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This module establishes a number of results concerning various L1 minimization algorithms designed for sparse signal recovery from noisy measurements. The results in this module apply to both bounded noise as well as Gaussian (or more generally, sub-Gaussian) noise.

The ability to perfectly reconstruct a sparse signal from noise-free measurements represents a promising result. However, in most real-world systems the measurements are likely to be contaminated by some form of noise. For instance, in order to process data in a computer we must be able to represent it using a finite number of bits, and hence the measurements will typically be subject to quantization error. Moreover, systems which are implemented in physical hardware will be subject to a variety of different types of noise depending on the setting.

Perhaps somewhat surprisingly, one can show that it is possible to modify

x ^ = arg min z z 1 subject to z B ( y ) .

to stably recover sparse signals under a variety of common noise models  [link] , [link] , [link] . As might be expected, the restricted isometry property (RIP) is extremely useful in establishing performance guarantees in noise.

In our analysis we will make repeated use of Lemma 1 from "Noise-free signal recovery" , so we repeat it here for convenience.

Suppose that Φ satisfies the RIP of order 2 K with δ 2 K < 2 - 1 . Let x , x ^ R N be given, and define h = x ^ - x . Let Λ 0 denote the index set corresponding to the K entries of x with largest magnitude and Λ 1 the index set corresponding to the K entries of h Λ 0 c with largest magnitude. Set Λ = Λ 0 Λ 1 . If x ^ 1 x 1 , then

h 2 C 0 σ K ( x ) 1 K + C 1 Φ h Λ , Φ h h Λ 2 .


C 0 = 2 1 - ( 1 - 2 ) δ 2 K 1 - ( 1 + 2 ) δ 2 K , C 1 = 2 1 - ( 1 + 2 ) δ 2 K .

Bounded noise

We first provide a bound on the worst-case performance for uniformly bounded noise, as first investigated in  [link] .

(theorem 1.2 of [link] )

Suppose that Φ satisfies the RIP of order 2 K with δ 2 K < 2 - 1 and let y = Φ x + e where e 2 ϵ . Then when B ( y ) = { z : Φ z - y 2 ϵ } , the solution x ^ to [link] obeys

x ^ - x 2 C 0 σ K ( x ) 1 K + C 2 ϵ ,


C 0 = 2 1 - ( 1 - 2 ) δ 2 K 1 - ( 1 + 2 ) δ 2 K , C 2 = 4 1 + δ 2 K 1 - ( 1 + 2 ) δ 2 K .

We are interested in bounding h 2 = x ^ - x 2 . Since e 2 ϵ , x B ( y ) , and therefore we know that x ^ 1 x 1 . Thus we may apply [link] , and it remains to bound Φ h Λ , Φ h . To do this, we observe that

Φ h 2 = Φ ( x ^ - x ) 2 = Φ x ^ - y + y - Φ x 2 Φ x ^ - y 2 + y - Φ x 2 2 ϵ

where the last inequality follows since x , x ^ B ( y ) . Combining this with the RIP and the Cauchy-Schwarz inequality we obtain

Φ h Λ , Φ h Φ h Λ 2 Φ h 2 2 ϵ 1 + δ 2 K h Λ 2 .


h 2 C 0 σ K ( x ) 1 K + C 1 2 ϵ 1 + δ 2 K = C 0 σ K ( x ) 1 K + C 2 ϵ ,

completing the proof.

In order to place this result in context, consider how we would recover a sparse vector x if we happened to already know the K locations of the nonzero coefficients, which we denote by Λ 0 . This is referred to as the oracle estimator . In this case a natural approach is to reconstruct the signal using a simple pseudoinverse:

x ^ Λ 0 = Φ Λ 0 y = ( Φ Λ 0 T Φ Λ 0 ) - 1 Φ Λ 0 T y x ^ Λ 0 c = 0 .

The implicit assumption in [link] is that Φ Λ 0 has full column-rank (and hence we are considering the case where Φ Λ 0 is the M × K matrix with the columns indexed by Λ 0 c removed) so that there is a unique solution to the equation y = Φ Λ 0 x Λ 0 . With this choice, the recovery error is given by

x ^ - x 2 = ( Φ Λ 0 T Φ Λ 0 ) - 1 Φ Λ 0 T ( Φ x + e ) - x 2 = ( Φ Λ 0 T Φ Λ 0 ) - 1 Φ Λ 0 T e 2 .

Questions & Answers

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
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Akash Reply
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
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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.
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
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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?
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for screen printed electrodes ?
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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
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I'm interested in nanotube
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Ramkumar Reply
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Sravani Reply
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Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
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Smarajit Reply
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Source:  OpenStax, An introduction to compressive sensing. OpenStax CNX. Apr 02, 2011 Download for free at http://legacy.cnx.org/content/col11133/1.5
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