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-   Chapter 2 - Training versus Testing (http://book.caltech.edu/bookforum/forumdisplay.php?f=109)
-   -   Exercise 2.13 (http://book.caltech.edu/bookforum/showthread.php?t=4531)

mahaitao 10-23-2014 08:43 PM
Exercise 2.13
 
In Exercise 2.13 (a), Prove dvc(H)<=log_2M. How to think this problem? M is the number of hypotheses, what is the relationship between dvc and M?

(b) What does dvc(\cap H_k) and \cap H_k the intersection of hypotheses mean? How can we intersect hypotheses?

htlin 10-24-2014 11:54 PM
Re: Exercise 2.13
 
For (a), maybe it is worth thinking about the dichotomies that can be generated by M hypotheses?

For (b), the intersection and union of hypothesis "sets" are simply "set" intersection and union.

http://en.wikipedia.org/wiki/Union_%28set_theory%29

Hope this helps.

mahaitao 10-26-2014 05:50 PM
Re: Exercise 2.13
 
Professor Lin,
I think that I did not represent my question clear.

In perceptron example, we have only one H that is a set of infinite lines in the plane. My question is if we consider that H is a union of some subsets, what are they? They are subsets of these infinite lines? How to distinguish them?

And how about intersection of these subsets?

htlin 10-29-2014 02:56 AM
Re: Exercise 2.13
 
For instance, the union of "positive rays" and "negative rays" is "positive or negative rays" which is simply 1-D perceptron. Similarly, you can have perceptrons with w_0 \ge 0, and perceptrons with w_0 \le 0. Their union is all perceptrons; their intersection is perceptrons with w_0 = 0---that is, perceptrons that pass the origin. Hope this helps.

mahaitao 10-30-2014 05:28 PM
Re: Exercise 2.13
 
I got it. Thanks for your patience, professor Lin.


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