Sparse Autoencoders: Pairwise Interactions Lift Rank Fidelity to 0.7790

Sparse autoencoders (SAEs) have become a core tool of mechanistic interpretability: they decompose model activations into sparse features that decode linearly. A new paper, featured in Hugging Face Daily Papers, tests whether the behavioral effects of those features also combine linearly. The answer is: partly. The authors introduce a metric called SAE Additivity Residuals (SAR) and show, across 18 model-behavior combinations, that SAE coordinates are more additive than a random basis. Pairwise interactions are still needed, though: in a combined steering experiment, rank fidelity rose from 0.5874 to 0.7790, and ordering error dropped from 0.1444 to 0.0606.

Linear decomposition, nonlinear consequences

The SAE decomposition itself is additive by construction: an activation equals the sum of its feature contributions. But the model's later layers sit between the activation and the behavior, and those layers are nonlinear. If you switch on two features at once, the effect on the output does not have to equal the sum of each feature's individual effect. This matters in practice. When a researcher explains a behavior with a set of features, or steers a model by mixing in several features at once, they implicitly count on addition. Without a check, that assumption remains a hypothesis.

What SAR actually measures

SAR estimates the share of behavioral variation that remains unexplained above a chosen interaction order. Order 1 is a purely additive model, order 2 adds pairwise terms, and so on. When feature subsets are sampled independently, the population SAR is exactly equal to the normalized energy of interactions above that order. It also bounds the coalition ordering error, meaning the error in ranking sets of features by the strength of their effect, for the optimal low-order approximation.

To separate the contribution of the SAE itself from that of the model's underlying computation, the authors compared the trained coordinates with random rotations of the same chosen subspace that preserve the sum. If additivity were determined only by the model's computation, there would be no difference. There is a difference.

The results in numbers

• In all 18 settings (model plus behavior), the trained SAE coordinates have a lower first-order residual than the randomly rotated ones.
• Pairwise terms improve coalition ordering quality on held-out data in every setting.
• Pairwise terms outperform parameter-matched models built on randomly chosen third-order terms. The gain comes from the structure of the pairs, not from the parameter count.
• In the combined steering experiment, rank fidelity rose from 0.5874 to 0.7790.
• In the same experiment, ordering error fell from 0.1444 to 0.0606, a drop of more than half.

What practitioners should do

The takeaway for people working with SAEs is twofold. First, additive descriptions of behavior through SAE features are justified noticeably better than in a random basis. For quick estimates and initial analysis, a linear model of effects may be enough. Second, if you need "faithful" joint explanations or reliable steering with several features at once, individual contributions are not enough. Pairwise corrections give a tangible gain in ranking accuracy, while triplet terms on random samples do not.

In practice, this means that when selecting feature combinations for steering, you should measure SAR and check pairwise effects rather than trust the sum of single-feature effects. There is a caveat: the abstract does not reveal the specific models, the SAE dictionary sizes, or the types of behaviors, so you should be careful about transferring these numbers to your own stack. The code is published in the OpenSAE/SAE-Behavioral-Compositionality repository, so you can run the metric on your own SAEs and see how large the residual is for your features.