Diffusion Approximations to Schrödinger Bridges and the Convergence of Entropic Potentials

Source abstract

Consider the Monge-Kantorovich optimal transport problem between two Euclidean densities $μ$ and $ν$ and quadratic cost. The $\varepsilon$-Schrödinger bridge is the solution to the entropic regularized problem with regularization parameter $\varepsilon$. By taking the conditional expectation of the second coordinate given the first under this coupling, we obtain the $\varepsilon$-entropic Brenier map. As $\varepsilon$ goes down to zero, it is known that the entropic Brenier map converges to the quadratic cost optimal transport map between the two measures, i.e., the Brenier map. We show that, under some smoothness and log-concavity constraints on the marginals, the difference between the entropic Brenier map and the Brenier map is equal to $\varepsilon$ times one-half of the score function of the first marginal plus an error that is $o(\varepsilon)$ in $\mathbf{L}^2(μ)$. This expansion holds irrespective of the second marginal $ν$. The proof relies on an approximation of the Schrödinger bridge by a noisy version of the McCann interpolation. Under additional assumptions we show a second approximation to the Schrödinger bridge via so-called Mirror Langevin diffusions, which are Langevin diffusions on the Hessian manifold generated by the Brenier map. These two approximations, that appear quite different at first glance, are nonetheless shown to be closely connected. Our proofs utilize a random surface and a novel stochastic operation called the tangent Markov projection.