The restricted isometry property meets nonlinear approximation with redundant frames

Details The restricted isometry property meets nonlinear approximation with redundant frames The restricted isometry property meets nonlinear approximation with redundant frames

2011-02-25

arxiv.org/abs/1102.5324v1

Mathematics

Functional Analysis

This work has been submitted for possible publication

Scientific paper

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 new 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) the proof that the RIP implies Bernstein inequalities. As a result, we obtain that in most overcomplete random Gaussian dictionaries with fixed aspect ratio, just as in any orthonormal basis, the error of best $m$-term approximation of a vector decays at a certain rate if, and only if, the vector admits a compressible expansion in the dictionary. Yet, 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 perturbation of the dictionary.

Affiliated with

Also associated with

No associations

Rating

The restricted isometry property meets nonlinear approximation with redundant frames does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The restricted isometry property meets nonlinear approximation with redundant frames, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The restricted isometry property meets nonlinear approximation with redundant frames will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-301559