wassersteinDistance_sq · Samplinglib

Plain-English statement

Squaring the extended W2 value recovers the quadratic Kantorovich cost.

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)^2 = transportCost((x,y) maps to ||x-y||^2,mu,nu).

Intuition

This is the source display that relates distance notation to the coupling optimization problem.

Conditions

the ENNReal W2 definition

Why these conditions cannot be dropped

extended-real algebra keeps the equality valid even when the transport cost is infinite

Proof route

rewrite natural square as real rpow two
combine rpow exponents one half and two

Lean interface notes

the proof uses ENNReal.rpow_two and ENNReal.rpow_mul
attainment is not used

Lean learning studio · mathematics → formal proof

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Lean statement

theorem wassersteinDistance_sq
    {E : Type*} [NormedAddCommGroup E] [MeasurableSpace E]
    (μ ν : Measure E) :
    wassersteinDistance μ ν ^ 2 =
      Transport.transportCost (quadraticCost (E := E)) μ ν := by
  rw [wassersteinDistance, ← ENNReal.rpow_two, ← ENNReal.rpow_mul]
  norm_num

/-- Chewi Definition 1.3.12: a probability measure in `P₂,ac` has finite
second moment and is absolutely continuous with respect to Lebesgue volume.

The generic finite-dimensional real inner-product space specializes to
Euclidean `R^d` in the source. -/

Open AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:36published source at 7bcd37294df1