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...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.展开更多
基金supported by HKBU 179356,NSFC 12201530 and HKRGC ECS 22302123supported by NSFC/RGC grant N-HKBU214-19 and NORCE Kompetanseoppbygging programsupported by HKRGC GRF grants CityU1101120,CityU11309922,CRF grant C1013-21GF,and HKRGC-NSFC Grant N CityU214/19.
文摘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.