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Superposition -- when a neural network represents more ``features'' than it has dimensions -- seems to pose a serious challenge to mechanistically interpreting current AI systems. Existing theory work studies \emph{representational} superposition, where superposition is only used when passing information through bottlenecks. In this work, we present mathematical models of \emph{computation} in superposition, where superposition is actively helpful for efficiently accomplishing the task. We first construct a task of efficiently emulating a circuit that takes the AND of the $\binom{m}{2}$ pairs of each of $m$ features. We construct a 1-layer MLP that uses superposition to perform this task up to $\varepsilon$-error, where the network only requires $\tilde{O}(m^{\frac{2}{3}})$ neurons, even when the input features are \emph{themselves in superposition}. We generalize this construction to arbitrary sparse boolean circuits of low depth, and then construct ``error correction'' layers that allow deep fully-connected networks of width $d$ to emulate circuits of width $\tilde{O}(d^{1.5})$ and \emph{any} polynomial depth. We conclude by providing some potential applications of our work for interpreting neural networks that implement computation in superposition., Comment: 28 pages, 5 figures. Published at the ICML 2024 Mechanistic Interpretability (MI) Workshop

The deployment of language models brings challenges in generating reliable information, especially when these models are fine-tuned using human preferences. To extract encoded knowledge without (potentially) biased human labels, unsupervised probing techniques like Contrast-Consistent Search (CCS) have been developed (Burns et al., 2022). However, salient but unrelated features in a given dataset can mislead these probes (Farquhar et al., 2023). Addressing this, we propose a cluster normalization method to minimize the impact of such features by clustering and normalizing activations of contrast pairs before applying unsupervised probing techniques. While this approach does not address the issue of differentiating between knowledge in general and simulated knowledge - a major issue in the literature of latent knowledge elicitation (Christiano et al., 2021) - it significantly improves the ability of unsupervised probes to identify the intended knowledge amidst distractions., Comment: 30 pages, 35 figures