# Phase Retrieval, Self-Calibration, Random Matrices, and Convex Optimization

# Phase Retrieval, Self-Calibration, Random Matrices, and Convex Optimization

I will demonstrate how two important, but seemingly unrelated, problems, namely Phase Retrieval and Self-Calibration, can be solved by using methods from random matix theory and convex optimization. Phase retrieval is the century-old problem of reconstructing a function, such as a signal or image, from intensity measurements, typically from the modulus of a diffracted wave. Phase retrieval problems -- which arise in numerous areas including X-ray crystallography, astronomy, diffraction imaging, and quantum physics -- are notoriously difficult to solve numerically. They also pervade many areas of mathematics, such as numerical analysis, harmonic analysis, algebraic geometry, combinatorics, and differential geometry. Self-calibration is an increasingly important concept, since the need for precise calibration of sensing devices manifests itself as a major roadblock in many scientific and technological endeavors. The idea of self-calibration is to equip a hardware device with a smart algorithm that can compensate automatically for the lack of calibration. I will demonstrate how both phase retrieval and self-calibration can be treated efficiently via convex programming by "lifting" the assoicated non-linear inverse problem to an underdetermined linear problem. Using tools from random matrix theory and compressive sensing, we will see that for certain types of random measurements both problems can be solved exactly via a a convenient semidefinite program. Applications in x-ray crystallography, array calibration, and wireless communications will be discussed.