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

Phase retrieval aims to recover a signal $x \in \mathbb{C}^{n}$ from its amplitude measurements $|<x, a_i > |^2$, $i=1,2,...,m$, 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 $X = x x^*$. 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, $a_i$'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.