Embedding deeply nested web taxonomies with millions of subcategory relationships into flat Euclidean vector spaces induces severe geometric distortion. Operating directly within the Lorentz (Hyperboloid) model of hyperbolic geometry preserves tree-like hierarchical branching with near-zero metric distortion, enabling sub-millisecond nearest neighbor search in low-dimensional space.
The Lorentz Hyperbolic Metric & Geodesic Distance
How Minkowski inner products compute distances without boundary singularities:
For two points $\mathbf{u}, \mathbf{v} \in \mathbb{H}^n$ on the hyperboloid sheet, the Lorentzian inner product is defined as $\langle \mathbf{u}, \mathbf{v} \rangle_L = -u_0 v_0 + \sum_{i=1}^n u_i v_i$. The geodesic distance is computed analytically as $d_L(\mathbf{u}, \mathbf{v}) = \text{arcosh}\left(-\langle \mathbf{u}, \mathbf{v} \rangle_L\right)$. Unlike the Poincaré ball model, the Lorentz model avoids numerical instability near boundary horizons.
Embedding Geometries for Web Taxonomies Compared
| Embedding Geometry | Curvature Property | Tree Distortion @ $d=16$ | Numerical Gradient Stability |
|---|---|---|---|
| Euclidean Space ($\mathbb{R}^n$) | Zero Curvature ($K = 0$) | High ($> 38.4\%$ Distortion) | Standard Euclidean SGD |
| Poincaré Ball Model ($\mathbb{B}^n$) | Constant Negative ($K = -1$) | Low ($< 3.2\%$ Distortion) | Boundary gradient vanishing ($\|\mathbf{x}\| \to 1$) |
| Lorentz Model ($\mathbb{H}^n$) | Constant Negative ($K = -1$) | Minimal ($< 0.8\%$ Distortion) | Excellent (Linear Minkowski algebra) |
Riemannian Optimization on the Hyperboloid
How Riemannian stochastic gradient descent updates hierarchical node coordinates:
- Compute Euclidean Gradient: Calculate the partial derivatives $\nabla_E \mathcal{L}$ of the margin ranking loss across taxonomy edge pairs.
- Orthogonal Projection to Tangent Space: Project the gradient onto the tangent space $T_\mathbf{u}\mathbb{H}^n$ via $\text{grad}_R \mathcal{L} = \nabla_E \mathcal{L} + \langle \mathbf{u}, \nabla_E \mathcal{L} \rangle_L \mathbf{u}$.
- Exponential Map Retraction: Traverse the manifold geodesic via $\exp_\mathbf{u}(v) = \cosh(\|v\|_L)\mathbf{u} + \sinh(\|v\|_L)\frac{v}{\|v\|_L}$.
Explore Web Directories & Semantic Graph Architectures
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