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We present a novel compressed sensing recovery algorithm - termed BayesianOptimal Structured Signal Approximate Message Passing (BOSSAMP) - that jointlyexploits the prior distribution and the structured sparsity of a signal thatshall be recovered from noisy linear measurements. Structured sparsity isinherent to group sparse and jointly sparse signals. Our algorithm is based onapproximate message passing that poses a low complexity recovery algorithmwhose Bayesian optimal version allows to specify a prior distribution for eachsignal component. We utilize this feature in order to establish aniteration-wise extrinsic group update step, in which likelihood ratios ofneighboring group elements provide soft information about a specific groupelement. Doing so, the recovery of structured signals is drastically improved.We derive the extrinsic group update step for a sparse binary and a sparseGaussian signal prior, where the nonzero entries are either one or Gaussiandistributed, respectively. We also explain how BOSSAMP is applicable toarbitrary sparse signals. Simulations demonstrate that our approach exhibitssuperior performance compared to the current state of the art, while it retainsa simple iterative implementation with low computational complexity.

Bayesian Optimal Approximate Message Passing to Recover Structured Sparse Signals