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Old 08-06-2012, 11:17 AM
Andrs Andrs is offline
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Default Question about the union of hypothesis Q10?

I am having problems to understand the practical meaning of the union of hypthesis sets. What does the Union mean for the classification task? If we look into one example: There is one hypothesis set Ha (ha1,ha2,ha3...han) that defines a 2d perceptron. There is another hypothesis set Hb (hb1,hb2,hb3...hbn)that defines another 2d perceptron .
Now we create a set H = Ha U Hb that contains the hypothesis (ha1,ha2,ha3...han,hb1,hb2,hb3...hbn). The hypothesis set H contains more hypothesis but they are still of the same kind (according to the Union the components of the two sets are merged into one set ). What does H define? Does it define a 2d perceptron. Or does the H hypothesis define TWO 2d perceptrons that correspond to two lines? It is not clear to me how to get concrete view of the Union. Any suggestions....?
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Old 08-06-2012, 11:56 AM
bzlearn bzlearn is offline
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Default Re: Question about the union of hypothesis Q10?

I believe the former is correct.
If H1 = all the 2d perceptrons
and H2 = all the 2d perceptrons
H1 U H2 = H1 = H2

If however,
H1 = all 2d perceptrons with positive or zero slope
H2 = all 2d perceptrons with negative slope
H1 U H2 = all 2d perceptrons with any slope
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Old 08-06-2012, 01:45 PM
Andrs Andrs is offline
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Default Re: Question about the union of hypothesis Q10?

Thanks bzlearn!
you are right in your first assumption! I do some paraphrasing for my own understanding.
If we assume that the hypothesis set for the 2d perceptron can handle any slope and/position, the two hypothesis set should be the same. According to the set operations, the union of two identical sets is another set equal to the components. That is, there should not be any value added to the union of two 2d perceptrons.
However, if we have two different hypothesis set, e.g. a 2d perceptron (H1) and a 3d perceptron (H2), we should get a value added for the union set (H = H1 U H2). The classification result based on H should belong to either H1 or H2. It will affect the bounds for the growth function for the union set H and its vc dim.
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