PROMP: A sparse recovery approach to lattice-valued signals

Optimizing Sparsity over Lattices and Semigroups

“…The ℓ 0 -norm minimization problem over reals is central in the theory of the classical compressed sensing, where a linear programming relaxation provides a guaranteed approximation [8,9]. Support minimization for solutions to Diophantine equations is relevant for the theory of compressed sensing for discrete-valued signals [11,12,17]. There is still little understanding of discrete signals in the compressed sensing paradigm, despite the fact that there are many applications in which the signal is known to have discrete-valued entries, for instance, in wireless communication [22] and the theory of error-correcting codes [7].…”

Recovery of Binary Sparse Signals From Compressed Linear Measurements via Polynomial Optimization

“…To fill this gap, in the last years, novel CS strategies have been proposed, which exploit the knowledge of A. These strategies are shown to significantly improve the performance with respect to algorithms unaware of the discrete structure, see, e.g., [18,11,14] and the references therein.…”

“…CS strategies tailored for BCS generally require less measurements than classical CS techniques. Different approaches are proposed, e.g., Bayesian models in [31], bi-partite graph models in [25], 0 minimization in [23], ∞ minimization in [1], 1 minimization in [29,2], greedy methods in [11], and non-convex models in [13].…”

Iterative Signal Detection for Large-Scale GSM-MIMO Systems

“…The steps with highest complexity in the proposed algorithm are the s-update step (14), which involves an t t N N × matrix inversion (although it is only computed in the beginning of the algorithm), and the x-update step (15) (for the VAC-ADMM algorithm version). In Table I we present the worst-case complexities (i.e., no earlier termination of the algorithm occurs) in terms of real-valued floating point operations (flops) of the proposed algorithms, as well as of MLD, OB-MMSE [9], BPDN [13] and sMMP [16] (T is the number of child candidates expanded at each iteration of sMMP).…”

“…For example, in [12] the authors applied and evaluated the basis pursuit denoising (BPDN) formulation from [13] to the detection of GSM-MIMO signals. However, directly applying conventional CS algorithm to problems defined over discrete sets, such as in GSM-MIMO, will have a performance which is far from optimal unless knowledge of the discrete nature of the signal is directly exploited inside the reconstruction method [14] [15]. Therefore in [16], a greedy algorithm named multipath matching pursuit with slicing (sMMP) was presented which combines the use of an inner integer slicing step with the adoption of multiple promising candidates for minimizing the residual.…”