Sparse autoencoder features do not simply add up: pairwise terms lift coalition rank fidelity from 0.5874 to 0.7790

A sparse autoencoder (SAE) writes a model activation as sparse, linearly decoded features, and knowing each feature's effect does not mean summing them predicts their joint effect. The study measures this additivity question on downstream behavior rather than on activation reconstruction. Earlier SAE evaluations characterized reconstruction, sparsity and interpretability, but did not say when independently characterized features can be composed into faithful joint explanations or interventions. Across 57 model, layer and behavior combinations the first-order residual median is positive, while affine controls stay at numerical zero, so additivity is not exact under these interventions.

SAE Additivity Residuals (SAR) quantify the behavioral variance unexplained above a chosen interaction order, and under independent subset sampling they connect to explicit feature interactions and coalition ordering error. Linear decoding makes SAE coordinates additive for activation reconstruction, not for downstream effects; later computation can make feature effects reinforce, suppress or depend on one another. Under independent Bernoulli coalition sampling, SAR is exactly the share of interaction energy above that order and bounds the ordering error of the optimal low-order approximation.

To separate compositionality coming from the SAE coordinates from that induced by downstream computation, the study compares learned SAE coordinates with sum-preserving random rotations of the same selected subspace. A rotation keeps the selected subspace, its total contribution and the downstream model fixed, changing only the coordinate decomposition. Learned SAE coordinates have a lower first-order residual median in all 18 model and behavior combinations, for example 0.02179 versus 0.1196 on Mistral-7B subject-verb agreement.

Pairwise terms recover coalition structure that the first-order model misses, improving prediction in both explanation and steering settings. A first-order model uses only individual feature effects and cannot describe interactions between feature pairs. Pairwise models improve rank fidelity in all 18 combinations, raising the average from 0.9782 to 0.9959 and cutting selection regret from 0.0272 to 0.0051, stable across all 20 repeated splits; on 1,024 steering combinations in Llama-3.1 8B, rank fidelity rises from 0.5874 to 0.7790 and ordering error falls from 0.1444 to 0.0606.

The conclusions are limited to the selected feature contributions, the specific behavioral readouts and the specific intervention distribution, and the authors explicitly do not treat them as a global property of an SAE dictionary. The main comparisons use Gemma-3 4B, Mistral-7B and Llama-3.1 8B across simple copying, subject-verb agreement, gendered pronoun preference, CounterFact, IOI and greater-than comparison; the steering experiment uses 16 published CorrSteer features in Llama-3.1 8B and 1,400 MMLU-Pro questions.

The authors note that the fitted second-order residual is not a high-precision estimate of its population value: going from 224 to 256 fitting coalitions changes second-order rank fidelity by 0.14% while the normalized second-order residual changes by about 21%. Steering effects are heterogeneous across domains, and in some domains the second-order held-out residual can exceed the first-order one. Part of the depth trend comes from the remaining downstream computation, random rotations show the same first-order decrease in eight of nine comparisons, and Llama profiles are nonmonotonic near the final sampled layer.