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Phase retrieval aims to recover a signal $x \in \mathbb{C}^{n}$ from itsamplitude measurements $|<x, a_i > |^2$, $i=1,2,...,m$, where $a_i$'s areover-complete basis vectors, with $m$ at least $3n -2$ to ensure a uniquesolution up to a constant phase factor. The quadratic measurement becomeslinear in terms of the rank-one matrix $X = x x^*$. Phase retrieval is then arank-one minimization problem subject to linear constraint for which a convexrelaxation based on trace-norm minimization (PhaseLift) has been extensivelystudied recently. At $m=O(n)$, PhaseLift recovers with high probability therank-one solution. In this paper, we present a precise proxy of rank-onecondition via the difference of trace and Frobenius norms which we callPhaseLiftOff. The associated least squares minimization with this penalty asregularization is equivalent to the rank-one least squares problem under a mildcondition on the measurement noise. Stable recovery error estimates are validat $m=O(n)$ with high probability. Computation of PhaseLiftOff minimization iscarried out by a convergent difference of convex functions algorithm. In ournumerical example, $a_i$'s are Gaussian distributed. Numerical results showthat PhaseLiftOff outperforms PhaseLift and its nonconvex variant(log-determinant regularization), and successfully recovers signals near thetheoretical lower limit on the number of measurements without the noise.

PhaseLiftOff: an Accurate and Stable Phase Retrieval Method Based on Difference of Trace and Frobenius Norms