On minimal representations of shallow ReLU networks

• URL: http://arxiv.org/abs/2108.05643v1
• Date: Thu, 12 Aug 2021 10:22:24 GMT
• Title: On minimal representations of shallow ReLU networks
• Authors: S. Dereich and S. Kassing
• Abstract summary: We show that the minimal representation for $f$ uses either $n$, $n+1$ or $n+2$ neurons. In particular, where the input layer is one-dimensional, minimal representations always use at most $n+1$ neurons but in all higher dimensional settings there are functions for which $n+2$ neurons are needed.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The realization function of a shallow ReLU network is a continuous and piecewise affine function $f:\mathbb R^d\to \mathbb R$, where the domain $\mathbb R^{d}$ is partitioned by a set of $n$ hyperplanes into cells on which $f$ is affine. We show that the minimal representation for $f$ uses either $n$, $n+1$ or $n+2$ neurons and we characterize each of the three cases. In the particular case, where the input layer is one-dimensional, minimal representations always use at most $n+1$ neurons but in all higher dimensional settings there are functions for which $n+2$ neurons are needed. Then we show that the set of minimal networks representing $f$ forms a $C^\infty$-submanifold $M$ and we derive the dimension and the number of connected components of $M$. Additionally, we give a criterion for the hyperplanes that guarantees that all continuous, piecewise affine functions are realization functions of appropriate ReLU networks.

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