Langevin Diffusion Approximation to Same Marginal Schrödinger Bridge

TL;DR

This work develops a Langevin-diffusion-based approximation to the same-marginal Schrödinger bridge, establishing precise low-temperature expansions. The entropic Brenier map $\mathcal{B}_{\rho,\varepsilon}$ approaches the identity with a leading correction $\frac{\varepsilon}{2}\nabla\log\rho$, yielding $\frac{\mathcal{B}_{\rho,\varepsilon}(x)-x}{\varepsilon} \to \tfrac{1}{2}\nabla\log\rho(x)$ in $L^2(\rho)$, and $\varepsilon^{-2}\|\mathcal{B}_{\rho,\varepsilon}-\mathrm{Id}\|_{L^2(\rho)}^{2} \to \tfrac{1}{4}I(\rho)$. The authors show that, under mild regularity, the pair of joint laws $(\ell_{\rho,\varepsilon},\pi_{\rho,\varepsilon})$ are exponentially close in the sense of relative entropy, with a bound scaling as $\varepsilon^2$, and hence the Jensen–Shannon divergence is $o(\varepsilon^2)$ (and $O(\varepsilon^4)$ in the Gaussian case). They further demonstrate that, for smooth compactly supported test functions $\xi$, the SB conditional expectation $\mathrm{E}_{\pi_{\rho,\varepsilon}}[\xi(Y)|X]$ admits a first-order expansion whose derivative is the Langevin generator $L$, implying an approximate semigroup structure for SB operators at low temperature. The main technical contributions include a probabilistic proof framework, an explicit link between SB and Langevin dynamics via entropic interpolations, and a mechanism to approximate tangent vectors in Wasserstein space using entropic Brenier maps, with implications for discretizations in optimal transport and diffusion-model contexts.

Abstract

We introduce a novel approximation to the same marginal Schrödinger bridge using the Langevin diffusion. As $\varepsilon \downarrow 0$, it is known that the barycentric projection (also known as the entropic Brenier map) of the Schrödinger bridge converges to the Brenier map, which is the identity. Our diffusion approximation is leveraged to show that, under suitable assumptions, the difference between the two is $\varepsilon$ times the gradient of the marginal log density (i.e., the score function), in $\mathbf{L}^2$. More generally, we show that the family of Markov operators, indexed by $\varepsilon > 0$, derived from integrating test functions against the conditional density of the static Schrödinger bridge at temperature $\varepsilon$, admits a derivative at $\varepsilon=0$ given by the generator of the Langevin semigroup. Hence, these operators satisfy an approximate semigroup property at low temperatures.

Theorems & Definitions (22)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof : Proof of Theorem \ref{['thm:ld-sb-general-g']}
• corollary 1
• proof
• Theorem 2
• Theorem 3