An Approach to Complex Bayesian-optimal Approximate Message Passing

In this work we aim to solve the compressed sensing problem for the case of a complex unknown vector by utilizing the Bayesian-optimal structured signal approximate message passing (BOSSAMP) algorithm on the jointly sparse real and imaginary parts of the unknown. By introducing a latent activity variable, BOSSAMP separates the tasks of activity detection and value estimation to overcome the problem of detecting different supports in the real and imaginary parts. We complement the recovery algorithm by two novel support detection schemes that utilize the updated auxiliary variables of BOSSAMP. Simulations show the superiority of our proposed method against approximate message passing (AMP) and its Bayesian-optimal sibling (BAMP), both in mean squared error and support detection performance

Bayesian Optimal Approximate Message Passing to Recover Structured Sparse Signals

We present a novel compressed sensing recovery algorithm - termed Bayesian Optimal Structured Signal Approximate Message Passing (BOSSAMP) - that jointly exploits the prior distribution and the structured sparsity of a signal that shall be recovered from noisy linear measurements. Structured sparsity is inherent to group sparse and jointly sparse signals. Our algorithm is based on approximate message passing that poses a low complexity recovery algorithm whose Bayesian optimal version allows to specify a prior distribution for each signal component. We utilize this feature in order to establish an iteration-wise extrinsic group update step, in which likelihood ratios of neighboring group elements provide soft information about a specific group element. Doing so, the recovery of structured signals is drastically improved. We derive the extrinsic group update step for a sparse binary and a sparse Gaussian signal prior, where the nonzero entries are either one or Gaussian distributed, respectively. We also explain how BOSSAMP is applicable to arbitrary sparse signals. Simulations demonstrate that our approach exhibits superior performance compared to the current state of the art, while it retains a simple iterative implementation with low computational complexity.Comment: 13 pages, 9 figures, 1 table. Submitted to IEEE Transactions on Signal Processin