Georgia Institute of Technology College of Sciences

We propose a new convex relaxation for the problem of solving (random) quadratic equations known as phase retrieval. The main advantage of the proposed method is that it operates in the natural domain of the signal. Therefore, it has significantly lower computational cost than the existing convex methods that rely on semidefinite programming and competes with the recent non-convex methods. In the proposed formulation the quadratic equations are relaxed to inequalities describing a "complex polytope". Then, using an *anchor vector* that itself can be constructed from the observations, a simple convex program estimates the ground truth as an (approximate) extreme point of the polytope. We show, using classic results in statistical learning theory, that with random measurements this convex program produces accurate estimates. I will also discuss some preliminary results on a more general class of regression problems where we construct accurate and computationally efficient estimators using anchor vectors.