Sampling and Reconstructing Signals From a Union of Linear Subspaces

In this paper, we study the problem of sampling and reconstructing signals which are assumed to lie on or close to one of several subspaces of a Hilbert space. Importantly, we here consider a very general setting in which we allow infinitely many subspaces in infinite dimensional Hilbert spaces. This general approach allows us to unify many results derived recently in areas such as compressed sensing, affine rank minimization, analog compressed sensing and structured matrix decompositions. Our main contribution is to show that a conceptually simple Projected Landweber Algorithm is able to recover signals from a union of subspaces whenever the sampling operator satisfies a bi-Lipschitz embedding condition. Importantly, this result holds for all Hilbert spaces and unions of subspaces, as long as the sampling procedure satisfies this condition for the set of subspaces considered.