3.4 The nullspace property

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We begin with a property of the null space $N$ which is at the heart of proving results on instance-optimality.

We say that $N$ has the Null Space Property if for all $\eta \in N$ and all $T$ with $\text{#}T\le k$ we have ${\parallel \eta \parallel }_{X}\le {c}_{1}{\parallel {\eta }_{{T}^{C}}\parallel }_{X}$

Intuitively, NSP implies that for any vector in the nullspace the energy will not be concentrated in a small number of entries.

The following are equivalent formulations for NSP $X$ for $k$ :

• ${\parallel \eta \parallel }_{X}\le {c}_{1}{\sigma }_{k}\left(\eta \right)$
• ${\parallel {\eta }_{T}\parallel }_{X\text{}}\le {c}_{1}^{\prime }{\parallel {\eta }_{{T}^{C}}\parallel }_{X}$ where $\eta ={\eta }_{T}+{\eta }_{{T}^{C}}$ .

Note also that the triangle inequality can be used as follows

${\parallel \eta \parallel }_{X}={\parallel {\eta }_{T}+{\eta }_{{T}^{C}}\parallel }_{X}\le {\parallel {\eta }_{T}\parallel }_{X}+{\parallel {\eta }_{{T}^{C}}\parallel }_{X}$

which shows that (b) is equivalent to NSP.

• If $\left(\Phi ,\Delta \right)$ is instance optimal on $X$ for the value $k$ , then $\Phi$ satisfies the NSP for $2k$ on $X$ with an equivalent constant.
• If $\Phi$ has the NSP for $X$ and $2k$ then $\exists \Delta$ s.t. $\Phi$ has the instance optimal property for $k$ .

We will prove a slightly weaker version of this to save time. We first prove that instance optimality for $k$ implies NSP $X$ for $k$ (hence this is slightly weaker than advertised) . Let $\eta \in N$ and set $z=\Delta \left(0\right)$ then

$\begin{array}{cc}\begin{array}{c}\begin{array}{c}\parallel \eta -z\parallel \le {c}_{0}{\sigma }_{k}\left(\eta \right)\\ \parallel \eta +z\parallel \le {c}_{0}{\sigma }_{k}\left(\eta \right)\text{\hspace{0.17em}}\end{array}\\ \parallel \eta \parallel \le \mathrm{max}\left\{\parallel \eta -z\parallel ,\parallel \eta +z\parallel \right\}\le {c}_{0}{\sigma }_{k}\left(\eta \right)\end{array}& \begin{array}{c}\begin{array}{c}\text{instance optimal property}\\ -z\in \mathcal{N}\end{array}\\ \text{triangle inequality}\end{array}\end{array}$

We now prove 2. Suppose $\Phi$ has the NSP for $2k$ . Given $y$ , $ℱ$ $\left(y\right)=\left\{x:\Phi \left(x\right)=y\right\}$ . Let us define the decoder $\Delta$ by $\Delta \left(y\right):=\mathrm{arg}\mathrm{min}\left\{{\sigma }_{K}{\left(x\right)}_{X}:x\in \text{}ℱ\text{}\left(y\right)\right\}$ , then

$\begin{array}{l}\begin{array}{l}\begin{array}{l}{\parallel x-\mathrm{\Delta }\left(\mathrm{\Phi }\left(x\right)\right)\parallel }_{X}={\parallel x-{x}^{\prime }\parallel }_{X}\phantom{\rule{0ex}{0ex}}\\ \le {c}_{1}{\sigma }_{2K}\left(x-{x}^{\prime }\right)\end{array}\\ \le {c}_{1}\left({\sigma }_{K}\left(x\right)-{\sigma }_{K}\left({x}^{\prime }\right)\right)\text{}\phantom{\rule[-0.0ex]{3.0em}{0.0ex}}\text{specific 2K term approximation}\end{array}\\ \le 2{c}_{1}{\sigma }_{K}\left(x\right)\end{array}\phantom{\rule{0ex}{0ex}}$

QED.

Note that the instance optimal property automatically gives reproduction of $K$ -sparse signals.

At this stage the challenge is to create $\Phi$ with this instance optimal property. For this we shall use the restricted isometry property as introduced earlier and which we now recall.

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