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This paper introduces SAE Additivity Residuals (SAR) to quantify the non-additive behavioral compositionality of sparse autoencoder (SAE) features, finding that while SAE coordinates are more compositional than random projections, pairwise interactions are still crucial for accurate behavioral explanations and interventions. By measuring unexplained behavioral variation above a given interaction order, SAR provides a principled way to bound coalition ordering error and assess the degree of additivity in SAE representations. The study demonstrates that incorporating pairwise SAE feature interactions significantly improves rank fidelity and reduces ordering error in behavioral steering experiments, highlighting the necessity of higher-order terms for faithful joint explanations.
SAE features are more compositional than random projections, but pairwise interactions are still essential for accurate behavioral steering.
Sparse autoencoders (SAEs) represent model activations with sparse, linearly decoded features, but the downstream effects of those features need not compose additively. We introduce SAE Additivity Residuals (SAR), which measure the behavioral variation left unexplained above a chosen interaction order. Under independent feature subset sampling, population SAR is exactly the normalized energy of interactions above that order and bounds coalition ordering error for the optimal low order approximation. To distinguish compositionality associated with the SAE representation from that induced by downstream computation, we compare learned SAE coordinates with sum preserving random rotations of the same selected subspace and find lower first order residuals for the learned coordinates across all 18 model and behavior settings. The coordinates are nevertheless not fully additive: pairwise terms improve held out coalition ordering in every setting and outperform parameter matched models using randomly selected third order terms. In a combined steering experiment, pairwise modeling raises rank fidelity from 0.5874 to 0.7790 and reduces ordering error from 0.1444 to 0.0606. Together, these results show that SAE coordinates support comparatively compositional behavioral descriptions while faithful joint explanations and interventions still require interactions beyond individual feature effects. Code is available at https://github.com/OpenSAE/SAE-Behavioral-Compositionality.