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

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Compressive sensing. OpenStax CNX. Sep 21, 2007 Download for free at http://cnx.org/content/col10458/1.1
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