A Numerical DEs Perspective on Unfolded Linearized ADMM Networks for Inverse Problems

Many research works show that the continuous-time Differential Equations (DEs) allow for a better understanding of traditional Alternating Direction Multiplier Methods (ADMMs). And many unfolded algorithms directly inherit the traditional iterations to build deep networks. Although they obtain a faster convergence rate and superior practical performance, there is a lack of an appropriate explanation of the unfolded network architectures. Thus, we attempt to explore the connection between the existing unfolded Linearized ADMM (LADMM) and numerical DEs, and propose efficient unfolded network design schemes. First, we present an unfolded Euler LADMM scheme as a by-product, which originates from the Euler method for solving first-order DEs. Then inspired by the trapezoid method in numerical DEs, we design a new more effective network scheme, called unfolded Trapezoid LADMM scheme. Moreover, we analyze that the Trapezoid LADMM scheme has higher precision than the Euler LADMM scheme. To the best of our knowledge, this is the first work to explore the connection between unfolded ADMMs and numerical DEs with theoretical guarantees. Finally, we instantiate our Euler LADMM and Trapezoid LADMM schemes into ELADMM and TLADMM with the proximal operators, and ELADMM-Net and TLADMM-Net with convolutional neural networks. And extensive experiments show that our algorithms are competitive with state-of-the-art methods.