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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 4

### Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

### PhaseLiftOff: An accurate and stable phase retrieval method based on difference of trace and Frobenius norms

Pages: 1033 – 1049

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n4.a10

#### Authors

#### Abstract

Phase retrieval aims to recover a signal $x \in \mathbb{C}^n$ from its amplitude measurements ${\vert \langle x, a_i \rangle \rvert}^2 , i=1, 2, \cdots , m, \;$ 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, $a_i$’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.

#### Keywords

phase retrieval, phase lift regularized by trace minus Frobenius norms, equivalent minimization problems, Karush-Kuhn-Tucker conditions, difference of convex functions algorithm, convergence

#### 2010 Mathematics Subject Classification

65K05, 90C22, 94A12

Published 12 March 2015