Approximation theory of structured neural networks

由于深度神经网络在处理各种实用领域的大数据方面的强大应用，人们迫切需要深度学习的数学理论。主要的困难在于为特定学习任务设计的网络的结构和体系结构。无论是经典的逼近理论还是最近的RELU神经网络深度逼近理论，都不能应用于RELU神经网络，这是因为处理大维数据(如数万维的自然图像)所强加的结构。本项目旨在研究结构化神经网络的逼近理论。我们计划用深度卷积神经网络、算子学习和谱图网络来建立反卷积的数学理论。

Mathematical theory for deep learning has been desired due to the power applications of deep neural networks to deal with big data in various practical domains. The main difficulty lies in the structures and architectures imposed to networks designed for specific learning tasks. Neither the classical approximation theory nor the recent one for depths of ReLU neural networks can be applied due to the structures imposed for processing large dimensional data such as natural images of tens of thousands of dimensions. This project aims at an approximation theory for structured neural networks. We plan to establish mathematical theories for deconvolution with deep convolutional neural networks, operator learning, and spectral graph networks.