  We consider the estimation of an i.i.d. (possibly non-Gaussian) vector $\xbf
\in \R^n$ from measurements $\ybf \in \R^m$ obtained by a general cascade model
consisting of a known linear transform followed by a probabilistic
componentwise (possibly nonlinear) measurement channel. A novel method, called
adaptive generalized approximate message passing (Adaptive GAMP), that enables
joint learning of the statistics of the prior and measurement channel along
with estimation of the unknown vector $\xbf$ is presented. The proposed
algorithm is a generalization of a recently-developed EM-GAMP that uses
expectation-maximization (EM) iterations where the posteriors in the E-steps
are computed via approximate message passing. The methodology can be applied to
a large class of learning problems including the learning of sparse priors in
compressed sensing or identification of linear-nonlinear cascade models in
dynamical systems and neural spiking processes. We prove that for large i.i.d.
Gaussian transform matrices the asymptotic componentwise behavior of the
adaptive GAMP algorithm is predicted by a simple set of scalar state evolution
equations. In addition, we show that when a certain maximum-likelihood
estimation can be performed in each step, the adaptive GAMP method can yield
asymptotically consistent parameter estimates, which implies that the algorithm
achieves a reconstruction quality equivalent to the oracle algorithm that knows
the correct parameter values. Remarkably, this result applies to essentially
arbitrary parametrizations of the unknown distributions, including ones that
are nonlinear and non-Gaussian. The adaptive GAMP methodology thus provides a
systematic, general and computationally efficient method applicable to a large
range of complex linear-nonlinear models with provable guarantees.
