Kernelsin

Pattern Recognition

A Langur - BaboonBinary Problem

• http://www.tribuneindia.com/2006/20060712/himplus4.jpg

• … HA HA HA …

• http://www.sickworld.net/db4/00381/sickworld.net/_uimages/baboons.jpg

Representation of Binary Data

Concept of Kernels

• Idea proposed by Aizerman in 1964.

• Feature … space … dimensionality … transformation such that

• The dot product exists {i.e. is not infinite} in higher dimension &

• Data is linearly separable.

Dot Product

• The scalar value signifies the amount of projection of a in the direction of b

• The scalar value also signifies the degree of similarity between a and b

• Adopted from http://www.netcomuk.co.uk/~jenolive/vect6.html

A Geometrical Interpretation Mapping

• Mapping data from low dimension to high dimension.

• Data is linearly separable in higher dimension.

• Separable hyperplane defined by a normal or weight vector.

Cross Product

• Normal vector or Weight vector i.e. perpendicular to the hyperplane. http://www.netcomuk.co.uk/~jenolive/vect8.html

• Area covered while moving a to b in counterclockwise direction moves the vector upwards ... Like tightening of a screw

• This vector is perpendicular to the plane in which a and b lie.

Importance of dot product&

kernel == dot product• Classification requires computation of

dot product between normal of hyperplane and test point.

• Often, normal is expressed as a linear combination of points in higer dimension.

• Dot products signify on which side of the hyperplane the test point lies – act of classification

• Dot product computation expensive and transformation not easy to find, so propose a kernel function, whose scalar value is equivalent to the dot product in higer dimensional plane.

Geometrical Interpretation ofImportance of dot product

&kernel == dot product

How does a kernel look like?A Planner View from Top

How does a kernel look like?An Isometric View from different Side angles

The End

Vapnick proposesSupport Vector Machines

An Apple – OrangeBinary Problem

• http://en.wikipedia.org/wiki/Image:Apples.jpg

• http://en.wikipedia.org/wiki/Image:Ambersweet_oranges.jpg

Representation of Binary Data

Separable Case

The Lagrangian

• Optimize• Subject to

• Differentiate w.r.t• w weight vector• b the constant• alpha Lagrangian

parameter

Non-Separable Case

The Lagrangian

• Optimize• Subject to

• Differentiate w.r.t• w weight vector• b the constant• alpha Lagrangian

parameter• xi another Lagrangian

paramer

Finally … after some mental mathematical harrasment we get:

• Optimized values of weight vector and b values.

• And Then

• Use it to classify new test examples …

In The End

If SVMs can’t help classify…

then DITCH them and classify apples and oranges by eating them yourself ...