From Flat to Hierarchical : Extracting Sparse Representations with Matching Pursuit
Valérie Costa, Thomas Fel, Ekdeep S Lubana, Bahareh Tolooshams, Demba Ba
Motivated by the hypothesis that neural network representations encode abstract, interpretable features as linearly accessible and approximately orthogonal directions, sparse autoencoders (SAEs) have become a popular tool in the interpretability literature. However, recent work has revealed properties of model representations that fall outside this hypothesis, showing evidence of hierarchical, nonlinear, and multi-dimensional features. This mismatch raises a fundamental question: can standard SAEs recover meaningful features when the latent geometry of representations deviates from linear or quasi-orthogonal assumptions? If not, what would it mean to design an architecture whose inductive bias aligns with the actual structure observed in modern representations? To explore this, we adopt a construction-based approach and introduce MP-SAE as a direct implementation of a revised structural assumption: conditional orthogonality, which has emerged from recent empirical studies. By reinterpreting the Matching Pursuit (MP) algorithm from sparse coding through this new lens, we develop an SAE that unrolls its encoder into a sequence of residual-guided steps, enabling it to capture hierarchical and nonlinearly accessible features. Comparing this architecture with existing SAEs across synthetic and natural data settings, we find that (i) hierarchical concepts induce conditionally orthogonal features, which standard SAEs fail to capture accurately, and (ii) the nonlinear encoding mechanism of MP-SAE recovers highly meaningful features. This allows us to uncover shared structure in seemingly divergent representation spaces, such as those found in vision-language models, and challenges the assumption that useful features are solely linearly accessible. We also show that the sequential encoder of MP-SAE provides an additional benefit: adaptive sparsity during inference, which may be of independent interest. Ultimately, our work reflects a broader principle: interpretability should begin with the observed structure of neural representations, allowing appropriate methods to emerge from inductive biases that are aligned with that structure.