Basis Pursuit and Linear Programming Equivalence: A Performance Comparison in Sparse Signal Recovery

Basis pursuit (BP) with $\ell_{1}$-norm criterion received much attention in the past. One of its obvious applications is the discrete-time sparse signal acquisition. In this work, two alternative forms of the BP optimization are presented. Both are intended to perform the same task as the BP but are expressed as linear programming (LP) frameworks. The performance of the LP expressions, which are equivalent to the BP, is observed and then compared to that given by the typical BP. It is found that the error performance of the equivalent BP methods in terms of LP is the same as that of the BP algorithm. One of the BP-equivalent LP problems takes the same computational time as the BP, while another lasts longer in computation. In the same manner, the first BP-equivalent LP problem consumes nearly the same amount of required memory as the BP, whereas another occupies significantly more memory space during the computation.