THE SAMPLING COMPLEXITY ON NONCONVEX SPARSE PHASE RETRIEVAL PROBLEM

This paper discusses the k-sparse complex signal recovery from quadratic measurements via the 4-minimization model, where 0 < p < 1. We establish the 4 restricted isometry property over simultaneously low-rank and sparse matrices, which is a weaker restricted isometry property to guarantee the successful recovery in the 4 case. The main result is to demonstrate that Lp-minimization can recover complex k-sparse signals from m > k+pklog(n/k) complex Gaussian quadratic measurements with high probability. The resulting sufficient condition is met by fewer measurements for smaller p and reaches m> k when p turns to zero. Furthermore, an iteratively-reweighted algorithm is proposed. Numerical experiments also demonstrate that 4 minimization with 0 < p <1 performs better than L1 minimization.