Mathematical Models of Computation in Superposition

TL;DR

This paper introduces mathematical models demonstrating how superposition can be actively used for efficient computation in neural networks, challenging the view that superposition only hinders interpretability.

Contribution

The work constructs neural network models that leverage superposition for efficient circuit emulation, extending to deep networks and providing insights for interpretability.

Findings

Single-layer MLP emulates pairwise AND with superposition using O(m^{2/3}) neurons

Deep networks with error correction layers emulate low-depth circuits of width O(d^{1.5})

Potential applications for interpreting superposition-based neural computation

Abstract

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.