Introduction of Deep ADMM-Net

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

Deep ADMM-Net for FastMRI

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.

ADMM solver
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:

Deep ADMM-Net

  • 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