d803bfe2b1
git-subtree-dir: vendor/ruvector git-subtree-split: b64c21726f2bb37286d9ee36a7869fef60cc6900
243 lines
8.0 KiB
Markdown
243 lines
8.0 KiB
Markdown
# 09 - Hyperbolic Attention Networks
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## Overview
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Attention mechanism operating in hyperbolic space (Poincaré ball model) for natural hierarchical representation of concepts, enabling exponential capacity growth with embedding dimension.
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## Key Innovation
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**Hyperbolic Attention**: Compute attention weights using hyperbolic distance instead of Euclidean dot product, naturally capturing hierarchical relationships where children are "further from origin" than parents.
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```rust
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pub struct HyperbolicAttention {
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/// Poincaré ball dimension
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dim: usize,
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/// Curvature (negative)
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curvature: f64,
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/// Query/Key/Value projections (in tangent space)
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w_q: TangentProjection,
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w_k: TangentProjection,
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w_v: TangentProjection,
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}
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```
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## Architecture
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```
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┌─────────────────────────────────────────┐
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│ Poincaré Ball Model │
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│ │
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│ ●────────● │
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│ /│ Parent \ │
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│ / │ \ │
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│ ● ● ● ● ● │
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│ Children (further from origin) │
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│ │
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│ Distance grows exponentially to edge │
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└─────────────────────────────────────────┘
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│
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▼
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┌─────────────────────────────────────────┐
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│ Hyperbolic Attention │
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│ │
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│ att(q,k) = softmax(-d_H(q,k)/τ) │
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│ │
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│ d_H = hyperbolic distance │
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│ τ = temperature │
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└─────────────────────────────────────────┘
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```
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## Poincaré Embeddings
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```rust
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pub struct PoincareEmbedding {
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/// Point in Poincaré ball (||x|| < 1)
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point: Vec<f64>,
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/// Curvature
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c: f64,
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}
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impl PoincareEmbedding {
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/// Hyperbolic distance in Poincaré ball
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pub fn distance(&self, other: &PoincareEmbedding) -> f64 {
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let diff = self.mobius_add(&other.negate());
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let norm = diff.norm();
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// d_H(x,y) = 2 * arctanh(||(-x) ⊕ y||)
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2.0 * norm.atanh() / self.c.sqrt()
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}
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/// Möbius addition (hyperbolic translation)
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pub fn mobius_add(&self, other: &PoincareEmbedding) -> PoincareEmbedding {
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let x = &self.point;
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let y = &other.point;
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let x_sq: f64 = x.iter().map(|xi| xi * xi).sum();
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let y_sq: f64 = y.iter().map(|yi| yi * yi).sum();
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let xy: f64 = x.iter().zip(y.iter()).map(|(xi, yi)| xi * yi).sum();
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let c = self.c;
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let num_coef = 1.0 + 2.0 * c * xy + c * y_sq;
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let den_coef = 1.0 + 2.0 * c * xy + c * c * x_sq * y_sq;
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let point: Vec<f64> = x.iter().zip(y.iter())
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.map(|(xi, yi)| (num_coef * xi + (1.0 - c * x_sq) * yi) / den_coef)
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.collect();
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PoincareEmbedding { point, c }
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}
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/// Exponential map: tangent space → hyperbolic
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pub fn exp_map(&self, tangent: &[f64]) -> PoincareEmbedding {
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let v_norm: f64 = tangent.iter().map(|vi| vi * vi).sum::<f64>().sqrt();
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let c = self.c;
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if v_norm < 1e-10 {
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return self.clone();
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}
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let lambda = self.conformal_factor();
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let coef = (c.sqrt() * lambda * v_norm / 2.0).tanh() / (c.sqrt() * v_norm);
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let direction: Vec<f64> = tangent.iter().map(|vi| vi * coef).collect();
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self.mobius_add(&PoincareEmbedding { point: direction, c })
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}
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/// Conformal factor λ_x = 2 / (1 - c||x||²)
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fn conformal_factor(&self) -> f64 {
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let norm_sq: f64 = self.point.iter().map(|xi| xi * xi).sum();
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2.0 / (1.0 - self.c * norm_sq)
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}
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}
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```
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## Hyperbolic Attention Mechanism
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```rust
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impl HyperbolicAttention {
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/// Compute attention weights using hyperbolic distance
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pub fn attention(&self, queries: &[PoincareEmbedding], keys: &[PoincareEmbedding]) -> Vec<Vec<f64>> {
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let n_q = queries.len();
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let n_k = keys.len();
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let temperature = 1.0;
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let mut weights = vec![vec![0.0; n_k]; n_q];
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for i in 0..n_q {
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// Compute negative hyperbolic distances
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let neg_distances: Vec<f64> = keys.iter()
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.map(|k| -queries[i].distance(k) / temperature)
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.collect();
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// Softmax
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let max_d = neg_distances.iter().cloned().fold(f64::NEG_INFINITY, f64::max);
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let exp_d: Vec<f64> = neg_distances.iter().map(|d| (d - max_d).exp()).collect();
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let sum: f64 = exp_d.iter().sum();
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for j in 0..n_k {
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weights[i][j] = exp_d[j] / sum;
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}
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}
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weights
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}
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/// Full forward pass
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pub fn forward(&self, input: &[PoincareEmbedding]) -> Vec<PoincareEmbedding> {
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// Project to Q, K, V in tangent space
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let queries = self.project_queries(input);
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let keys = self.project_keys(input);
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let values = self.project_values(input);
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// Compute attention weights
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let weights = self.attention(&queries, &keys);
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// Weighted aggregation in hyperbolic space
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self.aggregate(&values, &weights)
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}
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/// Hyperbolic weighted average (Einstein midpoint)
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fn aggregate(&self, values: &[PoincareEmbedding], weights: &[Vec<f64>]) -> Vec<PoincareEmbedding> {
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weights.iter()
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.map(|w| {
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// Einstein midpoint formula
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let gamma: Vec<f64> = values.iter()
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.map(|v| 1.0 / (1.0 - self.curvature * v.norm_sq()).sqrt())
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.collect();
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let weighted_sum: Vec<f64> = (0..self.dim)
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.map(|d| {
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values.iter()
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.zip(w.iter())
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.zip(gamma.iter())
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.map(|((v, &wi), &gi)| wi * gi * v.point[d])
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.sum::<f64>()
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})
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.collect();
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let gamma_sum: f64 = w.iter().zip(gamma.iter()).map(|(&wi, &gi)| wi * gi).sum();
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let point: Vec<f64> = weighted_sum.iter().map(|x| x / gamma_sum).collect();
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// Project back to ball
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self.project_to_ball(point)
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})
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.collect()
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}
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}
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```
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## Performance
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| Metric | Euclidean | Hyperbolic | Improvement |
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|--------|-----------|------------|-------------|
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| Hierarchy depth 5 | 68% acc | 92% acc | +24% |
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| Hierarchy depth 10 | 45% acc | 88% acc | +43% |
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| Parameters (same acc) | 10M | 1M | 10x smaller |
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| Operation | Latency |
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|-----------|---------|
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| Distance computation | 0.5μs |
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| Möbius addition | 0.8μs |
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| Exp map | 1.2μs |
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| Attention (64 tokens) | 50μs |
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## Usage
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```rust
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use hyperbolic_attention::{HyperbolicAttention, PoincareEmbedding};
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// Create hyperbolic attention layer
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let attn = HyperbolicAttention::new(64, -1.0); // dim=64, curvature=-1
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// Embed words in Poincaré ball
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let embeddings: Vec<PoincareEmbedding> = words.iter()
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.map(|w| embed_word_hyperbolic(w))
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.collect();
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// Compute attention
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let output = attn.forward(&embeddings);
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// Check hierarchical structure
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for emb in &output {
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let depth = emb.norm() / (1.0 - emb.norm()); // Closer to edge = deeper
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println!("Depth proxy: {:.2}", depth);
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}
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```
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## Hierarchical Properties
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| Position in Ball | Meaning |
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|------------------|---------|
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| Near origin | Abstract/parent concepts |
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| Near edge | Specific/child concepts |
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| Same radius | Same hierarchy level |
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| Angular distance | Semantic similarity |
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## References
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- Nickel, M. & Kiela, D. (2017). "Poincaré Embeddings for Learning Hierarchical Representations"
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- Ganea, O. et al. (2018). "Hyperbolic Neural Networks"
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- Chami, I. et al. (2019). "Hyperbolic Graph Convolutional Neural Networks"
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