<< Chapter < Page Chapter >> Page >

Support vector machines

This set of notes presents the Support Vector Machine (SVM) learning algorithm. SVMs are among the best (and many believe are indeed the best) “off-the-shelf”supervised learning algorithm. To tell the SVM story, we'll need to first talk about margins and the idea of separating data with a large “gap.” Next, we'll talk about the optimal marginclassifier, which will lead us into a digression on Lagrange duality. We'll also see kernels, which give a way to apply SVMs efficiently in very highdimensional (such as infinite-dimensional) feature spaces, and finally, we'll close off the story with the SMO algorithm, which gives an efficient implementationof SVMs.

Margins: intuition

We'll start our story on SVMs by talking about margins. This section will give the intuitions about margins and about the “confidence” of our predictions; these ideaswill be made formal in "Functional and geometric margins" .

Consider logistic regression, where the probability p ( y = 1 | x ; θ ) is modeled by h θ ( x ) = g ( θ T x ) . We would then predict “1” on an input x if and only if h θ ( x ) 0 . 5 , or equivalently, if and only if θ T x 0 . Consider a positive training example ( y = 1 ). The larger θ T x is, the larger also is h θ ( x ) = p ( y = 1 | x ; w , b ) , and thus also the higher our degree of “confidence” that the label is 1. Thus, informally we can think of ourprediction as being a very confident one that y = 1 if θ T x 0 . Similarly, we think of logistic regression as making a very confident prediction of y = 0 , if θ T x 0 . Given a training set, again informally it seems that we'd have found a good fit to thetraining data if we can find θ so that θ T x ( i ) 0 whenever y ( i ) = 1 , and θ T x ( i ) 0 whenever y ( i ) = 0 , since this would reflect a very confident (and correct) set of classifications for all the training examples. This seems to be a nice goal toaim for, and we'll soon formalize this idea using the notion of functional margins.

For a different type of intuition, consider the following figure, in which x's represent positive training examples, o's denote negative training examples,a decision boundary (this is the line given by the equation θ T x = 0 , and is also called the separating hyperplane ) is also shown, and three points have also been labeled A, B and C.

a data set broken into two types with a line in between them

Notice that the point A is very far from the decision boundary. If we are asked to make a predictionfor the value of y at A, it seems we should be quite confident that y = 1 there. Conversely, the point C is very close to the decision boundary, and while it's on theside of the decision boundary on which we would predict y = 1 , it seems likely that just a small change to the decision boundary could easily have caused our prediction to be y = 0 . Hence, we're much more confident about our prediction at A than at C. The point B liesin-between these two cases, and more broadly, we see that if a point is far from the separating hyperplane, then we may be significantly more confident in our predictions.Again, informally we think it'd be nice if, given a training set, we manage to find a decision boundary that allows us to make all correct and confident (meaning far from thedecision boundary) predictions on the training examples. We'll formalize this later using the notion of geometric margins.

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
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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
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
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
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
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Machine learning' conversation and receive update notifications?