SAR measures how much of a model’s downstream behavior is left unexplained when you treat Sparse Autoencoder (SAE) features as independent additive contributions, and shows that learned SAE coordinates are more additive than random rotations of the same subspace but still need pairwise interaction terms for faithful joint predictions.
Sparse Autoencoder (SAE)s are the current workhorse for interpretability: you decompose a model’s internal activation into a sparse set of human-readable features, then try to build explanations or steer behavior by combining several features at once. The implicit assumption in most of this work is that if you know what each feature does on its own, you can predict what they’ll do together by adding effects up. That assumption is load-bearing for feature circuits, attribution graphs (Marks et al. 2025 (Sparse Feature Circuits)), and multi-feature steering.
The problem: SAE decoders are linear for reconstructing the activation, but after that activation flows through the rest of the network, non-linear layers can make features reinforce, cancel, or gate each other. Nobody had a direct measurement of whether “sum the individual effects” is actually a good approximation for the thing you care about, which is downstream behavior. Prior SAE evaluations like SAEBench score reconstruction, sparsity, and interpretability, but not compositionality of behavioral effects.
The core experiment is a subset-ablation game. For one prompt, pick the top 16 active SAE features at some layer. Each feature contributes a decoded vector to the activation. Now sample random subsets of those 16: keep the chosen contributions, zero out the rest, hold everything else about the activation fixed (including the SAE reconstruction error), then run the rest of the model and record some scalar behavior (say, logit of the correct token). This gives a function g(S) over subsets.
If feature effects were truly additive, g(S) would be well-fit by a sum of per-feature coefficients. The authors fit that first-order model, measure held-out squared error, and normalize by behavioral variance. That ratio is SAR_1, the SAE Additivity Residual at order 1: the fraction of behavioral variance that per-feature effects fail to explain. Add pairwise terms, you get SAR_2. Under uniform random subset sampling, they prove SAR_r equals exactly the normalized energy of feature interactions above order r, so it has a clean interaction-decomposition interpretation rather than being just a regression residual.
To tell apart “the SAE basis is special” from “the downstream computation happens to be nearly linear here,” they compare each SAE basis against sum-preserving random rotations of the same selected subspace: rotate the 16 decoder directions by a random orthogonal matrix that fixes the all-ones vector, so the subspace and the total contribution are unchanged but the coordinate axes are scrambled. If the SAE basis has lower SAR_1 than the rotations, the compositionality is a property of the chosen coordinates, not just of the subspace or the rest of the network.
for prompt in prompts:
N = top_k_sae_features(prompt, k=16)
deltas = [z_i * d_i for i in N] # decoded contributions
h0 = activation(prompt) - sum(deltas) # fixed remainder
for S in sample_subsets(N, m=384):
h_tilde = h0 + sum(deltas[i] for i in S)
g[S] = behavior(run_from_layer(h_tilde, prompt))
fit a1 on 256 subsets; fit a2 (pairs) on same 256
report SAR_1, SAR_2, rank fidelity on held-out 128
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First-order additivity is never exact. Across 57 model-layer-behavior combinations spanning Gemma 3 4B, Mistral 7B, and Llama-3.1-8B-Instruct on tasks like CounterFact and International Olympiad in Informatics, the prompt-median SAR_1 is positive everywhere, while synthetic affine controls sit at numerical zero (confirming the estimator works).
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Learned SAE coordinates beat random rotations in all 18 model-behavior cells tested. Example: Mistral on subject-verb agreement, median SAR_1 is 0.0218 for the SAE versus 0.1196 for rotations. The subspace and downstream network are identical, so coordinate choice itself carries compositional structure.
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Pairs help a lot. Adding pairwise interaction terms improves held-out coalition rank fidelity in all 18 cells, and beats a parameter-matched control that spends the same number of coefficients on randomly chosen third-order terms. That rules out “more parameters = better fit” as the explanation.
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The steering test is the cleanest practical demo. Using 16 published CorrSteer steering directions in Llama and 1,024 random on/off combinations evaluated on MMLU-Pro, pairwise modeling raises rank fidelity from 0.5874 to 0.7790 and cuts behavior-weighted ordering error from 0.1444 to 0.0606. Predicting combined steering outcomes from individual-direction effects alone leaves a lot on the table.
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SAR captures something different from reconstruction. Pooling six SAEs at a fixed Gemma site, SAR correlates positively with sign error and selection regret, but Fraction of Variance Unexplained does not. A better-reconstructing SAE is not automatically a more behaviorally-additive one.
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If you’re building a feature-circuit explanation or combining multiple SAE features for an intervention, treat the “sum of individual effects” prediction as a baseline, not an answer. The paper’s evidence is that pairwise correction is meaningful across every setting they tried, and in steering it moves rank fidelity by ~0.19. Worth fitting a pairwise model on a sample of coalitions before trusting the additive story.
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If you’re choosing between SAEs for a downstream interpretability task that involves combining features, add SAR_1 to your evaluation. The six-SAE comparison shows reconstruction quality (FVU) and behavioral additivity disagree, so picking by reconstruction alone can mislead you about which SAE will support cleanest joint explanations.
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If you’re comparing SAEs to alternative decompositions, the sum-preserving random rotation is a cheap, principled control: it isolates “did the SAE find good axes” from “is this subspace just easy.”
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The pipeline needs internal activation access and the ability to re-run forward from a patched layer. It does not work through a hosted inference API that only exposes logits. Code is released. •
SAR is defined relative to a specific choice of selected features (top-K active, K=16 here), a specific behavioral readout, and a specific intervention distribution (uniform subset sampling at p=0.5). It is not a global property of an SAE dictionary, and the sensitivity table shows SAR_1 shifts as you change K.
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The fitted second-order residual itself has roughly 18% within-prompt coefficient of variation across fresh coalition draws; the comparative conclusions (pairs beat singletons, SAE beats rotations) are far more stable than the absolute numbers.
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The intervention keeps the SAE reconstruction error pinned in the activation. An alternative game that reconstructs from retained features alone would mix in SAE reconstruction quality, which the authors argue would confound the measurement; readers who care about that mode need a different setup.
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Lower SAR_1 at later layers partly reflects having less downstream computation left, not just the SAE basis; the depth comparisons show the same trend in random rotations.