  Recent work has proposed the linear representation hypothesis: that language
models perform computation by manipulating one-dimensional representations of
concepts ("features") in activation space. In contrast, we explore whether some
language model representations may be inherently multi-dimensional. We begin by
developing a rigorous definition of irreducible multi-dimensional features
based on whether they can be decomposed into either independent or
non-co-occurring lower-dimensional features. Motivated by these definitions, we
design a scalable method that uses sparse autoencoders to automatically find
multi-dimensional features in GPT-2 and Mistral 7B. These auto-discovered
features include strikingly interpretable examples, e.g. circular features
representing days of the week and months of the year. We identify tasks where
these exact circles are used to solve computational problems involving modular
arithmetic in days of the week and months of the year. Finally, we provide
evidence that these circular features are indeed the fundamental unit of
computation in these tasks with intervention experiments on Mistral 7B and
Llama 3 8B, and we find further circular representations by breaking down the
hidden states for these tasks into interpretable components.
