Sparse autoencoders (SAEs) have emerged as a workhorse of modern mechanistic interpretability, but leading SAE approaches with top-$k$ style activation functions lack theoretical grounding for selecting the hyperparameter $k$. SAEs are based on the linear representation hypothesis (LRH), which assumes that the representations of large language models (LLMs) are linearly encoded, and the superposition hypothesis (SH), which states that there can be more features in the model than its dimensionality. We show that, based on the formal definitions of the LRH and SH, the magnitude of sparse feature vectors (the latent representations learned by SAEs of the dense embeddings of LLMs) can be approximated using their corresponding dense vector with a closed-form error bound. To visualize this, we propose the ZF plot, which reveals a previously unknown relationship between LLM hidden embeddings and SAE feature vectors, allowing us to make the first empirical measurement of the extent to which feature vectors of pre-trained SAEs are over- or under-activated for a given input. Correspondingly, we introduce Approximate Feature Activation (AFA), which approximates the magnitude of the ground-truth sparse feature vector, and propose a new evaluation metric derived from AFA to assess the alignment between inputs and activations. We also leverage AFA to introduce a novel SAE architecture, the top-AFA SAE, leading to SAEs that: (a) are more in line with theoretical justifications; and (b) obviate the need to tune SAE sparsity hyperparameters. Finally, we empirically demonstrate that top-AFA SAEs achieve reconstruction loss comparable to that of state-of-the-art top-k SAEs, without requiring the hyperparameter $k$ to be tuned. Our code is available at: https://github.com/SewoongLee/top-afa-sae.

本研究针对当前稀疏自编码器（SAE）方法在超参数选择方面缺乏理论基础的问题，提出了一种新颖的方法，通过近似特征激活来评估和设计SAE。研究发现，提出的近似特征激活（AFA）方法不仅能有效测量稀疏特征向量的大小，还可引入全新SAE架构top-AFA SAE，从而避免手动调整稀疏性超参数且在重建损失上表现优异。

通过近正交性的近似评估与设计稀疏自编码器