If we have a sample dataset $S = \{(x_1, y_i),\dots,(x_n,y_n)\}$ where $y_i = \{0,1\}$, how can we tune $\sigma$ such that there is no error on $S$ from a classifier using the Laplacian kernel?

Laplacian Kernel is

$$ K(x,x') = \exp\left(-\dfrac{\| x - x'\|}{\sigma}\right) $$

If this is true, does it mean that if we run hard-SVM with the Laplacian kernel and $\sigma$ from the above on $S$, we can find no error separating classifier also?

My solution:

For the no error on Laplacian kernel, my approach was to tune $\sigma$, to show that any polynomial predictor over the original space can be learned by using the Laplacian Kernel. Then show that VC dimension of the class all polynomial predictors is infinite, meaning that any labeling can be classified without any error.

For the hard-SVM, I need to find the supporting vectors using this sigma. If the supporting vectors can be correctly found then its ok.

Is this the right approach, please advise.

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