Hyperbolic Busemann Neural Networks

Ziheng Chen; Bernhard Schölkopf; Nicu Sebe

cs.LGFeb 21, 2026

Hyperbolic Busemann Neural Networks

Authors: Ziheng Chen, Bernhard Schölkopf, Nicu Sebe

Organizations: University of Trento · MPI-IS

Abstract

Hyperbolic spaces provide a natural geometry for representing hierarchical and tree-structured data due to their exponential volume growth. To leverage these benefits, neural networks require intrinsic and efficient components that operate directly in hyperbolic space. In this work, we lift two core components of neural networks, Multinomial Logistic Regression (MLR) and Fully Connected (FC) layers, into hyperbolic space via Busemann functions, resulting in Busemann MLR (BMLR) and Busemann FC (BFC) layers with a unified mathematical interpretation. BMLR provides compact parameters, a point-to-horosphere distance interpretation, batch-efficient computation, and a Euclidean limit, while BFC generalizes FC and activation layers with comparable complexity. Experiments on image classification, genome sequence learning, node classification, and link prediction demonstrate improvements in effectiveness and efficiency over prior hyperbolic layers. The code is available at https://github.com/GitZH-Chen/HBNN.

Figures & tables

Figure 1: Illustration: red curves are different horospheres of

B^{v}

Euclidean	Hyperbolic
Straight line	Geodesic
Parallel lines	Asymptotic geodesic rays
Inner product $\left\langle v,x\right\rangle$	Busemann function $-B^{v}(x)$
Hyperplane	Horosphere
Parallel hyperplanes with $v\in\mathbb{S}^{n-1}$	Horospheres with $v\in\mathbb{S}^{n-1}\cap T_{e}\mathcal{H}^{n}_{K}$

Table 1: Generalization: Euclidean vs. hyperbolic.

Method	Logit $u_{k}(x),\ \forall k\in\{1,\ldots,C\}$	Space	Dist	#Params	Compact params	FLOPs	Batch efficiency
Euclidean MLR	$\left\langle a_{k},x\right\rangle+b_{k}$ , with $a_{k}\in\mathbb{R}^{n},b_{k}\in\mathbb{R}$	$\mathbb{R}^{n}$	Real	$C(n+1)$	✓	$C(2n)$	✓
Poincaré MLR [ 23 , Eq. (25)]	$\displaystyle\frac{\lambda_{p_{k}}^{K}\left\\|a_{k}\right\\|}{\sqrt{-K}}\sinh^{-1}\left(\frac{2\sqrt{-K}\left\langle-p_{k}\oplus_{\mathrm{M}}x,a_{k}\right\rangle}{\left(1+K\left\\|-p_{k}\oplus_{\mathrm{M}}x\right\\|^{2}\right)\left\\|a_{k}\right\\|}\right)$ , with $p_{k}\in\mathbb{P}^{n}_{K},a_{k}\in T_{p_{k}}\mathbb{P}^{n}_{K}$	$\mathbb{P}^{n}_{K}$	Real	$C(2n)$	✗	$C(19n+29)$	✗
Poincaré MLR [ 51 , Eq. (6)]	$\displaystyle\frac{2}{\sqrt{-K}}\alpha_{k}\sinh^{-1}\left(\alpha-\beta\right)$ , $\alpha=\lambda_{x}^{K}\sqrt{-K}\left\langle x,v_{k}\right\rangle\cosh\left(2\sqrt{-K}b_{k}\right)$ , $\beta=\left(\lambda_{x}^{K}-1\right)\sinh\left(2\sqrt{-K}b_{k}\right)$ , with $\alpha_{k}>0,v_{k}\in\mathbb{S}^{n-1},b_{k}\in\mathbb{R}$	$\mathbb{P}^{n}_{K}$	Real	$C(n+2)$	✓	$C(4n+52)$	✓
Pseudo-Busemann MLR [ 45 , Cor. 4.3]	$\displaystyle-d(x,p_{k})\frac{B^{v_{k}}\left(-p_{k}\oplus_{\mathrm{M}}x\right)}{\left\\|-p_{k}\oplus_{\mathrm{M}}x\right\\|}$ , with $p_{k}\in\mathbb{P}^{n}_{K},v_{k}\in\mathbb{S}^{n-1}$	$\mathbb{P}^{n}_{K}$	Pseudo	$C(2n)$	✗	$C(19n+34)$	✗
Lorentz MLR [ 4 , Eq. (12)]	$\displaystyle\frac{1}{\sqrt{-K}}\operatorname{sign}(\alpha)\beta\left\|\sinh^{-1}\left(\sqrt{-K}\frac{\alpha}{\beta}\right)\right\|$ , $\alpha=\cosh\left(\sqrt{-K}b_{k}\right)\left\langle z_{k},x_{s}\right\rangle-\sinh\left(\sqrt{-K}b_{k}\right)$ , $\beta=\sqrt{\left\\|\cosh(\sqrt{-K}b_{k})z_{k}\right\\|^{2}-(\sinh(\sqrt{-K}b_{k})\left\\|z_{k}\right\\|)^{2}}$ , with $z_{k}\in\mathbb{R}^{n},\ b_{k}\in\mathbb{R}$	$\mathbb{L}^{n}_{K}$	Real	$C(n+1)$	✓	$C(4n+52)$	✓
BMLR	$-\alpha_{k}B^{v_{k}}(x)+b_{k}$ , with $\alpha_{k}>0,v_{k}\in\mathbb{S}^{n-1},b_{k}\in\mathbb{R}$	$\mathbb{P}^{n}_{K}$ $\mathbb{L}^{n}_{K}$	Real	$C(n+2)$	✓	$\mathbb{P}^{n}_{K}:C(6n+12)$ $\mathbb{L}^{n}_{K}:C(2n+12)$	✓

Table 2: Comparison of

C

-class MLR. In Dist, Real means the point-to-hyperplane distance is the real distance, obtained by

\inf_{y\in H}\operatorname{d}(x,y)

, where

H

is a hyperplane and

\operatorname{d}

is the geodesic distance; Pseudo denotes a surrogate that coincides with the real distance only in Euclidean geometry. Compact params indicate whether each logit avoids an additional manifold-valued parameter. Batch efficiency indicates whether the MLR can avoid inefficient per-class loops in implementation (see App. C ). In #Params, we highlight the heaviest in red . In FLOPs, we mark the slowest in red and the fastest in green .

Method	$\mathcal{F}:\mathcal{H}^{n}_{K}\ni x\mapsto y\in\mathcal{H}^{m}_{K}$	Space	Methodology	Parameters	#Params	FLOPs
Möbius [ 23 , Eq. 27]	$\displaystyle\frac{1}{\sqrt{-K}}\tanh\left(\frac{\left\\|Wx\right\\|}{\left\\|x\right\\|}\tanh^{-1}\left(\sqrt{-K}\left\\|x\right\\|\right)\right)\frac{Wx}{\left\\|Wx\right\\|}$	$\mathbb{P}^{n}_{K}$	Tangent	$W\in\mathbb{R}^{m\times n}$	$mn$	$2nm+2n$ $+2m+24$
Poincaré FC [ 51 , Eq. (7)]	$\displaystyle y=\frac{\omega}{1+\sqrt{1-K\left\\|\omega\right\\|^{2}}},\ \omega_{k}=\frac{\sinh\left(\sqrt{-K}u_{k}(x)\right)}{\sqrt{-K}}$ , with $u_{k}(x)$ in Tab. 2	$\mathbb{P}^{n}_{K}$	Poincaré geometry	$\alpha_{k}>0,\ v_{k}\in\mathbb{S}^{n-1}$ , $b_{k}\in\mathbb{R}$ , for $k=1,\ldots,m$	$m(n+2)$	$4nm+71m+4$
Lorentz FC [ 11 , Eq. (3)]	$\displaystyle y=\begin{bmatrix}\sqrt{\left\\|\psi(Wx,v)\right\\|^{2}-1/K}\\ \psi(Wx,v)\end{bmatrix}$ , $\displaystyle\psi(Wx,v)=\lambda\sigma\left(v^{\top}x+b^{\prime}\right)\frac{W\phi(x)+b}{\left\\|W\phi(x)+b\right\\|}$ with $\phi$ and $\sigma$ as the activation and sigmoid	$\mathbb{L}^{n}_{K}$	Ambient Minkowski	$W\in\mathbb{R}^{m\times(n+1)}$ , $v\in\mathbb{R}^{n+1}$ , $b\in\mathbb{R}^{m}$ , $b^{\prime}\in\mathbb{R}$ , $\lambda>0$	$m(n+1)+m$ $+(n+1)+2$	$2nm+8m$ $+2n+10$
BFC	$\displaystyle y=\frac{\omega}{1+\sqrt{1-K\left\\|\omega\right\\|^{2}}}$ , $\displaystyle\omega=\frac{\sinh\left(\sqrt{-K}u(x)\right)}{\sqrt{-K}}$ ; $\displaystyle y_{s}=\frac{1}{\sqrt{-K}}\sinh\left(\sqrt{-K}u(x)\right)$ , $\displaystyle y_{t}=\sqrt{\frac{1}{-K}+\left\\|y_{s}\right\\|^{2}}$ , with $\displaystyle u_{k}(x)=\phi(-\alpha_{k}B^{v_{k}}(x)+b_{k})$	$\mathbb{P}^{n}_{K}$ $\mathbb{L}^{n}_{K}$	Busemann	$\alpha_{k}>0,\ v_{k}\in\mathbb{S}^{n-1}$ , $b_{k}\in\mathbb{R}$ , for $k=1,\ldots,m$	$m(n+2)$	$6nm+29m+4$ $2nm+30m+2$

Table 3: Comparison of hyperbolic FC layers. For simplicity, BFC layers do not involve the gyroaddition and assume

\phi

is the identity map, which is in line with the Möbius and Lorentz FC layers.

Space

Method

CIFAR-10 (Num. classes: 10)

CIFAR-100 (Num. classes: 100)

Tiny-ImageNet (Num. classes: 200)

ImageNet-1k (Num. classes: 1000)

Acc

Fit Time

#Params

Acc

Fit Time

#Params

Acc

Fit Time

#Params

Acc

Fit Time

#Params

\mathbb{R}^{n}

MLR

95.14 ± 0.12

10.66

5.13K

77.72 ± 0.15

10.60

51.30K

65.19 ± 0.12

69.17

102.60K

71.87

2263.12

513K

\mathbb{P}^{n}_{K}

PMLR

95.04 ± 0.13

11.94

5.14K

77.19 ± 0.50

12.11

51.40K

64.93 ± 0.38

71.90

102.80K

71.77

2300.11

514K

PBMLR-P

95.23 ± 0.08

21.92

10.24K

77.78 ± 0.15

76.84

102.40K

65.43 ± 0.27

336.58

204.80K

71.46

3907.12

1024K

BMLR-P

95.32 ± 0.14

12.01

5.14K

78.10 ± 0.35

12.13

51.40K

66.16 ± 0.19

71.98

102.80K

73.36

2300.77

514K

\mathbb{L}^{n}_{K}

LMLR

94.98 ± 0.12

11.55

5.13K

78.03 ± 0.21

11.72

51.30K

65.63 ± 0.10

69.27

102.60K

72.46

2277.17

513K

Table 4: Top-1 image classification accuracy (%) of MLR methods on the ResNet-18 backbone. Best results within each hyperbolic model are in bold . Fit time (s/epoch) highlights: fastest hyperbolic MLR in green , slowest MLR in red . #Params denotes the number of parameters in the MLR head, with the largest marked in red .

Figure 2: Validation accuracy curves on ImageNet-1k.

Benchmark	Task	Dataset	Num.	$\mathbb{P}^{n}_{K}$			$\mathbb{L}^{n}_{K}$
Benchmark	Task	Dataset	classes	PMLR	PBMLR-P	BMLR	LMLR	BMLR-L
TEB	Retrotransposons	LTR Copia	2	75.34 ± 1.02	74.37 ± 1.48	76.73 ± 1.08	73.01 ± 1.07	75.86 ± 1.52
		LINEs	2	85.54 ± 0.61	85.92 ± 0.65	86.05 ± 1.08	83.14 ± 0.80	86.72 ± 0.58
		SINEs	2	95.30 ± 0.85	95.34 ± 1.58	95.99 ± 0.74	96.70 ± 0.87	96.29 ± 0.59
	DNA transposons	CMC-EnSpm	2	83.39 ± 0.56	83.62 ± 1.00	84.03 ± 0.71	81.78 ± 1.05	84.15 ± 1.00
	DNA transposons	hAT-Ac	2	89.38 ± 0.90	89.86 ± 0.54	89.62 ± 0.74	88.94 ± 0.69	90.70 ± 0.51

Table 5: Genomic MCC of MLR methods under the CNN backbone. Best results within each hyperbolic model are in bold .

Dataset	$\mathbb{P}^{n}_{K}$			$\mathbb{L}^{n}_{K}$
Dataset	PMLR	PBMLR-P	BMLR	LMLR	BMLR-L
LTR Copia	3.96	5.11	4.06	3.92	3.77
LINEs	5.80	6.90	5.73	5.50	5.32
SINEs	1.36	1.80	1.39	1.38	1.28
CMC-EnSpm	3.11	4.38	3.04	2.89	2.82
hAT-Ac	4.37	5.37	4.38	4.13	3.96

Table 6: Fit time (s/epoch) on genome sequence learning. The fastest times are in green and the slowest ones are in red .

Space	Method	Methodology	Disease	Airport	PubMed	Cora
Space	Method	Methodology	$\delta=0$	$\delta=1$	$\delta=3.5$	$\delta=11$
$\mathbb{P}^{n}_{K}$	Möbius	Tangent	76.35 ± 1.83	93.31 ± 0.41	94.93 ± 0.06	90.80 ± 0.56
	Poincaré FC	Poincaré geometry	79.45 ± 1.01	94.31 ± 0.16	94.24 ± 0.25	88.21 ± 0.72
	BFC-P	Busemann	80.45 ± 0.93	94.88 ± 0.39	94.85 ± 0.07	91.94 ± 0.32
$\mathbb{L}^{n}_{K}$	LTFC	Tangent	71.32 ± 5.36	92.68 ± 0.35	94.85 ± 0.17	89.37 ± 0.64
	Lorentz FC	Ambient Minkowski	72.78 ± 2.04	92.99 ± 0.33	94.20 ± 0.10	92.06 ± 0.62

Table 7: Comparison of hyperbolic FC layers on link prediction. Best results within each hyperbolic model are in bold .

Space	Method	Disease	Airport	PubMed	Cora
Space	Method	$\delta=0$	$\delta=1$	$\delta=3.5$	$\delta=11$
$\mathbb{P}^{n}_{K}$	HGCN	86.87 ± 2.58	85.34 ± 1.16	76.29 ± 0.98	76.56 ± 0.81
	HGCN-PMLR	88.98 ± 1.96	84.78 ± 1.48	76.02 ± 1.09	77.47 ± 1.15
	HGCN-PBMLR-P	89.05 ± 0.78	85.04 ± 0.97	75.89 ± 0.78	77.90 ± 1.00
	HGCN-BMLR-P	92.45 ± 0.96	86.02 ± 0.53	77.36 ± 0.73	78.48 ± 1.52
$\mathbb{L}^{n}_{K}$	HGCN	87.83 ± 0.77	84.94 ± 1.40	76.49 ± 0.88	77.37 ± 1.72

Table 8: Node classification F1 scores of hyperbolic MLRs on the HGCN backbone, where

\delta

denotes the graph hyperbolicity (lower is more hyperbolic). The best results within each hyperbolic model are highlighted in bold .

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Notation	Description
$\mathcal{M}$	Riemannian manifold
$T_{x}\mathcal{M}$	Tangent space at $x$
$\operatorname{Exp}_{x}(v)$	Exponential map at $x$
$\operatorname{Log}_{x}(y)$	Logarithmic map at $x$
$\operatorname{PT}_{x\rightarrow y}(v)$	Parallel transport of $v$ from $x$ to $y$
$g_{x}(u,v)$	Riemannian metric at $x$

Appendix

Table 9: Summary of notation.

Operator	Poincaré ball $\mathbb{P}^{n}_{K}$	Lorentz $\mathbb{L}^{n}_{K}$
Definition	$\displaystyle\mathbb{P}^{n}_{K}=\{x\in\mathbb{R}^{n}\mid\left\\|x\right\\|^{2}<-1/K\}$	$\displaystyle\mathbb{L}^{n}_{K}=\{x\in\mathbb{R}^{n+1}\mid\left\langle x,x\right\rangle_{\mathcal{L}}=1/K,x_{t}>0\}$
$g_{x}(w,v)$	$\displaystyle\left(\lambda_{x}^{K}\right)^{2}\left\langle w,v\right\rangle,\quad\lambda_{x}^{K}=\frac{2}{1+K\left\\|x\right\\|^{2}}$	$\displaystyle\left\langle w,v\right\rangle_{\mathcal{L}}=\left\langle v_{s},w_{s}\right\rangle-v_{t}w_{t}$
$\operatorname{d}(x,y)$	$\displaystyle\frac{2}{\sqrt{\|K\|}}\tanh^{-1}\left(\sqrt{\|K\|}\left\\|-x\oplus_{\mathrm{M}}y\right\\|\right)$	$\displaystyle\frac{1}{\sqrt{\|K\|}}\cosh^{-1}\left(K\left\langle x,y\right\rangle_{\mathcal{L}}\right)$
$\operatorname{Log}_{x}y$	$\displaystyle\frac{2}{\sqrt{\|K\|}\lambda_{x}^{K}}\tanh^{-1}\left(\sqrt{\|K\|}\left\\|-x\oplus_{\mathrm{M}}y\right\\|\right)\frac{-x\oplus_{\mathrm{M}}y}{\left\\|-x\oplus_{\mathrm{M}}y\right\\|}$	$\displaystyle\frac{\cosh^{-1}(\beta)}{\sqrt{\beta^{2}-1}}(y-\beta x),\quad\beta=K\left\langle x,y\right\rangle_{\mathcal{L}}$
$\operatorname{Exp}_{x}v$	$\displaystyle x\oplus_{\mathrm{M}}\left(\tanh\left(\sqrt{\|K\|}\frac{\lambda_{x}^{K}\left\\|v\right\\|}{2}\right)\frac{v}{\sqrt{\|K\|}\left\\|v\right\\|}\right)$	$\displaystyle\cosh(\alpha)x+\frac{\sinh(\alpha)}{\alpha}v,\quad\alpha=\sqrt{\|K\|}\left\\|v\right\\|_{\mathcal{L}}$
$\operatorname{PT}_{x\rightarrow y}(v)$	$\displaystyle\frac{\lambda_{x}^{K}}{\lambda_{y}^{K}}\operatorname{gyr}[y,-x]v$	$\displaystyle v-\frac{K\left\langle y,v\right\rangle_{\mathcal{L}}}{1+K\left\langle x,y\right\rangle_{\mathcal{L}}}(x+y)$

Appendix

Table 10: Riemannian operators on Poincaré ball and Lorentz (

K<0

Method	Hyperplane	Formulation	Applied manifolds	Compact params
Euclidean MLR	Euclidean	$\left\{x\in\mathbb{R}^{n}\mid\left\langle a,x\right\rangle+b=0\right\}$ $a\in\mathbb{R}^{n},\ b\in\mathbb{R}$	$\mathbb{R}^{n}$	✓
Poincaré MLR [ 23 ]	Geodesic [ 23 , Def. 3.1]	$\left\{x\in\mathbb{P}^{n}_{K}\mid\left\langle\operatorname{Log}_{p}\left(x\right),a\right\rangle_{p}=0\right\}$ $p\in\mathbb{P}^{n}_{K},\ a\in T_{p}\mathbb{P}^{n}_{K}$	$\mathbb{P}^{n}_{K}$	✗
Pseudo-Busemann MLR [ 45 ]	Busemann & gyro [ 45 , Def. 4.1]	$\left\{x\in\mathbb{P}^{n}_{K}\mid B^{v}\left(-p\oplus_{\mathrm{M}}x\right)=0\right\}$ $v\in\mathbb{S}^{n-1},\ p\in\mathbb{P}^{n}_{K}$	$\mathbb{P}^{n}_{K}$	✗
Lorentz MLR [ 4 ]	Ambient Minkowski [ 4 , Eq. (7)]	$\left\{x\in\mathbb{L}^{n}_{K}\mid\left\langle w,x\right\rangle_{\mathcal{L}}=0\right\}$ $p\in\mathbb{L}^{n}_{K},\ w\in T_{p}\mathbb{L}^{n}_{K}$	$\mathbb{L}^{n}_{K}$	✗
BMLR	Horosphere	$\left\{x\in\mathcal{H}^{n}_{K}\mid-\alpha B^{v}\left(x\right)+b=0\right\}$ $\alpha>0,\ v\in\mathbb{S}^{n-1},\ b\in\mathbb{R}$	$\mathbb{P}^{n}_{K}$ , $\mathbb{L}^{n}_{K}$	✓

Appendix

Table 11: Comparison of hyperplanes. Compact params indicates whether the parameterization requires an additional manifold-valued point.

Method	Point-to-hyperplane distance	Applied manifolds	Dist
Euclidean MLR	$\displaystyle\frac{\|\left\langle a,x\right\rangle+b\|}{\left\\|a\right\\|}$	$\mathbb{R}^{n}$	Real
Poincaré MLR [ 23 ]	$\displaystyle\frac{1}{\sqrt{-K}}\sinh^{-1}\left(\frac{2\sqrt{-K}\left\|\left\langle-p\oplus_{\mathrm{M}}x,a\right\rangle\right\|}{\left(1+K\left\\|-p\oplus_{\mathrm{M}}x\right\\|^{2}\right)\left\\|a\right\\|}\right)$ [ 23 , Thm. 5]	$\mathbb{P}^{n}_{K}$	Real
Pseudo-Busemann MLR [ 45 ]	$\displaystyle\operatorname{d}(x,p)\frac{B^{v}\left(-p\oplus_{\mathrm{M}}x\right)}{\left\\|-p\oplus_{\mathrm{M}}x\right\\|}$ [ 45 , Cor. 4.3]	$\mathbb{P}^{n}_{K}$	Pseudo
Lorentz MLR [ 4 ]	$\displaystyle\frac{1}{\sqrt{-K}}\left\|\sinh^{-1}\left(\sqrt{-K}\frac{\left\langle v,x\right\rangle_{\mathcal{L}}}{\left\\|v\right\\|_{\mathcal{L}}}\right)\right\|$ [ 4 , Eq. (44)]	$\mathbb{L}^{n}_{K}$	Real
BMLR	$\displaystyle\frac{\left\|-\alpha B^{v}(x)+b\right\|}{\alpha}$	$\mathbb{P}^{n}_{K}$ , $\mathbb{L}^{n}_{K}$	Real

Appendix

Table 12: Comparison of point-to-hyperplane distances. Real means the point-to-hyperplane distance is the real distance, obtained by

\inf_{y\in H}\operatorname{d}(x,y)

with

H

as a hyperplane and

\operatorname{d}

as the geodesic distance. Instead, Pseudo means the point-to-hyperplane distance is a surrogate, which only equals to the real distance under the Euclidean geometry.

Hyperparameter	CIFAR-10/100 Tiny-ImageNet	ImageNet-1k
Epochs	200	100
Batch size	128	256
Initial learning rate	$0.1$	$0.1$
LR schedule	$60,120,160;\gamma=0.2$	$30,60,90;\gamma=0.1$
Weight decay	$5e^{-4}$	$1e^{-4}$
Optimizer	SGD	SGD

Appendix

Table 13: Summary of hyperparameters used in the image classification task.

Task	Species	Datasets	Num. classes	Max length	Train / Dev / Test
Retrotransposons	Plant	LTR Copia	2	500	7666 / 682 / 568
		LINEs		1000	22502 / 2030 / 1782
		SINEs		500	21152 / 1836 / 1784
DNA transposons	Plant	CMC-EnSpm	2	200	19912 / 1872 / 1808
		hAT-Ac		1000	17322 / 1822 / 1428
Pseudogenes	Human	processed	2	1000	17956 / 1046 / 1740

Appendix

Table 14: Summary statistics for TEB.

Task	Species	Dataset	Num. classes	Length	Train / Dev / Test
Core promoter detection	Human	tata	2	70	4904 / 613 / 613
		notata			42452 / 5307 / 5307
		all			47356 / 5920 / 5920
Promoter detection	Human	tata	2	300	4904 / 613 / 613
		notata			42452 / 5307 / 5307
		all			47356 / 5920 / 5920

Appendix

Table 15: Summary statistics for the adopted GUE datasets.

Batch size	100
Epochs	100
Optimizer	Adam
$\beta_{1}$ , $\beta_{2}$	0.9, 0.999
Initial learning rate	$1e^{-4}$
LR schedule	$60,85$ ; $\gamma=0.1$
Weight decay	$0.1$

Appendix

Table 16: Hyperparameters for genome sequence learning.

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

Appendix

Table 17: Summary statistics for the node classification datasets.

Space	Weight decay	Dropout
$\mathbb{P}^{n}_{K}$	$1e-4,1e-5,1e-3,1e-3$	$0.3,0,0,0.2$
$\mathbb{L}^{n}_{K}$	$1e-4,5e-5,1e-3,1e-3$	$0,0,0,0.3$

Appendix

Table 18: Hyperparameters for node classification on Disease, Airport, PubMed, and Cora.

Space	Method	Disease		Airport		PubMed		Cora
Space	Method	Fit Time	#Params	Fit Time	#Params	Fit Time	#Params	Fit Time	#Params
$\mathbb{P}^{n}_{K}$	Möbius	0.0200	464	0.0535	480	0.1120	8288	0.0229	23216
	Poincaré FC	0.0198	528	0.0536	544	0.1176	8352	0.0248	23280
	BFC-P	0.0201	528	0.0512	544	0.1123	8352	0.0231	23280
$\mathbb{L}^{n}_{K}$	LTFC	0.0343	464	0.0818	480	0.1633	8288	0.0370	23216
	Lorentz FC	0.0232	563	0.0715	580	0.1537	8876	0.0261	24737

Appendix

Table 19: Efficiency comparison: fit time (s/epoch) and parameter count. Slowest results and largest parameter counts are in red .

Space	$\phi$	Airport	Cora
$\mathbb{P}^{n}_{K}$	None	94.57 ± 0.29	87.55 ± 0.34
$\mathbb{P}^{n}_{K}$	tanh	94.88 ± 0.39	91.94 ± 0.32
$\mathbb{L}^{n}_{K}$	None	94.91 ± 0.24	87.45 ± 1.39
$\mathbb{L}^{n}_{K}$	tanh	95.37 ± 0.17	92.28 ± 0.12

Appendix

Table 20: Ablation on activation

\phi

in BFC layers.

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Sep 28, 2026

Building Transformation Layers for Riemannian Neural Networks

Recently, deep neural networks on manifold-valued representations have garnered significant attention across various machine learning applications. One recent focus is the generalization of Euclidean fully connected (FC) and convolutional layers to non-Euclidean geometries. However, previous approaches typically focus on a few selected manifolds and rely on specific properties of the target manifold. In contrast, this work proposes a framework for constructing FC and convolutional layers over computationally tractable Riemannian spaces. This framework incorporates several previous FC layers across different geometries as special cases and is instantiated on ten representative manifolds, including three hyperbolic models, five geometries of the symmetric positive definite (SPD) manifold, and two Grassmannian perspectives. Experiments on different manifolds demonstrate the effectiveness and applicability of our approach. Code can be found at https://github.com/GitZH-Chen/RieTrans.

Ziheng Chen

University of Trento · MPI for Intelligent Systems, Tübingen

Geometric Representation Learning

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