LFD Book Forum - View Single Post - Help in understanding proof for VC-dimension of perceptron.

MaciekLeks · #8 05-06-2016, 03:34 AM

Quote:

Originally Posted by ntvy95

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).