Learning with convolution and pooling operations in kernel methods
This provides theoretical insights into hierarchical convolutional kernels for image classification, but it is incremental as it focuses on a stylized setting rather than practical applications.
The paper tackles the challenge of understanding how convolution and pooling operations affect approximation and generalization in kernel methods, by characterizing the RKHS and computing sharp asymptotics of generalization error for single-layer convolutional kernels on uniformly distributed image pixels, showing gains in sample complexity from locality and translation invariance.
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 their success is that these architectures 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 characterize exactly the RKHS of kernels composed of single layers of convolution, pooling, and downsampling operations. We use this characterization to compute sharp asymptotics of the generalization error for any given function in high-dimension. In particular, we quantify the gain in sample complexity brought by enforcing locality with the convolution operation and approximate translation invariance with average pooling. Notably, these results provide a precise description of how convolution and pooling operations trade off approximation with generalization power in one layer convolutional kernels.