# 0.6 Dimensionality reduction

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This collection reviews fundamental concepts underlying the use of concise models for signal processing. Topics are presented from a geometric perspective and include low-dimensional linear, sparse, and manifold-based signal models, approximation, compression, dimensionality reduction, and Compressed Sensing.

Recent years have seen a proliferation of novel techniques for what can loosely be termed “dimensionality reduction.”Like the tasks of approximation and compression discussed above, these methods involve some aspect in which low-dimensionalinformation is extracted about a signal or collection of signals in some high-dimensional ambient space.Unlike the tasks of approximation and compression, however, the goal of these methods is not always to maintain a faithfulrepresentation of each signal. Instead, the purpose may be to preserve some criticalrelationships among elements of a data set or to discover information about a manifold on which the data lives.

In this section, we review two general methods for dimensionality reduction. [link] begins with a brief overview of techniques for manifold learning. [link] then discusses the Johnson-Lindenstrauss (JL) lemma, which concerns the isometric embedding of a cloud points as it isprojected to a lower-dimensional space. Though at first glance the JL lemma does not pertain to any of the low-dimensional signalmodels we have previously discussed, we later see in Connections with dimensionality reduction that the JL lemma plays a critical role in the core theory of CS, and we also employ the JL lemma indeveloping a theory for isometric embeddings of manifolds.

## Manifold learning

Several techniques have been proposed for solving a problem known as manifold learning in which certain properties of a manifold are inferred from a discrete collection of points sampled from that manifold. A typical manifold learning setup is as follows: an algorithm is presented with a set of $P$ points sampled from a $K$ -dimensional submanifold of ${\mathbb{R}}^{N}$ . The goal of the algorithm is to produce an mapping of these $P$ points into some lower dimension ${\mathbb{R}}^{M}$ (ideally, $M=K$ ) while preserving some characteristic property of the manifold.Example algorithms include ISOMAP  [link] , Hessian Eigenmaps (HLLE)  [link] , and Maximum Variance Unfolding (MVU)  [link] , which attempt to learn isometric embeddings of the manifold (thus preserving pairwise geodesic distances in ${\mathbb{R}}^{M}$ ); Locally Linear Embedding (LLE)  [link] , which attempts to preserve local linear neighborhood structures among the embedded points;Local Tangent Space Alignment (LTSA)  [link] , which attempts to preserve local coordinates in each tangent space; and a methodfor charting a manifold  [link] that attempts to preserve local neighborhood structures.

The internal mechanics of these algorithms differs depending on the objective criterion to be preserved, but as an example, the ISOMAP algorithm operates by first estimating the geodesic distance between each pair of points on the manifold (by approximating geodesic distance as the sum of Euclidean distances between pairs of the available sample points). After the $P×P$ matrix of pairwise geodesic distances is constructed, a technique known as multidimensional scaling uses an eigendecomposition of the distance matrix to determine the proper $M$ -dimensional embedding space. An example of using ISOMAP to learn a $2$ -dimensional manifold is shown in [link] .

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
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
what is biological synthesis of nanoparticles
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
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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