Monotone rearrangement theorem · W₁ and W₂ distances · Sinkhorn algorithm for discrete OT
Optimal transport finds the cheapest way to move mass from distribution μ to ν.
W₁ = ∫|F_μ⁻¹(t)−F_ν⁻¹(t)|dt (quantile coupling — monotone rearrangement, cheapest for convex costs on ℝ).
W₂² = ∫(T(x)−x)² dμ(x) where T=∇φ is a gradient of a convex function (Brenier theorem).
Sinkhorn: add entropic regularization ε·H(P), solve via alternating row/column normalization.
Right: discrete OT coupling matrix between 8-point distributions.