用三维神经网络高效逼近解析函数和L^p函数，提升模型效率。
Efficient Approximation to Analytic and $L^p$ functions by Height-Augmented ReLU Networks
- 引入三维增强的ReLU网络架构，优化函数逼近结构。
- 对解析函数实现指数级更快逼近速度，参数更少。
- 首次给出L^p函数高阶逼近的量化非渐近结果，适合理论研究者。
本文解决神经网络逼近理论中的两个根本局限。我们证明，三维网络结构能显著更高效地表示锯齿函数，这是逼近解析函数和L^p函数的核心基础。首先，我们为若干重要类别的解析函数建立了显著改进的指数逼近速率，并提出参数高效的网络设计。其次，首次实现了对一般L^p函数的高阶逼近的定量且非渐近分析。本方法推进了神经网络在基本函数空间中逼近的理论理解，为设计更参数高效的网络提供了理论依据。
原文摘要 · Abstract (English)
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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