Sparse Autoencoders: Feature Dictionaries for Mechanistic Interpretability

Let $x \in \mathbb{R}^d$ be an activation vector from one layer of a model, often the residual stream of a transformer.

Encode to sparse latents, decode back to the activation

An SAE maps $x$ into a sparse latent code $z \in \mathbb{R}^m$, usually with $m$ much larger than $d$, and reconstructs it as $\hat x$:

The columns of $W_{\text{dec}}$ form a learned dictionary of feature directions. If $z_j>0$, the $j$th direction contributes $z_j$ times itself to the reconstruction. Subtracting $b_{\text{pre}}$ before encoding and adding it back after decoding follows Bricken et al. (2023).

Reconstruction plus sparsity

The classic objective balances faithfulness and simplicity:

• $\lVert x - \hat x \rVert_2^2$ asks the dictionary to explain the real activation.
• $\lambda \lVert z \rVert_1$ penalizes large and numerous active latents.

The decoder columns are usually constrained to unit norm. Without that, the model could shrink $z$ and grow the columns to escape the penalty. If $\lambda$ is too small, the code is dense. If it is too large, reconstruction worsens and some latents stop firing on any input ("dead" latents). The $L_1$ penalty also pulls every active value toward zero, an effect called shrinkage.

Top-k sparse coding

Gao et al. (2025) replace the $L_1$ penalty with a hard rule that keeps only the $k$ largest latents, the $k$-sparse autoencoder of Makhzani and Frey (2014):

Here $\mathrm{TopK}(a, k)$ keeps the $k$ largest nonnegative activation scores and sets the rest to zero.

Now the number of active latents, $\lVert z\rVert_0$, is set directly to (at most) $k$ instead of being tuned through $\lambda$. Gao et al. report that this removes the shrinkage caused by the $L_1$ penalty and gives lower reconstruction error at the same sparsity. They write the encoder as $z=\mathrm{TopK}(W_{\text{enc}}(x-b_{\text{pre}}))$, without $b_{\text{enc}}$, and apply a ReLU in their code so that activations stay nonnegative.