Title: The Geometry and Topology of Circuits: the Manifolds of Modular Addition
Primary area: interpretability and explainable AI
Keywords: mechanistic interpretability, representation learning, geometry, topology, manifolds, universality, platonic representation hypothesis

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 the Clock and the Pizza circuits 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.

TL;DR: We quantitatively and qualitatively show that the manifolds learned by neural networks trained on modular addition are universally the entire input space manifold or projections of it.