  Approximate message passing (AMP) is an effective iterative sparse recovery
algorithm for linear system models. Its performance is characterized by the
state evolution (SE) which is a simple scalar recursion. However, depending on
a measurement matrix ensemble, AMP may face a convergence problem. To avoid
this problem, orthogonal AMP (OAMP), which uses de-correlation linear
estimation and divergence-free non-linear estimation, was proposed by Ma and
Ping. They also provide the SE analysis for OAMP. In their SE analysis, the
following two assumptions were made: (i) The estimated vector of the
de-correlation linear estimator consists of i.i.d. zero-mean Gaussian entries
independent of the vector to be estimated and (ii) the estimated vector of the
divergence-free non-linear estimator consists of i.i.d. entries independent of
the measurement matrix and the noise vector. In this paper, we derive a simple
scalar recursion to characterize iterative sparse recovery algorithms with
divergence-free estimators without such assumptions of independence of messages
by using the generating functional analysis (GFA), which allows us to study the
dynamics by an exact way in the large system limit.
