Compressed sensing in Hilbert spaces

In many linear inverse problems, we want to estimate an unknown vector belonging to a high-dimensional (or infinite-dimensional) space from few linear measurements. To overcome the ill-posed nature of such problems, we use a low-dimension assumption on the unknown vector: it belongs to a low-dimensional model set. The question of whether it is possible to recover such an unknown vector from few measurements then arises. If the answer is yes, it is also important to be able to describe a way to perform such a recovery. We describe a general framework where appropriately chosen random measurements guarantee that recovery is possible. We further describe a way to study the performance of recovery methods that consist in the minimization of a regularization function under a data-fit constraint.