# Multi-Prototype Hyperbolic Learning Guided by Class Hierarchy

Paul Berg, Léo Buecher, Björn Michele, Minh-Tan Pham, Laetitia Chapel,

Nicolas Courty

► To cite this version:

Paul Berg, Léo Buecher, Björn Michele, Minh-Tan Pham, Laetitia Chapel, et al. Multi-Prototype Hyperbolic Learning Guided by Class Hierarchy. International Journal of Computer Vision, 2025, 10.1007/s11263-025-02571-8. hal-05222772

HAL Id: hal-05222772

https://hal.science/hal-05222772v1

Submitted on 5 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Multi-Prototype Hyperbolic Learning Guided by Class Hierarchy

Paul Berg<fnref n="2" />, ◙ · Léo Buecher<fnref n="1" /> · Björn Michele<fnref n="1" />,<fnref n="3" /> · Minh-Tan Pham<fnref n="1" /> · Laetitia Chapel<fnref n="1" />,<fnref n="4" /> · Nicolas Courty<fnref n="1" />

Received: 6 March 2025 / Accepted: 11 August 2025

© The Author(s) 2025

## Abstract

In many computer vision applications, datasets often exhibit an underlying taxonomy within the label space. To adhere to this hierarchical structure, hyperbolic spaces have emerged as an effective manifold for representation learning, thanks to their ability to encode hierarchical relationships, with little distortion, even for low-dimensional embeddings. Hyperbolic prototypical learning, where class labels are represented by prototypes, has recently demonstrated strong potential in this setting. However, existing methods generally assume a uniform distribution of prototypes, overlooking the hierarchical organization of labels that may be available for a given task. To better exploit this prior knowledge, we propose a hierarchically informed method for prototype positioning. Our approach leverages the Gromov-Wasserstein distance to align the hierarchical relationships between labels with the initial uniform spherical distribution of prototypes, leading to more structured and semantically meaningful representations. Additionally, within a deep learning framework, we propose an alternative characterization of decision boundaries using horospheres, which are level sets of the Busemann function. Geometrically, horospheres correspond to spheres tangent to the boundary of hyperbolic space at a virtual point analogous to a prototype, which makes them a compliant tool in the prototypical learning context. Accordingly, we define a new horospherical layer that can be adapted to any neural network backbone. This layer is particularly advantageous when the number of prototypes exceeds the number of classes, offering enhanced flexibility to the classifier. Through our experiments, we demonstrate that the combination of proper initialization and optimized prototype positioning significantly enhances baseline performance for image classification on hierarchical datasets. Additionally, we validate our approach in two semantic segmentation tasks, using both image and point cloud datasets, confirming its effectiveness in leveraging hierarchical label structures for improved classification performance.

**Keywords** Hyperbolic Spaces · Prototypical Learning · Representation Learning

Communicated by Carlos Moreno-Garcia.

Work done while at IRISA.

<figure><img src="image_2.png" /></figure>

Paul Berg
paul.berg@bfh.ch

Léo Buecher
leo.buecher@irisa.fr

Björn Michele
bjoern.michele@valeo.com

Minh-Tan Pham
minh-tan.pham@irisa.fr

Laetitia Chapel
laetitia.chapel@irisa.fr

Nicolas Courty
nicolas.courty@irisa.fr

## 1 Introduction

Euclidean representations have been the ubiquitous choice for deep representation learning methods. Recently, other embedding spaces have emerged as attractive alternatives. Indeed, embedding samples into a different metric space can improve the representation capabilities (Bronstein et al., 2017). Spherical representations, for instance, have had a large success in self-supervised representation learning (Chen et al., 2020; Wang & Isola, 2020; Caron et al., 2020) and supervised learning (Mettes et al., 2019) leveraging the so-called cosine similarity as a measure of alignment between samples representations. Other notable examples

<footnote n="1">¹ IRISA, Université de Bretagne Sud, Vannes, France</footnote>

<footnote n="2">² Bern University of Applied Sciences, Biel, Switzerland</footnote>

<footnote n="3">³ valeo.ai, Paris, France</footnote>

<footnote n="4">⁴ L'Institut Agro Rennes-Angers, Rennes, France</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1** Horospherical learning with smart prototypes. We propose to position ideal prototypes following the label hierarchy and then optimize their position during training. We observe that the similarity matrix</figcaption></figure>

for prototypes initialized with our hierarchical approach exhibits similar patterns than that of the tree similarity matrix

include the cone of Positive Semi-Definite (PSD) matrices (e.g. Pennec (2020)) used for covariance matrices or symmetric spaces such as the Siegel space (López et al., 2021) that naturally embed graphs.

Among those alternative spaces, Hyperbolic spaces have recently emerged as an alternative to Euclidean spaces to learn data embeddings. Indeed, they were shown to be efficient at embedding trees with minimal distortion Sarkar (2011) compared to Euclidean spaces. As a result, Hyperbolic spaces have successfully been used in several key machine learning tasks such as natural language processing Nickel and Kiela (2017), computer vision Khrulkov et al. (2020) and classification with hyperbolic support vector machines Cho et al. (2019); Fan et al. (2024).

Prototype learning can be a very useful tool when performing Machine Learning on such general manifolds. Indeed, if it is straightforward to design boundary decisions in Euclidean spaces, through the use of hyperplanes, the question is much less obvious in more atypical spaces. Prototypes offer a very flexible approach to tackle this problem Snell et al. (2017); Mettes et al. (2019). Those are special points living in the embedding space that are assigned to classes. A data point is then labelled according to the closest prototype, given a certain metric or similarity measure. When working with the Hyperbolic space, one can consider ideal prototypes living

at the infinite boundary of the space and compare them to data points with the Busemann function. This yields decision boundaries called Horospheres that can be seen as a generalisation of hyperplanes in the Hyperbolic space Ghadimi Atigh et al. (2021); Fan et al. (2024).

It is also often possible to derive a hierarchy from the labels of an image dataset. For example, the ImageNet dataset Deng et al. (2009) labels are directly derived from the WordNet Miller (1995) hierarchy. However, label hierarchies are often unused by deep learning models which discriminate only among leaf nodes in the hierarchy (i.e. labels). We consider that adding hierarchical information in the training process can be beneficial for the resulting performance. Especially, when a particular dataset exhibits a correlation between the hierarchical similarity of labels and the visual similarity of corresponding samples.

In this article, we build upon the ability of hyperbolic spaces to embed trees with low distortion to better model hierarchical classification problems. Our contributions are the following. First, we introduce a hyperbolic classification layer based on trainable horospheres which are parameterized by their prototype location and a bias term. Second, we extend the framework of prototypical learning so that each class can be represented by multiple prototypes. Third, we propose a hierarchically informed scheme to initially pos-
tion these prototypes respecting the label hierarchy present in
the dataset. Finally, we provide experiments on image clas-
sification and semantic segmentation which show that the
proposed approach improves performance on hierarchical
datasets compared to other hyperbolic and prototype-based
baselines.

## 2 Related Work

**Prototypical Networks.** Prototypical networks rely on comparing the position of samples with prototypes and learning to project samples as close to the corresponding prototype as possible. While prototypical networks have been used with success in Euclidean spaces Snell et al. (2017) by computing prototypes as the centroid of representations for each class, research has expanded them to other embedding manifolds such as graphs Vincent-Cuaz et al. (2022), hyperspheres Mettes et al. (2019) or hyperbolic spaces Ghadimi Atigh et al. (2021); Hamzaoui et al. (2022) where prototypes are positioned independently from the data or have their positions updated during training. The initial prototype positioning can sometimes be directed by prior information such as word embeddings of class labels Mettes et al. (2019); Liu et al. (2020). In a similar fashion, we consider prototypes which are not defined as centroids but instead live as points in the latent representation space.

Prototypical learning can also be employed with multi-
ple prototypes per class. Allen et al. (2019) has for instance
explored this approach in the context of few-shot learning,
relying on the theory of mixture models. In their work, pro-
totypes are defined as the means of clusters among a class,
and data points are then assigned to the label of the closest
prototype. Other approaches considering a number of proto-
types different from the number of classes include what can
be called template learning. In this approach, the distances
of a point to all the templates, which are homologous to pro-
totypes, are stored in a vector to get a new representation that
can then be used for classification. This has been considered
as early as in 1991 in Chen et al. (1991), where the templates
are learnt in an unsupervised way. Other works have devel-
oped similar approaches that consist in taking the templates
among the training points Balcan et al. (2008); Duin and
Pekalska (2012). More recent and more applied examples of
the use of template methods come from graph learning (Chen
et al., 2021; Vincent-Cuaz et al., 2022), where it is leveraged
to embed graphs in Euclidean spaces.

**Hierarchical Learning.** Another active area of research in computer vision is leveraging label hierarchies to improve downstream performance Bertinetto et al. (2020) because it can cause the model to make less important mistakes. For example, misclassifying between birds of the same biological family may be less critical than between different biologi-

cal orders Chang et al. (2021). The hierarchical information
can be used either to design the classification layer itself
or as a regularisation by post-processing predictions hover
labels during training Dhall et al. (2020). Hou et al. Hou et
al. (2016) leverage the Wasserstein distance to penalize pre-
dictions which do not respect the class ordering which can
be seen as a sort of hierarchy. Similarly, Garnot et Landrieu.
Garnot and Landrieu (2020) propose to use the hierarchy
present in datasets combined with prototypical networks in
Euclidean space by regularizing the training to limit the dis-
tortion between prototypes and the hierarchy. Ghadimi Atigh
et al. Ghadimi Atigh et al. (2022) propose to compute prob-
abilities for each node in the label hierarchy as posteriors for
the actual class prediction with a hierarchical softmax.

**Hyperbolic Spaces.** Hyperbolic spaces have successfully
been used in a number of text Nickel and Kiela (2017); Tifrea
et al. (2018); Ganea et al. (2018); Dhingra et al. (2018), data
classification Cho et al. (2019); Fan et al. (2024), and vision
Ghadimi Atigh et al. (2022, 2021); Ermolov et al. (2022);
Ge et al. (2023) related tasks as an alternative to Euclidean
or other embedding spaces. Compared to Euclidean spaces,
a tree can be embedded in hyperbolic space with minimum
distortion Sarkar (2011). A number of hyperbolic neural net-
work layers have been proposed Peng et al. (2021) to replace
their Euclidean counterparts such as linear layers Ganea et
al. (2018), convolutional layers Shimizu et al. (2020) or even
entire hyperbolic networks van Spengler et al. (2023) where
all activations are in hyperbolic space. In particular, hyper-
bolic representations show promising results in hierarchical
tasks such as action recognition Long et al. (2020); Surís et
al. (2021) or image segmentation Ghadimi Atigh et al. (2022)
where label hierarchies are well-defined. Yu et al. Yu et al.
(2022) use the hyperbolic space to classify skin diseases with
the help of prototypes which live in this same embedding
space and which are learnt jointly, leveraging hierarchical
knowledge.

Most related to us, is Ghadimi Atigh et al. Ghadimi Atigh
et al. (2021) who introduces Busemann learning with ideal
prototypes. We build upon their method by proposing a hier-
archically informed positioning scheme for ideal prototypes
as well as introducing a trainable variant of the Busemann
prototype classification layer.

## 3 Background on Hyperbolic Spaces and Horospheres

### 3.1 Hyperbolic Spaces

While Euclidean spaces are often the embedding space of
choice for representation learning, hyperbolic spaces have
recently emerged as an attractive alternative embedding
space Nickel and Kiela (2017); Khrulkov et al. (2020).
Hyperbolic spaces are Riemannian manifolds with constant negative curvature. There are multiple models to represent the hyperbolic spaces, in this paper we leverage the Poincaré Ball model which is defined as:

$$ \mathcal{H}_d = \left\{ x \in \mathbb{R}^d \mid \|x\|^2 \le \frac{1}{c} \right\}, \qquad (1) $$

where $c > 0$ controls the negative curvature of the hyperbolic space, and $\| \cdot \|$ must be understood as the euclidean norm of $\mathbb{R}^d$. In the following we assume $c = 1$. On the Poincaré ball, the hyperbolic distance between two points $x, y \in \mathcal{H}^d$ is defined as:

$$ d_{\mathcal{H}}(x, y) = \cosh^{-1} \left( 1 + 2 \frac{\|x - y\|^2}{(1 - \|x\|^2)(1 - \|y\|^2)} \right). \qquad (2) $$

Such a space locally resembles a Euclidean space, and we can define, at any point $x$ of this manifold, the tangent space in $x$, which is the first order approximation of $\mathcal{H}_d$ in this point. It is denoted $T_x \mathcal{H}_d$ and results to be a Euclidean space of the same dimension. It is possible to map a tangent space to the Poincaré ball and vice versa through the exponential and the logarithm maps. The exponential map $\exp_x : \mathbb{R}^d \to \mathcal{H}_d$ in some point $x \in \mathcal{H}_d$ is defined as such:

$$ \exp_x(v) = x \oplus \left( \tanh \left( \frac{\lambda_x \|v\|}{2} \right) \frac{v}{\|v\|} \right), \qquad (3) $$

where $\oplus$ corresponds to the Möbius addition, a transformation analogous to vector addition in Euclidean space but in hyperbolic space and $\lambda_x = 2/(1 - \|x\|^2)$ is the conformal factor at point $x$. The Möbius addition between $x \in \mathcal{H}_d$ and $y \in \mathcal{H}_d$ is defined as

$$ x \oplus y = \frac{(1 + 2\langle x, y \rangle + \|y\|^2)x + (1 - \|x\|^2)y}{1 + 2\langle x, y \rangle + 2\|x\|^2\|y\|^2}. \qquad (4) $$

In this work, we mostly consider the exponential map in the origin of the Poincaré ball, which has a simplified expression:

$$ \exp_0(v) = \tanh \left( \frac{v}{\|v\|} \right). \qquad (5) $$

## 3.2 Horospheres and the Busemann function

In Euclidean spaces, linear layers are often used as the last layer of a model to discriminate samples among different classes. A linear layer is composed of a set of hyperplanes,

each responsible for an output feature of the layer. The value for each feature is computed from the distance to the hyperplane. Ganea et al. (2018) proposed a counterpart for Euclidean hyperplanes in the hyperbolic space, more specifically the Poincaré ball model, called hyperbolic gyroplanes. They are the image of hyperplanes through the exponential map, or, said differently, they are the union of the geodesics orthogonal to some given normal vector in some given point. An illustration of gyroplanes in the context of classification on the Poincaré disk can be seen in Figure 3.

Another method to perform hyperbolic classification relies on ideal prototypes Ghadimi Atigh et al. (2021). An ideal prototype $p$ is a fictive point positioned at the infinite of the hyperbolic space. In the Poincaré ball model, these so-called ideal prototypes are located on the surface of the corresponding hypersphere $p \in \{x \in \mathbb{R}^d \mid \|x\| = 1\}$. The Busemann function can then be used to measure a distance between $p$ and any point $z \in \mathcal{H}_d$, or rather a dissimilarity, that tends toward $-\infty$ as $z$ gets closer to $1$-p, and thus be used for classification.

More generally, in a completely geodesic Riemannian manifold $\mathcal{M}$, the Busemann function is defined, for some unit-speed geodesic $\gamma : \mathbb{R} \to \mathcal{M}$, by

$$ B_{\gamma}(z) = \lim_{t \to +\infty} d_{\mathcal{M}}(\gamma(t), z) - t, \quad \forall z \in \mathcal{M}, \qquad (6) $$

where $d_{\mathcal{M}}$ is the Riemannian distance, when such limit exists. In the case of the Euclidean space $\mathbb{R}^d$, one can show that the Busemann function is well defined and is expressed, for a geodesic $\gamma_w : t \mapsto tw$ with $w \in \mathbb{S}^{d-1} := \{u \in \mathbb{R}^d \mid \|u\| = 1\}$, by $B_{\gamma_w} : x \mapsto -\langle w, x \rangle$. Its level sets are hyperplanes. It is therefore natural to consider this function to generalize the Euclidean hyperplanes and the linear features to other manifolds.

In the case of the hyperbolic space, the Busemann function is well defined too. Consider a unit-speed geodesic $\gamma_p(t)$ passing through the origin a time $t = 0$, which can be parametrized by the ideal prototype $p \in \mathbb{S}^{d-1}$ it tends toward$^1$, and whose expression is given by $\gamma_p(t) = \exp_0(t \frac{p}{2})$, $\forall t \in \mathbb{R}$. Then the Busemann function for this geodesic is given by

$$ B_p(x) = \log \left( \frac{\|p - x\|^2}{1 - \|x\|^2} \right), \quad \forall x \in \mathcal{H}_d. \qquad (7) $$

As mentioned earlier, it can simultaneously be seen as a dis-similarity to the ideal prototype $p \in \mathbb{S}^{d-1}$ located at the boundary of the hyperbolic space and as a generalization of the linear feature $x \mapsto \langle x, w \rangle$ in the hyperbolic space, making it an ideal tool for classification on this manifold.

<footnote n="1">¹ For the Euclidean norm of the Poincaré model.</footnote>
The level sets of the Busemann function are called horospheres and are defined, for an ideal prototype $p \in \mathbb{S}^{d-1}$ and a bias term $a \in \mathbb{R}$, by

$$H_{p,a} = \{x \in \mathcal{H}_d | -B_p(x) + a = 0\}. \qquad (8)$$

Therefore, from the previous discussion, they can be seen as a generalization of the Euclidean hyperplanes to the hyperbolic space. They have been considered in Fan et al. (2024) to adapt SVM to the hyperbolic setting. In the Poincaré ball model, their expression is rather simple as they appear as spheres tangent to the boundary of the ball, as stated by the following proposition. Figure 2 illustrates the values of the Busemann function on the Poincaré ball as well as the corresponding horospheres.

**Proposition 1** For $p \in \mathbb{S}^{d-1}$ and $a \in \mathbb{R}$, the horosphere $H_{p,a}$ of the Poincaré ball $\mathcal{H}_d$ is a hypersphere of center $(1-\sigma(a))p$ and radius $\sigma(a)$ where $\sigma$ is the logistic function $\sigma: x \in \mathbb{R} \mapsto \frac{1}{1+e^{-x}}$.

**Proof**

$$-B_p(x) + a = 0 \iff \log \left( \frac{\|p - x\|^2}{1 - \|x\|^2} \right) = a \qquad (9) \\ \iff (e^a + 1) \|x\|^2 - 2\langle p, x \rangle = e^a - 1 \qquad (10)$$

$$\iff \|x\|^2 - 2 \left\langle \frac{p}{e^a + 1}, x \right\rangle + \frac{1}{(e^a + 1)^2} = \frac{e^a - 1}{e^a + 1} + \frac{1}{(e^a + 1)^2} \qquad (11)$$

$$\iff \|x - m\|^2 = R^2 \qquad (12)$$

with $m = \frac{p}{e^a+1} = (1-\sigma(a))p$ and $R^2 = \frac{(e^a-1)(e^a+1)+1}{(e^a+1)^2} = \sigma(a)^2$. □

The distance to a horosphere $H_{p,a}$ can be directly obtained by the Busemann function (see Fan et al. (2024), Proposition 3.1):

$$d(x, H_{p,a}) := \inf_{y \in H_{p,a}} d(x, y) = |B_p(x) - a|, \quad \forall x \in \mathcal{H}_d. \qquad (13)$$

As a consequence, the set of points at some fixed distance $\delta \in \mathbb{R}$ of a horosphere $H_{p,a}$ is again a horosphere. This makes another common point with Euclidean hyperplanes. Moreover, this feature $d(x, H_{p,a})$ results having a very simple expression in the Poincaré model, unlike its gyroplane counterpart Ganea et al. (2018).

<figure><img src="image_4.png" /><figcaption>**Fig. 2** Illustration of the values of the Busemann function on the Poincaré ball and its level sets called horospheres. The Busemann function is here computed relatively to the prototype located in (1, 0) and represented by an orange dot. Level sets corresponding to $B_p(x) = -2, 0, 2$ are highlighted in black. The values of the Busemann function go from $-\infty$, near the prototype, to $+\infty$, far from the prototype</figcaption></figure>

# 4 Hierarchically Informed Horospherical Classification

## 4.1 Horospherical Classifiers

In the article introducing hyperbolic classification based on ideal prototypes Ghadimi Atigh et al. (2021), authors left the prototypes fixed during training, thus giving a lot of importance to the prototype initialisation step. In our work, we propose to fully learn the horospheres $H_{p,a}$ that underlie the decision function by optimizing the prototypes $p$ as well the the corresponding bias terms $a$.

For each horosphere, we define the score function $\xi_{p,a}: \mathcal{H}_d \to \mathbb{R}$, which is then used to perform classification between samples embedded in hyperbolic space:

$$\xi_{p,a}(x) = -B_p(x) + a. \qquad (14)$$

The logit $\xi_{p,a}$ can be used to perform binary classification by computing $P(\hat{y} = 1) = \sigma(\xi_{p,a}(x))$, with $\sigma: \mathbb{R}^d \to [0, 1]$ being the logistic function. In this case, the binary decision boundary would be equivalent to the associated horosphere level-set $H_{p,a}$. Moreover, horospheres can be adapted in a multi-class classification setting by defining an horosphere per class and using a softmax normalization to obtain class membership probabilities. An horospherical layer consists of $K$ horospheres and we define the probability of a sample $x \in \mathcal{H}_d$ with respect to one of the $K$ classes as the softmax
<figure><img src="image_5.png" /><figcaption>**Fig. 3** Illustration of three hyperbolic classification methods on a simple 3-classes toy dataset. The first row depicts Gyroplanes Ganea et al. (2018) (first column), that are arc of circles orthogonal to the boundaries of the Poincaré ball, Busemann prototypes Ghadimi Atigh et al. (2021) (second column), represented in Blue, that are fixed points located at infinity, and our Horospherical classifiers (third column), depicted as circles tangent to the boundary in one infinite prototypical point. The second row illustrates the resulting decision function.</figcaption></figure>

over the K horosphere logits:

$$P(\hat{y} = k | x, p, a) = \frac{\exp(\xi_{p_k, a_k}(x))}{\sum_{l=1}^{K} \exp(\xi_{p_l, a_l}(x))}, \forall k \in \{1..K\}. \quad (15)$$

This horospherical multi-class classifier can be trained using the negative log-likelihood loss over the training data x, y:

$$\mathcal{L}_{p,a}(x, y) = \frac{1}{N} \sum_{i=1}^{N} - \log P(\hat{y} = y_i | x_i, p, a). \quad (16)$$

We can replace the last linear layer of a Euclidean classification fully-connected layer with an exponential map to project in the Poincaré ball followed by an evaluation of the horospherical layer in hyperbolic space. As with Euclidean linear layers, during inference, the model's prediction can be chosen as the class whose predicted probability is highest among all classes.

The position of prototypes is updated during training using Riemannian gradient descent Bonnabel (2013) so that prototypes are able to move on the boundary of the hyperbolic space (i.e. $S^{d-1}$ in the case of the Poincaré ball model).

Our proposed horospherical classifier has similarity with the HoroSVM Fan et al. (2024) method. One of the main differences lies in the optimization of the classification layer. Fan et al. (2024) propose to use the hinge loss and a one-versus-rest scheme for multi-class classification whereas our horospherical classifier is using a softmax normalization combined with a negative log-likelihood loss. Additionally,

in our experiments we focus on the task of building a full image classification (resp. segmentation) pipeline whereas the HoroSVM method is tested only on classification among samples already embedded in hyperbolic space, and is generally not adapted to also learn representations through a neural network backbone in an end-to-end fashion.

## 4.2 Uniformly Distributed Ideal Prototypes

The horospherical classifier presented in Section 4.1 is initialized with a set of ideal prototypes. In order to leverage the embedding space, we initialize the layer with prototypes uniformly distributed around the hypersphere $S^{d-1} = \{x \in \mathbb{R}^d | \|x\| = 1\}$. While uniformly distributing around the circle can be done in a closed-form by simply dividing the circle in K slices, such a simple solution does not exist in higher dimensions. Computing uniform positions for a set of points on the hypersphere can be done in different manners Wang and Isola (2020); Bonet et al. (2023). We chose to optimize a set of K points $\{p_i \in S^{d-1} | i = 1, \dots, K\}$ on the sphere using the uniform loss proposed by Wang and Isola (2020). This uniform loss is based on the pairwise Gaussian potential. It effectively maximises the pairwise distance between each pairs $p_i, p_j$:

$$\mathcal{L}_{\text{unif}}(p) = \log \left( \frac{1}{K(K-1)} \sum_{i \neq j} \exp(-\|p_i - p_j\|^2) \right). \quad (17)$$

After optimization, this set of points can be used as ideal prototypes in the Poincaré ball model. Whereas the Busemann classifier Ghadimi Atigh et al. (2021) randomly assigns
prototypes to class labels, in the following section, we consider the problem of assigning a class label to each prototype in a hierarchically informed manner to improve the performance of the classification model.

## 4.3 Hierarchical Prototypes Assignment

Given a classification task, a label hierarchy encodes the semantic relationship between classes. The hierarchy can be represented as a tree, where ancestors in the tree correspond to super-classes. The depth of the hierarchy corresponds to the longest path from the root to any leaves. Label hierarchies can be built from the semantic and lexical relationships between labels as is the case in the WordNet hierarchy Miller (1995) which is the base for the ImageNet Deng et al. (2009) label hierarchy. In this section, we investigate how to leverage a label hierarchy to position ideal prototypes to minimize the distortion between the prototypes and the tree induced by the hierarchy.

Our goal is to assign a prototype for each leaf node in the label hierarchy. This assignment should be established in such a way that the distance in the tree between the labels should match a distance between the corresponding prototypes. However, labels are placed in the hierarchy tree whereas prototypes are located on the hypersphere. These two spaces are incomparable, in the sense that one can not embed isometrically a tree onto a sphere due to their differing topological and geometric properties. A tree is a discrete, one-dimensional structure with branches, while the hypersphere is a continuous, compact manifold with positive curvature (see Chapter 8, Goodman and O'Rourke (1997) for instance). Moreover, from a practical point of view, any assignment procedure would require to have access to a common metric between the leaves and the prototype, which is not accessible.

Garnot and Landrieu (2020) work around this problem by introducing a scale-free variant of their comparison loss. Instead, in such a context of incomparable spaces, we choose to resort to the Gromov-Wasserstein (GW) distance Mémoli (2011) (see also Section 10.6, Peyré et al. (2017) or Section 2.2, Vayer (2020)). The GW distance is used to compute a transport plan P between samples from two empirical distributions whose supports are embedded in incomparable spaces (i.e. for which a distance measure between the hierarchical labels tree and the prototypes spaces is not available). The GW transport plan is computed by comparing the inner metrics from each distribution. Therefore, the transport plan is based on the inner structure from each distribution instead of on the pairwise distances between samples from each distribution.

Given two empirical distributions $\mu = \sum_{i=1}^{n} a_i \delta_{x_i} \in \mathcal{P}(\mathcal{X})$ and $\nu = \sum_{j=1}^{n'} b_j \delta_{y_j} \in \mathcal{P}(\mathcal{Y})$ over two metric

spaces $(\mathcal{X}, d_{\mathcal{X}})$ and $(\mathcal{Y}, d_{\mathcal{Y}})$ respectively with $a$ and $b$, two simplex histograms of $n$ and $n'$ bins respectively such that $\sum_{i=1}^{n} a_i = 1$ and $\sum_{j=1}^{n'} b_j = 1$. $\delta_{\mathcal{X}}$ is the Dirac measure in $\mathcal{X}$. $M_1$ and $M_2$ are the matrices of pairwise distance between atoms of $\mu$ and $\nu$ respectively. The Gromov-Wasserstein distance between two empirical distributions can be defined as:

$$ GW_{p}^{p}(M_1, M_2, a, b) = \min_{P \in \mathcal{U}(a,b)} \sum_{i,j,k,l} |(M_1)_{i,k} - (M_2)_{j,l}|^p P_{i,j} P_{k,l}, \quad (18) $$

where $U(a, b)$ is the set of linear transport constraints defined by $a$ and $b$. That is, the set of doubly stochastic matrices which respect the marginal constraints $a$ and $b$, $U(a, b) = \{P | P \ge 0, P1_{|a|} = a, P^{\top}1_{|b|} = b\}$.

The Gromov-Wasserstein problem seeks to assign points $(x_i)_i$ to points $(y_j)_j$ so that their inner distances match as much as possible. Its formulation enables source points masses to be split and assigned to several target points, leading to a potential soft assignment $P$. Nonetheless, whenever the two empirical distributions are uniform with the same number of Diracs, and when $M_1$ and $M_2$ are both conditionally negative definite (CND), the optimal coupling $P$ can be chosen as a permutation matrix, i.e. a one to one assignment (see Theorems 2 and 3, Sejourne et al. (2021) Equation (1.2), Dumont et al. (2024), and Proposition 2.1, Peyré et al. (2019) for details). Figure 4 illustrates the Gromov-Wasserstein assignment in the case of distributions on the sphere and on a tree.

Therefore, we place prototypes uniformly on the hypersphere and assign the prototype using the Gromov-Wasserstein transport plan between the tree metric and the metric of distances on the sphere. Given $M_T$ and $M_S$, the distance metrics and the tree and on the sphere respectively, one can compute the optimal transport plan between atoms from $T$ and $S$:

$$ P^* = \underset{P \in \mathcal{U}(a,b)}{\operatorname{argmin}} \sum_{i,j,i',j'} |(M_T)_{i,i'} - (M_S)_{j,j'}|^2 P_{i,j} P_{i',j'} \quad (19) $$

where $a_i = b_i = 1/K$ are uniform marginal distributions for both labels in the hierarchy and prototypes on the sphere respectively.

The tree metric is taken to be the length of the path between two nodes in the hierarchy. Therefore, two leaves $i$ and $j$ with the same parent will have a distance $(M_T)_{i,j} = 2$. For the sphere metric, we use the cosine distance $(M_S)_{i,j} = 1 - \langle p_i, p_j \rangle$ which is an increasing function of the length of the arc between $p_i$ and $p_j$ and $0 \le (M_S)_{i,j} \le 2$.

Since both $M_S$ and $M_T$ have the same size and are CND, the optimal transport plan $P^*$ can be chosen as a permutation matrix which can be used to assign prototypes to nodes in the hierarchy. Therefore, label $i$ will be assigned to prototype $j^*$.
<figure><img src="image_6.png" /><figcaption>**Fig. 4** Illustration of the Gromov-Wasserstein assignment. The assignment is done in order to preserve "as much as possible" the distances. Its formulation enables a point to be split and assigned to several points from the other distribution. But in general, when the numbers of points in each side are equal and that the masses are uniformly distributed, each point is assigned a unique other point.</figcaption></figure>

as such:

$$j^* = \underset{j \in \{1..K\}}{\operatorname{argmax}} P_{i,j}^*. \qquad (20)$$

Additionally, we also consider the simpler case of positioning hierarchical prototypes on the Poincaré disk in two dimensions. Firstly, uniformly distributing points on the circle can be done in a closed form by slicing the circle in arcs of equal length $2\pi/K$ as explained in Section 4.2. The hierarchical structure can also be embedded on the circle without having to compute an optimal transport plan, by using the graph visualization algorithm used in Figure 1 called twopi introduced by Wills Wills (1999). We rely on this method in 2 dimensions. A comparison of the prototype positions yielded by the two schemes can be seen in Figure 5.

In order to visualize the impact of this positioning scheme, we can compare the pairwise distance matrices between the tree hierarchy, uniform prototypes and the prototypes positioned using Gromov-Wasserstein in Figure 1. Before training, the uniformly distributed prototypes do not exhibit any structure in the metric whereas after optimization, the distance between prototypes appear less noisy. Before optimization, the hierarchically positioned prototypes already exhibit a block structure in the spherical metric which by construction has similarities with the tree metric because their relative distortion is minimized.

## 4.4 Multiprototypical Extension

Additionally from the horospherical classifier introduced in section 4.1, we show that it can be extended to the more general setting of having multiple prototypes per class in the dataset. The membership score associated with each class is computed from the mean of scores from each prototypes in the class. When a class is represented by a single prototype, this extension falls back to the method proposed earlier.

Let us first consider the simple setting where we assign $m \in \mathbb{N}$ prototypes to each of the $K$ classes. We initialize $q = mK$ assigned prototypes ($p_{k,i}$) $1 \le k \le K, 1 \le i \le m$ on the

hypersphere, and define the new logits as

$$\xi_k(x) = -\frac{1}{m} \sum_{i=1}^{m} B_{p_{k,i}}(x) + a_k \qquad (21)$$

$$= [AB(x)]_k + a_k \qquad (22)$$

where $B(x)$ is the vector ($B_{p_{k,i}}(x)_{k,i}$ of the Busemann similarities to all the prototypes and $A \in \mathbb{R}^{K \times q}$ is the assignment matrix: for $j = (k', i) \in \{1..K\} \times \{1..m\}$ and $k \in \{1..K\}$, $A_{k,j}$ is $\frac{1}{m}$ if and only if $p_j$ is assigned to class $k$, i.e. iff $k' = k$, and zero otherwise. The remaining question is then how to assign the prototypes to the classes, which is to say how to define this matrix $A$.

The Gromov-Wasserstein transport problem (19) is still well-defined when the support of the two measures have different cardinals. We only need to renormalize the weights $b_i$ of the measure $\nu$ (corresponding to the locations of the prototypes), choosing $b_i = \frac{1}{mK}$. Similarly to the case $m = 1$ from previous section, the solution set of the Gromov-Wasserstein problem includes vertices of the convex polytope $U(a, b)$. So using a reasoning akin to the proof of Birkhoff's theorem Birkhoff (1946), the optimal plan can be chosen so as to guarantee that each prototype is assigned a unique class. The assignment matrix can then simply be defined as $A = KP^*$.

We see from the expression (22) and the formulation of the Gromov-Wasserstein transport problem that it is possible to extend this variant to the case where $q$ is not a multiple of the number $K$ of classes. This implies that the prototypes may no longer be assigned to a single class each, or, said differently, the assignment matrix $A$ may have columns with several positive entries. However, $A$ is guaranteed to be sparse, with $K + q - 1$ or fewer positive entries (see e.g. Proposition 3.4, Peyré et al. (2017)), which means that most of the prototypes will be assigned to a single class and only few prototypes may be assigned to several, generally 2, classes.

To showcase the expressiveness of the multiprototypical model, we illustrate in Figure 6 the decision boundaries on a small toy dataset with 3 classes, in dimension 2, similar to the one from Figure 3. The data points are sampled from
<figure><img src="image_7.png" /><figcaption>**Table 1** Accuracies on image classification tasks. The mean and standard deviation of the accuracies evaluated over 3 runs are reported here. The best metric per setting is reported in bold while the second best is</figcaption></figure>

underlined. Empty cells correspond to cases for which we faced numerical issues

<table><thead><tr><th rowspan="2">Dataset<br>Dimension d</th><th colspan="4">CIFAR100 Krizhevsky et al. (2009)</th></tr><tr><th>2</th><th>3</th><th>4</th><th>50</th></tr></thead><tbody><tr><td>Euclidean</td><td>45.9±0.5</td><td>53.3±0.8</td><td>56.2±0.2</td><td>62.5±0.3</td></tr><tr><td>Hyperbolic Neural NetworksGanea et al. (2018)</td><td>38.7±0.7</td><td>43.9±3.8</td><td>44.5±0.2</td><td>55.9±0.5</td></tr><tr><td>Metric Guided PrototypesGarnot and Landrieu (2020)</td><td>54.2±0.8</td><td>57.9±0.6</td><td>58.9±0.4</td><td>62.4±0.4</td></tr><tr><td>Busemann - Uniform Ghadimi Atigh et al. (2021)</td><td>49.7±0.7</td><td>48.7±2.3</td><td>50.2±1.5</td><td>63.3±0.6</td></tr><tr><td>Busemann - Smart</td><td><strong>55.7±0.5</strong></td><td>54.8±0.4</td><td>55.4±0.3</td><td><strong>63.3±0.3</strong></td></tr><tr><td>Horospherical - Smart (m = 1)</td><td>53.7±0.8</td><td><u>58.7 ± 0.3</u></td><td><strong>60.0±0.1</strong></td><td>60.9±0.1</td></tr><tr><td>Horospherical - Smart (m = 5)</td><td><u>55.7 ± 0.6</u></td><td><strong><u>59.4±0.4</u></strong></td><td><u>59.9 ± 0.5</u></td><td>61.0±0.3</td></tr></tbody><thead><tr><th rowspan="2">Dataset<br>Dimension d</th><th colspan="4">CUB200 Wah et al. (2011)</th></tr><tr><th>2</th><th>3</th><th>4</th><th>50</th></tr></thead><tbody><tr><td>Euclidean</td><td>24.3±1.3</td><td>46.8±0.7</td><td>50.5±1.5</td><td>51.8±0.8</td></tr><tr><td>Hyperbolic Neural NetworksGanea et al. (2018)</td><td>24.4±6.0</td><td>22.3±12.0</td><td>16.9±1.5</td><td>33.6±2.1</td></tr><tr><td>Metric Guided PrototypesGarnot and Landrieu (2020)</td><td>31.0±2.8</td><td>48.8±1.4</td><td>53.2±1.9</td><td><u>57.3 ± 0.7</u></td></tr><tr><td>Busemann - Uniform Ghadimi Atigh et al. (2021)</td><td>27.6±7.4</td><td>40.0±2.0</td><td>45.0±1.7</td><td>49.5±3.7</td></tr><tr><td>Busemann - Smart</td><td><strong>53.3±1.6</strong></td><td>54.3±0.4</td><td><strong>58.0±1.1</strong></td><td>50.4±0.6</td></tr><tr><td>Horospherical - Smart (m = 1)</td><td>38.7±4.3</td><td><u>56.4 ± 1.6</u></td><td>56.3±0.8</td><td>56.6±0.6</td></tr><tr><td>Horospherical - Smart (m = 5)</td><td><u>41.6 ± 8.6</u></td><td><strong><u>57.4±0.7</u></strong></td><td><u>57.3 ± 1.1</u></td><td><strong><u>59.2±0.5</u></strong></td></tr></tbody><thead><tr><th rowspan="2">Dataset<br>Dimension d</th><th colspan="4">Stanford CarsKrause et al. (2013)</th></tr><tr><th>2</th><th>3</th><th>4</th><th>50</th></tr></thead><tbody><tr><td>Euclidean</td><td>40.6±1.7</td><td>72.6±0.8</td><td>78.0±0.6</td><td>82.1±0.3</td></tr><tr><td>Hyperbolic Neural NetworksGanea et al. (2018)</td><td>-</td><td>70.5±0.7</td><td>76.7±2.6</td><td><u>83.7 ± 0.4</u></td></tr><tr><td>Metric Guided PrototypesGarnot and Landrieu (2020)</td><td>67.3±0.7</td><td>77.4±2.5</td><td>80.0±1.5</td><td><strong>84.4±0.6</strong></td></tr><tr><td>Busemann - Uniform Ghadimi Atigh et al. (2021)</td><td><u>76.4 ± 0.6</u></td><td>74.0±0.4</td><td>77.7±0.7</td><td>-</td></tr><tr><td>Busemann - Smart</td><td><strong>80.0±0.7</strong></td><td><u>77.6 ± 0.3</u></td><td>79.7±1.3</td><td>-</td></tr><tr><td>Horospherical - Smart (m = 1)</td><td>75.6±0.7</td><td>77.2±1.5</td><td><u>80.4 ± 0.3</u></td><td>82.4±0.3</td></tr><tr><td>Horospherical - Smart (m = 5)</td><td>76.1±0.7</td><td><strong>80.2±0.6</strong></td><td><strong><u>80.7±0.6</u></strong></td><td>82.9±0.1</td></tr></tbody></table>
<table><caption>**Table 2** Results on semantic segmentation (%mIoU). The mean and standard deviation of the performances evaluated over 3 runs are reported here. The best metric per setting is reported in bold while the second best is underlined.</caption><thead><tr><th rowspan="2">Dataset<br>Dimension</th><th colspan="3">NuScenes Caesar et al. (2020)</th><th colspan="3">Cityscapes Cordts et al. (2016)</th></tr><tr><th>d = 2</th><th>d = 3</th><th>d = 128</th><th>d = 2</th><th>d = 3</th><th>d = 128</th></tr></thead><tbody><tr><td>Euclidean</td><td><u>54.0 ± 2.4</u></td><td><u>67.4 ± 0.3</u></td><td><b>70.5±0.5</b></td><td>35.9 ± 0.0</td><td><u>60.9 ± 4.6</u></td><td><b>78.8±0.4</b></td></tr><tr><td>HIS Ghadimi Atigh et al. (2022)</td><td>40.2±5.4</td><td>58.0±2.7</td><td><u>69.5 ± 0.2</u></td><td><u>41.3 ± 4.8</u></td><td>45.1±7.7</td><td>77.9±0.2</td></tr><tr><td>Horospherical - Smart (m = 1)</td><td><b>68.4±0.1</b></td><td><b>68.7±0.0</b></td><td>69.2±0.3</td><td><b>73.5±0.4</b></td><td><b>76.1±0.1</b></td><td><u>78.2 ± 0.4</u></td></tr></tbody></table>

<table><caption>**Table 3** Ablation study by freezing prototypes on our proposed horospherical method on semantic segmentation. Each result is computed over 1 run. The best metric per setting is reported in bold.</caption><thead><tr><th rowspan="2">Prototypes</th><th rowspan="2">Frozen</th><th colspan="3">NuScenes Caesar et al. (2020)</th><th colspan="3">Cityscapes Cordts et al. (2016)</th></tr><tr><th>d = 2</th><th>d = 3</th><th>d = 128</th><th>d = 2</th><th>d = 3</th><th>d = 128</th></tr></thead><tbody><tr><td>Uniform</td><td>✓</td><td>59.6</td><td>64.9</td><td><b>67.2</b></td><td>63.2</td><td>66.8</td><td><b>76.9</b></td></tr><tr><td>Smart</td><td>✓</td><td><b>64.7</b></td><td><b>65.9</b></td><td>67.1</td><td><b>66.4</b></td><td><b>72.9</b></td><td>74.8</td></tr></tbody></table>

bimodal distributions, with modes potentially far from each other. This illustrates cases where classes are heterogeneous and the classifier struggles to embed points from a same class in a single location. The decision boundaries are depicted for different values of $q = Km$, which are not necessarily multiples of the number $K$ of classes. We see that in this configuration, the single prototype classifier is unable to correctly classify one of the modes of the blue class. Although the classifier with $q = 5$ prototypes still slightly struggles to draw clear decision areas around the samples, it shows an increased flexibility, while the model using 2 prototypes per class tackles the task with success.

# 5 Experiments

## 5.1 Image Classification

To evaluate the performance of our hyperbolic classification method, we first conduct experiments on hierarchical image classification datasets. We choose the CUB-200 Wah et al. (2011) which contains 200 bird species. It is composed of 5994 training images and a test set of 5794 images. Each species belongs to a single family which in turn belongs to an order. This gives us a hierarchy of depth 3 with 252 nodes, which can be seen on Figure 1. We also experiment on the CIFAR100 (Krizhevsky et al., 2009) dataset which contains 100 classes of common visual objects.

Finally, we also test our method on the Stanford Cars dataset Krause et al. (2013) which contains images of cars, including 8144 train samples and 8041 test ones, assigned to 196 classes corresponding to car models, that we manually divide into 9 metaclasses, similarly to Liu et al. (2016): cab, convertible, coupe, hatchback, minivan, sedan, SUV, van and wagon. Those metaclasses serve as intermediary level in the class hierarchy.

We compare the performance of the horospherical classifier with Hyperbolic Busemann learning (Ghadimi Atigh et al., 2021), with either uniformly distributed prototypes and hierarchical prototypes. We also compare them with the Hyperbolic neural networks (Ganea et al., 2018) as well as the Metric Guided Euclidean prototype learning (Garnot & Lanbrieu, 2020). We test different dimensions $d$ for the embedding space. For the CIFAR100, we use an available hierarchy which has 9 levels in its deepest branch. Unlike the CUB-200 and the Stanford Cars hierarchies, the branches do not all have the same length but for ease of use we insert inert nodes where needed to extend branches which are too short to the desired length of the longest branch.

As the backbone, we use a ResNet32 He et al. (2016). Models are trained for 1110 epochs for the CIFAR100 dataset, 2110 for the CUB200 dataset, as done in Ghadimi Atigh et al. (2021), and 2001 for the Stanford Cars dataset. The Riemannian variant Bécigneul and Ganea (2018) of the Adam optimizer Kingma and Ba (2014) implemented in the Geoopt toolbox Kochurov et al. (2020) is used for optimizing the parameters for the Horospherical method. For other methods without trainable parameters on manifolds, we use the Adam optimizer. A curvature of $c = 1.0$ is set for all hyperbolic methods. The Gromov-Wasserstein transport plan for the hierarchical positioning is computed using the Python Optimal Transport toolbox Flamary et al. (2021). All experiments are done with 3 runs and we report the best metric measured during the training in order to limit the impact of the number of epochs between the different methods.

Results can be seen in Table 1. We can see that the performance is improved with hierarchically positioned prototypes and horospherical classifier, especially when the embedding dimension is small such as with $d \le 4$. On its own, the performance is improved for the Busemann method when initialized with hierarchically-positioned prototypes. We also observe that the performance of our proposed method is
increased when using more prototypes for each class compared to a single one.

## 5.2 Image and Point Cloud Segmentation

In this section, we evaluate the proposed method on semantic segmentation tasks, for both 2D image and 3D point cloud datasets. We compare with the Euclidean baseline as well as the Hyperbolic Image Segmentation (HIS) method from Ghadimi Atigh et al. (2022).

For image segmentation, we consider the Cityscapes dataset Cordts et al. (2016) which contains 19 classes divided in 7 super-classes (flat, construction, object, nature, sky, human, and vehicle). We use a DeepLab-v3+ Chen et al. (2018) backbone trained with the official split for 240 epochs. We set $\lambda = 0.5$. For all hyperbolic methods, c is set to 1.0. Experiments are done with 3 runs and results can be observed in Table 2.

For point cloud segmentation, we perform experiments on the NuScenes Caesar et al. (2020) dataset which contains 40k frames, sampled in Boston and Singapore, with a rotating LiDAR. Points are classified into 16 semantic classes and 1 ignore class. We use the official train split for training, and report the results obtained on the publicly available validation set. For the class hierarchy we leverage the NuScenes class description and split into 5 branches (movable object, vehicle, pedestrian, flat and static). We additionally split the vehicle branch into 4-wheeled and 2-wheeled. This leads to a hierarchy with 3 layers. As a point cloud processing backbone, we use the commonly used sparse-voxel Minkowski U-Net Choy et al. (2019), with a 10 cm voxel size. We set $\lambda = 0.1$. Results can be seen in Table 2.

In low embedding dimensions ($d = 2, 3$) our horospherical prototype method outperforms both the Euclidean and the HIS classifiers for both dense semantic segmentation tasks. Our classifier closes or significantly minimizes the performance gap between low and high embedding dimensions. For the NuScenes dataset we also investigated an increased number of prototypes, but $m = 1$ was already sufficient to obtain a maximal performance. In high dimensional embeddings ($d = 128$) the Euclidean classifier is then on par, even slightly outperforming both classifier relying on hyperbolic representations.

## 5.3 Ablation Study

In order to evaluate the impact of the prototype positions during training, we compare the performance with freezing them during training. This means that the last layer is, akin to Ghadimi Atigh et al. (2021), parameter-less. In this setting, we expect that the initial prototypes positions will have a larger impact than when the prototypes are trained.

<table><caption>**Table 4** Ablation study by changing the initialization scheme between uniformly distributed and smart prototypes for our proposed horospherical classifier. The mean and standard deviation of the performances evaluated over 3 runs are reported here. The best metric per setting is reported in bold.</caption><thead><tr><th rowspan="2">Dataset<br>Dimension<br>Metric</th><th rowspan="2">d = 2<br>%Acc</th><th colspan="3">CUB200 Wah et al. (2011)</th><th colspan="3">NuScenes Caesar et al. (2020)</th><th colspan="3">Cityscapes Cordts et al. (2016)</th></tr><tr><th>d = 3<br>%Acc</th><th>d = 4<br>%Acc</th><th>d = 50<br>%Acc</th><th>d = 2<br>%mIoU</th><th>d = 3<br>%mIoU</th><th>d = 128<br>%mIoU</th><th>d = 2<br>%mIoU</th><th>d = 3<br>%mIoU</th><th>d = 128<br>%mIoU</th></tr></thead><tbody><tr><td>Horospherical - Uniform</td><td>21.5±3.3</td><td>43.2±2.4</td><td>46.3±1.9</td><td>55.9±0.8</td><td>67.4</td><td><strong>68.9</strong></td><td>68.8</td><td>73.3</td><td>75.7</td><td>78.3</td></tr><tr><td>Horospherical - Smart</td><td><strong>38.7±4.3</strong></td><td><strong>56.4±1.6</strong></td><td><strong>56.3±0.8</strong></td><td><strong>56.6±0.6</strong></td><td><strong>68.4</strong></td><td>68.7</td><td><strong>69.2</strong></td><td><strong>73.8</strong></td><td><strong>76.0</strong></td><td><strong>78.6</strong></td></tr></tbody></table>
<table><caption>**Table 5** Accuracies on image classification tasks, with varying number *m* of prototypes per class. The mean and standard deviation of the accuracies evaluated over 3 runs are reported here. The best metric per setting is reported in bold</caption><thead><tr><th rowspan="2">Dataset<br>Dimension</th><th colspan="3">CIFAR100 Krizhevsky et al. (2009)</th></tr><tr><th>d = 2</th><th>d = 3</th><th>d = 50</th></tr></thead><tbody><tr><td>m = 1</td><td>53.7±0.8</td><td>58.7±0.3</td><td>60.9±0.1</td></tr><tr><td>m = 1.5</td><td>54.9±0.4</td><td>58.8±0.3</td><td>60.5±0.3</td></tr><tr><td>m = 2</td><td>54.7±0.4</td><td>58.9±0.5</td><td><b>61.0±0.3</b></td></tr><tr><td>m = 3</td><td>55.1±0.6</td><td><b>59.5±0.2</b></td><td><b>61.0±0.1</b></td></tr><tr><td>m = 5</td><td><b>55.7±0.6</b></td><td>59.4±0.2</td><td><b>61.0±0.3</b></td></tr></tbody></table><table><thead><tr><th rowspan="2">Dataset<br>Dimension</th><th colspan="3">CUB200 Wah et al. (2011)</th></tr><tr><th>d = 2</th><th>d = 3</th><th>d = 50</th></tr></thead><tbody><tr><td>m = 1</td><td>38.7±4.3</td><td>56.4±1.6</td><td>56.6±0.6</td></tr><tr><td>m = 1.5</td><td>34.3±5.0</td><td>56.8±0.5</td><td>56.4±0.5</td></tr><tr><td>m = 2</td><td>43.1±4.3</td><td>56.8±0.2</td><td>58.3±0.3</td></tr><tr><td>m = 3</td><td><b>45.5±5.5</b></td><td>56.7±0.4</td><td>58.4±0.5</td></tr><tr><td>m = 5</td><td>41.6±8.6</td><td><b>57.4±0.7</b></td><td><b>59.2±0.5</b></td></tr></tbody></table><table><thead><tr><th rowspan="2">Dataset<br>Dimension</th><th colspan="3">Stanford Cars Krause et al. (2013)</th></tr><tr><th>d = 2</th><th>d = 3</th><th>d = 50</th></tr></thead><tbody><tr><td>m = 1</td><td>75.6±0.7</td><td>77.2±1.5</td><td>82.4±0.3</td></tr><tr><td>m = 1.5</td><td><b>77.8±0.5</b></td><td>79.4±0.6</td><td>81.7±0.7</td></tr><tr><td>m = 2</td><td>76.2±1.4</td><td><b>80.4±0.1</b></td><td>82.7±0.6</td></tr><tr><td>m = 3</td><td>75.4±0.5</td><td>79.7±1.1</td><td><b>83.3±0.6</b></td></tr><tr><td>m = 5</td><td>76.1±0.7</td><td>80.2±0.6</td><td>82.9±0.1</td></tr></tbody></table>

This training setup offers similarities with Busemann learning Ghadimi Atigh et al. (2021) which also has frozen prototypes. The horospherical layer is different because it uses probabilities computed with Equation 15 along with the cross-entropy loss defined in Equation 16. Results from the ablation study can be seen in Table 3. We can see that the initialization has more impact when the representation dimension is small. We also compare the training performance of initializing our proposed horospherical classifier with uniformly distributed prototypes instead of hierarchically positioned prototypes. Results can be seen in Table 4. In this case, we can see that hierarchical prototypes perform better than uniformly distributed prototypes.

We finally study the impact of the number of prototypes in the multiprototypical variant from section 4.4. These ablation experiments are performed over the same image classification datasets, CIFAR100, CUB200 and Stanford Cars. The number *m* of prototypes per class is tested with several values as well as the final embedding dimension *d* ∈ {2, 3, 50}. Performances are reported in Table 5.

Using several prototypes per class is clearly beneficial as the cases with *m* > 1 outperform almost always the case with *m* = 1, or perform equally well. This can be interpreted

as the fact it helps tackle complex and heterogeneous data among a class, as seen in Figure 6. Figure 7 provides a qualitative example supporting this interpretation. It represents the prototypes and the embeddings of the validation set of the mushroom class from dataset CIFAR100 after training, in dimension 2 with 2 prototypes per class. The prototypes are far away from each other. We observe that one of them has been trained in an attempt to capture the dotted mushrooms that were being confused with leopards. This is a reasonable behaviour as mushrooms can indeed show a great diversity of shapes, colors and textures. However this case must be analysed cautiously as, first, the dotted mushrooms have been misclassified despite the presence of the prototype nearby, and, second, as this case is rather an exception. In general, it was hard for us to interpret the meaning and the roles of the different prototypes of a same class.

Moreover, in many cases, the prototypes of a same class end up very close after training, what we refer to as "merging". Table 6 shows the proportion of classes from CIFAR100 whose prototypes have not merged, for different angle thresholds (0.1°, 1°, 10°), in the two-dimensional setting with 5 prototypes. This shows that the prototypes tend to merge more easily as the dimension increases. This is consistent with the fact that the improvement in performances brought by the multiprototypical method decreases with the dimension, as observed in Table 5 for CIFAR100. Such observations are also dataset dependent. The Stanford Cars dataset, for instance, exhibits higher rates of merging.

Overall it is still not very clear under which conditions the multiprototypical extension adds a significant benefit, and how the optimal number of prototypes is determined. This is all the more true as the multiprototypical experiments we made on the segmentation tasks resulted being inconclusive. We leave this as future work.

# 6 Discussion

The different results from both image classification and dense segmentation tasks, exhibit a similar pattern. In lower dimensions ($d \le 4$), our method improves upon other hyperbolic baselines. The improvements can be attributed to smart prototypes when the hierarchy is meaningful such as in the CUB Wah et al. (2011). This is confirmed by the fact the Busemann classifier method scores quite well too in the lower dimensions when equipped with such prototypes.

In higher dimensions, hyperbolic methods remain competitive although they do not perform best anymore. This is a known limitation of methods leveraging hyperbolic geometry for deep learning (Ghadimi Atigh et al., 2022). Still, in high dimensions, our horospherical classifier with smart prototypes performs best among hyperbolic methods, making it
<figure><img src="image_8.png" /><figcaption>**Fig. 6** Illustration of the multiprototypical horospherical method on a toy dataset with 3 bimodal classes in dimension 2, for different numbers $m \in \{1, \frac{5}{3}, 2\}$ of prototypes per class, which is to say for a varying total number $q = Km$ of prototypes. The positions of the prototypes and their corresponding biases are optimized. In the case $q = 5$, second column, 2 prototypes are assigned to two different classes: they are represented with two colors, proportionally to the assignment weights</figcaption></figure>

corresponding to those classes (two third for one class and one third for the other). In our method, this multi-assignment is determined before the training process, through the GW assignment, based the hierarchical relations between classes. In the case $q = 6$, the two red prototypes are located at the same place, hence the impression that there is only one of it. The first row shows the 0-level set for the logits $\xi_k$ of each class. The second row displays the resulting decision boundaries for each class.

<figure><img src="image_9.png" /><figcaption>**Fig. 7** Embeddings of the mushroom images from the CIFAR100 dataset, and their corresponding prototypes, in the case $d = 2$ and $m = 2$. The prototypes are far away from each other (left subfigure). We see on the zoom picture (right) that one of them "tried" to capture the dotted mushrooms that are confused with leopards. It is an example of how the multiprototype method can be used to tackle heterogeneous</figcaption></figure>

<figure><img src="image_10.png" /></figure>

classes (here mushrooms, that can have very different aspects). Note that in this case, the dotted mushrooms have been nonetheless misclassified, like all the points scattered on the Poincaré disk far away from the prototypes (which means that we are in a case where the model performed poorly).
<table><caption>**Table 6** Proportion of classes having two prototypes at least distant from some given angles (0.1°, 1°, 10°), after optimization, for CIFAR100 with 5 prototypes. We observe that when the dimension increases, the prototypes from a same class tend to merge.</caption><thead><tr><th></th><th>&gt; 0.1°</th><th>&gt; 1°</th><th>&gt; 10°</th></tr></thead><tbody><tr><td>d = 2</td><td>76%</td><td>55%</td><td>36%</td></tr><tr><td>d = 3</td><td>59%</td><td>29%</td><td>4.7%</td></tr><tr><td>d = 50</td><td>7.0%</td><td>2.0%</td><td>0.0%</td></tr></tbody></table>

a method of choice among methods for classification within the hyperbolic space.

The use of the hyperbolic space as embedding space introduces a greater computational complexity, although this can be mitigating by efficient implementations of the exponential map Kochurov et al. (2020). While this overhead is justified by the significant performance gains in low dimensions, the previous discussion suggests that it may no longer be justified in higher dimensions. Future works could therefore focus on scaling our classifier to those higher dimensions.

## 7 Conclusion

Our model is built upon prototypical learning with ideal prototypes. We introduce several key contributions. First, we optimize the position of the prototypes during training using a Riemanannian gradient descent. We further leverage the embedding space by performing a hierarchical initial prototype assignment, aligning them with the class taxonomy. Finally, to further improve classifier flexibility, we propose a multiprototypical extension that allows multiple prototypes per class. By jointly optimizing hyperbolic embeddings, multiple prototypes per class, and the classifier, we empirically demonstrate that our approach outperforms state-of-the-art Euclidean and hyperbolic classification methods, particularly in low-dimensional embeddings.

While in low dimensions, our method can achieve indeed much better performances than those based on Euclidean representations, this performance gap gets smaller, or even vanishes, in higher dimensions. Future works could therefore focus on exploring how to maintain this performance advantage in higher dimensions so as to justify the greater computational complexity introduced by the projection in the hyperbolic space. Another future direction could be to investigate natural extensions of the prototypical method, such as a free positioning of the prototypes within the hyperbolic space instead of forcing them to live on the boundary.

**Supplementary Information** The online version contains supplementary material available at https://doi.org/10.1007/s11263-025-02571-8.

**Funding** Open access funding provided by Université de Bretagne Sud. This work is funded by the ANR AI chair OTTOPIA under reference ANR-20-CHIA-0030. It was granted access to the HPC resources of IDRIS under the allocations 2023-AD011013839R1 and 2024-AD011013514R2 made by GENCI.

**Data Availability** All datasets used for experiments in this paper (CIFAR100 Krizhevsky et al. (2009), CUB200 Wah et al. (2011), Stanford Cars Krause et al. (2013), NuScenes Caesar et al. (2020) and Cityscapes Cordts et al. (2016)) are publicly available in the internet. CIFAR100 Krizhevsky et al. (2009) is available at https://www.cs.toronto.edu/~kriz/cifar.html. CUB200 Wah et al. (2011) is available at https://www.vision.caltech.edu/datasets/cub_200_2011/. Stanford Cars Krause et al. (2013) is available at https://ai.stanford.edu/~jkrause/cars/car_dataset.html or https://www.kaggle.com/code/subhangaupadhaya/pytorch-stanfordcars-classification/input?select=cars_test_annos_withlabels+. NuScenes Caesar et al. (2020) is available at https://www.nuscenes.org/nuscenes. Cityscapes Cordts et al. (2016) is available at https://www.cityscapes-dataset.com.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## References

* Allen, K., Shelhamer, E., Shin, H., & Tenenbaum, J. (2019). Infinite Mixture Prototypes for Few-shot Learning. In: Proceedings of the 36th International Conference on Machine Learning, pp. 232–241. PMLR, ???. ISSN: 2640-3498. https://proceedings.mlr.press/v97/allen19b.html Accessed 2024-10-17

* Balcan, M.-F., Blum, A., & Srebro, N. (2008). A theory of learning with similarity functions. Machine Learning, 72(1), 89–112. https://doi.org/10.1007/s10994-008-5059-5. Accessed 2024-05-27.

* Bécigneul, G., & Ganea, O.-E. (2018). Riemannian adaptive optimization methods. arXiv preprint arXiv:1810.00760

* Bertinetto, L., Mueller, R., Tertikas, K., Samangooei, S., & Lord, N.A. (2020). Making better mistakes: Leveraging class hierarchies with deep networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (pp. 12506–12515)

* Birkhoff, G. (1946). Tres observaciones sobre el algebra lineal. Univ. Nac. Tucuman, Ser. A 5, 147–154

* Bonet, C., Berg, P., Courty, N., Septier, F., Drumetz, L., & Pham, M.-T. (2023). Spherical sliced-wasserstein. In: International Conference on Learning Representations

* Bonnabel, S. (2013). Stochastic gradient descent on riemannian manifolds. IEEE Transactions on Automatic Control, 58(9), 2217–2229.

* Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., & Vandergheynst, P. (2017). Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34(4), 18–42.

* Caesar, H., Bankiti, V., Lang, A.H., Vora, S., Liong, V.E., Xu, Q., Krishnan, A., Pan, Y., Baldan, G., & Beijbom, O. (2020). nuScenes: A
multimodal dataset for autonomous driving. In: 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)

Caron, M., Misra, I., Mairal, J., Goyal, P., Bojanowski, P., & Joulin, A. (2020). Unsupervised learning of visual features by contrasting cluster assignments. *Advances in Neural Information Processing Systems*, 33, 9912–9924.

Chang, D., Pang, K., Zheng, Y., Ma, Z., Song, Y.-Z., & Guo, J. (2021). Your" flamingo" is my" bird" fine-grained, or not. In: *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*, (pp. 11476–11485).

Chen, B., Bécigneul, G., Ganea, O.-E., Barzilay, R., & Jaakkola, T. (2021). Optimal Transport Graph Neural Networks. arXiv. arXiv:2006.04804 [cs, stat] version: 6. https://doi.org/10.48550/arXiv:2006.04804. http://arxiv.org/abs/2006.04804 Accessed 2024-06-11

Chen, T., Kornblith, S., Norouzi, M., & Hinton, G. (2020). A simple framework for contrastive learning of visual representations. In: International Conference on Machine Learning, pp. 1597–1607. PMLR

Chen, L.-C., Zhu, Y., Papandreou, G., Schroff, F., & Adam, H. (2018). Encoder-decoder with atrous separable convolution for semantic image segmentation. In: Proceedings of the European Conference on Computer Vision (ECCV), (pp. 801–818)

Chen, S., Cowan, C. F. N., & Grant, P. M. (1991). Orthogonal least squares learning algorithm for radial basis function networks. *IEEE Transactions on Neural Networks*, 2(2), 302–309. https://doi.org/10.1109/72.80341. Conference Name: IEEE Transactions on Neural Networks. Accessed 2024-06-13.

Cho, H., DeMeo, B., Peng, J., & Berger, B. (2019). Large-margin classification in hyperbolic space. In: The 22nd International Conference on Artificial Intelligence and Statistics, pp. 1832–1840. PMLR

Choy, C., Gwak, J., & Savarese, S. (2019). 4d spatio-temporal convnets: Minkowski convolutional neural networks. In: CVPR

Cordts, M., Omran, M., Ramos, S., Rehfeld, T., Enzweiler, M., Benenson, R., Franke, U., Roth, S., & Schiele, B. (2016). The cityscapes dataset for semantic urban scene understanding. In *Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*, (pp. 3213–3223)

Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., & Fei-Fei, L. (2009). Imagenet: A large-scale hierarchical image database. In: 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 248–255. Ieee

Dhall, A., Makarova, A., Ganea, O., Pavllo, D., Greeff, M., & Krause, A. (2020). Hierarchical image classification using entailment cone embeddings. In *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops*, (pp. 836–837).

Dhingra, B., Shallue, C.J., Norouzi, M., Dai, A.M., & Dahl, G.E. (2018). Embedding text in hyperbolic spaces. arXiv preprint arXiv:1806.04313

Duin, R. P. W., & Pekalska, E. (2012). The dissimilarity space: Bridging structural and statistical pattern recognition. *Pattern Recognition Letters*, 33(7), 826–832. https://doi.org/10.1016/j.patrec.2011.04.019. Accessed 2024-06-18.

Dumont, T., Lacombe, T., & Vialard, F.-X. (2024). On the Existence of Monge Maps for the Gromov–Wasserstein Problem. Foundations of Computational Mathematics, 1–48 https://doi.org/10.1007/s10208-024-09643-0 . Company: Springer Distributor: Springer Institution: Springer Label: Springer Publisher: Springer US. Accessed 2025-02-10

Ermolov, A., Mirvakhabova, L., Khrulkov, V., Sebe, N., & Oseledets, I. (2022). Hyperbolic vision transformers: Combining improvements in metric learning. In *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*, (pp. 7409–7419)

Fan, X., Yang, C.-H., & Vemuri, B. (2024). Horospherical decision boundaries for large margin classification in hyperbolic space. *Advances in Neural Information Processing Systems* 36

Flamary, R., Courty, N., Gramfort, A., Alaya, M. Z., Boisbunon, A., Chambon, S., Chapel, L., Corenflos, A., Fatras, K., & Fournier, N. (2021). Pot: Python optimal transport. *The Journal of Machine Learning Research*, 22(1), 3571–3578.

Ganea, O., Bécigneul, G., & Hofmann, T. (2018). Hyperbolic entailment cones for learning hierarchical embeddings. In *International Conference on Machine Learning*, (pp. 1646–1655). PMLR

Ganea, O., Bécigneul, G., & Hofmann, T. (2018). Hyperbolic neural networks. Advances in neural information processing systems 31

Garnot, V.S.F., & Landrieu, L. (2020). Leveraging class hierarchies with metric-guided prototype learning. arXiv preprint arXiv:2007.03047

Ge, S., Mishra, S., Kornblith, S., Li, C.-L., & Jacobs, D. (2023). Hyperbolic contrastive learning for visual representations beyond objects. In *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*, (pp. 6840–6849).

Ghadimi Atigh, M., Keller-Ressel, M., & Mettes, P. (2021). Hyperbolic busemann learning with ideal prototypes. *Advances in Neural Information Processing Systems*, 34, 103–115.

Ghadimi Atigh, M., Schoep, J., Acar, E., Van Noord, N., & Mettes, P. (2022). Hyperbolic image segmentation. In *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*, (pp. 4453–4462).

Goodman, J. E., & O'Rourke, J. (Eds.). (1997). *Handbook of Discrete and Computational Geometry*. USA: CRC Press Inc.

Hamzaoui, M., Chapel, L., Pham, M.-T., & Lefèvre, S. (2022). A hierarchical prototypical network for few-shot remote sensing scene classification. In *International Conference on Pattern Recognition and Artificial Intelligence*, (pp. 208–220). Springer

He, K., Zhang, X., Ren, S., & Sun, J. (2016). Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 770–778

Hou, L., Yu, C.-P., & Samaras, D. (2016). Squared earth mover's distance-based loss for training deep neural networks. arXiv preprint arXiv:1611.05916

Khrulkov, V., Mirvakhabova, L., Ustinova, E., Oseledets, I., & Lempitsky, V. (2020). Hyperbolic image embeddings. In *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*, (pp. 6418–6428).

Kingma, D.P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980

Kochurov, M., Karimov, R., & Kozlukov, S. (2020). Geoopt: Riemannian Optimization in PyTorch

Krause, J., Stark, M., Deng, J., & Fei-Fei, L. (2013). 3d object representations for fine-grained categorization. In: 2013 IEEE International Conference on Computer Vision Workshops, pp. 554–561. IEEE, ??? https://openaccess.thecvf.com/content_iccv_workshops_2013/W19/papers/Krause_3D Object Representations_2013_ICCV_paper.pdf Accessed 2025-02-10

Krizhevsky, A., Hinton, G., et al. (2009). Learning multiple layers of features from tiny images

Liu, S., Chen, J., Pan, L., Ngo, C.-W., Chua, T.-S., & Jiang, Y.-G. (2020). Hyperbolic visual embedding learning for zero-shot recognition. In *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*, (pp. 9273–9281).

Liu, M., Yu, C., Ling, H., & Lei, J. (2016). Hierarchical Joint CNN-Based Models for Fine-Grained Cars Recognition. In *Sun, X., Liu, A., Chao, H.-C., Bertino, E. (eds.) Cloud Computing and Security*, (pp. 337–347). Springer, Cham. https://doi.org/10.1007/978-3-319-48674-1_30

Long, T., Mettes, P., Shen, H.T., & Snoek, C.G. (2020). Searching for actions on the hyperbole. In *Proceedings of the IEEE/CVF
*Conference on Computer Vision and Pattern Recognition*, (pp. 1141–1150).

López, F., Pozzetti, B., Trettel, S., Strube, M., & Wienhard, A. (2021). Symmetric spaces for graph embeddings: A finsler-riemannian approach. In: Meila, M., Zhang, T. (eds.) Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event. Proceedings of Machine Learning Research, vol. 139, (pp. 7090–7101).

Mémoli, F. (2011). Gromov-wasserstein distances and the metric approach to object matching. *Foundations of computational mathematics*, 11, 417–487.

Mettes, P., Pol, E., & Snoek, C. (2019). Hyperspherical prototype networks. Advances in neural information processing systems 32

Miller, G. A. (1995). Wordnet: a lexical database for english. *Communications of the ACM*, 38(11), 39–41.

Nickel, M., & Kiela, D. (2017). Poincaré embeddings for learning hierarchical representations. Advances in neural information processing systems 30

Peng, W., Varanka, T., Mostafa, A., Shi, H., & Zhao, G. (2021). Hyperbolic deep neural networks: A survey. *IEEE Transactions on pattern analysis and machine intelligence*, 44(12), 10023–10044.

Pennec, X. (2020). 3-manifold-valued image processing with spd matrices. In: Pennec, X., Sommer, S., Fletcher, T. (eds.) Riemannian Geometric Statistics in Medical Image Analysis, pp. 75–134. Academic Press, ???

Peyré, G., & Cuturi, M. (2019). Computational optimal transport: With applications to data science. *Foundations and Trends® in Machine Learning* 11(5-6), 355–607

Peyré, G., Cuturi, M., et al. (2017). Computational optimal transport. Center for Research in Economics and Statistics Working Papers (2017-86)

Sarkar, R. (2011). Low distortion delaunay embedding of trees in hyperbolic plane. In: International Symposium on Graph Drawing, (pp. 355–366). Springer

Sejourne, T., Vialard, F.-X., & Peyré, G. (2021). The unbalanced gromov wasserstein distance: Conic formulation and relaxation. In: Ranzato, M., Beygelzimer, A., Dauphin, Y., Liang, P.S., Vaughan, J.W. (eds.) Advances in Neural Information Processing Systems, vol. 34, pp. 8766–8779. Curran Associates, Inc., ??? . https://proceedings.neurips.cc/paper_files/paper/2021/file/4990974d150d0de5e6e15a1454fe6b0f-Paper.pdf

Shimizu, R., Mukuta, Y., & Harada, T. (2020). Hyperbolic neural networks++. arXiv preprint arXiv: 2006.08210

Snell, J., Swersky, K., & Zemel, R. (2017). Prototypical networks for few-shot learning. *Advances in neural information processing systems* 30

Spengler, M., Berkhout, E., & Mettes, P. (2023). Poincar\'e resnet. arXiv preprint arXiv:2303.14027

Surís, D., Liu, R., & Vondrick, C. (2021). Learning the predictability of the future. In *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*, (pp. 12607–12617)

Tifrea, A., Bécigneul, G., & Ganea, O.-E. (2018). Poincar\'e glove: Hyperbolic word embeddings. arXiv preprint arXiv:1810.06546

Vayer, T.: A contribution to Optimal Transport on incomparable spaces. phdthesis, Université de Bretagne Sud (Lorient Vannes) (UBS), FRA. (November 2020). https://hal.science/tel-03102632
Accessed 2025-02-20

Vincent-Cuaz, C., Flamary, R., Corneli, M., Vayer, T.,& Courty, N. (2022). Template based graph neural network with optimal transport distances. In *Advances in Neural Information Processing Systems*, vol. 35, (pp. 11800–11814).

Wah, C., Branson, S., Welinder, P., Perona, P., & Belongie, S. (2011). The caltech-ucsd birds-200-2011 dataset

Wang, T., & Isola, P. (2020). Understanding contrastive representation learning through alignment and uniformity on the hypersphere. In *International Conference on Machine Learning*, (pp. 9929–9939). PMLR

Wills, G. J. (1999). Nicheworks—interactive visualization of very large graphs. *Journal of computational and Graphical Statistics*, 8(2), 190–212.

Yu, Z., Nguyen, T., Gal, Y., Ju, L., Chandra, S. S., Zhang, L., Bonnington, P., Mar, V., Wang, Z., & Ge, Z. (2022). Skin lesion recognition with class-hierarchy regularized hyperbolic embeddings. In L. Wang, Q. Dou, P. T. Fletcher, S. Speidel, & S. Li (Eds.), *Medical Image Computing and Computer Assisted Intervention - MICCAI 2022* (pp. 594–603). Cham: Springer.

**Publisher's Note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.