Hilbert-Schmidt Independence Criterion (HSIC)

Given two kernels of the feature representations $K=k(x,x)$ and $L=l(y,y)$, HSIC is defined as¹²

$$ \operatorname{HSIC}(K, L) = \frac{1}{(n-1)^2} \operatorname{tr}( K H L H ), $$

where

• $x$, $y$ are the representations of features,
• $n$ is the dimension of the representation of the features,
• $H$ is the so-called [[centering matrix]] Centering Matrix Useful when centering a vector around its mean .

We can choose different kernel functions $k$ and $l$. For example, if $k$ and $l$ are linear kernels, we have $k(x, y) = l(x, y) = x \cdot y$. In this linear case, HSIC is simply $\parallel\operatorname{cov}(x^T,y^T) \parallel^2_{\text{Frobenius}}$.