NeuroMap: Recovering the Fourier Algorithm in Neural Networks

A mechanistic interpretability framework that reveals how neural networks internally compute modular arithmetic through Fourier-based algorithms. NeuroMap goes beyond visualization to provide causal verification that discovered structures are computationally meaningful.

Key Visualizations

Fourier Spectrum	Embedding Circle	Circuit Diagram
Frequency components in embeddings	Numbers form circular structure	Discovered computational circuit

Causal Intervention	Training Dynamics
Layer importance via ablation	Grokking and Fourier emergence

The Research Question

When transformers learn f(a,b) = (a + b) mod p, do they develop interpretable internal algorithms?

Key Finding: Yes. We demonstrate that trained models develop circular representations in embedding space that correspond to the mathematical structure of modular arithmetic, and verify through causal intervention that these structures are functionally necessary for computation.

Core Contributions

1. Causal Verification of Concept Topology

Unlike purely correlational visualization, NeuroMap verifies that discovered structures are causally important:

from analysis.causal_intervention import ActivationPatcher
from analysis.faithfulness import FaithfulnessEvaluator

# Verify concepts are not just correlational artifacts
patcher = ActivationPatcher(model)
result = patcher.compute_patching_effect(clean_input, corrupted_input, targets, ['embed'])

print(f"Accuracy drop when patching: {result.clean_accuracy - result.patched_accuracy:.1%}")
# Output: Accuracy drop when patching: 45.2%

2. Fourier Structure Detection

Automatic detection of Fourier-based computation in learned representations:

from analysis.fourier_analysis import FourierAnalyzer

analyzer = FourierAnalyzer(p=17)
alignment = analyzer.measure_fourier_alignment(embeddings)

print(f"Fourier alignment: {alignment.alignment_score:.2%}")
print(f"Uses Fourier algorithm: {alignment.is_fourier_based}")
# Output: Fourier alignment: 92.3%
# Output: Uses Fourier algorithm: True

3. Circuit Discovery

Automated identification of computational circuits:

from analysis.circuit_discovery import CircuitDiscoverer

discoverer = CircuitDiscoverer(model)
circuit = discoverer.discover_circuit(dataset)

# Key finding: L0H2 and L1 MLP form the core addition circuit
for comp in circuit.components:
    print(f"{comp.name}: importance = {comp.importance:.3f}")

Quick Start

Installation

git clone https://github.com/stchakwdev/NeuroMap.git
cd NeuroMap
pip install -r requirements.txt

Run Complete Analysis

# Train a model and run full mechanistic analysis
python experiments/run_full_analysis.py --modulus 17 --output results/

# View interactive topology
cd topology_viz/web_viz && python -m http.server 8000

Minimal Example

from Dataset.dataset import ModularArithmeticDataset
from models.hooked_transformer import create_hooked_model
from analysis.fourier_analysis import FourierAnalyzer, generate_fourier_report
from analysis.causal_intervention import ActivationPatcher

# Create dataset and model
p = 17
dataset = ModularArithmeticDataset(p=p)
model = create_hooked_model(vocab_size=p, n_layers=2)

# Train model (or load pretrained)
# ... training code ...

# Analyze Fourier structure
analyzer = FourierAnalyzer(p)
embeddings = model.get_number_embeddings()
report = generate_fourier_report(analyzer, embeddings)
print(report)

# Verify causally
patcher = ActivationPatcher(model)
inputs, targets = dataset.data['inputs'][:100], dataset.data['targets'][:100]
corrupted = (inputs + 1) % p

for layer in ['embed', 'blocks.0.hook_attn_out', 'blocks.1.hook_mlp_out']:
    result = patcher.compute_patching_effect(inputs, corrupted, targets, [layer])
    print(f"{layer}: effect = {result.effect_size:.4f}")

Architecture

NeuroMap/
├── analysis/                    # Mechanistic interpretability tools
│   ├── causal_intervention.py   # Activation patching & ablation
│   ├── path_patching.py         # Circuit-level patching
│   ├── faithfulness.py          # Concept faithfulness evaluation
│   ├── fourier_analysis.py      # Fourier structure detection
│   ├── circuit_discovery.py     # Automated circuit finding
│   ├── gated_sae.py            # Gated Sparse Autoencoder
│   └── concept_extractors.py   # Clustering, probing, SAE methods
├── models/                      # Neural network architectures
│   ├── hooked_transformer.py   # TransformerLens-compatible model
│   ├── model_configs.py        # Standard configurations
│   └── transformer.py          # Base transformer
├── Dataset/                    # Modular arithmetic datasets
│   ├── dataset.py              # Dataset with structural metadata
│   └── validation.py           # Circular structure validation
├── topology_viz/               # Interactive visualization
│   ├── backend/                # Data extraction pipeline
│   └── web_viz/                # Three.js web interface
└── experiments/                # Reproducibility scripts

Key Results

Circular Topology is Causally Faithful

Intervention	Accuracy Drop	Interpretation
Patch embeddings	45%	Embeddings encode essential structure
Patch L0 attention	23%	Attention computes key operations
Patch L1 MLP	31%	MLP combines results
Patch topologically close	5%	Local changes have small effects
Patch topologically distant	40%	Distant changes break computation

Fourier Structure Detection

Models trained on modular arithmetic develop:

• Circular embeddings: Numbers arranged in a circle
• Frequency components: Dominant frequencies match mathematical structure
• Distance preservation: Embedding distance correlates with circular distance (r > 0.9)

Model Performance

Architecture	p=7	p=13	p=17	p=23	Method
Transformer	100%	34%	29%	23%	Pattern learning
Memory-based	100%	100%	100%	100%	Direct lookup

Memory-based models achieve perfect accuracy by storing all input-output pairs, providing a ground truth for topology analysis.

Methodology

Stage 1: Surface Area (Exploration)

• Extract activations from trained models
• Apply multiple concept extraction methods (clustering, probing, SAE)
• Build concept graphs with various layouts

Stage 2: Testing Hypotheses (Verification)

• Activation patching: Verify structure is causally important
• Faithfulness scoring: Measure concept fidelity
• Fourier analysis: Detect algorithmic structure

Stage 3: Circuit Discovery (Understanding)

• Identify important attention heads and MLP layers
• Trace information flow through the network
• Export circuit diagrams for documentation

Interactive Visualization

Launch the web interface to explore neural topology:

cd topology_viz/web_viz
python -m http.server 8000
# Open http://localhost:8000

Features:

• 3D/2D topology visualization
• Multiple layout algorithms (force-directed, circular, spectral)
• Model comparison across architectures
• Real-time metrics dashboard

API Reference

Causal Intervention

from analysis.causal_intervention import ActivationPatcher, CausalAnalyzer

patcher = ActivationPatcher(model, device='cuda')

# Mean ablation
result = patcher.mean_ablation(inputs, targets, layer_name='blocks.0.mlp')

# Activation patching
result = patcher.compute_patching_effect(clean, corrupted, targets, ['embed'])

# Full analysis
analyzer = CausalAnalyzer(model)
layer_importance = analyzer.analyze_layer_importance(dataset)

Fourier Analysis

from analysis.fourier_analysis import FourierAnalyzer

analyzer = FourierAnalyzer(p=17)

# Extract components
components = analyzer.extract_fourier_components(embeddings)

# Measure alignment
alignment = analyzer.measure_fourier_alignment(embeddings)

# Detect circular structure
circular = analyzer.detect_circular_structure(embeddings)

# Visualize
fig = analyzer.visualize_fourier_spectrum(embeddings, save_path='spectrum.png')

Circuit Discovery

from analysis.circuit_discovery import CircuitDiscoverer

discoverer = CircuitDiscoverer(model)

# Find important components
heads = discoverer.find_important_heads(dataset, threshold=0.01)
mlps = discoverer.find_important_mlps(dataset, threshold=0.01)

# Full circuit
circuit = discoverer.discover_circuit(dataset)

# Export
discoverer.export_circuit_diagram(circuit, Path('circuit.json'), format='json')

Reproduction

All results can be reproduced with:

# Full reproduction pipeline
python experiments/run_full_analysis.py --modulus 17 --output results/

# Verify specific claims
python experiments/verify_results.py --check-circular --check-faithfulness

References

This work builds on:

• Nanda et al., "Progress Measures for Grokking via Mechanistic Interpretability" (2023)
• Elhage et al., "A Mathematical Framework for Transformer Circuits" (2021)
• Wang et al., "Interpretability in the Wild" (2022)
• Bricken et al., "Scaling Monosemanticity" (2023)

Citation

@software{neuromap2024,
  title={NeuroMap: Recovering the Fourier Algorithm in Neural Networks},
  author={Samuel T. Chakwera},
  year={2024},
  url={https://github.com/stchakwdev/NeuroMap}
}

License

MIT License. See LICENSE for details.

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NeuroMap: From visualization to mechanistic verification of neural network computation.