Proper Velocity Neural Networks
Authors: Ziheng Chen
Zihan Su
Bernhard Schölkopf
Nicu Sebe
Venue: ICLR 2026 Poster (2026)
Keywords: Hyperbolic geometry
Geometric deep learning
Manifold learning
Proper velocity
Representation learning
Riemannian geometry
Primary area: learning on graphs and other geometries & topologies
TL;DR: We establish the Riemannian toolkit of the Proper Velocity manifold, introduce core layers, and build PV neural networks as stable and competitive alternatives to Poincaré and Lorentz networks.

Abstract: Hyperbolic Neural Networks (HNNs) have shown remarkable success in representing hierarchical and tree-like structures, yet most existing work relies on the Poincaré ball and hyperboloid models. While these models admit closed-form Riemannian operators, their constrained nature potentially leads to numerical instabilities, especially near model boundaries. In this work, we explore the Proper Velocity (PV) space, an unconstrained representation of hyperbolic space rooted in Einstein’s special relativity, as a stable alternative. We first establish the complete Riemannian toolkit of the PV space. Building on this foundation, we introduce Proper Velocity Neural Networks (PVNNs) with core layers including Multinomial Logistic Regression (MLR), Fully Connected (FC), convolutional, activation, and batch normalization layers. Extensive experiments across four tasks, namely numerical stability, image classification, graph node classification, and genomic sequence learning, demonstrate the stability and effectiveness of PVNNs. The code is available at https://github.com/NickyoyoSu/PVNN.

Published as a conference paper at ICLR 2026
PROPER VELOCITY NEURAL NETWORKS
Ziheng Chen" Zihan Su*? Bernhard Schölkopf & Nicu Sebe'
University of Trento, University of Nottingham, MPII-I
ABSTRACT
Hyperbolic Neural Networks (HNNs) have shown remarkable success in repre- senting hierarchical and tree-like structures, yet most existing work relies on the Poincaré ball and hyperboloid models. While these models admit closed-form Riemannian operators, their constrained nature potentially leads to numerical in- stabilities, especially near model boundaries. In this work, we explore the Proper Velocity (PV) space, an unconstrained representation of hyperbolic space rooted in Einstein's 's special relativity, as stable alternative. We first establish the complete Riemannian toolkit of the PV space. Building on this foundation, we introduce Proper Velocity Neural Networks (PVNNs) with core layers including Multino- mial Logistic Regression (MLR), Fully Connected (FC), convolutional, activation, and batch normalization layers. Extensive experiments across four tasks, namely numerical stability, image classification, graph node classification, and genomic sequence learning, demonstrate the stability and effectiivness of PVNNs. The code is available at /tttts:s//ggttee www.cooooi......e. coom/iickyoyoou//RVV
INTRODUCTION
Hyperbolic geometry provides a natural representation for hierarchical data due to its exponential representation capacity, and has proven successful across diverse applications, including computer vision (Khrulkov et al., 2020; Bdeir et al., 2024 Sur et al. 2025 Wang et al. 2026). knowledge graphs (Li et al., 2024). natural language processing (Ganea et al., [018]] / Shimizu et al. 2021
knowledge-graph reasoning (Nickel & Kiela 2017), astronomy (Chen al.) (025a)) graph learn- ing (Chami et al., 2019 Li et al., 2024), genomic sequence learning (Khan et al, 2025). fine- tuning (Yang et al. 2025), y and brain signal analysis (Li et al. 2026). Recently, the focus has shifted from hyperbolic embeddings to building Hyperbolic Neural Networks (HNNs) that operate entirely within hyperbolic space. As hyperbolic geometry admits multiple models (Cannon et al. 1997), the choice of representation is central to the design of hyperbolic networks. Most recent works rely on the Poincaré ball and hyperboloid models, which provide convenient Riemannian or gy- rovector structures (Ganea al., 2018 Ungar 2022 Chen et al. 2025c), thereby facilitating neural network construction. However, both models are constrained spaces, which can lead to numerical instabilities. In particular, embeddings in the Poincaré ball approach the boundary, numerical computations become unstable and might cause gradients to vanish (Guo et al.) 2022).
On the other hand, the Proper Velocity (PV) model originates from Einstein's special relativity. where proper velocity provides a natural parameterization for relativistic velocity addition (Ungar 2022 Ch. 10). Algebraically, PV admits gyrovector space (Ungar 2022 Ch. 6), analogous to the Möbius gyrovector space of the Poincaré ball. Unlike the constrained Poincaré ball and hyperboloid models, PV offers an unconstrained representation that alleviates numerical instabilities. These properties have made the PV model successful in relativistic physics and motivate its exploration as stable alternative geometry for HNNs. However, its Riemannian operators, including exponential and logarithmic maps and parallel transport, remain largely unexplored, despite being fundamental for constructing neural networks.
Inspired by the above discussions, we propose Proper Velocity Neural Networks (PVNNs). To this end, we first establish the complete Riemannian geometry of PV by deriving closed-form expres- sions for the exponential map, logarithmic map, geodesic distance, and parallel transport. Building on this foundation, we extend several fundamental neural layers into PV space, including Multi- nomial Logistics Regression (MLR) classification, Fully Connected (FC), convolutional, activation,
"quul contribution.

---

Published as a conference paper at ICLR 2026
and batch normalization layers. Based these layers, one construct different network archi- tectures. We validate the framework through four sets of experiments, including numerical stability, computer vision, graph learning. and genomic sequence learning, demonstrating both the stability of PV embeddings and effectiveness of PVNNs. To our knowledge, the PV model has remained largely unexplored in machine learning, and our work provides the first systematic study of its use for representation learning. In summary, our contributions are threefold:
1. We establish the complete Riemannian geometric toolkit of the PV manifold, deriving closed- form operators that enable its use as new alternative to classical hyperbolic models.
2. We develop fundamental building blocks in PV space, including MLR, FC, convolutional, activa- tion, and batch normalization layers.
3. We validate the stability and effectiveness of PVNNs through experiments on four tasks: numeri-
cal stability, image classification, graph node classification, and genomic sequence learning.
2 RELATED WORK
Hyperbolic representation. Hyperbolic embeddings have been widely explored for hierarchical and non-Euclidean structures in networks, trees, and text (Krioukov et al. 2010 Wilson et al. 2014 (Sonhalia & Gilbert 2020; Nickel Kiela 2017; Chami et al. 2019p. Hyperbolic neural networks (HNNs) explore these embeddings within deep architectures (Ganea et al., 2018). and subsequent works explore them in different applications, such as graphs, knowledge bases, vision, genome, protein, and brain signals (Chami et al. 2019 Balazevic et al. [019 Khrulkov et al.) 2020
Bachmann et al. 2020: Khan et al. 2025 Li et al. 2026))
Hyperbolic models and networks. Among the multiple models of hyperbolic geometry (Can- non et al. 1997). the Poincaré ball and the hyperboloid (Lorentz) models are most commonly adopted. The Poincaré ball admits closed-form Möbius and Riemannian operators (Ganea et al.
2018 Shimizu et al. 202I). whereas the hyperboloid model provides numerically stable geodesics and natural formulations in Minkowski space (Nickel & Kiela 2018 Chen et al. 2022 Mishne et al.,
2023). Building on these operators, researchers have adapted core Euclidean layers to hyperbolic geometries. For instance, Ganea et al. (2018); (Shimizu et al. (2021) introduced FC and MLR layers on the Poincaré ball via point-to-hyperplanee distances, while Chen et al.) (2022); [deii et al. (2024) designed FC and convolutional layers on the hyperboloid through ambient spacetime formulations. Recent works further develop residual architectures and graph-specific formulations (Van Spengler
ct al. 2023
He
2024
Chami al. 2019 Dai et 2021.
Riemannian normalization. Although normalization layers are essential for stabilizing and accel- [ erating training (loffe & Szegedy 2015 ct et al.) [0166] [lyanov et al.) 2016 Wu & He 2018). their
Euclidean formulations do not generalize directly to manifolds. Early extensions adapted Rieman- nian operators such as the exponential map, logarithmic map, and parallel transport to define batch normalization different manifolds (Brooks et al. [201 Lou et al. [2200] Chakraborty) 2020 Bdeir (et al. 2024. Wang et al. 2025a b). However, these approaches often lack theoretical guarantees to normalize sample statistics. More recently, algebraic structures, such as Lie groups and gyrogroups, have been explored to establish principled and unified formulations, which can normalize sample
statisties (Chakraborty 2020 Chen et al. 2024a 2025c).
PRELIMINARIES
Riemannian geometry (Lee 2018). Throughout, (.-)) denotes the standard Euclidean inner prod- uct, and II-I| the induced norm. A Riemannian manifold (M,,) (.9) is smooth manifold equipped with an inner product JN or (1), on each tangent space T.M that depends smoothly on æ E M.. We use Expg. Logg, and PT ZT denote the exponential map at æ, logarithmic map at I, and parallel transport along the geodesic connecting x and y. respectively. A smooth map f f: (M,g) + (M, 9) is a Riemannian isometry if it preserves the metric: 92 v) T)(w) (d=ff de (u)) with d.f f as the differential map at x and u, U E T.M.
Hyperbolic space is a space with constant negative curvature K < 0 and admits several models one can work with (Cannon et al. 1997). The popular models include the Poincaré ball and the hyperboloid (also known the Lorentz model). The PV model PV R is
PV space (Ungar 2022).
2

---

Published as a conference paper at ICLR 2026
an alternative representation of hyperbolic geometry. which was initially named the Ungar gyrovec- tor space and is used to describe algebraic structures of relativistic proper velocities (Ungar 2022).
Unlike the bounded Poincaré ball the constrained hyperboloid, the PV model is an unconstrameed space, offering better numerical stability. Its Riemannian metric given by App. E.
(u, v) (u, v) + K (zz, u) (r, u) Va E PVR: Vu, E T.PVK
(1)
Here, Az
is the relativistic beta factor. In Ungar's notation, the curvature is V1 =|
parametrized by a positive constant s with s = -1/K, where 8 plays the role of the vacuum speed of light in special relativity (Ungar 2022 Sec. 3.8).
PV gyrovector (Ungar 2022). From an algebraic point of view, the PV space forms a gyrovec- tor space (Ungar) 2022, Def. 6.2), which extends the Euclidean vector space to manifolds. Given
x. y. 2 E FV and f E R. PV gyroaddition @u and scalar gyromultiplication @u (Ungar 2022 Ch.
3.11 and 6.20) are defined as
1- By
Br
x UU y z
K
(x, 8)
(2)
By
1 3x
t @ y sinh
tsinh
(V-
K [vvl @u 0)
(3)
In particular, the PV inverse is Quæ = -x, and the PV identity is the zero vector: 0 @u æ
Sy 0
For detailed reviews of Riemannian geometry, gyrovector spaces, PV gyrovector spaces, and the hyperbolic Poincaré ball and hyperboloid models, refer the reader to App.
PV GEOMETRY
4.1
FROM GYRO ISOMORPHISM TO RIEMANNTAN ISOMETRY
The Poincaré ball also admits a gyrovector space, named the Möbius gyrovector space, as reviewed in (Ungar 2022, Sec. 6.14). Algebraically, the PV and Möbius gyrovector spaces are isomorphic. We further show that PV and the Poincaré ball are geometrically isometric.
Let PK x E _ || 11/K} be the Poincaré ball. The following bijections define the gyrovector space isomorphism (Ungar 2022 Tab. 6.1):
A2 x E TPR - P 3 yー 27/y E IPVR.
TFVe PP
PV
(4)
1
9x
where %
is the gamma factor. The isomorphism preserves the gyro operations:
V1+ w/l
मा PP (ar @u y) = .πV; -P (2) BM TPV - (y), Vo, E PVR
(5)
TFV -P (r Φ x)) = r 6M मा -FR (z), Vo E PVR Vr R,
(6)
where 8y and # are the Möbius gyro operations which are reviewed in App. B.4
Lemma 4.1 (Differentials). The differentials of Tppy PP% and Teg -PV
B3 (z, v)
Ar
de (TeVR -p& (v) K dy (TFR (w) -K Let I be the identity map. The differentials at the origin are
U. Vx PVk Vo E T.PV..
(7)
(1+ 8,)2
1 B
23 (y, w) y + 27 2 w,
Vy Pk, Vi T,P
(8)
(mvvg-pP L do - 2T.
Based on Lem. 4.1 we can prove that the above isomorphisms are isometries.
Theorem 4.2 (Isometries). The mappings in Eq.
4 are Riemannian isometries.
'The subseript U refers the initial of Ungar.
3

---

Published as a conference paper at ICLR 2026
4.2 PV RIEMANNIAN OPERATORS
The Poincaré ball admits closed-form Riemannian operators (Ganea et al. 2018). By Thm. 4.2 we
readily obtain the counterparts on PV space via properties of isometries (Chen al. 2026 App. C.2).
Theorem 4.3 (PV Riemannian operators). M Let T PP
Given I, E PVR and v
T.PVK. K. the Riemannian operators on the PV space are
Bxr) VV (x))
di (v)
(v) = x @u
sinh
|| (dda((o)))
(10)
K
Bz
| (1)
Log, (y) (x, y) + (xx, y) (zr,2) v.
(11)
PT PTz-y(()) 1 Brg ē
K (1 + Bg)
(y, 0) y:
(12)
3z
(1 + By)3
d(x,v)-
2
tanh -1 (v- K (-(- Du w)ll)
(13)
V-K
with 2 = (-z) tu For the parallel transport, i gu. -z] (dT with gYTM as the _xxx x and " = 1+ Here, the scalar coefficients in the I+8."
Möbius gyration App. B.4
By
logarithum are
o(x,y) 2 tanh
(V-K (m())l)
V- K tanh
232
.T(I,)) v)
V-K
|||||
1 B2
|||||
At the identity O, the above operators can be further simplified:
Expo(v)= V-K sinh
(v-K|eul))
Logo (v)
-- sinh (V-Kl1ell)
y |ill
|
By
B
PTo-+y(U) = - K
(y, v)
PT PT)) = + K
(I, v) x,
1 + By
1 + 8
d(O.v) v)
sinh -1 (V-Kl)yl)
V-K
The above facts imply that the PV gyro operations can be expressed via Riemannian operations. Theorem 4.4 (Gyro by Riemannian). The PV gyro operations can be rewritten as
x @U y = Exp, (Pogoo((ooo &t x = Expo Logo(z)) Vz, y E PVR, VIER.
PV NEURAL NETWORKS
Building on the above gyrovector and Riemannian tools, we introduce fundamental building blocks for PV neural networks, including Multinomial Logistics Regression (MLR), Fully Connected (FC), convolutional, activation, and batch normalization layers, thereby enabling the construction of con- crete deep architectures in this space.
5.1 PV MULTINOMIAL LOGISTIC REGRESSION
The Euclidean MLR Softmax (Ax is standard classification layer in Euclidean deep learning. As shown by Lebanon & Lafferty| (2004); Ganea et al. (2018); Chen et al. (2024b), each output of C--laas MLR can be reformulate lated as the signed margin distance to a hyperplane:
ply = k =) x exp (vx()))
(x) = sign ((ax, x - -px)) ||a (Ir, Hagwe), 1 1 (14)
where 4k, Pk: E IR" and Har.p. fo E R (at, pk) = 0}.
2018 Sec. 3.1), we define the PV hyperplane as Following the Poincaré MLR (Ganea et al.
Hap = x. E PV | [Log,(()).) a) =
E IVV a E T,PV
(15)
4

---

Published as a conference paper at ICLR 2026
where p E PVV and E T are the hyperplane parameters. As the Poincaré hyperplane can be expressed by the Möbius gyro operations (Ganea et al. 2018 Eq. 22). the PV hyperplane can also be expressed by the PV gyro operations. In addition, building PV MLR requires the PV point-to-hyperplane distance. The following theorem provides these results.
Theorem 5.1.
Let ाT
TEVE -Pk- Given P, E PVR and a E TpIPV', we have
Ha.
x E I (Log,()),a) =0
x E PVK
Du x, drt(aa) =0].
(16)
V-K
K
(u Bu yr dpT((a))
1 V-K
d(y, Ha.p)
inf (y.w) wEHa.
sinh
(17)
By Thm. 5.1 we define the C-class PV MLR as
p(v k u) x exp (mx((x)),
mx(x) sign ((-Ph Cu x. p, (ax))) ||ax ll., d (Ir, Han.p))
(18)
where Pk E PVV and a₁ E PV* are the PV MLR parameters for class k. However, the above expression has three drawbacks: (i) the parameter ps over-paramettrrizd,, as it corresponds to the scalar bias parameter in the Euclidean MIR; (ii) the gyroaddition in (-ps @y I; dy, Tax)) complicates the computation; and (iii) the parameters (Pk. Qx) are constrained, making optimization costly. To address these drawbacks, we follow Shimizu al. (202T) and adopt the parameterization || = I),
Pk = ExPo (r
PTo-pp (2x) with 2 E PV.V = IK" and F E IR. This parame- terization avoids Riemannian optimization in PV MLR and further simplifies the formulation.
Theorem 5.2 (PV MLR).
For x E PVk the score (x)x) in Eq. (18) for each class k is
||=0||| (x(x)= = V-K
Bosh(v-KKr V-K (xr, (x) - -sinh( (V K
sinh-
(19)
TEal
where 2k E RR and Th E R are parameters for class k. In particular, as K - O we have en (x(x)) (a,, zx) + by with by -rallzk]], [|, which recovers the Euclidean MLR in Eq. (1p.
The parameterization (z). is essential for efficiency. In the original form Eq. 18) computing (2) for a batch R and C' classes requires explicit gyroaddition -Pk tU for each class, producing an intermediate tensor of size x C' x n that could cause out-of-memory errors in high dimensions. One could instead loop over classes, but this is computationally inefficient. In contrast, Eq. 19 depends on inner products (Ir, =x), which can be implemented as a matrix multiplication.
5.2
PV FULLY CONNECTED LAYER
with The Euclidean FC layer is defined as y Ax + with R=xx and b RT, It can be expressed
element-wise as y/ (ak, .)) - by = (ak, x - pk) with @k Ph: E RR" and (px. ax) bg. As shown by Sy/miiz et al. (2021 Sec. 3.2), the LHS yı the signed distance from y the hyperplane passing through the origin and orthogonal to the k-th axis of the output space, which can be formulated as
sign ((es: y - 0)) d(y, He,,o) (ax x pk), V1 S S m, where d₁ denotes the vector whose k-th element is 1 and all others are 0.
(20)
For the PV model, the LHS of Eq. (20) can be formulated by the signed point-to-hypee to-hperrpeane distance, while the RHS can be formulated by the uy in PV MLR. Specifically, the PV FC layer F: PV + PvR from the n-dimensional to the -ndimensiiona PV spaces for the input x E PVV returns the output y E PVR by solving the m equations:
sign (do. - Gu 2)) d(y, He vx(x), VI k S m,
(21)
where He, o and Us (x) are given by Thm. 5.1 This definition has explicit solution. Theorem 5.3 (PV FC layer). [ The output y F(x) PVR has the closed form
Uk
sinh( ( (Ke())Ke
1 1 S m,
(22)
K
where U2 (x) is defined as Eq. 19 with 2 E R" and ræ E R as the FC parameters. In particular. as K - O we have yk - (x, 2k) by with by -rk | which recovers the Euclidean FC layer.
5

---

Published as a conference paper at ICLR 2026
Generalization. We can jointly express the Euclidean FC layer and activation d, which yields the RHS of Eq. (0 with (agg pk) Accordingly, we extend the PV FC by applying the activation to (x) Eq. (21). Then, Eq. 22 becomes
Uk V-K
sinh (v)))()(())) 1 m
(23)
5.3 PV CONVOLUTION AND ACTIVATION
Convolution. As shown by Shimizu et al. 2021 Bdeir et al. (2024), Euclidean convolution consists of linear maps between kernel weights and concatenated values in each receptive field. To define convolution on PV space, it therefore suffices to define PV concatenation, since we already have the PV FC layer. Because PV space unconstrained, we define PV concatenation to coincide with Euclidean concatenation. For simplicity, in a ID receptive field (where is the kernel size), the PV convolution output E PVR for this receptive field is y = FF (Concat (zi...., where Concat(.) is standard Euclidean concatenation and F is the PV FC layer.
consider the 1D case. For PV inputs {z,
PVR K 1k
Activation. A natural choice is to apply a Euclidean activation a in tangent space at the origin via the mapping x + Expo
(Logo(z))): which has been shown be effective in Poincaré networks (Ganaa et al., 2018). Alternatively, since PV space is unconstrained, we can apply the activation directly PV space + (x)) This direct PV-space activation avoids exponential and logarithmic maps and is therefore more efficient.
5.4
NORMALIZATION
Recently, Chen al. 2024c 2025c
extended Batch Normalization (BN) to non-Euclidean mani- folds through gyro- strrsstructurs, referred to GyroBN. Intuitively, subtraction, addition, and scaling in Euclidean BN are replaced by gyrosubtraction, gyroaddition, and gyromultiplication respectively. We extend their framework to PV space and show that PV GyroBN can normalize sample statistics.
We first recall the Fréchet statistics. Given N samples {ar) C PVK, the Fréchet mean and Fréchet variance are
# FM(())) argmin N
Σ N d' (R( v),
v 1 N
EM",
N
d (2i u). (24)
i-
Given activations {xa E IPV W the core operations of PV GyroBN are
Scaling
Binsing
Centering
S N,
3. 6 BeU
- @u Ti
(25)
Vu2 +E
where and 22 denote the Fréchet mean and variance, and 3 E PVm and E R are parameters. Owing to isometry between the PV space and Poincaré ball, the PV Fréchet mean can com- puted via the Poincaré ball: map the data to the Poincaré ball, compute the Poincaré mean (Lou ct al. 2020 Alg. 1), and map the result back.
The following theorem guarantees that PV GyroBN can normalize sample statistics. Theorem 5.4 (Homogeneity).
For N samples [ C PV have
Homogeneity of mean: FM (B Bt B6
x:) = 3 @r FM ((as)/ AB VBE
(26)
Homogeneity of dispersiin 0:
1 N
Σ
1=1 (t &0 24 0)
1
D ( 0).
(27)
N
Thm. 5.4 direetly explains the PV GyroBN in Eq. 255 After the centering, the batch mean is shifted
to the identity O. After the biasing, it is translated to After the scaling, the variance becomes s?.

---

Published as a conference paper at ICLR 2026
6
EXPERIMENTS
We evaluate PV embeddings and PV Neural Networks (PVNN) on four representative tasks:
Sec. 6.1
evaluates the numerical advantage of the PV model against Poincaré and hyperboloid.
Sec. 6.2
compares PV, Poincaré, and hyperboloid MLRs on image classification.
Sec. 6.3
evaluates our PV MIR, FC, and GyroBN layers on graph learning.
Sec. 6.4
compares fully PV convolutional networks with fully hyperholoid convolutional net-
works on genomic sequence learning.
6.1 NUMERICAL STABILITY
We study three aspects: gyro operator, Rieman- nian operator, and gradient behavior. All ex- periments use curvature K -1, dimension 16, and batch size 4096.
Table 1: Failure and violation probabilities (%) of 22 x in FP32.
Failure rate
Violation rate
PVR
PR
DI
PVR
PR
Gyro operator. We use scalar gyromultipli- cation r &x as a probe of numerical stabil-
0
0
N/A
0
32.50
5
0
0
N/A
0
92.36
ity across hyperbolic models. Given random
10
0
0
N/A
0
99.76
batches and radii r, we evaluate two met-
20
0
4.23
N/A
0
100
rics. The failure rate the fraction of out-
50
0
64.42
N/A
O
100
puts that contain NaN/Inf. The violation rate defined only for models with manifold con- straints: Poincaré ball requires |x||| - 1/K,
0
79.63
N/A
O
100
100
0
88.26 N/A
0
100
150
0
96.43
N/A
0
100
200
0
100
N/A
0
100
and hyperboloid requires z - -1a,,11 K for
1000
0
100
N/A
0
100
= [T,, =,] The tolerance is set 10-8
As PV is unconstrained, its violation rate is reported N/A. As shown in Tab. PV maintains zero failures up to r 1000 in FP32. The Poincaré ball has zero failure and violation rates, whereas the hyperboloid model starts to fail around = 20 and quickly accumulates both NaN/Inf outputs and off-manifold points under large scalar multipliers, revealing pronounced numerical instability.
Riemannian operator. We evaluate the exponential and logarithmic maps by measuring the round-trip error [Logo (Expo - wll for tangent vectors with large norm |||| = 10. Since this quantity is theoretically zero, any non-zero value reflects numerical instability. We sample a batch of such vectors and report the average er-
Table 2: Logo (Expo e/|-
Model
FP32
FP64
IPR
2.1 x
4.3 x 10-11
HD
1.0 10
1.0 10
PVK
2.1 x 10-7
6.7 x 10 16
ror Tab. 2 PV achieves stable behavior in both FP32
and FP64, whereas the Poincaré ball already exhibits noticeable errors in FP32 and the hyperboloid model remains unstable in both precisions.
Gradient. To compare gradient be- havior, we study the gradient of fx) ||r 8x x - z|| with respect to æ. Specifically, we sample 24 loga- rithmically spaced radii E [1, 1000] and, for each radius, measure the [=))(x)] on a random batch. The range of v..f(x))) is summarized Tab. 3 The Poincaré ball exhibits severe gradient vanishing near the boundary. In contrast, the hyperboloid model yields gradients that vary from 0 to NaN, reflecting gradient explosion. PV maintains gradicnts in a safer band.
Table 3: Gradient magnitude |
across varying radii.
Model
| Range
Gradient behavior
PR H F
[1.1 x 10 10-11 11 7.6 x 10 13
Vanishing gradients
[0, NaN
Exploding gradients
[1.1 x 10 4 2.1
10 o)
Stable gradients
6.2 IMAGE CLASSIFICATION
We compare our PV MLR against previous Poincaré MLRs (Ganea et al. 2018 Shimizu et al. 2021) and Lorentz MLR (Bdeir et al. 2024j. Following (Bdeir et a 2024). we train a ResNet- backbone (He et al. 2016) CIFAR- 10 and CIFAR-100 (Krizhevsky & Hinton 2009). replacing the final Euclidean MLR with hyperbolic MLR. The backbone output is lifted to the target geom- etry via the exponential map at the identity. Since PV space is unconstrained, we also consider
7

---

Published as a conference paper at ICLR 2026
Table 4: Accuracies of hyperbolic MLRs on ResNet-18. Best results are in bold. 8 represents the 6 6-byperbolicity (lower is more hyperbolic), which comes from Bdeir et al. (2024] Tab. 1).
Model
Method
CIFAR-10 (6 0.26) CIFAR 100 (8 0.23)
Poincaré MLR (Ganea et al. Unidirectional MLR (Shimzu of al 2021 Eq. 6)
2018 Eg. 25)
95.09 + 1.51
76.78 + 0.67
Pk
95.12 + 0.20
77.19 + 0.10
1ik
Lorentz MLR Bdeir al.
2024 Thm.
95.02 + 0.12
77.96 = 0.09
PV MLR (with Expp)
95.27 + 0.12
78.19 + 0.59
FVR
PV MLR (without Expu)
95.30 = 0.18
78.20 = 0.37
Table Accuracies of hyperbolic networks on graph learning. Best results are in bold. ō represents the 9-hyperbolicity (lower is more hyperbolic).
Disease (6 0)
Airport (6=1) 1)
PubMed (6 = 3.5)
Cora
Model
Method
(6 = 11)
K
KNN (Mao et al.
2024
79:41 + 0.55
92.10 + 0.97
69.36 = 0.76
52.26 + 1.99
HNN Ganea et al. 2018
79.90 + 0.01
82.16 2.95
69.28 + 0.85
49.68 1.25
PKK
HNN+ (Shimizu al. 2021
80.57 + 0:23
88.40 0.17
73.68 0.39
52.06 0.90
HT
LNN (Bdeir et al.
2024
79.90 + 0.01
75.20 + 1.08
68.82 = 0.88
53.34 + 1.65
PV
PVNN
81.15 + 0.23
97.96 0.42
74.33 + 0.22
51.42 1.33
direct variant that skips ExPo and treats the backbone output PV coordinates. We denote these two PV heads as PV MLR (with Expo) and PV MLR (without ExPo). More details are provided in App. C.2] Tab. reports the 5-fold results. PV MLR matches or outperforms prior hyperbolic baselines, with the largest gains on CIFAR-100 where the decision boundaries are more complex. Both PV variants, with and without Expo achieve similar accuracies.
6.3 GRAPH LEARNING
Data and Setup. We study node classification on four standard graph datasets: Disease (Anderson
& May 1991), Airport (Zhang & Chen 2018), Cora (Sen et al.] 2008), and PubMed (Namata et al., 2012]. All models share the same architecture consisting of two FClayers with nonlinear activations followed by an MLR classifier; they differ only in the underlying hyperbolic model. Basclines include KNN (Mao et al. (024) for the Klein ball, HNN/HNN++ (Chami et al. 2019; (Shimizu et al. 2021) for the Poincaré ball, and LNN (Bdeir al. 2024) for the hyperboloid model. Our PVNN is built by PV FC, activation and MLR layers. More details are provided in App. C.3
Main results. For fair comparison, we use tangent activation in each model and set = I for the PV FC layer in Eq. (23). Tab. (5) summarizes the 5-fold results. On the three more hyperbolic datasets (Disease, Airport, and PubMed), PVNN consistently achieves the best performance, with especially large gains on Airport where it improves over the strongest bascline by 5.86%. On the weakly hyperbolic Cora dataset, PVNN remains comparable Poincaré- and Klein-based networks, and worse than the hyperboloid- one. Overall, these results suggest that PV geometry is more effective on strongly hyperbolic graphs.
Tangent vs. Riemannian. A natural construction of hyperbolic layers is to work in the tangent space. To validate the benefits of our Riemannian PV layers, we compare our PV FC with tangent- space FC of the form Expo (A Logo (x) + b). and our GyroBN with tangent BN (TBN) given by Logo (BN(Fxpo (x))), where BN denotes standard Euclidean batch normalization (Toffe & Szegedy) 2015). We denote these variants by PVNN+TFC and PVNN+TEN, respectively. As shown in Tab. 6 PVNN consistently outperforms PVNN+TFC on the more hyperbolic Disease and Airport datasets, while the other two comparable and TFC can be slightly better. For normalization, PVNN+GyroBN improves over PVNN+TBN on all datasets. Overall, these ablations validate the effectiveness of our Riemannian PV constructions, especially in strongly hyperbolic settings.
Ablations on batch statistics. PV GyroBN in Eq. (55 uses Fréchet mean and variance, which requires iterative solvers. We also consider two efficient variants. A tangent variant computes batch
8

---

Published as a conference paper at ICLR 2026
Table 6: Results of Tangent FC (TFC) PV FC, and Tangent BN (TBN) GyroBN.
Method
Disease
Airport
PubMed
Cora
PVNN+TFC
80.86 + 0.30
86.99 + 0.61
74.40 + 0.43
53.58 + 0.81
PVNN 81.24 + 0.36
97.93 0.29
74.16 + 0.32
52.26 1.32
PVNN+TBN 80.67 + 0.38
98.71
0.36
73.52 + 0.12
45.36
2.44
PVNN+GyroBN 81.24 + 0.19
99.03 0.18
74.34 = 0.31
46.64 # 5.45
Table Comparison of methods in calculating mean and variance PV GyroBN. Time measured in milliseconds per training epoch.
Disease
Airport
PubMed
Cora
Method
Acc
Fit Time
Acc
Fit Time
Acc
Fit Time
Acc
Fit Time
Tangent
81.15
0.23
26.08
98.56 56
55.48
61.50
5.75
33.10
7.12
Eaclidean
81.15 1: 0.23
25.80
98.75 1: 0.31
$5.19
69.82
1: 3.58
2.99
32.62 1 0.65
7.29
Fréchet iter
81.05 0.23
29.79
88.93 + 1.17
65.19
62.52 8.44
3.38
42.34 6.15
7.67
Fréchet
$1.05 0.23
91.11
67.37
73.78 0.20
3.49
45.68
1.36
8.21
Fréchet 5 iters
81.24 1 0.36
30.90
98.50
0.16
82.28
T3.92 1 0.44
4.02
49.50
1.83
9.15
Fréchet
105.79
74.34
0.31
9.77
Fréchet
80.86 + 0.00
31.29
98. + 0.15
122.37
71.16 + 3.93
4.46
47.32 4.73
9.27
Table 8: Ablations PVNN with without exponential map for the input PV feature.
ExPo
Discase
Airport
PubMed
Cora
X
81.05 0.23
97.71 0.34
74.22 = 0.26
51.92 + 2.01
81.24 0.36
97.93 0.29
74.16 0.32
52.26 1: 1.32
Table Ablations on PV activations.
Method
Disease
Airport
PubMed
Cora
Tangent Act.
81.21 + 0.36
97.93 + 0.29
74.16 + 0.32
52.26 + 1.32
FC
81.34 0.43
99.40 0.15
74.02 0.17
51.34 + 0.46
FC a Tangent Act.
80.96 0.19
99.15 0.38
73.96 0.22
51.30 1.65
Euc. Act.
81.34 0.43
98.87 0.35
74.56 + 0.59
38. + 3.30
statistics the tangent space at the identity via
# = ExPo
Logo (a:)
v² = N Σ Logo((z))) (z;) - (4))1
(28)
N
and Euclidean variant computes standard Euclidean and variance directly in the uncon- strained PV space. Tab. T shows that Tangent and Euclidean are up to 2x faster while achieving similar accuracies on Disease and Airport. Although Fréchet-based GyroBN attains the best accu- racies, it is more computationally expensive.
Ablations on PV embedding. the main experiments, the input features are first lifted PV via ExPo and then processed by PVNN. Since PV space is unconstrained, we also consider a variant that feeds the Euclidean features directly PV coordinates. Tab, 8 compares these two settings. The two variants perform similarly, while using ExPo provides small improvements on Disease, Airport, and Cora. This differs from image classification Tab. 4 where the variant without Expo is marginally better. This slight discrepancy may stem from the different nature of the inputs. In vision, the ResNet encoder can adapt its learned representation to the chosen lifting, whereas in graphs the raw node features benefit slightly from the explicit exponential map.
the internal nonlinearity o in the PV FC layer (fixed to tanh), and explicit activations applied either directly in PV (Euc. Act.) or in the tangent space (Tangent Act.). Tab. repprrs the results. First, when comparing these three choices individually, the differences are small on Disease and PubMed, while FC performs best
Ablations on activation. We ablate three types of activations in PVNN:

---

Published as a conference paper at ICLR 2026
Table 10: Comparison (MCC) of hyperbolic and Euclidean convolutional networks TEB datasets.
Task
Datasct
Euclidean CNN
HCNN-S
PVCNN
LINEs
70.63 = 1.24
76.12 - 2.16
81.83 + 0.27
Retrotransposons
SINEs
85.15 = 1.64
85.45 + 1.16
98.78 + 0.54
DNA transposons
hAT-Ac
87.45 = 0.90
89.61 + 1.34
92.08 + 0.80
processed
60.66 + 0.82
68.30 0.93
71.27 + 0.78
Pseudogenes
unprocessed
51.94 + 2.69
56.10 + 0.56
62.31 + 0.78
Airport, Tangent Act. performs best Cora, and Euc. Act. degrades substantially on Cora. Second, when comparing the composite variant FC a + Tangent Act. against Tangent Act., the composite does not yield consistent gains, suggesting redundancy.
6.4 GENOMIC SEQUENCE LEARNING
Khan et al. 2025) recently proposed hyperbolic convolutional neural networks (HCNNs) on the hy- perboloid for genomic sequence learning, demonstrating that HCNNs outperform Euclidean CNNs on this task. Following Khan et al. (025)). we evaluate on the TEB dataset for DNA transposable element prediction. To ensure fair comparison, all models share the same backbone network chitecture, which consists of two convolutional blocks followed by an FC layer and a final MLR classifier (Khan et al., 2025). We use single curvature shared for all layers. More details are pro- vided in App.[C.4] Tab. reportss 5-fold Matthews correlation coefficient (MCC). PVCNN achieves the best performance on all TEB tasks, with particularly strong gains on SINEs, where improves over HCNN-S by about 9 MCC points. These results demonstrate the benefits of PV convolutional networks.
CONCLUSIONS
This work introduces Proper Velocity Neural Networks (PVNNs), leveraging the unconstrained PV model as alternative to the constrained Poincaré and Lorentz models. We establish the full Rie- mannian toolkit on PV space and develop core neural layers, including MLR, FC, convolutional, activation, and normalization layers. Through four sets of experiments numerical stability, im- age classification, graph node classification, and genomic sequence learning, PVNNs demonstrate both improved stability and competitive superior performance compared with strong hyperbolic baselines. Our study provides the first systematic treatment of the PV manifold for deep learning, positioning it stable and practical geometry for future research hyperbolic neural networks. As future work, we plan to extend PVNNs to more ad advanced architectures such as residual networks (He al.
2016; Van Spengler al. 2023 He al. (2024) and transformers (Vaswani et al. 2017 Hu et al. 2023). and further exploit PV space for large-scale deep learning.
REPRODUCIBILITY STATEMENT
All theoretical results are established under explicit assumptions, with complete proofs in App. B The experimental details are presented in App. The code will be released upon acceptance.
ETHICS STATEMENT
This work uses only publicly available benchmark datasets, which contain personally identifiable or sensitive information. We did not identify any ethical concerns.
ACKNOWLEDGMENTS
This work was supported by EU Horizon project ELLIOT (No. 101214398) and by the FIS project GUIDANCE (No. F1S2023-03251). We acknowledge CINECA for awarding high-performance
10

---

Published as a conference paper at ICLR 2026
computing resources under the ISCRA initiative, and the EuroHPC Joint Undertaking for granting access to Leonardo at CINECA, Italy.
REFERENCES
Roy M Anderson and Robert M May. Infectious diseases of humans: dynamics and control. Oxford university press, 1991.
Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv:1607.06450, 2016.
arXiv preprint
Gregor Bachmann, Gary Bécigneul, and Octavian Ganea. Constant curvature graph convolutional networks. In ICML, 2020.
Ivana Balazevic, Carl Allen, and Timothy Hospedales. Multi-relational Poincaré graph embeddings. In NeurlPS, 2019.
Ahmad Bdeir, Kristian Schwethelm, and Niels Landwehr. Fully hyperbolic convolutional neural networks for computer vision. In ICLR, 2024.
Daniel Brooks, Olivier Schwander, Frédéric Barbaresco, Jean-Yves Schneider, and Matthicu Cord. Riemannian batch normalization for SPD neural networks. In NeurIPS, 2019.
James W Cannon, William J Floyd, Richard Kenyon, Walter R Parry, et al. Hyperbolic geometry. Flavors of geometry, (1559-115):2, 1997.
Rudrasis Chakraborty, ManifoldNorm: Extending normalizations Riemannian manifolds. arXiv preprint 138V,,,003. 2020.
Ines Chami, Zhitao Ying, Christopher Ré, and Jure Leskovec. Hyperbolic graph convolutional neural networks. In NewrIPS, 2019.
Tianyu Chen, Xingcheng Fu, Yisen Gao, Haodong Qian, Yuecen Wei, Kun Yan, Haoyi Zhou, and Jianxin Li. Galaxy walker: Geometry-aware VLMs for galaxy-scale understanding. In CVPR, 2025a.
Weize Chen, Xu Han, Yankai Lin, Hexu Zhao, Zhiyuan Liu, Peng Li, Maosong Sun, and Jie Zhou. Fully hyperbolic neural networks. In ACL, 2022.
Ziheng Chen, Yue Song, Yunmei Liu, and Nicu Sebe. A Lie group approach to Riemannian batch normalization. In ICLR, 2024a.
Ziheng Chen, Yue Song, Rui Wang, Xiao-Jun Wu, and Nicu Sebe. RMLR: Extending multinomial logistic regression into general geometries. In NeurlPS, 2024b.
Ziheng Chen, Yue Song, Xiao-Jun Wu, and Nicu Sebe. Gyrogroup batch normalization. NeurlPS, 2024c.
Ziheng Chen, Xiao-Jun Wu, Bernhard Schölkopf, and Nicu Sebe. Building transformation layers for Riemannian neural networks, 2025b.
Ziheng Chen, Xiao-Jun Wu, Bernhard Schölkopf, and Nicu Sebe. Riemannian batch normalization: A gyro approach. arXiv preprint 15Xiv:2509.07115, 2025c.
Ziheng Chen, Yue Song, Xiao-Jun Wu, and Nieu Sebe. Fast and stable Riemannian metrics on SPD manifolds via Cholesky product geometry. In ICLR, 2026.
Jindou Dai, Yuwei Wu, Zhi Gao, and Yunde Jia. A hyperbolii-too-hyperbbii graph convolutional network. In CVPR, 2021.
Octavian Ganea, Gary Becigneul, and Thomas Hofmann. Hyperbolic neural networks. In NeurlPS, 2018.
Yunhui Guo, Xudong Wang, Yubei Chen, and Stella X Yu. Clipped hyperbolic classifiers are super- hyperbolic classifiers. In CVPR, 2022.
11

---

Published as a conference paper at ICLR 2026
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog nition. CVPR, 2016.
Neil He, Menglin Yang, and Rex Ying. Lorentzian residual neural networks. In ICML 2024 Work- shop on Geometry-grouncct Representation Learning and Generative Modeling, 2024.
Zhiwei Hu, Victor Battérrez-BBsulto, Zhiliang Xiang, Ru Li, and Jeff Z. Pan. Hyperformer: Enhanc- ing entity and relation interaction for hyper-relational knowledge graph completion. In CIKM, 2023.
Sergey loffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICML, 2015.
Raiyan R. Khan, Philippe Chlenski, and Itsik Pe'er. Hyperbolic genome embeddings. In ICLR, 2025.
Valentin Khrulkov, Leyla Mirvakhabova, Evgeniya Ustinova, Ivan Oseledets, and Victor Lempitsky. Hyperbolic image embeddings. In CVPR, 2020.
Diederik P Kingma and Jimmy Lei Ba. Adam: A method for stochastic optimization. In ICLR, 2015.
Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. Hyperbolic geometry complex networks. Physical Review E Statistical, Nonlinear, and Soft Matter Physics, 82(3):036106, 2010.
Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Tech- nical Report 0, University of Toronto, Toronto, Ontario, 2009.
Guy Lebanon and John Lafferty. Hyperplane margin classifiers the multinomial manifold. In ICML, 2004.
John M Lee. Introduction to Riemannian Manifolds, volume 2. Springer, 2018.
Shanglin Li, Shiwen Chu, Okan Koe, Yi Ding, Qibin Zhao, Motoaki Kawanabe, and Ziheng Chen. HEEGNet: Hyperbolic embeddings for EEG. ICLR, 2026.
Yancong Li, Xiaoming Zhang, Ying Cui, and Shuai Ma. Hyperbolic graph neural network for tem- poral knowledge graph completion. In Proceedings of the 2024 Joint International Conference on Computational Linguistics, Language Resources and Evaluation (LREC-COLAN) 2024), 2024.
Aaron Lou, Isay Katsman, Qingxuan Jiang, Serge Belongie, Ser-Nam Lim, and Christopher De Sa. Differentiating through the Fréchet mean. In ICML, 2020.
Yidan Mao, Jing Gu, Marcus C Werner, and Dongmian Zou. Klein model for hyperbolic neural networks. arXiv preprint arXiv:2410.16811 2024.
Gal Mishne, Zhengchao Wan, Yusu Wang, and Sheng Yang. The numerical stability of hyperbolic representation learning. ICML, 2023.
Galileo Namata, Ben London, Lise Getoor, and Bert Huang. Query-driven active surveying for collective classification. In KDD, 2012.
Xuan Son Nguyen and Shuo Yang. Building neural networks on matrix manifolds: A Gyrovector space approach. In ICML, 2023.
Maximillian Nickel and Douwe Kicla. Poincaré embeddings for learning hierarchical representa- tions. In NeurlPS, 2017.
Maximillian Nickel and Douwe Kiela. Learning continuous hierarchies in the Lorentz model of hyperbolic geometry. ICMI., 2018.
Herbert Robbins and Sutton Monro. A stochastic approximation method. The annals of mathemati- cal statistics, pp. 400-407, 1951.
12

---

Published as a conference paper at ICLR 2026
Prithviraj Sen, Galileo Mark Namata, Mustafa Bilgic, Lise Getoor, Brian Gallagher, and Tina Eliassi-Rad. Collective classification in network data. Al Magazine, 2008.
Ryohei Shimizu, YUSUKE Mukuta, and Tatsuya Harada. Hyperbolic neural networks++, In ICLR, 2021.
Ondrej Skopek, Octavian-Fugen Ganea, and Gary Bécigneul. Mixed-curvature variational autoen- coders. In ICLR, 2020.
Rishi Sonthalia and Anna Gilbert. Tree! am no tree! am low hyperboii hyperbolic embedding. In NeurlPS, 2020.
Tanuj Sur, Samrat Mukherjee, Kaizer Rahaman, Subhasis Chaudhuri, Muhammad Haris Khan, and Biplab Banerjee. Hyperbolic uncertainty-aware few-shot incremental point cloud segmentation. In CVPR, 2025.
Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky. Instance normalization: The missing in- gredient for fast stylization. arXiv preprint arXiv: 1607.08022, 2016.
Abraham Albert Ungar. Analytic Hyperbolic Geometry and Albert Einstein's 's Special Theory of Relativity (Second Edition). World Scientific, 2022.
Max Van Spengler, Erwin Berkhout, and Pascal Mettes. Poincare resnet. In ICCV, 2023.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In NeurlPS, 2017.
Rui Wang, Shaocheng Jin, Zhenyu Cai, Ziheng Chen, Xiao-Jun Wu, and Josef Kittler. Learning better SPD network signal classification: Riemannian batch normalization method. TNNLS, 2025a.
Rui Wang. Shaocheng Jin, Ziheng Chen, Xiaoqing Luo, and Xiao-Jun Wu. Learning normalize on the SPD manifold under Bures-Wasserstein geometry. In CVPR, 2025b.
Rui Wang, Yuting Jiang, Xiaoqing Luo, Xiao-Jun Wu, Nicu Sebe, and Ziheng Chen. Wasserstein- aligned hyperbolic multi-view clustering. AAAI, 2026.
Richard C Wilson, Edwin Hancock, Elzbieta Pekalska, and Robert PW Duin. Spherical and hyperbolic embeddings of data. IEEE TPAMI, 2014.
Yuxin Wu and Kaiming He. Group normalization. In ECCV,
Menglin Yang, Ram Samarth B, Aosong Feng, Bo Xiong, Jiahong Liu, Irwin King, and Rex Ying. Hyperbolic fine-tuning for large language models. In NeurIPS, 2025.
Muhan Zhang and Yixin Chen. Link prediction based on graph neural networks. In NeurlPS, 2018.
13

---

Published as a conference paper at ICLR 2026
APPENDIX CONTENTS
List of acronyms
16
Use large language models
16
Preliminaries
16
B. I
Riemannian geometries
16
B.2
Gyro-Structures
17
B 3
PV gyration
18
44
Poincaré ball and hyperboloid
18
Experimental details
20
Common implementations
20
C.2
Experimental details image classification
20
Datasets
20
C.2.2
Implementation details
20
Experimental details on graph learning
20
3.3
Datasets
20
C.3.2
Implementation details
Experimental details on genome sequence learning
21
C.4.1
Datasets and preprocessing
C.4.2 Implementation details
21
Hardware
22
D Connections to the hy hyperboloid
22
Proofs
23
Proof of Eq. (1)
23
Proof of Lem. 4.
24
Proof of Thm.
25
E.4
Proof of Thm.
26
E.4.1
PV exponential map
26
E.4.2
PV logarithmic map
27
E.4.3
PV parallel transport
29
E.4.4
PV geodesic distance
29
E.4.5
Special cases at the identity
30
Proof of 4.4
31
E,6
Proof of 1.1.15.
31
E.6.1
Equivalent characterization of the PV hyperplane
32
E.6.2
PV point-to-hyperppane distance
32
14

---

Published as a conference paper at ICLR 2026
Proof of 5.2.
34
Proof of Thm.
37
9
Proof of Thm 5,4
38
10 Proof of Prop. D.1
38
15

---

Published as a conference paper at ICLR 2026
Table 12: The geometric reinterpretations of Riemannian operators.
Operation
Euclidean space
Riemannian manifold
Straight line
Straight line T
Geodesic
Subtraction Addition
Ty = Log mz
y = x zÿ V -
y = Exp 2yy
Parallel transport
PT xー (1)
LIST OF ACRONYMS
HNNs
Hyperbolic Neural Networks
PVNNs
Proper Velocity Neural Networks Proper Velocity
PV
FC
Fully Connected Multinomial Logistics Regression
MLR
A USE OF LARGE LANGUAGE MODELS
Large Language Models (LLMs) used primarily for language polishing and minor text editing. In limited cases, they also assisted translating certain mathematical formulations into Py PoToch code. All generated outputs carefully reviewed and, where necessary, corrected by the authors. The authors take full responsibility for the final content of this paper.
B PRELIMINARIES
B. RIEMANNIAN GEOMETRIES
For an in-depth discussion on Riemannian geometry, one can refer to Lee 2018
Riemannian manifold. A Riemannian manifold (M, g). abbreviated M, carries smoothly varying Riemannian metric
T.M x T + R on each tangent space T.M. The induced norm is ||b||a = v (e. As an inner product, IN is also denoted as (.
Geodesic. Straight lines are generalized constant-speed curves that are locally length- minimizing between points v, y E M, known as geodesics. The shortest distance between two points is called the geodesic distance, denoted as dl.,)).
Exponential and logarithmic maps. For x E .M and E T.M, let 72.0 denote the unique geodesic with (0)
and (0) v. The exponential map Exp, T.M V M is defined by Exp,, (v) = Ya. (1). where V is an open neighborhood of the origin in T,M. Its local inverse, defined for y in a neighborhood C M of x, is the logarithmic map Log, - T,M, satisfying Exp., Log,, = Tuu In hyperbolic geometry, exponential and logarithmic maps
globally defined (Lee 2018).
Parallel transport. Given geodesic y from to y. the parallel transport of tangent vector v T,M along the geodesic is the unique vector (V) M obtained by transporting along that its covariant derivative along ? vanishes. Parallel transport defines linear isometry between T.M and M.
Tab. 12 compares the corresponding operators in Euclidean and Riemannian geometries.
Isometry. Isometries generalize bijections in Riemannian geometry. If (M,g) and (M,ğ) both Riemannian manifolds, smooth map f : > M called a (Riemannian) isometry is diffeomorphism that satisfies
Gx. (v. w) (dv) (v),ds((w))
16

---

Published as a conference paper at ICLR 2026
where dzf(+): T. [(a is the differential map of f at M, and v, T are two tangent vectors.
B.2
This subsection briefly reviews the gyrovector space (Ungar 2022). which generalizes the vector structure to manifolds. It has shown great success in building hyperbolic neural networks (Ganea
al. 2018 Chami et al., 2019: Shimizu al. 2021).
We start from the gyrogroup. Intuitively, gyrogroups are natural generalizations of groups. Unlike groups, gyrogroups are non-associative but have gyroassociativity characterized by gyrations.
Definition B.1 (Gyrogroups (Ungar )022)). Given nonempty G with a binary operation @ GxG + G, (G,0) forms gyrogroup if its binary operation satisfies the following axioms for any U, G
(GI) There is at least one element E G called a left identity (or neutral element) such that e x.
(G2) There is element Ou G called left inverse of i such that ex
(G3) There is an automorphism gyr [r. vl G - G for each x, y E G such that
(y + =) (ar y) @ gyrlæ, y]z
(Left Gyroassociative Law).
The automorphism gyr(z, y) is called the gyroautomorphism, or the gyration of G' generated by æ, (G4) Left reduction law: gyr(z, y] gyr(r @ y. g].
Definition B.2 (Gyrocommutativee Gyrogroups (Ungar 2022 A gyrogroup (G, @) is gyrocom- mutative it satisfies
I + y = gyr[a, M1 xx)
(Gyrocommutative Law).
Similarly, gyrovector space generalizes a vector space.
Definition B.3 (Gyrovector Spaces (Chen et al. 2025c A gyrocommutative gyrogroup (G, +)
equipped with scalar gyromultiplication @ D::.
G G' is called gyrovector space if it satisfies
the following axioms for 8, E R and x, 2
(V1) Identity Scalar Multiplication: 1 (
(V2) Scalar Distributive Law: (s + @ (0lr=s00000 2@20/er.
(V3) Scalar Associative Law: (st)
(t@x).
(V4) Gyroautomorphism: gyr(z, (t @ 2) = gog[[z..
(V5) Identity Gyroautomorphism: gyr(s @ x, x:
where I is the identity map.
Definition B.4 (Real Inner Product Gyrovector Spaces (Ungar 2022 Let (G, (,, @) be gyrovec- tor space and let (:,-) denote the Euclidean inner product on R" with associated norm ||-Il- We call (G, B @ +)) a real inner product gyrovector space the following conditions hold.
(V6) G C I.n and inherits the inner product (-,-) and norm |-|||
(V7) Inner product gyroinvariance: (gyr(z, y)u, gyr(z, ylu) = =(u, v), Va,y,u,vEG.
|a|8z (V8) Scaling property: [|
[ Yx G (0), Vs R. {
(V10) Let |G|| = [+|]]]]
I E G} C IR. The set |G]] forms one-dimensional real vector space with respect to the vector addition and scalar multiplication induced by @ and G.
(V11) Homogeneity property: ||s @ æ]| = [z||] Vr E G, R.
(V12) Gyrotriangle inequality: || @ vll S || Bllll. Vr,y
When gyrovector space (G, B &) is subset of the real inner product vector space R" and satisfies additional axioms with respect to || forms a real inner gyrovector space. This is analogous to the relationship between inner product spaces and vector spaces.
Definition B.5 (Gyrovector Space Isomorphisms (Ungar
2022
Let (G1, @1) and
(G2, D2 82) be real inner product gyrovector spaces. A map @ G + G2 is gyrovector space isomorphism if it is bijective and satisfies
o(x t1 y) = o(x) O2 p(v). Vo,y G1,
(29)
(1 x) 602 p(x), Va Gre VEE
(30)
17

---

Published as a conference paper at ICLR 2026
Table 13: Riemannian operators on the Poincaré ball and the hyperboloid (K' 0).
Operator
Poincaré ball Fx
Hyperboloid [
Definition
P
(x E R"
=| < -1/K3
H& (x R++1 (z. 1/K, 0}
(x(w,)
A (w,,). v) A + K||r!
(0, c (v,, 1,W1
2 tanh -1 (V -I #M wll)
d(z,y)
cosh -1 (K (ar, ().
K
K cosh (3)
Logg
tanh (Vixll -I txs all)
Em y || tus Gal vll
3zr) B K (x.1))
[K]
8
X |ull
cosh(a)= sinh(a)
Exp, v
9M
tanh
v k
v,
ViKllelle
v ||7||||||
A (u,
K (v, v)e (x + v)
PT.--(())
gYT
A
1 K (1,
and it keeps the inner product of unit gyrovectors invariant,
(o(r), d(v))
(z. v)
Va, E G1 with x + 0, + 0.
(31)
( [v)
a ||l
A useful property is that gyrovector space isomorphisms preserve the gyration, inverse, and identity. Proposition B.6. Let (G1, B1 @1) and (G2, L2 @2) be real inner product gyrovector spaces with gyrations gyr, and gyI2, respectively. If G1 G2 is gyrovector space isomorphism, then for all x. y, 2 E G1,
① (gyn [)) = sv1))(()), (r,
(32)
d(e1) e2:
(33)
(01x)) = θ2δ(z);
(34)
where e, and 62 are the gyro identities in G and Gz, respectively.
Proof. The gyration properties have been shown by Ungar 2022 Ch. 6.21). The proofs for the gyro identity and gyroinverse follow directly from the isomorphism and the uniqueness of inverse and identity Ungar
2022 Thm. 2.10).
B.3 PV GYRATION
As shown by Ungar 2022 Eqs. 3.220 and 3.221), the PV gyration for any x, v, E PVK W is given by
gyr(2, v]= = Ar + By
(35)
where the coefficients are
A (1 (xx) (y, 2) + 2833KK3 (x,w) (y. 8,)3,8, (x,2),
B (1 3
D = (1 B2) + (1- (z, + B,8,)
Here, 3r V is the relativistic beta factor.
B.4 POINCARÉ BALL AND HYPERBOLOID
The Poincaré ball is defined as PR = fir E R || -1/K3 with sectional curvature K < 0. The hyperboloid, also known as the Lorentz model, is defined as
H = x gn+1 (x, = 1/K, ax > 0).
(36)
18

---

Published as a conference paper at ICLR 2026
where (z, v) 2t + (Ie. Us) is the Lorentz inner product. Here, at R and 2g denote the time component and space components. The induced norm 1.llc is the Lorentz norm. Let H& E { Given z, H' and tangent vectors v, H. Tab. 13 summarizes the Riemannian operators.
R"
the hyperbolic space can be defined by its Riemannian operators (Ganea et al. 2018 Chen et al. 2025c). Let e 0 for the Poincaré ball and e = = // for hyperboloid. Given x, 1/2 Hg and E I. the gyroaddition and gyromultiplicaioo are defined as
The gyro-structure
th Exp, (PTe-z (Log, v))
(37)
88 Exp, Log. x),
(38)
gyr(z Ir, y/=
Su (x @N v) Or (a @n (v @n z)),
(39)
On the Poincaré ball Pk, such gyro-structure is known the Möbius gyrovector space Unnar
2022
Ch. 6.14):
x BM y= (1 [1-2K( (x, v) - |1 x + (1 K |x|13)
(40)
2K 2x(x, 8/ + K 13111111111
tanh (tannh - (VVIK(11=))))
x
1 8M x
(41)
VIKi
||=||
gyTM (r, u/2 =z D
(Ax: + By).
(42)
with
A = K² (z. 2) 11v11 K (y, 2) 2K² (x. (v,2),
(43)
B - K2 (v, 2) |x112
+ K (xx, 2).
(44)
D 2K (x, 1) |1 ||
(45)
Here, GMT = -1 8M = -r is the gyroinverse and O is the gyro identity; 0 @M x = x, Vor E
Pk-
Interestingly, the Möbius gyration has similar expression as the PV gyration.
As shown by Chen et al. 2025c Props. 24-25), the hyperboloid gyroaddition and gyromultipicatioon
also admit closed-form expressions:
I,
y = 0,
0,
x Dr y
D K- |K| D+KN
(46)
Otherwise.
MAa,, +...+ D-KN
0,
t 1=00= 0
cosh (10) sinh (t0)
(47)
Otherwise,
VIKI
||xs||| Fx
Here, 0 = coshh" cosh-'(v following notation:
K K(tt),
A, =ob? - 2KKbzg
Kan, and Ay = b(a² Kn,) with the
+ v K| 1+ VK
Ng = || n, = =118=112 32y = (Ke.
(48)
D a 2b² 2Kabs + K2nnnog
N a² ny + 2abs bbng.
In partieular, the gyro identity is 0 and the gyroinverse is O1. -1 60, [Me, x T
19

---

Published as a conference paper at ICLR 2026
Table 14: Summary statistics for the node classification datasets.
Dataset
#Nodes
#Edges
#Classes
#Features
Disease
1044
1043
2
1000
Airport 3188
18631
4
4
PubMed
19717
44338
3
500
Cora
2708
5429
7
1433
C EXPERIMENTAL DETAILS
C.I COMMON IMPLEMENTATIONS
As we use trivialization tricks in our MIR, FC, and GyroBN layers, all parameters in PVNN lie in Euclidean space and are optimized using standard Euclidean optimizers.
C.2 EXPERIMENTAL DETAILS ON IMAGE CLASSIFICATION
C.2.1 DATASETS
CIFAR-10 10 and CIFAR- 100 Krizhevsky Hinton 2009) datasets contain 60K 32x32 colored im- ages from 10 and 100 different classes, respectively. We use the dataset split implemented in Py- Torch, which has 50K training images and 10K testing images. Following [deir et al. (2024), we use data augmentation that includes random cropping with padding of pixels and random horizontal flipping.
C.2.2 IMPLEMENTATION DETAILS
We implement the experiments using the official code of Bdeir al. (2024). ). All models share common backhone, which consists of ResNet-18 encoder followed by a hyperbolic MLR classifier. The output embedding of the ResNet-15 18 backbone is mapped to the target hyperbolic space via the exponential map the identity that is, Exp, (z). Here, 0 for the Poincaré and PV spaces, and for the hyperboloid model. All models are trained from scratch. Optimization is performed using SGD (Robbins & Monro, 195T) with an initial learning rate of 0.1, momentum of 0.9, and a weight decay learning rate is decayed by factor of 0.2 epochs 60, 120, and 160. The curvature for the PV space is set as K -.5..
x 10 Training is conducted with a batch size 128 for 200 epochs. The
C.3 EXPERIMENTAL DETAILS ON GRAPH LEARNING
C.3.1 DATASETS
Disease (Anderson & May 1991. It represents disease propagation tree, simulating the SIR disease transmission model, with each node representing either an infection or a non-infection state.
Airport (Zhang & Chen 2018). It is transductive dataset where nodes represent airports and edges represent the airline routes from OpenFlightssorgg
PubMed (Namata et al., 2012). This is standard benchmark describing citation networks where nodes represent scientific papers in the area of medicine, edges are citations between them, and node labels are academic (sub)areas.
citation network where nodes represent scientific papers in the area of machine learning, edges are citations between them, and node labels are academic (sub)areas.
Cora (Sen al. 2008).
Tab. summarizes the statistics of the datasets.
20

---

Published as a conference paper at ICLR 2026
Table 15: Hyperparameters for PVNN that vary across graph datasets.
Hyperparameter
Disease
Airport
PubMed
Cora
Learning rate
0.01
0.01
0.05
0.05
Dropout 0.4
0.4
0.6
0.6
Curvature
-0.3
-0.3
-1.0
-1.0
Table 16: Hyperparameters for PVNN that shared across graph datasets.
Setting
Epochs
Batch size
Weight decay
Value 2000
128
5 10-1
C.3.2 IMPLEMENTATION DETAILS
We adopt the official code of HGCN Chami et al. 2019 to conduct experiments. The features of each node are embedded into the hyperbolic space via the exponential map the identity. The hyperbolic network consists of two FC layers: the first maps the input feature dimension to 16- dimensional hidden representation, and the second maps from to 16. Each FC layer is followed by activation function. An MLR layer is then used for classification. All models are trained using the Adam optimizer (Kingma Ba, 2013). evaluate performance every 10 epochs and employ early stopping with a patience of 200 evaluations, restoring the checkpoint with the best test accuracy. Tabs. [5 and 16 summarize the hyperparameters for PVNN. For KNN (Mao et al.) (024), HNN (Ganea al.) 2018), HNN++ (Shimizu al. (022), and LNN (Bdeir al. (2024), follow their original papers to implement the experiments. Tab. summarizes the hyperbolic layers used in each model.
C.4 EXPERIMENTAL DETAILS ON GENOME SEQUENCE LEARNING
C.4.1 DATASETS AND PREPROCESSING
We the Transposable Elements Benchmark (TEB) datasets (Khan al. 2025). This benchmark provides seven DNA sequence classification datasets spanning three prediction tasks: retrotrans- posons, DNA transposons, and pseudogenes. We focus five among them, summarized in We follow their original train/validation/test splits and preprocessing.
Tab. 18
Table 18: Statistics for the TEB datasets.
Prediction task
Species
Max length
Dataset
Train / Val / Test
Retrotransposons
Plant
1000
LINEs
22502 / 2030/ 1782
500
SINES
21152 1836 / 1784
DNA Transposons
Plant
1000
hAT-Ac
17322 1822 1428
Pseudogenes
Human
1000
processed
17956 1046 1740
1000
unprocessed
12938 766 884
C.4.2 IMPLEMENTATION DETAILS
For the Euclidean CNN and the hyperbolic CNN baseline (HCNN-S). we directly use the results reported the original paper (Khan et al. 2025 Tab. 2). Our PV CNN architecture follows their implementation" Each DNA sequence is represented as a length I. sequence with 4 input channels. We first apply a PV convolution that maps the 4 input channels tangent batch normalization and tanh tangent activation. A second PV convolution layer is then applicd. The final PV feature is concatenated and passed through an FC layer, and finally classified
channels, followed by PV
https:///githbb..cmm/mbbbb
21

---

Published as a conference paper at ICLR 2026
Table 17: Summary of the hyperbolic layers used in the graph node classification models.
Model
FC layer
Activation
MLR
PVNN
PV FC in Thm. 5.3
(eExoo (ax))(a)))
PV MLR in Thm. 5.2
KNN
Logo (W ExPo [T)
Logg (Expo (a:])
Enclidean MLR after Ex
HNN
Logg ExPo (e))
Loga (Exppll (x))
Poincaré MLR Ganea al.
2018
HNN+
Poincaré FC Shimizu al
2021
Loga (<xDa (al))
Poincuré MLR Shimiu
LNNN
Lorentz KC
Chen cl gl
Lorentz activation (Bdeir et al. 2024
Lorentz MI. Bidee ct at
Table 19: Hyperparameters for TEB.
Setting
Value
Optimizer
Adam
Learning rate
le 4
Weight decay
2e-2
Batch size
100
Dropout
0.1
Adam (31, 92)
(0.9, 0.999)
with PV MLR head. The curvature is initialized K -0.5 and learned during training. We train for 100 epochs with a step learning-rate schedule, using milestones at epochs 60 and 85 with decay factor 0.1. For the PV FC layer, in Eq. (23) is set tanh. All other hyperparameters are summarized in Tab. 19
C.5 HARDWARE
All experiments are conducted on an NVIDIA A6000 GPU.
D CONNECTIONS TO THE HYPERBOLOID
This section discusses the connections between the PV model and the hyperboloid model. We first show the isometry between the two models. Then, we show that several current hyperboloid network layers can be rewritten PV layers.
Proposition D.1 (PV-hyperboloid isometries). ! The following maps are Riemannian isometries between the hyperboloid model Hi% and the PV model Fvr::
mag - :H
Zt Jz
F. E TPV
(49)
|x/| - E Hj.
mFV3 -11g PPV K 3 x
(50)
Implications.
The PV-hyoobboloid isometries in Prop. D.I imply that several standard layers in hyperboloid networks can be rewritten as PV layers composed with map -rve and Teve -Hn-
22

---

Published as a conference paper at ICLR 2026
The Lorentz activation [Beiir al., 2024 Eq. 13), Lorentz FC layer (Chen al. 2022 Sec. 3.1)
and Lorentz concatenation (Bdeir et al.
2024 Eq. 32) are
LAct
Zt [Ts]]
(=(o((=>)1 (
K
(51)
VlWa. + +bll
LFC
(52)
[ea]]
xx +
N
N-1
i== Fie
HCat( (CCat(f2))=11=
31.8
E H
(53)
NN
where x = xx,, E H& and n; = [m "..."" E Hi for 1 s N. Then Prop. D.1 implies
that the above Lorentz layers can be rewritten in terms of PV layers follows:
LAct(x) -P - (T (x))),
(54)
LFC(x) =P -Fg (W कम P (x) + b),
(55)
HCat((=:))= TPV: EER ( -PVR (x1)...... ...., TER - (IN))
(56)
These identities show that many hyperboloid constructions effectively operate by mapping to PV space, applying Euclidean building blocks there, and mapping back through - This per- spective naturally motivates desigming networks directly in PV space, instead repeatedly switch- ing between equivalent models. Moreover, even if one follows the pattern HITk + PVV - PPV + HI construct FC layers, the intermediate map should be the PV FC layer from Thm. 5.3 rather than a naive linear map, since PV is non-linear Riemannian manifold and intrinsic layers must respect its geometry.
PROOFS
E.
PROOF Eq.
The PV line element at I E PVR can be written in terms of the curvature parameter K < 0 as
(u(u) =|uu|² + KK (z, u) Vu e TyPVI R"
(57)
where B, 1/1-K||=|1"
This is equivalent the expression in Ungar 2022 Eq. 7.76) after
substituting 8² 1/K. Given u, E T.PVA. K. the bilinear form (=(", v) is obtained by the polarization identity:
(u, v)= = (Qa Qu - v)). v))-
(58)
We first expand the terms in the polarization identity:
Q=(u + v) = =IIu + u112 + K33 (ar, u + v)
= |||u| + (u, u) + |||1|| + K89 (a, u)² + (z, w) (x, 0) + (cr, 0)),
(59)
Qo(u- - v) ||u u|| + K3 (x, u)²
||u| - 2 (u, u) +||0||2 K33 (z, u)² 2 2 u) (x, + (2, u)).
Taking the difference yields
(u Q - 4 (u, w) + 4K32 (z, u) (I,
(60)
Substituting this expression into the polarization identity, we obtain
(u, u, v) 1 (Q
(w)
(61)
(u, v) K8 (w., u) (I,1),
which coincides with the expression of the PV metric in Eq.
23

---

Published as a conference paper at ICLR 2026
E.2 PROOF OF LEM. 4.1
Proof. Differential of TPV% - Consider the curve e: (-8,e) + PVR which satisfies c(0) = x
and c'(0) v. By definition of the differential,
dz TOVR PPx (v)
x PP (c()).
(62)
de 1=
Using TPVp (x)
8 I+3,"
with 3g Vi-K| we write
Be(e)
TIVR-PR (c(1)) = 1(())(((),
R(1)) 1 Bo(t)
(63)
Let r(1) (r)(((((((())))))))) =|[c(()]] so that 8:(0) = (1- -rrr(t))-144 Then
(0) = 2 (a, u) Bao0 = {1 - (r(00) 3/2
K3 (r, v).
(64)
Differentiating (t()) at t = 0 gives
{'(0))
(0( (1 + 3,
(1 + Bx) (x, v)
(65)
Finally, differentiating h(t))(t) at t = 0 yields
dz (MPVR~>P) PP (v) 11(() + M()e
(66)
K (1 + B,)
(z, 1) +
1 B.
In particular, at x = 0 one has 90 = and (x. 0) = 0. Thus, we have
9o
do(Tvv-- PPR (v) 1 900
7 2 v.
(67)
Differential of -PP-- Consider the curve e (-8,6) + IP which satisfies c(0) y and c(()) w. By definition of the differential,
d
dy (Tp x -PVp (w)
Tpp -PVp (c(1))-
(68)
dt r=
Using the explicit expression Map -FVR (1) = 27 with 7/-
we obtain
V1+ klyy
T2h -FVR (c(()) = (x
(69)
Let rtt) r(t)=[[c(t)]] =| so that 22(00 (1 Kr(tt) Then
d
Kr'(0)
(0) = 2 (y, w))
2 Pe(e)
(1 + Kr(0))= -2K (M, w).
(70)
dl
1=
Differentiating 27 (t) at yields
dy(Tpp -pvp (w) 2 d
2o 27
dt
(71)
1=
=-4K 1 (y, w) y + 2 w.
In particular, at y = O we have "o = and (y, w.) = 0. Thus, we have
(w) = 2w.
(72)
24

---

Published as a conference paper at ICLR 2026
E.3 PROOF OF THM. 4.2
Proof. It suffices to show that for any a E IPV and U, w E T.PVK.
oy de TFVE -PP (w), (TevR-Pp (w))
-08 (v, (v,w).. w),
(73)
where y -PV. PR (I)).
We first recall the following equations from Eq. T App. B.4 and Lem. 4.1
Ix p (v. w) = (u, w) + K K (2) o) (o, w), Vr E PVK Vu, w E T,PVR
0u, 2) (25) (u, vy Vu, 2 E TyPk.
Br.
(74)
Tw (x)
Vo E IVV
1 1 Bz
3
dy (1o7 (v)
+ K-
(c, x, Vo E PVR Vo E T.PVI.
1 1 Bz
(1 + B2)
Let
a
K
(75)
+ Br
(1 + 3x)
Then dr
= av + (x, u) and d, (TVV -PR (w) aa + (x, w) x. Thus.
dx ( +P (w). de - (w)
gy
(2%)
2 (av + b x, aw + b (z, w) x)
(76)
(
(a² (u, w) ab (z, w) (v,x) + ab (z, v) (z, w) + 6² (xx (x (x,x))
(x4) ? a² (u. w) (2ab 63 11111))) (or, u) (z, (x:
Using y MyV. -rg (I) and the relation between A Br. and from App. B.4 coefficients. First,
simplify the
8n
8n
K||yl12 K
1 Br
1
B
B2
K
|
(77)
(1 + ),))
32 1
(1 + 32)2²
where we use 82 1-K Hence
B2 -2-1
2,8
+ K ||
+
(78)
(1+ (1+,,)²
1 BB
which implies A/
I+f 3,
Therefore
2 (xK)
11 Bx
2
Ba
2
a²
(79)
3r
+
3x
Next, we compute
3n K
83
2K 34
2ab = 2 1 + Br
(1 + 32)
(1 + 1,)3)
(80)
86
32 -1)
|/|10|12 = x²
|| = K
(1+ 3r)
(1+ 3,)
25

---

Published as a conference paper at ICLR 2026
which brings to
2ab 11|1=]1
(2 + Bx) + 8/ -1)
(1 + )z
(81)
(3,, 1)2 K
3
(1 + Bx)
(1 + )²)
Multiplying by (25) (x5)²= =11+,0)) yields
(x5) (2ab b² 1211=11²)
1 + 32
K
B4
K3
(82)
+ 3,)²
Substituting these identities into the expression for ge gives
by (oe (NeVX-PR) (w), dx (TVVX +F% -Px
(w)) = (w, w) + K3 (c. v) (c., w) = (0, (v,w). w). (83)
E.4 PROOF THM.
Following the notation in the main theorem, we further denote:
(x:
K
= E Pk. dr,(u) E TT
(84)
Recalling Eq. and Lem.
4.1
we have the following:
Az
T(x)= +
x,
Vx PVR.
(85)
B2
n (T) 2750.
Vi E Fk.
(86)
g
B2
dT (v) = K
(1 + 32)² (I, x
U,
Vz E P Vo E T.PVY,
(87)
1 Br
(w) = - 4K (ū. w) + 2, V Yu E TyPk
(88)
Next, we derive the expressions for each PV operator.
E.4.1 PV EXPONENTIAL MAP
We recall from Tab. 13 that the Riemannian exponential on the Poincaré ball is
K
K
(x) θm
tanh
(89)
V-K
2
|
By the Riemannian isometry and the gyrovector isomorphism of æ, for any x PVR and v e have
T, T.P
K
Exp, (v) ! -- (Expr(T)
( tanh
V-KA
(90)
@u n.
2
|
The above equalities follow from the following facts.
(1) Isometry.
Gyrovector isomorphism.
26

---

Published as a conference paper at ICLR 2026
Let
K |,
u
K
tanh
||8||
(91)
We have
V K |0|
||]||
tanh
(92)
V-K K
2
Lett V
so that v K |u| = tanh(t). Then
2
(u)
1 K ||/l
tanh (f)
+ K |||u|]
V
K
|||||
1
tanh(t)
1 tanh?(t) V-K
||0|
2
tanh(t)
V-K tanh²(t)) |0|
①1)
2
tanh(t) cosh cosh"
V-K
K
2
sinh(t) cosh ((t)
(93)
V-K K cosh(t)
|
2
sinh(t) cosh(t)
V-K
K
|0||
(
1
(2sinh cos)(())
T
K
1
sinh (21)
V-K
K
|
1
sinh
(
K
K |81)
$
T
V-K
K
V-K(1 + 3R)
8
sinh
||
3z
|
The above equalities use:
(1) - -tanh?(t) 1/ cosh 2 (t);
(2) sinh (2t) = 2 sinh(t) cosh(t);
(3) 1+0. 8=
E.4.2 PV LOGARITHMIC MAP
We recall from Tab.
13 that the Riemannian logarithm on the Poincaré ball is
2 tanh -1
(V-K|E=1)
Log
2.
2 z= == PM 0.
(94)
K-KAK
|
where A
We define
2
(-x) @ y, == m())
(95)
27

---

Published as a conference paper at ICLR 2026
By the Riemannian isometry of и,, we have
Log, (v) dæ d (Log
tanh
1|
K
2
= =d((==1)
(96)
V- K
||
= =d( (2),
where
tanh -1
(V-K|
2
a(x v) V- K
(97)
|||||
The differential of π at 8 π (ar) is
4h E TzIPk d (h) -KKK (z. h) 2 2
(98)
where 7/2
V++ Using I = 1+ x and the relation
- K||00||
D2 we have
1
82
82
82 12-17 1
B2
K K| 2 K (1 + B₂)²
|0/1
(1
B₂)
32
=
(1 + Pz)
(99)
B2
233
1 1
K
| 2 1 +
(1 + B,)
1 Br
Hence
1
1+ 3x
1 Bz 2,3
(1 B,)²
2/
7/2
(100)
1 K |
2,3
4,2
Substituting these into (= yields
(1 Ba)²
1+ BO
dz -4K
(2,h)
4,32
2,3
=-K (1+ B²)²
1 Ba
(7, (2,1)
(101)
32
8
1 + Br
h K (x. x.
Bz
where the last equality uses that Ø
I+B x. Applying Eq. 1011 with = and using that
Br
E = (T()) is collinear with (-x) @uu y, we obtain
Log, (3) a(x 8) (ddx))
+ Br
(102)
a
K (,,
3r
4 and are collinear and Since x=2) and is given by Eq.
32
pz,
(103)
1+ 32
which also implies (x, = p (x, z). Substituting these into Eq. 1002 yields
1 + Br
Log, (v) = a(x, v)
pz
Kp (x, 2)
32
1 Br
(104)
= a(x, v)
+ (-Ka(I, v)p) (z, 2) x.
BN
T(1,8)
Oxx, y)
28

---

Published as a conference paper at ICLR 2026
Using the definition of a2 y) in Eq. (97 together with A ward simplification yields
1+ and p 1+8."
straightfor-
tanh -1
[V=K||
tanh -1
||
2
25
(I. V-K
((z
(105)
|=||
1 Br
||
Thus,
Log, (v) = (x, u) + [2, (1 (2,,2) x.
(106)
E.4.3
PV PARALLEL TRANSPORT
We recall from Tab.
13 that the parallel transport on the Poincaré ball is
A 25
PT p
gyTMlū. -ww),, with E TgIP..
(107)
We have
PT
①1) dg(# 1)
(TT - (dTx
A gyrralu - x (t))
dg 1)
1
2 AK
dg (T 11]
[gyTMll: - (dT (dTz())))
X
(108)
3
K
1 + By
gyrMlu. - (d)z(v)) - K (..gTTMl. -x] (dma(w)))
A
By
(4)
(1 + B2y
1 By
gyI -i] (dT=(()) - K (D: (. 1.. - (d
(1 + 3,
By
+ Az
KYTM [7, x] (dTx( - K (1+ B2)By ( (U.KYTM 1. - (dxx )
Bi
(1 + B,)
The above equalities use:
(1) the isometry property of ,;
(2) lincarity of dg(a=
(3) Eq. (IO)).
(4) Using the relation between Al and B. in the proof of Thm. 4.2
1
Br
1 + Bg
A
A
(109)
B
9
By
E.4.4 PV GEODESIC DISTANCE
We recall from Tab. 1 that the geodesic distance on the Poincaré ball F is
2 tanh [V-K](-v)) @M Bell)) V1. E PR.
d (y1. U2)
(110)
By isometry and isomorphism, the PV geodesic distance is
d(z, y) = d (T(2), T()))
2
tanh (V (V-K (- (x) BM T (())l)
V-K K
(1II)
2
K tanh
[--K|Ta(Iee e vill.
29

---

Published as a conference paper at ICLR 2026
E.4.5 SPECIAL CASES THE IDENTITY
Exponential map at the identity.
v
V-K(L K[
Expo(u) (v)
sinh
|11||
K
fn
(112)
1
sinh
V-KKleol)
K
||o||
The above comes from the following.
(1) 0 is the gyro identity;
(2) 30 and d (v) = u.
Logarithmic map at the identity. As x = -0 tu y y, we have
Logo (v) or o0, +0, (0, 2)
= =0 y)y
(113)
2 tanh (V-X||
V-K
K
y.
Using (vy)
8y T+
y, obtain From π (v)
By 1+ y and By VI K||l|
we obtain
By V- K K|1b/1
K [()]]|] = 1 + By
(114)
Let = V- Klylll and 8 = v1 + 12, so that By
Define
V1 - Kllll
3g
(115)
1 + Bg s+1 1
Using the hyperbolic double-angle identity, we have
tanh (2 tanh -1 (a)
2a
+ a²
2t/(s 1)
1+
21(s +1)
(s + 1)² +
21(s + 1) 3 +2 + 1 +
(116)
+((++1) 1)
21 +
t(s 1(s+1)
1+++8
1(s+1)
32 +8
t
s +
Denoting u sinh (t). we have
cosh(u) = / + sinh?(u) = / +
(117)
Therefore,
tanh (inh ((() tanh(u) = sinh(u) t
(118)
cush(u) V1+
30

---

Published as a conference paper at ICLR 2026
Since tanh is strictly increasing on R. this implies that
By 1 By
2 tanh -1
sinh -1 -().
(119)
Substituting this identity back gives
sinh (V-R 11sll)
1
(0,
(120)
V-K
|
and therefore
(y)
1
sinh
y (V-Kloll)
(121)
V-K
Parallel transport from the identity. The gyration satisfies gyTM (.. gyTulg. 0] I. Substi- tuting this into Eq. (108) gives
B0
(1 + 80) (y.dro(v))
(To-+(U))
doo(v)
K
90
(1 + 9,)30
28,
= 11 K
(y, u)
(122)
2
1 + By
2
By
K
(y, v) y-
+ By
Parallel transport to the identity. Taking y 0 in Eq. (108) and using gyTM -ā] I yiclds
1 Br
PT,=o(())
doT((1)
Br
1+ Be
B3
B.
K
(x, v) x +
(123)
B₂
(1 BB)²
+ B
B2
U K
(x, u)
1 B3
Distance from the identity. This can be directly obtained by gyro identity.
a(o, v)
V-K tanh (V-KII (w)))
2
(124)
Using the same identity as above with /- K [yl] yields
d(O.v)
v sinh -1 (V-K(1e1)
1
(125)
E.5 PROOF OF THM. 4.4
Proof. As shown by (Chen et al. 2025c Prop. 21), the Möbius gyroaddition and gyromultiplication can be written by the Riemannian operators. Besides, the isometry TyVg -77 preserves the identity:
TrVg -gg (0) = 0. By Nguyen & Yang 2023 Lems 2. 2-122, one can directly obtain the results.
E.6 PROOF THM. 5.1
We first establish the PV hyperplane equivalence and then derive the distance formula.
31

---

Published as a conference paper at ICLR 2026
E.6.1 EQUIVALENT CHARACTERIZATION OF THE PV HYPERPLANE
We first review useful lemma from Chen et al. 2025b Lem. J.1).
Lemma E.1. We assume that the manifold M admits gyrogroup defined by
+ y Exp= (PTc-x (Loge (U)) Vz, y M.
(126)
where e M is the origin of the manifold. Then, we have the following
(Logp()))a),
(Ep (ep 3 2), PT (a))) Vx,p M and E TpM.
(127)
Now, we ready prove Thm. 5.1
Proof of PV hyperplane. Thm. 4.4 indicates that the assumption of Lem. B.1 holds with M PVV.
+ = @u and e = 0. Then, the PV hyperplane
(L08,(0),0)-- =0}
Ha
x E
PV
(128)
can be rewritten as
Ha. {x E PVR (Logol-p @u x), PT pTp-)((a)) = =0]
(129)
Using the explicit PV operators in Thm. 4.3 and the PV metric in Eq. (T) we have
Logo(-p @u x) a(-p tu x), for some scalar 2 0, (a)) = (d),(a), for some scalar /3 > 0, (o v) (u,
(130)
As a is trivial, we only consider the case >> 0:
(Logol-p @u x)) PT
xr, d,)((a)))) =0. 0.
(131)
E.6.2 PV POINT-TO-HYPEEPPANE DISTANCE
We first prove a lemma on the isometry and point-to-hyperplane distance, which will be used to derive the PV point-to-hyperplane distance.
Lemma E.2 (Isometry and point-to-hye hhypetl-hyperpllnn distance). Let (M,g) and (M, g) be Riemannian manifolds and let & M - M be a Riemannian isometry. For p E M and a E T,M, define the hyperplane
Ha f Z 1p (Log, =0] -
(2),,
(132)
Let p = ((p) and = d, E Tr,M, and define the corresponding hyperplane on M by
Hap = - Fp (Log,(t), = 0)
(133)
Then é maps Ha.p onto Ha.p. that is,
ò (Ho.p) Ha.
(134)
Moreover, for every x E M we have
dm (x. Ha,p) = d (0(x), Hl.s)
(135)
when the poin-tto-hyeerplane distance exists. Here, dm and d denote the Riemannian distances M and M, respectively.
32

---

Published as a conference paper at ICLR 2026
Proof. Since ø is Riemannian isometry, have
9p (LoRg(t),a) Do (1p5 (Lo8g(e)) drd(a)) - 9p (Logg (0(x)), a).
(136)
Therefore,
Dv (Logp(0), =0
gp (Lög, (d(x)) a) 0,
(137)
which shows that Ha. if and only if (xx) Ha.p and hence & (Ha.)) Ha.p. For the ceint-to-hyperplee distance, recall that for a subset S C MM the distance from x to
dm(r, S) = inf dm(a, sES
(138)
For the point-to-hyperplam distance, we have
dm (z. Hap) = inf d2(x,2)
=EHu..
inf dm (d(x), x(z))
=EHl..
(139)
inf dm (x(x),
sefla,p
=da Ha,))
Next, we review the Poincaré hyperplane and point-t--hyperpaame distance Ganea et al. 2018 Sec. 3.1).
Poincaré point-to-hyperplanne distance. For point Fn and a normal vector E Tp k the Poincaré point-to-hyperplane distance is given by Ganea et al. (2018) Thm. 5):
(Log)(Z), = =0} = for Px 1 (-p x, a2 =0]
H 0.P
x E PR
(140)
2 2/-K BrM y, v,a)l
d' (y, Happ)
sinh-1
(141)
V-K
(1 Kl1-p BMM v |aal
Proof of the PV point-to-hsperplanee distance. Let
p -==
(p)
Pk, = da) TTPR k: ύ= x Pk-
(142)
By Lem. E.2 the point-to-hyperppanee distances satisfy
d (v. Haw)) = d (ū. Ha,))
(143)
Applying the Poincaré distance formula 141 with p. a = ā. and gives
2V-K | P DM V. a)
d' (ū. Ha.p) =
sinh -1
(144)
K
(1 [1+K]| K| CDM 17 51²
|a|l
The gyrovector isomorphism & implies
BM y = (-pp @uu y)-
(145)
Denote -p @u y. From Sec. 4 we have the explicit expression
32
(2)) =
(146)
1 1 B
33

---

Published as a conference paper at ICLR 2026
with 3g 0. Since B₂ V1-K||=|12
obtain
B3
<||[<<>)|>
=1)
||=11 2
1+ 3x
111
-1+
(1 3,)²
8.(1 359
+
(1)
(147)
82 -1
+
(1+ 3,)²
3.-1
+
1 8g
2
1+ B
The above yields
2v-K (TT(),, a)
V-K az,
(148)
(1 + [1=]|] ||ā||
||a||
Therefore,
V-K K
( (-p ou y. dor((a))
d (y. Hap) = V-K
sinh -1
(149)
E.7 PROOF OF THM.
Proof of PV MLR. For clarity, fix class index l and omit in the notation whenever possible. We denote π = TPVg PPR as in Thm. 5.I
Step 1: From hyperplane distance to signed score. The PV MLR in Eq. (18) associated with parameters (p, a) for x E PV
(x(x)
siign (<-pe @u x. dm, (at))) |laellaa d (xr, Har.pa)
V-K (-pR Ph Ou x, dr {axx))
||ax V-K
sign (( ((-ps pk Bu x, dp (ax)))
sinh
(150)
[l (ax)ll
|aallpe
Pk tu x, dpe (ax))
V-K sinh
ddy (ax))
The above comes from the following.
(1) Thm. 5.1
(2) sinh 1 is odd and strictly increasing.
Step 2: Trivialization and reduction single direction. We adopt the unidirectional parame- terization in Sec. 5.I.
Ps = Expo (r*[=x])
a% = PTo-ppa (2x),
[2x] = 2
(151)
||
with z E PVP ~ IR" and r% E R.
34

---

Published as a conference paper at ICLR 2026
As parallel transport is an isometry, we have
|| lpp 11=x/lo |=x||-
(152)
Moreover, and are collinear, because Expo in Thm. 4.3 preserves directions at the origin.
Using the explicit expression of PTo- the origin in Thm. 4.3 we see that PTo-pp, maps 25. to a linear combination of 2µ and Therefore, is also collinear with Eg.
The differential dpp π in Lem. 4.1 the form
dpp π (v) age + 9k (Pk: v) Pk. > 0, Bk E R.
(153)
so dp, maps any vector in span(=+) into the same one-dimensional subspace. Consequently, there exists a scalar d > 0 such that
dpu (ap) = 2
(154)
The sign of A can be absorbed into 26 by redefining 21 +- - if necessary. Without loss of generality we may assume Aς > Putting Eq. 151p. Eq. 152). Eq. (154) and || x (ax))l A into Eq. (150) yields
| V-K
VV-K |El||
(-pa Bu x, 2,
(x(x)
=
sinh
(155)
Step 3: Eliminating the gyroaddition. The remaining task is to expand the gyro-additive term in PV gyroaddition is given by
Eq. (55). From Sec.
3
Ou v u +
1-
B.
K
Bu
(u, (u.
Bw
1
(156)
3
1 Bu
V1
K k|
Setting Pp and yields
1 1-
Bpe
Pk tu x = Pk + x
K
(-pke x) (-pe).
(157)
Bx
1 + Sy
Taking the inner product with z& gives
1 1-B2 Bx
Sp
( (-Du (-Pks 2nd
(-ph @U x, 2x) (-Pes 2x) + (x,, 2x) +
K
Br
I + Spx
(158)
1- 3r
3pm
(x, 2x)
1
K
( 1-Pk. 2))
( (-Des 2x).
Oz
1
Bpn
Next, we rewrite the above expression using the unidirectional parameterization of pk. From Eq. (15l) and the explicit PV exponential at the origin in Thm. 4.3 we have
p = Expo (r*[z*])
sinh (V-Kig Krk) V-K
2k
(159)
||
Thus,
Pkk 2n)
sinh (V-KrK) ||
(160)
V-K Moreover, since pk and 2y share the same direction, any admits the decomposition
(I, 24)
21 + HL.
I1
1 1
Zk.
(1) = 0,
(161)
which implies
(-pk, xr) = (-Pk- Pk. mn)
=
(x, 2xt
|| 2 1|2
(-pk, 2x) = V sinh (V-Krz)
(x, 2x)
1z
(162)
The beta factor at p': is
1
sech (V-Kre).
Ppn V1 K
(163)
1
35

---

Published as a conference paper at ICLR 2026
where we used | - sinh? (v-Krk) and the identity 1 + sinh"(t) = cosh (t).
Using (-pk: 2k), < Pka 2), and Spe. we have
(-pk UUU 24, 2kx
Bz
Spe
(x) +
1
K
(-pks 23)
(-pks znd
3z
+ Ppz
1
3px
(xr +
K
(-Du, 2t)
(-Pks 2x)
8z
1 + 3pa
sinh
(VKTx))
1
+ -2,
||=E||
Bx
V-K K
(164)
Byr
sinh
V-Krk)
(x,, axx)
sinh (V-KTs)
K
||
1 Ppn
V-K
11
V-K K
sinh V V-Kr())
Krk
Spa sinh?
(v- Kroa)
(v-Kre)
||zØ|
(ar, zx)
(2e, zxd
V-K
Br
1 + Spu
pk sinh?
(V-Kre)
sinh (VKrr)
||
(20,2x)
+ Bpe
V-K
Br
Since 3r, sech
Kr= and + sinli (V-Krs) cosh?
(V-Kra) 1/Bp we have
Sp. sinh? (V-Kr&)
1 1 Spp + Spe sinh?
VVKrr
+
1 + 3pk
1 Bpr
+ Spa cosh? 2 (V-Krk)
(165)
1 3px
+ 1/30z
cosh
(V-Kra)
1 + Ap
3pm
which implies
sinh (V-Krk)
(-pk Pk @U x, 24) cosh (V-Krk) (0, 2x)
||
(166)
V-K
B
Recalling that B2 = 11--/1-KKaaa//l K| we obtain
sinh (V-Krz)
(-pk @u F, 2x) cosh (V-Krk) (x,, 2x)
||
K
(167)
V-K
Substituting Eq. 167 into Eq. 155
arrive
(x) 2 sinh
cosh V (V-Kk))
V-K
(x,, 2x)
sinh (V-KTe) V1-K||z] K|
||za||
(168)
36

---

Published as a conference paper at ICLR 2026
Proof of PV MLR limits.
By Taylor expansions, we have Kr
cosh
(V-Krz
Krk)
+ +((K))
sinh
(VKrkk
Vrk O (1-K)//2
(169)
K|
K |||
+O(K3).
2
The argument of sinh (-) in Eq. 168 be simplified
sinh
cosh
(v-Krs)
K
(or, (sr,2k) - sinh (V-Krz) V1 K |//1
|
{1-- Kr
+0(R2)
V-K
=sinh"
(Ir, (1, 2k)
Krk +0 ((K))//)))
(1-
1
K 1
+ OO(K))
(170)
2
(x, [x)
8(( (1
sinh
V-K
rk)
+ O
|2ael
(ar,
+0 +0 ((-K))//2)
= V-K
||z
Substituting this into Eq. 168) gives
||
VVK (1,,2k)
Tk/ +0 +0 =(-x))//2)
Uk (x)
V-K
||*1||
(I, 2b)
|,
Tk + +0(-K)
(171)
|| ||
(x,, 2k)
rk |< (-(K))
K-+0, (ar, =x) - -r
E.8 PROOF OF THM. 5.3
Proof of PV FC layer. Specializing Thm. 5.1 p and ex and using that -0 -Ou y gives the LHS
sign (do. (ex). - @u 2) d (y. Her,o) V-K sinh (v- VV-Kys)
(172)
with 8/ (y. ex). Then, we obtain
yk V-K
1
sinh (V-Keu(z)) k = 1..... ......m
(173)
Proof of PV FC limits. By Thm. 5.2 as K + 0 we have
un (xxx) + {xr, 24) + bE. by = -rr =
(174)
For K < 0 and (x(x) A O, rewrite y/
sinh (VKox((r)
9k = (x)
(175)
V- -eKu((z)
37

---

Published as a conference paper at ICLR 2026
and we define the fraction to be 1 when 24 (2) = O. Since V-K + 0 and fx(x) converges to
finite limit, have (x) 0. Using the standard limit lima- sinh(u)/u 1. it follows that
sinh (V (V- (V)))(())
+1 K 0
(176)
Vu
Combining the above limits yields
lim K-0
yk
lim ux(x) (xx, 2x) + bk-
(177)
K-+0
E.9 PROOF OF THM. 5.4
Proof. The result is first established in the Poincaré ball model (Chen et al. 2025c Thms. 14 and 16). Since a : PVtooP is Riemannian isometry, it intertwines the key geometric operators used in the proof:
du o ExpE = Expx(e) odTz, PV x 0 y) (x) 0 (1))
dug Log PV = Logx(z)) odTT,
(178)
(179)
Hence the homogeneity identities, written purely in terms of Exp. Log and the gyroaddition Cr. are preserved under л. Therefore the same theorem holds for the PV model (Chen et al., 2025c Lem. 11).
E.10
PROOF OF PROP. D.I
Proof. We first recall the isometries between the Poincaré ball and the hyperboloid (Skopek et al.
2020 Sec. 2)1):
-am--P -PR (I)
(180)
1 + / K|
1 K K|
VIKi
K K| 2
TP - (v)
(181)
2y
1 |lyl
Hence, the following are Riemannian isometries:
THe -P T PPV T
TEVR-ER
-Hg
TTVR -Pk
(182)
It remains derive the explicit formulas.
For = [TT Hg, we first map to the Poincaré ball:
= k (x)
(183)
1 K 112
Applying Tेд -PVR from Eq. 4 yields
The -VV (x) Tpg pVp (v) 27
7g
(184)
1 K K|bBl1
Using y = x1/(1 /K (1)). we compute
(1 / [K
2
K K|<...1'
||
+ K| 1
(185)
(1 (1+ K
2
(1
38

---

Published as a conference paper at ICLR 2026
Since Hi& satisfies (x, 1/K, have
(x, -x² + +100811 K
|0,1| xt K
(186)
Substituting this into the numerator gives
(1+ vk(ss) 2 + 111=0111 = (1 + viKl + +K o + K
-(1+ viKl.a) Ka² +
(187)
11 2 KK + |K|0, + Kx +
2 1 v1K(
Therefore,
2 2 v 100
2
1 K |y/l
(188)
(1+ v (x,
+ VI | 10
and hence
1
14 + |K
2/
(189)
+ K |111/1
Finally,
THy PPV (z) 27 2
1
V | D
2s
(190)
2
1 14
K Jarz.
For - take a E PVR and map to the Poincaré ball by Eq.
(4)
8r
y == mPV (x)
T,
B2
(191)
1+ 82
V K [| 12
Applying meg - we obtain
1 K K1ll11
VIKI + Klvlll 2
TEVE - (z)
TTp
Hg (v)
2y
(192)
+
||
We now simplify the spatial and temporal components separately. We write
3
3
|
|| 2
(193)
1+ 8
1 3
Using 33 =1(11 K|| ||r])]) we obtain
32
83
By 1
K K||y/|² = K 11
(1 + Bz)²
(1 + Bz)
(1 + 3z)
(194)
2,3
2
1 K|
K
K||b81
+ 3x
1 3x
The spatial component of गाyz +Hg (z) is
8g T.
2y
(195)
+ K K|yy||
20,
1+8.
and the temporal component is
1 K|
2
K|| K|
I+9.
(196)
K 1 K||b|
28_
V
K
I+9.
39

---

Published as a conference paper at ICLR 2026
Thus,
-
TFVR - (4)
(197)
40


Reviews:
[{"review_id": "U2zSHPkVrO", "reviewer": "Reviewer_36hh", "summary": "This paper proposes some building blocks for neural networks on hyperbolic space based on proper velocity (PV) model. Specifically, the authors propose multinomial logistic regression (MLR), fully-connected (FC), batch normalization (BN), convolutional, activation layers. These layers are validated on graph node classification, image classification, and genomic sequence learning.", "questions": "The paper aims at constructing some building blocks for HNNs. However, the authors simply use the approaches in [Shimizu et al., ICLR 2021; Nguyen and Yang, ICML 2023; Nguyen et al., ICLR 2024; Chen et al., NeurIPS 2024] rather than developing new approaches for their construction. Although the PV model is less studied in previous works for HNNs, its geometry is well-understood, making it straightforward to adapt existing approaches to this geometry.  \n\nThe exposition of the paper can  be significantly improved. For instance, the tables look messy with small texts and numbers in some tables and too big ones in others. \n\nExperimental results are not convincing and important baselines are missing:\n\n- Some experimental settings are given in Appendix, but these are very sketchy. For instance, for node classification experiments, it is crucial to provide and discuss settings for the dimension of node embeddings. However, I did not find this setting in the main paper and Appendix. \n\n- The Hyperboloid model is used for comparison on node classification experiments (Table 2, Bdeir et al., 2024) but is not used for comparison on image classification experiments (Table 6). It is noted that the Hyperboloid model outperforms the PV model on both CIFAR-10 and CIFAR-100 datasets in terms of mean accuracy (see also my first question below).\n\nQuestions:\n\n1. The authors argue that the PV model is an unconstrained representation of hyperbolic space which can lead to more stable results for HNNs. I am not sure why this is the case ? It seems that the Lorentz models is also another unconstrained representation of hyperbolic space, since the time dimension is computed w.r.t. the space dimensions. Why the PV model can be more stable than the Hyperboloid model in the considered applications ?\n\n2. In the introduction, the authors claim that the new representation space can be a good alternative for existing models of hyperbolic spaces due to its numerical stability. However, its performance on datasets with high hyperbolicity is inferior to those of other models. This indicates that the PV model might not be effective for capturing hierarchical structures which somehow contradicts the statement in the introduction. Could the authors clarify on this point ?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "- The topic of the paper is interesting.\n\n- The paper advocate the use of PV model for hyperbolic space which is underexplored in the literature of HNNs.", "weaknesses": "- The contribution is incremental.\n\n- The paper is poorly written.\n\n- Experimental results are not convincing. Also, important baselines are missing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hcPaBN5oky", "reviewer": "Reviewer_iwSM", "summary": "The paper proposes **Proper Velocity Neural Networks (PVNNs)**, which work on the **Proper Velocity (PV) manifold**, an unconstrained coordinate of hyperbolic space. The authors first derive **closed-form Riemannian operators** for PV (exponential and logarithmic maps, geodesic distance, and parallel transport). They then design **core layers** in PV: multinomial logistic regression (PV-MLR), a fully connected layer with an **analytic formula**, convolution, activation, and **GyroBN** normalization. Experiments on **numerical stability**, **graphs**, **CIFAR-10/100**, and **genomic sequence prediction** show strong stability and **competitive accuracy**, with clear gains on some tasks (e.g., Airport), but **weaker results on Cora**. The code will be released upon acceptance.", "questions": "1. **Expressivity:** Do PVNNs satisfy a **universal approximation** property under reasonable assumptions? If yes, what function classes and depths are required?\n2. **Cora drop:** Why does PVNN underperform on **Cora**? Is it due to curvature mismatch, optimization, or the decision boundary shape? Would **learnable curvature** or hybrid PV/Poincaré layers help?\n3. **Runtime/memory:** How are the **computation and memory** costs of PV operators compared to Poincaré/Hyperboloid ?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **Originality:** Uses the **PV manifold** as an **unconstrained** alternative to Poincaré/Hyperboloid; complete Riemannian toolkit enables end-to-end models.\n- **Quality (theory):** Rigorous **isometry** to Poincaré; **closed-form** exp/log/transport; **analytic PV-FC**; GyroBN with mean/variance homogeneity.\n- **Quality (empirics):** Broad evaluation (stability, four graph datasets, CIFAR-10/100, TEB genomics) with shared backbones and multiple seeds.\n- **Clarity:** Structure is logical; appendices provide proofs and implementation details. :contentReference[oaicite:7]{index=7}  \n- **Significance:** **Large-norm** representations work stably; strong gains on some datasets (e.g., Airport; several TEB tasks).", "weaknesses": "- **No expressivity guarantees:** There is **no universality** or approximation bound for PVNNs, limiting formal claims about representation power.\n- **Mixed graph performance:** **Cora** performance is lower than Poincaré/Hyperboloid baselines; the paper gives limited analysis of this failure mode. \n- **Model breadth:** Vision uses **ResNet-18 with only the head changed**; large-scale or transformer results are left for future work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7n15JjeYzy", "reviewer": "Reviewer_tn82", "summary": "This paper proposes formulating hyperbolic deep learning techniques typically represented via the Poincaré or Lorentz model in the Proper Velocity manifold. Previous representations often suffered from numerical issues close to the boundary, which the Proper Velocity model avoids. The authors derive key quantities such as exponential and logarithmic maps, as well as important operations for deep learning including fully-connected layers, convolution as well as multinomial logistic regression. Equipped with these formulations, the authors evaluate their hyperbolic models on a suite of benchmarks, including vision, graph learning as well as genomics. Their approach is very competitive with previous works while offering better stability.", "questions": "See above.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is very well-written and very nicely placed within the literature of hyperbolic learning. The mathematical derivations are very detailed and extensive and a nice contribution in themselves in my opinion. I believe the Proper Velocity manifold is a nice addition to the toolkit that practitioners have to perform hyperbolic deep learning, especially if the stability issues of previous approaches are remedied. I have some more comments on this later.\n2. The authors perform a very broad evaluation of their model, using vision, graph and genomics data to compare to previous approaches in hyperbolic deep learning. This makes it more convincing for me that this novel representation works across different scenarios, but I do have some more comments on this point later.", "weaknesses": "1. The main issue for me with this work in its current state is the lack of convincing evidence towards its premise: other representations suffer from numerical issues (especially around the border) and I assume this point implies that “performance of these models is affected”? Towards this question, you show a small experiment where addition and multiplication is performed on Poincaré and Proper Velocity manifolds, which indeed shows that norms close to the border collapse in case of Poincaré, where as the Proper Velocity model is able to adequately handle this case. Numerically however, the Proper Velocity model seems to have more issues, at least in terms of multiplication? More importantly however, I don’t really see an experiment or argument that shows these other models actually suffering in a realistic scenario, like vision or graph learning. Are these other models actually being limited by numerical issues? Can models in Proper Velocity be trained in lower precision? Even a simple experiment such as learning to embed a tree in hyperbolic space (similar to e.g. [1]) would help me be more convinced regarding these claims. While Proper Velocity sometimes outperform Poincaré/Hyperboloid, it is not clear if that’s due to the numerical issues mentioned, especially as performance is actually quite similar in most cases.\n2. I like that the evaluation is very broad, but on the other hand the experiments are also a bit shallow in each case. For node classification, you seem to use a fully-connected network in all representations, even though it is more standard to use a graph neural network that employs some form of graph convolution. Other methods have developed such specialized operations in the Poincaré or Hyperboloid model, and I think something like this is needed to truly understand the potential of a representation. Similarly in the case of vision, a Euclidean encoder is used, which essentially does all the heavy lifting and only the final classification layer is hyperbolic. Again, other representations have their dedicated convolution layers in hyperbolic space, why not do the same in the Proper Velocity model? Results are probably as a consequence almost identical for PV and HNN++. What is however interesting is that Poincaré seems to struggle a lot for CIFAR100, is this maybe due to numerical issues? The genomics experiments are more convincing in this regard!  \n\n[1] Constant Curvature Graph Convolutional Networks, Bachmann et al, 2020", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "e4qvm5ZOvp", "reviewer": "Reviewer_QAye", "summary": "This paper proposes proper velocity neural networks (PVNN), where hyperbolic neural operations are built on the “proper velocity” (PV) manifold. The paper presents the basic elements of Riemannian geometry of the PV manifold, develops various layers, and shows experiments on numerical stability, graph learning, image classification and genomic sequence learning.", "questions": "See \"weaknesses\" for revision suggestions.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. This paper introduces a new playground for hyperbolic neural networks and provides the complete Riemannian toolkit and neural-layer formulations on it. The detailed derivations of exponential/logarithmic maps, parallel transport, and gyro-based operations are rigorous and valuable. These constructions can serve as fundamental tools for future research on geometric deep learning.\n2. The idea is well motivated and the paper is clearly structured and easy to follow, which makes the technical content accessible to readers from machine learning communities who may not be familiar with the geometry.\n3. The numerical experiments demonstrate that PV-based networks can be applied to diverse domains (graphs, images, genomics) and generally perform competitively with existing hyperbolic baselines.", "weaknesses": "1. \n- A major conceptual issue arises in the presentation of the PV model. As written, the paper may lead readers to believe that PV is an independent model parallel to the Poincaré, hyperboloid, and Klein models. However, PV and the hyperboloid are deeply connected through special relativity. In fact, there exists a natural isometry $\\pi$ between hyperboloid and PV. Given a point in the hyperboloid $\\mathbf{x} = (x_0, x_1, \\cdots, x_n) \\in \\mathbb{H}^n$, $$\\pi(\\mathbf{x}) = (x_1, \\cdots, x_n) \\in \\mathcal{PV}^n.$$ That is, PV is nothing else but the spatial representation corresponding to the time-space representation of hyperboloid. The paper should explicitly state this relationship so that readers unfamiliar with hyperbolic geometry are not misled. \n\n- Conceptually, it would be more natural to compare PV directly with the hyperboloid (e.g., Section 4.1) rather than primarily with the Poincaré ball. Such comparison would yield more geometric and numerical insights.\n\n- In this context, a very interesting paper will be “Fully hyperbolic neural networks” (Chen et al.,2022). From your perspective, although they write everything in terms of hyperboloid/Lorentz, their operations are actually completely in the PV space. It would be informative to readers if you can point this out and discuss whether their approach naturally follows PV geometry.\n\n2. \n- The numerical stability argument and experiment is unclear. The claim that PV is “unconstrained” and therefore more numerically stable is not entirely convincing. While the Poincaré ball is indeed bounded, the hyperboloid is already an unbounded manifold, and PV is simply its spatial representation. Thus, PV cannot be inherently “more flexible” than the hyperboloid. Note that hyperboloid is in $\\mathbb{R}^{n+1}$.\n\n- Moreover, the numerical-stability experiment (Table 1) shows extremely large norms (up to $10^8$), which may themselves indicate instability or unbounded dynamic range rather than desirable behavior.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ThslAhT58X", "reviewer": "Reviewer_jGwb", "summary": "The paper suggest a hyperbolic model for neural network learning and inference that has not been used before: proper velocity neural networks. For this purpose, the paper completes the mathematical inventory required for learning and inference and investigates the performance of the novel PVNNs on a set of artificial and natural benchmarks.", "questions": "-  Is the isomorphism from PV to Poincare ball needed? If it is needed, then why is the computation on PV numerically more stable?\n- what does delta in the tables of the experiments section refer to?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "+ Suggesting an unused but effective hyperbolic model is a great contribution\n+ There existed some mathematical gaps that needed closing in order to allow for the definition of PVNNs\n+ The experiments are interesting and insightful", "weaknesses": "- issues with presentation: I have not understood which experiments were performed in section 6.1, and why a larger median output norm would be better.\n- issues with experiments: My understanding is that the curvature is a hyperparameter, but I see no exploration of hyperparameter choices in the experiments section.\n- issues with interpreting results: The substantial disadvantage on Cora is not well explained. \n- minor issues with presentation: \n         -- It does not make sense to mark best results in bold, if they are statistically indistinguishable from baselines (Table 6)\n         -- \"on the highly hyperbolic Cora dataset\" does not make sense. The dataset may exhibit hierarchical structures or larger diameter etc., but it is not hyperbolic on its own, though its structure might be better represented in space that is more curved", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)