On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary

Among the many ways to model signals, a recent approach that draws considerable attention is sparse representation modeling. In this model, the signal is assumed to be generated as a random linear combination of a few atoms from a prespecified dictionary. In this work, two Bayesian denoising algorithms are analyzed for this model-the maximum a posteriori probability (MAP) and the minimum-mean-squared-error (MMSE) estimators, both under the assumption that the dictionary is unitary. It is well known that both these estimators lead to a scalar shrinkage on the transformed coefficients, albeit with a different response curve. We derive explicit expressions for the estimation-error for these two estimators. Upper bounds on these errors are developed and tied to the expected error of the so-called oracle estimator, for which the support is assumed to be known. This analysis establishes a worst-case gain-factor between the MAP/MMSE estimation errors and that of the oracle.