A recent line of work in mechanistic interpretability has focused on reverse-engineering the computation performed by neural networks trained on the binary operation of finite groups. We investigate the internals of one-hidden-layer neural networks trained on this task, revealing previously unidentified structure and producing a more complete description of such models in a step towards unifying the explanations of previous works (Chughtai et al., 2023; Stander et al., 2024). Notably, these models approximate equivariance in each input argument. We verify that our explanation applies to a large fraction of networks trained on this task by translating it into a compact proof of model performance, a quantitative evaluation of the extent to which we faithfully and concisely explain model internals. In the main text, we focus on the symmetric group S5. For models trained on this group, our explanation yields a guarantee of model accuracy that runs 3x faster than brute force and gives a >=95% accuracy bound for 45% of the models we trained. We were unable to obtain nontrivial non-vacuous accuracy bounds using only explanations from previous works.

Amino acid substitution rate matrices are fundamental to statistical phylogenetics and evolutionary biology. Estimating them typically requires reconstructed trees for massive amounts of aligned proteins, which poses a major computational bottleneck. In this paper, we develop a near-linear time method to estimate these rate matrices from multiple sequence alignments (MSAs) alone, thereby speeding up computation by orders of magnitude. Our method relies on a near-linear time cherry reconstruction algorithm which we call FastCherries and it can be easily applied to MSAs with millions of sequences. On both simulated and real data, we demonstrate the speed and accuracy of our method as applied to the classical model of protein evolution. By leveraging the unprecedented scalability of our method, we develop a new, rich phylogenetic model called SiteRM, which can estimate a general site-specific rate matrix for each column of an MSA. Remarkably, in variant effect prediction for both clinical and deep mutational scanning data in ProteinGym, we show that despite being an independent-sites model, our SiteRM model outperforms large protein language models that learn complex residue-residue interactions between different sites. We attribute our increased performance to conceptual advances in our probabilistic treatment of evolutionary data and our ability to handle extremely large MSAs. We anticipate that our work will have a lasting impact across both statistical phylogenetics and computational variant effect prediction. FastCherries and SiteRM are implemented in the CherryML package https://github.com/songlab-cal/CherryML.