# 3.6 Summary

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## Review

Last time we proved that for each $k\le {c}_{0}\frac{n}{logN/n}$ , there exists an $n×N$ matrix $\Phi$ and a decoder $\Delta$ such that

• ## (a)

$\parallel x-\Delta \Phi \left(x\right){\parallel }_{{\ell }_{1}}\le {c}_{0}{\sigma }_{k}\left(x{\right)}_{{\ell }_{1}}$
• ## (b)

$\parallel x-\Delta \Phi \left(x\right){\parallel }_{{\ell }_{2}}\le {c}_{0}\frac{{\sigma }_{k}\left(x{\right)}_{{\ell }_{1}}}{\sqrt{k}}$
Recall that we can find such a $\Phi$ by setting the entries $\left[\Phi {\right]}_{j,k}={\phi }_{j,k}\left(\omega \right)$ to be realizations of independent and identically distributed Gaussian random variables.

## Decoding is not implementable

Our decoding “algorithm” is:

$\Delta \left(y\right):={argmin}_{x\in \mathcal{F}\left(y\right)}{\sigma }_{k}{\left(x\right)}_{{\ell }_{1}}$
where $\mathcal{F}\left(y\right):=\left\{x:\Phi \left(x\right)=y\right\}$ . In general, this algorithm is not implementable. This deficiency, however, iseasily repaired. Specifically, define
${\Delta }_{1}\left(y\right):={argmin}_{x\in \mathcal{F}\left(y\right)}{\parallel x\parallel }_{{\ell }_{1}}.$
Then (a) and (b) hold for ${\Delta }_{1}$ in place of $\Delta$ . This decoding algorithm is equivalent to solving a linear programmingproblem, thus it is tractable and can be solved using techniques such as the interior point method or the simplex method. Ingeneral, these algorithms have computational complexity $O\left({N}^{3}\right)$ . For very large signals this can become prohibitive, and hencethere has been a considerable amount of research in faster decoders (such as decoding using greedy algorithms).

## We cannot generate such φ

The construction of a $\Phi$ from realizations of Gaussian random variables is guaranteed to work with high probability.However, we would like to know, given a particular instance of $\Phi$ , do (a) and (b) still hold. Unfortunately, this is impossible to check (since, to show that $\Phi$ satisfies the MRIP for $k$ , we need to consider all possible submatrices of $\Phi$ ). Furthermore, we would like to build $\Phi$ that can be implemented in circuits. We also might wantfast decoders $\Delta$ for these $\Phi$ . Thus we also may need to be more restrictive in building $\Phi$ . Two possible approaches that move in this direction are as follows:

• Find $\Phi$ that we can build such that we can prove instance optimality in ${\ell }_{1}$ for a smaller range of $k$ , i.e.,
$\parallel x-\Delta \Phi \left(x\right){\parallel }_{{\ell }_{1}}\le {c}_{0}{\sigma }_{k}\left(x{\right)}_{{\ell }_{1}}$
for $k . If we are willing to sacrifice and let $K$ be smaller than before, for example, $K\approx \sqrt{n}$ , then we might be able to prove that ${\Phi }_{T}^{t}{\Phi }_{T}$ is diagonally dominant for all $T$ such that $♯T=2k$ , which would ensure that $\Phi$ satisfies the MRIP.
• Consider $\Phi \left(\omega \right)$ where $\omega$ is a random seed that generates a $\Phi$ . It is possible to show that give $x$ , with high probability, $\Phi \left(\omega \right)\left(x\right)=y$ encodes $x$ in an ${\ell }_{2}$ -instance optimal fashion:
$\parallel x-\overline{x}{\parallel }_{{\ell }_{2}}\le 2{\sigma }_{k}\left(x{\right)}_{{\ell }_{2}}$
for $k\le {c}_{0}\frac{n}{\left(logN/n{\right)}^{5/2}}$ . Thus, by generating many such matrices we can recover any $x$ with high probability.

## Encoding signals

Another practical problem is that of encoding the measurements $y$ . In a real system these measurements must be quantized. This problem was addressed by Candes, Romberg,and Tao in their paper Stable Signal Recovery from Incomplete and Inaccurate Measurements. They prove that if $y$ is quantized to $\overline{y}$ , and if $x\in U\left({\ell }_{p}\right)$ for $p\le 1$ , then we get optimal performance in terms the number of bits requiredfor a given accuracy. Notice that their result applies only to the case where $p\le 1$ . One might expect that this argument could be extended to $p$ between 1 and 2, but a warning is in order at this stage:

Fix $1 . Then there exist $\Phi$ and $\Delta$ satisfying

$\parallel x-\Delta \Phi \left(x\right){\parallel }_{{\ell }_{p}}\le {C}_{0}{\sigma }_{k}\left(x{\right)}_{{\ell }_{p}}$
if
$k\le {c}_{0}{N}^{\frac{2-2/p}{1-2/p}}{\left(\frac{n}{logN/n}\right)}^{\frac{p}{2-p}}.$
Furthermore, this range of k is the best possible (save for the $log$ term).

Examples:

• $p=1$ , we get our original results
• $p=2$ , we do not get instance optimal for $k=1$ unless $n\approx N$
• $p=\frac{3}{2}$ , we only get instance optimal if $k\le {c}_{0}{N}^{-2}{\left(\frac{n}{logN/n}\right)}^{3}$

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in general
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