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-   -   On Q9 (http://book.caltech.edu/bookforum/showthread.php?t=2606)

skyw2012 10-24-2012 09:54 PM

On Q9
 
Trying to solve this question (sadly I failed) I realized that I didn't quite understand the shattering correctly in the beginning. In the meantime I've clarified this but still I don't know how to answer this question.
I imagine that must be a particular way of either selecting the right sets of points from the options of the question to prove they can be shattered or to identify the break point for this case to ensure that any set of points equal or greater can't be shattered.
Any suggestions on how to approach the solution? Thank you

yaser 10-24-2012 10:56 PM

Re: On Q9
 
Quote:

Originally Posted by skyw2012 (Post 6959)
Trying to solve this question (sadly I failed) I realized that I didn't quite understand the shattering correctly in the beginning. In the meantime I've clarified this but still I don't know how to answer this question.
I imagine that must be a particular way of either selecting the right sets of points from the options of the question to prove they can be shattered or to identify the break point for this case to ensure that any set of points equal or greater can't be shattered.
Any suggestions on how to approach the solution? Thank you

There is no systematic way of doing this, but here are some tips.

The choice of where to place the points should be done so as to maximize the chances that they will be shattered. Since the triangle is a convex region, having one of the points being 'internal' to other points will preclude the possibility of shattering (same situation as in the case of convex regions discussed in the lecture). Therefore, choosing the points at the perimeter of a circle (for example) would be a good starting point.

Now, imagine all kinds of triangles trying to split these points into different dichotomies, and look for how many points you can afford to have while being able to split them in every possible way using triangles. It's an interesting puzzle to work on.

melipone 01-26-2013 11:05 AM

Re: On Q9
 
Can it be several overlapping triangles (like in picture) or just one triangle?
http://imgur.com/T95H3o5.png

ezreal 01-26-2013 11:30 AM

Re: On Q9
 
Since this thread doesn't say *Answer* in the title, I don't think we can discuss any hints further than the notion of clever point placement.

Quote:

Originally Posted by melipone (Post 9007)
Can it be several overlapping triangles (like in picture) or just one triangle?
http://imgur.com/T95H3o5.png

I believe it should only be one triangle.

MartinT 01-26-2013 01:44 PM

Re: On Q9
 
One word in addition - with triangles we can't afford the convenience of having arbitrarily many vertices (as with convex sets in general - just stating the obvious). It might therefore be useful to keep in mind that the triangle's vertices don't have to rest on the circle, though the data points better do. As the question states, for a point to be covered (h(x)=+1) it suffices if it lies within the triangle, not necessarily directly on its edges or vertices.

Elroch 04-17-2013 06:14 PM

Re: On Q9
 
It's interesting to consider the relationship between the parametrisation of hypotheses and VC dimension. A triangle requires a fixed finite number of parameters (count them :) ), more general convex sets can require an unbounded number of parameters. I wouldn't expect a hypothesis set with a finite parametrisation to have an infinite VC dimension (although the two dimensions are only loosely related) but I haven't investigated the details much.


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