View Single Post
  #8  
Old 05-06-2016, 04:34 AM
MaciekLeks MaciekLeks is offline
Member
 
Join Date: Jan 2016
Location: Katowice, Upper Silesia, Poland
Posts: 17
Default Re: Help in understanding proof for VC-dimension of perceptron.

Quote:
Originally Posted by ntvy95 View Post
Instead of choosing y_{i} = 1 or y_{i} = -1 explicitly, the proof let the choice for the value of y_{i} depends on the value of a_{i}: y_{i} = sign(a_{i}), because for whatever the real value of a_{i} is, sign(a_{i}) only has value of 1 or -1, hence y_{i} = 1 or y_{i} = -1. In my understanding, this dependence does not make the chosen dichotomy invalid.
That's the point. This part is crucial: "the proof let the choice for the value of y_{i} depends on the value of a_{i}". The only certain correlation is y_{i}=sign(\mathbf{w}^{T}\mathbf{x}_{i})). How do we know (definition, theorem, lemma,...) that sign(a_{i}) works the same as sign(\mathbf{w}^{T}\mathbf{x}_{i})? IMHO it cannot be drawn from the \mathbf{x}_{d+2}=\sum_{i=1}^{d+1}a_{i}\mathbf{x}_{i}, (where not all a_{i} coefficients are zeros).
Reply With Quote