Source-linked AI summary
On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint
Yongxin Chen, Tryphon Georgiou, Michele Pavon
TL;DR
The paper addresses how optimal mass transport and Schrödinger bridges are related beyond the standard zero-diffusivity connection. Using stochastic control, it derives fluid-dynamic formulations for both problems and for optimal transport with a prior, showing that the latter arises as a zero-noise limit for Markovian prior evolutions, including a Gaussian case and Brownian-particle numerical example.
Problem
The paper examines the relationship between optimal mass transport and Schrödinger bridges, whose stochastic-control connections are described as richer and deeper than existing accounts.
Method
The paper uses stochastic control to derive fluid-dynamic formulations of optimal mass transport, Schrödinger bridges, and optimal transport with a prior.
Results
Optimal transport with a prior is obtained as the zero-noise limit of Schrödinger bridges with any Markovian prior evolution, with the Gaussian case worked out and a Brownian-particle numerical example provided.
Takeaways & Limitations
The time-symmetric bridge formulation connects Schrödinger bridges with optimal transport even without zero-noise limits, while the prior formulation extends the correspondence to general Markovian evolutions.
Abstract
from arXiv · showhide
We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from the stochastic control perspective. We show that the connections are richer and deeper than described in existing literature. In particular: a) We give an elementary derivation of the Benamou-Brenier fluid dynamics version of the optimal transport problem; b) We provide a new fluid dynamics version of the Schrödinger bridge problem; c) We observe that the latter provides an important connection with optimal transport without zero noise limits; d) We propose and solve a fluid dynamic version of optimal transport with prior; e) We can then view optimal transport with prior as the zero noise limit of Schrödinger bridges when the prior is any Markovian evolution. In particular, we work out the Gaussian case. A numerical example of the latter convergence involving Brownian particles is also provided.
I. INTRODUCTION
The paper unifies optimal mass transport and Schrödinger bridges through stochastic control, deriving fluid-dynamic formulations and connecting optimal transport with prior to zero-noise bridge limits.
- The Schrödinger bridge control problem is computationally challenging because it involves two nonlinearly coupled partial differential equations with boundary-value coupling.
- The paper develops a unifying stochastic-control view of the relationship between optimal mass transport and Schrödinger bridges.
- It gives an elementary derivation of the Benamou-Brenier fluid-dynamics formulation of optimal mass transport.
- It introduces a time-symmetric fluid-dynamic formulation of the Schrödinger bridge problem, revealing a connection with optimal transport without taking a zero-noise limit.
- It formulates and solves optimal transport with a prior, then studies its relation to Schrödinger bridges with general Markovian prior evolutions.
- The paper works out the Gaussian case and reports numerical examples involving Wiener-measure mean shifting and two-dimensional overdamped Brownian particles.
II. OPTIMAL MASS TRANSPORT AS A STOCHASTIC CONTROL PROBLEM
The paper reformulates optimal mass transport as a stochastic control problem and derives the Benamou–Brenier fluid-dynamic formulation through probability-density flows and velocity fields.
- The relaxed Kantorovich problem seeks a coupling of two endpoint distributions whose marginals are the prescribed measures.
- Optimal transport is equivalently represented by probability measures on paths with fixed initial and final marginals.Disintegrating path measures by endpoint positions yields couplings between the endpoint distributions.
- The particle formulation has a hydrodynamic counterpart using continuous feedback controls that steer each initial point to its prescribed endpoint.
- For endpoint densities, admissible trajectories induce a flow satisfying the continuity equation, expressing conservation of probability mass.
- This reduction yields the Benamou–Brenier fluid-dynamic optimal transport problem, with optimization over density flows and feedback velocity fields.
- A Hamilton–Jacobi construction produces an optimal feedback field from a density flow satisfying the endpoint condition.
- The formulation requires solving a two-point boundary-value problem, and classical solutions cannot generally be expected; viscosity solutions may be necessary.
A. Finite energy diffusions
The paper develops the finite-energy diffusion framework using Wiener measures, equivalent path distributions, forward and backward drifts, and relative entropy.
- The framework uses the continuous-path space Ω = C([0, 1], R^n) and Wiener measures as reference processes.
- Distributions equivalent to Wiener measure admit forward and backward stochastic representations under Girsanov’s theorem.
- Relative entropy H(Q, P) compares an alternative path distribution Q with a reference distribution P, with endpoint marginal terms appearing in the decomposition.
- Forward and backward drift relations are established through conditional expectations and localization of stochastic integrals.
B. The Schr¨odinger bridge problem
The Schrödinger bridge is formulated as relative-entropy minimization over path distributions with prescribed endpoint densities, and its solution connects stochastic control, harmonic functions, and optimal transport.
- Given positive endpoint densities, the Schrödinger bridge minimizes relative entropy over path distributions whose endpoint marginals are fixed.
- When a finite-entropy feasible path distribution exists, the Schrödinger bridge has a unique minimizer.
- The bridge can be analyzed through endpoint disintegrations and joint initial–final distributions under the prior and optimized processes.
- For a Markovian diffusion prior, the bridge preserves a Markovian structure and its one-time density factors into forward and backward harmonic components.
- The paper derives a fluid-dynamic stochastic control formulation whose optimal control is the gradient of the logarithm of a positive space-time harmonic function.
- The bridge solution is an h-path process in both forward and backward time directions.
- The optimal feedback is u*(x, t) = ∇ln ϕ(x, t), and the value function is S(x, t) = −ln ϕ(x, t) = inf_u J(u).
V. A TIME-SYMMETRIC FORMULATION
The paper develops a time-symmetric stochastic-control formulation of the Schrödinger bridge problem using current and osmotic drifts. The bridge is characterized variationally through incremental kinetic energies.
- The formulation defines current and osmotic drifts for each admissible process.
- The Schrödinger bridge minimizes the sum of the two incremental kinetic energies when boundary relative entropies are fixed.
- A suitable pair of current and osmotic drifts makes the variational functional equal to zero and is therefore optimal.
- The resulting drifts agree with the earlier bridge characterization, and a two-control variational analysis can be developed analogously.
VI. A FLUID DYNAMIC FORMULATION OF THE SCHR ¨ODINGER BRIDGE PROBLEM
The paper derives a fluid-dynamic formulation of the Schrödinger bridge problem from stochastic control. In the Wiener-prior case, the resulting functional differs from optimal transport by a time-integrated Fisher-information term, linking the problems without a zero-noise limit.
- For a stationary Wiener prior, the forward and backward drift components vanish, corresponding to no prior information.
- The Schrödinger bridge is obtained by minimizing a stochastic-control functional over processes with fixed endpoint marginals.
- Restricting to Markovian processes and using Nelson’s duality expresses the bridge problem through current and osmotic drift fields.
- Compared with the optimal-transport functional, the two formulations differ by a multiple of the time integral of Fisher information.
- This formulation reveals a relation between Schrödinger bridges and optimal transport that does not require zero-noise limits.
VII. OPTIMAL TRANSPORT WITH A “PRIOR”
The paper introduces optimal transport with a prior evolution by penalizing deviations from a prescribed velocity field while matching endpoint marginals. The formulation recovers standard optimal transport for a stationary prior and supports zero-noise analysis for general Markovian evolutions.
- The prior-transport problem seeks an evolution between new endpoint marginals that stays close to the evolution generated by a previous velocity field.
- The prior evolution is represented by a continuous vector field v, with admissible density flows constrained by the continuity equation and endpoint densities.
- If the prior flow already has the prescribed endpoint densities, it solves the problem; setting v ≡ 0 recovers standard optimal transport.
- The particle version uses the Lagrangian L(t, x, ˙x) = ∥˙x − v(x, t)∥2, penalizing velocity deviations from the prior.
- A limitation is that explicitly calculating the induced cost c(x, y) for nonzero prior fields is nontrivial.
- The formulation is proposed as a natural setting for studying zero-noise limits of Schrödinger bridges with general Markovian prior evolutions.
- The optimal feedback control has the form v∗ = v + ∇ψ, where the density-control pair satisfies the prior-constrained transport problem.
VIII. GAUSSIAN CASE
The Gaussian section analyzes zero-noise Schrödinger-bridge limits for linear prior dynamics and Gaussian marginals. It shows that the limiting process solves optimal transport with a linear prior velocity field.
- As the reference noise decreases, the Schrödinger bridge is analyzed through a slowed reference evolution and its zero-noise limit.
- When A(t) ≡ 0, the Schrödinger bridge solution process converges to the classical optimal-mass-transport solution.
- For general A(t), the limiting process yields an optimal solution to transport with prior.
- Theorem 8.1 identifies the optimal velocity as ˜v(x, t) = (A(t) − Π0(t))x + m(t), with prior velocity v(x, t) = A(t)x.
- The proof verifies the terminal Gaussian boundary condition and the gradient correction required by the Hamilton–Jacobi equation.
IX. EXAMPLE: SHIFTING THE MEAN OF NORMAL DISTRIBUTIONS
The Gaussian Schrödinger bridge is analyzed for shifting a normal distribution's mean, yielding an explicit current drift and its zero-noise optimal-transport limit.
- The example considers the Schrödinger bridge on [0, 1] with a Brownian prior and prescribed endpoint marginals.
- The bridge has a forward differential representation obtained from the general theory.
- The interpolating mean is m_t = t, satisfying m(0) = 0 and m(1) = 1.
- The current drift of the Schrödinger bridge is computed explicitly for this Gaussian setting.
- As σ^2 ↘ 0, the bridge coefficients converge so that its drift approaches the optimal control of the corresponding optimal transport problem.
X. NUMERICAL EXAMPLE
The numerical example studies Brownian-particle interpolation between observed Gaussian endpoint distributions. As noise decreases, Schrödinger-bridge paths approach optimal transport with the prior.
- The example uses highly overdamped Brownian motion in a force field and the Smoluchowski model as the configuration-space dynamics.
- The particle follows dx(t) = −∇V(x(t))dt + √ϵdw(t), with −∇V(x) = Ax and w a standard two-dimensional Wiener process.
- The observed endpoint distributions are normal, and the experiment interpolates the particle density at intermediate times.
- Figures 1–3 show Schrödinger-bridge flows for ϵ = 9, ϵ = 4, and ϵ = 0.01, respectively.
- As ϵ ↘ 0, bridge paths resemble those of the corresponding optimal-transport process, represented by the optimal-transport-with-prior limit.
- Optimal transport without a prior provides a comparison interpolation given by constant-speed translation.