How to use real-valued sparse recovery algorithms for complex-valued sparse recovery?

Finding the sparse solution of an underdetermined system of linear equations (the so called sparse recovery problem) has been extensively studied in the last decade because of its applications in many different areas. So, there are now many sparse recovery algorithms (and program codes) available. However, most of these algorithms have been developed for real-valued systems. This paper discusses an approach for using available real-valued algorithms (or program codes) to solve complex-valued problems, too. The basic idea is to convert the complex-valued problem to an equivalent real-valued problem and solve this new real-valued problem using any real-valued sparse recovery algorithm. Theoretical guarantees for the success of this approach will be discussed, too. On the other hand, a widely used sparse recovery idea is finding the minimum ℓ
¹
norm solution. For real-valued systems, this idea requires to solve a linear programming (LP) problem, but for complex-valued systems it needs to solve a second-order cone programming (SOCP) problem, which demands more computational load. However, based on the approach of this paper, the complex case can also be solved by linear programming, although the theoretical guarantee for finding the sparse solution is more limited.