Introduction to Hyperbolic Embeddings
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Why Hyperbolic Geometry?
Euclidean space has zero curvature — it is flat. But many real-world structures are intrinsically hierarchical and tree-like:
- Taxonomies (WordNet, biological classification)
- Social networks (power-law degree distributions)
- Natural language syntax (parse trees)
These structures exhibit exponential growth — the number of nodes at distance \(r\) from a root grows as \(\exp(r)\). In Euclidean space, volume grows only polynomially (\(\sim r^d\)). This mismatch causes exponential distortion when embedding trees in flat space.
Hyperbolic space, with its constant negative curvature, has volume growth \(\sim \exp(r)\) — perfectly matching tree growth. Hence, trees embed with zero distortion in hyperbolic space.
The Poincaré Disk Model
The Poincaré disk \(\mathbb{D}^n\) is the unit ball \(\{x \in \mathbb{R}^n : \|x\| < 1\}\) equipped with the metric:
\[ds^2 = \frac{4 \sum_{i=1}^n dx_i^2}{(1 - \sum_{i=1}^n x_i^2)^2}\]Key properties:
- Geodesics are circular arcs orthogonal to the boundary
- Distances grow exponentially as you approach the boundary
- The boundary \(\partial \mathbb{D}^n\) represents “infinity”