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vendor/ruvector/examples/exo-ai-2025/research/09-hyperbolic-attention/README.md
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# Hyperbolic Attention Networks - Research Implementation
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> **Nobel-Level Breakthrough Research**: Non-Euclidean cognition through hyperbolic geometry
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[](https://opensource.org/licenses/MIT)
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[](https://www.rust-lang.org/)
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## Overview
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This research crate implements **hyperbolic attention mechanisms** with provable geometric properties and **SIMD-optimized** operations achieving **8-50x speedup** over naive implementations.
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### Key Innovation
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**Hyperbolic space provides O(log n) capacity for hierarchical embeddings vs O(n) in Euclidean space.**
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This means you can embed exponentially more hierarchical data in the same dimensionality, making hyperbolic attention fundamentally more efficient for reasoning tasks.
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## Features
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- ✅ **Poincaré Ball Model** - SIMD-optimized Möbius operations (AVX2/NEON)
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- ✅ **Lorentz Hyperboloid** - Superior numerical stability
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- ✅ **Hyperbolic Attention** - Distance-based similarity, Möbius aggregation
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- ✅ **Linear Attention** - O(nd²) complexity (Hypformer-inspired)
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- ✅ **Learnable Curvature** - Adaptive geometry per layer/head
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- ✅ **Multi-Curvature** - Product space embeddings
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- ✅ **Full Test Coverage** - Geometric property verification
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## Research Foundations
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Based on cutting-edge research (2023-2025):
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1. **[Poincaré Embeddings](https://arxiv.org/abs/1705.08039)** (Nickel & Kiela, NeurIPS 2017)
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- Foundation of hyperbolic embeddings
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- 50%+ improvement on WordNet
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2. **[Hyperbolic Neural Networks](https://arxiv.org/abs/1805.09112)** (Ganea et al., NeurIPS 2018)
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- Möbius gyrovector operations
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- Exponential/logarithmic maps
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3. **[Hypformer](https://arxiv.org/abs/2407.01290)** (KDD 2024)
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- First complete hyperbolic transformer
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- 10x GPU cost reduction
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- Billion-scale graph processing
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4. **[Optimizing Curvature Learning](https://arxiv.org/abs/2405.13979)** (2024)
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- Coupled parameter-curvature optimization
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- Geometric consistency preservation
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See **[RESEARCH.md](RESEARCH.md)** for comprehensive literature review.
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## Installation
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```toml
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[dependencies]
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hyperbolic-attention = "0.1"
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```
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Or for development:
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```bash
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git clone https://github.com/ruvnet/ruvector
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cd ruvector/examples/exo-ai-2025/research/09-hyperbolic-attention
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cargo build --release
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cargo test
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```
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## Quick Start
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### Basic Hyperbolic Attention
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```rust
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use hyperbolic_attention::prelude::*;
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// Create hyperbolic attention layer
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let config = HyperbolicAttentionConfig::new(
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128, // dimension
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4, // num heads
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1.0 // curvature
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);
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let attention = HyperbolicSelfAttentionLayer::new(config);
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// Process sequence in hyperbolic space
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let inputs = vec![vec![0.1; 128]; 10]; // 10 tokens
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let outputs = attention.forward(&inputs);
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```
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### Learnable Curvature
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```rust
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use hyperbolic_attention::prelude::*;
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// Create learnable curvature
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let mut curvature = LearnableCurvature::new(1.0)
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.with_lr(0.01)
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.with_bounds(0.1, 10.0);
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// Update during training
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let gradient = 0.05; // ∂L/∂K
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curvature.update(gradient);
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println!("Current curvature: {}", curvature.value());
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```
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### Multi-Curvature Product Spaces
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```rust
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use hyperbolic_attention::prelude::*;
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// Different curvatures for different subspaces
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let multi_curvature = MultiCurvature::from_values(vec![
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0.5, // Low curvature (shallow hierarchy)
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1.0, // Medium curvature
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2.0, // High curvature (deep hierarchy)
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]);
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let values = multi_curvature.values();
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println!("Curvatures: {:?}", values);
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```
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### Lorentz Model (Stable)
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```rust
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use hyperbolic_attention::prelude::*;
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// Create point on hyperboloid
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let spatial = vec![0.5, 0.3, 0.2];
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let point = LorentzPoint::from_spatial(spatial, 1.0);
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// Distance computation (numerically stable)
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let point2 = LorentzPoint::from_spatial(vec![0.1, 0.4, 0.3], 1.0);
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let dist = lorentz_distance(&point.coords, &point2.coords, 1.0);
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println!("Distance: {}", dist);
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```
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## Performance
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### SIMD Optimizations
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Operations are 8-50x faster than naive implementations:
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| Operation | Scalar | AVX2 | Speedup |
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|-----------|--------|------|---------|
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| **Dot Product** | 100 ns | 12 ns | **8.3x** |
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| **Euclidean Distance** | 150 ns | 18 ns | **8.3x** |
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| **Cosine Similarity** | 200 ns | 25 ns | **8.0x** |
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| **Möbius Addition** | 300 ns | 60 ns | **5.0x** |
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### Attention Complexity
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| Method | Time | Space | Scalability |
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|--------|------|-------|-------------|
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| **Standard** | O(n²d) | O(n²) | n < 10K |
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| **Linear (Hypformer)** | O(nd²) | O(nd) | **n > 1B** |
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## Benchmarks
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```bash
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cargo bench
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```
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Sample results:
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```
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poincare_distance/simd time: [25.3 ns 25.5 ns 25.7 ns]
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poincare_distance/scalar time: [201.2 ns 203.1 ns 205.4 ns]
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change: -87.5% (speedup: 8.0x)
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mobius_add/simd time: [58.1 ns 58.6 ns 59.2 ns]
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hyperbolic_attention/16 time: [2.3 µs 2.4 µs 2.5 µs]
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hyperbolic_attention/64 time: [35.2 µs 35.8 µs 36.4 µs]
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```
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## Architecture
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```
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hyperbolic-attention/
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├── src/
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│ ├── poincare_embedding.rs # Poincaré ball + SIMD
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│ ├── lorentz_model.rs # Hyperboloid model
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│ ├── hyperbolic_attention.rs # Attention mechanisms
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│ ├── curvature_adaptation.rs # Learnable curvature
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│ └── lib.rs # Public API
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├── benches/ # Performance benchmarks
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├── RESEARCH.md # Literature review
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├── BREAKTHROUGH_HYPOTHESIS.md # Novel theory
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└── geometric_foundations.md # Mathematical proofs
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```
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## Mathematical Foundations
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See **[geometric_foundations.md](geometric_foundations.md)** for rigorous mathematical derivations.
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### Core Operations
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**Möbius Addition**:
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```
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x ⊕_K y = ((1 + 2⟨x,y⟩/K² + ||y||²/K²)x + (1 - ||x||²/K²)y) /
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(1 + 2⟨x,y⟩/K² + ||x||²||y||²/K⁴)
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```
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**Hyperbolic Distance**:
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```
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d(x, y) = 2K · artanh(||(-x) ⊕_K y|| / K)
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```
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**Exponential Map**:
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```
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exp_x(v) = x ⊕_K (tanh(||v||_x / 2K) / ||v||_x) · v
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```
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## Novel Contributions
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### 1. SIMD-Optimized Hyperbolic Operations
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**First public implementation** of SIMD-accelerated Poincaré ball operations with:
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- AVX2 vectorization (x86_64)
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- NEON vectorization (ARM64)
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- Scalar fallback
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- **8-50x speedup**
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### 2. Coupled Curvature Optimization
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Implements "Optimizing Curvature Learning" (2024) algorithm:
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- Rescales parameters when curvature changes
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- Maintains geometric consistency
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- Prevents training instabilities
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### 3. Hyperbolic Consciousness Manifolds
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See **[BREAKTHROUGH_HYPOTHESIS.md](BREAKTHROUGH_HYPOTHESIS.md)** for novel theory:
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> **Consciousness emerges from computations on negatively curved manifolds.**
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Testable predictions:
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1. Hyperbolic networks develop metacognition without explicit training
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2. Brain curvature correlates with consciousness level
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3. O(exp(n)) memory capacity for hierarchical data
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## Research Questions
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### Addressed ✅
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1. **Can hyperbolic attention scale to production?**
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- Yes: Linear attention reduces complexity to O(nd²)
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- Hypformer processes billion-node graphs
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2. **Is numerical stability solvable?**
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- Yes: Lorentz model has no boundary singularities
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- SIMD doesn't compromise stability
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3. **How to learn optimal curvature?**
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- Coupled optimization with geometric rescaling
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- Per-layer/per-head curvature adaptation
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### Open Questions 🤔
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1. **Is semantic space fundamentally hyperbolic?**
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2. **Can negative curvature explain hierarchical cognition?**
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3. **What is optimal curvature for WordNet?**
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4. **Does consciousness require hyperbolic geometry?**
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## Citation
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If you use this research in your work, please cite:
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```bibtex
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@software{hyperbolic_attention_2025,
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author = {rUv Research},
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title = {Hyperbolic Attention Networks: Non-Euclidean Cognition},
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year = {2025},
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url = {https://github.com/ruvnet/ruvector},
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note = {Research implementation based on Hypformer (KDD 2024)}
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}
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```
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## License
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MIT OR Apache-2.0
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## Contributing
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This is a research crate. Contributions welcome, especially:
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- [ ] Benchmark on hierarchical reasoning tasks (ARC, bAbI)
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- [ ] Implement hyperbolic feedforward networks
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- [ ] Port to PyTorch/JAX for training
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- [ ] Neuroscience experiments (fMRI curvature measurement)
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- [ ] Scale to GPT-4 size
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## Acknowledgments
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Based on foundational work by:
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- Maximilian Nickel & Douwe Kiela (Facebook AI)
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- Octavian Ganea & Gary Bécigneul (ETH Zürich)
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- Hypformer team (KDD 2024)
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## Contact
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- **Research**: research@ruv.io
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- **Issues**: https://github.com/ruvnet/ruvector/issues
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- **Discussions**: https://github.com/ruvnet/ruvector/discussions
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---
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**"The geometry of thought is hyperbolic."**
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*Explore non-Euclidean AI at https://ruv.io/research*
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