Foundation Lab

Optimal Transport & Wasserstein Distance

Wasserstein distance explains why WGANs are more stable than vanilla GANs—it provides gradients even when distributions do not overlap

Concept 38 of 100TheoryPhase 8

#38OT/WassersteinTheory

key equation

W_p(\mu, \nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int \|x - y\|^p \, d\gamma(x, y) \right)^{1/p}

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Concept 38 of 100OT/WassersteinTheory / Phase 8: Scaling, theory & multimodal

Current question

KL divergence is infinite when supports do not overlap; Wasserstein is finite and measures "how far to move mass"

W_p(\mu, \nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int \|x - y\|^p \, d\gamma(x, y) \right)^{1/p}

PredictionCommit before tracing the equation.

Ask what should change under a concrete input, then trace that expectation through the equation.

EvidenceCompare the equation and source.

Use the key equation and canonical papers as the available witnesses, without implying that a runnable panel exists.

InvariantName what survives notation changes.

The useful learning product is the reusable mechanism you can carry into another model, paper, or engineering tradeoff.

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Use prerequisites, dependents, and semantic links to repair the next gap without leaving the object behind.

Wasserstein distance explains why WGANs are more stable than vanilla GANs—it provides gradients even when distributions do not overlap
Flow matching and rectified flows are built on OT: they learn the optimal transport map directly
OT provides a principled way to measure distance between distributions that respects geometry

What is still poorly explained in textbooks and papers:

KL divergence is infinite when supports do not overlap; Wasserstein is finite and measures "how far to move mass"
The "earth mover" intuition: imagine distributions as piles of dirt, OT finds the cheapest way to reshape one into the other
Entropic regularization (Sinkhorn) makes OT tractable: adds −εH(γ) to get differentiable, GPU-friendly algorithms

If you want intuition first, start with the key equation and cited sources. Come back here for the full walkthrough.

Key Equation

W_p(\mu, \nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int \|x - y\|^p \, d\gamma(x, y) \right)^{1/p}

Optimal transport finds the minimum-cost way to move mass from distribution $\mu$ to $\nu$ :

W_p(\mu, \nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int \|x - y\|^p \, d\gamma(x, y) \right)^{1/p}

where $\Gamma(\mu, \nu)$ is the set of joint distributions with marginals $\mu, \nu$ .

Kantorovich duality (key for computation):

W_1(\mu, \nu) = \sup_{\|f\|_L \leq 1} \mathbb{E}_{x \sim \mu}[f(x)] - \mathbb{E}_{y \sim \nu}[f(y)]

Brenier's theorem: For absolutely continuous $\mu$ , the optimal map is $T = \nabla \phi$ for convex $\phi$ .

Peyré & Cuturi2019Foundations and Trends in ML

Arjovsky, Chintala, Bottou2017ICML

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