Two-Dimensional Deep ReLU CNN Approximation for Korobov Functions: A Constructive Approach
Information and Engineering College, Dalian University（大连大学信息与工程学院）；School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学数学学院及上海当代应用数学重点实验室）；School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学学院）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学学院）
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.
Department of Applied Mathematics, Hong Kong Polytechnic University（应用数学系，香港理工大学）；Department of Mathematics, National University of Singapore（数学系，新加坡国立大学）；Department of Mathematics and Statistics, Old Dominion University（数学与统计学系，老 Dominion 大学）
