SUPER: Sparse signals with unknown phases efficiently recovered

Compressive phase retrieval algorithms attempt to reconstruct a “sparse high-dimensional vector” from its “low-dimensional intensity measurements”. Suppose x is any length-n input vector over ℂ with exactly k non-zero entries, and A is an m × n (k <; m ≪ n) phase measurement matrix over ℂ. The decoder is handed m “intensity measurements” (|A₁x|, ..., |A_mx|) (corresponding to component-wise absolute values of the linear measurement Ax) - here A_i´s correspond to the rows of the measurement matrix A. In this work, we present a class of measurement matrices A, and a corresponding decoding algorithm that we call SUPER, which can reconstruct x up to a global phase from intensity measurements. The SUPER algorithm is the first to simultaneously have the following properties: (a) it requires only O(k) (order-optimal) measurements, (b) the computational complexity of decoding is O(k log k) (near order-optimal) arithmetic operations, (c) it succeeds with high probability over the design of A. Our results hold for all k ∈ {1, 2, ..., n}.