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We begin with a property of the null space N N which is at the heart of proving results on instance-optimality.

We say that N N has the Null Space Property if for all η N η in N and all T T with # T k italic "#" T<= k we have η X c 1 η T C X

Intuitively, NSP implies that for any vector in the nullspace the energy will not be concentrated in a small number of entries.

The following are equivalent formulations for NSP X X for k k :

  • η X c 1 σ k ( η )
  • η T X c 1 η T C X where η = η T + η T C .

Note also that the triangle inequality can be used as follows

η X = η T + η T C X η T X + η T C X

which shows that (b) is equivalent to NSP.

  • If ( Φ , Δ ) \( {Φ , Δ} \) is instance optimal on X X for the value k k , then Φ Φ satisfies the NSP for 2 k 2 k on X X with an equivalent constant.
  • If Φ Φ has the NSP for X X and 2 k 2 k then Δ exists Δ s.t. Φ Φ has the instance optimal property for k k .

We will prove a slightly weaker version of this to save time. We first prove that instance optimality for k k implies NSP X X for k k (hence this is slightly weaker than advertised) . Let η N η in N and set z = Δ ( 0 ) z =Δ { \( 0 \)} then

η z c 0 σ k ( η ) η + z c 0 σ k ( η ) η max { η z , η + z } c 0 σ k ( η ) instance optimal property z N triangle inequality

We now prove 2. Suppose Φ Φ has the NSP for 2 k 2 k . Given y y , ( y ) = { x : Φ ( x ) = y } { \( y \)} ={ lbrace {x : Φ { \( x \)} = y} rbrace} . Let us define the decoder Δ Δ by Δ ( y ) : = arg min { σ K ( x ) X : x ( y ) } , then

x Δ ( Φ ( x ) ) X = x x X c 1 σ 2 K ( x x ) c 1 ( σ K ( x ) σ K ( x ) ) specific 2K term approximation 2 c 1 σ K ( x )


Note that the instance optimal property automatically gives reproduction of K K -sparse signals.

At this stage the challenge is to create Φ Φ with this instance optimal property. For this we shall use the restricted isometry property as introduced earlier and which we now recall.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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
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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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