Efficient Approximation to Analytic and $L^p$ functions by Height-Augmented ReLU Networks

• URL: http://arxiv.org/abs/2603.11128v1
• Date: Wed, 11 Mar 2026 15:38:57 GMT
• Title: Efficient Approximation to Analytic and $L^p$ functions by Height-Augmented ReLU Networks
• Abstract summary: This work addresses two fundamental limitations in neural network approximation theory.<n>A three-dimensional network architecture enables a significantly more efficient representation of sawtooth functions.<n>For the first time, we derive a quantitative and non-asymptotic approximation of high orders for general $Lp$ functions.
• Score: 34.433370923614966
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This work addresses two fundamental limitations in neural network approximation theory. We demonstrate that a three-dimensional network architecture enables a significantly more efficient representation of sawtooth functions, which serves as the cornerstone in the approximation of analytic and $L^p$ functions. First, we establish substantially improved exponential approximation rates for several important classes of analytic functions and offer a parameter-efficient network design. Second, for the first time, we derive a quantitative and non-asymptotic approximation of high orders for general $L^p$ functions. Our techniques advance the theoretical understanding of the neural network approximation in fundamental function spaces and offer a theoretically grounded pathway for designing more parameter-efficient networks.

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