Learning dynamics of kernel-based deep neural networks in manifolds

Convolutional neural networks (CNNs) obtain promising results via layered kernel convolution and pooling operations, yet the learning dynamics of the kernel remain obscure. We propose a continuous form to describe kernel-based convolutions through integration in neural manifolds. The status of spatial expression is proposed to analyze the stability of kernel-based CNNs. We divide CNN dynamics into the three stages of unstable vibration, collaborative adjusting, and stabilized fluctuation. According to the system control matrix of the kernel, the kernel-based CNN training proceeds via the unstable and stable status and is verified by numerical experiments.