# 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).

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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