LFD Book Forum Exercises and Problems

#1
03-24-2012, 11:25 PM
 yaser Caltech Join Date: Aug 2009 Location: Pasadena, California, USA Posts: 1,477
Exercises and Problems

Please comment on the chapter problems in terms of difficulty, clarity, and time demands. This information will help us and other instructors in choosing problems to assign in our classes.

Also, please comment on the exercises in terms of how useful they are in understanding the material.
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#2
07-02-2012, 01:36 PM
 tadworthington Member Join Date: Jun 2012 Location: Chicago, IL Posts: 32
ERRATA: Small mistake in description of Exercise 2.1

I won't "mathify" this correction, as I don't know how in this forum (my LaTex has escaped me after years of neglect!). It's a minor point, but I feel like it should be corrected. In the wording for Exercise 2.1 on page 45:

ERROR: "Verify that m_H(n) < 2^k"
CORRECTION: "Verify that m_H(k) < 2^k"

Thanks!
#3
07-02-2012, 02:58 PM
 yaser Caltech Join Date: Aug 2009 Location: Pasadena, California, USA Posts: 1,477
Re: ERRATA: Small mistake in description of Exercise 2.1

Quote:
 Originally Posted by tadworthington I won't "mathify" this correction, as I don't know how in this forum (my LaTex has escaped me after years of neglect!). It's a minor point, but I feel like it should be corrected. In the wording for Exercise 2.1 on page 45: ERROR: "Verify that m_H(n) < 2^k" CORRECTION: "Verify that m_H(k) < 2^k" Thanks!
Thank you for catching this!
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#4
08-24-2012, 08:20 AM
 vsthakur Member Join Date: Jun 2012 Posts: 14
Problem 2.9 : Growth function of perceptron, seems incorrect

The problem says that in case of perceptron in d-dimensional space, growth function is equal to B(N,k). Consider the following case :

d = 2, implies dvc = 2+1 = 3
N = 4
B(N,k) = 15
but the maximum no. of dichotomies possible in this case is only 14

Can someone please comment if i am missing something.

Thanks.

Vishwajeet.
#5
08-24-2012, 12:17 PM
 magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 595
Re: Problem 2.9 : Growth function of perceptron, seems incorrect

Thanks for catching this erratum. The problem shows the upper bound based on the VC dimension. The actual growth function is given by:

.

Our appologies.

Quote:
 Originally Posted by vsthakur The problem says that in case of perceptron in d-dimensional space, growth function is equal to B(N,k). Consider the following case : d = 2, implies dvc = 2+1 = 3 N = 4 B(N,k) = 15 but the maximum no. of dichotomies possible in this case is only 14 Can someone please comment if i am missing something. Thanks. Vishwajeet.
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#6
08-25-2012, 08:04 PM
 vsthakur Member Join Date: Jun 2012 Posts: 14
Re: Problem 2.9 : Growth function of perceptron, seems incorrect

Thank you
#7
08-29-2012, 01:39 AM
 vsthakur Member Join Date: Jun 2012 Posts: 14
Re: Problem 2.9 : Growth function of perceptron, seems incorrect

Sorry for the delayed response here, but i still find that

is not the case for a perceptron in d-dimensional space.

When N=6 and d=2, this equation says , while i was able to get 38 dichotomies (by picking 6 equidistant points on the circumference of a circle).

If i am missing something, then can you please point me to the proof.

Thank you.
#8
08-29-2012, 07:09 PM
 htlin NTU Join Date: Aug 2009 Location: Taipei, Taiwan Posts: 601
Re: Problem 2.9 : Growth function of perceptron, seems incorrect

Quote:
 Originally Posted by vsthakur Sorry for the delayed response here, but i still find that is not the case for a perceptron in d-dimensional space. When N=6 and d=2, this equation says , while i was able to get 38 dichotomies (by picking 6 equidistant points on the circumference of a circle). If i am missing something, then can you please point me to the proof. Thank you.
I checked the case you are describing, and the number of dichotomies in the case is . (Hint: did you double-calculate the case of 3-positive and 3-negative?) Hope this helps.
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#9
08-29-2012, 02:10 AM
 vsthakur Member Join Date: Jun 2012 Posts: 14
Problem 2.10

To prove :

As this is a generic statement, it has to apply to every growth function. But all we know about the growth functions (in general) is their bound, in terms of and .

Also, we know that if then is an increasing function whose value is . But, if , then we can only say that is non-decreasing and is bounded by .

I guess my question is that how can we prove the generic statement above. Kindly shed some light on the proof strategy.

Thank you,

Vishwajeet.
#10
08-29-2012, 11:44 AM
 yaser Caltech Join Date: Aug 2009 Location: Pasadena, California, USA Posts: 1,477
Re: Problem 2.10

Quote:
 Originally Posted by vsthakur But all we know about the growth functions (in general) is their bound
Actually, we also know the definition of growth functions, and this may be the key to answering the question.
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