A null-space-based genetic algorithm for constrained l1 minimization

Abstract

Compressive sensing (CS) theory enables linking the sensing effort, that is, the volume of data that a sensor produces, to the amount of information this data conveys, rather than to the desired bandwidth, as prescribed by classical sampling theory. As a consequence, in the typical CS scenario, one ends up with a set of m measurements and the objective is to recover a signal whose dimensionality is n » m, typically under the assumption of being sparse. Recovering the sparsest solution that satisfies the measurements is an NP-hard problem and a common workaround is relaxing it to a linearly-constrained li minimization. In this work we introduce a novel algorithm for solving this problem that exhibits the structure of a genetic algorithm, but fully operates in null-space domain. This allows reducing the dimensionality of the chromosomes to the minimum, i. e., n-m. Crossover follows a deterministic scheme, with adjustable number of parents and children per pairing. Furthermore, mutations are not random, but guided along the direction of the negative gradient of the fitness function. Numerical simulation revealed that the proposed algorithm performs better than comparable alternatives in terms of reconstruction error when the number of iterations is to be kept very low. This, together with its high parallelization potential, paves the way for faster CS reconstruction.