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Cartan Networks: Group theoretical Hyperbolic Deep Learning

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proposes Cartan networks, hyperbolic neural networks whose layers are solvable-Lie-group homomorphisms composed with isometries, and shows they match Euclidean and Poincaré baselines on benchmarks.

desk verdict Promising group-theoretic framing undercut by a broken core formula and test-set selection; Theorem 3.1 is the solid part. read the letter →

arxiv 2505.24353 v1 pith:E6TJG3LQ submitted 2025-05-30 cs.LG

Federico Milanesio , Matteo Santoro , Pietro G. Fré , Guido Sanguinetti This is my paper
classification cs.LG
keywords hyperbolicdeeplearningLiegroupssymmetricspacessolvableCartannetworksgrouphomomorphismsisometriesneural
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to build neural networks on hyperbolic geometry by exploiting the fact that hyperbolic space is not only a Riemannian manifold but also a solvable Lie group. The authors propose Cartan networks, in which each layer transforms a solvable group into another by a group homomorphism followed by an isometry, and they supply an explicit coordinate formula for the hyperbolic case. Their central claim is that these intrinsically geometric layers become more expressive as depth increases, even without pointwise activation functions, because the group-theoretic exponential and logarithmic maps introduce nonlinearity. On synthetic regression tasks and image classification benchmarks, the authors report that Cartan networks match or beat comparable Euclidean and Poincaré-ball networks when enough depth and width are used. A reader should care because the construction suggests a general recipe for deep learning on any non-compact symmetric space, not just hyperbolic space.

What carries the argument

The central object is the solvable Lie group structure of hyperbolic space. A non-compact symmetric space like M[1,1+q] ≃ $H^{{q+1}}$ is metrically equivalent to a solvable Lie group Exp(Solv), whose elements are represented by upper-triangular matrices and parametrized by solvable coordinates Υ=(Υ1, Υ2). Υ1 is the Cartan coordinate and Υ2 the paint coordinates; the group operation is matrix multiplication, with the explicit law Ψ∗Υ=(Υ1+Ψ1, Υ2+$e^{{-Υ1}}$Ψ2). The machinery is the alternating composition in each layer of a group homomorphism h(W,b)(Υ)=(Υ1, WΥ2+(1-$e^{{-Υ1}}$)b), which changes the dimension of the manifold, with an isometry φ(β,u,Q)=Ru(β∗(QΥ)), where Ru is the fiber rotation that mixes Cartan and paint coordinates. This composition carries the expressive power of the network: the fiber rotation and the next layer's exponentiation introduce intrinsic nonlinearity, and the authors show that stacking layers increases expressivity even without pointwise activations such as their diffeomorphic DiLU.

What would settle it

Evaluate the first line of Eq. 11 at Υ=(0,0) and u=(±1,0,...,0): the bracket inside the logarithm becomes -1, producing ln(-1), a non-real number, so the claimed real isometry fails at the identity for those parameters. A corrected expression that is real at the identity, or a numerical check that training never visits these parameters, would settle the issue.

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Extended reading notes

Core claim

The paper's discovery is a parametric family of hyperbolic neural network layers built entirely from two geometric ingredient classes: homomorphisms between solvable Lie groups and isometries of the target space. Concretely, the hyperbolic space M[1,1+q] is identified with a solvable Lie group whose group law is Ψ∗Υ = (Υ1+Ψ1, Υ2+$e^{{-Υ1}}$Ψ2); a homomorphism h(Υ)=(Υ1, WΥ2+(1-$e^{{-Υ1}}$)b) takes the group to another such group, possibly of different dimension, and the isometry φ includes a paint rotation Q∈SO(q), a group translation β, and a fiber rotation Ru. Stacking these layers yields the Cartan network, whose linear layer is flin(Υ)=Ru(β ∗ (Υ1, WΥ2+b)). The authors argue that repeated application of the fiber rotation exponentiates the Cartan coordinate, so depth alone increases expressivity, and they report experiments where activation-free Cartan networks improve with depth and reach accuracies comparable to Euclidean and Poincaré baselines.

Load-bearing premise

The load-bearing premise is that the fiber rotation Ru is a well-defined real isometry of M[1,1+q] for every solvable coordinate and every unit vector u; at Υ=(0,0) with u0=±1, the logarithm in Eq. 11 receives the argument -1 and the map is non-real, so as printed the layer is undefined there.

Editorial extensions

If this is right

  • A Cartan network can change the dimension of the representation between layers because the homomorphism's weight matrix W need not be square, giving a natural way to compress or expand hyperbolic embeddings.
  • Because the layer operations are group homomorphisms and isometries, the architecture uses only intrinsic geometric operations, and component-wise nonlinearities on tangent spaces are unnecessary for depth-driven expressivity.
  • Setting u′=0 and β1=0 in every layer recovers a stack of Euclidean linear layers, so the construction contains ordinary fully connected networks as a special case.
  • The same combination of solvable-group homomorphisms and isometries applies to every non-compact symmetric space, so the framework is not specific to hyperbolic space.
  • On the tested tasks, activation-free Cartan networks show accuracy that increases with depth, and with activations they match Euclidean and Poincaré-ball networks at comparable width and depth.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is to train Cartan networks on larger image or graph tasks with depth held large and width small, checking whether the depth-driven expressivity predicted by the paper translates into consistent accuracy gains over Euclidean networks; the current results hint at this but are limited to a few datasets.
  • The same solvable-group construction should transfer immediately to the symmetric spaces SO(r,r+p)/(SO(r)×SO(r+p)) with r>1, where the paint group is larger; if the authors' claim about generality is right, one can define Cartan layers there without new design choices.
  • The fiber-rotation formula at the identity appears to produce a non-real coordinate when u0=±1, so a corrected definition, for instance using a different branch of the logarithm or an alternative parametrization of SO(q+1), is needed before the architecture is fully well-defined; this is an internal consistency issue rather than a question about the empirical results.
  • One could test the authors' 'depth without activations' thesis by measuring the functional complexity, such as the number of linear regions, of randomly initialized Cartan networks as depth grows; the paper only reports accuracy, not a direct expressivity measure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper introduces 'Cartan networks', a class of hyperbolic neural networks built on the solvable Lie group structure of hyperbolic space. Each layer is defined as a composition of a Lie-group homomorphism between solvable groups and an isometry of the target space, expressed in 'solvable coordinates' Υ = (Υ1, Υ2). Theorem 3.1 characterizes all such homomorphisms as maps of the form h(Υ) = (Υ1, WΥ2 + (1 − e^{−Υ1})b). The authors also propose a hyperbolic softmax based on totally geodesic hyperplanes and report experiments on synthetic regression and image classification tasks, comparing against Euclidean and Poincaré-ball networks. The paper claims that this construction yields a novel class of expressive hyperbolic architectures with competitive benchmark performance.

Significance. If the construction were correct, the paper would offer a genuinely new route to hyperbolic deep learning: instead of gyrovector or exponential/logarithm operations, layers would be built from intrinsic Lie-group homomorphisms and isometries. The homomorphism characterization in Theorem 3.1 is clean and potentially useful, and the paper makes its code available. However, the central fiber-rotation map is not a well-defined self-map of the hyperbolic space at the identity, and the classification experiments use test labels for early stopping and report the best test accuracy, so the two central claims—a well-defined new architecture and promising empirical performance—are not supported as written.

major comments (3)
  1. [§3.2 and App. D.2, Eq. (11)/(44)] The fiber rotation R_u is not a well-defined self-map of M[1,1+q]. Substituting Υ = (0,0) into the first component of Eq. (11) gives −log(−1): the argument equals −1/2(e^0(1+0)+e^0)(1+u_0)+e^0 u_0 = −(1+u_0)+u_0 = −1 for every u_0 ∈ [−1,1]. The real logarithm is therefore undefined, so R_u(0) is not a point of M[1,1+q]. Because Eq. (13) defines the linear layer as R_u composed with group translation and the homomorphism, and Eq. (19) stacks such layers, the architecture is undefined at any input whose image under the preceding operations is the identity, including the initial embedding (Eq. (20)) when x = 0 and β = 0. This is a formal defect in the central construction, not a numerical or tuning issue; the definition of the Cartan layer must be corrected before the claimed architecture can be evaluated.
  2. [App. G.3] The experimental protocol for classification uses test labels for model selection: early stopping is applied 'on the test loss' and 'maximum reached accuracy' is reported. This means the reported accuracies in Tables 2 and 3 and Fig. 3 are selected on the test set, so they do not estimate generalization performance and cannot support the paper's claims of competitive or better performance. The authors should use a validation split for early stopping and report the performance of the final model on a held-out test set.
  3. [§3.4] The claim that setting u'_ℓ = 0 and β^ℓ_1 = 0 for all ℓ reduces the architecture to a stack of Euclidean linear layers is not supported by Eq. (11). With u' = 0, the unit-norm constraint gives u_0 = ±1, and the logarithm in the first component of Eq. (11) is −log(−e^{Υ1}(1+|Υ2|^2)) for u_0 = 1 or −log(−e^{−Υ1}) for u_0 = −1, both non-real for generic Υ. Additionally, the second component of Eq. (11) divides by 1+u_0, which vanishes at u_0 = −1. Thus the fiber rotation does not reduce to the identity in this limit; this is a consequence of the same defect in Eq. (11).
minor comments (3)
  1. [§2] The notation around Eq. (4) and Eq. (5) is inconsistent: M[1,1+p] has dimension p+1, so the statement 'for r = 1 we realize the hyperbolic space H^{p−1}' appears to be a typo and should be H^{p+1} or the indices should be harmonized.
  2. [§6 and App. G.1] The repository URLs differ between Sec. 6 (github.com/FedericoMilanesio/CartanNetworks) and App. G.1 (github.com/CartanNetworks/Cartan_Networks); the authors should ensure one link is correct and that the code matches the printed equations.
  3. [App. D.2] The derivation of Eq. (11) is not given in the paper; the text relies on the co-authored preprint [23] for the isometry classification and the solvable-coordinate formalism. Since Eq. (11) is load-bearing, the paper should either provide a self-contained derivation or clearly state the provenance of each step so that the formula can be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the network derivation rests on an independent Lie-algebra proof and external benchmarks, not on a fitted or self-referential reduction.

full rationale

No significant circularity. The paper's central mathematical claim, Theorem 3.1, is proven directly from Lie algebra homomorphisms, and the performance claims are empirical comparisons against Euclidean and Poincaré baselines on standard datasets; no fitted parameter is later renamed as a prediction. The same-author citations, chiefly to [23], supply the solvable-coordinate and isometry machinery for hyperbolic symmetric spaces. That cited material is parameter-free mathematical support whose assumptions do not include the present Cartan-network architecture or its benchmark numbers, so under the stated rules it counts as independent support rather than as a circularity. I flag two concerns without scoring them as circular: (i) App. D.2 delegates the derivation of the fiber rotation formula (Eq. 11/44) to [23] instead of proving it in the text; and (ii) as printed, Eq. 11 appears undefined at Υ = (0,0) for u0 = ±1, since the logarithm receives a negative or zero argument, which is a formal correctness defect in the core layer rather than an input-output reduction. Neither issue makes Eq. 19 equivalent to Eq. 11 or to any fitted quantity by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The mathematical background is largely imported from the self-cited preprint [23], and the paper's own additions are Thm 3.1 and the layer construction. The main unstated assumption is the validity of the fiber rotation formula, which appears to have a sign error. No new physical or structural entities are postulated.

free parameters (3)
  • DiLU slope alpha = learned (per layer)
    Introduced in Eq. 21 as a trainable parameter of the activation function; benchmark performance depends on it.
  • Learning rate = 0.01 (regression), 1e-4 (classification)
    Hand-chosen separately for task families without a validation split; the central empirical comparison depends on these values.
  • Early stopping buffer = 15 epochs on test loss
    A hand-chosen hyperparameter that uses test labels for model selection; affects reported test accuracies.
assumptions (4)
  • domain assumption Metric equivalence M[1,1+q] ~= Exp(Solv[q])
    The paper relies on this equivalence to define solvable coordinates and the group operation (Eq. 7), citing the self-authored preprint [23].
  • standard math Lie's theorem: solvable algebras are triangularizable
    Used in Sec. 2 to motivate the upper-triangular matrix representation of solvable groups.
  • standard math Hall's theorem 5.6: homomorphisms of simply connected Lie groups lift from algebra to group
    Used in the proof of Thm 3.1 in App E.
  • ad hoc to paper The fiber rotation map Ru (Eq. 11/44) is a well-defined isometry of M[1,1+q] for all real coordinates
    The entire layer definition depends on this map; as printed, the argument of the logarithm is negative at the origin, so the map is not real-valued. This is the core unstated assumption whose validity is questionable.

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Cite this review

Pith. "Pith review of Cartan Networks: Group theoretical Hyperbolic Deep Learning." pith.science (2026). https://pith.science/paper/E6TJG3LQ

@misc{pith2026250524353,
  author       = {Pith},
  title        = {Pith review of: Cartan Networks: Group theoretical Hyperbolic Deep Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6TJG3LQ}},
  note         = {Machine review of arXiv:2505.24353}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2505.24353 by the authors.

Figure 1
Figure 1. Structure of Cartan network (binary classification). This figure illustrates the composi￾tion of the proposed Cartan networks between symmetric spaces. By alternating homomorphisms and isometries, our networks parametrize a larger class of maps while only using geometrically motivated functions. 3.2 Maps between hyperbolic spaces Isometries. The set of isometries of M[1, 1+q] into itself is given by SO(1, 1 + q) (th… view at source ↗
Figure 2
Figure 2. Hyperplanes in M[1,2] ≃ Hn. This figure illustrates an example of the hyperplanes that divide the hyperbolic space. For q = 1, they correspond to the set of all the geodesics. (a) In the Poincarè disk model, the geodesics consist of all arcs of Euclidean circles orthogonal to the disk boundary, plus all the disk diameters. (b) Geodesics obtained by applying the isometry given by left multiplication (Eq. 7) to the wh… view at source ↗
Figure 3
Figure 3. Network performance for depth and hidden layer size on classification datasets. Boxplot showing reached accuracy during training as detailed in Sec. 4 for Euclidean, Cartan, and Poincaré neural networks. The rightmost column of each subplot depicts the accuracy of Cartan networks without nonlinearities. Different colors represent hidden layer sizes, while the network depth increases from left to right within a colum… view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: For regression tasks (Tab. 1), models were trained on a single machine equipped with two [PITH_FULL_IMAGE:figures/full_fig_p020_3.png]

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Forward citations

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.