Question

In kernel trick method we do not need the coordinates of the data in the feature space false true

Answer 1

Answer:

No, we do not. The prerequisite to apply the kernel trick is that wherever the algorithm uses a data point x, it is used in an expression of the form xTz, where z is some other point. For instance, in the SVM dual formulation, the optimization problem is the following:

maxαi∑iαi−12∑i∑jαiαjyiyjxTixj

s.t.

0≤αi≤C

∑iαiyi=0

The corresponding prediction function, once α’s are known, is given by:

y=f(x)=∑iαiyixTix

Note that both the objective function and the prediction function have x’s in the form xTz. Therefore, you can map the points x to ϕ(x) where you don’t need to know ϕ(x) explicitly. As long as you can compute ϕ(x)Tϕ(z), you can replace all occurrences of xTz by ϕ(x)Tϕ(z) and implicitly work in the higher-dimensional space.