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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. Distance to Hyperplane. x. x \'. Selection of Classifiers. ?. Which classifier is the best?

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classification iv

Classification IV

Lecturer: Dr. Bo Yuan

E-mail: [email protected]

overview
Overview
  • Support Vector Machines
linear classifier
Linear Classifier

w

w·x + b =0

w·x + b <0

w·x + b >0

selection of classifiers
Selection of Classifiers

?

Which classifier is the best?

All have the same training error.

How about generalization?

unknown samples
Unknown Samples

B

A

Classifier B divides the space more consistently (unbiased).

margins
Margins

Support Vectors

Support Vectors

margins1
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!
    • Others can be discarded!
  • Select the classifier with the maximum margin.
    • Linear Support Vector Machines (LSVM)
  • Works very well in practice.
  • How to specify the margin formally?
margins2
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
Objective Function
  • Correctly classify all data points:
  • Maximize the margin
  • Quadratic Optimization Problem
    • Minimize
    • Subject to
lagrange multipliers
Lagrange Multipliers

Dual Problem

Quadratic problem again!

solutions of w b
Solutions of w & b

inner product

an example
An Example

x2

(1, 1, +1)

x1

(0, 0, -1)

soft margin

e11

e2

wx+b=1

e7

wx+b=0

wx+b=-1

Soft Margin
feature space
Feature Space

x2

x22

Φ: x→ φ(x)

x1

x12

feature space1
Feature Space

x2

Φ: x→ φ(x)

x1

quadratic basis functions
Quadratic Basis Functions

Constant Terms

Number of terms

Linear Terms

Pure Quadratic Terms

Quadratic Cross-Terms

kernel trick
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
String Kernel

Similarity between text strings: Car vs. Custard

more maths
More Maths …

Quadratic Optimization

Lagrange Duality

Karush–Kuhn–Tucker Conditions

reading materials
Reading Materials
  • 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
  • Wikipedia & Google
review
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
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
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