On the Theoretical Foundation of Sparse Dictionary Learning in Mechanistic Interpretability

Abstract

As AI models achieve remarkable capabilities across diverse domains, understanding what representations they learn and how they process information has become increasingly important for both scientific progress and trustworthy deployment. Recent works in mechanistic interpretability have shown that neural networks represent meaningful concepts as directions in their representation spaces and often encode diverse concepts in superposition. Various sparse dictionary learning (SDL) methods, including sparse autoencoders, transcoders, and crosscoders, address this by training auxiliary models with sparsity constraints to disentangle these superposed concepts into monosemantic features. These methods have demonstrated remarkable empirical success but have limited theoretical understanding. Existing theoretical work is limited to sparse autoencoders with tied-weight constraints, leaving the broader family of SDL methods without formal grounding. In this work, we develop the first unified theoretical framework considering SDL as one optimization problem. We demonstrate how diverse methods instantiate the theoretical framework and provide rigorous analysis of the optimization landscape. We provide novel theoretical explanations for empirically observed phenomena, including feature absorption and dead neurons. We design the Linear Representation Bench, a benchmark that strictly follows the Linear Representation Hypothesis, to evaluate SDL methods with fully accessible ground-truth features. Motivated by our theory and findings, we develop feature achoring, a novel technique applicable for all SDL methods, to enhance their feature recovery capabilities.