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And in a scenario that I want to consider, sometimes Phi of X will be very high dimensional, and in fact sometimes Phi of X – so for example, Phi of X may contain very high degree polynomial features. Sometimes Phi of X will actually even be an infinite dimensional vector of features, and the question is if Phi of X is an extremely high dimensional, then you can’t actually compute to these inner products very efficiently, it seems, because computers need to represent an extremely high dimensional feature vector and then take [inaudible] inefficient.

It turns out that in many important special cases, we can write down – let’s call the kernel function, denoted by K, which will be this, which would be inner product between those feature vectors. It turns out there will be important special cases where computing Phi of X is computationally very expensive – maybe is impossible.

There’s an infinite dimensional vector, and you can’t compute infinite dimensional vectors. There will be important special cases where Phi of X is very expensive to represent because it is so high dimensional, but nonetheless, you can actually compute a kernel between XI and XJ. You can compute the inner product between these two vectors very inexpensively.

And so the idea of the support vector machine is that everywhere in the algorithm that you see these inner products, we’re going to replace it with a kernel function that you can compute efficiently, and that lets you work in feature spaces Phi of X even if Phi of X are very high dimensional. Let me now say how that’s done. A little bit later today, we’ll actually see some concrete examples of Phi of X and of kernels. For now, let’s just think about constructing kernels explicitly. This best illustrates my example.

Let’s say you have two inputs, X and Z. Normally I should write those as XI and XJ, but I’m just going to write X and Z to save on writing. Let’s say my kernel is K of X, Z equals X transpose Z squared. And so this is – right? X transpose Z – this thing here is X transpose Z and this thing is X transpose Z, so this is X transpose Z squared. And that’s equal to that. And so this kernel corresponds to the feature mapping where Phi of X is equal to – and I’ll write this down for the case of N equals free, I guess.

And so with this definition of Phi of X, you can verify for yourself that this thing becomes the inner product between Phi of X and Phi of Z, because to get an inner product between two vectors is – you can just take a sum of the corresponding elements of the vectors. You multiply them. So if this is Phi of X, then the inner product between Phi of X and Phi of Z will be the sum over all the elements of this vector times the corresponding elements of Phi of Z, and what you get is this one.

And so the cool thing about this is that in order to compute Phi of X, you need [inaudible] just to compute Phi of X. If N is a dimension of X and Z, then Phi of X is a vector of all pairs of XI XJ multiplied of each other, and so the length of Phi of X is N squared. You need order N squared time just to compute Phi of X.

Questions & Answers

what is the stm
Brian Reply
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
How can I make nanorobot?
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
how can I make nanorobot?
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.
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Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
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