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Phase Retrieval for Sparse Signals

Abstract

The aim of this paper is to build up the theoretical framework for the recovery of sparse signals from the magnitude of the measurement. We first investigate the minimal number of measurements for the success of the recovery of sparse signals without the phase information. We completely settle the minimality question for the real case and give a lower bound for the complex case. We then study the recovery performance of the β„“1\ell_1 minimization. In particular, we present the null space property which, to our knowledge, is the first sufficient and necessary condition for the success of β„“1\ell_1 minimization for kk-sparse phase retrievable.Comment: 14 page

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