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Old 04-24-2013, 03:39 PM
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yaser yaser is offline
Join Date: Aug 2009
Location: Pasadena, California, USA
Posts: 1,478
Default Re: Perceptron VC dimension

Nice argument! (and thank you for the compliment)

Originally Posted by Elroch View Post
There is a non-trivial linear relation on these points:

\sum\limits_{x\in X}a_x x = 0

Rearrange so all the coefficients are positive (changing labels for convenience)

\sum\limits_{x\in X_A}a_x x =\sum\limits_{x\in X_B}b_x x

X_A and X_B must be non-empty subsets of X because the relation is non-trivial, and the first co-ordinate of all the points is 1.
Just to elaborate, the coefficients are not all zeros so because of the constant 1 coordinate, there has to be at least one positive and one negative coefficient, hence the rest of the argument even if all other coefficients are zeros.
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