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prof. dr. Lambert Schomaker
Shattering two binary dimensionsover 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
0
0
1
1
f1
f2
4
2-D feature space, 2 classes
0
0
1
1
f1
f2
5
the other class…
0
0
1
1
f1
f2
6
2 left vs 2 right
0
0
1
1
f1
f2
7
top vs bottom
0
0
1
1
f1
f2
8
right vs left
0
0
1
1
f1
f2
9
bottom vs top
0
0
1
1
f1
f2
10
lower-right outlier
0
0
1
1
f1
f2
11
lower-left outlier
0
0
1
1
f1
f2
12
upper-left outlier
0
0
1
1
f1
f2
13
upper-right outlier
0
0
1
1
f1
f2
14
etc.
0
0
1
1
f1
f2
15
2-D feature space
0
0
1
1
f1
f2
16
2-D feature space
0
0
1
1
f1
f2
17
2-D feature space
0
0
1
1
f1
f2
18
XOR configuration A
0
0
1
1
f1
f2
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XOR configuration B
0
0
1
1
f1
f2
20
2-D feature space, two classes: 16 hypotheses
f1=0f1=1f2=0f2=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
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2-D feature space, two classes, 16 hypotheses
f1=0f1=1f2=0f2=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
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XOR, a possible non-linear separation
0
0
1
1
f1
f2
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XOR, a possible non-linear separation
0
0
1
1
f1
f2
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2-D feature space, three classes, # hypotheses?
f1=0f1=1f2=0f2=1
0 1 2 3 4 5 6 7 8
…………
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2-D feature space, three classes, # hypotheses?
f1=0f1=1f2=0f2=1
0 1 2 3 4 5 6 7 8
…………
34 = 81 possible hypotheses
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Maximum, discrete space
Four classes: 44 = 256 hypotheses Assume that there are no more classes than
discrete cells Nhypmax = ncellsnclasses
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2-D feature space, three classes…
0
0
1
1
f1
f2
In this example, is linearly separatablefrom the rest, as is .
But is not linearly separatable from the rest of the classes.
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2-D feature space, four classes…
0
0
1
1
f1
f2 Minsky & Papert:simple tablelookup or logic will do nicely.
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2-D feature space, four classes…
0
0
1
1
f1
f2 Spheres or radial-basisfunctions may offer a compact classencapsulation in case of limited noise andlimited overlap
(but in the end the datawill tell: experimentationrequired!)
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