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-   -   Exercise problem 2.4 (http://book.caltech.edu/bookforum/showthread.php?t=4030)

cls2k 02-25-2013 01:27 PM

Exercise problem 2.4
 
I'm stuck at the exercise problem 2.4 behind the book despite the hint. My approach is to characterize the B(N+1,K) >= recursion as an upper bound on the lower order (in N) terms and then follow the approach of proving the Sauer lemma. However I'm stuck on constructing the "specific set" of dichotomies. I fail to see how the special property to this set (limiting the number of -1 as hinted) can make this proof go easier.

I'm not very good at mathematical proofs so any additional hints will be greatly appreciated.

magdon 02-25-2013 01:57 PM

Re: Exercise problem 2.4
 
Here is a hint. Lets consider showing B(5,2)>=1+5; It suffices to list 6 dichotomies on 5 points such that no subset of size 2 is shattered.

Consider the following 6 dichotomies on 5 points:

[1 1 1 1 1] (zero -1s)

[-1 1 1 1 1] (5 ways of having one -1)
[1 -1 1 1 1]
[1 1 -1 1 1]
[1 1 1 -1 1]
[1 1 1 1 -1]

Can you show that no subset of 2 points is shattered? Hint: are there any two points that are classified -1,-1?

So, in general, you can guarantee that no subset of size k can be shattered if no subset of size k is classified all -1. This means that at most k-1 points are classified -1 by any dichotomy.

Quote:

Originally Posted by cls2k (Post 9532)
I'm stuck at the exercise problem 2.4 behind the book despite the hint. My approach is to characterize the B(N+1,K) >= recursion as an upper bound on the lower order (in N) terms and then follow the approach of proving the Sauer lemma. However I'm stuck on constructing the "specific set" of dichotomies. I fail to see how the special property to this set (limiting the number of -1 as hinted) can make this proof go easier.

I'm not very good at mathematical proofs so any additional hints will be greatly appreciated.


cls2k 02-25-2013 09:40 PM

Re: Exercise problem 2.4
 
Thank you! this is such an awesome and elegant insight and now the proof is easy.

BojanVujatovic 02-27-2013 07:53 AM

Re: Exercise problem 2.4
 
Professor, thank you for this amazingly helpful hint!

k_sze 03-05-2018 07:30 AM

Re: Exercise problem 2.4
 
I still don't get it after reading this hint. :clueless:

Could somebody PM me to help me out so I don't ruin it for everybody else?

k_sze 03-15-2018 08:50 AM

Re: Exercise problem 2.4
 
Never mind. I got it.

Put another way: think of a way to systematically enumerate a set of dichotomies that won't shatter k out of N points. That's your minimum B(N, k).


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