Kernel methods are powerful tools to capture nonlinear patterns behind given data but often lead to poor performance on complicated tasks compared to convolutional neural networks. The reason is that kernel methods are still shallow and fully connected models, failing to reveal hierarchical features and local interdependencies. In this paper, to acquire hierarchical and local knowledge, we incorporate kernel methods with deep architectures and convolutional operators in a spectral kernel learning framework. Based on the inverse Fourier transform and Rademacher complexity theory, we provide the generalization error bounds for the proposed model and prove that under suitable initialization, deeper networks lead to tighter error bounds. Inspired by theoretical findings, we finally completed the convolutional spectral kernel network (CSKN) with two additional regularizers and an initialization strategy. Extensive ablation results validate the effectiveness of non-stationary spectral kernel, multiple layers, additional regularizers, and the convolutional filters, which coincide with our theoretical findings. We further devise a VGG-type 8-layers CSKN, and it outperforms the existing kernel-based networks and popular CNN models on the medium-sized image classification tasks.

内核方法是捕获给定数据背后的非线性模式的强大工具，但与卷积神经网络相比，通常会导致复杂任务的性能不佳。原因是内核方法仍然是浅层和完全连接的模型，无法揭示层次特征和局部相互依赖关系。在本文中，为了获取分层和局部知识，我们将核方法与深度架构和卷积算子结合到谱核学习框架中。基于逆傅里叶变换和 Rademacher 复杂性理论，我们为所提出的模型提供了泛化误差范围，并证明在适当的初始化下，更深的网络会导致更严格的误差范围。受理论发现的启发，我们最终完成了卷积谱核网络（CSKN）带有两个额外的正则化器和一个初始化策略。广泛的消融结果验证了非平稳谱核、多层、附加正则化器和卷积滤波器的有效性，这与我们的理论发现一致。我们进一步设计了一个 VGG 类型的 8 层CSKN，它在中型图像分类任务上优于现有的基于内核的网络和流行的 CNN 模型。