Sparse phase retrieval using partial nested fourier samplers

The problem of sparse phase retrieval is considered where the measurement vectors are deterministic and Fourier-like. Under some mild assumptions, O(slog N) measurements are proved to be sufficient to recover an s-sparse complex vector of dimension N from its phaseless measurement via convex programming. The key contribution is to show that unlike existing work in sparse phase retrieval, the so-called "collision-free" condition is not needed for the proposed approach, and hence, there is no upper bound on s for which the sparse vector can be recovered. Even for non sparse complex data, the number of measurements needed by this approach almost attains the lower bound conjectured in current literature. The algorithms developed in this work are based on a newly introduced class of Fourier samplers, namely Partial Nested Fourier Samplers, which can naturally avoid the "collision-free" condition by performing a novel decoupling of quadratic terms arising in the phaseless measurements.