Hilbert-Schmidt Independence Criterion (HSIC): definition #
hsicDef P := MMD²(P, P_X ⊗ P_Y) where P_X and P_Y are the marginals
of P on X × Y. Nonnegative; zero when P factors as an independent
product.
noncomputable def
hsicDef
{X : Type u_1}
[MeasurableSpace X]
{Y : Type u_2}
[MeasurableSpace Y]
{H : Type u_3}
[NormedAddCommGroup H]
[InnerProductSpace ℝ H]
[CompleteSpace H]
[RKHS ℝ H (X × Y) ℝ]
(P : MeasureTheory.Measure (X × Y))
:
Hilbert-Schmidt Independence Criterion: MMD²(P, P_X ⊗ P_Y).
Equations
- hsicDef P = mmdSq P ((MeasureTheory.Measure.map Prod.fst P).prod (MeasureTheory.Measure.map Prod.snd P))
Instances For
theorem
hsicDef_nonneg
{X : Type u_1}
[MeasurableSpace X]
{Y : Type u_2}
[MeasurableSpace Y]
{H : Type u_3}
[NormedAddCommGroup H]
[InnerProductSpace ℝ H]
[CompleteSpace H]
[RKHS ℝ H (X × Y) ℝ]
(P : MeasureTheory.Measure (X × Y))
:
HSIC is nonnegative.
theorem
hsicDef_eq_zero_of_independent
{X : Type u_1}
[MeasurableSpace X]
{Y : Type u_2}
[MeasurableSpace Y]
{H : Type u_3}
[NormedAddCommGroup H]
[InnerProductSpace ℝ H]
[CompleteSpace H]
[RKHS ℝ H (X × Y) ℝ]
(P : MeasureTheory.Measure (X × Y))
(hind : P = (MeasureTheory.Measure.map Prod.fst P).prod (MeasureTheory.Measure.map Prod.snd P))
:
If the joint distribution is an independent product, HSIC vanishes.