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Hyperbolic Busemann Neural Networks
Ziheng Chen, Bernhard Schölkopf, Nicu Sebe
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Summary
This paper introduces Hyperbolic Busemann Neural Networks, proposing two new layers: Busemann Multinomial Logistic Regression (BMLR) and Busemann Fully Connected (BFC) layers. These layers lift core neural network components into hyperbolic space using Busemann functions, offering intrinsic geometric interpretations, compact parameterization, and batch-efficient computation. The approach generalizes Euclidean MLR and FC layers to both Poincaré ball and Lorentz models, demonstrating improved effectiveness and efficiency in tasks like image classification, genome sequence learning, node classification, and link prediction.
Entities (12)
Relation Signals (10)
Busemann Function → defines → Horosphere
confidence 95% · The level sets of a Busemann function are called horospheres...
Busemann Multinomial Logistic Regression → operatesin → Poincaré ball
confidence 95% · The hyperbolic BMLRs converge to the Euclidean MLR: (Poincaré) ... (Lorentz) ...
Busemann Multinomial Logistic Regression → operatesin → Lorentz model
confidence 95% · The hyperbolic BMLRs converge to the Euclidean MLR: (Poincaré) ... (Lorentz) ...
Busemann Multinomial Logistic Regression → uses → Busemann Function
confidence 95% · We introduce BMLR, deriving intrinsic logits directly from Busemann functions...
Busemann Multinomial Logistic Regression → appliedto → Image Classification
confidence 90% · Experiments on image classification... demonstrate improvements...
Busemann Multinomial Logistic Regression → appliedto → Genome Sequence Learning
confidence 90% · Experiments on... genome sequence learning... demonstrate improvements...
Busemann Multinomial Logistic Regression → appliedto → Node Classification
confidence 90% · Experiments on... node classification... demonstrate improvements...
Busemann Multinomial Logistic Regression → appliedto →
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Abstract
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 this https URL.
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- Source: https://arxiv.org/abs/2602.18858v2
- Canonical: https://arxiv.org/abs/2602.18858v2
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Hyperbolic Busemann Neural Networks Ziheng Chen1,2, Bernhard Schölkopf2 & Nicu Sebe1 1 University of Trento, 2 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. 1 Introduction Hyperbolic representations have recently delivered strong performance across different applications because the exponential volume growth of negatively curved manifolds enables low-distortion embeddings of tree-like and hierarchical structure [46]. These advantages have been validated in computer vision [24, 34, 20, 56, 25, 4, 31, 3, 54, 42, 57], graph learning [9, 2, 22, 53], multimodal learning [18, 48], recommendation systems [60], astronomy [10], genome sequence learning [33], natural language processing [46, 23, 47, 27, 30, 58], and brain signal decoding [40]. These empirical successes highlight the need for principled and efficient hyperbolic neural network components that fully leverage hyperbolic geometry. Among canonical formulations, the Poincaré ball and the Lorentz model are the most widely used, owing to their closed-form Riemannian operators. To support hyperbolic deep learning, several key building blocks in neural networks have been generalized to Poincaré or Lorentz spaces, including attention [27, 11, 59], batch normalization [43, 4, 13, 15], linear feed-forward layers [23, 51, 11], activation [23, 4], residual blocks [56, 32, 31], Multinomial Logistics Regression (MLR) [51, 4, 45], and graph convolution [9, 41, 2, 16]. Among these components, MLR classification and Fully Connected (FC) layers play a fundamental role in feature transformation and final decision-making. Recently, hyperplanes and point-to-hyperplane distances [38] have been adopted to construct hyperbolic MLR in both Poincaré [23, 51, 45] and Lorentz [4] models. Ganea et al. [23, Sec. 3.1] introduced the first Poincaré MLR based on Poincaré hyperplanes, but the formulation suffers from over-parameterization and lacks batch efficiency. Shimizu et al. [51, Sec. 3.1] alleviated these issues through a re-parameterization strategy. Building on these ideas, Bdeir et al. [4, Sec. 4.3] proposed a Lorentz MLR. However, its hyperplane is defined by the ambient Minkowski space, which is model-specific and may distort Lorentzian geometry. For hyperbolic FC layers, three main formulations exist. Ganea et al. [23, Sec. 3.2] introduced Möbius matrix-vector multiplication through the tangent space on the Poincaré ball. Shimizu et al. [51, Sec. 3.2] further proposed the Poincaré FC layer, defined intrinsically but restricted to the Poincaré model. On the Lorentz model, Chen et al. [11, Sec. 2.2] constructed a Lorentz FC layer by applying linear transformations in the ambient Minkowski space followed by projection onto the Lorentz model. Thus, Möbius and Lorentz FC rely on flat-space (tangent or ambient) approximations that could distort intrinsic geometry, whereas Poincaré FC is intrinsic but model-specific. On the other hand, the Busemann function and its level sets, horospheres, have emerged as powerful intrinsic tools for hyperbolic learning. They enjoy convenient metric properties [6, Ch. I.8] and admit closed-form expressions on both the Poincaré and Lorentz models [5, Prop. 9]. These operators have supported several hyperbolic algorithms, including SVM [21], PCA [8], Sliced Wasserstein distances [5], and prototype learning [26]. We also note that Nguyen et al. [45, Cor. 4.3] proposed a Poincaré MLR based on the Busemann function. However, its induced point-to-hyperplane distance is pseudo, coincides with the true distance only in Euclidean geometry, remains over-parameterized, and is not batch efficient. These observations motivate intrinsic and batch-efficient formulations for MLR and FC layers that can operate on both the Poincaré ball and the Lorentz model. To address this need, we propose Busemann Multinomial Logistics Regression (BMLR) and Busemann Fully Connected (BFC) layers, two Busemann-based components for hyperbolic networks. Our contributions are summarized as follows: • We introduce BMLR, deriving intrinsic logits directly from Busemann functions with a point-to-horosphere distance interpretation. BMLR uses a compact per-class parameterization, eliminates manifold-valued parameters in prior MLRs, remains batch-efficient, and recovers Euclidean MLR as curvature tends to zero. • We develop BFC layers by generalizing the FC and activation layers through the Busemann function, providing intrinsic constructions on both the Poincaré and Lorentz models. BFC preserves comparable complexity and parameter counts, and recovers Euclidean FC in the zero curvature limit. • We provide empirical validation across image classification, genome sequence learning, node classification, and link prediction. BMLR and BFC generally outperform existing hyperbolic layers. BMLR shows particularly large gains as the number of classes increases, and the Lorentz BMLR is the fastest among all hyperbolic MLRs. 2 Preliminaries Hyperbolic geometry. An n-dimensional hyperbolic space is a Riemannian manifold with constant negative sectional curvature K<0K<0. There exist five models of hyperbolic space [7, Sec. 7]. Among them, the Poincaré ball and Lorentz models are the most widely used. The Poincaré ball model [39, p. 62] is ℙKn=x∈ℝn:‖x‖2<−1/KP^n_K=\x ^n: \|x \|^2<-1/K\ with Riemannian metric gx(u,v)=(λxK)2⟨u,v⟩g_x(u,v)=( _x^K)^2 u,v , where λxK=21+K‖x‖2 _x^K= 21+K \|x \|^2. Here ⟨⋅,⋅⟩ ·,· and ∥⋅∥ \|· \| denote the Euclidean inner product and norm. The origin is the zero vector ∈ℙKn0 ^n_K. The Lorentz model [39, p. 62], also called the hyperboloid model, is Kn=x∈ℝn+1:⟨x,x⟩ℒ=1/K,xt>0L^n_K=\x ^n+1: x,x _L=1/K,\ x_t>0\, where ⟨x,y⟩ℒ=−xtyt+⟨xs,ys⟩ x,y _L=-x_ty_t+ x_s,y_s is the Lorentzian inner product and ‖x‖ℒ=⟨x,x⟩ℒ \|x \|_L= x,x _L is the induced Lorentzian norm. Following the convention of special relativity [49, §3.1], we write x=[xt,xs⊤]⊤x=[x_t,x_s ] with temporal component xt∈ℝx_t and spatial component xs∈ℝnx_s ^n. The canonical origin is ¯=[1/|K|,0,…,0]⊤∈Kn 0=[ 1/|K|,0,…,0] ^n_K. Throughout, we write ℋKn∈ℙKn,KnH^n_K∈\P^n_K,L^n_K\ for a hyperbolic space. Geodesic, geodesic ray, and asymptote. Geodesics are length-minimizing curves. We consider unit speed geodesics. A geodesic ray is a geodesic defined for t∈[0,∞)t∈[0,∞). Two geodesic rays are asymptotic if the distance between corresponding points remains bounded as t tends to infinity [6, I. Def. 8.1]. This notion generalizes Euclidean parallel lines in two respects: (i) Euclidean parallels have constant separation, whereas asymptotic rays have bounded separation; (i) Euclidean parallels never intersect, and asymptotic rays do not intersect as well [6, I. Prop. 8.2]. Busemann function. Let γ be a geodesic ray and let d(⋅,⋅)d(·,·) denote the geodesic distance. The Busemann function [6, I. Def. 8.17] associated with γ is defined as ∀x∈ℋKn,Bγ(x)=limt→∞(d(x,γ(t))−t).∀ x ^n_K, B^γ(x)= _t→∞ (d (x,γ(t) )-t ). (1) In hyperbolic spaces, this limit exists [6, I. Lem. 8.18]. In Euclidean space, the Busemann function associated with the geodesic γ(t)=tvγ(t)=tv that starts at 0 with unit direction v∈n−1=v∈ℝn∣‖v‖=1v ^n-1=\v ^n \|v \|=1\ is ∀x∈ℝn,Bv(x)=−⟨x,v⟩,∀ x ^n, B^v(x)=- x,v , (2) which coincides, up to a sign, with the inner product. Hence, the Busemann function provides an intrinsic generalization of the inner product to manifolds. Up to an additive constant, it is equivalent among asymptotic geodesic rays [6, I. Cor. 8.20], independent of the starting point. Consequently, we write Bv(x)B^v(x) for the ray that emanates from the origin e∈ℋKne ^n_K with unit velocity v∈n−1⊂TeℋKnv ^n-1⊂ T_eH^n_K. For v∈n−1v ^n-1 and x∈ℋKnx ^n_K, closed forms of the Poincaré and Lorentz Busemann functions [5, Prop. 9] are ℙKn:Bv(x)=1−Klog(‖v−Kx‖21+K‖x‖2), ^n_K:\ B^v(x)= 1 -K ( \|v- -Kx \|^21+K \|x \|^2 ), (3) Kn:Bv(x)=1−Klog(−K(xt−⟨xs,v⟩)). ^n_K:\ B^v(x)= 1 -K ( -K (x_t- x_s,v ) ). (4) Figure 1: Illustration: red curves are different horospheres of BvB^v. Horosphere. The level sets of a Busemann function are called horospheres, the hyperbolic counterpart of Euclidean hyperplanes. In Euclidean space, for a unit direction v, the hyperplanes Hτv=x∈ℝn∣⟨x,v⟩=τH^v_τ=\x ^n x,v =τ\ with τ∈ℝτ are parallel. Analogously, for fixed v, the horospheres Hτv=x∈ℋKn∣Bv(x)=τH^v_τ=\x ^n_K B^v(x)=τ\ are equidistant (see Thm.˜3.3). Fig.˜1 illustrates such horospheres, and Tab.˜1 summarizes the correspondence of the above concepts between Euclidean and hyperbolic geometries. Table 1: Generalization: Euclidean vs. hyperbolic. Euclidean Hyperbolic Straight line Geodesic Parallel lines Asymptotic geodesic rays Inner product ⟨v,x⟩ v,x Busemann function −Bv(x)-B^v(x) Hyperplane Horosphere Parallel hyperplanes with v∈n−1v ^n-1 Horospheres with v∈n−1∩TeℋKnv ^n-1∩ T_eH^n_K Riemannian & gyrovector operators. The hyperbolic space ℋKnH^n_K provides closed-form expressions for standard Riemannian operators. It also supports a gyrovector structure that extends vector space operations to hyperbolic geometry. On the Poincaré ball ℙKnP^n_K, this structure corresponds to the Möbius gyrovector space [55, Ch. 6.14], denoted as ℙKn,⊕M,⊙M\P^n_K, _M, _M\. As shown by Chen et al. [15, Props. 24 and 25], the Lorentz model also admits a closed-form gyrovector space, denoted as Kn,⊕,⊙\L^n_K, _L, _L\. Please refer to Sec.˜B.4 for detailed expressions. 3 Busemann multinomial logistic regression We begin by reformulating the Euclidean Multinomial Logistics Regression (MLR), then lift it to hyperbolic space via the Busemann function, introducing Busemann MLR (BMLR). We also present a point-to-hyperplane interpretation. Finally, we compare BMLR with existing hyperbolic MLRs, highlighting our advantages in geometric fidelity, parameterization, and computational efficiency. 3.1 Formulation The Euclidean MLR softmax(Ax+b)softmax(Ax+b) computes the multinomial probability for each class k∈1,…,Ck∈\1,…,C\ given an input x∈ℝnx ^n. It admits the inner product form: ∀k,p(y=k∣x)=exp(⟨ak,x⟩+bk)∑j=1Cexp(⟨aj,x⟩+bj), llasection to20.74pt eq:euc-mlr eq:euc-mlr∀ k,\ p(y=k x)= ( a_k,x +b_k) _j=1^C ( a_j,x +b_j), (5) where ak∈ℝna_k ^n and bk∈ℝb_k are the weight and bias for the class k. We write p(y=k∣x)∝exp(uk(x))p(y=k x) (u_k(x) ) with uk(x)=⟨ak,x⟩+bku_k(x)= a_k,x +b_k. Decomposing the weight vector into a magnitude αk=‖ak‖>0 _k= \|a_k \|>0 and a unit direction vk=ak‖ak‖∈n−1v_k= a_k \|a_k \| ^n-1, each logit is uk(x)=αk⟨vk,x⟩+bk. llasection to20.74pt eq:vk-decomp eq:vk-decompu_k(x)= _k v_k,x +b_k. (6) As shown in Tab.˜1, the Busemann function naturally generalizes the Euclidean inner product. Analogously to Eq.˜6, we define the hyperbolic logits via the Busemann function, yielding BMLR: ∀k,p(y=k∣x) ∀ k,\ p(y=k x) =exp(uk(x))∑j=1Cexp(uj(x)), = (u_k(x)) _j=1^C (u_j(x)), (7) uk(x) llasection to20.74pt eq:b-logits eq:b-logitsu_k(x) =−αkBvk(x)+bk, =- _kB^v_k(x)+b_k, (8) with αk>0 _k>0, vk∈n−1v_k ^n-1, and bk∈ℝb_k as parameters. The following results show that, as K→0−K→ 0^-, both the Poincaré and Lorentz BMLRs reduce to the Euclidean MLR. Theorem 3.1 (Limits of BMLRs). As K→0−K→ 0^-, the hyperbolic Busemann functions converge to the Euclidean inner product: (Poincaré) Bv(x)→K→0−2⟨v,x⟩, B^v(x) K→ 0^--2 v,x , (9) (Lorentz) Bv(x)→K→0−⟨v,xs⟩. B^v(x) K→ 0^-- v,x_s . (10) The hyperbolic BMLRs converge to the Euclidean MLR: (Poincaré) uk(x)→K→0−2αk⟨vk,x⟩+bk, u_k(x) K→ 0^-2 _k v_k,x +b_k, (11) (Lorentz) uk(x)→K→0−αk⟨vk,xs⟩+bk, u_k(x) K→ 0^- _k v_k,x_s +b_k, (12) Proof. The proof is provided in Sec.˜F.1. ∎ Remark 3.2 (Intuition). On the Poincaré ball, letting K→0−K→ 0^- recovers Euclidean geometry [52, App. A.4.2]. For the Lorentz model, as K→0−K→ 0^-, the temporal coordinate diverges while the spatial component approaches ℝnR^n, making KnL^n_K converge to a Euclidean space. Consistently, the Poincaré and Lorentz Busemann functions and the associated BMLR logits reduce to their Euclidean counterparts, providing a natural generalization of Euclidean MLR. Table 2: Comparison of C-class MLR. In Dist, Real means the point-to-hyperplane distance is the real distance, obtained by infy∈Hd(x,y) _y∈ Hd(x,y), where H is a hyperplane and dd 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 Logit uk(x),∀k∈1,…,Cu_k(x),\ ∀ k∈\1,…,C\ Space Dist #Params Compact params FLOPs Batch efficiency Euclidean MLR ⟨ak,x⟩+bk a_k,x +b_k, with ak∈ℝn,bk∈ℝa_k ^n,b_k ℝnR^n Real C(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‖) _p_k^K \|a_k \| -K ^-1 ( 2 -K -p_k _Mx,a_k (1+K \|-p_k _Mx \|^2 ) \|a_k \| ), with pk∈ℙKn,ak∈TpkℙKnp_k ^n_K,a_k∈ T_p_kP^n_K ℙKnP^n_K Real C(2n)C(2n) ✗ C(19n+29)C(19n+29) ✗ Poincaré MLR [51, Eq. (6)] 2−Kαksinh−1(α−β) 2 -K _k ^-1 (α-β ), α=λxK−K⟨x,vk⟩cosh(2−Kbk)α= _x^K -K x,v_k (2 -Kb_k ), β=(λxK−1)sinh(2−Kbk)β= ( _x^K-1 ) (2 -Kb_k ), with αk>0,vk∈n−1,bk∈ℝ _k>0,v_k ^n-1,b_k ℙKnP^n_K Real C(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‖ -d(x,p_k) B^v_k (-p_k _Mx ) \|-p_k _Mx \|, with pk∈ℙKn,vk∈n−1p_k ^n_K,v_k ^n-1 ℙKnP^n_K Pseudo C(2n)C(2n) ✗ C(19n+34)C(19n+34) ✗ Lorentz MLR [4, Eq. (12)] 1−Ksign(α)β|sinh−1(−Kαβ)| 1 -Ksign(α)β | ^-1 ( -K αβ ) |, α=cosh(−Kbk)⟨zk,xs⟩−sinh(−Kbk)α= ( -Kb_k ) z_k,x_s - ( -Kb_k ), β=‖cosh(−Kbk)zk‖2−(sinh(−Kbk)‖zk‖)2β= \| ( -Kb_k)z_k \|^2-( ( -Kb_k) \|z_k \|)^2, with zk∈ℝn,bk∈ℝz_k ^n,\ b_k KnL^n_K Real C(n+1)C(n+1) ✓ C(4n+52)C(4n+52) ✓ BMLR −αkBvk(x)+bk- _kB^v_k(x)+b_k, with αk>0,vk∈n−1,bk∈ℝ _k>0,v_k ^n-1,b_k ℙKnP^n_K KnL^n_K Real C(n+2)C(n+2) ✓ ℙKn:C(6n+12)P^n_K:C(6n+12) Kn:C(2n+12)L^n_K:C(2n+12) ✓ 3.2 Geometric interpretation As shown by Lebanon and Lafferty [38, Sec. 5], the Euclidean MLR can be reformulated by point-to-hyperplane distances: uk(x)=sign(⟨ak,x⟩+bk)‖ak‖d(x,Hak,bk), llasection to20.74pt eq:euc-point-to-hyperplane eq:euc-point-to-hyperplaneu_k(x)=sign ( a_k,x +b_k ) \|a_k \|d (x,H_a_k,b_k ), (13) with ak∈ℝna_k ^n and bk∈ℝb_k . Here, Hak,bk=x∈ℝn:⟨ak,x⟩+bk=0H_a_k,b_k=\x ^n: a_k,x +b_k=0\ denotes the margin hyperplane, and the point-to-hyperplane distance is d(x,Hak,bk)=|⟨ak,x⟩+bk|‖ak‖d (x,H_a_k,b_k )= | a_k,x +b_k | \|a_k \|. We next show that our BMLR also admits such an interpretation through point-to-horosphere distances. We first introduce these distances in Hadamard spaces, metric spaces with favorable properties (see Sec.˜B.2.2), where Euclidean and hyperbolic geometries arise as special cases. Theorem 3.3 (Hadamard horosphere distance). Let (,d)(X,d) be a Hadamard space, and Bγ:→ℝB^γ:X be the Busemann function associated with a geodesic ray γ:[0,∞)→γ:[0,∞) . For any τ1,τ2∈ℝ _1, _2 , define the horospheres by Hτiγ=x∈∣Bγ(x)=τi,i=1,2.H^γ_ _i= \x B^γ(x)= _i \,\ i=1,2. (14) The distance between these horospheres is constant: d(Hτ1γ,Hτ2γ)=d(Hτ2γ,Hτ1γ)=|τ2−τ1|. llasection to20.74pt eq:equidistance eq:equidistanced (H^γ_ _1,H^γ_ _2 )=d (H^γ_ _2,H^γ_ _1 )= | _2- _1 |. (15) In particular, the point-to-horosphere distance is d(x,Hτγ)=|Bγ(x)−τ|,∀x∈.d (x,H^γ_τ )= |B^γ(x)-τ |,\ ∀ x . (16) Proof. A Hadamard space does not need to be a manifold. It extends manifolds with nonpositive curvature to the broader setting of metric spaces. A brief review of metric geometry and the complete proof are given in Secs.˜B.2 and F.2. ∎ Corollary 3.4 (Point-to-horosphere distance). In a hyperbolic space ℋKn∈ℙKn,KnH^n_K∈\P^n_K,L^n_K\ with nonpositive curvature K≤0K≤ 0, the point-to-horosphere distance is d(x,Hτv)=|Bv(x)−τ|,d (x,H^v_τ )= |B^v(x)-τ |, (17) where Hτv=x∣Bv(x)=τH^v_τ= \x B^v(x)=τ \ denotes the horosphere with respect to the direction v∈n−1v ^n-1. A Euclidean hyperplane Ha,bH_a,b can be parameterized as Hv,α,b=x∈ℝn:α⟨v,x⟩+b=0H_v,α,b= \x ^n:α v,x +b=0 \ with v∈n−1v ^n-1, α>0α>0, and b∈ℝb . Similarly, we parameterize a hyperbolic horosphere as Hv,α,b=x∈ℋKn∣−αBv(x)+b=0, llasection to20.74pt eq:horosphere_param eq:horosphere_paramH_v,α,b= \x ^n_K -α B^v(x)+b=0 \, (18) with v∈n−1v ^n-1, α>0α>0, and b∈ℝb . With this parameterization, we extend Eq.˜13 to hyperbolic spaces: uk(x)=signkαkd(x,Hvk,αk,bk) llasection to20.74pt eq:horosphere-mlr eq:horosphere-mlru_k(x)=sign_k _kd (x,H_v_k, _k,b_k ) (19) where signk=sign(−αkBvk(x)+bk)sign_k=sign (- _kB^v_k(x)+b_k ), and vk∈n−1,αk>0,bk∈ℝ\v_k ^n-1, _k>0,b_k \ are parameters for class k. By Sec.˜3.2, the point-to-horosphere distance is d(x,Hv,α,b)=|−αBv(x)+b|α. llasection to20.74pt eq:point-to-horosphere-dist-param eq:point-to-horosphere-dist-paramd (x,H_v,α,b )= |-α B^v(x)+b |α. (20) By Eq.˜20, Eq.˜19 equals the exact BMLR logit in Eq.˜8. Remark 3.5 (Generality). Since Bv(x)=−⟨v,x⟩B^v(x)=- v,x in Euclidean geometry, Eqs.˜18, 20 and 19 naturally generalize to their Euclidean counterparts. We also acknowledge Fan et al. [21, Eq. (2) and Prop. 3.1], who used horospheres and point-to-horosphere distances to construct a hyperbolic SVM. However, they considered only the unit Poincaré ball with the specific curvature K=−1K=-1, which is a special case of our Eqs.˜18 and 20. 3.3 Comparison Based on the point-to-hyperplane reformulation in Eq.˜13, recent work extended MLR to the Poincaré [23, 51, 45] and Lorentz [4] models. Ganea et al. [23, Sec. 3.1] introduced the first Poincaré MLR by replacing the Euclidean point-to-hyperplane distance with its hyperbolic counterpart, where the hyperplane is defined by geodesics and the resulting distance is the real point-to-hyperplane distance, obtained as an infimum over the hyperplane. However, the formulation is not batch efficient (see App.˜C). It also requires per-class parameters ak∈TpkℙKna_k∈ T_p_kP^n_K and pk∈ℙKnp_k ^n_K, which leads to over-parameterization. Shimizu et al. [51, Sec. 3.1] alleviated such issues via re-parameterization. Bdeir et al. [4, Sec. 4.3] further developed a Lorentz MLR, but its hyperplanes are defined by the ambient Minkowski space, which is tailored to the Lorentz model and does not fully respect the intrinsic hyperbolic geometry. Moreover, Nguyen et al. [45, Cor. 4.3] proposed a Poincaré MLR based on the Busemann function. We refer to it as Pseudo-Busemann MLR, as the induced point-to-hyperplane distance is pseudo, coinciding with the real point-to-hyperplane distance only in Euclidean geometry. It also suffers from over-parameterization and is not batch efficient. As summarized in Tab.˜2111Relative to [45, Def. 4.2, Cor. 4.3, and App. B.1.2], the Pseudo-Busemann MLR written here includes an additional sign −-; this is intentional and matches their official implementation., BMLR unifies advantages that prior hyperbolic MLRs offer only partially. In particular, BMLR respects the real point-to-horosphere distance, uses compact parameters without an additional manifold-valued point, attains the lowest FLOPs on KnL^n_K and a competitive cost on ℙKnP^n_K, and supports batch-efficient computation. On KnL^n_K, its FLOPs are even close to those of the Euclidean MLR. 4 Busemann fully connected layer We first revisit the Euclidean Fully Connected (FC) layer from a geometric perspective, then propose the hyperbolic Busemann FC (BFC) layer with its manifestations in the Poincaré and Lorentz models. 4.1 Formulation Euclidean FC layers. An FC affine transformation ℱ:ℝn∋x↦y=Ax+b∈ℝmF:R^n x y=Ax+b ^m can be expressed element-wise as yk=⟨ak,x⟩+bky_k= a_k,x +b_k with ak∈ℝna_k ^n and bk∈ℝb_k . As shown by Shimizu et al. [51, Sec. 3.2] and Chen et al. [14, Sec. 3.1], the LHS yky_k can be interpreted as the signed distance from y to the hyperplane passing through the origin and orthogonal to the k-th axis of the output space. Hence, the FC layer can be expressed as d¯(y,Hek,0)=⟨ak,x⟩+bk,∀1≤k≤m, llasection to20.74pt eq:euc-fc-p2h eq:euc-fc-p2h d (y,H_e_k,0 )= a_k,x +b_k,\ ∀ 1≤ k≤ m, (21) where d¯(y,Hek,0)=sign(⟨ek,y⟩)d(y,Hek,0) d (y,H_e_k,0 )=sign ( e_k,y )d (y,H_e_k,0 ) is the signed point-to-hyperplane distance, and Hek,0=y∈ℝm∣⟨ek,y⟩=0H_e_k,0=\y ^m e_k,y =0\ is a hyperplane with ek∈ℝme_k ^m as the vector whose k-th element is 1 and all others are 0. Lifting to hyperbolic space. To extend Eq.˜21 into hyperbolic space, the RHS can be readily replaced by Eq.˜8, as it generalizes the term ⟨ak,x⟩+bk a_k,x +b_k. For the LHS, a natural idea is to use the signed point-to-horosphere distance. However, as detailed in App.˜D, this may fail to admit a solution for y. We therefore follow the point-to-hyperplane distance in [23, Thm. 5] for the Poincaré model and the one in [4, Eq. (44)] for the Lorentz model. Given x∈ℋKnx ^n_K, the hyperbolic BFC layer ℱ:ℋKn∋x↦y∈ℋKmF:H^n_K x y ^m_K is given by solving y via the following m equations: d¯(y,Hek,e)=uk(x),∀1≤k≤m, llasection to20.74pt eq:hyp-fc-p2h eq:hyp-fc-p2h d (y,H_e_k,e )=u_k(x),\ ∀ 1≤ k≤ m, (22) where uk(x)=−αkBvk(x)+bku_k(x)=- _kB^v_k(x)+b_k with αk>0,vk∈n−1,bk∈ℝ\ _k>0,v_k ^n-1,b_k \ as parameters. Here, d¯(y,Hek,e) d (y,H_e_k,e ) is the hyperbolic signed distance from y to the hyperplane passing through the origin e∈ℋKne ^n_K. Next, we show that the above implicit definition has an explicit solution for the output y. Theorem 4.1. Given an input x∈ℙKnx ^n_K, the Poincaré BFC layer ℱ:ℙKn→ℙKmF:P^n_K ^m_K is given by y=ω1+1−K‖ω‖2,ω=[sinh(−Kuk(x))−K]k=1m,y= ω1+ 1-K \|ω \|^2,\ ω= [ ( -Ku_k(x) ) -K ]_k=1^m, where uk(x)=−αkBvk(x)+bku_k(x)=- _kB^v_k(x)+b_k with αk>0,vk∈n−1,bk∈ℝ\ _k>0,v_k ^n-1,b_k \ as parameters for k=1,…,m\k=1,…,m\. Proof. The proof, together with Hek,eH_e_k,e and d¯(y,Hek,e) d (y,H_e_k,e ), is provided in Sec.˜F.3, which is inspired by Shimizu et al. [51, App. D.3]. ∎ Theorem 4.2. Given an input x∈Knx ^n_K, the Lorentz BFC layer ℱ:Kn→KmF:L^n_K ^m_K is given by y=[ytys]=[1−K+‖ys‖21−Ksinh(−Ku(x))]y= bmatrixy_t\\ y_s bmatrix= bmatrix 1-K+ \|y_s \|^2\\[6.0pt] 1 -K ( -Ku(x) ) bmatrix where u(x)=(u1(x),…,um(x))⊤u(x)= (u_1(x),…,u_m(x) ) with uk(x)=−αkBvk(x)+bku_k(x)=- _kB^v_k(x)+b_k. Here, αk>0,vk∈n−1,bk∈ℝ\ _k>0,v_k ^n-1,b_k \ are parameters for k=1,…,m\k=1,…,m\. Proof. The proof, along with the hyperplane and point-to-hyperplane distance, is provided in Sec.˜F.4. ∎ Analogously to Thm.˜3.1, our BFC layers converge to their Euclidean counterparts as K→0−K→ 0^-. Theorem 4.3 (Limits of BFC layers). As K→0−K→ 0^-, the hyperbolic BFC layer ℋKn∋x↦y∈ℋKmH^n_K x y ^m_K reduces to a Euclidean FC layer: (Poincaré) yk→K→0−αk⟨vk,x⟩+12bk, y_k K→ 0^- _k v_k,x + 12b_k, (23) (Lorentz) (ys)k→K→0−αk⟨vk,xs⟩+bk. (y_s)_k K→ 0^- _k v_k,x_s +b_k. (24) Proof. The proof is provided in Sec.˜F.5. ∎ Table 3: Comparison of hyperbolic FC layers. For simplicity, BFC layers do not involve the gyroaddition and assume ϕφ is the identity map, which is in line with the Möbius and Lorentz FC layers. Method ℱ:ℋKn∋x↦y∈ℋKmF:H^n_K x y ^m_K Space Methodology Parameters #Params FLOPs Möbius [23, Eq. 27] 1−Ktanh(‖Wx‖x‖tanh−1(−K‖x‖))Wx‖Wx‖ 1 -K ( \|Wx \| \|x \| ^-1 ( -K \|x \| ) ) Wx \|Wx \| ℙKnP^n_K Tangent W∈ℝm×nW ^m× n mnmn 2nm+2n2nm+2n +2m+24+2m+24 Poincaré FC [51, Eq. (7)] y=ω1+1−K‖ω‖2,ωk=sinh(−Kuk(x))−K y= ω1+ 1-K \|ω \|^2,\ _k= ( -Ku_k(x) ) -K, with uk(x)u_k(x) in Tab.˜2 ℙKnP^n_K Poincaré geometry αk>0,vk∈n−1 _k>0,\ v_k ^n-1, bk∈ℝb_k , for k=1,…,mk=1,…,m m(n+2)m(n+2) 4nm+71m+44nm+71m+4 Lorentz FC [11, Eq. (3)] y=[‖ψ(Wx,v)‖2−1/Kψ(Wx,v)] y= bmatrix \|ψ(Wx,v) \|^2-1/K\\ ψ(Wx,v) bmatrix, ψ(Wx,v)=λσ(v⊤x+b′)Wϕ(x)+b‖Wϕ(x)+b‖ ψ(Wx,v)=λσ (v x+b ) Wφ(x)+b \|Wφ(x)+b \| with ϕφ and σ as the activation and sigmoid KnL^n_K Ambient Minkowski W∈ℝm×(n+1)W ^m×(n+1), v∈ℝn+1v ^n+1, b∈ℝmb ^m, b′∈ℝb , λ>0λ>0 m(n+1)+m(n+1)+m +(n+1)+2+(n+1)+2 2nm+8m2nm+8m +2n+10+2n+10 BFC y=ω1+1−K‖ω‖2 y= ω1+ 1-K \|ω \|^2, ω=sinh(−Ku(x))−K ω= ( -Ku(x) ) -K; ys=1−Ksinh(−Ku(x)) y_s= 1 -K ( -Ku(x) ), yt=1−K+‖ys‖2 y_t= 1-K+ \|y_s \|^2, with uk(x)=ϕ(−αkBvk(x)+bk) u_k(x)=φ(- _kB^v_k(x)+b_k) ℙKnP^n_K KnL^n_K Busemann αk>0,vk∈n−1 _k>0,\ v_k ^n-1, bk∈ℝb_k , for k=1,…,mk=1,…,m m(n+2)m(n+2) 6nm+29m+46nm+29m+4 2nm+30m+22nm+30m+2 4.2 Generalization When incorporating an activation function ϕ:ℝ→ℝφ:R , we can jointly express the Euclidean FC and activation layers, which yields Eq.˜21 with uk(x)=ϕ(⟨ak,x⟩+bk)u_k(x)=φ ( a_k,x +b_k ). Accordingly, we extend the hyperbolic BFC by inserting the activation into Eq.˜22: d¯(y,Hek,e)=ϕ(uk(x)),∀1≤k≤m. d (y,H_e_k,e )=φ (u_k(x) ),\ ∀ 1≤ k≤ m. (25) This is reflected in Thms.˜4.2 and 4.3 by replacing every uk(x)u_k(x) with ϕ(−αkBvk(x)+bk)φ (- _kB^v_k(x)+b_k ). Moreover, inspired by the Poincaré Möbius transformation [23, Sec. 3.2], a BFC transformation could be further followed by a gyroaddition ⊕ℋ _H: ℋKn∋x↦ℱ(x)⊕ℋb∈ℋKmH^n_K x (x) _Hb ^m_K with b∈ℋKmb ^m_K as a gyro bias. For example, the Lorentz BFC layer is generalized as Kn∋x↦y=[1−K+‖ys‖21−Ksinh(−Ku(x))]⊕b∈Km,L^n_K x y= bmatrix 1-K+ \|y_s \|^2\\[6.0pt] 1 -K ( -Ku(x) ) bmatrix _Lb ^m_K, where uk(x)=ϕ(−αkBvk(x)+bk)u_k(x)=φ (- _kB^v_k(x)+b_k ) with parameters αk>0,vk∈n−1,bk∈ℝk=1m\ _k>0,v_k ^n-1,b_k \^m_k=1 and b∈Kmb ^m_K. 4.3 Comparison Tab.˜3 compares BFC with prior hyperbolic FC layers. BFC faithfully respects hyperbolic geometry, whereas the Möbius and Lorentz FC layers apply Euclidean transformations in the tangent or ambient Minkowski space, which can distort intrinsic geometry. BFC also offers flexibility across models, while Poincaré FC and Lorentz FC are tailored to their respective models. In addition, BFC uses a comparable parameterization and maintains (nm)O(nm) FLOPs. On KnL^n_K, its FLOPs are (2mn)O(2mn), matching the fastest layers. 5 Experiments We first compare BMLRs with prior hyperbolic MLRs on three architectures: ResNet-18 (image classification), CNN (genome sequences), and HGCN (node classification). We then compare BFC with prior hyperbolic FC layers on link prediction. All experiments use both the Poincaré and Lorentz models. 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. 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 ℝnR^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 ℙKnP^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 KnL^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 BMLR-L 95.25 ± 0.02 11.08 5.14K 78.07 ± 0.26 11.22 51.40K 65.99 ± 0.14 69.19 102.80K 73.24 2276.53 514K Figure 2: Validation accuracy curves on ImageNet-1k. Table 5: Genomic MCC of MLR methods under the CNN backbone. Best results within each hyperbolic model are in bold. Benchmark Task Dataset Num. ℙKnP^n_K KnL^n_K 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 hAT-Ac 2 89.38 ± 0.90 89.86 ± 0.54 89.62 ± 0.74 88.94 ± 0.69 90.70 ± 0.51 Pseudogenes processed 2 72.45 ± 1.49 71.99 ± 2.04 73.09 ± 1.66 73.71 ± 1.76 73.32 ± 1.65 unprocessed 2 75.37 ± 2.27 71.99 ± 1.47 75.71 ± 1.89 74.54 ± 1.98 76.15 ± 1.61 GUE Core Promoter Detection tata 2 80.95 ± 1.47 79.32 ± 2.44 80.29 ± 1.63 80.90 ± 1.15 81.76 ± 1.16 notata 2 70.02 ± 0.52 70.60 ± 0.75 70.48 ± 0.35 71.26 ± 0.56 70.43 ± 0.39 all 2 67.64 ± 0.77 68.02 ± 0.63 68.50 ± 0.61 67.63 ± 0.56 68.36 ± 1.07 Promoter Detection tata 2 80.30 ± 1.59 80.27 ± 2.71 82.83 ± 1.69 83.27 ± 1.95 82.55 ± 1.54 notata 2 92.63 ± 0.36 93.05 ± 0.32 92.75 ± 0.51 91.74 ± 0.57 92.60 ± 0.49 all 2 90.53 ± 0.50 90.79 ± 0.77 90.20 ± 0.65 89.34 ± 0.40 89.82 ± 0.45 Covid Variant Classification Covid 9 74.09 ± 0.25 70.84 ± 0.80 73.40 ± 0.30 64.07 ± 0.51 72.45 ± 0.21 Species Classification Virus 20 67.24 ± 2.10 59.17 ± 3.32 77.12 ± 1.23 71.34 ± 2.05 77.21 ± 1.04 Fungi 25 15.06 ± 1.32 18.75 ± 1.77 30.01 ± 0.76 15.07 ± 1.88 30.14 ± 2.48 5.1 Image classification Setup. Following Bdeir et al. [4, Sec. 5.1] and Guo et al. [28, Sec. 4], we use a hybrid architecture with a ResNet-18 [29] backbone and an MLR head. We compare Euclidean MLR with hyperbolic variants in both models. In Poincaré, we evaluate Poincaré MLR (PMLR) re-parameterized by Shimizu et al. [51], Pseudo Busemann MLR (PBMLR-P) [45], and our BMLR-P. In Lorentz, we evaluate Lorentz MLR (LMLR) [4] and our BMLR-L. For hyperbolic MLRs, we map the ResNet-18 features to the target hyperbolic space before classification. We evaluate on CIFAR-10 [36], CIFAR-100 [36], Tiny-ImageNet [37], and ImageNet-1k [17]. On the first three datasets, we conduct five-fold experiments. More details appear in Sec.˜E.1. Results. Tab.˜4 reports top-1 validation accuracy, fit time per epoch, and classifier-head parameters. Fig.˜2 presents the ImageNet-1k accuracy curves. Overall, BMLR-P and BMLR-L consistently outperform prior hyperbolic MLRs with comparable parameters. Within each hyperbolic model, the accuracy margin over prior hyperbolic MLRs increases with the number of classes, from CIFAR-10 to CIFAR-100 and Tiny-ImageNet, with the largest gains on ImageNet-1k. This demonstrates the advantage of BMLR as task complexity increases. Besides, PBMLR-P uses approximately double the head parameters and is markedly slower due to complex batch-inefficient computation (see App.˜C), whereas BMLR-L achieves the fastest fit time among all hyperbolic MLRs. 5.2 Genome sequence learning Table 6: Fit time (s/epoch) on genome sequence learning. The fastest times are in green and the slowest ones are in red. Dataset ℙKnP^n_K KnL^n_K 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 processed 5.02 5.94 4.89 4.68 4.58 unprocessed 3.30 3.90 3.29 3.05 2.95 CPD-tata 0.72 1.35 0.70 0.67 0.62 CPD-notata 6.13 12.40 6.00 5.73 5.71 CPD-all 6.85 14.35 6.59 6.35 6.33 PD-tata 0.98 1.20 0.97 1.04 0.94 PD-notata 8.33 10.53 8.29 8.11 7.84 PD-all 9.40 12.07 9.19 9.06 8.83 Covid 28.96 45.52 27.97 27.58 26.67 Virus 25.28 29.57 25.67 25.12 24.85 Fungi 6.41 8.96 6.41 6.25 6.25 Setup. Recently, Khan et al. [33] demonstrated the effectiveness of hyperbolic embedding in genome sequence learning. Following their settings, we adopt a CNN backbone, which consists of three convolutional blocks and an MLR head. Similar to Sec.˜5.1, we compare our BMLR against previous hyperbolic MLR heads by replacing the final Euclidean MLR with a hyperbolic MLR. We validate on two benchmarks: Transposable Element Benchmark (TEB) [33] and Genome Understanding Evaluation (GUE) [62], covering a total of 16 datasets. More details are provided in Sec.˜E.2. Results. Tab.˜5 summarizes 5-fold average Matthews Correlation Coefficient (MCC) across TEB and GUE. Compared with other hyperbolic MLRs, our BMLR-P and BMLR-L achieve higher MCC in most tasks. Similar to Sec.˜5.1, the gains are more pronounced on complex datasets with more classes, e.g., Virus (20 classes) and Fungi (25 classes), demonstrating the effectiveness of our approach. Tab.˜6 reports fit time per epoch, where PBMLR-P is consistently the slowest due to batch inefficiency, and BMLR-L is the fastest. Table 7: Comparison of hyperbolic FC layers on link prediction. Best results within each hyperbolic model are in bold. Space Method Methodology Disease Airport PubMed Cora δ=0δ=0 δ=1δ=1 δ=3.5δ=3.5 δ=11δ=11 ℙKnP^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 KnL^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 BFC-L Busemann 78.36 ± 0.51 95.37 ± 0.17 94.90 ± 0.04 92.28 ± 0.12 Table 8: Node classification F1 scores of hyperbolic MLRs on the HGCN backbone, where δ denotes the graph hyperbolicity (lower is more hyperbolic). The best results within each hyperbolic model are highlighted in bold. Space Method Disease Airport PubMed Cora δ=0δ=0 δ=1δ=1 δ=3.5δ=3.5 δ=11δ=11 ℙKnP^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 KnL^n_K HGCN 87.83 ± 0.77 84.94 ± 1.40 76.49 ± 0.88 77.37 ± 1.72 HGCN-LMLR 89.72 ± 1.51 82.61 ± 1.01 75.44 ± 1.17 69.91 ± 3.61 HGCN-BMLR-L 90.80 ± 1.15 85.27 ± 1.17 77.30 ± 0.41 77.65 ± 2.10 5.3 Node classification Setup. Following Nguyen et al. [45], we adopt the HGCN [9] backbone to evaluate our BMLR on graph datasets, including Disease [1], Airport [61], PubMed [44], and Cora [50]. The HGCN backbone consists of a hyperbolic Graph Convolutional Network (GCN) and an MLR as the final classification layer. Both the GCN and the MLR are built on the hyperbolic space. The vanilla HGCN uses a tangent MLR, which maps features into the tangent space via LogeLog_e and applies a Euclidean MLR. We replace this with different hyperbolic MLRs. More details are provided in Sec.˜E.3. Results. Tab.˜8 reports average F1 scores. Our BMLRs consistently outperform prior hyperbolic MLRs within each hyperbolic model. As graphs become less hyperbolic, that is, for larger δ, existing hyperbolic heads could underperform the vanilla tangent-based MLR, for example, PBMLR-P on PubMed, and LMLR on Airport, PubMed, and Cora. Especially on Cora, which has the largest δ, LMLR lags the tangent baseline by a large margin (69.91 vs. 77.37). In contrast, BMLR remains the top performer across all δ values, indicating that Busemann-based decoding robustly strengthens HGCN over a broader range of graph hyperbolicity. 5.4 Link prediction Setup. We compare our BFC layers with prior hyperbolic FC layers, including the Möbius layer [23] that operates via the tangent space, the Lorentz FC layer [11] that operates through the ambient Minkowski space, and the Poincaré FC layer [51]. Mimicking the Möbius layer, we further implement a Lorentz tangent FC layer, Log¯(MLog¯(x))Log_ 0(MLog_ 0(x)), referred to as LTFC. Following Chami et al. [9], we evaluate on four graph datasets for the link prediction task: Disease [1], Airport [61], PubMed [44], and Cora [50]. Following the HNN implementation [23, 9], all methods share the same backbone with two FC layers. For a fair comparison, all hyperbolic FC layers are followed by a gyroaddition biasing. For BFC, we set ϕ=tanhφ= on Airport and Cora, which yields better performance, while we use the identity map on the other two datasets. More details are provided in Sec.˜E.4. Results. Tab.˜7 reports 5-fold test AUC. Our BFC layers generally outperform prior hyperbolic FC layers. The gains are most pronounced on Disease, which is the most hyperbolic (δ=0δ=0), where Busemann-based decoding is markedly more effective than tangent or ambient methods, indicating a better capture of intrinsic hyperbolic geometry. This observation aligns with geometric intuition, since tangent space or ambient space approximations inherently struggle to represent curved manifolds in highly non-Euclidean cases. Sec.˜E.4.2 summarizes fit time and parameter counts. LTFC is the slowest due to costly logarithmic and exponential maps, and LFC uses the largest number of parameters among Lorentz variants. In contrast, our BFC layers achieve training time and model size comparable to those of existing layers. 6 Conclusion We introduce BMLR and BFC as intrinsic components for hyperbolic neural networks on the Poincaré and Lorentz models, both built from the Busemann function. BMLR provides compact parameters, a point-to-horosphere interpretation, batch-efficient computation, and a limit that recovers Euclidean MLR. BFC extends FC and activation layers with practical O(nm)O(nm) complexity. Experiments across image classification, genome sequence learning, node classification, and link prediction validate the effectiveness and efficiency of our approaches. BMLR shows increasingly strong performance as the number of classes increases, and replacing existing hyperbolic FC layers with BFC yields additional gains. These results indicate that Busemann geometry offers unified and effective mathematical tools for building hyperbolic neural networks. Acknowledgements This work was supported by EU Horizon project ELLIOT (No. 101214398) and by the FIS project GUIDANCE (No. FIS2023-03251). We acknowledge CINECA for awarding high-performance computing resources under the ISCRA initiative, and the EuroHPC Joint Undertaking for granting access to Leonardo at CINECA, Italy. References [1] R. M. Anderson and R. M. May (1991) Infectious diseases of humans: dynamics and control. Oxford University Press. Cited by: §E.3.1, §5.3, §5.4. [2] G. Bachmann, G. Bécigneul, and O. Ganea (2020) Constant curvature graph convolutional networks. In ICML, Cited by: §1, §1. [3] A. Bdeir, J. Burchert, L. Schmidt-Thieme, and N. Landwehr (2025) Robust hyperbolic learning with curvature-aware optimization. In NeurIPS, Cited by: §1. [4] A. Bdeir, K. Schwethelm, and N. Landwehr (2024) Fully hyperbolic convolutional neural networks for computer vision. In ICLR, Cited by: Table 11, Table 11, Table 12, Table 12, §E.1.2, §E.1.2, §1, §1, §1, §3.3, Table 2, §4.1, §5.1, footnote 2. [5] C. Bonet, L. Drumetz, and N. Courty (2025) Sliced-Wasserstein distances and flows on Cartan-Hadamard manifolds. JMLR. Cited by: §1, §2. [6] M. R. Bridson and A. Haefliger (2013) Metric spaces of non-positive curvature. Vol. 319, Springer Science & Business Media. Cited by: §B.2.2, §B.2, §1, §2, §2, §2, §2. [7] J. W. Cannon, W. J. Floyd, R. Kenyon, W. R. Parry, et al. (1997) Hyperbolic geometry. Flavors of geometry 31 (59-115), p. 2. Cited by: §2. [8] I. Chami, A. Gu, D. P. Nguyen, and C. Ré (2021) HoroPCA: hyperbolic dimensionality reduction via horospherical projections. In ICML, Cited by: §1. [9] I. Chami, Z. Ying, C. Ré, and J. Leskovec (2019) Hyperbolic graph convolutional neural networks. In NeurIPS, Cited by: §E.3.2, §E.4.1, §1, §1, §5.3, §5.4. [10] T. Chen, X. Fu, Y. Gao, H. Qian, Y. Wei, K. Yan, H. Zhou, and J. Li (2025) Galaxy walker: geometry-aware vlms for galaxy-scale understanding. In CVPR, Cited by: §1. [11] W. Chen, X. Han, Y. Lin, H. Zhao, Z. Liu, P. Li, M. Sun, and J. Zhou (2022) Fully hyperbolic neural networks. In ACL, Cited by: §E.4.1, §1, §1, Table 3, §5.4. [12] Z. Chen, Y. Song, X. Wu, and N. Sebe (2024) RMLR: extending multinomial logistic regression into general geometries. In NeurIPS, Cited by: Appendix C. [13] Z. Chen, Y. Song, X. Wu, and N. Sebe (2025) Gyrogroup batch normalization. In ICLR, Cited by: §1. [14] Z. Chen, X. Wu, B. Schölkopf, and N. Sebe (2025) Building transformation layers for Riemannian neural networks. External Links: Link Cited by: §4.1. [15] Z. Chen, X. Wu, and N. Sebe (2025) Riemannian batch normalization: a gyro approach. arXiv preprint arXiv:2509.07115. Cited by: Definition B.15, §B.4, §B.4, §1, §2. [16] J. Dai, Y. Wu, Z. Gao, and Y. Jia (2021) A hyperbolic-to-hyperbolic graph convolutional network. In CVPR, Cited by: §1. [17] J. Deng, W. Dong, R. Socher, L. Li, K. Li, and L. Fei-Fei (2009) ImageNet: a large-scale hierarchical image database. In CVPR, Cited by: §E.1.1, §5.1. [18] K. Desai, M. Nickel, T. Rajpurohit, J. Johnson, and S. R. Vedantam (2023) Hyperbolic image-text representations. In ICML, Cited by: §1. [19] M. P. Do Carmo and J. Flaherty Francis (1992) Riemannian geometry. Vol. 6, Springer. Cited by: §B.1. [20] A. Ermolov, L. Mirvakhabova, V. Khrulkov, N. Sebe, and I. Oseledets (2022) Hyperbolic vision transformers: combining improvements in metric learning. In CVPR, Cited by: §1. [21] X. Fan, C. Yang, and B. Vemuri (2023) Horospherical decision boundaries for large margin classification in hyperbolic space. In NeurIPS, Cited by: §1, Remark 3.5. [22] X. Fu, Y. Gao, Y. Wei, Q. Sun, H. Peng, J. Li, and X. Li (2024) Hyperbolic geometric latent diffusion model for graph generation. In ICML, Cited by: §1. [23] O. Ganea, G. Bécigneul, and T. Hofmann (2018) Hyperbolic neural networks. In NeurIPS, Cited by: §B.4, item 3, Table 11, Table 11, Table 12, Table 12, §E.4.1, §1, §1, §1, §1, §3.3, Table 2, §4.1, §4.2, Table 3, §5.4. [24] Z. Gao, Y. Wu, Y. Jia, and M. Harandi (2021) Curvature generation in curved spaces for few-shot learning. In ICCV, Cited by: §1. [25] Z. Gao, C. Xu, F. Li, Y. Jia, M. Harandi, and Y. Wu (2023) Exploring data geometry for continual learning. In CVPR, Cited by: §1. [26] M. Ghadimi Atigh, M. Keller-Ressel, and P. Mettes (2021) Hyperbolic busemann learning with ideal prototypes. In NeurIPS, Cited by: §1. [27] C. Gulcehre, M. Denil, M. Malinowski, A. Razavi, R. Pascanu, K. M. Hermann, P. Battaglia, V. Bapst, D. Raposo, A. Santoro, et al. (2019) Hyperbolic attention networks. In ICLR, Cited by: §1, §1. [28] Y. Guo, X. Wang, Y. Chen, and S. X. Yu (2022) Clipped hyperbolic classifiers are super-hyperbolic classifiers. In CVPR, Cited by: §E.1.2, §5.1. [29] K. He, X. Zhang, S. Ren, and J. Sun (2016) Deep residual learning for image recognition. In CVPR, Cited by: §5.1. [30] N. He, R. Anand, H. Madhu, A. Maatouk, S. Krishnaswamy, L. Tassiulas, M. Yang, and R. Ying (2025) HELM: hyperbolic large language models via mixture-of-curvature experts. In NeurIPS, Cited by: §1. [31] N. He, M. Yang, and R. Ying (2025) Lorentzian residual neural networks. In KDD, Cited by: §1, §1. [32] I. Katsman, E. Chen, S. Holalkere, A. Asch, A. Lou, S. N. Lim, and C. M. De Sa (2024) Riemannian residual neural networks. In NeurIPS, Cited by: §1. [33] R. R. Khan, P. Chlenski, and I. Pe’er (2025) Hyperbolic genome embeddings. In ICLR, Cited by: §E.2.1, §E.2.2, §1, §5.2. [34] V. Khrulkov, L. Mirvakhabova, E. Ustinova, I. Oseledets, and V. Lempitsky (2020) Hyperbolic image embeddings. In CVPR, Cited by: §1. [35] D. P. Kingma (2015) Adam: a method for stochastic optimization. In ICLR, Cited by: §E.3.2, §E.4.1. [36] A. Krizhevsky, G. Hinton, et al. (2009) Learning multiple layers of features from tiny images. Toronto, ON, Canada. Cited by: §E.1.1, §5.1. [37] Y. Le and X. Yang (2015) Tiny ImageNet visual recognition challenge. Cited by: §E.1.1, §5.1. [38] G. Lebanon and J. Lafferty (2004) Hyperplane margin classifiers on the multinomial manifold. In ICML, Cited by: §1, §3.2. [39] J. M. Lee (2018) Introduction to riemannian manifolds. Vol. 2, Springer. Cited by: §2. [40] S. Li, C. Shiwen, O. Koç, Y. Ding, Q. Zhao, M. Kawanabe, and Z. Chen (2026) HEEGNet: hyperbolic embeddings for EEG. In ICLR, Cited by: §1. [41] Q. Liu, M. Nickel, and D. Kiela (2019) Hyperbolic graph neural networks. In NeurIPS, Cited by: §1. [42] Y. Liu, Z. He, and K. Han (2025) Hyperbolic category discovery. In CVPR, Cited by: §1. [43] A. Lou, I. Katsman, Q. Jiang, S. Belongie, S. Lim, and C. De Sa (2020) Differentiating through the Fréchet mean. In ICML, Cited by: §1. [44] G. Namata, B. London, L. Getoor, B. Huang, and U. Edu (2012) Query-driven active surveying for collective classification. In 10th International Workshop on Mining and Learning with Graphs, Vol. 8, p. 1. Cited by: §E.3.1, §5.3, §5.4. [45] X. S. Nguyen, S. Yang, and A. Histace (2025) Neural networks on symmetric spaces of noncompact type. In ICLR, Cited by: Table 11, Table 11, Table 12, Table 12, §E.1.2, §E.3.2, §E.3.2, §1, §1, §1, §3.3, Table 2, §5.1, §5.3, footnote 1. [46] M. Nickel and D. Kiela (2017) Poincaré embeddings for learning hierarchical representations. In NeurIPS, Cited by: §1. [47] M. Nickel and D. Kiela (2018) Learning continuous hierarchies in the Lorentz model of hyperbolic geometry. In ICML, Cited by: §1. [48] A. Pal, M. van Spengler, G. M. D. di Melendugno, A. Flaborea, F. Galasso, and P. Mettes (2025) Compositional entailment learning for hyperbolic vision-language models. In ICLR, Cited by: §1. [49] J. G. Ratcliffe (2006) Foundations of hyperbolic manifolds. Springer. Cited by: §2. [50] P. Sen, G. Namata, M. Bilgic, L. Getoor, B. Galligher, and T. Eliassi-Rad (2008) Collective classification in network data. AI magazine 29 (3), p. 93–93. Cited by: §E.3.1, §5.3, §5.4. [51] R. Shimizu, Y. Mukuta, and T. Harada (2021) Hyperbolic neural networks++. In ICLR, Cited by: §E.4.1, §F.3, §F.3, §1, §1, §1, §3.3, Table 2, §4.1, §4.1, Table 3, §5.1, §5.4, footnote 2. [52] O. Skopek, O. Ganea, and G. Bécigneul (2020) Mixed-curvature variational autoencoders. In ICLR, Cited by: Remark 3.2. [53] Y. Sun, X. Cao, Y. Wang, W. Ye, J. Guo, and Q. Guo (2024) Geometry awakening: cross-geometry learning exhibits superiority over individual structures. In NeurIPS, Cited by: §1. [54] T. Sur, S. Mukherjee, K. Rahaman, S. Chaudhuri, M. H. Khan, and B. Banerjee (2025) Hyperbolic uncertainty-aware few-shot incremental point cloud segmentation. In CVPR, Cited by: §1. [55] A. A. Ungar (2022) Analytic hyperbolic geometry and albert einstein’s special theory of relativity (second edition). World Scientific. Cited by: Definition B.13, Definition B.14, §B.3.1, §B.3.1, §B.4, §2. [56] M. Van Spengler, E. Berkhout, and P. Mettes (2023) Poincaré resnet. In ICCV, Cited by: §1, §1. [57] R. Wang, Y. Jiang, X. Luo, X. Wu, N. Sebe, and Z. Chen (2026) Wasserstein-aligned hyperbolic multi-view clustering. In AAAI, Cited by: §1. [58] M. Yang, R. S. B. B, A. Feng, B. Xiong, J. Liu, I. King, and R. Ying (2025) Hyperbolic fine-tuning for large language models. In NeurIPS, Cited by: §1. [59] M. Yang, H. Verma, D. C. Zhang, J. Liu, I. King, and R. Ying (2024) Hypformer: exploring efficient transformer fully in hyperbolic space. In KDD, Cited by: §1. [60] X. Yang, X. Li, H. Chang, X. Yang, S. Tao, M. Shigeno, N. Chang, J. Wang, D. Yin, E. Min, et al. (2025) Hgformer: hyperbolic graph transformer for collaborative filtering. In ICML, Cited by: §1. [61] M. Zhang and Y. Chen (2018) Link prediction based on graph neural networks. In NeurIPS, Cited by: §E.3.1, §5.3, §5.4. [62] Z. Zhou, Y. Ji, W. Li, P. Dutta, R. Davuluri, and H. Liu (2024) DNABERT-2: efficient foundation model and benchmark for multi-species genome. In ICLR, Cited by: §E.2.1, §5.2. Supplementary Material Appendix Contents Appendix A Notation Tab.˜9 summarizes the notation used throughout the paper. Table 9: Summary of notation. Notation Description ℳM Riemannian manifold TxℳT_xM Tangent space at x Expx(v)Exp_x(v) Exponential map at x Logx(y)Log_x(y) Logarithmic map at x PTx→y(v)PT_x→ y(v) Parallel transport of v from x to y gx(u,v)g_x(u,v) Riemannian metric at x d(x,y)d(x,y) Geodesic distance γ(t)γ(t) Unit speed geodesic ray Bγ(x)B^γ(x) Busemann function associated with the geodesic ray γ (,d)(X,d) Metric space and its distance function ∂ Boundary at infinity of X [x,y][x,y] Geodesic segment joining x and y Δ(x,y,z) (x,y,z) Geodesic triangle in X (MKn,dK)(M^n_K,d_K) Model space of constant curvature K with distance dKd_K CAT(K)CAT(K) CAT(K)CAT(K) space HBτγ,HτγHB^γ_τ,\ H^γ_τ Horoball and horosphere of γ at level τ ℝnR^n Euclidean space of dimension n ⟨⋅,⋅⟩ ·,· , ∥⋅∥ \|· \| Euclidean inner product and norm 0 Zero vector in ℝnR^n ℋKnH^n_K Hyperbolic space, either ℙKnP^n_K or KnL^n_K K<0K<0 Constant sectional curvature ⊕ℋ _H and ⊙ℋ _H Gyroaddition and scalar gyromultiplication on ℋKnH^n_K e Origin in ℋKnH^n_K ℙKnP^n_K Poincaré ball model in ℝnR^n with curvature K λxK _x^K Conformal factor 21+K‖x‖2 21+K \|x \|^2 ⊕M,⊙M _M,\ _M Möbius gyroaddition and scalar gyromultiplication on ℙKnP^n_K KnL^n_K Lorentz model in ℝn+1R^n+1 with curvature K ⟨x,y⟩ℒ x,y _L, ‖x‖ℒ \|x \|_L Lorentzian inner product and norm x=[xt,xs⊤]⊤x= [x_t,x_s ] Temporal and spatial decomposition in the Lorentz ambient space ¯ 0 Origin of KnL^n_K ⊕,⊙ _L,\ _L Gyroaddition and scalar gyromultiplication on KnL^n_K n−1S^n-1 Unit sphere in ℝnR^n Bv(x)B^v(x) Busemann function associated with a unit direction v∈n−1v ^n-1 Hv,α,bH_v,α,b Horosphere x∈ℋKn∣−αBv(x)+b=0 \x ^n_K -α B^v(x)+b=0 \ d¯(y,H) d(y,H) Signed point-to-hyperplane distance softmaxsoftmax Softmax operator uk(x)u_k(x) Logit for class k ek∈ℝme_k ^m k-th standard basis vector ϕφ Activation function ℝ→ℝR Appendix B Preliminaries B.1 Riemannian geometry We briefly review Riemannian geometry. For in-depth discussions, please refer to [19]. Tangent space. Given a smooth manifold ℳM, the tangent space TxℳT_xM at x∈ℳx is a Euclidean space, consisting of velocities of smooth curves through x, namely v∈Txℳv∈ T_xM if there exists a smooth γ with γ(0)=xγ(0)=x and γ˙(0)=v γ(0)=v. Riemannian manifold. A Riemannian manifold is a smooth manifold ℳM endowed with a Riemannian metric g, that is, for each x∈ℳx an inner product gx(⋅,⋅)g_x(·,·) or ⟨⋅,⋅⟩x ·,· _x on the tangent space TxℳT_xM varying smoothly with x. The metric induces the length of a smooth curve γ:[0,1]→ℳγ:[0,1] as L(γ)=∫01gγ(t)(γ˙(t),γ˙(t))tL(γ)= _0^1 g_γ(t) ( γ(t), γ(t) )dt, and the geodesic distance d(x,y)d(x,y) is the infimum of lengths over all smooth curves joining x and y. Geodesic. Straight lines generalize to constant-speed curves that are locally length minimizing between points x,y∈ℳx,y , known as geodesics: γ∗=argminγL(γ)subject to γ(0)=x,γ(1)=y,‖γ˙(t)‖γ(t)=c>0.γ^*= _γL(γ) to γ(0)=x,γ(1)=y, \| γ(t) \|_γ(t)=c>0. (26) We focus on unit-speed geodesics, i.e., c=1c=1. In hyperbolic space, between any two points there exists a unique geodesic segment that is globally length minimizing. Exponential and Logarithmic Maps. For x∈ℳx and v∈Txℳv∈ T_xM, let γx,v _x,v denote the unique geodesic with γx,v(0)=x _x,v(0)=x and γ˙x,v(0)=v γ_x,v(0)=v. The exponential map Expx:Txℳ⊃→ℳExp_x:T_xM is defined by Expx(v)=γx,v(1)Exp_x(v)= _x,v(1), where V is an open neighborhood of the origin in TxℳT_xM. Its local inverse, defined for y in a neighborhood ⊂ℳU of x, is the logarithmic map Logx:→TxℳLog_x:U→ T_xM, satisfying Expx∘Logx=Exp_x _x= 1_U. In hyperbolic space, these maps are globally well-defined, that is, ExpxExp_x is defined on all of TxℳT_xM and Logx(y)Log_x(y) exists for every y∈ℳy . Parallel transport. Parallel transport moves tangent vectors along a curve while preserving the norm. Given a geodesic γ from x to y, the parallel transport of a tangent vector v∈Txℳv∈ T_xM is the unique vector PTx→y(v)∈TyℳPT_x→ y(v)∈ T_yM obtained by transporting v along γ so that its covariant derivative along γ vanishes. Parallel transport defines a linear isometry between TxℳT_xM and TyℳT_yM. B.2 Metric geometry We present a concise overview of metric geometry, which generalizes Riemannian concepts to metric spaces. The theory develops geodesics and curvature without smooth structure, providing the tools used in our analysis of point-to-horosphere distances. We follow [6, Ch. I.1, I.2, I.1, I.2 and I.8] for definitions and results. B.2.1 Geodesic metric spaces We begin by recalling some basic notions in metric spaces. Definition B.1 (Metric space). A metric space is a pair (,d)(X,d) where X is a set and the distance function d:×→ℝd:X×X satisfies, for all x,y,z∈x,y,z , Positivity: d(x,y)≥0,d(x,y)=0⇔x=y, (x,y)≥ 0, (x,y)=0 x=y, (27) Symmetry: d(x,y)=d(y,x), (x,y)=d(y,x), (28) Triangle inequality: d(x,z)≤d(x,y)+d(y,z) (x,z) (x,y)+d(y,z) (29) Geodesics, rays, and lines generalize unit-speed minimizing geodesics to metric spaces. Definition B.2 (Geodesic, geodesic ray and line). Let (,d)(X,d) be a metric space and let I=[0,l]⊆ℝI=[0,l] be a closed interval. A geodesic joining x to y is a map γ:I→γ:I with γ(0)=xγ(0)=x, γ(l)=yγ(l)=y such that d(γ(t),γ(t′))=|t−t′|for all t,t′∈I∩[0,l].d (γ(t),γ(t ) )=|t-t | all t,t ∈ I∩[0,l]. (30) A geodesic ray is a map γ:[0,∞)→γ:[0,∞) such that d(γ(t),γ(t′))=|t−t′|d (γ(t),γ(t ) )=|t-t | for all t,t′≥0t,t ≥ 0. A geodesic line is a map γ:ℝ→γ:R such that d(γ(t),γ(t′))=|t−t′|d (γ(t),γ(t ) )=|t-t | for all t,t′∈ℝt,t . Geodesic metric spaces extend the Riemannian premise that any two points can be joined by a geodesic to metric spaces. Definition B.3 (Geodesic metric space). The metric space (,d)(X,d) is a geodesic metric space (or, more briefly, a geodesic space) if every two points are joined by a geodesic. We say that (,d)(X,d) is uniquely geodesic if there is exactly one geodesic joining x to y for all x,y∈x,y . Convexity is defined through geodesics, generalizing linear convexity in Euclidean space and geodesic convexity on manifolds. Definition B.4 (Convex subset). Let (,d)(X,d) be a metric space. A subset C⊆C is convex if every pair x,y∈Cx,y∈ C can be joined by a geodesic in X and the image of every such geodesic is contained in C. B.2.2 CAT(0) spaces We next review concepts that extend nonpositive curvature from manifolds to metric spaces. As a starting point, we recall the model spaces, that is, manifolds of constant curvature, including hyperbolic, Euclidean, and spherical geometries. These serve as reference spaces for metric spaces. Definition B.5 (Model space MKnM^n_K). For K∈ℝK , the model space (MKn,dK)(M^n_K,d_K) is given by (MKn,dK)=(n, 1/−Kd),K>0,(ℝn,d),K=0,(n,1/Kd),K<0,(M^n_K,d_K)= cases(L^n,\ 1/ -Kd),&K>0,\\ (R^n,\ d),&K=0,\\ (S^n,1/ Kd),&K<0, cases (31) where nL^n and nS^n are the n-dimensional unit Lorentz and sphere manifolds, respectively, and dd is the geodesic distance in the corresponding manifold. The diameter of Mk2M_k^2 is denoted DkD_k, which is equal to π/Kπ/ K if K>0K>0, and ∞ otherwise. Definition B.6 (Comparison triangle). Let △(x,y,z) (x,y,z) be a geodesic triangle in X with side lengths a=d(y,z)a=d(y,z), b=d(x,z)b=d(x,z), and c=d(x,y)c=d(x,y). A comparison triangle for △(x,y,z) (x,y,z) is a triangle △(x¯,y¯,z¯) ( x, y, z) in MK2M^2_K with dK(y¯,z¯)=ad_K( y, z)=a, dK(x¯,z¯)=bd_K( x, z)=b, and dK(x¯,y¯)=cd_K( x, y)=c. When a+b+c<2DKa+b+c<2D_K, the comparison triangle exists. CAT(K) spaces encode curvature through triangle comparison with the model plane MK2M^2_K. Intuitively, a CAT(K)CAT(K) space is a metric space where triangles are "thinner" than the corresponding comparison triangles in the model space MK2M_K^2. Definition B.7 (CAT(K)CAT(K) space). Let X be a metric space and let K be a real number. Let Δ be a geodesic triangle in X with perimeter less than 2DK2D_K. Let Δ¯⊂MK2 ⊂ M_K^2 be a comparison triangle for Δ . Then, Δ is said to satisfy the CAT(K)CAT(K) inequality if for all x,y∈Δx,y∈ and all comparison points x¯,y¯∈Δ¯ x, y∈ , d(x,y)≤dK(x¯,y¯).d(x,y)≤ d_K( x, y). (32) Then, the CAT(K)CAT(K) space is defined as follows. • If K≤0K≤ 0, then X is called a CAT(K)CAT(K) space (more briefly, "X is CAT(K)CAT(K)") if X is a geodesic space all of whose geodesic triangles satisfy the CAT(K)CAT(K) inequality. • If K>0K>0, then X is called a CAT(K)CAT(K) space if X is DKD_K-geodesic and all geodesic triangles in X of perimeter less than 2DK2D_K satisfy the CAT(K)CAT(K) inequality. Here, DKD_K-geodesic means that for every pair of points x,y∈x,y with d(x,y)<DKd(x,y)<D_K there is a geodesic joining x to y. Definition B.8 (Hadamard space). A Hadamard space is a complete CAT(0)CAT(0) space. Proposition B.9 (Orthogonal projection). Let (,d)(X,d) be a CAT(0)CAT(0) space and let C⊆C be a convex subset that is complete in the induced metric. For every x∈x , there exists a unique point πC(x)∈C _C(x)∈ C such that d(x,πC(x))=infy∈Cd(x,y)=d(x,C).d (x, _C(x) )= _y∈ Cd (x,y )=d (x,C ). (33) If x′x belongs to the geodesic segment [x,πC(x)] [x, _C(x) ], then πC(x′)=πC(x). _C(x )= _C(x). (34) The map πC:→C _C:X→ C is a called an orthogonal projection, or simply a projection. With geodesics, we can extend the asymptote, Busemann function and horosphere in Sec.˜2 to metric spaces. Definition B.10 (Asymptote). Let (,d)(X,d) be a metric space. Two geodesic rays γ,η:[0,∞)→γ,η:[0,∞) are asymptotic if supt≥0d(γ(t),η(t))<∞. _t≥ 0d (γ(t),η(t) )<∞. (35) The set ∂ of boundary points of X, which we shall also call the points at infinity or ideal points, is the set of equivalence classes of geodesic rays: two geodesic rays being equivalent if and only if they are asymptotic. Definition B.11 (Busemann function, horoball, horosphere). Let (,d)(X,d) be a metric space and let γ:[0,∞)→γ:[0,∞) be a geodesic ray. The Busemann function associated with γ is Bγ(x)=limt→∞(d(x,γ(t))−t),x∈.B^γ(x)= _t→∞ (d (x,γ(t) )-t ), x . (36) Its sublevel sets HBτγ=x∈:Bγ(x)≤τHB^γ_τ= \x :B^γ(x)≤τ \ are horoballs and the level sets Hτγ=x∈:Bγ(x)=τH^γ_τ= \x :B^γ(x)=τ \ are horospheres. As shown by Bridson and Haefliger [6, Lem. I. 8.18], the limits in the Busemann function exist. Besides, Busemann functions in Hadamard spaces are invariant to the choice of asymptotic geodesic ray Bridson and Haefliger [6, Cor. I. 8.20]. Corollary B.12. If X is a Hadamard space, then the Busemann functions associated to asymptotic rays in X are equal up to addition of a constant. B.3 Gyrovector space In this subsection we first present the general definitions of gyrogroup and gyrovector space. We then instantiate these notions on concrete manifolds. B.3.1 General definition Classical vector spaces can be characterized as a commutative group together with a compatible scalar multiplication. By analogy, a gyrovector space is built from a gyrocommutative gyrogroup endowed with a compatible scalar gyromultiplication [55]. Definition B.13 (Gyrogroup [55]). Given a nonempty set G with a binary operation ⊕:G×G→G :G× G→ G, the pair (G,⊕)(G, ) is a gyrogroup if, for all x,y,z∈Gx,y,z∈ G, the following axioms hold: (G1) There is at least one element e∈Ge∈ G called a left identity (or neutral element) such that e⊕x=xe x=x. (G2) There is an element ⊖x∈G x∈ G called a left inverse of x such that ⊖x⊕x=e x x=e. (G3) There is an automorphism gyr[x,y]:G→Ggyr[x,y]:G→ G for each x,y∈Gx,y∈ G such that x⊕(y⊕z)=(x⊕y)⊕gyr[x,y]z(Left gyroassociative law).x (y z )= (x y ) [x,y]z (Left gyroassociative law). The map gyr[x,y]gyr[x,y] is the gyration of G generated by x and y. (G4) Left reduction law: gyr[x,y]=gyr[x⊕y,y]gyr[x,y]=gyr[x y,y]. Definition B.14 (Gyrocommutative gyrogroup [55]). A gyrogroup (G,⊕)(G, ) is gyrocommutative if x⊕y=gyr[x,y](y⊕x)(Gyrocommutative law).x y=gyr[x,y] (y x ) (Gyrocommutative law). Definition B.15 (Gyrovector space [15]). A gyrocommutative gyrogroup (G,⊕)(G, ) equipped with a scalar gyromultiplication ⊙:ℝ×G→G :R× G→ G is a gyrovector space if, for s,t∈ℝs,t and x,y,z∈Gx,y,z∈ G, the following axioms hold: (V1) Identity scalar multiplication: 1⊙x=x1 x=x. (V2) Scalar distributive law: (s+t)⊙x=s⊙x⊕t⊙x(s+t) x=s x t x. (V3) Scalar associative law: (st)⊙x=s⊙(t⊙x)(st) x=s (t x ). (V4) Gyroautomorphism homogeneity: gyr[x,y](t⊙z)=t⊙gyr[x,y]zgyr[x,y] (t z )=t [x,y]z. (V5) Identity gyroautomorphism: gyr[s⊙x,t⊙x]=gyr [s x,t x ]= 1, where 1 denotes the identity map. Intuition. Gyrogroups generalize groups: they are nonassociative, yet obey a controlled form of associativity governed by gyrations. Since gyrations in a group are the identity, every group is a gyrogroup. In the same spirit, gyrovector spaces extend vector spaces and provide an algebraic toolkit that has proved effective for modeling hyperbolic geometry [55]. B.4 Hyperbolic geometry Table 10: Riemannian operators on Poincaré ball and Lorentz (K<0K<0). Operator Poincaré ball ℙKnP^n_K Lorentz KnL^n_K Definition ℙKn=x∈ℝn∣‖x‖2<−1/K ^n_K=\x ^n \|x \|^2<-1/K\ Kn=x∈ℝn+1∣⟨x,x⟩ℒ=1/K,xt>0 ^n_K=\x ^n+1 x,x _L=1/K,x_t>0\ gx(w,v)g_x(w,v) (λxK)2⟨w,v⟩,λxK=21+K‖x‖2 ( _x^K )^2 w,v , _x^K= 21+K \|x \|^2 ⟨w,v⟩ℒ=⟨vs,ws⟩−vtwt w,v _L= v_s,w_s -v_tw_t d(x,y)d(x,y) 2|K|tanh−1(|K|‖−x⊕My‖) 2 |K| ^-1 ( |K| \|-x _My \| ) 1|K|cosh−1(K⟨x,y⟩ℒ) 1 |K| ^-1 (K x,y _L ) LogxyLog_xy 2|K|λxKtanh−1(|K|‖−x⊕My‖)−x⊕My‖−x⊕My‖ 2 |K| _x^K ^-1 ( |K| \|-x _My \| ) -x _My \|-x _My \| cosh−1(β)β2−1(y−βx),β=K⟨x,y⟩ℒ ^-1(β) β^2-1(y-β x), β=K x,y _L ExpxvExp_xv x⊕M(tanh(|K|λxK‖v‖2)v|K|‖v‖) x _M ( ( |K| _x^K \|v \|2 ) v |K| \|v \| ) cosh(α)x+sinh(α)αv,α=|K|‖v‖ℒ (α)x+ (α)αv, α= |K| \|v \|_L PTx→y(v)PT_x→ y(v) λxKλyKgyr[y,−x]v _x^K _y^Kgyr[y,-x]v v−K⟨y,v⟩ℒ1+K⟨x,y⟩ℒ(x+y) v- K y,v _L1+K x,y _L(x+y) Given x,y∈ℋKnx,y ^n_K and tangent vectors v,w∈TxℋKnv,w∈ T_xH^n_K, Tab.˜10 summarizes the Riemannian operators. Gyrovector operators. The gyro-structure over the hyperbolic space can be defined by its Riemannian operators [23, 15]. Given x,y,z∈ℋKnx,y,z ^n_K and t∈ℝt , the gyroaddition and gyromultiplication are defined as x⊕ℋy x _Hy =Expx(PTe→x(Logey)), =Exp_x (PT_e→ x (Log_ey ) ), (37) t⊙ℋx t _Hx =Expe(tLogex), =Exp_e (tLog_ex ), (38) gyr[x,y]z [x,y]z =⊖ℋ(x⊕ℋy)⊕ℋ(x⊕ℋ(y⊕ℋz)), = _H (x _Hy ) _H (x _H (y _Hz ) ), (39) where e denotes the origin in ℋKnH^n_K. On the Poincaré ball ℙKnP^n_K, such gyro-structure is known as the Möbius gyrovector space [55, Ch. 6.14]: x⊕My x _My =(1−2K⟨x,y⟩−K‖y‖2)x+(1+K‖x‖2)y1−2K⟨x,y⟩+K2‖x‖2‖y‖2, = (1-2K x,y -K\|y\|^2 )x+ (1+K\|x\|^2 )y1-2K x,y +K^2\|x\|^2\|y\|^2, t⊙Mx t _Mx =tanh(ttanh−1(|K|‖x‖))|K|x‖x‖ = (t ^-1( |K|\|x\|) ) |K| x\|x\| where ⊖Mx=−1⊙Mx=−x _Mx=-1 _Mx=-x is the gyroinverse and 0 is the gyro identity: ⊕Mx=x,∀x∈ℙKn0 _Mx=x,∀ x ^n_K. As shown by Chen et al. [15, Props. 24-25], the Lorentz gyroaddition and gyromultiplication also admit closed-form expressions: x⊕y=x,y=¯,y,x=¯,[1|K|D−KND+KN2(Asxs+Ayys)D+KN],Otherwise. x _Ly= casesx,&y= 0,\\ y,&x= 0,\\ bmatrix 1 |K| D-KND+KN\\ 2 (A_sx_s+A_yy_s )D+KN bmatrix,&Otherwise. cases (40) t⊙x=¯,t=0∨x=¯1|K|[cosh(tθ)sinh(tθ)‖xs‖xs],Otherwise, t _Lx= cases 0,&t=0 x= 0\\ 1 |K| bmatrix (tθ )\\ (tθ ) \|x_s \|x_s bmatrix,&Otherwise, cases (41) Here, θ=cosh−1(|K|xt)θ= ^-1( |K|x_t), As=ab2−2Kbsxy−KanyA_s=ab^2-2Kbs_xy-Kan_y and Ay=b(a2+Knx)A_y=b(a^2+Kn_x) with the following notation: a=1+|K|xt,b=1+|K|yt, a=1+ |K|x_t,b=1+ |K|y_t, (42) nx=‖xs‖2,ny=‖ys‖2,sxy=⟨xs,ys⟩, n_x= \|x_s \|^2,n_y= \|y_s \|^2,s_xy= x_s,y_s , D=a2b2−2Kabsxy+K2nxny, D=a^2b^2-2Kabs_xy+K^2n_xn_y, N=a2ny+2absxy+b2nx. N=a^2n_y+2abs_xy+b^2n_x. In particular, the gyro identity is ¯ 0 and the gyroinverse is ⊖x=−1⊙x=[xt,−xs⊤]⊤ _Lx=-1 _Lx=[x_t,-x_s ] . Appendix C Comparison with existing hyperbolic MLR Table 11: Comparison of hyperplanes. Compact params indicates whether the parameterization requires an additional manifold-valued point. Method Hyperplane Formulation Applied manifolds Compact params Euclidean MLR Euclidean x∈ℝn∣⟨a,x⟩+b=0 \x ^n a,x +b=0 \ a∈ℝn,b∈ℝa ^n,\ b ℝnR^n ✓ Poincaré MLR [23] Geodesic [23, Def. 3.1] x∈ℙKn∣⟨Logp(x),a⟩p=0 \x ^n_K _p (x ),a _p=0 \ p∈ℙKn,a∈TpℙKnp ^n_K,\ a∈ T_pP^n_K ℙKnP^n_K ✗ Pseudo-Busemann MLR [45] Busemann & gyro [45, Def. 4.1] x∈ℙKn∣Bv(−p⊕Mx)=0 \x ^n_K B^v (-p _Mx )=0 \ v∈n−1,p∈ℙKnv ^n-1,\ p ^n_K ℙKnP^n_K ✗ Lorentz MLR [4] Ambient Minkowski [4, Eq. (7)] x∈Kn∣⟨w,x⟩ℒ=0 \x ^n_K w,x _L=0 \ p∈Kn,w∈TpKnp ^n_K,\ w∈ T_pL^n_K KnL^n_K ✗ BMLR Horosphere x∈ℋKn∣−αBv(x)+b=0 \x ^n_K -α B^v (x )+b=0 \ α>0,v∈n−1,b∈ℝα>0,\ v ^n-1,\ b ℙKnP^n_K, KnL^n_K ✓ Table 12: Comparison of point-to-hyperplane distances. Real means the point-to-hyperplane distance is the real distance, obtained by infy∈Hd(x,y) _y∈ Hd(x,y) with H as a hyperplane and dd 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. Method Point-to-hyperplane distance Applied manifolds Dist Euclidean MLR |⟨a,x⟩+b|‖a‖ | a,x +b| \|a \| ℝnR^n Real Poincaré MLR [23] 1−Ksinh−1(2−K|⟨−p⊕Mx,a⟩|(1+K‖−p⊕Mx‖2)‖a‖) 1 -K ^-1 ( 2 -K | -p _Mx,a | (1+K \|-p _Mx \|^2 ) \|a \| ) [23, Thm. 5] ℙKnP^n_K Real Pseudo-Busemann MLR [45] d(x,p)Bv(−p⊕Mx)‖−p⊕Mx‖ (x,p) B^v (-p _Mx ) \|-p _Mx \| [45, Cor. 4.3] ℙKnP^n_K Pseudo Lorentz MLR [4] 1−K|sinh−1(−K⟨v,x⟩ℒ‖v‖ℒ)| 1 -K | ^-1 ( -K v,x _L \|v \|_L ) | [4, Eq. (44)] KnL^n_K Real BMLR |−αBv(x)+b|α |-α B^v(x)+b |α ℙKnP^n_K, KnL^n_K Real Hyperbolic MLRs follow a point-to-hyperplane formulation, of which Chen et al. [12, Eqs. 4-6] provides the Riemannian prototype. Therefore, the key difference lies in hyperplanes and point-to-hyperplane distances across methods. In addition to Tab.˜2, Tabs.˜11 and 12 further make this comparison. We draw the following three conclusions. 1. Hyperplanes. Our BMLR uses Busemann-based horospheres that simultaneously satisfy three desiderata: (i) compact parameterization without a per-class manifold-valued point, whereas other hyperbolic ones222We note that Shimizu et al. [51], Bdeir et al. [4] mitigate this issue through re-parameterization: a=PTe→p(z)a=PT_e→ p(z) and p=expe(bz‖z‖)p= _e (b z \|z \| ), where e∈ℋKne ^n_K denotes the origin, z∈ℝnz ^n, and b∈ℝb . Nevertheless, the underlying definitions are over-parameterized. are over-parameterized; (i) a natural generalization of Euclidean hyperplanes via horospheres, while the Lorentz MLR relies on the ambient Minkowski space, failing to fully respect the intrinsic geometry; and (i) applicability across hyperbolic models, whereas the Lorentz MLR is tailored to the Lorentz model. 2. Point-to-hyperplane distances. Although pseudo-Busemann MLR also exploits Busemann functions, it relies on a pseudo point-to-hyperplane distance that only coincides with the real distance in Euclidean geometry. In contrast, our BMLR calculates the real point-to-horosphere distance, ensuring geometric fidelity across hyperbolic models. 3. Batch efficiency. Recalling Tab.˜2, the Poincaré MLR [23] computes logits using ⟨−pk⊕Mx,ak⟩ -p_k _Mx,a_k and ‖−pk⊕Mx‖2 \|-p_k _Mx \|^2. For a batch X∈ℝbs×nX ^bs× n and C classes, evaluating −pk⊕MX-p_k _MX for every k yields an intermediate tensor of shape [bs,C,n][bs,C,n], and materializing this tensor can cause GPU out of memory (OOM) when n or C is large. The same limitation holds for the pseudo-Busemann MLR. Consequently, their official implementations compute per class in a for-loop, which is batch inefficient. By contrast, BMLR uses logits −αkBvk(x)+bk- _kB^v_k(x)+b_k, whose Busemann term reduces to class-wise inner products ⟨vk,x⟩ v_k,x (or ⟨vk,xs⟩ v_k,x_s ). With X∈ℝbs×nX ^bs× n and V=[v1,…,vC]∈ℝn×CV=[v_1,…,v_C] ^n× C, such inner products can be efficiently implemented as a single matrix multiplication XVXV without any [bs,C,n][bs,C,n] intermediate, yielding high throughput and low memory usage. Appendix D Busemann fully connected layers and point-to-horosphere distances An apparently natural attempt to define a hyperbolic FC layer is to replace the LHS of Eq.˜22 by the signed point-to-horosphere distance. The Euclidean hyperplane passing through the origin and orthogonal to ek∈ℝme_k ^m is Hek,0=y∈ℝm∣⟨ek,y⟩=0H_e_k,0= \y ^m e_k,y =0 \. The corresponding hyperbolic horosphere, following Eq.˜18, is Hek,1,0=y∈ℋKn∣−Bek(y)=0H_e_k,1,0=\y ^n_K -B^e_k(y)=0\. By Eq.˜20, the signed point-to-horosphere distance to Hek,1,0H_e_k,1,0 equals −Bek(y)-B^e_k(y). Accordingly, we can define an alternative FC mapping ℱ:ℋKn∋x↦y∈ℋKmF:H^n_K x y ^m_K via Bek(y)=αkBvk(x)−bk,k=1,…,m, llasection to20.74pt eq:busemann-fc-p2horosphere eq:busemann-fc-p2horosphereB^e_k(y)= _kB^v_k(x)-b_k, k=1,…,m, (43) where uk(x)=−αkBvk(x)+bku_k(x)=- _kB^v_k(x)+b_k, and αk>0 _k>0, vk∈n−1v_k ^n-1, bk∈ℝb_k are learnable. Although this only differs from Eq.˜22 on the LHS, the following discussion shows that such a definition is infeasible in general and fails to deliver a valid hyperbolic FC layer. D.1 Poincaré model Using Eq.˜3 with v=ekv=e_k, Eq.˜43 becomes 1−Klog(‖ek−Ky‖21+K‖y‖2)=−uk(x). llasection to20.74pt eq:pfc-busemann-eq eq:pfc-busemann-eq 1 -K ( \|e_k- -Ky \|^21+K \|y \|^2 )=-u_k(x). (44) Define tk=exp(−Kuk(x))>0t_k= (- -Ku_k(x) )>0. Exponentiating Eq.˜44 gives tk(1+K‖y‖2)=1−2−Kyk−K‖y‖2,k=1,…,m. llasection to20.74pt eq:pfc-comp-eq eq:pfc-comp-eqt_k (1+K \|y \|^2 )=1-2 -Ky_k-K \|y \|^2, k=1,…,m. (45) Writing R=‖y‖2R= \|y \|^2, Eq.˜45 yields an affine expression for each coordinate yk=ck+dkR,ck=1−tk2−K,dk=−K2(1+tk). llasection to20.74pt eq:pfc-affine eq:pfc-affiney_k=c_k+d_kR, c_k= 1-t_k2 -K, d_k= -K2 (1+t_k ). (46) Imposing R=∑k=1myk2=∑k=1m(ck+dkR)2R= _k=1^my_k^2= _k=1^m (c_k+d_kR )^2 gives a quadratic in R: A2R2+(A1−1)R+A0=0, llasection to20.74pt eq:pfc-R-quadratic eq:pfc-R-quadraticA_2R^2+ (A_1-1 )R+A_0=0, (47) where, denoting T=∑k=1mtkT= _k=1^mt_k and q=∑k=1mtk2q= _k=1^mt_k^2, A2=−K4(m+2T+q),A1=m−q2,A0=m−2T+q4(−K)≥0. llasection to20.74pt eq:pfc-coeffs eq:pfc-coeffsA_2= -K4 (m+2T+q ), A_1= m-q2, A_0= m-2T+q4(-K)≥ 0. (48) Its discriminant is ΔP _P =(A1−1)2−4A2A0 =(A_1-1)^2-4A_2A_0 (49) =(m−q2−1)2−4(−K4(m+2T+q))(m−2T+q4(−K)) = ( m-q2-1 )^2-4 ( -K4 (m+2T+q ) ) ( m-2T+q4(-K) ) =(m−2−q2)2−(m+2T+q)(m−2T+q)4 = ( m-2-q2 )^2- (m+2T+q ) (m-2T+q )4 =14[(m−2−q)2−((m+q)2−(2T)2)] = 14 [(m-2-q)^2- ((m+q)^2-(2T)^2 ) ] =14[((m−2−q)−(m+q))((m−2−q)+(m+q))+4T2] = 14 [ ((m-2-q)-(m+q) ) ((m-2-q)+(m+q) )+4T^2 ] =14[(−2−2q)(2m−2)+4T2] = 14 [(-2-2q)(2m-2)+4T^2 ] =T2−(m−1)(1+q). =T^2-(m-1) (1+q ). A real solution R exists only if ΔP≥0 _P≥ 0. In addition, feasibility requires 0≤R<−1/K0≤ R<-1/K so that y∈ℙKmy ^m_K. Such conditions can fail for generic uk(x)\u_k(x)\, in which case no y∈ℙKmy ^m_K satisfies Eq.˜43. D.2 Lorentz model Using Eq.˜4 with v=ekv=e_k, Eq.˜43 becomes 1−Klog(−K(yt−(ys)k))=−uk(x). llasection to20.74pt eq:lfc-busemann-eq eq:lfc-busemann-eq 1 -K ( -K (y_t-(y_s)_k ) )=-u_k(x). (50) Define tk=exp(−Kuk(x))>0t_k= (- -Ku_k(x) )>0 and, denoting T=∑k=1mtkT= _k=1^mt_k and q=∑k=1mtk2q= _k=1^mt_k^2, we obtain from Eq.˜50, (ys)k=yt−tk−K,k=1,…,m,⟹ys=yt−1−Kt. llasection to20.74pt eq:lfc-spatial eq:lfc-spatial(y_s)_k=y_t- t_k -K, k=1,…,m, y_s=y_t 1- 1 -Kt. (51) Enforcing the hyperboloid constraint ‖ys‖2−yt2=1/K \|y_s \|^2-y_t^2=1/K yields a quadratic in yty_t: (m−1)Kyt2+2−KTyt−(1+q)=0. llasection to20.74pt eq:lfc-yt-quadratic eq:lfc-yt-quadratic(m-1)Ky_t^2+2 -KTy_t- (1+q )=0. (52) The discriminant is ΔL _L =(2−KT)2−4(m−1)K(−(1+q)) = (2 -KT )^2-4(m-1)K (- (1+q ) ) (53) =4(−K)T2+4(m−1)K(1+q) =4(-K)T^2+4(m-1)K (1+q ) =4(−K)[T2−(m−1)(1+q)]. =4(-K) [T^2-(m-1) (1+q ) ]. Hence, a real yty_t exists only if T2−(m−1)(1+q)≥0T^2-(m-1) (1+q )≥ 0. In addition, feasibility requires yt>0y_t>0. Such conditions can fail for generic uk(x)\u_k(x)\, where no y∈Kmy ^m_K satisfies Eq.˜43. D.3 Summary Equating Busemann coordinates as in Eq.˜43 requires nontrivial inequalities on the responses uk(x)\u_k(x)\. These constraints are not guaranteed during learning, so the system can become infeasible and the output y undefined. This motivates our choice in Eq.˜22 to use signed point-to-hyperplane distances on the LHS, which admit closed-form solutions that are feasible for all inputs and parameters in both the Poincaré and Lorentz models. Appendix E Experimental details and additional results E.1 Image classification E.1.1 Datasets The CIFAR-10 [36] and CIFAR-100 [36] datasets each contain 60,000 32×3232× 32 color images from 10 and 100 classes, respectively. We use the standard PyTorch splits: 50,000 training images and 10,000 test images. Tiny-ImageNet [37] is a subset of ImageNet with 100,000 images from 200 classes, resized to 64×6464× 64. We use the official validation split for evaluation. The large-scale ImageNet-1k [17] dataset contains 1.28M training images, 50K validation images, and 100K test images distributed across 1k classes. Table 13: Summary of hyperparameters used in the image classification task. Hyperparameter CIFAR-10/100 Tiny-ImageNet ImageNet-1k Epochs 200 100 Batch size 128 256 Initial learning rate 0.10.1 0.10.1 LR schedule 60,120,160;γ=0.260,120,160;γ=0.2 30,60,90;γ=0.130,60,90;γ=0.1 Weight decay 5e−45e^-4 1e−41e^-4 Optimizer SGD SGD Precision 32-bit 32-bit #GPUs 1 (RTX A6000) 2 (RTX A100) Curvature K −1-1 −1-1 E.1.2 Implementation details For CIFAR-10/100 and Tiny-ImageNet, we follow Bdeir et al. [4, App. C.1]. For ImageNet-1k, we follow Guo et al. [28, Sec. 4]. Tab.˜13 summarizes the dataset-specific hyperparameters. For the hyperbolic MLR, before mapping into the hyperbolic space, we clip the feature vector by CLIP(x;r)=min1,r‖x‖x llasection to20.74pt app:eq:clip-hyper app:eq:clip-hyperCLIP (x;r )= \1, r \|x \| \x (54) where r>0r>0 is a hyperparameter. The clipped Euclidean embedding is projected via the exponential map to the target hyperbolic space: Expe(CLIP(x;r))Exp_e (CLIP(x;r) ). For the Lorentz model, the clipping parameter is r=1r=1 on CIFAR-10/100 and r=4r=4 on Tiny-ImageNet and ImageNet-1k. On the Poincaré ball, r=1r=1 on all four datasets. All methods are implemented in PyTorch and trained with cross-entropy loss. The results of MLR, PMLR, and LMLR on CIFAR-10/100 and Tiny-ImageNet are copied from Bdeir et al. [4, Tab. 1], while the ones of PBMLR-P on CIFAR-10/100 are copied from Nguyen et al. [45, Tab. 2]. The remaining results are obtained by our careful implementation. E.2 Genome sequence learning Table 14: Summary statistics for TEB. 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 unprocessed 12938 / 766 / 884 Table 15: Summary statistics for the adopted GUE datasets. 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 Covid variant classification Virus Covid 9 1000 77669 / 7000 / 7000 Species classification Fungi Fungi 25 5000 8000 / 1000 / 1000 Virus Virus 20 10000 4000 / 500 / 500 E.2.1 Datasets TEB. The Transposable Elements Benchmark (TEB) [33] comprises seven binary classification datasets that investigate transposable elements (TEs) across plant and human genomes. The seven datasets are LTR Copia, LINEs, and SINEs for plant retrotransposons; CMC-EnSpm and hAT-Ac for plant DNA transposons; and processed and unprocessed pseudogenes for human pseudogenes. TEB targets a less explored area of genome organization in genomics deep learning and provides a novel resource for benchmarking models. For each dataset, positive examples are sequences spanning annotated elements of interest, and negatives are randomly sampled, non-overlapping genomic segments outside these regions. We adopt chromosome-level training, validation, and test splits, using chromosomes 8 and 9 for validation and test in plant genomes, and chromosomes 20 to 22 and 17 to 19 for validation and test in human genomes, respectively. Summary statistics are provided in Tab.˜14. GUE. The Genome Understanding Evaluation (GUE) benchmark [62] is a recently published tool that contains seven biologically significant genome analysis tasks that span 28 datasets. GUE prioritizes genomic datasets that are challenging enough to discern differences between models. The datasets contain sequences ranging from 70 to 1000 base pairs in length and originating from yeast, mouse, human, and virus genomes. In our experiments, we select Core promoter detection and Promoter detection, and the multi-class tasks Covid variant classification and species classification, totaling nine datasets from GUE. Summary statistics are provided in Tab.˜15. E.2.2 Implementation details We mainly follow the official implementations of Khan et al. [33] for data processing, model architecture, and training. We adopt a simple CNN with three convolutional blocks followed by dense, ReLU activated layers to extract features [33, Fig. 4]. Before classification, the features are clipped and mapped to the target hyperbolic space as in Eq.˜54, then passed to either a prior hyperbolic MLR or our BMLR head. The clipping factor defaults to r=1r=1, with the following exceptions on GUE: on the Lorentz model, r=2.0r=2.0 for Covid variant classification and r=5.0r=5.0 for species classification; on the Poincaré ball, r=2.0r=2.0 for species classification. Following Khan et al. [33], we treat the curvature K as a learnable parameter initialized as −1-1. The remaining hyperparameters are listed in Tab.˜16. All models share these hyperparameters, except LMLR on Covid variant classification, for which we set the weight decay to 1e−31e^-3 to ensure convergence. All methods are implemented in PyTorch and trained with cross-entropy loss. Results are obtained from our reimplementation. Table 16: Hyperparameters for genome sequence learning. Batch size 100 Epochs 100 Optimizer Adam β1 _1, β2 _2 0.9, 0.999 Initial learning rate 1e−41e^-4 LR schedule 60,8560,85; γ=0.1γ=0.1 Weight decay 0.10.1 #GPUs 1 (RTX A6000) Initial curvature K −1-1 E.3 Node classification E.3.1 Datasets Table 17: 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 Disease [1]. It represents a disease propagation tree, simulating the SIR disease transmission model, with each node representing either an infection or a non-infection state. Airport [61]. It is a transductive dataset where nodes represent airports and edges represent the airline routes as from OpenFlights.org. PubMed [44]. This is a 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. Cora [50]. It is a 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. Tab.˜17 summarizes the statistics of the datasets. E.3.2 Implementation details We follow the official implementations of HGCN [9] and PBMLR [45] and conduct experiments on the Poincaré ball and the Lorentz model, respectively. We adhere to their experimental settings. The only changes are weight decay and dropout. We train with cross-entropy loss and the Adam optimizer [35] for 5000 epochs with a learning rate of 1e−21e^-2, curvature as −1-1, embedding dimension 16, and three GCN layers. We tune weight decay and dropout and report the values in Tab.˜18. All methods are implemented in PyTorch and run on a single RTX A6000 GPU. The results of HGCN-PMLR and HGCN-PBMLR-P on the Poincaré ball are taken from Nguyen et al. [45, Tab. 11]. Results for the remaining baselines are obtained from our reimplementation following the original settings. Table 18: Hyperparameters for node classification on Disease, Airport, PubMed, and Cora. Space Weight decay Dropout ℙKnP^n_K 1e−4,1e−5,1e−3,1e−31e-4,1e-5,1e-3,1e-3 0.3,0,0,0.20.3,0,0,0.2 KnL^n_K 1e−4,5e−5,1e−3,1e−31e-4,5e-5,1e-3,1e-3 0,0,0,0.30,0,0,0.3 E.4 Link prediction The datasets are identical to those used for node classification. We provide implementation details and additional ablation studies below. E.4.1 Implementation details We follow the official implementations of HNN [23], HNN++ [51], and HyboNet [11], and adopt the experimental protocol of Chami et al. [9] for link prediction. The encoder consists of two fully connected layers: the first maps the input features to 16, and the second maps 16 to 16. Each FC layer is instantiated as either our BFC or an existing hyperbolic FC layer. After each FC, we apply the activation Expe(ReLU(Loge(x)))Exp_e (ReLU(Log_e(x)) ), where e denotes the model origin. Following Chami et al. [9], this activation is disabled on Cora. As in the Möbius layer, we apply a gyro bias after each FC, that is, x⊕ℋbx _Hb. We train with Adam [35] at a learning rate of 1e−21e^-2 and tune weight decay and FC dropout. For BFC, we set ϕ=tanhφ= on Airport and Cora, which yields better performance, while we use the identity map on the other two datasets. E.4.2 Ablations on training time and parameter count Table 19: Efficiency comparison: fit time (s/epoch) and parameter count. Slowest results and largest parameter counts are in red. Space Method Disease Airport PubMed Cora Fit Time #Params Fit Time #Params Fit Time #Params Fit Time #Params ℙKnP^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 KnL^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 BFC-L 0.0244 528 0.0713 544 0.1525 8352 0.0280 23280 Tab.˜19 summarizes fit time per epoch and parameter counts. Our BFC layers achieve training time and model size comparable to existing layers. In particular, LTFC is the slowest due to costly logarithmic and exponential maps, and LFC uses the largest number of parameters among Lorentz variants. E.4.3 Ablations on the activation in BFC layers Table 20: Ablation on activation ϕφ in BFC layers. Space ϕφ Airport Cora ℙKnP^n_K None 94.57 ± 0.29 87.55 ± 0.34 tanh 94.88 ± 0.39 91.94 ± 0.32 KnL^n_K None 94.91 ± 0.24 87.45 ± 1.39 tanh 95.37 ± 0.17 92.28 ± 0.12 We find that ϕ=tanhφ= in our BFC layers (see Sec.˜4.2) is beneficial on Airport and Cora. This is validated by Tab.˜20, which shows that setting ϕ=tanhφ= consistently improves AUC over the identity activation on the Poincaré and Lorentz models. Appendix F Proofs F.1 Proof of Thm.˜3.1 Proof. Denoting κ2=−Kκ^2=-K with κ>0κ>0, we rewrite the Busemann functions in Eqs.˜3 and 4 as (Poincaré) Bv(x)=1κlog(‖v−κx‖21−κ2‖x‖2), B^v(x)= 1κ ( \|v-κ x \|^21-κ^2 \|x \|^2 ), (55) (Lorentz) Bv(x)=1κlog(κxt−κ⟨xs,v⟩). B^v(x)= 1κ (κ x_t-κ x_s,v ). (56) For the Poincaré case, ‖v−κx‖21−κ2‖x‖2=1−2κ⟨v,x⟩+κ2‖x‖21−κ2‖x‖2=1+−2κ⟨v,x⟩+2κ2‖x‖21−κ2‖x‖2. \|v-κ x \|^21-κ^2 \|x \|^2= 1-2κ v,x +κ^2 \|x \|^21-κ^2 \|x \|^2=1+ -2κ v,x +2κ^2 \|x \|^21-κ^2 \|x \|^2. (57) Using log(1+u)=u+O(u2) (1+u )=u+O (u^2 ) as u→0u→ 0 and (1−κ2‖x‖2)−1=1+O(κ2) (1-κ^2 \|x \|^2 )^-1=1+O (κ^2 ), we obtain Bv(x) B^v(x) =1κ[(−2κ⟨v,x⟩+2κ2‖x‖2)(1+O(κ2))+O(κ2)] = 1κ [ (-2κ v,x +2κ^2 \|x \|^2 ) (1+O (κ^2 ) )+O (κ^2 ) ] (58) =−2⟨v,x⟩+2κ‖x‖2+O(κ), =-2 v,x +2κ \|x \|^2+O (κ ), =−2⟨v,x⟩+O(κ). =-2 v,x +O (κ ). Therefore, Bv(x)→K→0−2⟨v,x⟩B^v(x) K→ 0^--2 v,x . For the Lorentz case, the hyperboloid constraint −xt2+‖xs‖2=−κ−2-x_t^2+ \|x_s \|^2=-κ^-2 and xt>0x_t>0 yield κxt=1+κ2‖xs‖2=1+12κ2‖xs‖2+O(κ4).κ x_t= 1+κ^2 \|x_s \|^2=1+ 12κ^2 \|x_s \|^2+O (κ^4 ). (59) Set z=−κ⟨xs,v⟩+12κ2‖xs‖2+O(κ4)z=-κ x_s,v + 12κ^2 \|x_s \|^2+O (κ^4 ). Then log(κxt−κ⟨xs,v⟩)=log(1+z)=z+O(z2)=−κ⟨xs,v⟩+O(κ2), (κ x_t-κ x_s,v )= (1+z )=z+O (z^2 )=-κ x_s,v +O (κ^2 ), (60) which implies Bv(x)=−⟨xs,v⟩+O(κ)B^v(x)=- x_s,v +O (κ ) and therefore Bv(x)→K→0−⟨xs,v⟩B^v(x) K→ 0^-- x_s,v . Substituting the above two limits into uk(x)=−αkBvk(x)+bku_k(x)=- _kB^v_k(x)+b_k gives the stated Euclidean limits of the logits. ∎ F.2 Proof of Thm.˜3.3 We first review two useful characterizations of Busemann functions in Hadamard spaces. Definition F.1. Let ℬB be the set of functions h:→ℝh:X on (,d)(X,d) satisfying: (i) h is convex; (i) 1-Lipschitz: |h(x)−h(y)|≤d(x,y)|h(x)-h(y)| (x,y) for all x,y∈x,y ; (i) for any x0∈x_0 and r>0r>0, the function h attains its minimum on the sphere Sr(x0)S_r(x_0) at a unique point y with h(y)=h(x0)−rh(y)=h(x_0)-r. Proposition F.2. For a function h:→ℝh:X , the following conditions are equivalent: (1) h is a Busemann; (2) h∈ℬh ; (3) h is convex, and for every t∈ℝt , the set h−1(−∞,t]h^-1(-∞,t] is nonempty; moreover, for each x∈x , the curve cx:[0,∞)→c_x:[0,∞) defined by t↦πh−1(−∞,h(x)−t](x)t _h^-1(-∞,h(x)-t](x) is a geodesic ray. Now, we are ready to prove Thm.˜3.3. Proof. As τ2=τ1 _2= _1 is trivial, we only consider τ2≠τ1 _2≠ _1. We assume τ2>τ1 _2> _1, and discuss the other direction at last. Step 1: symmetry. By definition, d(Hτ1γ,Hτ2γ)=infx∈Hτ1γ,y∈Hτ2γd(x,y)=infy∈Hτ2γ,x∈Hτ1γd(y,x)=d(Hτ2γ,Hτ1γ).d (H^γ_ _1,H^γ_ _2 )= _x∈ H^γ_ _1,y∈ H^γ_ _2d(x,y)= _y∈ H^γ_ _2,x∈ H^γ_ _1d(y,x)=d (H^γ_ _2,H^γ_ _1 ). (61) Step 2: lower bound. For any x∈Hτ2γx∈ H^γ_ _2 and y∈Hτ1γy∈ H^γ_ _1, the 11-Lipschitz property of BγB^γ gives |Bγ(x)−Bγ(y)|≤d(x,y). |B^γ(x)-B^γ(y) | (x,y). (62) With Bγ(x)=τ2B^γ(x)= _2 and Bγ(y)=τ1B^γ(y)= _1 this yields τ2−τ1≤d(x,y)∀y∈Hτ1γ. llasection to20.74pt eq:lipschitz-lb eq:lipschitz-lb _2- _1 (x,y) ∀ y∈ H^γ_ _1. (63) Taking infimum in Eq.˜63 over y∈Hτ1γy∈ H^γ_ _1 gives τ2−τ1≤d(x,Hτ1γ). llasection to20.74pt eq:lower-bound eq:lower-bound _2- _1 (x,H^γ_ _1 ). (64) Step 3: upper bound. For any x∈Hτ2γx∈ H^γ_ _2, by property (3) in Sec.˜F.2, the projection map cx(t)=πBγ≤Bγ(x)−t(x)=πHBτ2−tγ(x),t∈[0,∞),c_x(t)= _\B^γ≤ B^γ(x)-t\(x)= _HB^γ_ _2-t(x), t∈[0,∞), (65) is a unit-speed geodesic ray: d(x,cx(t))=td(x,c_x(t))=t. Let t=τ2−τ1>0t= _2- _1>0 and z=cx(t)∈HBτ1γz=c_x(t)∈ HB^γ_ _1. If Bγ(z)<τ1B^γ(z)< _1, then z lies in the interior of the horoball HBτ1γHB^γ_ _1. We can move slightly from z toward x along the geodesic segment xz¯ xz to obtain a point zεz_ with d(x,zε)<d(x,z)d(x,z_ )<d(x,z), contradicting the minimality of the projection. Hence, the projected point z=cx(t)z=c_x(t) indeed lies on Hτ1γH^γ_ _1: Bγ(z)=τ1B^γ(z)= _1. Then, we have the following: d(x,Hτ1γ)≤d(x,HBτ1γ)=d(x,z)=d(x,cx(t))=t=τ2−τ1. llasection to20.74pt eq:upper-bound eq:upper-boundd(x,H^γ_ _1) (x,HB^γ_ _1)=d(x,z)=d(x,c_x(t))=t= _2- _1. (66) Step 4: sandwich closure. Combining Eqs.˜64 and 66 gives d(x,Hτ1γ)=τ2−τ1.d(x,H^γ_ _1)= _2- _1. (67) Combining Eq.˜66 and Eq.˜64, d(x,Hτ1γ)=τ2−τ1,for every x∈Hτ2γ.d (x,H^γ_ _1 )= _2- _1, every x∈ H^γ_ _2. (68) The right-hand side does not depend on x, hence d(Hτ2γ,Hτ1γ)=τ2−τ1.d (H^γ_ _2,H^γ_ _1 )= _2- _1. (69) Step 5: opposite direction. If τ1>τ2 _1> _2, we can swap the roles of τ1,τ2 _1, _2 above due to the symmetry of the distance between horospheres, which brings d(Hτ2γ,Hτ1γ)=|τ2−τ1|.d (H^γ_ _2,H^γ_ _1 )=| _2- _1|. (70) ∎ F.3 Proof of Thm.˜4.1 Proof. The hyperplane and point-to-hyperplane distance are presented in Tabs.˜11 and 12. The origin of the Poincaré ball model is e=∈ℙKme=0 ^m_K. The specific ones w.r.t. the origin [51, Def. 1 and Eq. (56)] are Hek, H_e_k,0 =y∈ℙKm∣⟨ek,y⟩=0, =\y ^m_K e_k,y =0\, (71) d¯(y,Hek,) d (y,H_e_k,0 ) =1−Ksinh−1(2−Kyk1+K‖y‖2). = 1 -K ^-1 ( 2 -Ky_k1+K \|y \|^2 ). (72) Equating d¯(y,Hek,) d (y,H_e_k,0 ) with uk(x)u_k(x) from Eq.˜22 gives sinh−1(2−Kyk1+K‖y‖2)=−Kuk(x),∀1≤k≤m. llasection to20.74pt eq:poincare-fc-prf-1 eq:poincare-fc-prf-1 ^-1 ( 2 -Ky_k1+K \|y \|^2 )= -Ku_k(x),\ ∀ 1≤ k≤ m. (73) Note that Eq.˜73 takes the same form as Shimizu et al. [51, Eq. (56)], except that their responses are different. The below proof are inspired by their derivation. Applying sinh(⋅) (·) on both sides of Eq.˜73 yields 2−Kyk1+K‖y‖2=sinh(−Kuk(x)). 2 -Ky_k1+K \|y \|^2= ( -Ku_k(x) ). (74) Define ωk:=1−Ksinh(−Kuk(x)) _k:= 1 -K ( -Ku_k(x) ) and ω=[ωk]k=1mω=[ _k]_k=1^m. Then, we have 2yk=(1+K‖y‖2)ωk,∀k,2y_k= (1+K \|y \|^2 ) _k, ∀ k, (75) which is equivalent to the vector identity 2y=(1+K‖y‖2)ω. llasection to20.74pt eq:poincare-fc-vector-eq eq:poincare-fc-vector-eq2y= (1+K \|y \|^2 )ω. (76) Hence, y is collinear with ω. Write y=λωy=λω with λ≥0λ≥ 0. Substituting into Eq.˜76 and taking norms gives a quadratic in λ: K‖ω‖2λ2−2λ+1=0.K \|ω \|^2λ^2-2λ+1=0. (77) Solving and selecting the branch that satisfies y→0y→ 0 as ω→0ω→ 0 yields λ=1−1−K‖ω‖2K‖ω‖2=11+1−K‖ω‖2.λ= 1- 1-K \|ω \|^2K \|ω \|^2= 11+ 1-K \|ω \|^2. (78) Therefore, y=ω1+1−K‖ω‖2,ωk=sinh(−Kuk(x))−K,y= ω1+ 1-K \|ω \|^2, _k= ( -Ku_k(x) ) -K, (79) which proves the claim. One can check that y∈ℙKmy ^m_K. ∎ F.4 Proof of Thm.˜4.2 Proof. Recalling Tab.˜11, a Lorentz hyperplane is Hw,p=x∈Km∣⟨w,x⟩ℒ=0, with p∈Km,w∈TpKm.H_w,p=\x ^m_K w,x _L=0\, with p ^m_K,\ w∈ T_pL^m_K. (80) The canonical origin is ¯∈Km 0 ^m_K. The tangent space at the origin is T¯Km=[0,v⊤]⊤,v∈ℝm,T_ 0L^m_K=\[0,v ] ,v ^m\, (81) where each tangent vector has a zero time component. Therefore, the coordinate hyperplane through the origin and orthogonal to the k-th axis is He¯k,e H_ e_k,e =y∈Km∣⟨e¯k,y⟩ℒ=0 =\y ^m_K e_k,y _L=0\ (82) =y=(yt,ys)∈Km∣(ys)k=0, =\y=(y_t,y_s) ^m_K (y_s)_k=0\, where e¯k=[0,ek⊤]⊤∈T¯Km e_k=[0,e_k ] ∈ T_ 0L^m_K. From Tab.˜12, the associated signed point-to-hyperplane distance is d¯(y,He¯k,e) d (y,H_ e_k,e ) =sign(⟨e¯k,y⟩ℒ)d(y,He¯k,e) =sign ( e_k,y _L )d (y,H_ e_k,e ) (83) =1−Ksinh−1(−K(ys)k). = 1 -K ^-1 ( -K(y_s)_k ). Equating d¯(y,He¯k,e) d (y,H_ e_k,e ) with uk(x)u_k(x) from Eq.˜22 gives sinh−1(−K(ys)k)=−Kuk(x),1≤k≤m. llasection to20.74pt eq:lorentz-fc-prf-1 eq:lorentz-fc-prf-1 ^-1 ( -K(y_s)_k )= -Ku_k(x), 1≤ k≤ m. (84) Applying sinh(⋅) (·) to both sides of Eq.˜84 yields (ys)k=1−Ksinh(−Kuk(x)),1≤k≤m.(y_s)_k= 1 -K ( -Ku_k(x) ), 1≤ k≤ m. (85) Stacking the coordinates gives ys=1−Ksinh(−Ku(x)),u(x)=(u1(x),…,um(x))⊤.y_s= 1 -K ( -Ku(x) ), u(x)= (u_1(x),…,u_m(x) ) . (86) Since y∈Kmy ^m_K, the hyperboloid constraint ⟨y,y⟩ℒ=1/K y,y _L=1/K implies −yt2+‖ys‖2=1/K-y_t^2+ \|y_s \|^2=1/K. Taking the positive time component yields yt=1−K+‖ys‖2.y_t= 1-K+ \|y_s \|^2. (87) Combining the expressions for yty_t and ysy_s proves the claim. ∎ F.5 Proof of Thm.˜4.3 Proof. Set K=−κ2K=-κ^2 with κ>0κ>0. Poincaré case. Recall that y=ω1+1+κ2‖ω‖2,ωk=sinh(κuk(x))κ.y= ω1+ 1+κ^2 \|ω \|^2, _k= (κ u_k(x) )κ. (88) For any bounded scalar z, sinh(κz)=κz+κ3z33!+O(κ5)⇒sinh(κz)κ=z+κ2z36+O(κ4). (κ z )=κ z+ κ^3z^33!+O (κ^5 )\ \ (κ z )κ=z+ κ^2z^36+O (κ^4 ). (89) Applying this to z=uk(x)z=u_k(x) yields ωk=uk(x)+κ26uk(x)3+O(κ4)=uk(x)+O(κ2). _k=u_k(x)+ κ^26u_k(x)^3+O (κ^4 )=u_k(x)+O (κ^2 ). (90) Thus, ω=u(x)+O(κ2)ω=u(x)+O (κ^2 ) and ‖ω‖2=‖u(x)‖2+O(κ2) \|ω \|^2= \|u(x) \|^2+O (κ^2 ). For the denominator, 1+κ2‖ω‖2=1+12κ2‖ω‖2+O(κ4), 1+κ^2 \|ω \|^2=1+ 12κ^2 \|ω \|^2+O (κ^4 ), (91) which gives 1+1+κ2‖ω‖2=2+12κ2‖ω‖2+O(κ4).1+ 1+κ^2 \|ω \|^2=2+ 12κ^2 \|ω \|^2+O (κ^4 ). (92) Taking the reciprocal produces 11+1+κ2‖ω‖2=12+O(κ2). 11+ 1+κ^2 \|ω \|^2= 12+O (κ^2 ). (93) Multiplying with ω=u(x)+O(κ2)ω=u(x)+O (κ^2 ) gives y=12u(x)+O(κ2).y= 12u(x)+O (κ^2 ). (94) By Thm.˜3.1, uk(x)→2αk⟨vk,x⟩+bku_k(x)→ 2 _k v_k,x +b_k, hence yk→αk⟨vk,x⟩+12bk.y_k→ _k v_k,x + 12b_k. (95) Lorentz case. Recall that ys=1κsinh(κu(x)),yt=1κ2+‖ys‖2.y_s= 1κ (κ u(x) ), y_t= 1κ^2+ \|y_s \|^2. (96) Using the same expansion as above, ys,k=1κ(κuk(x)+κ33!uk(x)3+O(κ5))=uk(x)+O(κ2).y_s,k= 1κ (κ u_k(x)+ κ^33!u_k(x)^3+O (κ^5 ) )=u_k(x)+O (κ^2 ). (97) Therefore, ys=u(x)+O(κ2)y_s=u(x)+O (κ^2 ) and ‖ys‖2=‖u(x)‖2+O(κ2) \|y_s \|^2= \|u(x) \|^2+O (κ^2 ). Factor out κ−1κ^-1 and expand the square root: yt y_t =1κ1+κ2‖ys‖2 = 1κ 1+κ^2 \|y_s \|^2 (98) =1κ(1+12κ2‖ys‖2+O(κ4)) = 1κ (1+ 12κ^2 \|y_s \|^2+O (κ^4 ) ) =1κ+κ2‖ys‖2+O(κ3) = 1κ+ κ2 \|y_s \|^2+O (κ^3 ) =1κ+O(κ)→∞. = 1κ+O (κ )→∞. By Thm.˜3.1, uk(x)→αk⟨vk,xs⟩+bku_k(x)→ _k v_k,x_s +b_k. Using the spatial expansion yields (ys)k→αk⟨vk,xs⟩+bk.(y_s)_k→ _k v_k,x_s +b_k. (99) This completes the proof. ∎