Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the absence of the RIP in real-world applications of compressed
  sensing and the RIP in levels   The purpose of this paper is twofold. The first is to point out that the
Restricted Isometry Property (RIP) does not hold in many applications where
compressed sensing is successfully used. This includes fields like Magnetic
Resonance Imaging (MRI), Computerized Tomography, Electron Microscopy, Radio
Interferometry and Fluorescence Microscopy. We demonstrate that for natural
compressed sensing matrices involving a level based reconstruction basis (e.g.
wavelets), the number of measurements required to recover all $s$-sparse
signals for reasonable $s$ is excessive. In particular, uniform recovery of all
$s$-sparse signals is quite unrealistic. This realisation shows that the RIP is
insufficient for explaining the success of compressed sensing in various
practical applications. The second purpose of the paper is to introduce a new
framework based on a generalised RIP-like definition that fits the applications
where compressed sensing is used. We show that the shortcomings that show that
uniform recovery is unreasonable no longer apply if we instead ask for
structured recovery that is uniform only within each of the levels. To examine
this phenomenon, a new tool, termed the 'Restricted Isometry Property in
Levels' is described and analysed. Furthermore, we show that with certain
conditions on the Restricted Isometry Property in Levels, a form of uniform
recovery within each level is possible. Finally, we conclude the paper by
providing examples that demonstrate the optimality of the results obtained.

7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn