Towards a mathematical theory of superposition

Abstract:We develop a mathematical theory of superposition in neural networks using tools from frame theory and compressed sensing. In our model, a sparse binary vector \(x\) of active features is encoded through an overcomplete dictionary \(W\), and feature recovery is performed by applying \(\operatorname{ReLU}(W^\top W x+b)\) with an appropriate bias vector \(b\). We prove several recovery theorems for this model. In the random-support setting, we establish high-probability support recovery for nearly tight, low-coherence dictionaries, with guarantees when the expected sparsity is up to order \(d/\log n\). In the worst-case support setting, we give a sharp and computable criterion for which sparsity levels permit support recovery. We apply this criterion to Gaussian random matrices and equiangular tight frames. For real equiangular tight frames with \(n>d+1\), we determine the exact recovery threshold in terms of the coherence. The proof of this result for real equiangular tight frames relies on a novel characterization---which should be of independent interest to frame theorists---of the distribution of signs in the Gram matrix.

Machine Learning (stat.ML); Information Theory (cs.IT); Machine Learning (cs.LG); Combinatorics (math.CO)

arXiv:2608.27540 [stat.ML]

(or arXiv:2608.27540v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2608.27540 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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