Evaluating and Designing Sparse Autoencoders by Approximating Quasi-Orthogonality

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.