  Both theoretical analysis and empirical evidence confirm that the approximate
message passing (AMP) algorithm can be interpreted as recursively solving a
signal denoising problem: at each AMP iteration, one observes a Gaussian noise
perturbed original signal. Retrieving the signal amounts to a successive noise
cancellation until the noise variance decreases to a satisfactory level. In
this paper we incorporate the Stein's unbiased risk estimate (SURE) based
parametric denoiser with the AMP framework and propose the novel parametric
SURE-AMP algorithm. At each parametric SURE-AMP iteration, the denoiser is
adaptively optimized within the parametric class by minimizing SURE, which
depends purely on the noisy observation. In this manner, the parametric
SURE-AMP is guaranteed with the best-in-class recovery and convergence rate. If
the parameter family includes the families of the mimimum mean squared error
(MMSE) estimators, we are able to achieve the Bayesian optimal AMP performance
without knowing the signal prior. In the paper, we resort to the linear
parameterization of the SURE based denoiser and propose three different kernel
families as the base functions. Numerical simulations with the
Bernoulli-Gaussian, $k$-dense and Student's-t signals demonstrate that the
parametric SURE-AMP does not only achieve the state-of-the-art recovery but
also runs more than 20 times faster than the EM-GM-GAMP algorithm.
