On the Complexity of Neural Computation in Superposition

TL;DR

This paper establishes theoretical complexity bounds for neural networks computing in superposition, revealing fundamental limits and capabilities that influence model sparsification, capacity, and feature representation.

Contribution

It provides the first lower bounds for superposition computation in neural networks, showing the relationship between neurons, parameters, and features, and offers nearly tight upper bounds for logical operations.

Findings

Neural networks require at least (\u221a{m'     m'}) neurons for superposition features.

Networks can compute logical operations with (     m') neurons and parameters.

Capacity of networks is bounded by (n^2 / n) features, showing a subexponential limit.

Abstract

Superposition, the ability of neural networks to represent more features than neurons, is increasingly seen as key to the efficiency of large models. This paper investigates the theoretical foundations of computing in superposition, establishing complexity bounds for explicit, provably correct algorithms. We present the first lower bounds for a neural network computing in superposition, showing that for a broad class of problems, including permutations and pairwise logical operations, computing $m'$ features in superposition requires at least $\Omega(\sqrt{m' \log m'})$ neurons and $\Omega(m' \log m')$ parameters. This implies an explicit limit on how much one can sparsify or distill a model while preserving its expressibility, and complements empirical scaling laws by implying the first subexponential bound on capacity: a network with $n$ neurons can compute at most $O(n^2 / \log n)$ features. Conversely, we provide a nearly tight constructive upper bound: logical operations like pairwise AND can be computed using $O(\sqrt{m'} \log m')$ neurons and $O(m' \log^2 m')$ parameters. There is thus an exponential gap between the complexity of computing in superposition (the subject of this work) versus merely representing features, which can require as little as $O(\log m')$ neurons based on the Johnson-Lindenstrauss Lemma. Our work analytically establishes that the number of parameters is a good estimator of the number of features a neural network computes.