  The generalized approximate message passing (GAMP) algorithm under the
Bayesian setting shows advantage in recovering under-sampled sparse signals
from corrupted observations. Compared to conventional convex optimization
methods, it has a much lower complexity and is computationally tractable. In
the GAMP framework, the sparse signal and the observation are viewed to be
generated according to some pre-specified probability distributions in the
input and output channels. However, the parameters of the distributions are
usually unknown in practice. In this paper, we propose an extended GAMP
algorithm with built-in parameter estimation (PE-GAMP) and present its
empirical convergence analysis. PE-GAMP treats the parameters as unknown random
variables with simple priors and jointly estimates them with the sparse
signals. Compared with Expectation Maximization (EM) based parameter estimation
methods, the proposed PE-GAMP could draw information from the prior
distributions of the parameters to perform parameter estimation. It is also
more robust and much simpler, which enables us to consider more complex signal
distributions apart from the usual Bernoulli-Gaussian (BGm) mixture
distribution. Specifically, the formulations of Bernoulli-Exponential mixture
(BEm) distribution and Laplace distribution are given in this paper. Simulated
noiseless sparse signal recovery experiments demonstrate that the performance
of the proposed PE-GAMP matches the oracle GAMP algorithm. When noise is
present, both the simulated experiments and the real image recovery experiments
show that PE-GAMP is still able to maintain its robustness and outperform EM
based parameter estimation method when the sampling ratio is small.
Additionally, using the BEm formulation of the PE-GAMP, we can successfully
perform non-negative sparse coding of local image patches and provide useful
features for the image classification task.
