract an image embedding = ((x) E R", where n depends on the size of the chosen backbone. For "small", "base", and "large" ViT backbones, n 384, 768, and 1024, respectively.
Two projection heads h, R" - R4 reduce f to Euclidean embeddings h, h.()), with i {1,2), d1 = 128 and d2 = 2. Subsequently, given curvature scaler E R, apply the exponential map expx to each projection to obtain hyperbolic representation in the Poincaré ball Dé.
21 = exp"((i)) E 1D228
Z2 ex DD
The high-dimenssonal branch captures fine-grained, class- and domain-spocific features, whereas the low- dimensional branch introes an information bottleneck, encouraging domain-invariant features and facili- tating direct visualization of the embeddings (ef. Appendix A.1).
3.2.1 Domain-invariant cross branch consistency
Each hyperbolic embedding z, is then passed through Multiclass Logistic Regression (MLR) head produce logits y; MLR(2,), where class scores computed from geodesic distances in the Poincaré ball, defined in Equations 2 3. The resulting logit vectors have dimension equal to the number of classes for both branches = 2. During training, we apply the cross-entropy loss to both branches, but at test-time only the high-dimensional branch is used for inference. To promote domain-agnostic knowledge from the bottlenock branch into the main branch, we introduce a cross-branch consistency loss:
LKL 82) 2 KLL (a(5::))llo((;:1))
(6)
where KL the Kullback- Leibler divergence, and T E R a temperature scaler. Notably, this cross-branch regularization works the class-logit level, allowing information transfer between branches despite their differing embedding dimensions.
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3.2.2 Overall objective
The final training objective combines the cross-entropy losses from both branches and the consistency loss, weighted with E R:
C LCB( (CE((2.y) + (KK,(11,00
By merging domain-sensiin sensitive high-dimensional features with a low-dimensional, domain-agnostic repre- sentation 22, our method learns both global, domain-invariant features and local, domain-specific features, implicitly taking advantage of the structure the hyperbolic manifold. During inference, we only use the logits from the high- sigh-dimensional branch yı, as they capture more fine-grained details, thereby providing richer semantic information.
4 Experimental results
In this section, we empirically evaluate and compare the advantages of representing medical image data within hyperbolic manifold compared to the traditional Euclidean one. Our experiments are designed to assess three key aspects: (1) in-distribution (ID) image classification, (2) robustness against distribution shifts, and (3) the influence of manifold geometry and latent bottleneck size on robustness.
Across our experiments, we utilize a frozen pre-trained foundation model as a feature extractor, followed by a linear layer of dimension di = 128 (Ermolov al., 2022). This head is instantiated in three variants that form the basis of our evaluation. First, a Euclidean ERM is implemented using a standard Euclidean classification cross-entropy objective. Second, hyperbolic counterpart (HypERM) projects
head with a softmax- the linear layer onto the Poincaré ball and applies Multiclass Logistic Regression (MLR) (Ganca et al., 2018). Third, our proposed bros-bbaacc consistency approach (HypCBC) int an additional low-dimensional branch (d2 = 2) and a KL loss controlled by temperature T = 3 and weight X 0.2. For hyperbolic models, we fix the curvature at c= 1.0, a commonly used default value (Mettes et al., 2024). This isolates the effect of hyperbolic geometry while avoiding an additional dataset-specific hyperparameter that could distort comparisons.
Furthermore, we employ feature-clipping radius of r 1.0 to ensure numerical stability (Guo et al., 2022). All models are trained using the AdamW optimizer (Loshchilov & Hutter, 2019) with cross-entropy loss and cosine-annealing learning-rate schedule. We use an initial learning rate of 1x10-* with batch size of 64, and apply early stopping after 10 epochs without improvement. The sensitivity to the hyperparameters of HypCBC is reported in Table 4.
4.1 Medical image classification
We first evaluate the accuracy performance achieved across cleven real-world medical datasets from the dataset collection of Yang et al. (2023), with default train val test splits and resolution 224x224. These datasets, detailed in Table 1, span a diverse range of imaging modalities (9), sample sizes (102-103), and label gramularities (2-11). Notably, we exclude Chest from our experiments because of its multi-label setup.
We further evaluate three distinct ViT-based models: VIT-S (Dosovitskiy et al., 2021), pre-trained on ImageNet-21k following the training recipe of Steiner et al. (2022), DeiT3-S (Touvron et al., 2022), and DINOv2-S (Oquab et al., 2024). This allows us to assess whether the hyperbolic gains consistently transfer across models with different pre-training strategics. For each model and dataset, we report the average classification accuracy (over five seed runs) for both single-branch Euclidean (ERM) and hyperbolic classifiers (HypERM).
4.1.1 Results
The results reported Table demonstrate that, overall, hyperbolic representation learning consistently improves classification performance, although the magnitude of the gains varies depending on the specific dataset and model.

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With ViT, hyperbolic embeddings outperform Euclidean 8 of 11 tasks, with the largest gains OCT (+1.32%), Tissue (+1.18%), and Pneumonia (+1.12%), minor drop on Retina, and negligible changes on Blood and OrganA. With DeïI', hyperbolic models score first on 10 of 11 datasets, most notably on Pncumonia, Derma, and Breast (gains from +1.41% to +1.89%). With DINOv2, the best model overall, hyperbolic embeddings outperform Euclidean ones in 10 out of tasks, with the biggest increases on OCT (+2.70%), Path (+1.48%), and Tissue (+1.36%). To summarize, across all models and datascts, hyperbolic classifiers offer statistically significant gains over Euclidean ones (Wilcoxon signed-rank test, p < 0.05).
Table 1: Dataset details including data source, imaging modality, type of classification task (with number of classes), and predefined data splits. ML: Multi-Label, MC: Multi-Class, BC: Binary-Class, OR: Ordinary Regression. Table adapted from Doerrich et al. (2025).
Task
Number of Samples
Dataset
Source
Imaging Modality
(# Classes)
Train / Val / Test
Blood
Acevedo et al. (2020)
Blood Cell Microscope
MC (8)
11, 959 / 1,2 / 3, 421
Breast
Al-Dhabyani et al. (2020)
Breast Ultrasound
BC (2)
546 78 / 156
Chest
Wang et al. (2017)
Chest X-Ray
MT,-BC (2)
78,468 / 11,219 22,433
Tschandl et al. (2018)
Derma
Dermatoscope
MC (7)
7,007 1,003 / 2,005
Codella et al. (2018)
OCT
Kermany al. (2018)
Retinal OCT
MC (4)
97,477
10,832 1,000
Bilic et al. (2023)
OrganA
Abdominal CT
MC (11)
34,561
6,491 / 17,778
Xu et al. (2019)
Bilic et al. (2023)
OrganC
Abdominal CT
MC (11)
12,975 / 2,392
8,216
Xu et al. (2019)
Bilic et al. (2023)
OrganS
Abdominal CT
MC (11)
13,932 / 2,452
8,827
Xu et al. (2019)
Path
Kather et al. (2019)
Colon Pathology
MC (11)
89,996
10.004
7,180
Pneumonia
Kermany et al. (2018)
Chest X-Ray
BC (2)
4,708 / 524 / 624
Retina
Liu et al. (2022a)
Fundus Camera
OR (5)
1,080 / 120 / 400
Tissue
Ljosa et al. (2012)
Kidney Cortex Microscope
MC (8)
165,466
23,640
47,280
Table 2: Accuracy (averaged over five runs) of Euclidean (ERM) and hyperbolic (HypERM) classifiers on eleven medical datasets, evaluated with three ViT-based models. We highlight bold the best manifold across each model and dataset. Notably, the hyperbolic representation is significantly better than the Euclidean one across all experiments (Wilcoxon signed-rank test,, p 0.05).
Br
Pn
Re
De
B1
OrC
OrS OrA
Pa
Ti OCT
Avg
VIT
ERM
82.56
87.21
61.90
81.65
97.68
85.42
76.09
91.50
93.48
61.04
74.02
81.14
HypERM
83.08
88.33
61.45
82.19
97.61
85.72
77.07
91.45
93.54
62.22
75.34
81.64
DeiT3
ERM
82.56
88.59
59.15
77.90
96.17
83.33
72.18
89.44
90.78
59.34
78.54
79.82
HypERM
83.97
90.48
59.65
79.44
96.50
83.61
72.03
89.48
91.23
60.61
78.90
80.54
DINOv2
ERM
85.77
89.94
64.65
82.77
97.90
88.04
76.98
92.82
92.46
61.95
83.60
83.35
HypERM
85.38
90.38
65.40
83.59
97.91
88.32
77.24
92.88
93.94
63.31
86.30
84.06

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4.2 Domain Generalization
4.2.1
Datasets
We evaluate domain generalization performance under out-of-distribution (OOD) conditions on three estab- lished medical dataset collections.
Fitzpatrick111 7h (Groh et al., 2021) is dermatological dataset including 16,577 samples labeled with three discase classes and skin-tone information according to the Fitzpatrick scale. The Fitzpatrick skin-type scale categorizes human skin tones from Type I (very light)) to Type VT (deeply pigmented), providing a standardized measure of skin pigmentation from light to dark. We aggregate the skin tone information in three groups (i.c., domains), as in Dancshjou et al. (2022): (I-II, IIIIV, V-VT3. Given the limited number of domains, we assess the performance on a leave-one-domain-onl protocol (LODO).
Camelyon In-elyS (Bandi al., 2018; Koh al., 2021) is standard binary domain generalization benchmark for histopathology consisting of 422,394 images acquired from five hospitals. We follow the default splits, training three hospitals, validating one, and testing one.
For Retina, 5-class diabetic retinopathy classification task, we construct cross-datasct shifts using four widely adopted fundus imaging datasets (Che et al., 2023; Zhou et al., 2023). APTOS 2019 (Karthik & Dane, 2019) and DeepDR (Liu et al., 2022b) serve as the in-distribution training data, comprising a total of 4, 4,608 labeled samples. IDRiD (Porwal et al., 2018) forms the validation domain with 1,744 samples, while Messidor-2 (Decencière et al., 2014) provides the test domain with 7,000 samples. These datasets differ in acquisition devices, grading protocols, and patient cohorts, thereby inducing realistic distribution shifts.
C17-WILDS
Fitzpatrick177
Retina
I-II
II-IV
V-VI
APTOS
DeepDR
IDRiD
Messidor
Train
Val
Test
Figure 3: Overview of dataset domains. Top-Left: WemIeLynIT-WILDSS, showing representative "tumor" patches from each contributing hospital
Bottom-Left: Relina, showing representative fundus images (y = 4) from each dataset domain (APTOS 2019, DeepDR, DRiD, Messidor-2). Colored frames indicate the train (pink), validation (orange), and test (purple) subsets. Right: Fitzpatrick1 7kk, showing representative "malignant" skin-lesion images from each of the three skin-tone groups (I-II, III IV, V VT).
4.2.2 Methods
We compare HypCBC against standard empirical risk minimization (ERM) and its hyperbolic variant (Hy- pERM). In addition, evaluate Euclidean embeddings enhanced with data augmentations such Ran- dAugment (Cubuk et al, 2020), AugMix (Hendrycks al., 2019), and Med-C (Di Salvo et al., 2024). We also include established methods such as IRM (Arjovsky et al., 2019), GroupDRO (Sagawa et al., 2020), VREx (Krueger et al., 2021), DANN (Ganin et al., 2016), CDANN (Li et al., 2018c), CORAL (Sun & Saenko, 2016), MMD (Li et al., 2018b).
All methods use DINOv2 image embeddings and the default hyperparameters defined DomainBed (Gul- rajani & Lapez-Paz, 2021). We report the area under the receiver operating curve (AUC), averaged over five seeds to accommodate varying class imbalance and the generally more challenging (OOD) conditions.
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Table 3: AUC averaged over five runs. Fitzpatrick17k is evaluated using a LODO strategy, reporting results for each fold, together with their macro-average. For Camelyon17-WILDS and Retina, both val and test set results are reported. We highlight in bold the best two methods and underline the significantly best (paired t-test, p 0.05) on each datasct. Across all datasets, HypCBC is the significantly bost method (Wilcoxon signed-rank test, < 0.05).
F17k
C17-WILDS
Retina
OOD
Method
I
IV
V
VI
Avg
Val
Test
Val
Test
Avg
ERM
79.93.10.2
80.2
79.23.1.33
80.68
97.05.1.11
98.32.10.1
86.6511..2
78.40.11.3
86.07
Med-C
80.15:0.11
82.80
79.66:0.2
80.87
97.1410.1
98.22:10.0
84.8510.22
77.13.10.5
85.71
RandAng
80.04+0.1
82.86+0.5
79.24+0.2
80.71
96.74+0.0
98.15+0.1
87.85
78.82+0.5
86.24
AugMix
80.26+0.1
82.86+0.5
79.25+0.2
80.79
96.50+0.1
98.06+0.1
86.73+0.3
79.80:0.4
86.21
JRM
76.57=0.7
80.20:0.44
78.84=0.4
78.54
96.98=0.1
98.17+0.2
85.0410.4
75.6410.5
84.49
GROupDRO
80.15:0.2
82.54+0.2
78.60=0.6
80.43
97.50=0.1
98.22::0.1
85.71#1.1
77.46:0.7
85.74
VREx
09.16=0.1
82.35=0.6
79.66=0.3
80.39
97.36=0.1
98.35-0.1
87.05+0.3
78.67#0.4
86.09
DANN
79.90.10.1
82.47-0.33
79.69
80.69
97.00.1.11
98.22.10.1
86.5410.7
78.05:10.4
85.98
CDANN
79.7110.11
82.44
80.02.0.4
80.72
97.0410.0
98.23
85.7610.9
76.92.1..
85.73
MMD
78.82+0.1
82.15+0.4
79.56=0.3
80.18
97.02+0.2
98.27+0.1
87.42:10.2
79.07+0.44
86.05
CORAL
78.83+0.1
82.40+0.3
79.52+0.4
80.25
97.22+0.1
98.25++0.2
87.36+0.2
79.28+0.4
86.12
HypERM
81.9310.1
84.31
03.5
83.27
0.6
98.07+0.2
87.23+0.9
79.39+0.7
87.49
HypCRC
82.3440.3 36.28.10::284427210
0..
84.30
08.
38.3310.3
87.3440.7
80.48
88.15
4.2.3 Results
As shown in Table 3, Euclidean methods achieve comparable performance overall. Among augmentation methods, Med-C leads Fitzpatrick17k, while RandAugment and AugMix lead Retina. Representation- learning methods, such as VREx and MMD, also yield notable benefits on Camelyon17- Regarding the hyperbolic manifold, HypERM songle-branch) outperforms its Euclidean counterpart in all settings except ble results with the top-scoring VREx. Notably, while VREx utilizes domain labels during training, our method does not require such information. On the Retina test split, HypCBC outperforms the best Eu- clidean method, AugMix, by 0.68%. Furthermore, hyperbolic methods exhibit the largest. improvements Fitzpatrick17k. Specifically, HypERM alone significantly boosts AUC, and HypCBC achieves additional significant. gains (paired I-test, < 0.05). This pattern aligns with the magnitude of the shift in skin tone groups. Indeed, Fitzpatrick17k presents the most extreme shift (cf. Figure 3), while the shifts ob- served in Camelyon17-WIIDDS and Retina are milder. Overall, HypERM delivers superior generalization across a broad range of real-world shifts, with an average improvement of +1.42% over Euclidean embed- dings (ERM). Furthermore, HypCBC yields an additional statistically significant improvement of +0.66% (Wilcoxon signed-rank test, p 0.05).
and Retina.
1-WTIDS (test), where our proposed method, HypCBC, exhibits compara-
5
Ablation studies
5.1 Latent dimension and generalization
To quantify how the size and manifold of the low-dimensional branch govern the trade-off between domain invariance and label discrimination,
train single-brace classifiers (f.e., no consistency constraint) with embedding dimension d E (2, 16, 32, 64, 128} in both Euclidean and hyperbolic spaces. We evaluate on three benchmarks: Fitzpatrick177 17k (three skin-lone groups, slandard ID split), Camelyon17-WILDD (five hospitals), and the Retina cross-dataset (four sources). For TamelyonI7-WILDS splits and re-partition the data into a 70/10/20 stratified split, thereby ensuring that all domain groups appear in train, validation, and test splits and enabling more reliable invariance estimates.
7- WILDS and Retina, we merge
the original train/ crainvalldaaiion,
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After training each model under identical hyperparameters previous experiments, freeze its projection head and fit two linear classifiers on the resulting embeddings: one to predict domain labels (domain- classification AUC, lower means more invariance) and one to predict disease labels (label-classffiaation AUC, higher mcans more discriminative). Experiments with non-linear classifiers are available in Appendix A.3.
Fitzpatrick17k
Camelyon
Retina
Fitzpatrick17k
Conclvon1--WII
Retina
100
86
09.8
91
Donacn AlI
90
Labe, AUCTT
70
28
90
70
10.4
60
70
EBM
10.2
Boyent
10:32
128
16.32
34
000
16332
64
64
128
16.32
128
16.32
64
Labera dincussion
Lateni dinetrsion
Lattent dimensioo
Lateaa dimenssion
Laten: dinetesion
Latent dimension
(a) Domain AUC +
(b) Label AUC f
Figure 4: Figure 4a shows the domain AUC (L) vs. latent dimension d: lower better (more domain- invariant), while Figure 4b plots the label AUC (f) vs. d: higher better (more discriminative). The curves are shown for Euclidean (ERM) and hyperbolic embeddings (HypERM). Results are reported on in- distribution splits where all domains appear in vrin/val/test, yiclding 3, 5, and 4 domains for Fitzpatrick17k. Camelyon17-WII and Retina, respectively.
5.1.1 Results
As shown in Figure 4a, the domain-classifcation AUC is lowest at d = 2 for both manifolds and steadily increases with d, confirming that a smaller bottleneck enforces stronger domain invariance, Crucially, Fig- ure 4b demonstrates that even with a 2D bottleneck, label-classfficttio AUC remains high (up to 99% on Camelyon17) and continues to improve with larger d. Notably, Fuclidean embeddings plateau on Fitz- patrick17k and Camelyon17-WILDDS, with no further increase on larger d. However, hyporbolic emboddings gain additional accuracy up to d = 128, especially on Fitzpatrick117k, with gap of approximately 3% AUC. Taken together, these results suggest that (1) a very low-dimensional bottleneck provides strong domain invariance with only modest loss in discriminative power, and (2) hyperbolic embeddings achieve higher label AUC than their Euclidean counterparts, with the largest gains observed at higher dimensions.
5.2 Hyperbolic cross-branch consistency
To isolate the contribution of our two-branch consistency regularization, we fix the high-dimensionall branch di 128 and vary the low-dimensional bottleneck d2 {2,8, 16, 128). For each configuration, measure the AUC gain of the two-branch model over its single-branch counterpart on the same three benchmarks; CamelyonI7-WILDS (test hospital), Retina (test dataset), and Fitzpatrick177k (leave-no-dommainoott)) This is evaluated on both hyperbolic and Euclidean manifolds.
5.2.1 Results
Figure 5 shows that hyperbolic space, the largest improvements occur at d2 and decline steadily as the bottleneck size grows, confirming that performance gains arise from compact regularization rather than added capacity. Indeed, d2 128 slightly degrades hyperbolic performance. On Camelyon17, Euclidean cross-branch consistency (CBC) produces surprisingly negative AUC changes for all d2, while hyperbolic CBC (HypCBC) yields positive gains at every reasonable bottleneck sizc. On Retina, both manifolds benefit at d2 = 2. Although Euclidean gains appear larger at this point, the hyperbolic single-branch baseline itself is already 0.99% higher than Euclidean (ef. Table 3). On Fitzpatrick1 HypCBC consistently improves performance across all folds, achieving up a 2% AUC boost on the III-IV split versus only 0.5% for Euclidean. Overall, hyperbolic consistency regularization yields consistent, positive AUC gains across all datasets and bottleneck sizes, while Fuclidean consistency produces uneven improvements.
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Fitzpatrickl17
C17-WILDS (Test)
Rotina (Toss
2.0
CBC
I-11
1,5
HypCBC
I1-IV
1.5
MXX
V-VI
0.2
1U
AUC
1.0
0.5
0.0
0.0
0.5
-0.2
0..5
0.0
d2 16
128
16
128
16
126
d2
dg
Figure 5: Improvement in AUC of two-branch consistency regularization over single-branch baseline (AAUC) for Euclidean and hyperbolic manifolds, while varying bottleneck dimension d2. These are termed CBC and HypCBC, respectively. From left to right: (1) Fitzpatrick177 7k leave-one-domain-out folds (I-II, III-IV, V- VT), diffcrentiated with bar hatches, (2) Camelyon17-WILDS (test), and (3) Retina (test). While Euclidean regularization gains vary by dataset, hyperbolic regularization provides positive AAUC across every lask and reasonable (i.e., 128) bottleneck size.
5.3 Sensitivity analysis
Finally, we perform a sensitivity analysis over our two key hyperparameters: the KL weight and the tem- perature T. We sweep over values for À E (0.1, 0.5, 1.0) and T E (1.0, 5.0, 10.0) Fitzpatrick1 (I-II, II-IV, V-VT), Retina (test), and Camelyon17-WIIDS (test), reporting the mean AUC and its devia- tion across these settings (each averaged over five seeds).
5.3.1 Results
As indicated by the standard deviations reported in Table 4, performance varies only marginally with À and T (i.e., a E 0.71, This empirically demonstrates the robustness of our proposed method with respect to the chosen hyperparameters, thereby confirming its superiority against. Euclidean baselines.
Table 4: Average AUC and standard deviation across different combinations of À and T, evaluated Fitzpatrick17.. Maneeyonl7-WII... and Retina.
IIII FIIK(III-IV) F17k(V-VI) C17-WILDS Retina
81.96+0.4 85.55
83.81+0.4
98.30+0.1
80.19+0.2
6 Conclusion
6.1
Limitations
Our current implementation fixes the curvature parameter e and relies on frozen Euclidean backbone followed by lightweight hyperbolic MLR heads. While this design enables stable and efficient training, static curvature may not be optimal across datasets manifold dimensions. Allowing curvature to be learnable, or adopting data- or manifld-adappive curvature schedules, could more effectively capture the geometric structure of each task. Moreover, although freezing the Teature extractor offers substantial computations ional savings, fine-tuning the backbone or training a fully hyperbolic classifier may further amplify the benefits of the proposed approach, the cost of increased training complexity. Finally, our method evaluated only for singlo-label multi-class classification. Although the cross- branch consistency operates at the logit level and is calc extendable to multi-label settings, we do not explore this extension in this manuscript.
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6.2 Discussion
Our work presents a hyperbolic representation learning framework for medical imaging that leverages the inherent hierarchical structure of clinical data. Across diverse modalities and scales, replacing Euclidean embeddings with our hyperbolic projections consistently improves in-distribution accuracy. Our two-branch, domain-invariant hyperbolic cross-branch consistency scheme further boosts on three challenging benchmarks. Crucially, ablation studies confirm that these gains arise from the compact low-dimensional consistency regularization, not merely from added capacity, and that hyperbolic consistency outperforms its Euclidean counterpart. Overall, hyperbolic embeddings offer straightforward yet powerful alternative for building robust, generalizable medical Al systems.
performance
Ethical and Data Quality Considerations
This work uses only publicly available datasets and adheres to their respective licenses. Fitzpatrick177 is known contain substantial label noise (<x 22%) due to heterogeneous data sources and annotation procedures, documented by recent analyses (Gröger et al., 2025). The dataset also presents limitations related to data provenance, as it was compiled from web-scraped images with limited transparency regarding original patient consent, and is distributed under CC-BY-NC license. While these factors may affect absolute performance and raise broader ethical considerations, all methods in this study are evaluated under identical conditions, and Fitzpatrick17K remains a widely adopted dataset.
Acknowledgements
This study was funded through the Hightech Agenda Bayern (HTA) of the Free State of Bavaria, Germany,
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A Appendix
A.1 Hyperbolic visualizations
Our method includes an explicit 2D hyperbolic branch, enabling direct visualization of learned represen- tations without relying on post-hoc dimensionality reduction. Unlike techniques such as t-SNE or UMAP, which produce approximate low-dimensional embeddings and may distort neighborhood structure, our ap- proach yields representations that are intrinsically (wo-dimensional and geometrically meaningful.
Figure 6 visualizes the learned 2D hyperbolic embeddings for Cameeyon17-WILLDS (train), Retina (train), and Fitzpatrick17k (train, fold V-VI). For each datasct, embeddings are shown in the Poincaré disk and colored by class (left) and by domain (right). Across all benchmarks, class structure is preserved, whereas domain labels exhibit, substantial overlap and lack compact, locally separable clusters, suggesting reduced domain-specific organization.
On CamelyonI7-WILDS, which comprises two classes and three hospital domains, clear class separation is observed, whereas domain labels remain highly mixed. Notably, the training domains are imbalanced, with 131,696, 116,959, and 53,425 samples from hospital 4, 3, and 0, respectively. This explains the predominance of purple and orange points and the diffuse presence of the under-represented hospital 0 domain. Similar qualitative behavior is observed on Retina and Fitzpatrick17k. While Fitzpatrick 17k shows a clearer class separation, Retina exhibits higher class overlap (especially with class 4). This is expected given the progres- sively graded nature of diabetic retinopathy severity, making class separation inherently more challenging.
To quantify these observations, we compute the average local k-NN entropy (k= = 15) in hyperbolic space, mcasuring label diversity within local neighborhoods. Lower entropy indicates stronger local discrimination. On Camelyom!7-WILDSS, we observe a low class entropy (IH = 0.071) and substantially higher domain entropy (Hldomain = 0.731), confirming that neighborhoods are consiscon but nomain-mixedd This is facilitated by the large dataset size and its binary nature. Corresponding results for the remaining datasets are reported in Table 5. Higher class entropy on Fitzpatrick177 and Retina is expected due to increased task complexity, and for Fitzpatrick177 values are averaged across LODO splits, smoothing fold-specific effects. Overall, these results provide geometric evidence that the proposed representations preserve class structure while reducing domain-spocific clustcring.
Table 5: Average local R-NN entropy of 2D hyperbolic training embeddings, computed with respect to class labels (Hiclams 4) and domain labels (H 1) on Fitzpatrick17k (3 classes, 2 train domains), Camelyon17- WILDS (2 classes, 3 train domains), and Retina (5 classes, 2 train domains). Lower class entropy indicates stronger class separability, while higher domain entropy reflects reduced domain-specific structure. For Fitzpatrick17k, entropy averaged across the three training folds, while for 1--nel,, results are computed on a subset of 30, 000 samples due to computational overhead. The results are consistent with the learned embeddings preserving class information while exhibiting limited domain discrimination.
F17k
C17-WILDS
Retina
Hetaon (4)
H (t)
Hetnon (L)
Haomain He (L)
Haomsin (t)
0.414 + 0.04 0.554 = 0.07
0.071
0.731
0.259
0.605
A.2 Standardized evaluation for Camelyon17-WILDS
Following the standard WILDS (Koh et al., 2021) evaluation protocol for Camelyon! we addition- ally report average accuracy alongside AUC (cf. Table 3) for both validation and test splits (ef. Table 6). The trends observed in AUC are consistent with those in accuracy. HypCBC achieves the best validation performance and competitive test accuracy within the standard deviation of the top-performing methods. Importantly, unlike GroupDRO and VRFx, our method does not rely domain labels during training.
17

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Published in Transactions on Machine Learning Research (02/2026)
Colored by Class
Colored by Domacn
No tunce
Hi
(a) Camelyon17-WW WI-LLDS
Colored by Class
Colored Domain
Colored by Class
Colored by Domain
Aptow DegDR
Benign
Malignent
Non-tooplatti
(b) Retina
(c) Fitzpatrick177k (V-VI)
Figure 6: Visualization of 2D hyperbolic (training) embeddings learned by the proposed method, shown in the Poincaré disk. For each dataset, left: points colored by class label; right: the same embeddings colored by domain. While class separation observed, domain separation is minimal.
Table 6: AUC and accuracy averaged over five runs for CamelyonI7-WILDS val and test sets. We highlight in bold the best two methods.
AUC
Accuracy
Method
Val
Test
Val
Test
ERM
97.05
98.32:0.1
91.2310.2
94.02:0.2
Med-C
97.1410.1
98.22
91.5011.1
94.0910..
RandAug
96.74+0.0
98.15:1.11
90.4840.11
93.89-10.6
AugMix
96.50-10.1
98.06:1.11
90.12+0.11
92.73+11.0
IRM
96.99+0.1
98.17#0.2
90.56++.2
98.76:+0.2
GroupDRO
97.50+0.1
98.22#0.1
91.79+0.2
94.30+10.2
VREx
97.3610.11
98.35-10.1
91.4910.3
94.39-10.3
DANN
97.0010.11
98.2210.1
91.0710.2
93.53.10.4
CDANN
97.0440.0
98.2310.1
91.2011.2
93.6410.6
MMD
97.02+-022
98.27+0.1
91.22+0.3
94.21+0.2
CORAL
97.22++0.1
98.25+0..2
91.3240.3
91.04+0.2
HypERM
97.95.10.6
98.07-0.2
92.79.1.1
93.22++0.5
HypCBC
98.04110.3
08.3310.3
92.89.10.8
94.1941044
A.3 Non-linear evaluation of domain invariance
Batch effects (i.e., domain-specific acquisition annotation artifacts) may be encoded nonlinearly (Rahman et al., 2023), in which case linear probes can underostinate the amount of domain information present in learned representations. To account for this, we replicate the Nommiin-redictabilii experiment of Section 5.1, replacing the linear classifier with a two-layer MLP trained on frozen embeddings for 50 epochs using AdamW with a learning rate of 44
As shown in Figure 7b, the nonlinear probe yields slightly higher absolute domain AUC than the linear clas- sifier (Figure 7a), confirming that some domain information is indeed nonlinearly encoded. Crucially, the
18

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Published in Transactions on Machine Learning Research (02/2026)
Fitquutrick17k
Comelyon Banevoon7-WWTI. 1S
Retina
Fitepwrrok17k
Camelyon 17-WILDS
Resina
100
100
Domaai ALC 4
90
90
Domain AUC 1
30
90
20
80
70
80
70
00
ERM Hbyeltt
60
KIS HspKRM
+0
60
2
16.322
64
128
2 32
64
128
2
16:32 61
128
2
16
32
64
128
2
36
61
128
2616
16
64
128
Lateut dimsession
Latent dineneloo
Latent dimension
Laten: dimenston
Latent dimension
Latent dimension
(a) Linear
(b) MLP
Figure 7:
Figure 7a shows the domain AUC (L) vs. latent dimension d obtained with a linear classifier, while Figure 7b plots the domain AUC (1) vs. d obtained with an MLP. The curves are shown for Euclidean (ERM) and hyperbolic embeddings (HypERM). Results are reported on TD splits where all domains appear in train/ train/val/test test, yielding 3, 3,5, and 4 domains for Fitzpatrick 17,, Somnelyon77-WLDDS and Retina, respectively.
qualitative trends remain unchangod: domain predictability is minimized at low-dimensional bottlenecks, in- creases monolonically with embedding dimension, and exhibits consistent relative behavior across Euclidean (ERM) and hyperbolic (HypERM) manifolds.
We further observe dataset-dependent differences in the gap between linear and nonlinear probes. On Fitzpatrick17k and Camelyon WILDS, the difference in domain AUC between linear and MLP is marginal, suggesting that dominant domain confounders, such as skin tone and hospital-specific staining (ef. Figure 3), are largely texture-driven and close to linearly separable at higher dimensions. In contrast, the Retina benchmark exhibits a larger gap, likely due to more heterogeneous domain shifts arising from multiple acquisition devices and grading protocols, which induce more complex and less linearly separable domain structure. The absence of perfect domain scparability on Fitzpatrick17k. is also consistent with known ambiguity in skin-tone annotation.
Overall, while the degree of nonlinearity in domain information is dataset-dependent,, the qualitative domain- invariance trends induced by low-dimensional bottlenecks remain consistent.
A.4 Augmentations within the hyperbolic manifold
This section evaluates the interaction between HypCBC and common data augmentation strategies. We consider the same domain generalization benchmarks used throughout the paper, i.e., Fitzpatrick177k CamelyonI7-WILDDS, and Retina, and combine our methods with RandAugment and AugMix, which are among the strongest augmentation--aase basclines in prior work.
Table 7 shows that the impact of data augmentation
HypCBC is dalaset-dependent. While modest gains are observed on Fitzpatrick177k (up to +0.62%), no consistent improvements are seen on Camelyon17- WILDS or Retina. This indicates that the effectivencss of augmentations varies with dataset charactoristics. representations, the model may still benefit
Although the proposed bottleneck promotes from input diversity when such variability is informative.
Table 7: Average AUC for hyperbolic methods combined with Randengment and AugMix across three domain generalization benchmarks, namely Fitzpatrick177k, Camelyon17-WILDSS, and Retina.
F17k
C17-WILDS
Retina
Method
I II
III
IV
V VI
Avg
Val
Test
Val
Test
HypCBC
82.34 + 0.3 86.28 + 0.2 84.27 = 0.3 84.30 98.04 + 0.3 98.33 + 0.3 87.34 0.7 80.48 + 0.5
w/AugMix
82.53 0.1 86.87 1 0.2 85.37 1 0.3 84.92 97.77 0.3 98.52 0.1 86.37 1 0.4 80.08 1 0.5
w/RandAug
82.43 1 0.1 86.57 0.2 84.83 4 0.3 84.61 97.76 0.4 98.52 0.1 86.77 1 0.5 79.83 1 0.4
19

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Published in Transactions on Machine Learning Research (02/2026)
Geodesic the Poincaré Disk
Map
Eudidean Decision Boundary
Exp Map
Figure 8: Toy illustration of Euclidean and hyperbolic decision boundaries under exponential and logarithmic maps, highlighting geometric differences between Euclidean linear classifiers and hyperbolic hyperplanes.
A.5 Euclidean Hyperbolic Decision Boundaries
To provide geometric intuition for the difference between Euclidean and hyperbolic classifiers, include two-dimensional toy example illustrating how decision boundaries behave under the exponential and loga- rithmic maps.
Figure 8 contrasts four corresponding views of the same classification setup. The top-left panel shows a hyperbolic decision boundary represented as geodesic the Poincaré disk, separating two synthetic clusters. In hyperbolic space, linear classifiers correspond hyperbolic hyperplanes, which appear as circular ares orthogonal to the unit disk. The top-right panel shows the same hyperbolic boundary after applying the logarithmic map to the Euclidean (angent space at the origin, where it becomes a non-linear curve. This illustrates how hyperbolic linearity translates into non-linear decision structure in Euclidean space.
For comparison, the bottom-left panel shows a standard Euclidean lincar decision boundary in tangent space. The bottom-right panel visualizes this Euclidean boundary after mapping into hyperbolic space via the exponential map. While the mapped boundary remains smooth, it does not correspond a hyperbolic hyperplane, since hyperbolic decision boundaries are defined by geodesics in the manifold, whereas the exponential map of Euclidean linear separator does not generally preserve geodesicity. This highlights the geometric mismatch between Euclidean linear classifiers and hyperbolic decision surfaces.
20


Reviews:
[{"review_id": "xHj9OfpAeu", "reviewer": "Reviewer_5Qjm", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a hyperbolic representation learning framework for medical image analysis. The authors introduce an unsupervised two-branch hyperbolic distillation strategy that promotes domain-invariant features via a low-dimensional bottleneck. The authors demonstrated that hyperbolic embeddings consistently outperform Euclidean embeddings on Med-MNIST across three Vision Transformer backbones.\nOn three domain-generalization benchmarks, the author demonstrated that, in most cases, the proposed technique provides better results.", "claims_and_evidence_yn": "No", "claims_explanation": "Even though the technique shows better performance, it is not clear if these gains are due to \"out-of-domain\" robustness. \n\nTo be specific \n- The authors did not quantify that the boost in performance is due to the 'domain invariant' nature of  distillation\n\nThe empirical way to demonstrate is to compare the IN-domain and out-of-domain performance of different models across different baselines and quantify the drop in performance, i.e., \n\n$\\Delta AUC = AUC_{IN} - ACC_{OUT}$ \n\nAlso, the authors did not present either a theoretical or an intuitive explanation for why the proposed distillation would be domain-invariant.", "audience_yn": "Yes", "audience_explanation": "Out-of-domain generalization remains a major challenge in medical image analysis. At present, the authors have not sufficiently explained or demonstrated the effectiveness of their technique. However, once these issues are addressed, it would be of considerable interest to the TMLR audience.", "requested_changes": "1. Demonstrate the domain invariant nature of distillation. Please compare the IN-domain and out-of-domain performance of different models across different baselines and quantify the drop in performance.\n\n\n2. \" However, end-to-end hyperbolic networks can be unstable on large datasets (Ayubcha et al., 2024)\" -- I could not find in the reference where the instability in end-to-end training of hyperbolic networks is discussed. Please make it clear.\n\n3. I am not sure if equations 2 and 3 are used in the technique. If not, they may be removed\n\n4. It is not clear what type of data augmentations were done during training. The proposed distillation appears to be data augmentation-agnostic. How would the model perform if the augmentation bases, such as RandAug/Cutmix, are combined with the proposed distillation? if this is a bad idea, then please explain the reason. \n\n5. \" , we freeze its projection head and fit two linear classifiers on the resulting embeddings: one to predict domain labels (domain-classification AUC, lower means more invariance) and one to predict disease labels (label-classification AUC, higher means more discriminative).\" - This assumes that the batch effect can be identified via linear models. However, batch effects can be nonlinear [ see a]. In that case, it cannot be well predicted by the linear regressors. \n\n6. \"... with a 2D bottleneck, label-classification AUC remains high (up to 99% on Camelyon17) and continues to improve with larger d.\" -- This implies that the task is relatively easy. In fact, for both \"Fitzpatrick17k\" and \"Camelyon17-Wilds\", the performance remained pretty high even for a 2-dimensional latent space. And only for \"Retina\" do we see improvements when increasing the latent size (a difficult task, in a sense, as it requires a larger representation space for classification). And from Table 3 e can see the pattern that the propose technque is performing well for the classification tasks that inhirently require smaller latent vecctor. And performing worse compared to the baseline on Ratina, which requires a larger embedding. Please add an explanation for this. If the observation is true, then add more competitive baselines. \n\n7. The proposed technique has a limitation that it can not be applied to multi-label classification. I am not if it can be extended to segmentation. Please add clarification on this. \n\n\n\n\n\n\na. BEENE: deep learning-based nonlinear embedding improves batch effect estimation", "broader_impact_concerns": "No direct ethical concern is apparent to me.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "z9jcilZC2B", "reviewer": "Reviewer_BQiW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses the fragility of medical AI under distribution shifts by proposing HypDIST, an unsupervised domain generalization framework. The authors argue that Euclidean spaces fail to capture the hierarchical nature of medical data and propose using the Poincaré ball model of hyperbolic geometry.\\\nMethodology: The core innovation is a dual-branch architecture where a high-dimensional branch ($d=128$) is distilled from a low-dimensional hyperbolic bottleneck ($d=2$). This bottleneck forces the model to discard \"style\" (domain) information while retaining \"content\" (labels).\\\nKey Strengths: Extensive validation across 11 modalities; strong geometric motivation; achieves domain invariance without requiring explicit domain labels.\\\nKey Weaknesses: Use of non-standard evaluation metrics for the WILDS benchmark and omission of very recent, highly relevant hyperbolic DG literature.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The claims are supported by a rigorous benchmarking of 11 in-distribution datasets and 3 OOD benchmarks. The ablation studies (Figure 4) convincingly demonstrate that 2D hyperbolic space preserves label discriminability significantly better than 2D Euclidean space under extreme compression. However, the evidence is weakened by:\n\nMetric Misalignment: For Camelyon17-Wilds, the authors report AUC, whereas the standard evaluation metric is Average/Worst-group Accuracy. This makes comparison with the official leaderboard difficult.\n\nBaselines: The paper fails to compare against concurrent hyperbolic domain generalization methods (e.g., Bi et al., AAAI 2025), which limits the \"State-of-the-Art\" claim.", "audience_yn": "Yes", "audience_explanation": "This work intersects three major interests within the TMLR community:\n\nMedical AI: Practitioners seeking unsupervised ways to handle hardware and population shifts.\n\nGeometric Deep Learning: Researchers interested in the \"intrinsic dimension\" of medical data and the practical utility of non-Euclidean manifolds.\n\nAI Fairness: Those investigating technical pathways to mitigate demographic bias (e.g., skin tone) via representation learning.", "requested_changes": "1. Metric Standardization: Update the Camelyon17-Wilds results to include Average Accuracy and Worst-Group Accuracy to align with the WILDS benchmark standards.\n2. Ethical & Data Quality Statement: Add a section addressing the known issues with the Fitzpatrick17k dataset, including the ~22% label error rate and the CC-BY-NC licensing/provenance concerns.\n3. Comparison with Concurrent SOTA: Discuss and, if possible, quantitatively compare against Bi et al. (AAAI 2025) and Yang et al. (IEEE TIP 2025). These are direct hyperbolic competitors that should be acknowledged to contextualize the novelty of HypDIST.\n4. Embedding Visualization: Provide Poincaré disk visualizations of the 2D embeddings for Fitzpatrick17k. Color points by domain vs. class to qualitatively demonstrate the effectiveness of the domain-invariant bottleneck.\n5. Technical Clarity: Explicitly state the pre-training data for the backbones (ImageNet-1k vs. 21k) and provide a brief justification for fixing curvature c=1.0.", "broader_impact_concerns": "\"Fairness Washing\": There is a risk that \"domain invariance\" is interpreted as a complete solution for \"fairness.\" The authors should explicitly state that if training data is fundamentally biased (e.g., missing specific pathologies for certain skin tones), a robust model may still fail to represent those groups accurately.\n\nData Provenance: The paper relies on scraped data (Fitzpatrick17k). The authors should acknowledge the ethical limitations regarding patient consent in such datasets to promote responsible data curation practices in the community.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Zp4qwUHwrJ", "reviewer": "Reviewer_RNcD", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents a study of hyperbolic embeddings in the domain of medical images. First, the background theory required for hyperbolic embeddings is reviewed. The paper then proposes to use pretrained vision transformers followed by a trainable linear layer and the hyperbolic exponential map to obtain good hyperbolic embeddings at a low cost. The embeddings are trained via a classification loss. Further, a dual branch “distillation” approach is proposed where a bottleneck low dimensional embedding is also learned in addition to a high dimensional embedding. The training proceeds via a classification loss for both the low and high dimensional embeddings as well as a KL-divergence loss between the logits produced by the low and high dimensional embeddings. The aim of adding the bottleneck low dimensional embedding is to obtain domain invariant features.\n\nExperiments show first that hyperbolic embeddings outperform Euclidean ones and then that the proposed dual branch training scheme improves performance compared to only using hyperbolic embeddings. A broad set of ablations are carried out to evaluate the contributions of different components.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The main claims are supported by a series of experiments.\n\n1. Hyperbolic embeddings are shown to outperform Euclidean embeddings.\n2. The efficacy of the proposed 2-branch training strategy is demonstrated.\n3. Ablations show the impact of varying the dimensionality of the latent dimensions.", "audience_yn": "Yes", "audience_explanation": "I believe that the findings are of interest to the medical image analysis community as well as the hyperbolic deep learning community.", "requested_changes": "### Critical\n\n1. “Distillation” is not the best term to use for the proposed two-branch training strategy. Distillation typically refers to the training of a smaller network to mimic a frozen larger network. Unless there is prior work calling a similar strategy distillation, I recommend using a different term.\n2. The experiments use three different ViTs. I recommend not to refer to these as different architectures, as they use minor variations of the same ViT architecture. Instead these could be differentiated as different “models” or different “training recipes”.\n\n### Recommended for strengthening the paper\n\n3. The fact that a 2D bottleneck proves to be useful opens up opportunities for visualizations. It would be very interesting to use visualizations of the 2D latents to see whether an interpretable hierarchical structure is obtained, as this is the stated goal of using hyperbolic embeddings.\n4. The comparison between Euclidean embeddings and hyperbolic embeddings is interesting. It would be helpful for the reader if the difference between these was more explicitly described. The only two differences, as far as I understand, is the use of the exponential map for the hyperbolic embeddings and the use of hyperbolic hyperplanes for the classification rather than Euclidean hyperplanes. It should be possible to map the hyperbolic hyperplanes via the logarithmic map to the Euclidean tangent space, yielding curved decision surfaces in the Euclidean space. In the 2D case it would even be possible to visualize these 2D curves. Vice versa, one could visualize Euclidean decision surfaces (2D lines) after mapping them through the exponential map to obtain some curved decision surfaces in hyperbolic space, contrasting the hyperbolic hyperplanes in the Poincare disk model. \n\nVisualizations as proposed in the two points above would help the reader understand the benefit provided by hyperbolic embeddings.", "broader_impact_concerns": "N/A", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)