PolySAE: Modeling Feature Interactions in Sparse Autoencoders via Polynomial Decoding

Sparse autoencoders (SAEs) interpret neural network representations by decomposing activations into sparse combinations of dictionary atoms. However, SAEs assume features combine additively through linear reconstruction, an assumption that cannot capture compositional structure: linear models cannot distinguish whether ''Starbucks'' arises from the composition of ''star'' and ''coffee'' features or merely their co-occurrence. This forces SAEs to allocate monolithic features for compound concepts rather than decomposing them into interpretable constituents. We introduce PolySAE, which extends the SAE decoder with higher-order terms to model feature interactions while preserving the linear encoder essential for interpretability. Through low-rank tensor factorization on a shared projection subspace, PolySAE captures pairwise and triple feature interactions with small parameter overhead (3\% on GPT2). Across four language models and three SAE variants, PolySAE achieves an average improvement of $\sim$8\% in probing F1 while maintaining comparable reconstruction error, and produces 2--10$\times$ larger Wasserstein distances between class-conditional feature distributions. Critically, learned interaction weights exhibit negligible correlation with co-occurrence frequency ($r = 0.06$ vs $r = 0.82$ for SAE feature covariance), suggesting that polynomial terms capture compositional structure largely independent of surface statistics. Finally, the learned interaction directions causally steer model outputs toward the corresponding compositional semantics.

As AI language models grow more powerful, understanding their internal workings becomes urgent, both for diagnosing mistakes and for safety. The leading approach is to decompose a model's internal activity into a dictionary of recognizable concepts. But these tools share a hidden limitation: they can only add concepts together, never combine them in richer ways, which is counterintuitive since language doesn't work by addition. "Starbucks" is not simply "star" plus "coffee"; it's a brand name whose meaning emerges from the combination. Existing tools either miss such compositions or store each compound as its own concept, obscuring the actual building blocks of language. We introduce PolySAE, which lets concepts multiply and bind together, much like how prefixes attach to words or how two words form a phrase. The change costs only a 3% increase in size, but unlocks a much richer view of how models build meaning. Across four language models, PolySAE discovers genuine compositions, such as "financial investing" emerging from the suffix "-ing" combined with "stock" and "market", that go beyond mere statistical coincidence. We can further use these learned combinations to steer the model's output, providing a more faithful and controllable window into AI systems.