PhaseMax: Convex Phase Retrieval via Basis Pursuit

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 is a "non-lifting" relaxation that 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

Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone

By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.Comment: Expanding on the previous version with additional lower bounds, new images, corrections and improvements. Comparison with Reifenberg approac

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Inner approximation of convex cones via primal-dual ellipsoidal norms

We study ellipsoids from the point of view of approximating convex sets. Our focus is on finding largest volume ellipsoids with specified centers which are contained in certain convex cones. After reviewing the related literature and establishing some fundamental mathematical techniques that will be useful, we derive such maximum volume ellipsoids for second order cones and the cones of symmetric positive semidefinite matrices. Then we move to the more challenging problem of finding a largest pair (in the sense of geometric mean of their radii) of primal-dual ellipsoids (in the sense of dual norms) with specified centers that are contained in their respective primal-dual pair of convex cones