<< Chapter < Page Chapter >> Page >

We say that an n × N n times N matrix Φ Φ has the restricted isometry property (RIP) for k k if for each T { 1 , , N } T subseteq { lbrace {1 , dotslow , N} rbrace} such that # T k italic "#" T<= k , Φ T Φ_T (the matrix formed by choosing the columns of Φ Φ whose indices are in T T ) has the property

( 1 δ k ) x T 2 Φ T ( x ) 2 ( 1 + δ k ) x T 2 (RIP)

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

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

( c 1 ) 1 x T 2 Φ T ( x ) 2 c 1 x T 2 (MRIP)

We can now state the following theorem.

If Φ Φ satisfies MRIP for 2 k 2 k then Δ exists Δ s.t. ( Φ , Δ ) \( {Φ , Δ} \) is instance optimal for 1 N ℓ_1^N for K K .

This shows that whenever we have a matrix Φ Φ satisfying the MRIP for 2 k 2 k then it will perform well on encoding vectors (at least in the sense of 1 N ℓ_1^N accuracy). The question is how can we construct measurement matrices with this property? We can construct Φ Φ using Gaussian entries and then normalizing the columns.

exists constant c > 0 c>0 s.t. if k c n log ( N n ) k<= c n over {l { \( {N ∕ n} \)}} then with high probability Φ Φ satisfies RIP and MRIP for k k .

Given N N and n n , the range of k k in the above results reflects how accurately we can recover data. There is another constant c c^′ 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 = { Φ M × N , Φ has some prescribed property(eg. Toeplitz, circulant, etc.) } . What is the largest k k for which such a matrix can have the MRIP.

Questions & Answers

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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Compressive sensing. OpenStax CNX. Sep 21, 2007 Download for free at http://cnx.org/content/col10458/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Compressive sensing' conversation and receive update notifications?