Classification IV

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# Classification IV - PowerPoint PPT Presentation

Classification IV. Lecturer: Dr. Bo Yuan E-mail: [email protected] Overview. Support Vector Machines. Linear Classifier. w. w · x + b =0. w · x + b &lt;0. w · x + b &gt;0. Distance to Hyperplane. x. x \'. Selection of Classifiers. ?. Which classifier is the best?

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### Classification IV

Lecturer: Dr. Bo Yuan

E-mail: [email protected]

Overview
• Support Vector Machines
Linear Classifier

w

w·x + b =0

w·x + b <0

w·x + b >0

Selection of Classifiers

?

Which classifier is the best?

All have the same training error.

Unknown Samples

B

A

Classifier B divides the space more consistently (unbiased).

Margins

Support Vectors

Support Vectors

Margins
• The margin of a linear classifier is defined as the width that the boundary could be increased by before hitting a data point.
• Intuitively, it is safer to choose a classifier with a larger margin.
• Wider buffer zone for mistakes
• The hyperplane is decided by only a few data points.
• Support Vectors!
• Select the classifier with the maximum margin.
• Linear Support Vector Machines (LSVM)
• Works very well in practice.
• How to specify the margin formally?
Margins

“Predict Class = +1” zone

M=Margin Width

x+

X-

wx+b=1

“Predict Class = -1” zone

wx+b=0

wx+b=-1

Objective Function
• Correctly classify all data points:
• Maximize the margin
• Minimize
• Subject to
Lagrange Multipliers

Dual Problem

Solutions of w & b

inner product

An Example

x2

(1, 1, +1)

x1

(0, 0, -1)

e11

e2

wx+b=1

e7

wx+b=0

wx+b=-1

Soft Margin
Feature Space

x2

x22

Φ: x→ φ(x)

x1

x12

Feature Space

x2

Φ: x→ φ(x)

x1

Constant Terms

Number of terms

Linear Terms

Kernel Trick
• The linear classifier relies on dot products between vectors xi·xj
• If every data point is mapped into a high-dimensional space via some transformation Φ: x→ φ(x), the dot product becomes: φ(xi)·φ(xj)
• A kernel function is some function that corresponds to an inner product in some expanded feature space: K(xi, xj) = φ(xi)·φ(xj)
• Example: x=[x1,x2]; K(xi, xj) = (1 + xi ·xj)2
String Kernel

Similarity between text strings: Car vs. Custard

More Maths …

Lagrange Duality

Karush–Kuhn–Tucker Conditions

• Text Book
• NelloCristianini and John Shawe-Taylor, An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press, 2000.
• Online Resources
• http://www.kernel-machines.org/
• http://www.support-vector-machines.org/
• http://www.tristanfletcher.co.uk/SVM%20Explained.pdf
• http://www.csie.ntu.edu.tw/~cjlin/libsvm/
• A list of papers uploaded to the web learning portal
Review
• What is the definition of margin in a linear classifier?
• Why do we want to maximize the margin?
• What is the mathematical expression of margin?
• How to solve the objective function in SVM?
• What are support vectors?
• What is soft margin?
• How does SVM solve nonlinear problems?
• What is so called “kernel trick”?
• What are the commonly used kernels?
Next Week’s Class Talk
• Volunteers are required for next week’s class talk.
• Topic : SVM in Practice
• Hints:
• Applications
• Demos
• Multi-Class Problems
• Software
• A very popular toolbox: Libsvm
• Any other interesting topics beyond this lecture
• Length: 20 minutes plus question time