Optimal transport and Wasserstein distance

For probability measures μ and ν, a coupling is a probability measure π on pairs (x,y) whose first marginal is μ and second marginal is ν. A discrete coupling is a nonnegative matrix with prescribed row and column sums. Its entry πᵢⱼ tells how much probability mass is assigned from source xᵢ to target yⱼ. A coupling can split a source's mass across multiple destinations. Restricting to a deterministic transport map is a different problem and may exclude feasible assignments when source atoms cannot be divided into the required target masses.

After choosing a cost c(x,y), minimize its average over admissible couplings. For finite supports this is a linear program: the variables are masses, constraints conserve each marginal, and the objective is linear in those variables. The independent product coupling always supplies a feasible probability coupling, although its cost can be infinite on an unbounded space. Existence of a finite-cost optimizer in general spaces needs hypotheses such as appropriate tightness and lower semicontinuity. A colorful set of arrows is not enough; checking marginal constraints is the first certificate of admissibility.

Let X be a complete separable metric space with distance d, and take p≥1. On probability measures with finite pth moment relative to one, hence any, base point, W_p is the pth root of the least average d(x,y)^p. In this setting it is a genuine metric. Symmetry and zero distance follow from the ground metric; the triangle inequality uses compatible couplings and an Lp inequality. Replacing d^p by an arbitrary cost or forgetting the final root can destroy the metric properties one intended to use.

On the real line, matching ordered quantiles is optimal for the convex power cost. For empirical measures with equal numbers of equally weighted atoms, sort both lists and pair corresponding entries. This is a special low-dimensional shortcut, not a general permission to pair arbitrary labeled points in higher dimensions. W₁ additionally has a dual formulation involving functions with Lipschitz constant at most one under standard metric-space hypotheses. That dual view connects transport to test functions, but a neural network critic only approximates it if the function-class and Lipschitz restrictions are controlled.

Entropic regularization changes the objective by adding an entropy-related penalty scaled by a positive parameter. This can enable efficient scaling algorithms for positive discrete kernels, but the resulting objective is not automatically the original Wasserstein distance. It can assign a nonzero self-cost, depending on the convention. Debiased constructions address one issue without making every finite-parameter approximation identical to unregularized transport. Separate three errors: approximating distributions by finite samples, modifying the optimization problem, and solving that modified problem only to finite numerical accuracy. Each requires its own diagnostic.

The geometric choice also matters before any algorithm runs. Changing units changes d and hence the numerical transport distance. In feature spaces, an arbitrary embedding may assign distances that fail to represent the distinctions a scientific question needs. The optimization lab below follows gradient descent on x²/2 with adjustable step size and initial value. It demonstrates optimization stability only. It has no coupling constraints, transport matrix, Wasserstein gradient flow, or entropic kernel, so use it to question the solver step rather than to infer a transport answer.