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Consider the following words taken from the lecture notes:

The Hilbert-Schmidt Independence Criterion (HSIC) measures the dependence of the two random variables $X$ and $Y$. An empirical estimate of the HSIC is proportional to the $trace(KHLH)$, where

  • $K$ is a kernel matrix on $X$,
  • $L$ is a kernel on $Y$,
  • $H$ is a centering matrix with $H_{ij} = δ(i, j) − \frac{1}{n}$.

$HSIC(X, Y ) = 0$ if and only if $X$ and $Y$ are independent. The larger $HSIC(X, Y)$, the larger the dependence between $X$ and $Y$.

For me the only unclear part of this definition is the "[K] kernel matrix on a distribution [X]" sentence. Can someone explain what it means to apply the kernel on a distribution?

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