Compressed Computation is (probably) not Computation in Superposition

A new analysis challenges the Compressed Computation (CC) model as a valid example of computation in superposition, arguing that an unintended mathematical artifact, not superposition, drives its performance gains [1]. The study, submitted to arXiv on June 12, 2026, by Stefan Heimersheim, examines the CC toy model introduced by Braun et al. in 2025 [1]. The model appeared to compute 100 ReLU functions using only 50 neurons, achieving a loss lower than what representing just 50 functions would predict [1]. Heimersheim's work demonstrates that this apparent efficiency stems from a mixing matrix embedded in the model's noisy residual stream, which inadvertently mixes input data into the labels [1]. When the training objective is split into a ReLU term and a mixing term, performance gains scale directly with the magnitude of the mixing matrix and disappear entirely when the matrix is removed [1]. The learned neuron directions concentrate in the subspace associated with the top 50 eigenvalues of that mixing matrix, indicating the mixing term governs the solution [1]. A semi-non-negative matrix factorization (SNMF) baseline, derived solely from the mixing matrix, reproduces the qualitative loss profile and outperforms prior baselines, though it does not fully match the trained model [1]. These findings suggest CC is not a suitable toy model of computation in superposition [1]. The paper was posted on arXiv, an open-access repository that hosts over two million e-prints and receives about 24,000 submissions per month as of late 2024 [9]. The repository, founded in 1991, serves as a primary distribution channel for preprints in physics, mathematics, and computer science [9]. The research appears under the Machine Learning category, a field closely tied to large language models, which are machine learning models with many parameters trained on vast amounts of text [11]. The study's conclusion adds a cautionary note to interpretability research, where toy models are used to isolate hypothesized mechanisms in neural networks [1].