| name | hyperbolic-neural-mapping |
| description | HyNeuralMap framework for mapping visual semantics to neural hierarchies using hyperbolic Lorentz geometry. Provides cross-modal semantic alignment in negative-curvature space, outperforming Euclidean baselines for fMRI-visual representation learning. Use when working with: hyperbolic embeddings, vision-neural mapping, cross-subject fMRI alignment, hierarchical semantic organization, Lorentz model, geometric deep learning for neuroscience, or neural representation learning with non-Euclidean geometry. Trigger words: HyNeuralMap, hyperbolic neural, Lorentz embedding, cross-modal alignment, hierarchical neural representation. |
HyNeuralMap: Hyperbolic Mapping of Visual Semantics to Neural Hierarchies
Based on arXiv:2605.09392
Core Idea
Map visual stimuli and neural responses (fMRI) into shared hyperbolic space (Lorentz model) rather than Euclidean space. Hyperbolic geometry's negative curvature naturally encodes hierarchical structure and preserves fine-grained semantic relationships across modalities.
Why Hyperbolic > Euclidean
| Property | Euclidean | Hyperbolic |
|---|
| Hierarchical encoding | Poor (linear growth) | Excellent (exponential growth) |
| Semantic proximity | Distorts at scale | Preserves via geodesics |
| Cross-subject similarity | Flat, loses structure | Captures nested hierarchy |
Key Components
1. Lorentz Model
The Lorentz model of hyperbolic space uses the hyperboloid:
H^n = {x ∈ R^(n+1) | ⟨x, x⟩_L = -1, x_0 > 0}
where ⟨x, y⟩_L = -x_0·y_0 + Σ_i x_i·y_i is the Lorentzian inner product.
2. Hyperbolic Geometric Alignment
- Visual embeddings: Encode image semantics into hyperbolic space
- Neural embeddings: Encode fMRI response patterns into the same space
- Joint optimization: Geodesic distances in hyperbolic space preserve both semantic proximity AND hierarchical relationships
- Cross-subject generalization: Shared hyperbolic manifold enables alignment across subjects
3. Training Pipeline
Visual Features → Hyperbolic Embedding ← fMRI Features
↓
Geodesic Distance Loss
↓
Semantic Proximity Loss
↓
Joint Optimization
Implementation Guide
Hyperbolic Operations (Lorentz Model)
import torch
def lorentz_inner(x, y):
"""Lorentzian inner product: -x0*y0 + sum(xi*yi)"""
return -x[..., 0] * y[..., 0] + torch.sum(x[..., 1:] * y[..., 1:], dim=-1)
def lorentz_dist(x, y):
"""Geodesic distance in Lorentz model"""
inner = lorentz_inner(x, y)
return torch.acosh(torch.clamp(-inner, min=1.0 + 1e-7))
def exp_map_o(v, c=1.0):
"""Exponential map from origin to hyperboloid"""
v_norm = torch.sqrt(torch.clamp(lorentz_inner(v, v), min=1e-7))
return torch.stack([
torch.cosh(c * v_norm),
(torch.sinh(c * v_norm) / v_norm) * v[..., 1:]
], dim=-1)
def log_map_o(x, c=1.0):
"""Logarithmic map from hyperboloid to tangent space at origin"""
x0 = x[..., 0]
return torch.acosh(x0) / torch.sqrt(torch.clamp(x0**2 - 1, min=1e-7)) * x[..., 1:]
Alignment Loss
def hyperbolic_alignment_loss(vis_emb, neural_emb, semantic_labels):
"""Joint loss: geodesic proximity + semantic consistency"""
geo_dist = lorentz_dist(vis_emb, neural_emb)
same_label = (semantic_labels.unsqueeze(1) == semantic_labels.unsqueeze(0)).float()
semantic_loss = torch.mean(same_label * geo_dist + (1 - same_label) * torch.relu(margin - geo_dist))
manifold_loss = torch.mean((lorentz_inner(vis_emb, vis_emb) + 1)**2)
return semantic_loss + lambda_reg * manifold_loss
Activation
- When to use: Cross-modal neural-visual alignment, hierarchical semantic mapping, fMRI encoding models with geometric deep learning
- When NOT to use: Simple linear encoding models, non-hierarchical data, when Euclidean distance suffices
- Keywords: HyNeuralMap, hyperbolic neural mapping, Lorentz embedding, cross-modal alignment, hierarchical neural representation, geometric deep learning, fMRI encoding, visual semantics
arXiv Reference
