  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.
