Fully Hyperbolic Rotation (FHRE) Overview

• Fully Hyperbolic Rotation (FHRE) is a class of geometric and algebraic operations defined on hyperbolic spaces using Lorentz rotations without tangent space projections.
• FHRE achieves computational efficiency and precise geometric fidelity by preserving Lorentzian inner products and avoiding numerical instabilities from repeated exponential/logarithmic maps.
• Applications of FHRE span deep learning, knowledge graph embedding, and dynamical systems, enabling robust neural network layers and effective modeling of rotational dynamics.

Fully Hyperbolic Rotation (FHRE) refers to a class of geometric and algebraic mechanisms in which rotational operations are formulated intrinsically within hyperbolic spaces, and are implemented as isometries—specifically, as Lorentz rotations—without recourse to tangent space projections, logarithmic/exponential maps, or manifold/Eucledean transitions. FHRE originates in the Lorentz (hyperboloid) model of constant negative curvature and arises in multiple modern research areas, notably deep learning (knowledge representation, neural networks) and dynamical systems (rotational decomposition of surface homeomorphisms) (Liang et al., 2024, Chen et al., 2021, &&&2&&&). FHRE mechanisms are characterized by computational efficiency, exact geometric fidelity, and structural isometry at the core of their respective domains.

1. Mathematical Foundations and the Lorentz Model

FHRE is formulated in the $n$-dimensional Lorentz model $\mathcal{L}^n_c = (\mathcal{H}^n, g_\mathcal{L})$ with constant negative curvature $c<0$. The ambient coordinate representation is $$\mathbf{x} = (x_0, x_1, ..., x_n) \in \mathbb{R}^{n+1}$$, constrained by the Lorentzian pseudo-inner product

$$\langle \mathbf{x}, \mathbf{y}\rangle_\mathcal{L} = -x_0y_0 + \sum_{i=1}^n x_i y_i$$

so that $$\langle\mathbf{x}, \mathbf{x}\rangle_\mathcal{L} = -1$$ (for $c=-1$). The geodesic distance is

$$d_\mathcal{L}(\mathbf{x}, \mathbf{y}) = \operatorname{arccosh}(-\langle \mathbf{x}, \mathbf{y}\rangle_\mathcal{L}),$$

or in squared form,

$$d^2_\mathcal{L}(\mathbf{x}, \mathbf{y}) = 2 - 2\langle \mathbf{x}, \mathbf{y}\rangle_\mathcal{L}.$$

A Lorentz rotation $R \in O_\mathcal{L}(n+1)$ is a linear transformation that preserves the Lorentzian inner product and, for FHRE, typically leaves $x_0$ invariant and acts as a rotation on the spatial components via $\tilde{R} \in \mathrm{SO}(n)$. This intrinsic formulation eliminates the need for exponential/logarithmic maps traditionally necessary in hyperbolic neural or embedding models (Liang et al., 2024, Chen et al., 2021).

2. FHRE in Neural Networks and Representation Learning

Traditionally, hyperbolic neural networks operate by alternating between embedding operations in hyperbolic space and Euclidean transformations within tangent spaces, employing repeated use of exponential and logarithmic maps. FHRE, as formalized in "Fully Hyperbolic Neural Networks" (Chen et al., 2021), replaces all such steps with direct-in-space Lorentz rotations. A typical FHRE layer parametrizes a skew-symmetric generator $A\in \mathbb{R}^{n\times n}$, computes $\tilde{R} = \exp(A)$, constructs the block-diagonal $R$, and applies $y = R x = [x_0; \tilde{R} x_{1:n}]$. These layers:

• preserve the Lorentzian structure and remain completely on the hyperboloid
• form a subgroup $\mathrm{SO}^+(1,n)$ under composition and inverse
• exhibit numerical stability (avoiding repeated $\operatorname{arccosh}$ or projections)
• are compatible with Riemannian optimization algorithms

In networks, FHRE can substitute for standard linear/affine layers in both shallow (e.g., GNNs, embedding architectures) and deep models (transformers, MLPs), and can be extended by incorporating Lorentz boosts for full expressivity (Chen et al., 2021).

3. FHRE in Knowledge Graph Embedding

"Fully Hyperbolic Rotation for Knowledge Graph Embedding" (Liang et al., 2024) extends the FHRE paradigm to relational modeling. Consider a knowledge graph triplet $(h, r, t)$, with head, relation, and tail entities. Unlike prior models (e.g., RotH, MuRP) that require transitions between Euclidean tangent space and hyperbolic manifolds via $\exp/\log$ mappings, FHRE models each relation $r$ as a Lorentz rotation $R_r$, block-diagonal with an $\mathrm{SO}(n)$ spatial action. The relation operator is constructed so that

$\mathbf{v}_h' = R_r \mathbf{v}_h,$

with $\mathbf{v}_h$ and $\mathbf{v}_t$ as entity embeddings in $\mathcal{L}^d$. The model’s scoring function uses squared Lorentzian distance, optionally with entity-specific biases: $$s(h, r, t) = 2-2\langle\mathbf v_h', \mathbf v_t\rangle_\mathcal{L} + b_h + b_t.$$ Training is accomplished via negative sampling, binary cross-entropy on sigmoid-linked distances, and Riemannian Adam optimization. An initial exponential map places embeddings on the manifold, after which training uses exclusively manifold-respecting (rotation-only) operations (Liang et al., 2024).

FHRE achieves competitive or state-of-the-art results on KGC benchmarks (FB15k-237, WN18RR, CoDEx-s/m, Nations), with greater parameter efficiency and enhanced numerical stability compared to prior hyperbolic and mixed-space baselines. For example, FHRE (at $d=500$) attains MRR=0.374 (FB15k-237), outperforming HYBONET and requiring fewer model parameters (20.5M vs. 26M–81.9M) (Liang et al., 2024).

4. Rotational FHRE in Dynamical Systems

In the context of topological dynamics, "A rotational hyperbolic theory for surface homeomorphisms" (Guihéneuf, 18 Nov 2025) introduces the fully hyperbolic rotational decomposition (FHRE) for identity-isotopic homeomorphisms $f\colon S\to S$ on closed orientable surfaces. The theory classifies $f$-ergodic measures with nontrivial “rotation speed” into equivalence classes (rotational homoclinic classes or chaotic classes) via dynamically transverse chains or shared tracking sets. Each chaotic class supports a network of local hyperbolic sets (horseshoes) whose union realizes the full set of possible rotation vectors within convex rotation blocks $\rho_i$ in $H_1(S,\mathbb{R})$.

The construction proceeds by:

• constructing Markov rectangles encoding distinct periodic points and their associated deck transformations
• building a directed graph of transverse Markov intersections
• synthesizing strongly connected components (chaotic class clusters) whose collective rotation sets fully capture the ergodic rotational behavior of $f$
• establishing five equivalent notions of “heteroclinic connection” between chaotic classes

The main theorem asserts that any rational point in the global rotation set can be achieved by a periodic orbit with bounded deviation and that the full homological rotation behavior is faithfully modeled by a finite network of hyperbolic horseshoes and their interconnections, giving a combinatorial description of rotational dynamics via FHRE (Guihéneuf, 18 Nov 2025).

5. Computational and Theoretical Properties

FHRE exhibits several properties of central importance:

• Isometric action: Lorentz rotations are exact isometries of the Lorentzian hyperboloid, preserving pseudo-inner products and geodesic distances (Chen et al., 2021, Liang et al., 2024).
• Stability: Direct rotational operations free from repeated manifold/tangent projections avoid numerical pathologies (e.g., NaNs from unbounded $\operatorname{arccosh}$).
• Computational efficiency: The FHRE approach enables relation-specific rotations with only $O(d)$ cost for $d$-dimensional embeddings, as opposed to $O(d^2)$ for full linear isometries (Liang et al., 2024).
• Expressivity: While “pure” FHRE uses only rotations (fixing the time coordinate), incorporation of Lorentz boosts allows general isometries. However, in knowledge graph applications the rotation-only operation is sufficient to capture symmetry, inversion, and composition (Liang et al., 2024, Chen et al., 2021).
• Limitations: FHRE with block-rotational parametrization requires even embedding dimension (for coordinate pairing) and may be less expressive for certain scaling-type relations, but this constraint yields enhanced interpretability.

6. Broader Connections and Synthesis

FHRE unifies approaches across distinct mathematical and computational domains: representation learning with end-to-end hyperbolic models (Liang et al., 2024, Chen et al., 2021), theoretical dynamical systems via rotational decomposition (Guihéneuf, 18 Nov 2025), and group-theoretic characterizations of isometries in negatively curved spaces. In all settings, FHRE delivers parsimonious, geometrically faithful mechanisms for encoding, manipulating, and analyzing high-complexity data where hierarchical, compositional, or rotational symmetries prevail.

Empirical successes—including state-of-the-art performance in knowledge graph completion and robust periodic orbit realization in surface homeomorphism dynamics—attest to both the practical and theoretical significance of FHRE. Ongoing research continues to investigate fully-hyperbolic variants of neural architectures and the role of rotationally-defined hyperbolic sets in broader classes of dynamical systems (Liang et al., 2024, Chen et al., 2021, Guihéneuf, 18 Nov 2025).