How Optimality Structures Sparse Dictionaries: A Theory for Understanding SAE Representations

• URL: http://arxiv.org/abs/2606.02385v1
• Date: Mon, 01 Jun 2026 15:34:34 GMT
• Title: How Optimality Structures Sparse Dictionaries: A Theory for Understanding SAE Representations
• Abstract summary: Sparse Autoencoders (SAEs) have found success parsing neural representations into interpretable concepts.<n>Here, we ask simply what properties any dictionary learning optimum must satisfy.<n>We use these constraints to explain a range of observed SAE behaviours.
• Score: 0.8122270502556375
• License: http://creativecommons.org/licenses/by/4.0/
• 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 & 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 & 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.

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