  We propose a new method that uses deep learning techniques to accelerate the
popular alternating direction method of multipliers (ADMM) solution for inverse
problems. The ADMM updates consist of a proximity operator, a least squares
regression that includes a big matrix inversion, and an explicit solution for
updating the dual variables. Typically, inner loops are required to solve the
first two sub-minimization problems due to the intractability of the prior and
the matrix inversion. To avoid such drawbacks or limitations, we propose an
inner-loop free update rule with two pre-trained deep convolutional
architectures. More specifically, we learn a conditional denoising auto-encoder
which imposes an implicit data-dependent prior/regularization on ground-truth
in the first sub-minimization problem. This design follows an empirical
Bayesian strategy, leading to so-called amortized inference. For matrix
inversion in the second sub-problem, we learn a convolutional neural network to
approximate the matrix inversion, i.e., the inverse mapping is learned by
feeding the input through the learned forward network. Note that training this
neural network does not require ground-truth or measurements, i.e., it is
data-independent. Extensive experiments on both synthetic data and real
datasets demonstrate the efficiency and accuracy of the proposed method
compared with the conventional ADMM solution using inner loops for solving
inverse problems.
