# 6.8 Inference using compressive measurements  (Page 2/2)

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In the case where the class of signals of interest corresponds to a low dimensional subspace, a truncated, simplified sparse approximation can be applied as a detection algorithm; this has been dubbed as IDEA  [link] . In simple terms, the algorithm will mark a detection when a large enough amount of energy from the measurements lies in the projected subspace. Since this problem does not require accurate estimation of the signal values, but rather whether it belongs in the subspace of interest or not, the number of measurements necessary is much smaller than that required for reconstruction, as shown in [link] .

## Classification

Similarly, random projections have long been used for a variety of classification and clustering problems. The Johnson-Lindenstrauss Lemma is often exploited in this setting to compute approximate nearest neighbors, which is naturally related to classification. The key result that randomprojections result in an isometric embedding allows us to generalize this work to several new classification algorithms and settings  [link] .

Classification can also be performed when more elaborate models are used for the different classes. Suppose the signal/image class of interest can be modeled as a low-dimensional manifold in the ambient space. In such case it can be shown that, even under random projections, certain geometric properties of the signal class are preserved up to a small distortion; for example, interpoint Euclidean ( ${\ell }_{2}$ ) distances are preserved  [link] . This enables the design of classification algorithms in the projected domain. One such algorithm is known as the smashed filter  [link] . As an example, under equal distribution among classes and a gaussian noise setting, the smashed filter is equivalent to building a nearest-neighbor (NN) classifier in the measurement domain. Further, it has been shown that for a $K-$ dimensional manifold, $M=O\left(KlogN\right)$ measurements are sufficient to perform reliable compressive classification. Thus, the number of measurements scales as the dimension of the signal class, as opposed to the sparsity of the individual signal. Some example results are shown in [link] (a).

## Estimation

Consider a signal $x\in {\mathbb{R}}^{N}$ , and suppose that we wish to estimate some function $f\left(x\right)$ but only observe the measurements $y=\Phi x$ , where $\Phi$ is again an $M×N$ matrix. The data streaming community has previously analyzed this problem for many common functions, such as linear functions, ${\ell }_{p}$ norms, and histograms. These estimates are often based on so-called sketches , which can be thought of as random projections.

As an example, in the case where $f$ is a linear function, one can show that the estimation error (relative to the norms of $x$ and $f$ ) can be bounded by a constant determined by $M$ . This result holds for a wide class of random matrices, and can be viewed as a straightforward consequence of the same concentration of measure inequality that has proven useful for CS and in proving the JL Lemma  [link] .

Parameter estimation can also be performed when the signal class is modeled as a low-dimensional manifold. Suppose an observed signal $x$ can be parameterized by a $K-$ dimensional parameter vector $\theta$ , where $K\ll N$ . Then, it can be shown that with 0 $\left(KlogN\right)$ measurements, the parameter vector can be obtained via multiscale manifold navigation in the compressed domain  [link] . Some example results are shown in [link] (b).

#### Questions & Answers

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
characteristics of micro business
Abigail
Do somebody tell me a best nano engineering book for beginners?
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
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
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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?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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