Optimal Transport
Optimal transport measures the distance between distributions as the least cost of moving one distribution's mass onto another — meaningful even when supports are disjoint. This base covers the Monge / Kantorovich formulations, the Wasserstein distance, and the everywhere-positive Boltzmann coupling that entropic regularization (Sinkhorn) yields — the heart of EVIA / CryoWGEN.
Each cell is shaded by its Boltzmann weight exp(−cᵢⱼ/γ): small γ concentrates mass on the low-cost diagonal band, approaching a hard assignment; large γ smears the weights with entropy and the coupling spreads toward the independent product μ⊗ν.
Articles in this base 3 articles
Optimal transport & the Wasserstein distance
Measuring the distance between probability distributions by the minimal cost of moving mass from one to the other.
Entropic optimal transport & Sinkhorn
Adding an entropy / KL penalty to the transport cost yields a unique, everywhere-positive smooth coupling, solved efficiently by Sinkhorn matrix scaling.
Kantorovich duality & the dual potentials
Every transport problem has a dual played on prices instead of plans; the c-transform collapses it to a single potential, the W₁ case is exactly the Wasserstein-GAN critic, and Brenier's theorem recovers the optimal map.