Equivariant Sparse Autoencoders: Mechanistic Interpretability of Neural Networks on Symmetric Data

Machine learning (ML) models achieve remarkable performance but remain hard to interpret due to their scale and complexity. In particular, their activations entangle many concepts into fewer dimensions, a phenomenon known as superposition. Mechanistic interpretability methods such as sparse autoencoders (SAEs) can disentangle these dense activations into sparse sums of interpretable features, but SAEs suffer from unidentifiability: different explanations can fit the data equally well without necessarily being more interpretable or faithful to the underlying model. We show that this problem is exacerbated by data symmetries such as rotations that are prevalent in scientific domains. We extend the Linear Representation Hypothesis, the theory behind SAEs, to account for symmetries and show on synthetic as well as real-world scientific datasets and models that the resulting Equivariant SAEs can (1) avoid the pitfalls of existing SAEs on symmetric data and (2) discover features more useful for downstream tasks despite worse reconstructions. Our results show that incorporating the correct priors in SAEs can significantly improve their usefulness while highlighting that reconstruction quality can be inversely correlated with feature usefulness under symmetries, cautioning against its use as a key measure of interpretability. Code: https://github.com/ege-erdogan/equivariant-sae