# 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 $k\le {c}_{0}{N}^{-2}{\left(\frac{n}{logN/n}\right)}^{3}$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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