SoftSAE: Dynamic Top-K Selection for Adaptive Sparse Autoencoders

Recently, sparse autoencoders (SAEs) have emerged as a promising technique for interpreting activations in foundation models by disentangling features into a sparse set of concepts. However, identifying the optimal level of sparsity for each neuron remains challenging in practice: excessive sparsity can lead to poor reconstruction, whereas insufficient sparsity may harm interpretability. While existing activation functions such as ReLU and TopK provide certain sparsity guarantees, they typically require additional sparsity regularization or cherry-picked hyperparameters. We show in this paper that dynamically sparse attention mechanisms using sparsemax can bridge this trade-off, due to their ability to determine the activation numbers in a data-dependent manner. Specifically, we first explore a new class of SAEs based on the cross-attention architecture with the latent features as queries and the learnable dictionary as the key and value matrices. To encourage sparse pattern learning, we employ a sparsemax-based attention strategy that automatically infers a sparse set of elements according to the complexity of each neuron, resulting in a more flexible and general activation function. Through comprehensive evaluation and visualization, we show that our approach successfully achieves lower reconstruction loss while producing high-quality concepts, particularly in top-n classification tasks.

Sparse autoencoders (SAEs) have become a leading tool for interpreting the representations of vision foundation models, decomposing their polysemantic activations into a larger set of sparse, more monosemantic features. The Top-$k$ SAE, a now-standard variant, enforces sparsity architecturally through its activation function, retaining only the $k$ most active latents per input. Because it was designed precisely to avoid the $\ell_1$ penalty used by earlier SAEs and its known drawbacks, it has not been combined with an explicit sparsity regularizer. Yet the Top-$k$ SAE retains limitations of its own, and we hypothesize that a sparsity penalty acting before the selection could sharpen each latent's selectivity and make the code more interpretable, without reintroducing the drawbacks of the $\ell_1$ penalty. We introduce two sparsity regularizers compatible with the Top-$k$ architecture, both acting on the activations before the Top-$k$ selection: an $\ell_1$ penalty on the unselected (off-support) units, and a scale-invariant $\ell_1/\ell_2$-ratio penalty that concentrates the code onto fewer effective units. Both penalties are applied only to the batch-active units, those selected by the Top-$k$ operator at least once within the batch. Across two datasets, three vision foundation models, and a range of $k$, both regularizers consistently improve monosemanticity at no cost to reconstruction quality. The $\ell_1/\ell_2$ penalty further concentrates information into fewer latents, making reconstruction more robust to the inference-time choice of $k$ and improving small-budget linear probing. Our central finding is that hard architectural sparsity and soft sparsity regularization are complementary rather than mutually exclusive.

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.