FLOW MATCHING AND OPTIMAL TRANSPORT: AN ILLUSTRATED INTRODUCTION TO WASSERSTEIN–OTTO GEOMETRY

We recall and derive the equations underlying Flow Match- ing, making explicit the probability density function of the linearly in- terpolated random variable and relating it to the transporting veloc- ity field through the continuity equation. We visualise how the initial density is transported and compare the resulting path with a geodesic in Wasserstein–Otto geometry. We demonstrate that, for an isotropic Gaussian initial law, the Flow Matching velocity field is the minimum- energy representative of the density variation at each time, among fields satisfying the continuity equation, even though the path is generally not a geodesic under the Wasserstein–Otto metric. We explain the dis- tinction between this local minimisation and the minimisation of the action over paths with fixed endpoints using explicit calculations and illustrations.