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

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 2-D feature space f1f1 f2f2

4 2-D feature space, 2 classes f1f1 f2f2

5 the other class… f1f1 f2f2

6 2 left vs 2 right f1f1 f2f2

7 top vs bottom f1f1 f2f2

8 right vs left f1f1 f2f2

9 bottom vs top f1f1 f2f2

10 lower-right outlier f1f1 f2f2

11 lower-left outlier f1f1 f2f2

12 upper-left outlier f1f1 f2f2

13 upper-right outlier f1f1 f2f2

14 etc f1f1 f2f2

15 2-D feature space f1f1 f2f2

16 2-D feature space f1f1 f2f2

17 2-D feature space f1f1 f2f2

18 XOR configuration A f1f1 f2f2

19 XOR configuration B f1f1 f2f2

20 2-D feature space, two classes: 16 hypotheses f 1 =0 f 1 =1 f 2 =0 f 2 = “hypothesis” = possible class partioning of all data samples

21 2-D feature space, two classes, 16 hypotheses f 1 =0 f 1 =1 f 2 =0 f 2 = two XOR class configurations: 2/16 of hypotheses requires a non-linear separatrix

22 XOR, a possible non-linear separation f1f1 f2f2

23 XOR, a possible non-linear separation f1f1 f2f2

24 2-D feature space, three classes, # hypotheses? f 1 =0 f 1 =1 f 2 =0 f 2 = ……………………

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

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

27 2-D feature space, three classes… 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 2-D feature space, four classes… f1f1 f2f2 Minsky & Papert: simple table lookup or logic will do nicely.

29 2-D feature space, four classes… 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!)