2014/06/26 by Penghang Yin, Jack Xin, Yin, Penghang +1 · 1 citation
Computer Science · Materials Science · Physics and Astronomy · #Advanced X-ray Imaging Techniques #Electron and X-Ray Spectroscopy Techniques #FOS: Mathematics #Optical measurement and interference techniques #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1406.6761
openalex publication_date 2014/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Phase retrieval aims to recover a signal x ∈ ℂn from its amplitude measurements | |2, i=1,2,...,m, where ai's are over-complete basis vectors, with m at least 3n -2 to ensure a unique solution up to a constant phase factor. The quadratic measurement becomes linear in terms of the rank-one matrix X = x x^*. Phase retrieval is then a rank-one minimization problem subject to linear constraint for which a convex relaxation based on trace-norm minimization (PhaseLift) has been extensively studied recently. At m=O(n), PhaseLift recovers with high probability the rank-one solution. In this paper, we present a precise proxy of rank-one condition via the difference of trace and Frobenius norms which we call PhaseLiftOff. The associated least squares minimization with this penalty as regularization is equivalent to the rank-one least squares problem under a mild condition on the measurement noise. Stable recovery error estimates are valid at m=O(n) with high probability. Computation of PhaseLiftOff minimization is carried out by a convergent difference of convex functions algorithm. In our numerical example, ai's are Gaussian distributed. Numerical results show that PhaseLiftOff outperforms PhaseLift and its nonconvex variant (log-determinant regularization), and successfully recovers signals near the theoretical lower limit on the number of measurements without the noise.