Phase retrieval for sparse signals

The aim of this paper is to build up the theoretical framework for the recovery of sparse signals from the magnitude of the measurements. We first investigate the minimal number of measurements for the success of the recovery of sparse signals from the magnitude of samples. We completely settle the minimality question for the real case and give a bound for the complex case. We then study the recovery performance of the ℓ1ℓ1 minimization for the sparse phase retrieval problem. In particular, we present the null space property which, to our knowledge, is the first sufficient and necessary condition for the success of ℓ1ℓ1 minimization for k-sparse phase retrieval.