1. Support Vector Machines
Graphic generated with Lucent Technologies Demonstration 2-D Pattern Recognition Applet at

2. Separating Line (or hyperplane)
Class 1
Class -1
Goal: Find the best line (or hyperplane) to separate the training data. How to formalize?
In two dimensions, equation of the line is given by:
Better notation for n dimensions:

3. Simple Classifier The Simple Classifier:
Points that fall on the right are classified as “1”
Points that fall on the left are classified as “-1”
Therefore: using the training set, find a hyperplane (line) so that
This is a perceptron!
How can we improve on the perceptron?
Class 1
Class -1

4. Finding the Best Plane
Not all planes are equal. Which of the two following planes shown is better?
Class -1
Class 1
Both planes accurately classify the training set.
The green plane is the better choice, since it is more likely to do well on future test data.
The green plane is further away from the data.

5. Separating the planes Construct the bounding planes: Class -1 Class 1
Draw two parallel planes to the classification plane.
Push them as far apart as possible, until they hit data points.
The classification plane with bounding planes furthest apart is the best one.
Class -1
Class 1

6. Recap: Finding the Best Plane
Details
All points in class 1 should be to the right of bounding plane 1.
All points in class -1 should be to the left of bounding plane -1.
Pick yi to be +1 or -1 depending on the classification. Then the above two inequalities can be written as one:
The distance between bounding planes should be maximized.
The distance between bounding planes is given by:
Class 1
Class -1

7. The Optimization Problem
The previous slide can be rewritten as:
such that
This is a mathematical program.
Optimization problem subject to constraints
More specifically, this is a quadratic program
There are high powered software tools for solving this kind of problem (both commercial and academic)
No special algorithms are necessary (in theory...)
Just enter this problem and the associated data into a quadratic programming solver (like CPLEX), and let it find an answer.

8. Data Which is Not Linearly Separable
What if a separating plane does not exist?
error
Class -1
Class 1
Find the plane that maximizes the margin and minimizes the errors on the training points.
Take original inequality and add a slack variable to measure error:

9. The Support Vector Machine
Push the planes apart and minimize the error at the same time:
such that
C is a positive number that is chosen to balance these two goals.
This problem is called a Support Vector Machine, or SVM.
The SVM is one of many techniques for doing supervised machine learning
Others: Neural networks, decision trees, k-nearest neighbor

10. Terminology
Those points that touch the bounding plane, or lie on the wrong side, are called support vectors.
If all the data points except the support vectors were removed, the solution would turn out the same.
The SVM is mathematically equivalent to force and torque equilibrium (hence the name support vectors).

11. Research: Solving Massive SVMs
The standard SVM is solved using a canned quadratic programming (QP) solver. Problem:
Standard tools bring all the data into memory. If dataset is bigger than memory, out of luck.
How do other supervised learning techniques handle data that does not fit in memory?
Why not use virtual memory? Let the operating system manage which data the QP solver is using.
Answer: The QP solver accesses data in a random, not a continuous fashion. The cost to page data in and out of memory is enormous.

12. What about nonlinear surfaces?
Some datasets may not be best separated by a plane.
How can we do nonlinear separating surfaces?
Simple method: Map into a higher dimensional space, and do the same thing we have already done.
Generated with Lucent Technologies Demonstration 2-D Pattern Recognition Applet at

13. Finding nonlinear surfaces
How to modify algorithm to find nonlinear surfaces?
First idea (simple and effective): map each data point into a higher dimensional space, and find a linear fit there
Example: Find a quadratic surface for
Use new coordinates in regular linear SVM
A plane in this quadratic space is equivalent to a quadratic surface in our original space.

14. Problem & Solution
If dimensionality of space is high, lots of calculations
For a high polynomial space, combinations of coordinates explodes
Need to do all these calculations for all training points, and for each testing point
Infinite dimensional spaces impossible
Nonlinear surfaces can be used without these problems through the use of a kernel function.
Demonstration:

15. Example: Checkerboard

16. 5-Nearest Neighbor

17. Sixth degree polynomial kernel