Schr\"odinger-Bass Bridge Barycenter Models for Measure-Valued Data

1 INTRODUCTION

• Barycenter models average probability measures and support supervised and unsupervised analysis when observations are themselves distributions.The paper situates its contribution among Wasserstein and Schrödinger-bridge barycenter models.Source: §1 · p. 1
• SBB jointly controls drift and volatility, distinguishing systematic transport from stochastic spreading in measure-valued data.Unlike the classical Schrödinger Bridge, its volatility can adapt; unlike Bass martingale transport, it does not impose zero drift or endpoint convex order [Henry-Labordère et al., 2026, Hasenbichler et al., 2026].Source: §1 · p. 2
• The paper introduces encoder, decoder, symmetric, and Markovian SBB barycenters, establishes their existence, and derives an orientation-specific hybrid Wasserstein gradient particle scheme.It also derives marginal first variations and a Gaussian decoder result, and applies the methods to spatial deformation and blur and single-cell RNA reconstruction.Source: §1 · p. 2

Wasserstein barycenter and entropic regularization

• Entropic regularization can change the barycenter geometry when one marginal is optimized, while debiased Sinkhorn addresses the resulting entropic bias [Janati et al., 2020a].Chizat [2023] distinguishes transport-plan regularization from an entropy penalty directly on the barycenter.Source: §2 · p. 2
• Gaussian Wasserstein barycenters retain Gaussian structure, and prior work also provides Gaussian closure results for entropic transport [Agueh and Carlier, 2011, del Barrio and Loubes, 2020].Prior computations include fixed-point, first-order, and particle-descent methods [Álvarez-Esteban et al., 2016, Altschuler et al., 2021, Nguyen et al., 2025, Chizat, 2023].Source: §2 · p. 2

Schrödinger bridges and multi-marginal barycenters

• Multi-marginal stochastic averaging has been studied through tree-structured Schrödinger problems, TreeDSB barycenters, and bridge matching with entropic barycentres [Haasler et al., 2021, Noble et al., 2023, Howard et al., 2025].These methods couple multiple distributions through a shared stochastic representation.Source: §2 · p. 2

Bass martingales and weak optimal transport

• Bass martingale transport optimizes volatility with zero drift and requires endpoint convex order, whereas SBB permits nonzero drift and removes that endpoint restriction [Bass, 1983, Backhoff-Veraguas et al., 2020, Hasenbichler et al., 2026].SBB's static cost is also a weak transport cost.Source: §2 · p. 3

Schrödinger–Bass Bridge

• SBB's pairwise foundations include semimartingale formulations, duality, static weak-transport representations, and sample-based bridge estimation [Henry-Labordère et al., 2026, Hasenbichler et al., 2026, Alouadi et al., 2026a].Alouadi et al. [2026b] derive a one-dimensional PDE formulation.Source: §2 · p. 3

3 SCHRÖDINGER-BASS BRIDGE (SBB)

• The SBB process evolves as dX_t = α_t dt + σ_t dW_t, with drift α_t and volatility σ_t.The action is defined according to the law of the controlled process.Source: §3 · p. 3
• SBB jointly controls drift and volatility, unlike Schrödinger bridges with prescribed diffusion and Bass martingale transport with zero drift.This control interpolates between entropic diffusion transport and martingale transport without imposing convex order on endpoints [Hasenbichler et al., 2026].Source: §3 · p. 3
• For τ > 0, SBB(μ, H_T μ) = 0 but SBB(H_T μ, μ) ≠ 0, demonstrating that the SBB cost is asymmetric.Here H_T μ is μ convolved with a Gaussian of covariance τI_d; reversing the endpoints would require a zero-cost process whose diffusion instead produces H_2T μ.Source: §3 · p. 3

4 PROPOSED SBB BARYCENTERS MODELS

• The encoder barycenter is a common terminal law reached from all inputs, whereas the decoder learns a common source generating the inputs through domain-specific bridges.The symmetric formulation uses both directions and avoids asymmetry.Source: §4 · p. 4
• The Markovian formulation treats the two input measures as initial and final marginals with nonnegative weights.For two marginals, an intermediate-time marginal can be interpolated as their barycenter.Source: §4 · p. 4
• For inputs in P2(R^d), each of the decoder, encoder, symmetric, and Markovian barycenter problems has at least one minimizer in P2(R^d).The decoder case is proved in detail; the other cases follow essentially the same argument.Source: §4 · p. 4
• Uniqueness of the SBB barycenter minimizer is left open for future work.The existence result therefore does not establish that the minimizer is unique.Source: §4 · p. 4

5 A HYBRID WASSERSTEIN GRADIENT PARTICLE DESCENT SCHEME

• The method represents the barycenter with particles and uses Wasserstein gradient descent, with dual potentials providing the first variation needed to update them.The scheme applies when the relevant dual potential can be computed for each objective term.Source: §5 · p. 4, First Variation of SBB and Update Rules · p. 4

First Variation of SBB and Update Rules

• A pairwise SBB solver supplies either an inverse map or a score/drift field to recover the bridge's dual quantities.The implementation chooses LightSBB [2026a].Source: First Variation of SBB and Update Rules · p. 4
• For encoder terms, the first variation uses the terminal dual potential of each bridge µi → ρ; for decoder terms, it uses the HJB value for ρ → µi.The decoder derivation notes a sign change because the optimized marginal changes position in the dual objective.Source: First Variation of SBB and Update Rules · p. 5
• The symmetric objective uses the average of the decoder and encoder descent velocities.Source: First Variation of SBB and Update Rules · p. 5
• When an encoder component enters an update, evaluating its duality at T − δk may improve numerical stability by using the HJB's additional smoothing.The passage identifies the encoder terminal dual potential's spatial derivative as potentially less regular at t = T.Source: First Variation of SBB and Update Rules · pp. 4–5
• The outer loop fits ρk → µi bridges for decoder terms and µi → ρk bridges for encoder terms before updating particles.The updates use the formulation-specific rules, and the returned barycenter is the empirical measure of the final particles.Source: First Variation of SBB and Update Rules · p. 6

• Unlike Bass transport, SBB permits nonzero drift without imposing convex order; unlike the Schrödinger Bridge, its adaptive volatility prevents reduction to ordinary point-to-point transport.The Bass Gaussian-source discussion additionally states that a common Gaussian source requires equal terminal means and covariance bounded above by each terminal covariance.Source: §6 · p. 6
• The proof uses an equivalent three-variable static representation, distinguishing physical initial X, hidden terminal Y, and physical terminal Z.This representation is the key tool for the Gaussian projection argument.Source: §6 · p. 6
• Gaussian decoder inputs admit a Gaussian barycenter minimizer, reducing the search to finite-dimensional optimization over its mean and covariance.The result is established through a projection principle: replacing the source measure with its moment-matched Gaussian can only decrease SBB cost.Source: §6 · p. 7
• For Gaussian pairs, the SBB geometry separates into a Euclidean mean component and a covariance component, with an affine Gaussian process attaining the optimum.The covariance action is introduced using a Lyapunov equation; the passage states that the optimal bridge has affine Gaussian dynamics.Source: §6 · p. 7
• The Gaussian specialization replaces M barycenter particles with a single mean vector and covariance matrix, while retaining pairwise Gaussian SBB problems as inner problems.The pairwise outputs are summarized via matrices K.Source: §6 · p. 7
• Whether Gaussian closure holds for the encoder barycenter remains open, and an anisotropic Gaussian experiment provides numerical evidence that it may fail.Source: §6 · p. 8

7.1 Directionality of the SBB Barycenter

• Because SBB is asymmetric, encoder, decoder, and symmetric formulations can yield different barycenters; in the illustrated example, decoder is more concentrated.The particle illustration is intended to show directional effects, not compare convergence rates or numerical accuracy.Source: §7.1 · p. 8

7.2 Recovering a Sharp Spatial Measure

• Decoder SBB achieves the best performance on all three spatial-recovery metrics, with lower W2 and pixel ℓ1 errors and higher SSIM.Images are normalized as probability measures on the pixel plane; observations apply geometric transformations and anisotropic Gaussian blur to an unseen clean source.Source: §7.2 · p. 8, Table 1 · p. 8

7.3 Single-cell developmental trajectory reconstruction

• Decoder SBB achieves the best overall ancestor reconstruction, leading on all four serum metrics and on W2, MMD, and Signature MAE under stromal.Some baselines give slightly better cell-type TV under stromal; the day-8 MET target is held out from fitting.Source: §7.3 · p. 9, Table 2 · p. 9
• For intermediate-state reconstruction, SBB Markovian leads on W2 and MMD under serum, while SBB Decoder leads on W2 and MMD under 2i.Neither directional variant dominates every criterion; the Markovian construction uses the intermediate population as the terminal state of one bridge and initial state of the next.Source: §7.3 · pp. 9–10, Table 3 · p. 10

8 CONCLUSION

• SBB barycenters provide a directed framework for averaging measure-valued data while accounting jointly for systematic transport and stochastic dispersion.The conclusion identifies existence theory, a Wasserstein gradient particle scheme, Gaussian analysis, and latent-recovery experiments as contributions, and names particle-scheme convergence guarantees as future work.Source: §8 · p. 10

A MORE BACKGROUND ON SCHRÖDINGER BASS BRIDGE AND PROPERTIES

• Unlike the Bass martingale formulation, SBB requires no convex-order constraint; the Bass martingale feasibility condition is convex order.The appendix also notes translation and orthogonal invariance, but not affine invariance, and states that transport from μ to its heat-flow convolution can have zero cost.Source: A MORE BACKGROUND ON SCHRÖDINGER BASS… · pp. 14–16

B.1 Proof for the existence of the decoder SBB barycenter (Theorem 4.1)

• A minimizing sequence is uniformly tight in P2(Rd), so a weakly convergent subsequence and weak lower semicontinuity yield a decoder minimizer.The proof first bounds second moments using a Wasserstein bound and a fixed input measure with finite second moment.Source: §B.1 · pp. 17–18
• The encoder existence argument follows the same compactness steps, with continuity of SBB in its second marginal completing the minimization proof.The cited continuity result is attributed to Henry-Labordère et al. [2026].Source: §B.1 · p. 18
• The same proof strategy yields existence for symmetric and Markovian objectives by combining orientation-specific Wasserstein bounds with joint weak lower semicontinuity.The symmetric objective weights both orientations by λi/2; the stated Markovian objective uses the first encoder and second decoder terms.Source: §B.1 · p. 18

B.2.1 Proof for Proposition 6.1

• The weak-transport proof uses a conditional-mean constraint to relate conditional laws and constructs a feasible coupling for the marginal cost comparison.The passage states that the conditional laws κx and ηx agree μ-almost everywhere and that the resulting π belongs to Π(μ, ν).Source: §B.2.1 · p. 20

B.2.2 Proof for Theorem Theorem 6.2

• Replacing a candidate by its Gaussian counterpart with matching moments does not increase decoder SBB cost when the target is Gaussian, supporting a Gaussian decoder minimizer for Gaussian inputs.The proof compares the costs through a jointly Gaussian law with matching joint mean and covariance.Source: §B.2.2 · pp. 21–22, §B.2.3 · p. 23

B.2.3 Proof for Corollary 6.3

• The Gaussian replacement result implies that the decoder barycenter optimization can be restricted to Gaussian laws when the input measures are Gaussian.A Gaussian counterpart of any minimizer is also a minimizer.Source: §B.2.3 · p. 23

B.3 Proof for Theorem 6.4

• For Gaussian endpoints, the covariance-path construction attains the lower bound, establishing equality in the Gaussian SBB cost characterization.The constructed process remains Gaussian, with the required terminal mean and covariance.Source: §B.3 · p. 24

C.1 Dual Formulation of SBB

• The SBB dual formulation yields a computationally crucial identity for the optimal controls, under the stated regularity assumptions on dual solutions.The appendix leaves precise measure conditions ensuring those assumptions to future work.Source: §C.1 · p. 25, §C.2 · p. 25

Encoder Marginal

• For the encoder marginal, an active maximizing dual potential is a convex subgradient of the SBB objective.The first-variation derivation is presented under suitable regularity assumptions on the dual solutions.Source: Encoder Marginal · p. 25

Decoder Marginal

• For the decoder marginal, the negative initial-time dual value function is a subgradient, with the first-variation argument requiring local dual attainment and regularity conditions.The stated assumptions include finite values and attained duals, uniform spatial regularity, and stability with respect to the marginal.Source: Decoder Marginal · p. 26

C.3 Pseudocode

• The general particle algorithm selects decoder, encoder, symmetric, or Markovian bridges according to the objective and updates particles using the corresponding marginal directions.The pseudocode gives decoder bridges ρk → μi and encoder bridges μi → ρk, then combines directions for symmetric and Markovian modes.Source: §C.3 · p. 27

D.1 The general affine update

• The general affine-update section presents particle updates through a chosen descent velocity and derives corresponding mean and centered-particle relations.The supplied text gives the relations’ role but not their full displayed formulas.Source: §D.1 · p. 29
• For Gaussian inputs, an affine update preserves Gaussianity; for finite clouds, the mean and empirical covariance identities are exact with normalization 1/M.The identities hold for any input law with a finite second moment.Source: §D.1 · p. 29

D.2 Decoder SBB Gaussian Barycenter update

• The decoder Gaussian update uses the decoder sign, with its covariance coefficient determined by the fitted bridge rather than a Euclidean covariance gradient.The pairwise bridge is from N(m, S) to N(mi, Σi).Source: §D.2 · p. 29

E.1 Complete toy visualizations

• The toy comparison uses three anisotropic two-dimensional Gaussians with equal weights, shared samples and initialization, and common optimization settings across decoder, encoder, and symmetric objectives.The setup uses 512 samples per distribution and 512 barycenter particles.Source: §E.1 · p. 29

E.2 Spatial MNIST: protocol

• The MNIST experiment represents image mass over spatial locations, modeling diffusion by Gaussian convolution rather than independent noise on pixel values.Observations are generated from clean digit sources through geometric transformations, with the same transformations and rendering procedure used for all methods.Source: §E.2 · p. 31
• Decoder SBB reconstructions are generally sharper and preserve clean-source geometry more closely than the arithmetic mean, Wasserstein, Regularized Wasserstein, and debiased Sinkhorn barycenters.Figure E.4 compares methods across digit classes, with clean latent sources in the first row.Source: Figure E.4 · p. 32

E.3 Single-cell RNA: protocol

• Cells from each considered condition are normalized and embedded into common 30-dimensional PCA space; serum and 2i are analyzed separately, with equal cell subsampling when necessary to control computational cost.Source: §E.3 · p. 32

Hyperparameter Setting

• The appendix provides intermediate-state reconstruction result tables for one-, two-, and three-day endpoint intervals, but the supplied captions do not state their numerical outcomes.The interval-specific evaluation and model configurations are described in the protocol.Source: Hyperparameter Setting · p. 33, Table E.1 · p. 34, Table E.2 · p. 35

F RUNNING TIME

• The running-time table records selected training budgets and times for L outer iterations and G directed bridges per iteration.The supplied table caption does not provide time values.Source: Table F.1 · p. 37