A Dominant Diffuse Phase in the Sparse Autoencoder Phase Diagram

This paper investigates the behavior of sparse autoencoders (SAEs) when tasked with recovering a known synthetic dictionary of features. Specifically, it addresses the MAIS-O43 open problem, which seeks to map the phase diagram of SAEs—identifying when they successfully recover true features versus when they merge them—as a function of nesting fraction, sparsity penalty, and dictionary size.

The author implemented the MAIS-O43 protocol, which involves training SAEs on a synthetic dataset where features are organized into parent-child hierarchical relationships. The study evaluated 200 independently initialized models across a grid of parameters using a full-batch optimizer, followed by a larger-scale robustness check involving 3,300 additional fits using standard minibatch Adam. The study employed strict quantitative criteria for feature recovery, merging, and splitting to ensure objective evaluation.

Contrary to theoretical expectations of recovery or merging, the experiments revealed a dominant diffuse phase. In this state, SAEs achieve near-perfect reconstruction of the input data, yet the learned dictionary atoms remain largely unaligned with the true features (median best cosine similarity typically between 0.5 and 0.7). Furthermore, the learned sparse codes are significantly denser than the ground truth. This diffuse phase is highly reproducible, appearing across all tested initializations and grid cells. Crucially, the study demonstrates that this behavior is not an artifact of under-training or the specific optimizer used, as it persists even when the training budget is doubled or when standard minibatch training is employed.

These findings suggest a fundamental gap between the behavior of theoretical global minimizers of sparse-coding objectives and the practical behavior of trained, amortized SAEs. The results serve as a cautionary tale for mechanistic interpretability: high-quality reconstruction error does not guarantee that an SAE has successfully recovered the true, interpretable features of a neural network. This highlights the need for more robust validation methods beyond simple reconstruction metrics.