PolySAE: Modeling Feature Interactions in Sparse Autoencoders via Polynomial Decoding

TL;DR

PolySAE extends sparse autoencoders with polynomial decoding to model feature interactions, improving interpretability and capturing compositional structure in neural representations without relying on feature co-occurrence.

Contribution

Introduces PolySAE, a novel autoencoder variant with polynomial decoding that models feature interactions efficiently while maintaining interpretability.

Findings

Achieves 8% average improvement in probing F1 across models.

Captures feature interactions with minimal parameter overhead.

Learns interaction weights independent of feature co-occurrence.

Abstract

Sparse autoencoders (SAEs) have emerged as a promising method for interpreting neural network representations by decomposing activations into sparse combinations of dictionary atoms. However, SAEs assume that 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 approximately 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, such as morphological binding and phrasal composition, largely independent of surface statistics.