Theofanis Karaletsos

D−1\text{Sparsity}[\mathbf{c}]=\frac{\sqrt{D}-\frac{1}{||\mathbf{c}||_{2}}}{\sqrt{D}-1} for a DD-dimensional vector of split fractions 𝐜\mathbf{c} such that ∥𝐜∥1=1\lVert\mathbf{c}\rVert_{1}=1 and where cdc_{d} measures the fraction of times the ddth variable was used in a split (higher →\rightarrow more important). The sparsity score is zero when all features are used equally often. Sparsity increases markedly with both whitening and ICA. Model Concentration (↑\uparrow) Base 0.163 PCA 0.332 PCA + ICA 0.386 PCA + Rand 0.287 Table 4: Concentration of biological variation in the top 25% of features. However, this measure does not specifically show that the information being ignored as a result of the increased sparsity specifically has to do with the distinction between technical and biological variation. To assess this, we measure how well the information useful for predicting the biological effect is concentrated in the top k%k\% of predictors. Denote by 𝐳k%\mathbf{z}_{k\%} the top k%k\% most important features for the prediction of the biological effect yy of interest, and by 𝐳k%′\mathbf{z}_{k\%}^{\prime} the remaining features. Then, the concentration is given by Concentration​[y]=AUROC​[y;𝐳k%]AUROC​[y;𝐳k%′]−1\text{Concentration}[y]=\frac{\text{AUROC}[y;\mathbf{z}_{k\%}]}{\text{AUROC}[y;\mathbf{z}_{k\%}^{\prime}]}-1, i.e., the improvement in predicting perturbation yy from the top features versus the bottom. The concentration increases uniformly with whitening and with ICA (Table 4), even in the case of PLK1 guides where it does not confer a substantial gain in out-of-distribution AUROC. The results are not sensitive to reasonable values of kk (Appendix A.7.4). Interestingly, the results suggest that whitening alone biases the representation toward becoming axis-aligned even without ICA. 5 Discussion We have developed a theory of statistical near-identifiability of neural representations which is applicable to the internal representations of real-world self-supervised models. Notably, in contrast to prior work, our result requires few assumptions on the data-generating process, instead trading these off for assumptions on the model alone, and applies to a broad class of models including supervised learners, next-token predictors, and self-supervised learners. Additionally, we have shown that additional assumptions on the data-generating process can confer an even stronger result: namely, provable structural identifiability of the latent variables which generated the observables. We directly test our theory in real-world, off-the-shelf, pretrained self-supervised models. Furthermore, we leverage our theory to motivate the application of ICA to the latent spaces of self-supervised models and show that it can achieve state-of-the-art disentanglement results, including some of the first disentanglement results for out-of-distribution generalization in real-world data. Limitations & future work The primary limitation of our work is the difficulty in empirically testing the bi-Lipschitz assumptions necessary for our theory. Instead, we test the downstream effects of our theory in four sets of experiments, and offer arguments from prior work which show that common regularization techniques which enable training of practical-scale neural networks (often referred to as “dynamical isometry”, see also Appendix A.4) may lead to this condition. Interestingly, because the local bi-Lipschitz assumption is largely agnostic to data modality and model implementation details, it potentially applies to a broad class of both data-generating processes and models. As such, it may be an interesting lens for studying the phenomenon of cross-model representation convergence (Maiorca et al., 2023; Fumero et al., 2024), which is largely unaddressed by existing theory because existing identifiability results each require different assumptions on the data-generating process for different models (Huh et al., 2024; Reizinger et al., 2025a). Although we echo the calls of Reizinger et al. (2025a) for extensions to the practical regime (e.g. finite samples, imperfect optimization), we do not treat this case here, although it could be an extension of our framework. Additionally, because ours are the first identifiability results that apply to the intermediate layers of transformer-based next-token predictors, they may be useful for the interpretation of these models (Basile et al., 2025). In particular, Liu et al. (2025) show that discrete concept models learned atop last-layer GPT representations render the entire model end-to-end linearly identifiable, and the results of our paper suggest that this technique may work for intermediate-layer representations as well. Acknowledgments This work was supported by the Chan Zuckerberg Initiative (CZI) through the AI Residency Program. We thank CZI for the opportunity to participate in this program and the CZI AI Infrastructure Team for support with the GPU cluster used to train our models. References P. Alestalo, D. A. Trotsenko, and J. Väisälä (2001) Isometric approximation. Israel Journal of Mathematics 125 (1), pp. 61–82. External Links: ISSN 1565-8511, Document, Link Cited by: §A.3.2, §A.3.2, Lemma A.4, Remark. D. M. Ando, C. Y. McLean, and M. Berndl (2017) Improving phenotypic measurements in high-content imaging screens. bioRxiv. External Links: Document, Link, https://www.biorxiv.org/content/early/2017/07/10/161422.full.pdf