Exercise 2.13

[mahaitao]

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]

For (a), maybe it is worth thinking about the dichotomies that can be generated by 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]

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]

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 , and perceptrons with . Their union is all perceptrons; their intersection is perceptrons with ---that is, perceptrons that pass the origin. Hope this helps.

[mahaitao]

I got it. Thanks for your patience, professor Lin.