# 3.2 The restricted isometry property  (Page 2/3)

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Let $\Phi :{\mathbb{R}}^{N}\to {\mathbb{R}}^{M}$ denote a sensing matrix and $\Delta :{\mathbb{R}}^{M}\to {\mathbb{R}}^{N}$ denote a recovery algorithm. We say that the pair $\left(\Phi ,\Delta \right)$ is $C$ -stable if for any $x\in {\Sigma }_{K}$ and any $e\in {\mathbb{R}}^{M}$ we have that

${∥\Delta ,\left(\Phi ,x,+,e\right),-,x∥}_{2}\le C∥e∥.$

This definition simply says that if we add a small amount of noise to the measurements, then the impact of this on the recovered signal should not be arbitrarily large. [link] below demonstrates that the existence of any decoding algorithm (potentially impractical) that can stably recover from noisy measurements requires that $\Phi$ satisfy the lower bound of [link] with a constant determined by $C$ .

If the pair $\left(\Phi ,\Delta \right)$ is $C$ -stable, then

$\frac{1}{C}{∥x∥}_{2}\le {∥\Phi ,x∥}_{2}$

for all $x\in {\Sigma }_{2K}$ .

Pick any $x,z\in {\Sigma }_{K}$ . Define

${e}_{x}=\frac{\Phi \left(z-x\right)}{2}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{e}_{z}=\frac{\Phi \left(x-z\right)}{2},$

and note that

$\Phi x+{e}_{x}=\Phi z+{e}_{z}=\frac{\Phi \left(x+z\right)}{2}.$

Let $\stackrel{^}{x}=\Delta \left(\Phi x+{e}_{x}\right)=\Delta \left(\Phi z+{e}_{z}\right)$ . From the triangle inequality and the definition of $C$ -stability, we have that

$\begin{array}{cc}\hfill {∥x,-,z∥}_{2}& ={∥x,-,\stackrel{^}{x},+,\stackrel{^}{x},-,z∥}_{2}\hfill \\ & \le {∥x,-,\stackrel{^}{x}∥}_{2}+{∥\stackrel{^}{x},-,z∥}_{2}\hfill \\ & \le C∥{e}_{x}∥+C{∥{e}_{z}∥}_{2}\hfill \\ & =C{∥\Phi ,x,-,\Phi ,z∥}_{2}.\hfill \end{array}$

Since this holds for any $x,z\in {\Sigma }_{K}$ , the result follows.

Note that as $C\to 1$ , we have that $\Phi$ must satisfy the lower bound of [link] with ${\delta }_{K}=1-1/{C}^{2}\to 0$ . Thus, if we desire to reduce the impact of noise in our recovered signal then we must adjust $\Phi$ so that it satisfies the lower bound of [link] with a tighter constant.

One might respond to this result by arguing that since the upper bound is not necessary, we can avoid redesigning $\Phi$ simply by rescaling $\Phi$ so that as long as $\Phi$ satisfies the RIP with ${\delta }_{2K}<1$ , the rescaled version $\alpha \Phi$ will satisfy [link] for any constant $C$ . In settings where the size of the noise is independent of our choice of $\Phi$ , this is a valid point — by scaling $\Phi$ we are simply adjusting the gain on the “signal” part of our measurements, and if increasing this gain does not impact the noise, then we can achieve arbitrarily high signal-to-noise ratios, so that eventually the noise is negligible compared to the signal.

However, in practice we will typically not be able to rescale $\Phi$ to be arbitrarily large. Moreover, in many practical settings the noise is not independent of $\Phi$ . For example, suppose that the noise vector $e$ represents quantization noise produced by a finite dynamic range quantizer with $B$ bits. Suppose the measurements lie in the interval $\left[-T,T\right]$ , and we have adjusted the quantizer to capture this range. If we rescale $\Phi$ by $\alpha$ , then the measurements now lie between $\left[-\alpha T,\alpha T\right]$ , and we must scale the dynamic range of our quantizer by $\alpha$ . In this case the resulting quantization error is simply $\alpha e$ , and we have achieved no reduction in the reconstruction error.

## Measurement bounds

We can also consider how many measurements are necessary to achieve the RIP. If we ignore the impact of $\delta$ and focus only on the dimensions of the problem ( $N$ , $M$ , and $K$ ) then we can establish a simple lower bound. We first provide a preliminary lemma that we will need in the proof of the main theorem.

Let $K$ and $N$ satisfying $K be given. There exists a set $X\subset {\Sigma }_{K}$ such that for any $x\in X$ we have ${∥x∥}_{2}\le \sqrt{K}$ and for any $x,z\in X$ with $x\ne z$ ,

${∥x,-,z∥}_{2}\ge \sqrt{K/2},$

and

$log|X|\ge \frac{K}{2}log\left(\frac{N}{K}\right).$

We will begin by considering the set

$U=\left\{x,\in ,{\left\{0,,,+,1,,,-,1\right\}}^{N},:,{∥x∥}_{0},=,K\right\}.$

By construction, ${∥x∥}_{2}^{2}=K$ for all $x\in U$ . Thus if we construct $X$ by picking elements from $U$ then we automatically have ${∥x∥}_{2}\le \sqrt{K}$ .

Next, observe that $|U|=\left(\genfrac{}{}{0pt}{}{N}{K}\right){2}^{K}$ . Note also that ${∥x,-,z∥}_{0}\le {∥x,-,z∥}_{2}^{2}$ , and thus if ${∥x,-,z∥}_{2}^{2}\le K/2$ then ${∥x,-,z∥}_{0}\le K/2$ . From this we observe that for any fixed $x\in U$ ,

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