PHASELIFTOFF: AN ACCURATE AND STABLE PHASE RETRIEVAL METHOD BASED ON DIFFERENCE OF TRACE AND FROBENIUS NORMS

摘要

Phase retrieval aims to recover a signal x is an element of C-n from its amplitude measurements vertical bar|< x,a(i)>vertical bar(2), i = 1,2,..., m, where a(i)'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 = xx*. 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 the 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.