Entropic optimal transport: Geometry and large deviations
Espen Bernton, Promit Ghosal, Marcel Nutz
Abstract
We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the local exponential convergence rate as the regularization parameter vanishes. The exact rate function is determined in a general setting and linked to the Kantorovich potential of optimal transport. Our arguments are based on the geometry of the optimizers and inspired by the use of c-cyclical monotonicity in classical transport theory. The results can also be phrased in terms of Schrödinger bridges.
Topics & Concepts
Monotonic functionMathematicsRate of convergenceConvergence (economics)Large deviations theoryRegularization (linguistics)Rate functionExponential functionApplied mathematicsTransport theoryFunction (biology)Mathematical optimizationGeometryMathematical analysisStatistical physicsPhysicsComputer scienceStatisticsEconomicsBiologyArtificial intelligenceChannel (broadcasting)Evolutionary biologyComputer networkEconomic growthGeometric Analysis and Curvature FlowsMarkov Chains and Monte Carlo MethodsNonlinear Partial Differential Equations