1. SVMs Reprised Reading: Bishop, Sec 4.1.1, 6.0, 6.1, 7.0, 7.1

2. Administrivia HW1 back today μ=18.7 (of 25) σ=5.4

3. I’ve slept since then... Last time: LSE example Regression to linear discriminants Binary discriminants to multi-class Intro to nonlinearly separable data and nonlinear projections Today: Geometry of hyperplanes The maximum margin method The SVM optimization problem Quadratic programming (briefly)

4. Tides of history Last time Linear classification (briefly) Multi-classes from sets of hyperplanes SVMs: nonlinear data projections Today Vector geometry The SVM objective function Linear programming (briefly) Slack variables

5. Linear separators are nice... but what if your data looks like this:

6. Nonlinear data projection Suppose you have a “projection function”: Original feature space “Projected” space Usually Do learning w/ linear model in Ex:

7. Nonlinear data projection

10. Common projections Degree- k polynomials: Fourier expansions:

11. Example nonlinear surfaces SVM images from lecture notes by S. Dreiseitl

12. Example nonlinear surfaces SVM images from lecture notes by S. Dreiseitl

13. Example nonlinear surfaces SVM images from lecture notes by S. Dreiseitl

14. Example nonlinear surfaces SVM images from lecture notes by S. Dreiseitl

15. The catch... How many dimensions does have? For degree- k polynomial expansions: E.g., for k =4, d =256 (16x16 images), Yike! For “radial basis functions”,

16. Linear surfaces for cheap Can’t directly find linear surfaces in Have to find a clever “method” for finding them indirectly It’ll take (quite) a bit of work to get there... Will need different criterion than We’ll look for the “maximum margin” classifier Surface s.t. class 1 (“true”) data falls as possible on one side; class -1 (“false”) falls as far as possible on the other

17. Max margin hyperplanes Hyperplane Margin

18. Max margin is unique Hyperplane Margin

19. Exercise Given a hyperplane defined by a weight vector What is the equation for points on the surface of the hyperplane? What are the equations for points on the two margins? Give an expression for the distance between a point and the hyperplane (and/or either margin) What is the role of ?

20. 5 minutes of math... A dot product (inner product) is a projection of one vector onto another When the projection of x onto w is equal to w 0, then x falls exactly onto the w hyperplane w Hyperplane X

21. 5 minutes of math... BTW, are we sure that hyperplane is perpendicular to w ? Why?

22. 5 minutes of math... BTW, are we sure that hyperplane is perpendicular to w ? Why? Consider any two vectors, x 1 and x 2, falling exactly on the hyperplane, then: is perpendicular to any vector in the hyperplane is some vector in the hyperplane

23. 5 minutes of math... Projections on one side of the line have dot products >0... w Hyperplane x

24. 5 minutes of math... Projections on one side of the line have dot products >0...... and on the other, <0 w Hyperplane x

25. 5 minutes of math... What is the distance from any vector x to the hyperplane? w x r =?

26. 5 minutes of math... What is the distance from any vector x to the hyperplane? Write x as a point on plane + offset from plane w Unit vector in direction of w

27. 5 minutes of math... Now:

28. 5 minutes of math... Theorem: The distance, r, from any point x to the hyperplane defined by w and is given by: Lemma: The distance from the origin to the hyperplane is given by: Also: r>0 for points on one side of the hyperplane; r<0 for points on the other

29. Back to SVMs & margins The margins are parallel to hyperplane, so are defined by same w, plus constant offsets w b b

30. Back to SVMs & margins The margins are parallel to hyperplane, so are defined by same w, plus constant offsets Want to ensure that all data points are “outside” the margins w b b

31. Maximizing the margin So now we have a learning criterion function: Pick w to maximize b s.t. all points still satisfy Note: w.l.o.g. can rescale w arbitrarily (why?)

32. Maximizing the margin So now we have a learning criterion function: Pick w to maximize b s.t. all points still satisfy Note: w.l.o.g. can rescale w arbitrarily (why?) So can formulate full problem as: Minimize: Subject to: But how do you do that? And how does this help?

33. Quadratic programming Problems of the form Minimize: Subject to: are called “quadratic programming” problems There are off-the-shelf methods to solve them Actually solving this is way, way beyond the scope of this class Consider it a black box If a solution exists, it will be found & be unique Expensive, but not intractably so

34. Nonseparable data What if the data isn’t linearly separable? Project into higher dim space (we’ll get there) Allow some “slop” in the system Allow margins to be violated “a little” w

35. The new “slackful” QP The are “slack variables” Allow margins to be violated a little Still want to minimize margin violations, so add them to QP instance: Minimize: Subject to:

36. You promised nonlinearity! Where did the nonlinear transform go in all this? Another clever trick With a little algebra (& help from Lagrange multipliers), can rewrite our QP in the form: Maximize: Subject to:

37. Kernel functions So??? It’s still the same linear system Note, though, that appears in the system only as a dot product:

38. Kernel functions So??? It’s still the same linear system Note, though, that appears in the system only as a dot product: Can replace with : The inner product is called a “kernel function”

39. Why are kernel fns cool? The cool trick is that many useful projections can be written as kernel functions in closed form I.e., can work with K() rather than If you know K(x i,x j ) for every (i,j) pair, then you can construct the maximum margin hyperplane between the projected data without ever explicitly doing the projection!