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Old 04-23-2013, 07:07 AM
jlaurentum jlaurentum is offline
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Default d-dimensional Perceptrons and break points (related to Q4 of homework)

Hello:

In slide 9 of lecture 5 (minute 33:03), the Professor gives an example of 3 colinear points for which there can be no possible hypothesis. Still, "it doesn't bother us because we want the maximum bound of possible dichotomies", so k=3 is not considered as a breakpoint. My question is:

In a d-dimensional perceptron, it appears we would not consider a set of points lying in a (d-1)-dimensional hyperplane as candidates for giving an "impossible" dichotomy. Why? Is it because the probability of picking such a set of points from the input space that all lie in a (d-1) dimensional space is zero? (As in the case of picking 3 collinear points in a plane).
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Old 04-23-2013, 08:13 AM
IsidroHidalgo IsidroHidalgo is offline
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Default Re: d-dimensional Perceptrons and break points (related to Q4 of homework)

No, the probability isn't cero. The question is that we are interested in the maximum of points our hypothesis can shatter. So you must take a set of points that maximizes the probability of shatter the most...
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Old 04-23-2013, 09:48 AM
Elroch Elroch is offline
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Default Re: d-dimensional Perceptrons and break points (related to Q4 of homework)

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Originally Posted by jlaurentum View Post
Hello:

In slide 9 of lecture 5 (minute 33:03), the Professor gives an example of 3 colinear points for which there can be no possible hypothesis. Still, "it doesn't bother us because we want the maximum bound of possible dichotomies", so k=3 is not considered as a breakpoint. My question is:

In a d-dimensional perceptron, it appears we would not consider a set of points lying in a (d-1)-dimensional hyperplane as candidates for giving an "impossible" dichotomy. Why? Is it because the probability of picking such a set of points from the input space that all lie in a (d-1) dimensional space is zero? (As in the case of picking 3 collinear points in a plane).
It's worth observing that the set H_n of d-dimensional perceptrons, restricted to a (d-k)-dimensional subspace, is simply H_{n-k}, the set of (d-k)-dimensional perceptrons on that subspace. hence, the capabilities of H_n restricted to the subspace is the same as that of H_{n-k}.

It turns out that the power of the hypothesis set comprising perceptrons increases as the dimension of their domain increases. The three points are a good example. If co-linear, they cannot be shattered, regardless of what dimension space they are in. If not co-linear, they can always be shattered: this requires the domain to be at least 2-dimensional.
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Old 04-23-2013, 09:55 AM
jlaurentum jlaurentum is offline
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Default Re: d-dimensional Perceptrons and break points (related to Q4 of homework)

Ok. This 3 point set: +1 -1 +1 cannot be shattered if the 3 points are collinear, no matter what dimension the perceptron is. Why isnt three the break point for a 2-d perceptron (or a 3-d perceptron, for that matter)? What is the reason that we must consider point sets that are in the same dimension as the input space?
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Old 04-23-2013, 09:57 AM
Elroch Elroch is offline
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Default Re: d-dimensional Perceptrons and break points (related to Q4 of homework)

That's simply a matter of the definition!

The break point is the (minimum) value of n such that no set of n points can be shattered. To put it another way, it is the (minimum) value of n such that every set of n points fails to be shattered. Finding one set of n points that fails to be shattered is consistent with the existence of a break point, but you need to demonstrate all other sets of n points have the same property.
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Old 04-23-2013, 11:05 AM
jlaurentum jlaurentum is offline
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Default Re: d-dimensional Perceptrons and break points (related to Q4 of homework)

Now I'm confused. The break point for 2-d perceptrons is 4. In lecture 5, one example of a 4-point set is given that is not shatterable. However, there are other 4-point sets that are (shatterable). Likewise for positive rays, positive intervals, where the break point is 2 and 3 respectively.
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Old 04-23-2013, 11:53 AM
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yaser yaser is offline
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Default Re: d-dimensional Perceptrons and break points (related to Q4 of homework)

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Originally Posted by jlaurentum View Post
The break point for 2-d perceptrons is 4. In lecture 5, one example of a 4-point set is given that is not shatterable. However, there are other 4-point sets that are (shatterable).
Hi,

It is actually not possible to shatter any set of 4 points using the 2-dimensional perceptron. Perhaps we can discuss the set of points you have in mind and look for which dichotomies would be impossible there.
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