Γ-convergence of Schrödinger to Monge-Kantorovich

The Γ-convergence of the Schrödinger problem to the Monge-Kantorovich optimal transport problem is the rigorous mathematical statement that optimal transport is recovered as the zero-noise limit of the Schrödinger bridge problem. This result, central to Léonard (2014), establishes that the SB is a principled entropic regularization of OT.

The construction works by “slowing down” the reference Markov process. Given a reference measure $R$ with generator $L$, one considers the sequence $R^k$ with generators $L^k:=L/k$, $k\ge 1$. If $(R^k{)}_{k\ge 1}$ satisfies a large deviation principle in path space with speed ${\alpha }_k$ and rate function $C$ (the kinetic action), then the rescaled dynamic Schrödinger problems

$H(P|R^k)/{\alpha }_k\to \min ;P\in \mathrm{P}(\Omega ):P_0={\mu }_0,P_1={\mu }_1(S_{\text{dyn}}^k)$

Γ-converge to the dynamic Monge-Kantorovich problem:

${\int }_{\Omega }CdP\to \min ;P\in \mathrm{P}(\Omega ):P_0={\mu }_0,P_1={\mu }_1(M{K}_{\text{dyn}})$

Similarly, the static versions $(S^k)$ Γ-converge to $(MK)$.

Key Details

• Brownian case: ${\alpha }_k=k$, $C(\omega )={\int }_{[0,1]}|{\dot{\omega }}_t{|}^2/2dt$, $c(x,y)=|x-y{|}^2/2$ — recovers the quadratic Monge-Kantorovich problem
• Random walk case: ${\alpha }_k=\log k$, $C(\omega )=\text{number of jumps}$, $c=d_{\sim }$ (graph distance) — proved rigorously in Theorem 5.2
• Convergence of solutions: The bridges converge ${\lim }_{k\to \infty }{R}^{k,xy}={\delta }_{{\gamma }^{xy}}$ (concentrating on geodesics), minimizers ${\hat{P}}^k\to \hat{P}$ solving $(M{K}_{\text{dyn}})$, and couplings ${\hat{\pi }}^k\to \hat{\pi }$ solving $(MK)$
• Dual convergence: Schrödinger duals $(D^k)$ converge to the Kantorovich dual $(D^{\infty })$ via the Laplace-Varadhan integral lemma
• Entropic interpolations converge: $[{\mu }_0,{\mu }_1{]}^k\to [{\mu }_0,{\mu }_1]$ (displacement interpolation)
• Implication for ML: Justifies using Schrödinger bridges as tractable proxies for OT in generative modeling — the entropic regularization provides smoothness and uniqueness while approximating the OT plan

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