Title: Deep Neural Networks Divide and Conquer Dihedral Multiplication
Primary area: interpretability and explainable AI
Keywords: mechanistic interpretability, neural representations, manifold hypothesis, group multiplication, divide and conquer, cayley graph

Abstract: We find multilayer perceptrons and transformers both learn an instantiation of the same divide-and-conquer algorithm and solve dihedral multiplication with logarithmic feature efficiency. Applying principal component analyses to clusters of neurons where neurons have similar activation behavior reveals remarkably clear structure: *the neural representations correspond to Cayley graphs.* It's noteworthy that these Cayley graphs are the low-dimensional manifolds networks are predicted to learn under the manifold hypothesis, and thus each neural representation learned by networks corresponds to a manifold. To our knowledge, this is the first work that gives closed-form equations describing how every point in the dataset maps onto each of the possible neural representations that can be learned.

TL;DR: We show deep neural networks learn different implementations of one divide and conquer algorithm across random seeds.