kernels in pattern recognition. a langur - baboon binary problem m/2006/20060712/himplu s4.jpg …...
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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 ...
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