hlbg-hyperbolic-learning-brain-graphs

name	hlbg-hyperbolic-learning-brain-graphs
description	Hyperbolic Learning on Brain Graphs (HLBG) methodology for brain network analysis using Lorentzian hyperbolic space to model hierarchical ROI-community-whole-brain relationships. Introduces Graph-aware Mamba (GaMamba) for capturing long-range dependencies while preserving graph topology. Achieves SOTA on ABIDE-I and REST-MDD disorder diagnosis. Activation: hyperbolic learning, brain graphs, functional connectivity, disorder diagnosis, Lorentzian space, graph mamba, hierarchical brain networks, biomarker identification
tags	["hyperbolic-geometry","brain-graphs","functional-connectivity","disorder-diagnosis","graph-neural-networks","mamba"]

HLBG - Hyperbolic Learning on Brain Graphs

Paper: Navigating Hierarchy: Hyperbolic Learning on Brain Graphs for Disorder Diagnosis
arXiv: 2607.07077
Authors: Yapeng Li, Bo Jiang, Ziyan Zhang, Dongdong Chen, Zhengzheng Tu
Published: 2026-07-08
Categories: cs.CV, cs.AI

Core Contribution

Proposes HLBG framework that exploits hyperbolic geometry to model the hierarchical organization of functional brain networks (ROI → community → whole-brain) for disorder diagnosis and biomarker identification.

Key Innovations

1. Hyperbolic Space for Brain Hierarchy: Projects ROI, community, and whole-brain representations into Lorentzian hyperbolic space, naturally capturing hierarchical relationships
2. Geometric Entailment Constraints: Two constraints enforce multi-level hierarchy:
   • ROI-to-community entailment
   • Community-to-whole-brain entailment
3. Graph-aware Mamba (GaMamba): Novel architecture incorporating topology-derived structural prompts into Mamba for:
   • Long-range dependency capture
   • Graph topology preservation
4. Disorder Biomarker Discovery: Identifies disorder-relevant functional biomarkers through learned representations

Methodology

Pipeline

1. Brain Graph Construction: Build functional connectivity graphs from fMRI
2. Multi-level Representation: Extract ROI, community, whole-brain features
3. Hyperbolic Projection: Map all levels to Lorentzian space (Poincaré ball model)
4. Geometric Constraints: Apply entailment losses to preserve hierarchy
5. GaMamba Processing: Process with topology-aware Mamba blocks
6. Classification/Regression: Downstream disorder diagnosis

Hyperbolic Geometry

• Uses Lorentz model (hyperboloid) for numerical stability
• Exponential/logarithmic maps for tangent space operations
• Distance metric captures tree-like hierarchy naturally
• Entailment: parent representation "contains" children in hyperbolic space

GaMamba Architecture

Input Graph → Structural Prompt Extraction → Mamba Block → Output
                    ↓
          Topology-aware state space model
          (preserves graph structure in SSM)

Experimental Results

Datasets: ABIDE-I (autism), REST-MDD (depression)

• Outperforms SOTA methods on both datasets
• Identifies clinically meaningful biomarkers
• Ablation confirms each component contribution

Applications

• Clinical Diagnosis: Automated brain disorder detection
• Biomarker Discovery: Identifying disorder-specific network patterns
• Brain Network Analysis: Understanding hierarchical organization
• Treatment Monitoring: Tracking network changes over time

Technical Details

Lorentz Model Operations

# Lorentz inner product
def lorentz_inner_product(x, y):
    return -x[0]*y[0] + sum(x[i]*y[i] for i in range(1, dim))

# Exponential map (Euclidean → hyperbolic)
def exp_map(v, c=1.0):
    norm_v = sqrt(lorentz_inner_product(v, v))
    return cosh(sqrt(c)*norm_v) * v[0] + sinh(sqrt(c)*norm_v) * v[1:] / (sqrt(c)*norm_v)

# Entailment loss (parent should contain child)
def entailment_loss(parent, child):
    dist = hyperbolic_distance(parent, child)
    return max(0, dist - margin)

GaMamba Integration

• Structural prompts derived from graph Laplacian eigenvectors
• State space model modified to respect graph topology
• Selective scan mechanism for efficient long-range modeling

Limitations

• Requires pre-defined community structure (parcellation-dependent)
• Hyperbolic optimization can be numerically unstable
• Computational cost of hyperbolic operations
• Limited to static functional connectivity (no temporal dynamics)

Related Work

• Hyperbolic graph neural networks (Hyperbolic GCN, HAT)
• Brain graph learning (BrainNetGNN, GNN-FC)
• Mamba/state space models for graphs
• Functional connectivity analysis

Related Skills

• [[hyperbolic-gcn-brain-network]] - Earlier hyperbolic GCN for brain networks
• [[hyperbolic-learning-brain-graphs]] - Related hyperbolic brain analysis
• [[functional-connectivity-graph-neural-networks]] - FC-based GNN methods
• [[gnn-visual-decoding-brain-network]] - GNN for brain decoding