Data recovery on a manifold is an important problem in many applications. Many such problems, e.g. phase retrieval, matrix recovery, tensor recovery, and compressive sensing, involve solving a system of linear equations knowing that the unknowns lie on a known manifold. In this thesis, we studied the recovery of signals lying on a manifold from linear measurements. Particularly, we focus on the case where signals lying on an algebraic variety. In this thesis we give a framework to study the above problem and give general results for minimum measurement problem of manifold recovery. It is applied to a variety of linear manifold recovery problems and give minimum linear measurement numbers for different cases. Many of the above minimum measurements results can be proved to be sharp....[ Read more ]

Data recovery on a manifold is an important problem in many applications. Many such problems, e.g. phase retrieval, matrix recovery, tensor recovery, and compressive sensing, involve solving a system of linear equations knowing that the unknowns lie on a known manifold. In this thesis, we studied the recovery of signals lying on a manifold from linear measurements. Particularly, we focus on the case where signals lying on an algebraic variety. In this thesis we give a framework to study the above problem and give general results for minimum measurement problem of manifold recovery. It is applied to a variety of linear manifold recovery problems and give minimum linear measurement numbers for different cases. Many of the above minimum measurements results can be proved to be sharp.