The success of deep convolutional networks on on tasks involving high-dimensional data such as images or audio suggests that they are able to efficiently approximate certain classes of functions that are not cursed by dimensionality. In this paper, we study this theoretically and empirically through the lens of kernel methods, by considering multi-layer convolutional kernels, which have achieved good empirical performance on standard vision datasets, and provide theoretical descriptions of over-parameterized convolutional networks in certain regimes. We find that while expressive kernels operating on input patches are important at the first layer, simpler polynomial kernels can suffice in higher layers for good performance. For such simplified models, we provide a precise functional description of the RKHS and its regularization properties, highlighting the role of depth for capturing interactions between different parts of the input signal, and the role of pooling for encouraging smooth dependence on the global or relative positions of such parts.

本研究通过核方法的角度对卷积核网络进行了研究，发现其RKHS由补丁之间的交互项的加性模型组成，其范数通过汇聚层促进这些项之间的空间相似性，并提供了泛化界，以说明池化和补丁如何提高样本复杂度保证。

基于内核视角的深度卷积模型近似和学习