Sparse but Wrong: Incorrect L0 Leads to Incorrect Features in Sparse Autoencoders

Sparse Autoencoders (SAEs) are widely used for mechanistic interpretability in large language models, yet their formulation assigns each latent feature a single decoder direction, implicitly assuming features to be one-dimensional. We show that this assumption mismatches with the multi-dimensional structure of model features, provably inducing feature splitting through two distinct mechanisms. Geometrically, reconstructing a feature of intrinsic dimension $d_i \ge 2$ to error $\varepsilon$ with single-direction decoders forces a number of atoms that is exponential in $d_i$. From an end-to-end optimization perspective, this splitting is not merely possible but actively preferred. We prove that there exists a continuous path from the true $d_i$-dimensional basis to a strictly lower risk of the $\ell_1$-regularized SAE objective, whose descent directions drive any trained dictionary into that exponential regime. A single coherent feature is therefore fragmented across many near-collinear latents, producing spurious multiplicity and obscuring the intrinsic geometry. Motivated by this, we introduce Subspace-Aware Sparse Autoencoders (SASA), which replace single-vector decoders with learned decoder subspaces, enforce block sparsity via Top-$s$ group gating, and adapt each group's effective rank with a nuclear-norm regularizer. We then show that once the block size satisfies $r \ge d_i$, a single group not only can represent the entire feature slice but is the global minimizer of the SASA objective. This consolidation yields a sample complexity polynomial in $d_i$ rather than exponential -- a decisive advantage given that every training activation costs an LLM forward pass. Empirically, on GPT-2 and Mistral-7B, SASA reduces feature splitting and absorption, improves monosemanticity and interpretability, and matches or exceeds standard SAEs while training on roughly half the token budget.

Sparse autoencoders (SAEs) are proposed to extract numerous features from large language model (LLM) representations, yet explaining these features still relies primarily on external observation. This reliance leads to superficial explanations inferred from observed model behavior and computational inefficiency from collecting such behavioral evidence at scale. We introduce SAEVerbalizer, a framework that injects SAE decoder directions into an LLM's representations and fine-tunes the LLM's downstream layers to generate natural-language explanations of the injected features. Once trained, the resulting verbalizer explains SAE features directly from decoder directions, addressing both limitations. Our experiments show that the learned verbalization capability generalizes to unseen features, transfers across separately trained SAE dictionaries, and, with a lightweight adapter, extends to SAE features from different LLMs. Intervention experiments show that injecting multiple directions yields an explanation combining their meanings, while reversing individual directions produces corresponding meaning shifts.

Sparse Autoencoders (SAEs) have been seen as a promising avenue for exploring the internals of Large Language Models (LLMs) and for steering model output generation. When AxBench - a model steering benchmark - was introduced in Wu et al. (2025), SAEs did not seem to live up to their original hype due to poor steering performance relative to a set of simple baselines. This work serves as a partial rebuttal for Sparse Autoencoders and suggests that the results of Wu et al. (2025) did not do them full justice. We find that Sparse Autoencoders can, in fact, perform close to on par with the reference LoRA performance on the AxBench benchmark, when features are selected and labelled with our supervised pipeline. We also find that our pipeline selects features that are surprisingly causal of their identified labels when using only its interpretability-based components. Lastly, we present evidence that high sparsity (low l0) may not be crucial for successful steering based on interpretability, which is in contrast to the earlier findings in Wang et al. (2025).