{"ID":2923582,"CreatedAt":"2026-06-02T04:05:25.881865328Z","UpdatedAt":"2026-06-04T13:12:39.622923895Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.02385","arxiv_id":"2606.02385","title":"How Optimality Structures Sparse Dictionaries: A Theory for Understanding SAE Representations","abstract":"Sparse Autoencoders (SAEs) have found success parsing neural representations into interpretable concepts, providing a basis for understanding and control. However, what exactly SAEs extract, and, correspondingly, the scientific conclusions we can draw from them, are not obvious. Empirically, the proof is in the pudding: SAEs learn interpretable features. Theoretically, we lack a clear account of what properties a 'concept' must satisfy for an SAE to extract it. There has been extensive identifiability work studying the conditions under which sparse coding recovers ground-truth features; however, these approaches tends to focus on simple data-generating models (e.g. sparse independent features) which poorly approximate the internet-swallowing language-model representations on which SAEs are trained. Here, avoiding data-generating models, we ask simply what properties any dictionary learning optimum must satisfy. Concretely, we extend local optimality analyses (Gribonval \u0026 Schnass, 2010) to the nonnegative joint-optimisation problem that vanilla SAEs approximate, and derive constraints relating optimal SAE features to their distributions. We use these constraints to explain a range of observed SAE behaviours - hierarchical splitting \u0026 absorption, the structure of residuals, and dense antipodal features - each reflecting how L1+nonnegativity interact with data to structure optimal dictionaries. Finally, we construct a novel large-dictionary convex problem and explore the wide atom-per-datapoint limit. In sum, we hope to tease model assumptions from unexpected observations, letting us learn more from SAEs' successes and provide principles for designing their successors.","short_abstract":"Sparse Autoencoders (SAEs) have found success parsing neural representations into interpretable concepts, providing a basis for understanding and control. However, what exactly SAEs extract, and, correspondingly, the scientific conclusions we can draw from them, are not obvious. Empirically, the proof is in the pudding...","url_abs":"https://arxiv.org/abs/2606.02385","url_pdf":"https://arxiv.org/pdf/2606.02385v1","authors":"[\"William Dorrell\"]","published":"2026-06-01T15:34:34Z","proceeding":"q-bio.NC","tasks":"[\"q-bio.NC\",\"cs.LG\"]","methods":"[]","has_code":false}