  From a numerical analysis perspective, assessing the robustness of
l1-minimization is a fundamental issue in compressed sensing and sparse
regularization. Yet, the recovery guarantees available in the literature
usually depend on a priori estimates of the noise, which can be very hard to
obtain in practice, especially when the noise term also includes unknown
discrepancies between the finite model and data. In this work, we study the
performance of l1-minimization when these estimates are not available,
providing robust recovery guarantees for quadratically constrained basis
pursuit and random sampling in bounded orthonormal systems. Several
applications of this work are approximation of high-dimensional functions,
infinite-dimensional sparse regularization for inverse problems, and fast
algorithms for non-Cartesian Magnetic Resonance Imaging.
