构建2D深度卷积网络，高效逼近柯罗博夫函数。
Two-Dimensional Deep ReLU CNN Approximation for Korobov Functions: A Constructive Approach
- 用零填充与ReLU激活的多通道卷积层构造网络
- 在连续权值模型下逼近率接近最优
- 理论证明缓解维度灾难，适合函数逼近应用
本文研究二维深度卷积神经网络（2D CNN）在柯罗博夫函数逼近中的能力。考虑包含多通道卷积层（零填充）、ReLU激活及全连接层的2D CNN结构。提出一种完全可构造的方法，用于构建逼近柯罗博夫函数的2D CNN，并对所构造网络的复杂度进行严格分析。结果表明，在连续权值选择模型下，2D CNN可实现近似最优的逼近率，显著缓解了维度灾难问题。该工作为2D CNN提供了坚实的理论基础，并展示了其在函数逼近中的广泛应用潜力。
原文摘要 · Abstract (English)
This paper investigates approximation capabilities of two-dimensional (2D) deep convolutional neural networks (CNNs), with Korobov functions serving as a benchmark. We focus on 2D CNNs, comprising multi-channel convolutional layers with zero-padding and ReLU activations, followed by a fully connected layer. We propose a fully constructive approach for building 2D CNNs to approximate Korobov functions and provide a rigorous analysis of the complexity of the constructed networks. Our results demonstrate that 2D CNNs achieve near-optimal approximation rates under the continuous weight selection model, significantly alleviating the curse of dimensionality. This work provides a solid theoretical foundation for 2D CNNs and illustrates their potential for broader applications in function approximation.
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