Interpretability-vs-kurtosis/skew correlation analysis
Sparse Autoencoders Find Highly Interpretable Features in Language Models — introduced
An analysis correlating a dictionary feature's autointerpretability score with statistical moments of its activation distribution, finding weak positive correlations (0.19-0.24) with skew and kurtosis respectively. Used to argue that more heavy-tailed features are more interpretable, and to explain why ICA -- which explicitly maximizes non-Gaussianity -- is the strongest of the non-SAE baselines.
Some learned features consistently score higher on automated interpretability tests than others, and it would be easy to shrug that off as noise — except a statistical pattern explains a meaningful share of it. Features whose activations spike rarely and sharply, rather than varying smoothly, tend to score better, matching a theoretical prediction about what a real, cleanly-recovered feature should look like against interference from other overlapping ones.
The page works through what these two statistical measures capture, the correlations found across layers and dictionary sizes, and what this analysis is used to explain, from why one classical baseline outperforms its peers to why a particular way of sampling text for scoring was chosen in the first place.
Testing a theoretical link against real dictionary features
This analysis, run in Appendix C.4, asks a question the rest of the paper only gestures at: is there a measurable statistical reason some dictionary features score higher on the Autointerpretability score than others? It correlates each feature's score against several moments of its activation distribution across all layers and dictionary sizes studied, and finds the answer is not "features that fire more" or "features that fire more variably," but specifically features whose activations are heavy-tailed (sparse-autoencoders, §"C.4 INTERPRETABILITY SCORES CORRELATE WITH KURTOSIS AND SKEW OF ACTIVATION", p. 15). The paper offers an intuition for why that pattern should hold at all: interference from other features sharing a direction under Superposition should be roughly Gaussian by the central limit theorem, so a feature whose activations stand out from that Gaussian background is more likely to be a real, cleanly-recovered feature than noise (sparse-autoencoders, §"C.4 INTERPRETABILITY SCORES CORRELATE WITH KURTOSIS AND SKEW OF ACTIVATION", p. 15).
Skew and kurtosis: measuring how a distribution deviates from Gaussian
Skewness and kurtosis are the third and fourth standardized moments of a distribution, measuring respectively its asymmetry and the heaviness of its tails relative to a normal distribution: $$\mathrm{skew}(X) = \mathbb{E}\!\left[\left(\frac{X-\mu}{\sigma}\right)^{3}\right], \qquad \mathrm{kurt}(X) = \mathbb{E}\!\left[\left(\frac{X-\mu}{\sigma}\right)^{4}\right].$$ A Gaussian has skew 0 and kurtosis 3 (or excess kurtosis 0); large positive values mean a distribution with occasional extreme spikes rather than smoothly varying values, exactly the pattern a feature would show if it stayed near zero except for rare, sharp, meaningful activations. Table 2 reports correlations with score of -0.09 for mean and 0.02 for variance -- both negligible -- against 0.20 for skew and 0.15 for kurtosis (sparse-autoencoders, §"C.4 INTERPRETABILITY SCORES CORRELATE WITH KURTOSIS AND SKEW OF ACTIVATION", p. 16); the main text quotes slightly different figures for the same relationship, 0.19 and 0.24 respectively (sparse-autoencoders, §"C.4 INTERPRETABILITY SCORES CORRELATE WITH KURTOSIS AND SKEW OF ACTIVATION", p. 15).
What the correlation is used to explain
The paper motivates the analysis by citing Qu et al. (2019)'s theoretical result that searching for sparse overcomplete dictionaries can be reformulated as searching for directions that maximize the $\ell_4$ norm (L4-norm maximization (sparse dictionary search reformulation)), a quantity closely tied to kurtosis; the correlation found here is offered as an empirical fingerprint of that theory holding up on real features from Sparse dictionary learning / sparse coding (sparse-autoencoders, §"C.4 INTERPRETABILITY SCORES CORRELATE WITH KURTOSIS AND SKEW OF ACTIVATION", p. 15). It also supplies the paper's explanation for why Independent Component Analysis (ICA) -- a technique that explicitly maximizes non-Gaussianity -- is by far the strongest of the non-dictionary-learning baselines: it is optimizing for the same heavy-tailedness this analysis shows real dictionary features share (sparse-autoencoders, §"C.4 INTERPRETABILITY SCORES CORRELATE WITH KURTOSIS AND SKEW OF ACTIVATION", p. 16). And it retroactively justifies preferring Top-and-random vs. random-only interpretability scoring over the stricter random-only alternative: if features really are heavy-tailed, a small random sample should be expected to underestimate their true score.