The restricted isometry property meets nonlinear approximation with redundant frames

It is now well known that sparse or compressible vectors can be stably recovered from their low-dimensional projection, provided the projection matrix satisfies a Restricted Isometry Property (RIP). We establish new implications of the RIP with respect to nonlinear approximation in a Hilbert space with a redundant frame. The main ingredients of our approach are: a) Jackson and Bernstein inequalities, associated to the characterization of certain approximation spaces with interpolation spaces; b) a proof that for overcomplete frames which satisfy a Bernstein inequality, these interpolation spaces are nothing but the collection of vectors admitting a representation in the dictionary with compressible coefficients; c) a proof that the RIP implies Bernstein inequalities. Our main result is that most overcomplete random Gaussian dictionaries with fixed aspect ratio, just as any orthonormal basis, satisfy the RIP and consequently the error of best mm-term approximation of a vector decays at a certain rate if, and only if, the vector admits a compressible expansion in the dictionary. Yet, it turns out that Bernstein estimates are extremely fragile. For mildly overcomplete dictionaries with a one-dimensional kernel, we give examples where the Bernstein inequality holds, but the same inequality fails for even the smallest geometric perturbation of the dictionary.