Deep neural network is a powerful tool for many tasks. Understanding why it is so successful and providing a mathematical explanation is an important problem and has been one popular research direction in past years. In the literature of mathematical analysis of deep neural networks, a lot of works is dedicated to establishing representation theories. How to make connections between deep neural networks and mathematical algorithms is still under development. In this paper, we give an algorithmic explanation for deep neural networks, especially in their connections with operator splitting. We show that with certain splitting strategies, operator-splitting methods have the same structure as networks. Utilizing this connection and the Potts model for image segmentation, two networks inspired by operator-splitting methods are proposed. The two networks are essentially two operator-splitting algorithms solving the Potts model. Numerical experiments are presented to demonstrate the effectiveness of the proposed networks.

翻译：深度神经网络是解决众多任务的有力工具。理解其成功背后的原因并提供数学解释，是过去数年来的一个重要研究方向，也是热门研究课题之一。在深度神经网络数学分析的相关文献中，大量工作致力于建立表示理论。然而，如何将深度神经网络与数学算法联系起来仍处于发展阶段。本文为深度神经网络提供了算法层面的解释，特别关注其与算子分裂方法的联系。我们证明，通过特定的分裂策略，算子分裂方法与网络具有相同的结构。利用这一联系以及用于图像分割的Potts模型，我们提出了两种受算子分裂方法启发的网络。这两种网络本质上是求解Potts模型的两种算子分裂算法。通过数值实验验证了所提网络的有效性。