Recent empirical work has shown that hierarchical convolutional kernels inspired by convolutional neural networks (CNNs) significantly improve the performance of kernel methods in image classification tasks. A widely accepted explanation for the success of these architectures is that they encode hypothesis classes that are suitable for natural images. However, understanding the precise interplay between approximation and generalization in convolutional architectures remains a challenge. In this paper, we consider the stylized setting of covariates (image pixels) uniformly distributed on the hypercube, and fully characterize the RKHS of kernels composed of single layers of convolution, pooling, and downsampling operations. We then study the gain in sample efficiency of kernel methods using these kernels over standard inner-product kernels. In particular, we show that 1) the convolution layer breaks the curse of dimensionality by restricting the RKHS to `local' functions; 2) local pooling biases learning towards low-frequency functions, which are stable by small translations; 3) downsampling may modify the high-frequency eigenspaces but leaves the low-frequency part approximately unchanged. Notably, our results quantify how choosing an architecture adapted to the target function leads to a large improvement in the sample complexity.

本研究探讨了一层卷积、汇集和降采样操作组成的核的RKHS，并用它来计算高维函数的一般化误差尖锐的渐近值。结果表明，卷积和池化操作在一层卷积核中如何在逼近和泛化能力之间权衡。

卷积和池化在核方法中的学习