Unveiling bias compensation in turbo-based algorithms for (discrete) compressed sensing

Abstract

Abstract-In Compressed Sensing, a real-valued sparse vector has to be recovered from an underdetermined system of linear equations. In many applications, however, the elements of the sparse vector are drawn from a finite set. Adapted algorithms incorporating this additional knowledge are required for the discrete-valued setup. In this paper, turbo-based algorithms for both cases are elucidated and analyzed from a communications engineering perspective, leading to a deeper understanding of the algorithm. In particular, we gain the intriguing insight that the calculation of extrinsic values is equal to the unbiasing of a biased estimate and present an improved algorithm.

Bias Compensation in Iterative Soft-Feedback Algorithms With Applicationto (Discrete) Compressed Sensing

“…In order to ensure convergence, all algorithms perform 50 iterations. Besides IMS and (•)uIMS, also the results for noise-based unbiasing with average variances (TMS [14]) and for the BAMP algorithm [18] are shown.…”

“…, in the noise-based case. Note that the latter equations equal the ones used in [13,14,15] without any justification, in particular not the above given interpretation.…”

“…where, again using E XN {X B N } = σ 2 eB , the scaling factor is calculated by [12,14] (14) gives the unbiased estimate 3 for x…”

“…Due to the sparsity constraint and the discrete alphabet, the problem of estimating x based on (19) is non-convex. Different algorithms for the approximate solution of the problem are available in the literature; for a detailed discussion thereon, cf., e.g., [13,14].…”

“…Note that the TMS algorithm [14] (strongly related to OAMP or VAMP [15,17]) is similar to nuIMS, however using average variances instead of individual ones; thus, it does not benefit from the information about the reliability of the particular elements as does uIMS.…”