Emmanuel J. Candès

TL;DR: In this article, a nonconvex formulation of the phase retrieval problem was proposed and a concrete solution algorithm was presented. But the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements.

TL;DR: A smoothing technique and an accelerated first-order algorithm are applied and it is demonstrated that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems and is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters.

TL;DR: A second-order ordinary differential equation is derived, which is the limit of Nesterov's accelerated gradient method, and it is shown that the continuous time ODE allows for a better understanding of Nestersov's scheme.

TL;DR: The paper reviews recent work on the continuous ridgelet transform (CRT), ridgelet frames, ridgelet orthonormal bases, ridgelets and edges and describes a new notion of smoothness naturally attached to this new representation.

TL;DR: A condition on the measurement/sensing matrix is introduced, which is a natural generalization of the now well-known restricted isometry property, and which guarantees accurate recovery of signals that are nearly sparse in (possibly) highly overcomplete and coherent dictionaries.