Wasserstein distance

The Wasserstein distance of order $p\ge 1$ between probability measures $\mu ,\nu$ on a Polish metric space $(\mathcal{X},d)$ is defined as the $p$-th root of the optimal transport cost with power cost:

$W_p(\mu ,\nu )={\left({\inf }_{\pi \in \Pi (\mu ,\nu )}\int d(x,y{)}^{p}d\pi (x,y)\right)}^{1/p}$

The Wasserstein space $P_p(\mathcal{X})$ consists of all probability measures with finite $p$-th moment, equipped with $W_p$. When $\mathcal{X}$ is Polish, $P_p(\mathcal{X})$ is itself Polish (complete, separable). The embedding $x\mapsto {\delta }_x$ is an isometry: $W_p({\delta }_x,{\delta }_y)=d(x,y)$.

$W_p$ metrises weak convergence in $P_p$: $W_p({\mu }_k,\mu )\to 0$ iff ${\mu }_k\to \mu$ weakly and $p$-th moments converge. The two most important cases are $p=1$ (Kantorovich-Rubinstein distance, dual to Lipschitz functions, most flexible) and $p=2$ (reflects Riemannian geometry, scales well with dimension, natural for displacement interpolation).

Key Details

• Triangle inequality: proved via the Gluing Lemma + Minkowski inequality in $L^p$
• $W_p\le W_q$ for $p\le q$ (Hölder’s inequality)
• $W_1$ dual: $W_1(\mu ,\nu )={\sup }_{\Vert \psi {\Vert }_{\text{Lip}}\le 1}(\int \psi d\mu -\int \psi d\nu )$
• $P_p(\mathcal{X})$ compact iff $\mathcal{X}$ compact; but NOT locally compact if $\mathcal{X}$ is only locally compact
• Controlled by weighted total variation: $W_p(\mu ,\nu )\le 2^{1/p}(\int d(x_0,x{)}^{p}d|\mu -\nu |{)}^{1/p}$
• For $p>1$: displacement interpolation ${\mu }_t$ is a constant-speed geodesic in $(P_p,W_p)$

Textbook References

Optimal Transport Old and New (Villani, 2009)

• Definition 6.1 (p. 105): $W_p(\mu ,\nu )=({\inf }_{\pi }\int d^{p}d\pi {)}^{1/p}$
• Definition 6.4 (pp. 106-107): $P_p(\mathcal{X})=\{\mu :\int d(x_0,x{)}^{p}d\mu <\infty \}$
• Remark 6.5 (p. 107): Kantorovich-Rubinstein duality for $W_1$
• Theorem 6.9 (p. 108): $W_p$ metrises weak convergence in $P_p$
• Theorem 6.18 (pp. 116-117): $P_p(\mathcal{X})$ is Polish. Proof: separability via finite-support approximation; completeness via Prokhorov + Cauchy tightness (Lemma 6.14)
• Five reasons $W_p$ is preferred (pp. 110-111): handles large distances, natural for OT, rich duality, easy upper bounds via any coupling, encodes geometry

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