1. prof. dr. Lambert Schomaker Shattering two binary dimensions over a number of classes Kunstmatige Intelligentie / RuG

2. 2 Samples and classes  In order to understand the principle of shattering sample points into classes we will look at the simple case of  two dimensions  of binary value

3. 3 2-D feature space 0 0 1 1 f1f1 f2f2

4. 4 2-D feature space, 2 classes 0 0 1 1 f1f1 f2f2

5. 5 the other class… 0 0 1 1 f1f1 f2f2

6. 6 2 left vs 2 right 0 0 1 1 f1f1 f2f2

7. 7 top vs bottom 0 0 1 1 f1f1 f2f2

8. 8 right vs left 0 0 1 1 f1f1 f2f2

9. 9 bottom vs top 0 0 1 1 f1f1 f2f2

10. 10 lower-right outlier 0 0 1 1 f1f1 f2f2

11. 11 lower-left outlier 0 0 1 1 f1f1 f2f2

12. 12 upper-left outlier 0 0 1 1 f1f1 f2f2

13. 13 upper-right outlier 0 0 1 1 f1f1 f2f2

14. 14 etc. 0 0 1 1 f1f1 f2f2

15. 15 2-D feature space 0 0 1 1 f1f1 f2f2

16. 16 2-D feature space 0 0 1 1 f1f1 f2f2

17. 17 2-D feature space 0 0 1 1 f1f1 f2f2

18. 18 XOR configuration A 0 0 1 1 f1f1 f2f2

19. 19 XOR configuration B 0 0 1 1 f1f1 f2f2

20. 20 2-D feature space, two classes: 16 hypotheses f 1 =0 f 1 =1 f 2 =0 f 2 =1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 “hypothesis” = possible class partioning of all data samples

21. 21 2-D feature space, two classes, 16 hypotheses f 1 =0 f 1 =1 f 2 =0 f 2 =1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 two XOR class configurations: 2/16 of hypotheses requires a non-linear separatrix

22. 22 XOR, a possible non-linear separation 0 0 1 1 f1f1 f2f2

23. 23 XOR, a possible non-linear separation 0 0 1 1 f1f1 f2f2

24. 24 2-D feature space, three classes, # hypotheses? f 1 =0 f 1 =1 f 2 =0 f 2 =1 0 1 2 3 4 5 6 7 8 ……………………

25. 25 2-D feature space, three classes, # hypotheses? f 1 =0 f 1 =1 f 2 =0 f 2 =1 0 1 2 3 4 5 6 7 8 …………………… 3 4 = 81 possible hypotheses

26. 26 Maximum, discrete space  Four classes: 4 4 = 256 hypotheses  Assume that there are no more classes than discrete cells  Nhypmax = ncells nclasses

27. 27 2-D feature space, three classes… 0 0 1 1 f1f1 f2f2 In this example,   is linearly separatable  from the rest, as is .  But  is not linearly separatable from the rest of the classes.

28. 28 2-D feature space, four classes… 0 0 1 1 f1f1 f2f2 Minsky & Papert: simple table lookup or logic will do nicely.

29. 29 2-D feature space, four classes… 0 0 1 1 f1f1 f2f2 Spheres or radial-basis functions may offer a compact class encapsulation in case of limited noise and limited overlap (but in the end the data will tell: experimentation required!)