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## Wednesday, January 04, 2017

### PhaseMax: Convex Phase Retrieval via Basis Pursuit - implementation -

Christoph just sent me the following:

Hey Igor,

Happy New Year!

Tom Goldstein and I created a website for our recent convex phase retrieval method called PhaseMax that avoids lifting (unlike PhaseLift or PhaseCut):

The website contains links to our arXiv paper and to four other papers that are related to PhaseMax. There is also example code that implements PhaseMax using our solver FASTA.

It would be great if you could feature this on your blog.

Thanks a lot!
Christoph
Christoph Studer
Assistant Professor
School of ECE, Rhodes Hall 331
Cornell University
Ithaca, NY 14853, USA
Thanks Christoph ! Here is the paper: PhaseMax: Convex Phase Retrieval via Basis Pursuit by Tom Goldstein, Christoph Studer

We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax operates in the original signal dimension. We show that the dual problem to PhaseMax is Basis Pursuit, which implies that phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice.

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