The restricted isometry property meets nonlinear approximation with redundant frames

TL;DR

This paper explores how the Restricted Isometry Property (RIP) influences nonlinear approximation in Hilbert spaces with redundant frames, establishing new links between RIP, Bernstein inequalities, and compressible representations.

Contribution

It introduces new implications of RIP for nonlinear approximation, proving that RIP implies Bernstein inequalities and characterizing compressible representations in overcomplete dictionaries.

Findings

RIP implies Bernstein inequalities in overcomplete dictionaries.

Best m-term approximation error decays for vectors with compressible expansions.

Counterexamples show limitations of Bernstein inequalities in mildly overcomplete dictionaries.

Abstract

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.