Decomposing The Dark Matter of Sparse Autoencoders

TL;DR

This paper investigates the unexplained variance ('dark matter') in sparse autoencoders for language models, revealing that much of it is linearly predictable and that nonlinear components are fundamentally different, affecting model performance.

Contribution

It demonstrates that a significant portion of SAE dark matter is linearly predictable from initial activations and introduces methods to analyze and reduce nonlinear error components.

Findings

About half of SAE error can be linearly predicted from initial activations.

Larger SAEs struggle with the same contexts as smaller ones, showing predictable scaling.

Nonlinear error contains fewer learned features and impacts downstream performance.

Abstract

Sparse autoencoders (SAEs) are a promising technique for decomposing language model activations into interpretable linear features. However, current SAEs fall short of completely explaining model performance, resulting in "dark matter": unexplained variance in activations. This work investigates dark matter as an object of study in its own right. Surprisingly, we find that much of SAE dark matter -- about half of the error vector itself and >90% of its norm -- can be linearly predicted from the initial activation vector. Additionally, we find that the scaling behavior of SAE error norms at a per token level is remarkably predictable: larger SAEs mostly struggle to reconstruct the same contexts as smaller SAEs. We build on the linear representation hypothesis to propose models of activations that might lead to these observations. These insights imply that the part of the SAE error vector that cannot be linearly predicted ("nonlinear" error) might be fundamentally different from the linearly predictable component. To validate this hypothesis, we empirically analyze nonlinear SAE error and show that 1) it contains fewer not yet learned features, 2) SAEs trained on it are quantitatively worse, and 3) it is responsible for a proportional amount of the downstream increase in cross entropy loss when SAE activations are inserted into the model. Finally, we examine two methods to reduce nonlinear SAE error: inference time gradient pursuit, which leads to a very slight decrease in nonlinear error, and linear transformations from earlier layer SAE outputs, which leads to a larger reduction.