This paper considers the reconstruction of non-uniformly sparse signals from noisy linear observations. By non-uniformly sparse, we mean that the signal coefficients can be partitioned into subsets that differ in the rate at which the coefficients tend to be active (i.e., nonzero). Inspired by recent work of Donoho, Maleki, and Montanari, we design a minimax-optimal approximate message passing (AMP) algorithm and we analyze it using a state evolution (SE) formalism that applies in the limit of very large problem dimensions. For the noiseless case, the SE formalism implies a phase transition curve (PTC) that bisects the admissible region of the sparsity-undersampling plane into two sub-regions: one where perfect recovery is very likely, and one where it is very unlikely. The PTC depends on the ratios of the activity rates and the relative sizes of the coefficient subsets. For the noisy case, we show that the same PTC also bisects the admissible region of the sparsity-undersampling plane into two sub-regions: one where the noise sensitivity remains finite and characterizable, and one where it becomes infinite (as the problem dimensions increase). Furthermore, we derive the formal mean-squared error (MSE) for (sparsity,undersampling) pairs in the region below the PTC. Numerical results suggest that the MSE predicted by the SE formalism closely matches the empirical MSE throughout the admissible region of the sparsity-undersampling plane, so long as the dimensions of the problem are adequately large.1
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