Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 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.

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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: econ