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Old 03-25-2012, 12:25 AM
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yaser yaser is offline
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Default 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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Old 07-02-2012, 02:36 PM
tadworthington tadworthington is offline
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Default 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!
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Old 07-02-2012, 03:58 PM
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Default Re: ERRATA: Small mistake in description of Exercise 2.1

Quote:
Originally Posted by tadworthington View Post
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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Old 08-24-2012, 09:20 AM
vsthakur vsthakur is offline
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Default 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.
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Old 08-24-2012, 01:17 PM
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Default 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:

2\sum_{i=0}^d \left({N-1}\atop i\right).

Our appologies.

Quote:
Originally Posted by vsthakur View Post
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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Old 08-25-2012, 09:04 PM
vsthakur vsthakur is offline
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Default Re: Problem 2.9 : Growth function of perceptron, seems incorrect

Thank you
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Old 08-29-2012, 02:39 AM
vsthakur vsthakur is offline
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Default Re: Problem 2.9 : Growth function of perceptron, seems incorrect

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

m_{H}(N) = 2 \sum_{i=0}^d \left({N-1}\atop i\right)

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

When N=6 and d=2, this equation says m_{H}(n) = 32, 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.
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Old 08-29-2012, 08:09 PM
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Default Re: Problem 2.9 : Growth function of perceptron, seems incorrect

Quote:
Originally Posted by vsthakur View Post
Sorry for the delayed response here, but i still find that

m_{H}(N) = 2 \sum_{i=0}^d \left({N-1}\atop i\right)

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

When N=6 and d=2, this equation says m_{H}(n) = 32, 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 32. (Hint: did you double-calculate the case of 3-positive and 3-negative?) Hope this helps.
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Old 08-29-2012, 03:10 AM
vsthakur vsthakur is offline
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Default Problem 2.10

To prove : m_H(2N) \le m_H(N)^2

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 N and d_{vc}.

Also, we know that if N \le d_{vc} then m_H(N) is an increasing function whose value is 2^N. But, if N > d_{vc}, then we can only say that m_H(N) is non-decreasing and is bounded by N^{d_{vc}} + 1.

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.
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  #10  
Old 08-29-2012, 12:44 PM
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yaser yaser is offline
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Default Re: Problem 2.10

Quote:
Originally Posted by vsthakur View Post
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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