Project Overview
This project explores the hidden geometric structure of language model embeddings through the lens of topological data analysis (TDA). By applying advanced mathematical techniques to GPT-2's word embeddings, we uncover how semantic relationships are encoded in high-dimensional space and create tools for interpretable analysis of language model representations.
Methodological Approach
1. Mapper Algorithm Implementation
The Mapper algorithm creates a simplified representation of high-dimensional data:
- Dimensionality Reduction: UMAP projection to 2D/3D space
- Cover Construction: 40 overlapping hypercubes with 55% overlap
- Clustering: Agglomerative clustering within each hypercube
- Graph Generation: Connected components form semantic clusters
- Visualization: Interactive network showing token relationships
2. Persistent Homology Analysis
Persistent homology captures topological features across multiple scales:
- Sample Selection: MaxMin sampling of 10,000 representative points
- Filtration: Ripser-based persistence computation
- Feature Types: H₀ (components), H₁ (loops), H₂ (voids)
- Persistence Diagrams: Birth-death visualization of topological features
- Caching: Optimized computation for repeated analysis
3. Graph Structure Analysis
Comprehensive analysis of the resulting semantic graph:
- Pathfinding: BFS algorithms for semantic distance measurement
- Component Analysis: Connected component identification and characterization
- Network Metrics: Centrality measures, clustering coefficients
- Random Walks: Exploration of semantic neighborhoods
- Bridge Detection: Identification of multi-contextual tokens
Technical Implementation
Core Components
Embedding Processing (embedding.py)
class TokenGraph:
def __init__(self, embeddings, tokens):
self.mapper = MapperAlgorithm(
umap_params={'n_components': 2, 'n_neighbors': 15},
cover_params={'n_cubes': 40, 'overlap': 0.55}
)
self.graph = self.mapper.fit_transform(embeddings)
Topological Analysis (homology.py)
def compute_persistence(embeddings, sample_size=10000):
samples = maxmin_subsample(embeddings, sample_size)
diagrams = ripser(samples, maxdim=2)
return analyze_features(diagrams)
Interactive Visualization (visualize_samples.py)
def create_dashboard():
return plotly_3d_scatter(
base_points=token_embeddings,
h1_features=loop_features,
h2_features=void_features,
controls=['persistence_threshold', 'feature_toggle']
)
Key Discoveries
Semantic Organization
- Hierarchical Clustering: Related concepts form distinct regions
- Bridge Tokens: Multi-contextual words connect semantic domains
- Topological Persistence: Stable semantic structures across scales
Graph Properties
- 47 Connected Components: Distinct semantic regions
- Small World Structure: Short paths between related concepts
- Scale-Free Properties: Hub tokens with high connectivity
Topological Features
- Loops (H₁): Circular semantic relationships
- Voids (H₂): Higher-order semantic gaps
- Persistence: Feature stability across filtration scales
Interactive Tools
3D Visualization Dashboard
Advanced Plotly interface featuring:
- Base Embeddings: 50K+ token positions in 3D space
- Topological Features: Color-coded H₁ and H₂ features
- Interactive Controls: Persistence threshold slider
- Hover Information: Token details and topological properties
- Feature Filtering: Toggle visibility of different feature types
Graph Analysis Tools
Comprehensive analysis capabilities:
- Semantic Pathfinding: Find connections between any two tokens
- Component Exploration: Analyze semantic clusters
- Random Walk Generation: Explore semantic neighborhoods
- Network Statistics: Centrality and connectivity metrics
Applications and Insights
Language Understanding
- Semantic Similarity: Topological distance correlates with semantic similarity
- Polysemy Detection: Multi-component tokens indicate multiple meanings
- Conceptual Hierarchies: Persistent features reveal stable semantic structures
Model Interpretability
- Embedding Quality: Topological analysis reveals embedding structure quality
- Training Dynamics: Track how semantic organization emerges during training
- Bias Detection: Identify systematic biases in semantic space organization
Research Applications
- Comparative Analysis: Compare different language models' embedding structures
- Cross-lingual Studies: Analyze semantic organization across languages
- Domain Adaptation: Understand how embeddings adapt to specialized domains
Technical Achievements
Computational Optimization
- MaxMin Sampling: Reduced computation from O(n²) to O(n) complexity
- Caching System: Persistent storage of computed features
- Memory Efficiency: Optimized for large-scale embedding analysis
Visualization Innovation
- Interactive 3D Graphics: Real-time exploration of embedding space
- Multi-scale Analysis: Threshold controls for feature exploration
- Responsive Design: Handles large datasets with smooth interaction
Graph Analysis Framework
- Modular Design: Extensible framework for various graph algorithms
- Efficient Algorithms: Optimized pathfinding and analysis tools
- Export Capabilities: JSON serialization for further analysis
Future Directions
XAI Agent System
Comprehensive 28-week development plan for automated explainable AI:
- Agent Architecture: Multi-modal analysis agents
- Automated Insights: AI-driven pattern discovery
- Interactive Explanations: Natural language explanations of findings
Enhanced Analysis
- Dynamic Embeddings: Track embedding evolution during training
- Multi-model Comparison: Comparative topological analysis
- Task-specific Analysis: Embedding analysis for specific NLP tasks
This project demonstrates how advanced mathematical techniques can provide deep insights into AI system representations, creating new tools for understanding and interpreting language models.