深度卷积网络可高效逼近高阶混合函数，突破维度诅咒
Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces
- 用CNN近似高阶Korobov函数，利用稀疏网格基函数的近似表示
- 深度每增加1层，逼近精度提升至(m+1)阶（对数修正）
- 适合研究高维函数逼近与深度学习表达能力的学者
本文研究了使用带ReLU激活函数的深度卷积神经网络（CNN）对高阶Korobov函数在$L_p$范数下的逼近误差。对于每个方向上具有$(m+1)$阶混合导数的目标函数，本文将经典的二阶逼近率改进为$(m+1)$阶（对数因子修正），以网络深度为变量。分析的核心在于利用CNN近似表示高阶稀疏网格基函数。结果表明，CNN的高阶表达能力并未因维度增加而严重退化，即未受维度诅咒的严重影响。
原文摘要 · Abstract (English)
This paper investigates the $L_p$ approximation error for higher order Korobov functions using deep convolutional neural networks (CNNs) with ReLU activation. For target functions having a mixed derivative of order m+1 in each direction, we improve classical approximation rate of second order to (m+1)-th order (modulo a logarithmic factor) in terms of the depth of CNNs. The key ingredient in our analysis is approximate representation of high-order sparse grid basis functions by CNNs. The results suggest that higher order expressivity of CNNs does not severely suffer from the curse of dimensionality.
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