3.3 The restricted isometry property

 Page 1 / 1

We say that an $n×N$ matrix $\Phi$ has the restricted isometry property (RIP) for $k$ if for each $T\subseteq \left\{1,\dots ,N\right\}$ such that $\text{#}T\le k$ , ${\Phi }_{T}$ (the matrix formed by choosing the columns of $\Phi$ whose indices are in $T$ ) has the property

 $\begin{array}{c}\left(1-{\delta }_{k}\right){\parallel {x}_{T}\parallel }_{{\ell }_{2}}{\le }_{}{\parallel {\Phi }_{T}\left(x\right)\parallel }_{{\ell }_{2}}{\le }_{}\left(1+{\delta }_{k}\right){{\parallel {x}_{T}\parallel }_{{\ell }_{2}}}_{}\end{array}$ (RIP)

where $0<{\delta }_{k}<1$ . This useful definition is by Candes and Tao. The idea is that the embedding of a $k$ -dimensional space in $M$ -dimensional space almost preserves norm – like an isometry. Another way of looking at it is to consider the matrix ${\Phi }_{T}^{t}{\Phi }_{T}$ , of size $k×k$ . This matrix is symmetric, positive definite, and it’s eigen-values are between $1-{\delta }_{k}$ and $1+{\delta }_{k}$ .

I prefer the following modified condition (dubbed the MIRP), which is more convenient for mathematical analysis:

 ${\left({c}_{1}\right)}^{-1}{\parallel {x}_{T}\parallel }_{{\ell }_{2}}\le {\parallel {\mathrm{\Phi }}_{T}\left(x\right)\parallel }_{{\ell }_{2}}\le {c}_{1}{\parallel {x}_{T}\parallel }_{{\ell }_{2}}$ (MRIP)

We can now state the following theorem.

If $\Phi$ satisfies MRIP for $2k$ then $\exists \Delta$ s.t. $\left(\Phi ,\Delta \right)$ is instance optimal for ${\ell }_{1}^{N}$ for $K$ .

This shows that whenever we have a matrix $\Phi$ satisfying the MRIP for $2k$ then it will perform well on encoding vectors (at least in the sense of ${\ell }_{1}^{N}$ accuracy). The question is how can we construct measurement matrices with this property? We can construct $\Phi$ using Gaussian entries and then normalizing the columns.

$\exists$ constant $c>0$ s.t. if $k\le c\frac{n}{\mathrm{log}\left(N∕n\right)}$ then with high probability $\Phi$ satisfies RIP and MRIP for $k$ .

Given $N$ and $n$ , the range of $k$ in the above results reflects how accurately we can recover data. There is another constant ${c}^{\prime }$ that serves as a converse bound for Theorem 3. This converse can be derived using Gluskin widths.

The following generic problem is of great interest: Consider the class of matrices $ℳ=\left\{\Phi M×N,\mathrm{\Phi }\text{has some prescribed property(eg. Toeplitz, circulant, etc.)}\right\}$ . What is the largest $k$ for which such a matrix can have the MRIP.

Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!