Research Paper#Neural Networks, Deep Learning, Modular Arithmetic, Attention Mechanisms, Topology🔬 ResearchAnalyzed: Jan 3, 2026 06:22
Modular Addition Representations: Geometric Equivalence
Published:Dec 31, 2025 18:53
•1 min read
•ArXiv
Analysis
This paper challenges the notion that different attention mechanisms lead to fundamentally different circuits for modular addition in neural networks. It argues that, despite architectural variations, the learned representations are topologically and geometrically equivalent. The methodology focuses on analyzing the collective behavior of neuron groups as manifolds, using topological tools to demonstrate the similarity across various circuits. This suggests a deeper understanding of how neural networks learn and represent mathematical operations.
Key Takeaways
- •Different attention mechanisms (uniform vs. trainable) learn equivalent representations for modular addition.
- •The study uses topological tools to analyze the geometry of learned representations.
- •The findings suggest a common underlying algorithm for modular addition across different architectures.
Reference
“Both uniform attention and trainable attention architectures implement the same algorithm via topologically and geometrically equivalent representations.”