Transport-Relaxed Schrödinger Bridge

• Transport-Relaxed Schrödinger Bridge is defined as a family of formulations that relaxes classical hard endpoint constraints by penalizing deviations using entropic and divergence-based measures.
• It employs mechanisms such as soft marginal relaxation, terminal divergence penalties, and alternating projection methods to improve stability and well-posedness in empirical and discrete settings.
• The approach features explicit duality, convergence guarantees, and practical applications in robust routing, graph transformation, and semimartingale optimal transport.

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Transport-relaxed Schrödinger bridge denotes a family of Schrödinger bridge formulations in which one or more of the classical hard constraints are softened while retaining an entropic, divergence-based, or stochastic-control structure. In the classical static problem, one minimizes $H(\pi\mid\gamma)$ over couplings $\pi\in\Pi(\mu,\nu)$; in the dynamic problem, one minimizes path-space relative entropy with prescribed initial and terminal marginals. Transport relaxation modifies this template in several ways: by penalizing the transport distance between the bridge marginals and the prescribed marginals, by replacing exact terminal matching with a terminal divergence penalty, by alternating between reciprocal and Markov projections until both properties are recovered at convergence, or by interpolating between the classical Schrödinger bridge and martingale transport through diffusion control. These variants are motivated especially by empirical measures, discrete state spaces, weak target constraints, and semimartingale optimal transport (Jiang et al., 8 Feb 2026).

1. Classical baseline and the meaning of relaxation

The classical Schrödinger bridge problem is the maximum-entropy or minimum-relative-entropy interpolation between prescribed endpoint marginals under a reference process. In path-space form, one seeks

$$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$

where $P$ is the law of a reference diffusion. In static form, the same problem is equivalent to entropy-regularized optimal transport, with the regularization induced by the reference kernel (Bortoli et al., 2021). A complementary optimal-transport treatment emphasizes the dual problem, entropic potentials, and convergence of Sinkhorn iterations in both two-marginal and multi-marginal settings (Marino et al., 2019).

The adjective “transport-relaxed” does not refer to a single modification. In one line of work, the exact marginal constraints are relaxed directly. In another, the transport plan remains exact at the endpoints but the process class is temporarily relaxed during computation. In a third, the bridge is embedded in a larger semimartingale transport problem in which both drift and volatility are controlled. This suggests that transport relaxation is best understood as a controlled weakening of either endpoint constraints, path-class constraints, or volatility structure, rather than as a single canonical model.

A central reason for such relaxations is that the classical problem may be ill-posed in sample-based regimes. When $\mu$ and $\nu$ are empirical measures and the reference $\gamma$ is absolutely continuous, the feasible set $\{\pi\in\Pi(\mu,\nu):H(\pi\mid\gamma)<\infty\}$ is empty. That specific pathology motivates formulations in which marginal matching is softened by optimal transport penalties rather than imposed exactly (Jiang et al., 8 Feb 2026).

2. Principal relaxation mechanisms

Taken together, the literature suggests several recurrent mechanisms for relaxing the classical bridge.

Mechanism	Representative formulation	Structural consequence
Soft marginal relaxation	$OT(\mu,\pi_x)+OT(\nu,\pi_y)+H(\pi\mid\gamma)$	Well-posed for empirical data
Terminal divergence penalty	$D(P\|Q)+D(p_N\|\nu_N)$	Nonlinear terminal boundary condition
Alternating reciprocal/Markov projection	IMF or bridge-mixture projection	Constraints recovered only at convergence, or exact transport preserved per iterate
Weak targets with general divergence	Weak OT terminal constraint plus $\pi\in\Pi(\mu,\nu)$0	Generalized Schrödinger system

In the semi-discrete static relaxation, the objective

$\pi\in\Pi(\mu,\nu)$1

permits arbitrary marginals $\pi\in\Pi(\mu,\nu)$2, but penalizes their deviation from the prescribed data. A scaled version uses $\pi\in\Pi(\mu,\nu)$3, so that smaller $\pi\in\Pi(\mu,\nu)$4 enforces closer agreement with the target marginals (Jiang et al., 8 Feb 2026).

In robust network routing, the relaxation takes a different form: one fixes the initial distribution $\pi\in\Pi(\mu,\nu)$5, but replaces the hard terminal constraint by a penalty $\pi\in\Pi(\mu,\nu)$6, or more generally $\pi\in\Pi(\mu,\nu)$7, added to the path-space relative entropy $\pi\in\Pi(\mu,\nu)$8. The resulting generalized Schrödinger system differs from the classical one by a nonlinear terminal boundary condition $\pi\in\Pi(\mu,\nu)$9 (Chen et al., 2018).

A distinct mechanism appears in weak-target formulations on path space. There, the relative entropy penalty is replaced by a general convex divergence $$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$0, and the terminal constraint is imposed only through a weak optimal transport condition $$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$1. This broadens the admissible class substantially and yields a generalized Schrödinger system that includes the entropic case and non-entropic cases such as the chi-divergence (Hernández et al., 24 Dec 2025).

A common misconception is that transport relaxation always means softening both marginals. The literature shows otherwise. Some models relax only the terminal constraint, some relax the path-class constraint during iteration, and some replace the entire entropic bridge by a larger semimartingale transport problem whose limiting regimes recover the classical Schrödinger bridge and Bass transport.

3. Duality, existence, uniqueness, and asymptotics

The semi-discrete transport-relaxed bridge admits a particularly explicit dual theory. When

$$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$2

and $$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$3 is absolutely continuous, the dual reduces to a finite-dimensional concave maximization over $$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$4, with max-affine functions

$$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$5

The primal and dual both admit solutions, and the optimizers are unique up to the gauge symmetry $$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$6 if $$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$7 for all $$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$8 (Jiang et al., 8 Feb 2026).

The same work identifies the small-$$\Pi^\star=\arg\min_{\Pi} KL(\Pi\|P),\qquad \Pi_0=\mu_0,\ \Pi_T=\mu_T,$$9 limit. As the transport penalty blows up, the relaxed optimizer converges to a discrete Schrödinger bridge, namely the unique minimizer of $P$0 over $P$1, where $P$2 is induced by the reference density at the support points. The optimal value exhibits the asymptotic expansion

$P$3

The leading-order logarithmic divergence is therefore intrinsic and dimension-dependent (Jiang et al., 8 Feb 2026).

In the semimartingale interpolation between Schrödinger and Bass, the cost

$P$4

produces a stochastic control problem over drift and volatility. Despite the lack of coercivity in the diffusion component, strong duality and dual attainment hold. More precisely, for $P$5 and $P$6, the primal value, the dynamic dual, its closure, and a reduced dual all coincide, and optimizers exist. The reduced dual is formulated over terminal $P$7-convex potentials, with a Moreau-envelope operator $P$8 and a backward heat potential $P$9 (Henry-Labordere et al., 29 Mar 2026).

The weak-divergence framework extends this pattern further. It establishes well-posedness, convex duality, and explicit structural characterizations for path-space problems in which the entropy is replaced by a general convex divergence and terminal constraints are imposed weakly. In that sense, transport relaxation can be read not only as a softening of endpoint matching, but also as a replacement of the entropic regularizer itself (Hernández et al., 24 Dec 2025).

4. Structural representations of relaxed bridges

A central structural theme is that many relaxed bridges admit explicit factorized or transport-map representations. In the weak-divergence setting, the optimal path measure has an explicit density with respect to the reference diffusion: $\mu$0 For relative entropy, this reduces to an exponential density, recovering the classical Schrödinger bridge form; for the chi-divergence, the density becomes a positive-part affine factor. The corresponding marginal-flow formula characterizes $\mu$1 through forward and time-reversed control processes, recovering the usual additive gradient structure in the entropic case and a more intricate non-gradient structure for non-entropic divergences (Hernández et al., 24 Dec 2025).

The Schrödinger–Bass bridge provides a transport-relaxed interpolation between the classical Schrödinger bridge and Bass martingale transport. In one dimension, a PDE derivation starts from an HJB equation,

$\mu$2

then passes through a Legendre transform and exponentiation to obtain a linear heat equation for $\mu$3. The optimal process is a Stretched Schrödinger Bridge,

$\mu$4

where $\mu$5 is a Schrödinger bridge process. The limits $\mu$6 and $\mu$7 recover, respectively, the classical Schrödinger bridge and the Bass construction (Alouadi et al., 25 Jan 2026).

The higher-dimensional semimartingale theory gives a closely related coupled Schrödinger–Bass bridge system. The optimal semimartingale is described by a backward heat potential $\mu$8 and maps

$\mu$9

with $\nu$0 and $\nu$1. The map $\nu$2 is the gradient of a $\nu$3-convex function, and the optimal process may be viewed as a stretched Brownian motion extending the Bass construction from martingales to general semimartingales (Henry-Labordere et al., 29 Mar 2026).

The martingale Schrödinger bridge forms a neighboring relaxation in which the coupling is constrained to be a martingale. In continuous time, it is the continuous martingale with prescribed marginals that minimizes a weighted quadratic energy measuring the deviation from Brownian motion. In the irreducible case, it coincides with the Föllmer martingale, that is, the Doob martingale associated to a suitable Föllmer process (Backhoff et al., 1 Apr 2026).

5. Algorithms and computational regimes

Relaxation is not only a modeling device; it is also an algorithmic one. In robust routing on a finite directed graph, the generalized Schrödinger system is solved by an iterative algorithm on positive cones. The associated fixed-point map contracts the Hilbert projective metric with contraction ratio less than $\nu$4, which yields fast global convergence (Chen et al., 2018).

For the semi-discrete transport-relaxed bridge, two numerical schemes are established with linear convergence rates: gradient ascent on the finite-dimensional dual variables $\nu$5, and a Sinkhorn-type fixed-point iteration based on a soft-max approximation of the max-affine dual. The gradient method converges linearly in the $\nu$6 norm, while the Sinkhorn-type scheme is contractive in the $\nu$7 norm (Jiang et al., 8 Feb 2026).

In discrete state spaces, transport relaxation appears through alternating reciprocal and Markov projections. Discrete Diffusion Schrödinger Bridge Matching extends Iterative Markovian Fitting to continuous-time Markov chains on high-dimensional discrete spaces. The reciprocal projection conditions on an endpoint coupling and produces a mixture of Markov bridges; the Markov projection then minimizes reverse KL over Markov processes. The resulting sequence converges in law to the true Schrödinger bridge, and in the graph setting the induced cost corresponds to the graph edit distance (Kim et al., 2024).

A related continuous-state construction is the iterated diffusion bridge mixture procedure. Each iteration forms a mixture of reference diffusion bridges from the current endpoint coupling, then projects that non-Markovian mixture onto the class of diffusion processes matching all time marginals. A key difference from IPF-style methods is that every iterate is already a valid transport between the target marginals. The KL sequence satisfies

$\nu$8

and both the bridge-mixture process and its Markovianization converge in law to the Schrödinger bridge (Peluchetti, 2023).

For continuous generative modeling, Diffusion Schrödinger Bridge approximates IPF by alternating forward and backward score-based diffusion updates and is explicitly presented as the continuous state-space analogue of the Sinkhorn algorithm. In that context, transport relaxation is operationalized through iterative marginal correction rather than exact closed-form scaling (Bortoli et al., 2021).

6. Applications, neighboring generalizations, and scope

Transport-relaxed Schrödinger bridges have been developed in response to concrete application domains. In network routing under random link failures, the relaxed terminal penalty allows a final distribution that is close to, rather than exactly equal to, the desired terminal law; the interpretation is a maximum-entropy path measure with an extra terminal cost (Chen et al., 2018). In graph transformation, the discrete CTMC formulation supports molecular optimization with minimal graph transformation and no need for paired data, precisely because the coupling emerges through iterative fitting rather than being supplied externally (Kim et al., 2024). In the semimartingale setting, the Schrödinger–Bass bridge is presented as a unified framework encompassing entropic and martingale optimal transport and yielding a variational foundation for data-driven diffusion models (Henry-Labordere et al., 29 Mar 2026).

The broader bridge literature contains several closely related extensions that are not always labeled “transport-relaxed” but illuminate the same design space. The LQR-Schrödinger bridge replaces the pathwise entropy cost by a sum of quadratic functions comprising potential terms and kinetic terms; under Gaussian boundary marginals it admits closed-form forward and backward propagation through dual Riccati equations, produces a non-homogeneous Gaussian Markov process, and extends Bures transport to geometries with negative curvature (Lambert, 12 Jun 2025). Schrödinger bridges on sub-Riemannian manifolds introduce noise aligned only with control directions, obtain a Sinkhorn-type algorithm under bracket-generating hypotheses, and recover deterministic sub-Riemannian optimal transport as the noise level vanishes (Adu et al., 12 May 2026). A quadratic state-cost regularization yields an exactly solvable reaction-diffusion bridge with a closed-form Mehler-type kernel for arbitrary endpoint distributions with finite second moments (Teter et al., 2024).

A further machine-learning variant exploits the entire entropic regularization spectrum between standard Schrödinger bridges and deterministic optimal transport. Rectified Schrödinger Bridge Matching uses a single parameter $\nu$9, proves invariance of the conditional velocity-field functional form across $\gamma$0, and shows that reducing $\gamma$1 linearly decreases the conditional velocity variance. This does not redefine transport relaxation in the semi-discrete sense, but it does show how varying entropic regularization can be used to balance multimodal coverage and path straightness in few-step inference (Luan et al., 7 Apr 2026).

Overall, transport-relaxed Schrödinger bridge is best regarded as a class of bridge problems rather than a single model. Across its variants, the recurring objective is the same: retain the Schrödinger bridge’s entropic or divergence-based regularization, while relaxing the parts of the classical formulation that become brittle in empirical, discrete, martingale, weak-target, or controlled-diffusion settings.