wassersteinDistance · Samplinglib

Plain-English statement

W2 is the positive square root of the cheapest expected squared distance among couplings.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

W2(mu,nu) = (inf_gamma integral ||x-y||^2 dgamma)^(1/2).

Intuition

Couplings pair mass from the two laws; the quadratic cost measures how far paired mass moves.

Conditions

a measurable real normed additive state space
two measures
the compiled coupling transport cost

Why these conditions cannot be dropped

ENNReal retains infinite transport values
finite-second-moment hypotheses are needed later to prove finiteness and metric structure

Proof route

specialize transportCost to squared norm
apply ENNReal rpow one half

Lean interface notes

quadraticCost uses ENNReal.ofReal
no optimal plan or metric theorem is bundled

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Lean statement

noncomputable def wassersteinDistance
    {E : Type*} [NormedAddCommGroup E] [MeasurableSpace E]
    (μ ν : Measure E) : ℝ≥0∞ :=
  (Transport.transportCost (quadraticCost (E := E)) μ ν) ^ (1 / 2 : ℝ)

/-- Chewi display (1.3.5): the square of `W₂` is the infimum of the
quadratic costs over all couplings. -/

Open AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:29published source at 7bcd37294df1