Small coherence implies the weak Null Space Property

TL;DR

This paper demonstrates that low coherence in measurement matrices implies a weak Null Space Property, providing new insights into compressed sensing guarantees and establishing bounds on singular value perturbations.

Contribution

It establishes that small coherence implies a weak Null Space Property and provides singular value perturbation bounds, extending understanding of matrix properties in compressed sensing.

Findings

Small coherence implies a weak Null Space Property for most index subsets.

The paper provides bounds on singular value perturbations.

Results may be useful for other applications in matrix analysis.

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. $\Vert h_T\Vert_2 \le C \ \Vert h_{T^c}\Vert_1/\sqrt{s}$ 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.