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-   -   Positive rays, Positive Intervals, Convex sets and the growth function (http://book.caltech.edu/bookforum/showthread.php?t=4231)

jlaurentum 04-21-2013 09:38 PM

Positive rays, Positive Intervals, Convex sets and the growth function
 
Hello,

From what I understood in lecture 5 as the discussion progresses from positive rays to positive intervals to convex sets, the idea is to put an upper bound on all possible dichotomies. As convex sets, with an upper bound of 2^N on the growth function, are the most general scenario possible, then this is the upperbound on the growth function for perceptrons, which fall somewhere in between the positive ray- positive interval-convex sets continuum. Is my understanding correct?

Due to the political unrest in my country (Venezuela), I've not been able to get started with lecture 6 at this time, so I don't know if what I'm about to ask is addressed later on.

I imagine rotating a perceptron to the point where the boundary line is horizontal and all points fall above or bellow this boundary line:

https://dl.dropboxusercontent.com/u/...ositiveray.png

If this is possible (the rotation), then wouldn't the growth function for linearly separable data with perceptrons be limited to N+1, as in the case of positive rays?

yaser 04-21-2013 10:33 PM

Re: Positive rays, Positive Intervals, Convex sets and the growth function
 
Quote:

Originally Posted by jlaurentum (Post 10545)
Hello,

From what I understood in lecture 5 as the discussion progresses from positive rays to positive intervals to convex sets, the idea is to put an upper bound on all possible dichotomies. As convex sets, with an upper bound of 2^N on the growth function, are the most general scenario possible, then this is the upperbound on the growth function for perceptrons, which fall somewhere in between the positive ray- positive interval-convex sets continuum. Is my understanding correct?

Due to the political unrest in my country (Venezuela), I've not been able to get started with lecture 6 at this time, so I don't know if what I'm about to ask is addressed later on.

I imagine rotating a perceptron to the point where the boundary line is horizontal and all points fall above or bellow this boundary line:

https://dl.dropboxusercontent.com/u/...ositiveray.png

If this is possible (the rotation), then wouldn't the growth function for linearly separable data with perceptrons be limited to N+1, as in the case of positive rays?

The points on which to generate dichotomies are arbitrary but fixed. When you rotate the plane, the points are no longer fixed, but move with every new hypothesis, so it is not the same notion.

Stay safe.


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