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### How the data range in different dimension will affect Gaussian Process fitting

We should scale them. We won't find "clear evidence" because usually, most applications are with a single feature in which case the range shouldn't matter too much as we optimise the length ...
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The problem with your approach is that the transformation you use from $\pmb u$ to $\pmb x$ is non continuous: basically it chooses one of the samples $\pmb x_i$ based on the first component of \$\pmb ...