Convex recovery from interferometric measurements

This note formulates a deterministic recovery result for vectors x from quadratic measurements of the form (Ax)i overline(Ax)j for some left-invertible A. Recovery is exact, or stable in the noisy case, when the couples (i,j) are chosen as edges of a well-connected graph. One possible way of obtaining the solution is as a feasible point of a simple semidefinite program. Furthermore, we show how the proportionality constant in the error estimate depends on the spectral gap of a data-weighted graph Laplacian. Such quadratic measurements have found applications in phase retrieval, angular synchronization, and more recently interferometric waveform inversion.