I am currently working in Math AI lab, and Deep ADMM Net is one of our main targets recently depends on papers:

• Yang, Y., Sun, J., Li, H., & Xu, Z. (2016). Deep ADMM-Net for compressive sensing MRI
• Yang, Y., Sun, J., Li, H., & Xu, Z. (2017). ADMM-Net: A Deep Learning Approach for Compressive Sensing MRI

## A deep dive into Deep ADMM-Net

#### Compressive Sensing MRI Model and ADMM Algorithm

General CS-MRI Model problem

where $A=P F \in \mathbb{R}^{N^{\prime} \times N}$ is a measurement matrix, $P \in \mathbb{R}^{N^{\prime} \times N}$ is a under-sampling matrix and $F$ is a Fourier transform.

By introducing $z=\left\{z_{1}, z_{2}, \cdots, z_{L}\right\}$,
the problem equals to

and its augmented Lagrangian function is

where $α = {α_l}$ are Lagrangian multipliers and $ρ = {ρ_l}$ are penalty parameters. ADMM alternatively optimizes ${x, z, α}$ by solving the following three subproblems:

• Reconstruction layer
• Convolution layer
• Nonlinear Transform layer
• Multiplier update layer

### Network Training

Given pairs of training data, the loss between the network output and ground truth is deﬁned as

#### Gradient Computation by Backpropagation over Data Flow Graph

• Multiplier update layer
• Nonlinear Transform layer
• Convolution layer
• Reconstruction layer