Fast and robust compressive phase retrieval with sparse-graph codes

In this paper, we tackle the compressive phase retrieval problem in the presence of noise. The noisy compressive phase retrieval problem is to recover a K-sparse complex signal s ∈ ℂⁿ, from a set of m noisy quadratic measurements: y_i = |a_i^Hs|² + w_i; where a_i^H ∈ ℂⁿ is the ith row of the measurement matrix A ∈ ℂ^m×n, and w_i is the additive noise to the ith measurement. We consider the regime where K = βn^δ, δ ∈ (0; 1). We use the architecture of PhaseCode algorithm [1], and robustify it using two schemes: the almost-linear scheme and the sublinear scheme. We prove that with high probability, the almost-linear scheme recovers s with sample complexity¹ Θ(K log(n)) and computational complexity Θ(n log(n)), and the sublinear scheme recovers s with sample complexity Θ(K log³(n)) and computational complexity Θ(K log³(n)). To the best of our knowledge, this is the first scheme that achieves sublinear computational complexity for compressive phase retrieval problem. Finally, we provide simulation results that support our theoretical contributions.