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

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


  • $x$, $y$ 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 $\parallel\operatorname{cov}(x^T,y^T) \parallel^2_{\text{Frobenius}}$.

  1. Gretton A, Bousquet O, Smola A, Schölkopf B. Measuring Statistical Dependence with Hilbert-Schmidt Norms. Algorithmic Learning Theory. Springer Berlin Heidelberg; 2005. pp. 63–77. doi:10.1007/11564089_7 ↩︎

  2. Kornblith S, Norouzi M, Lee H, Hinton G. Similarity of Neural Network Representations Revisited. arXiv [cs.LG]. 2019. Available: http://arxiv.org/abs/1905.00414 ↩︎

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L Ma (2021). 'Hilbert-Schmidt Independence Criterion (HSIC)', Datumorphism, 11 April. Available at: https://datumorphism.leima.is/cards/machine-learning/measurement/hilbert-schmidt-independence-criterion/.