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.