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The Euclidean quadratic Wasserstein space

Definition

For an integer $d \geq 1$ , let $R^{d}$ have its Euclidean norm and Borel sigma-algebra. Define $P_{2} (R^{d})$ to be the Borel probability measures $μ$ with $\int ∣ x ∣^{2} d μ (x) < \infty$ . A coupling of $μ, ν \in P_{2} (R^{d})$ is a Borel probability measure $π$ on $R^{d} \times R^{d}$ whose first and second coordinate marginals are $μ$ and $ν$ ; write $Γ (μ, ν)$ for the set of couplings. Define $W_{2} (μ, ν) = {(\inf_{π \in Γ (μ, ν)} \int_{R^{d} \times R^{d}} ∣ x - y ∣^{2} d π (x, y))}^{1 / 2} .$

Definition references

def borel probability law on a polish space

def probability measure

def expectation of a nonnegative or integrable random variable

def product sigma algebra and finite product sigma algebras

lem borel product of second countable spaces

def polish space

Filippo Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, §§5.1–5.3

Lénaïc Chizat and Francis Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport, Appendix A.1, PDF pp. 12–13

Facts & Assumptions

Given: An integer $d \geq 1$ and $μ, ν \in P_{2} (R^{d})$ .

[F1]

The product of two probability measures exists, is a probability, and has the two given marginals. The product measure of two sigma-finite measure spaces For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique

[F2]

For a nonnegative function on a product probability space, Tonelli identifies its integral with either iterated integral; $∣ x - y ∣^{2} \leq 2 ∣ x ∣^{2} + 2 ∣ y ∣^{2}$ . Tonelli's theorem for nonnegative measurable functions on a sigma-finite product Expectation of a nonnegative or integrable random variable

Proof

1.1givenF1

The product law $μ \otimes ν$ is a Borel probability on $R^{d} \times R^{d}$ with marginals $μ, ν$ by [F1], so it belongs to $Γ (μ, ν)$ . Thus the coupling set in the definition is nonempty.

2.1givenF1F2step 1.1∎

By the inequality in [F2] and the finite second moments of $μ, ν$ , Tonelli gives $\int ∣ x - y ∣^{2} d (μ \otimes ν) \leq 2 \int ∣ x ∣^{2} d μ + 2 \int ∣ y ∣^{2} d ν < \infty .$ All coupling costs are nonnegative, so their infimum is finite and nonnegative; its square root in the displayed definition is therefore a finite real number.

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Sources

Filippo Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, §§5.1–5.3 (standard reference, not scraped)
Lénaïc Chizat and Francis Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport, Appendix A.1, PDF pp. 12–13 (standard reference, not scraped)