cs.LGFeb 21, 2026

Hyperbolic Busemann Neural Networks

Authors: Ziheng Chen, Bernhard Schölkopf, Nicu Sebe

Organizations: University of Trento · MPI-IS

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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 BvB^{v} .
EuclideanHyperbolic
Straight lineGeodesic
Parallel linesAsymptotic geodesic rays
Inner product ⟨v,x⟩\left\langle v,x\right\rangleBusemann function −Bv(x)-B^{v}(x)
HyperplaneHorosphere
Parallel hyperplanes with v∈Sn−1v\in\mathbb{S}^{n-1}Horospheres with v∈Sn−1∩TeHKnv\in\mathbb{S}^{n-1}\cap T_{e}\mathcal{H}^{n}_{K}
Table 1: Generalization: Euclidean vs. hyperbolic.
MethodLogit uk(x), ∀k∈{1,…,C}u_{k}(x),\ \forall k\in\{1,\ldots,C\}SpaceDist#ParamsCompact paramsFLOPsBatch efficiency
Euclidean MLR⟨ak,x⟩+bk\left\langle a_{k},x\right\rangle+b_{k} , with ak∈Rn,bk∈Ra_{k}\in\mathbb{R}^{n},b_{k}\in\mathbb{R}Rn\mathbb{R}^{n}RealC(n+1)C(n+1)✓C(2n)C(2n)✓
Poincaré MLR [ 23 , Eq. (25)]λpkK∥ak∥−Ksinh⁡−1(2−K⟨−pk⊕Mx,ak⟩(1+K∥−pk⊕Mx∥2)∥ak∥)\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 pk∈PKn,ak∈TpkPKnp_{k}\in\mathbb{P}^{n}_{K},a_{k}\in T_{p_{k}}\mathbb{P}^{n}_{K}PKn\mathbb{P}^{n}_{K}RealC(2n)C(2n)✗C(19n+29)C(19n+29)✗
Poincaré MLR [ 51 , Eq. (6)]2−Kαksinh⁡−1(α−β)\displaystyle\frac{2}{\sqrt{-K}}\alpha_{k}\sinh^{-1}\left(\alpha-\beta\right) , α=λxK−K⟨x,vk⟩cosh⁡(2−Kbk)\alpha=\lambda_{x}^{K}\sqrt{-K}\left\langle x,v_{k}\right\rangle\cosh\left(2\sqrt{-K}b_{k}\right) , β=(λxK−1)sinh⁡(2−Kbk)\beta=\left(\lambda_{x}^{K}-1\right)\sinh\left(2\sqrt{-K}b_{k}\right) , with αk>0,vk∈Sn−1,bk∈R\alpha_{k}>0,v_{k}\in\mathbb{S}^{n-1},b_{k}\in\mathbb{R}PKn\mathbb{P}^{n}_{K}RealC(n+2)C(n+2)✓C(4n+52)C(4n+52)✓
Pseudo-Busemann MLR [ 45 , Cor. 4.3]−d(x,pk)Bvk(−pk⊕Mx)∥−pk⊕Mx∥\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 pk∈PKn,vk∈Sn−1p_{k}\in\mathbb{P}^{n}_{K},v_{k}\in\mathbb{S}^{n-1}PKn\mathbb{P}^{n}_{K}PseudoC(2n)C(2n)✗C(19n+34)C(19n+34)✗
Lorentz MLR [ 4 , Eq. (12)]1−Ksign⁡(α)β∣sinh⁡−1(−Kαβ)∣\displaystyle\frac{1}{\sqrt{-K}}\operatorname{sign}(\alpha)\beta\left|\sinh^{-1}\left(\sqrt{-K}\frac{\alpha}{\beta}\right)\right| , α=cosh⁡(−Kbk)⟨zk,xs⟩−sinh⁡(−Kbk)\alpha=\cosh\left(\sqrt{-K}b_{k}\right)\left\langle z_{k},x_{s}\right\rangle-\sinh\left(\sqrt{-K}b_{k}\right) , β=∥cosh⁡(−Kbk)zk∥2−(sinh⁡(−Kbk)∥zk∥)2\beta=\sqrt{\left\|\cosh(\sqrt{-K}b_{k})z_{k}\right\|^{2}-(\sinh(\sqrt{-K}b_{k})\left\|z_{k}\right\|)^{2}} , with zk∈Rn, bk∈Rz_{k}\in\mathbb{R}^{n},\ b_{k}\in\mathbb{R}LKn\mathbb{L}^{n}_{K}RealC(n+1)C(n+1)✓C(4n+52)C(4n+52)✓
BMLR−αkBvk(x)+bk-\alpha_{k}B^{v_{k}}(x)+b_{k} , with αk>0,vk∈Sn−1,bk∈R\alpha_{k}>0,v_{k}\in\mathbb{S}^{n-1},b_{k}\in\mathbb{R}PKn\mathbb{P}^{n}_{K} LKn\mathbb{L}^{n}_{K}RealC(n+2)C(n+2)✓PKn:C(6n+12)\mathbb{P}^{n}_{K}:C(6n+12) LKn:C(2n+12)\mathbb{L}^{n}_{K}:C(2n+12)✓
Table 2: Comparison of CC -class MLR. In Dist, Real means the point-to-hyperplane distance is the real distance, obtained by inf⁡y∈Hd⁡(x,y)\inf_{y\in H}\operatorname{d}(x,y) , where HH is a hyperplane and d⁡\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 .
MethodF:HKn∋x↦y∈HKm\mathcal{F}:\mathcal{H}^{n}_{K}\ni x\mapsto y\in\mathcal{H}^{m}_{K}SpaceMethodologyParameters#ParamsFLOPs
Möbius [ 23 , Eq. 27]1−Ktanh⁡(∥Wx∥∥x∥tanh⁡−1(−K∥x∥))Wx∥Wx∥\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\|}PKn\mathbb{P}^{n}_{K}TangentW∈Rm×nW\in\mathbb{R}^{m\times n}mnmn2nm+2n2nm+2n +2m+24+2m+24
Poincaré FC [ 51 , Eq. (7)]y=ω1+1−K∥ω∥2, ωk=sinh⁡(−Kuk(x))−K\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 uk(x)u_{k}(x) in Tab. 2PKn\mathbb{P}^{n}_{K}Poincaré geometryαk>0, vk∈Sn−1\alpha_{k}>0,\ v_{k}\in\mathbb{S}^{n-1} , bk∈Rb_{k}\in\mathbb{R} , for k=1,…,mk=1,\ldots,mm(n+2)m(n+2)4nm+71m+44nm+71m+4
Lorentz FC [ 11 , Eq. (3)]y=[∥ψ(Wx,v)∥2−1/Kψ(Wx,v)]\displaystyle y=\begin{bmatrix}\sqrt{\left\|\psi(Wx,v)\right\|^{2}-1/K}\\ \psi(Wx,v)\end{bmatrix} , ψ(Wx,v)=λσ(v⊤x+b′)Wϕ(x)+b∥Wϕ(x)+b∥\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 ϕ\phi and σ\sigma as the activation and sigmoidLKn\mathbb{L}^{n}_{K}Ambient MinkowskiW∈Rm×(n+1)W\in\mathbb{R}^{m\times(n+1)} , v∈Rn+1v\in\mathbb{R}^{n+1} , b∈Rmb\in\mathbb{R}^{m} , b′∈Rb^{\prime}\in\mathbb{R} , λ>0\lambda>0m(n+1)+mm(n+1)+m +(n+1)+2+(n+1)+22nm+8m2nm+8m +2n+10+2n+10
BFCy=ω1+1−K∥ω∥2\displaystyle y=\frac{\omega}{1+\sqrt{1-K\left\|\omega\right\|^{2}}} , ω=sinh⁡(−Ku(x))−K\displaystyle\omega=\frac{\sinh\left(\sqrt{-K}u(x)\right)}{\sqrt{-K}} ; ys=1−Ksinh⁡(−Ku(x))\displaystyle y_{s}=\frac{1}{\sqrt{-K}}\sinh\left(\sqrt{-K}u(x)\right) , yt=1−K+∥ys∥2\displaystyle y_{t}=\sqrt{\frac{1}{-K}+\left\|y_{s}\right\|^{2}} , with uk(x)=ϕ(−αkBvk(x)+bk)\displaystyle u_{k}(x)=\phi(-\alpha_{k}B^{v_{k}}(x)+b_{k})PKn\mathbb{P}^{n}_{K} LKn\mathbb{L}^{n}_{K}Busemannαk>0, vk∈Sn−1\alpha_{k}>0,\ v_{k}\in\mathbb{S}^{n-1} , bk∈Rb_{k}\in\mathbb{R} , for k=1,…,mk=1,\ldots,mm(n+2)m(n+2)6nm+29m+46nm+29m+4 2nm+30m+22nm+30m+2
Table 3: Comparison of hyperbolic FC layers. For simplicity, BFC layers do not involve the gyroaddition and assume ϕ\phi is the identity map, which is in line with the Möbius and Lorentz FC layers.
SpaceMethodCIFAR-10 (Num. classes: 10)CIFAR-100 (Num. classes: 100)Tiny-ImageNet (Num. classes: 200)ImageNet-1k (Num. classes: 1000)
AccFit Time#ParamsAccFit Time#ParamsAccFit Time#ParamsAccFit Time#Params
Rn\mathbb{R}^{n}MLR95.14 ± 0.1210.665.13K77.72 ± 0.1510.6051.30K65.19 ± 0.1269.17102.60K71.872263.12513K
PKn\mathbb{P}^{n}_{K}PMLR95.04 ± 0.1311.945.14K77.19 ± 0.5012.1151.40K64.93 ± 0.3871.90102.80K71.772300.11514K
PBMLR-P95.23 ± 0.0821.9210.24K77.78 ± 0.1576.84102.40K65.43 ± 0.27336.58204.80K71.463907.121024K
BMLR-P95.32 ± 0.1412.015.14K78.10 ± 0.3512.1351.40K66.16 ± 0.1971.98102.80K73.362300.77514K
LKn\mathbb{L}^{n}_{K}LMLR94.98 ± 0.1211.555.13K78.03 ± 0.2111.7251.30K65.63 ± 0.1069.27102.60K72.462277.17513K
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.
BenchmarkTaskDatasetNum.PKn\mathbb{P}^{n}_{K}LKn\mathbb{L}^{n}_{K}
classesPMLRPBMLR-PBMLRLMLRBMLR-L
TEBRetrotransposonsLTR Copia275.34 ± 1.0274.37 ± 1.4876.73 ± 1.0873.01 ± 1.0775.86 ± 1.52
LINEs285.54 ± 0.6185.92 ± 0.6586.05 ± 1.0883.14 ± 0.8086.72 ± 0.58
SINEs295.30 ± 0.8595.34 ± 1.5895.99 ± 0.7496.70 ± 0.8796.29 ± 0.59
DNA transposonsCMC-EnSpm283.39 ± 0.5683.62 ± 1.0084.03 ± 0.7181.78 ± 1.0584.15 ± 1.00
hAT-Ac289.38 ± 0.9089.86 ± 0.5489.62 ± 0.7488.94 ± 0.6990.70 ± 0.51
Table 5: Genomic MCC of MLR methods under the CNN backbone. Best results within each hyperbolic model are in bold .
DatasetPKn\mathbb{P}^{n}_{K}LKn\mathbb{L}^{n}_{K}
PMLRPBMLR-PBMLRLMLRBMLR-L
LTR Copia3.965.114.063.923.77
LINEs5.806.905.735.505.32
SINEs1.361.801.391.381.28
CMC-EnSpm3.114.383.042.892.82
hAT-Ac4.375.374.384.133.96
Table 6: Fit time (s/epoch) on genome sequence learning. The fastest times are in green and the slowest ones are in red .
SpaceMethodMethodologyDiseaseAirportPubMedCora
δ=0\delta=0δ=1\delta=1δ=3.5\delta=3.5δ=11\delta=11
PKn\mathbb{P}^{n}_{K}MöbiusTangent76.35 ± 1.8393.31 ± 0.4194.93 ± 0.0690.80 ± 0.56
Poincaré FCPoincaré geometry79.45 ± 1.0194.31 ± 0.1694.24 ± 0.2588.21 ± 0.72
BFC-PBusemann80.45 ± 0.9394.88 ± 0.3994.85 ± 0.0791.94 ± 0.32
LKn\mathbb{L}^{n}_{K}LTFCTangent71.32 ± 5.3692.68 ± 0.3594.85 ± 0.1789.37 ± 0.64
Lorentz FCAmbient Minkowski72.78 ± 2.0492.99 ± 0.3394.20 ± 0.1092.06 ± 0.62
Table 7: Comparison of hyperbolic FC layers on link prediction. Best results within each hyperbolic model are in bold .
SpaceMethodDiseaseAirportPubMedCora
δ=0\delta=0δ=1\delta=1δ=3.5\delta=3.5δ=11\delta=11
PKn\mathbb{P}^{n}_{K}HGCN86.87 ± 2.5885.34 ± 1.1676.29 ± 0.9876.56 ± 0.81
HGCN-PMLR88.98 ± 1.9684.78 ± 1.4876.02 ± 1.0977.47 ± 1.15
HGCN-PBMLR-P89.05 ± 0.7885.04 ± 0.9775.89 ± 0.7877.90 ± 1.00
HGCN-BMLR-P92.45 ± 0.9686.02 ± 0.5377.36 ± 0.7378.48 ± 1.52
LKn\mathbb{L}^{n}_{K}HGCN87.83 ± 0.7784.94 ± 1.4076.49 ± 0.8877.37 ± 1.72
Table 8: Node classification F1 scores of hyperbolic MLRs on the HGCN backbone, where δ\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

NotationDescription
M\mathcal{M}Riemannian manifold
TxMT_{x}\mathcal{M}Tangent space at xx
Exp⁡x(v)\operatorname{Exp}_{x}(v)Exponential map at xx
Log⁡x(y)\operatorname{Log}_{x}(y)Logarithmic map at xx
PT⁡x→y(v)\operatorname{PT}_{x\rightarrow y}(v)Parallel transport of vv from xx to yy
gx(u,v)g_{x}(u,v)Riemannian metric at xx
Appendix
Table 9: Summary of notation.
OperatorPoincaré ball PKn\mathbb{P}^{n}_{K}Lorentz LKn\mathbb{L}^{n}_{K}
DefinitionPKn={x∈Rn∣∥x∥2<−1/K}\displaystyle\mathbb{P}^{n}_{K}=\{x\in\mathbb{R}^{n}\mid\left\|x\right\|^{2}<-1/K\}LKn={x∈Rn+1∣⟨x,x⟩L=1/K,xt>0}\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\}
gx(w,v)g_{x}(w,v)(λxK)2⟨w,v⟩,λxK=21+K∥x∥2\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}}⟨w,v⟩L=⟨vs,ws⟩−vtwt\displaystyle\left\langle w,v\right\rangle_{\mathcal{L}}=\left\langle v_{s},w_{s}\right\rangle-v_{t}w_{t}
d⁡(x,y)\operatorname{d}(x,y)2∣K∣tanh⁡−1(∣K∣∥−x⊕My∥)\displaystyle\frac{2}{\sqrt{|K|}}\tanh^{-1}\left(\sqrt{|K|}\left\|-x\oplus_{\mathrm{M}}y\right\|\right)1∣K∣cosh⁡−1(K⟨x,y⟩L)\displaystyle\frac{1}{\sqrt{|K|}}\cosh^{-1}\left(K\left\langle x,y\right\rangle_{\mathcal{L}}\right)
Log⁡xy\operatorname{Log}_{x}y2∣K∣λxKtanh⁡−1(∣K∣∥−x⊕My∥)−x⊕My∥−x⊕My∥\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\|}cosh⁡−1(β)β2−1(y−βx),β=K⟨x,y⟩L\displaystyle\frac{\cosh^{-1}(\beta)}{\sqrt{\beta^{2}-1}}(y-\beta x),\quad\beta=K\left\langle x,y\right\rangle_{\mathcal{L}}
Exp⁡xv\operatorname{Exp}_{x}vx⊕M(tanh⁡(∣K∣λxK∥v∥2)v∣K∣∥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)cosh⁡(α)x+sinh⁡(α)αv,α=∣K∣∥v∥L\displaystyle\cosh(\alpha)x+\frac{\sinh(\alpha)}{\alpha}v,\quad\alpha=\sqrt{|K|}\left\|v\right\|_{\mathcal{L}}
PT⁡x→y(v)\operatorname{PT}_{x\rightarrow y}(v)λxKλyKgyr⁡[y,−x]v\displaystyle\frac{\lambda_{x}^{K}}{\lambda_{y}^{K}}\operatorname{gyr}[y,-x]vv−K⟨y,v⟩L1+K⟨x,y⟩L(x+y)\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<0K<0 ).
MethodHyperplaneFormulationApplied manifoldsCompact params
Euclidean MLREuclidean{x∈Rn∣⟨a,x⟩+b=0}\left\{x\in\mathbb{R}^{n}\mid\left\langle a,x\right\rangle+b=0\right\} a∈Rn, b∈Ra\in\mathbb{R}^{n},\ b\in\mathbb{R}Rn\mathbb{R}^{n}✓
Poincaré MLR [ 23 ]Geodesic [ 23 , Def. 3.1]{x∈PKn∣⟨Log⁡p(x),a⟩p=0}\left\{x\in\mathbb{P}^{n}_{K}\mid\left\langle\operatorname{Log}_{p}\left(x\right),a\right\rangle_{p}=0\right\} p∈PKn, a∈TpPKnp\in\mathbb{P}^{n}_{K},\ a\in T_{p}\mathbb{P}^{n}_{K}PKn\mathbb{P}^{n}_{K}✗
Pseudo-Busemann MLR [ 45 ]Busemann & gyro [ 45 , Def. 4.1]{x∈PKn∣Bv(−p⊕Mx)=0}\left\{x\in\mathbb{P}^{n}_{K}\mid B^{v}\left(-p\oplus_{\mathrm{M}}x\right)=0\right\} v∈Sn−1, p∈PKnv\in\mathbb{S}^{n-1},\ p\in\mathbb{P}^{n}_{K}PKn\mathbb{P}^{n}_{K}✗
Lorentz MLR [ 4 ]Ambient Minkowski [ 4 , Eq. (7)]{x∈LKn∣⟨w,x⟩L=0}\left\{x\in\mathbb{L}^{n}_{K}\mid\left\langle w,x\right\rangle_{\mathcal{L}}=0\right\} p∈LKn, w∈TpLKnp\in\mathbb{L}^{n}_{K},\ w\in T_{p}\mathbb{L}^{n}_{K}LKn\mathbb{L}^{n}_{K}✗
BMLRHorosphere{x∈HKn∣−αBv(x)+b=0}\left\{x\in\mathcal{H}^{n}_{K}\mid-\alpha B^{v}\left(x\right)+b=0\right\} α>0, v∈Sn−1, b∈R\alpha>0,\ v\in\mathbb{S}^{n-1},\ b\in\mathbb{R}PKn\mathbb{P}^{n}_{K} , LKn\mathbb{L}^{n}_{K}✓
Appendix
Table 11: Comparison of hyperplanes. Compact params indicates whether the parameterization requires an additional manifold-valued point.
MethodPoint-to-hyperplane distanceApplied manifoldsDist
Euclidean MLR∣⟨a,x⟩+b∣∥a∥\displaystyle\frac{|\left\langle a,x\right\rangle+b|}{\left\|a\right\|}Rn\mathbb{R}^{n}Real
Poincaré MLR [ 23 ]1−Ksinh⁡−1(2−K∣⟨−p⊕Mx,a⟩∣(1+K∥−p⊕Mx∥2)∥a∥)\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]PKn\mathbb{P}^{n}_{K}Real
Pseudo-Busemann MLR [ 45 ]d⁡(x,p)Bv(−p⊕Mx)∥−p⊕Mx∥\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]PKn\mathbb{P}^{n}_{K}Pseudo
Lorentz MLR [ 4 ]1−K∣sinh⁡−1(−K⟨v,x⟩L∥v∥L)∣\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)]LKn\mathbb{L}^{n}_{K}Real
BMLR∣−αBv(x)+b∣α\displaystyle\frac{\left|-\alpha B^{v}(x)+b\right|}{\alpha}PKn\mathbb{P}^{n}_{K} , LKn\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∈Hd⁡(x,y)\inf_{y\in H}\operatorname{d}(x,y) with HH as a hyperplane and d⁡\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.
HyperparameterCIFAR-10/100 Tiny-ImageNetImageNet-1k
Epochs200100
Batch size128256
Initial learning rate0.10.10.10.1
LR schedule60,120,160;γ=0.260,120,160;\gamma=0.230,60,90;γ=0.130,60,90;\gamma=0.1
Weight decay5e−45e^{-4}1e−41e^{-4}
OptimizerSGDSGD
Appendix
Table 13: Summary of hyperparameters used in the image classification task.
TaskSpeciesDatasetsNum. classesMax lengthTrain / Dev / Test
RetrotransposonsPlantLTR Copia25007666 / 682 / 568
LINEs100022502 / 2030 / 1782
SINEs50021152 / 1836 / 1784
DNA transposonsPlantCMC-EnSpm220019912 / 1872 / 1808
hAT-Ac100017322 / 1822 / 1428
PseudogenesHumanprocessed2100017956 / 1046 / 1740
Appendix
Table 14: Summary statistics for TEB.
TaskSpeciesDatasetNum. classesLengthTrain / Dev / Test
Core promoter detectionHumantata2704904 / 613 / 613
notata42452 / 5307 / 5307
all47356 / 5920 / 5920
Promoter detectionHumantata23004904 / 613 / 613
notata42452 / 5307 / 5307
all47356 / 5920 / 5920
Appendix
Table 15: Summary statistics for the adopted GUE datasets.
Batch size100
Epochs100
OptimizerAdam
β1\beta_{1} , β2\beta_{2}0.9, 0.999
Initial learning rate1e−41e^{-4}
LR schedule60,8560,85 ; γ=0.1\gamma=0.1
Weight decay0.10.1
Appendix
Table 16: Hyperparameters for genome sequence learning.
Dataset#Nodes#Edges#Classes#Features
Disease1044104321000
Airport31881863144
PubMed19717443383500
Cora2708542971433
Appendix
Table 17: Summary statistics for the node classification datasets.
SpaceWeight decayDropout
PKn\mathbb{P}^{n}_{K}1e−4,1e−5,1e−3,1e−31e-4,1e-5,1e-3,1e-30.3,0,0,0.20.3,0,0,0.2
LKn\mathbb{L}^{n}_{K}1e−4,5e−5,1e−3,1e−31e-4,5e-5,1e-3,1e-30,0,0,0.30,0,0,0.3
Appendix
Table 18: Hyperparameters for node classification on Disease, Airport, PubMed, and Cora.
SpaceMethodDiseaseAirportPubMedCora
Fit Time#ParamsFit Time#ParamsFit Time#ParamsFit Time#Params
PKn\mathbb{P}^{n}_{K}Möbius0.02004640.05354800.112082880.022923216
Poincaré FC0.01985280.05365440.117683520.024823280
BFC-P0.02015280.05125440.112383520.023123280
LKn\mathbb{L}^{n}_{K}LTFC0.03434640.08184800.163382880.037023216
Lorentz FC0.02325630.07155800.153788760.026124737
Appendix
Table 19: Efficiency comparison: fit time (s/epoch) and parameter count. Slowest results and largest parameter counts are in red .
Spaceϕ\phiAirportCora
PKn\mathbb{P}^{n}_{K}None94.57 ± 0.2987.55 ± 0.34
tanh94.88 ± 0.3991.94 ± 0.32
LKn\mathbb{L}^{n}_{K}None94.91 ± 0.2487.45 ± 1.39
tanh95.37 ± 0.1792.28 ± 0.12
Appendix
Table 20: Ablation on activation ϕ\phi in BFC layers.

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