TL;DR
Cartan networks introduce a novel hyperbolic deep learning framework leveraging the group structure of hyperbolic spaces, enabling efficient hierarchical data embeddings and promising benchmark performance.
Contribution
This work presents Cartan networks, a new class of hyperbolic deep learning algorithms that integrate Lie group homomorphisms with metric-preserving diffeomorphisms, exploiting the solvable group structure of hyperbolic spaces.
Findings
Promising results on benchmark datasets
New class of hyperbolic deep learning architectures
Leveraging group-theoretic properties of hyperbolic spaces
Abstract
Hyperbolic deep learning leverages the metric properties of hyperbolic spaces to develop efficient and informative embeddings of hierarchical data. Here, we focus on the solvable group structure of hyperbolic spaces, which follows naturally from their construction as symmetric spaces. This dual nature of Lie group and Riemannian manifold allows us to propose a new class of hyperbolic deep learning algorithms where group homomorphisms are interleaved with metric-preserving diffeomorphisms. The resulting algorithms, which we call Cartan networks, show promising results on various benchmark data sets and open the way to a novel class of hyperbolic deep learning architectures.
Peer Reviews
Decision·Submitted to ICLR 2026
1. The proposed Cartan networks posses intrinsic Hyperbolic architecture which avoids the need for tangent-space approximations (as in Poincaré networks), making computations geometrically consistent and potentially more expressive. 2. Cartan networks allow stacking of homomorphisms and isometries, giving architectural flexibility and they are compatible with standard deep learning techniques (convolutions, batch normalization, dropout). 3. Cartan networks outperform Euclidean and standard hyp
1. The proposed architectures seem complex and require careful handling of solvable coordinates, fiber rotations, and Lie group operations, which may be non-trivial to implement. Also, it is not clear if the the proposed architecture can be applied to large datasets and extended to larger architectures. 2. Although parameter-efficient compared to some Euclidean layers, the matrix multiplications and exponential/fiber rotations could introduce extra computational cost. 3. The use of solvable g
1. The proposed hyperbolic linear and regression layers based on group isomorphisms/homorphisms and series of solvable groups is interesting 2. The proposed layers are defined by operations that are rather intrinsic, which is a potential useful way to improve hyperbolic neural network operations 3. Experiments show incorporating activation in the model leads to improvement performance, which is consistent with the theory presented in the paper
1. The experiments show more or less comparable performance against the Euclidean and Poincare model, with the few cases of noticeable improvements being synthetic datasets. This shows relatively weak motivation for the need for the model and whether the proposed method is actually effective 2. The baseline Poincare model is rather naive and simple. Since Ganea et al., 2018, many works have proposed hyperbolic linear and activation layers that lead to significant improvements (e.g. Chen et al. 2
- The text of the paper seems well written. - If the new formulation is correct, then it could be an interesting new perspective on hyperbolic learning. - Again, if the formulation is correct, then automatically being able to generalize this kind of learning to any non-compact symmetric space could be very interesting as well.
I want to start by saying that I do not have a sufficiently strong background in Lie group theory for understanding the theoretical part of this paper, which spans most of it. As a result, I am incapable of providing any meaningful review of the theoretical part of the text. I do have significant experience with hyperbolic representation learning and computer vision, so I believe that I can judge the experimental part of this paper. There are, in my opinion, a few weaknesses to point out in th
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