Small coherence implies the weak Null Space Property

Abstract: 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. $$\Vert h_T\Vert_2 \le C \ \Vert h_{T<sup>c}\Vert_1/\sqrt{s}$$ 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.