  Approximate message passing (AMP) is a low-cost iterative signal recovery
algorithm for linear system models. When the system transform matrix has
independent identically distributed (IID) Gaussian entries, the performance of
AMP can be asymptotically characterized by a simple scalar recursion called
state evolution (SE). However, SE may become unreliable for other matrix
ensembles, especially for ill-conditioned ones. This imposes limits on the
applications of AMP.
  In this paper, we propose an orthogonal AMP (OAMP) algorithm based on
de-correlated linear estimation (LE) and divergence-free non-linear estimation
(NLE). The Onsager term in standard AMP vanishes as a result of the
divergence-free constraint on NLE. We develop an SE procedure for OAMP and show
numerically that the SE for OAMP is accurate for general unitarily-invariant
matrices, including IID Gaussian matrices and partial orthogonal matrices. We
further derive optimized options for OAMP and show that the corresponding SE
fixed point coincides with the optimal performance obtained via the replica
method. Our numerical results demonstrate that OAMP can be advantageous over
AMP, especially for ill-conditioned matrices
