why in SVM we have different indices for dot product?

Question

I am confused by Lagrangian method in SVM, I can not understand why we use different indices in dot product.

Suppose with using Lagrangian W is :

$ W_{i}=\sum_{i}L_{i}y_{i}x_{i} $

In SVM objective function we have

$min : 0.5 ( W_{i}^{2} )$

So I think we must have

$W_{i}^{2} =\sum_{i}L_{i}^{2}y_{i}x_{i}.x_{i} $

why we have

$W_{i}^{2} =\sum_{i}L_{i}L_{j}y_{j}y_{i}x_{i}.x_{j} $

these are very different in the second one we multiply all elements of x together which it does not make sense to me.

Answer 1

This is because dot product is a bi-linear form. suppose you have 2 observation $x_{1},x_{2}$ and 3 variables $ w_{1},w_{2},w_{3}$ then you have

$w_{1}=L_{1}y_{1}x_{11}+L_{2}y_{2}x_{21}$

$w_{2}=L_{1}y_{1}x_{12}+L_{2}y_{2}x_{22}$

$w_{3}=L_{1}y_{1}x_{13}+L_{2}y_{2}x_{23}$

Now in $W.W$ we have

$W.W=w_{1}w_{1}+w_{2}w_{2}+w_{3}w_{3}$

But notice that

$w_{1}w_{1}=(L_{1}y_{1}x_{11}+L_{2}y_{2}x_{21}).(L_{1}y_{1}x_{11}+L_{2}y_{2}x_{21})$