6.4 Hilbert-Schmidt Independence Criterion
Definition
6.17
HSIC via product-space MMD
Lean: hsicDef
For a joint measure \(P\) on \(X \times Y\) with marginals \(P_X\), \(P_Y\),
\[ \mathrm{HSIC}(P) \; :=\; \mathrm{MMD}^2\bigl(P,\; P_X \otimes P_Y\bigr). \]
This is the modern unified formulation [ 20 , Thm 24 ] ; originally introduced as a cross-covariance-operator Hilbert-Schmidt norm [ 12 ] .
Theorem
6.18
Non-negativity of HSIC
Lean: hsicDef_nonneg
\(\mathrm{HSIC}(P) \ge 0\) [ 12 ] .
Theorem
6.19
HSIC vanishes on independent measures
Lean: hsicDef_eq_zero_of_independent
If \(P = P_X \otimes P_Y\) then \(\mathrm{HSIC}(P) = 0\) (direct consequence of \(\mathrm{MMD}^2(P, P) = 0\)) [ 12 ] .