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For a Gaussian kernel, what is the sigma value, and how is it calculated?

$K(\mathbf{x}_i,\mathbf{x}_j) = \exp{-\frac{\|\mathbf{x}_i-\mathbf{x}_j\|^2}{\sigma^2}}$ ?

Is it the covariance of the entire data set?

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For a Gaussian kernel, what is the sigma value, and how is it calculated?

As @SleepWithStatistics pointed in the comments, it's not calculated: it's a parameter set prior to the kernel computations.

Is it the covariance of the entire data set?

No, but heuristics exist to set the parameter based on the variance/covariance structure in the data.

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