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Linear classifiers

Suppose F = {linear classifiers in R d }, then we have

V F = d + 1 , f ^ n = arg min f F R ^ n ( f )
E [ R ( f ^ n ) ] - inf f F R ( f ) 4 ( d + 1 ) log ( n + 1 ) + log 2 n .

Generalized linear classifiers

Normally, we have a feature vector X R d . A hyperplane in R d provides a linear classifier in R d . Nonlinear classifiers can be obtained by a straightforward generalization.

Let φ 1 , , φ d ' , d ' d be a collection of functions mapping R d R . These functions, applied to a feature X R d , produce a generalized set of features, φ = ( φ 1 ( X ) , φ 2 ( X ) , , φ d ' ( X ) ) ' . For example, if X = ( x 1 , x 2 ) ' , then we could consider d ' = S and φ = ( x 1 , x 2 , x 1 x 2 , x 1 2 , x 2 2 ) ' R 5 . We can then construct a linear classifier in the higher dimensional generalized feature space R d ' .

The VC bounds immediately extend to this case, and we have for F ' = { generalized linear classifiers based on maps φ : R d R d ' },

E [ R ( f ^ n ) ] - inf f F ' R ( f ) 4 ( d ' + 1 ) log ( n + 1 ) + log 2 n .

Half-space classifiers

Theorem

Steele '75, dudley '78

Let G be a finite-dimensional vector space of real-valued functions on R d . The class of sets A = { { x : g ( x ) 0 } : g G } has VC dimension dim( G ).

It is sufficient to show that no set of n = d i m ( G ) + 1 points can be shattered by A . Take any n points and for each g G , define the vector V g = ( g ( x 1 ) , , g ( x n ) ) .

The set { V g : g G } is a linear subspace of R n of dimension dim ( G ) = n - 1 . Therefore, there exists a non-zero vector α = ( α 1 , , α n ) R n such that i = 1 n α i g ( x i ) = 0 . We can assume that at least one of these α i S is negative (if all are positive, just negate the sum). We can then re-arrange thisexpression as i : α i 0 α i g ( x i ) = i : α i < 0 - α i g ( x i ) .

Now suppose that there exists a g G such that the set { x : g ( x ) 0 } selects precisely the x i S on the left-hand side above. Then all terms on the left are non-negative and allthe terms on the right are non-positive. Since α is non-zero, this is a contradiction. Therefore, x 1 , , x n cannot be shattered by sets in { x : g ( x ) 0 } , g G .  6.375pt0.0pt6.375pt

Consider half-spaces in R d of the form A = { x R d : x i b , i { 1 , , d } , b R } . Each half-space can be described by

g ( x ) = 0 , , 0 , 1 , 0 , , 0 x 1 x d - b
d i m ( G ) = d + 1 , V A d + 1 .

Tree classifiers

Let

T k = r e c u r s i v e r e c t a n g u l a r p a r t i t i o n s o f R d w i t h k + 1 c e l l s

Let T T k . Each cell of T results from splitting a rectangular region into two smaller rectangles parallel to one ofthe coordinate axes.

T T 3 , d = 2 .

Each additional split is analogous to a half-space set. Therefore, each additional split can potentially shatter d + 1 points. This implies that

V T k ( d + 1 ) k .

d = 1 .

k = 1 split shatters two points.

k = 2 splits shatters three points < 4 .

Vc bound for tree classifiers

F k = { t r e e c l a s s i f i e r s w i t h k + 1 l e a f s o n R d }
E [ R ( f ^ n ) ] - inf f F k R ( f ) 4 ( d + 1 ) k log n + log 2 n .

How can we decide what dimension to choose for a generalized linear classifier?

How many leafs should be used for a classification tree?

Complexity Regularization using VC bounds!

Structural risk minimization (srm)

SRM is simply complexity regularization using VC type bounds in place of Chernoff's bound or other concentration inequalities.

The basic idea is to consider a sequence of sets of classifiers F 1 , F 2 , ... , of increasing VC dimensions V F 1 V F 2 ... . Then for each k = 1 , 2 , ... we find the minimum empirical risk classifier

f ^ n ( k ) = arg min f F k R ^ n ( f )

and then select the final classifier according to

k ^ = arg min k 1 R ^ n ( t ^ n ( k ) ) + 32 V F k ( log n + 1 ) n

and f ^ n f ^ n ( k ^ ) is the final choice.

The basic rational is that we know

R n ( f ^ n ( k ) ) - inf f F k R ( f ) C ' V F k log n n

where C ' is a constant.

The end result is that

E [ R ( f ^ n ) ] min k 1 min f F k R ( f ) + 16 V F k log n + 4 2 n

analogous to our pervious complexity regularization results, except thatcodelengths are replaced by VC dimensions.

In order to prove the result we use the VC probability concentration bound and assume that = k 1 V F k < . This enables a union bounding argument and leads to a risk bound of the form given above.

Key point of vc theory

Complexity of classes depends on richness (shattering capability) relative to a set of n arbitrary points. This allows us to effectively “quantize" collections of functions in a slightlydata-dependent manner.

Application to trees

Let

F k = k l e a f d e c i s i o n t r e e s i n R d , V F k ( d + 1 ) ( k + 1 )
f ^ n ( k ) = arg min f F k R ^ n ( f )
k ^ = arg min k 1 min f F k R ( f ) + 32 ( d + 1 ) ( k - 1 ) ( log n + 1 ) n

Then

f ^ n = f ^ n ( k ^ )

satisfies

E [ R ( f ^ n ) ] min k 1 min f F k R ( f ) + 16 ( d + 1 ) ( k - 1 ) log n + 4 2 n

compare with

E [ R ( f ^ n ) ] min k 1 min f d y a d i c k l e a f t r e e s R ( f ) + ( 3 k - 1 ) log 2 + 1 2 log n 2 n

from Lecture 11 .

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Statistical learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
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