A Unified Theory of Sparse Dictionary Learning in Mechanistic Interpretability: Piecewise Biconvexity and Spurious Minima

As AI models achieve remarkable capabilities across diverse domains, understanding what representations they learn and how they encode concepts has become increasingly important for both scientific progress and trustworthy deployment. Recent works in mechanistic interpretability have widely reported that neural networks represent meaningful concepts as linear directions in their representation spaces and often encode diverse concepts in superposition. Various sparse dictionary learning (SDL) methods, including sparse autoencoders, transcoders, and crosscoders, are utilized to address this by training auxiliary models with sparsity constraints to disentangle these superposed concepts into monosemantic features. These methods are the backbone of modern mechanistic interpretability, yet in practice they consistently produce polysemantic features, feature absorption, and dead neurons, with very limited theoretical understanding of why these phenomena occur. Existing theoretical work is limited to tied-weight sparse autoencoders, leaving the broader family of SDL methods without formal grounding. We develop the first unified theoretical framework that casts all major SDL variants as a single piecewise biconvex optimization problem, and characterize its global solution set, non-identifiability, and spurious optima. This analysis yields principled explanations for feature absorption and dead neurons. To expose these pathologies under full ground-truth access, we introduce the Linear Representation Bench. Guided by our theory, we propose feature anchoring, a novel technique that restores SDL identifiability, substantially improving feature recovery across synthetic benchmarks and real neural representations.