Linear-Size Neural Network Representation of Piecewise Affine Functions in $\mathbb{R}^2$

It is shown that any continuous piecewise affine (CPA) function $\mathbb{R}^2\to\mathbb{R}$ with $p$ pieces can be represented by a ReLU neural network with two hidden layers and $O(p)$ neurons. Unlike prior work, which focused on convex pieces, this analysis considers CPA functions with connected but potentially non-convex pieces.

Publication Details

Journal
ArXiv.org
Published
2025-03-17
Primary Topic
Neural Networks and Applications
Type
article
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article

Linear-Size Neural Network Representation of Piecewise Affine Functions in $\mathbb{R}^2$

ArXiv.org
Neural Networks and Applications
article

Linear-Size Neural Network Representation of Piecewise Affine Functions in $\mathbb{R}^2$

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Abstract

It is shown that any continuous piecewise affine (CPA) function $\mathbb{R}^2\to\mathbb{R}$ with $p$ pieces can be represented by a ReLU neural network with two hidden layers and $O(p)$ neurons. Unlike prior work, which focused on convex pieces, this analysis considers CPA functions with connected but potentially non-convex pieces.

ArXiv.org
Reduced inequalities
Openalex Percentile: Top 74%
Neural Networks and Applications
Open Access PDFOpenAlex
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Linear-Size Neural Network Representation of Piecewise Affine Functions in $\mathbb{R}^2$ · ArXiv.org (2025) | TGRS Research Map | TGRS