On the geometry and topology of representations: the manifolds of modular addition

TL;DR

This paper demonstrates that different neural architectures for modular addition, whether uniform or learnable attention, produce topologically equivalent representations, revealing a shared underlying geometric structure.

Contribution

It introduces a topological and geometric analysis method to compare neural representations, showing that different architectures implement the same algorithm through equivalent manifolds.

Findings

Uniform and trainable attention architectures are topologically equivalent.

Learned representations form manifolds that can be statistically compared.

Different architectures naturally develop similar modular addition circuits.

Abstract

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.