On the geometry and topology of representations: the manifolds of modular addition
This resolves a debate in interpretability for neural networks, showing that architectural differences may not lead to fundamentally different learned circuits, which is incremental but clarifies foundational understanding.
The paper tackled the problem of whether different neural architectures implement distinct algorithms for modular addition, showing that both uniform and trainable attention architectures implement the same algorithm via topologically and geometrically equivalent representations, with statistical analysis across hundreds of circuits demonstrating similarity.
The Clock and Pizza interpretations, associated with architectures differing in either uniform or learnable attention, were introduced to argue that different architectural designs can yield distinct circuits for modular addition. In this work, we show that this is not the case, and that both uniform attention and trainable attention architectures implement the same algorithm via topologically and geometrically equivalent representations. Our methodology goes beyond the interpretation of individual neurons and weights. Instead, we identify all of the neurons corresponding to each learned representation and then study the collective group of neurons as one entity. This method reveals that each learned representation is a manifold that we can study utilizing tools from topology. Based on this insight, we can statistically analyze the learned representations across hundreds of circuits to demonstrate the similarity between learned modular addition circuits that arise naturally from common deep learning paradigms.