Orthogonal AMP

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.