Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Small coherence implies the weak Null Space Property   In the Compressed Sensing community, it is well known that given a matrix $X
\in \mathbb R^{n\times p}$ with $\ell_2$ normalized columns, the Restricted
Isometry Property (RIP) implies the Null Space Property (NSP). It is also well
known that a small Coherence $\mu$ implies a weak RIP, i.e. the singular values
of $X_T$ lie between $1-\delta$ and $1+\delta$ for "most" index subsets $T
\subset \{1,\ldots,p\}$ with size governed by $\mu$ and $\delta$. In this short
note, we show that a small Coherence implies a weak Null Space Property, i.e.
$\Vert h_T\Vert_2 \le C \ \Vert h_{T^c}\Vert_1/\sqrt{s}$ for most $T \subset
\{1,\ldots,p\}$ with cardinality $|T|\le s$. We moreover prove some singular
value perturbation bounds that may also prove useful for other applications.

36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics