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

Abstract

Sparse Autoencoders (SAEs) have proven effective at decomposing neural representations into interpretable concepts, thereby laying the groundwork for greater model interpretability and control. Yet, the precise nature of what SAEs extract—and the scientific inferences we can legitimately derive from them—remains ambiguous. While empirical evidence confirms that SAEs successfully learn interpretable features, theoretical frameworks detailing the specific properties a "concept" must possess to be extracted by an SAE are currently lacking.

Previous research on identifiability has investigated the conditions under which sparse coding can recover ground-truth features. However, these studies often rely on simplified data-generating models, such as those involving sparse independent features, which fail to accurately approximate the complex representations of large language models that SAEs are typically trained on. Moving beyond such data-generating assumptions, this work investigates the fundamental properties that any dictionary learning optimum must satisfy.

Specifically, we expand upon local optimality analyses (Gribonval & Schnass, 2010) to address the nonnegative joint-optimization problem approximated by standard SAEs. This extension allows us to derive constraints that link optimal SAE features to their underlying distributions. We apply these constraints to elucidate several observed SAE phenomena, including hierarchical splitting and absorption, residual structures, and the presence of dense antipodal features. Each of these behaviors illustrates how the interaction between L1 regularization, nonnegativity constraints, and the data shapes optimal dictionaries.

Furthermore, we introduce a novel convex problem involving large dictionaries and examine the limit where atoms per data point become abundant. Ultimately, our goal is to distinguish between model assumptions and unexpected empirical observations, enabling us to extract more insight from the successes of SAEs and establish guiding principles for the development of their successors.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC