LFD Book Forum Exercise problem 2.4
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#1
02-25-2013, 01:27 PM
 cls2k Junior Member Join Date: May 2012 Posts: 5
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.
#2
02-25-2013, 01:57 PM
 magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 595
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 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.
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#3
02-25-2013, 09:40 PM
 cls2k Junior Member Join Date: May 2012 Posts: 5
Re: Exercise problem 2.4

Thank you! this is such an awesome and elegant insight and now the proof is easy.
#4
02-27-2013, 07:53 AM
 BojanVujatovic Member Join Date: Jan 2013 Posts: 13
Re: Exercise problem 2.4

Professor, thank you for this amazingly helpful hint!
#5
03-05-2018, 07:30 AM
 k_sze Member Join Date: Dec 2016 Posts: 12
Re: Exercise problem 2.4

I still don't get it after reading this hint.

Could somebody PM me to help me out so I don't ruin it for everybody else?
#6
03-15-2018, 08:50 AM
 k_sze Member Join Date: Dec 2016 Posts: 12
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 out of points. That's your minimum .

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