Maximum Mean Discrepancy (MMD²)

mmdSq P Q := ‖μ_P − μ_Q‖² where μ_· is the kernel mean embedding. Basic properties: nonnegativity, mmdSq P P = 0, symmetry.

noncomputable def mmdSq {X : Type u_1} [MeasurableSpace X] {H : Type u_2} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] [RKHS ℝ H X ℝ] (P Q : MeasureTheory.Measure X) : ℝ

Squared MMD between two measures P and Q via their kernel mean embeddings.

Equations

• mmdSq P Q = ‖kernelMeanEmbedding P - kernelMeanEmbedding Q‖ ^ 2

theorem mmdSq_nonneg {X : Type u_1} [MeasurableSpace X] {H : Type u_2} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] [RKHS ℝ H X ℝ] (P Q : MeasureTheory.Measure X) : 0 ≤ mmdSq P Q

MMD² is nonnegative.

theorem mmdSq_self {X : Type u_1} [MeasurableSpace X] {H : Type u_2} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] [RKHS ℝ H X ℝ] (P : MeasureTheory.Measure X) : mmdSq P P = 0

MMD²(P, P) = 0.

theorem mmdSq_comm {X : Type u_1} [MeasurableSpace X] {H : Type u_2} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] [RKHS ℝ H X ℝ] (P Q : MeasureTheory.Measure X) : mmdSq P Q = mmdSq Q P

MMD² is symmetric.