Schrödinger-Bass Bridge Barycenter Models for Measure-Valued Data

Ngoc-Hai Nguyen; Thuan Nguyen; Prakash Ishwar; Shuchin Aeron

stat.MLOct 21, 2026

Schrödinger-Bass Bridge Barycenter Models for Measure-Valued Data

Authors: Ngoc-Hai Nguyen, Thuan Nguyen, Prakash Ishwar, Shuchin Aeron

Organizations: Department of Electrical and Computer Engineering Tufts University · Department of Engineering, Engineering Technology East Tennessee State University · Department of Electrical and Computer Engineering Boston University · Department of Electrical and Computer Engineering Tufts University NSF AI Institute for Artificial Intelligence and Fundamental Interactions

Abstract

Barycenter models, based on stochastic and deterministic optimal transport bridges between probability measures, have proven to be useful for supervised and unsupervised learning for (probability) measure-valued data. In this paper, we utilize the directional asymmetry and dual drift-diffusion control arising in the recently proposed Schrödinger-Bass Bridge (SBB) between two measures, and propose SBB-encoder, SBB-decoder, SBB-symmetric, and SBB-Markovian barycenter models. For each of these novel models, our first theoretical result establishes the existence of the SBB-barycenter under mild assumptions. Our second theoretical result characterizes the decoder SBB-barycenter of Gaussians where we show that for Gaussian input measures, the decoder formulation admits a Gaussian minimizer, reducing barycenter computation to a finite-dimensional optimization and yielding a compact algorithm. For general input measures, we derive a hybrid Wasserstein gradient particle descent scheme for all four formulations using the SBB dual representation. Finally, we utilize these algorithms for solving distributional-inverse/interpolation problems arising in the context of spatial images and single-cell RNA sequencing data, where we show that the SBB-decoder and Markovian barycenter allow us to interpolate between successive time points. Results show that our method outperforms existing barycenter models for measure interpolation and inference.

Figures & tables

Barycenter Method	$W_{2}$ error $\downarrow$	Pixel $\ell_{1}$ error $\downarrow$	SSIM $\uparrow$
Arithmetic	$3.21\pm 0.20$	$1.35\pm 0.09$	$0.9808\pm 0.005$
Wasserstein	$1.01\pm 0.11$	$0.62\pm 0.07$	$0.9964\pm 0.0013$
RegWB	$2.08\pm 0.04$	$1.15\pm 0.01$	$0.9727\pm 0.0015$
Debiased Sinkhorn	$0.99\pm 0.11$	$0.62\pm 0.06$	$0.9964\pm 0.0012$
Decoder SBB	$0.79\pm 0.09$	$0.45\pm 0.07$	$0.9975\pm 0.0010$

Table 1: MNIST latent-source recovery. Report paired averages across latent digits.

serum condition
Barycenter Method	$W_{2}\downarrow$	MMD $\downarrow$	Cell-type TV $\downarrow$	Signature MAE $\downarrow$
Arithmetic	$19.92\pm 0.31$	$0.235\pm 0.003$	$0.073\pm 0.006$	$0.080\pm 0.012$
Wasserstein	$15.99\pm 0.03$	$0.241\pm 0.003$	$0.068\pm 0.016$	$0.078\pm 0.011$
RegWB	$16.06\pm 0.10$	$0.515\pm 0.002$	$0.091\pm 0.007$	$0.075\pm 0.012$
Waddington-OT	$15.87\pm 0.05$	$0.254\pm 0.003$	$0.070\pm 0.010$	$0.086\pm 0.013$
SBB Decoder (ours)	$15.85\pm 0.05$	$0.214\pm 0.003$	$0.064\pm 0.013$	$0.068\pm 0.004$

Table 2: Ancestor reconstruction results. Lower values indicate better agreement for all four metrics. Serum: a culture medium containing serum, which provides components that support cell growth. Serum + stromal: negative-control experiment that adds the stromal population to the four serum-condition descendant populations

serum condition
Interpolation Method	$W_{2}\downarrow$	MMD $\downarrow$	Cell-type TV $\downarrow$	Signature MAE $\downarrow$
Arithmetic mixture	$11.9916\pm 0.1855$	$0.4044\pm 0.0120$	$0.0588\pm 0.0289$	$0.0364\pm 0.0062$
Wasserstein	$10.9134\pm 0.2175$	$0.4189\pm 0.0155$	$0.0859\pm 0.0284$	$0.0359\pm 0.0008$
RegWB	$11.0323\pm 0.2059$	$0.4154\pm 0.0151$	$0.0892\pm 0.0171$	$0.0370\pm 0.0029$
TrajectoryNet	$14.6255\pm 0.2413$	$0.5539\pm 0.0123$	$0.0658\pm 0.0119$	$0.0329\pm 0.0043$
Waddington-OT	$10.6621\pm 0.0752$	$0.4034\pm 0.0080$	$0.2383\pm 0.0491$	$0.0385\pm 0.0036$

Table 3: Intermediate-state reconstruction one-day intervals result. 2i: a serum-free culture medium containing two inhibitors.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

\bar{m}\leftarrow\sum_{i=1}^{N}\lambda_{i}m_{i}.

Appendix

Algorithm C.2 Gaussian Specialization of Decoder SBB Barycenter

Figure E.1: Decoder Barycenter visualization

Figure E.2: Encoder Barycenter visualization

Figure E.3: Symmetric Barycenter visualization

Figure E.4: Qualitative latent-source recovery on MNIST. Each column corresponds to one digit class, from $0$ to $9$ . The first row shows the clean latent source, while the remaining rows show the barycenters reconstructed by the arithmetic mean, Wasserstein barycenter, Regularized Wasserstein Barycenter, debiased Sinkhorn barycenter, and decoder SBB barycenter, respectively. The decoder SBB reconstruction is generally sharper and more closely preserves the geometry of the clean source, consistent with the quantitative results in Table 1

serum condition
Interpolation Method	$W_{2}\downarrow$	MMD $\downarrow$	Cell-type TV $\downarrow$	Signature MAE $\downarrow$
Arithmetic mixture	$11.9916\pm 0.1855$	$0.4044\pm 0.0120$	$0.0588\pm 0.0289$	$0.0364\pm 0.0062$
Wasserstein	$10.9134\pm 0.2175$	$0.4189\pm 0.0155$	$0.0859\pm 0.0284$	$0.0359\pm 0.0008$
RegWB	$11.0323\pm 0.2059$	$0.4154\pm 0.0151$	$0.0892\pm 0.0171$	$0.0370\pm 0.0029$
TrajectoryNet	$14.6255\pm 0.2413$	$0.5539\pm 0.0123$	$0.0658\pm 0.0119$	$0.0329\pm 0.0043$
Waddington-OT	$10.6621\pm 0.0752$	$0.4034\pm 0.0080$	$0.2383\pm 0.0491$	$0.0385\pm 0.0036$

Appendix

Table E.1: Intermediate-state reconstruction 1-day intervals result.

Panel of Figure 10 — Figure E.5: Some qualitative figures for reconstruction of intermediate cell populations using 1-day intervals.

serum condition
Interpolation Method	$W_{2}\downarrow$	MMD $\downarrow$	Cell-type TV $\downarrow$	Signature MAE $\downarrow$
Arithmetic mixture	$12.5039\pm 0.4085$	$0.5773\pm 0.0140$	$0.2077\pm 0.0502$	$0.2725\pm 0.0218$
Wasserstein	$12.2979\pm 0.4909$	$0.6103\pm 0.0145$	$0.1699\pm 0.0052$	$0.3029\pm 0.0146$
RegWB	$12.4198\pm 0.4824$	$0.6058\pm 0.0139$	$0.1654\pm 0.0216$	$0.2993\pm 0.0222$
TrajectoryNet	$14.1794\pm 0.5810$	$0.6762\pm 0.0221$	$0.3835\pm 0.0329$	$0.3046\pm 0.0040$
Waddington-OT	$12.2477\pm 0.3275$	$0.6125\pm 0.0102$	$0.1660\pm 0.0254$	$0.3613\pm 0.0135$

Appendix

Table E.2: Intermediate-state 2-day intervals reconstruction result.

Panel of Figure 12 — Figure E.6: Some qualitative figures for reconstruction of intermediate cell populations using 2-day intervals.

serum condition
Interpolation Method	$W_{2}\downarrow$	MMD $\downarrow$	Cell-type TV $\downarrow$	Signature MAE $\downarrow$
Arithmetic mixture	$12.4681\pm 0.3015$	$0.4484\pm 0.0148$	$0.1458\pm 0.0283$	$0.0377\pm 0.0124$
Wasserstein	$11.8049\pm 0.2348$	$0.4653\pm 0.0114$	$0.1992\pm 0.0676$	$0.0354\pm 0.0019$
RegWB	$11.9078\pm 0.2552$	$0.4611\pm 0.0112$	$0.1634\pm 0.0511$	$0.0349\pm 0.0026$
TrajectoryNet	$13.9434\pm 0.2421$	$0.5383\pm 0.0125$	$0.3945\pm 0.0234$	$0.0584\pm 0.0050$
Waddington-OT	$12.1768\pm 0.1722$	$0.4949\pm 0.0046$	$0.1133\pm 0.0263$	$0.0422\pm 0.0029$

Appendix

Table E.3: Intermediate-state 3-day intervals reconstruction result.

Panel of Figure 14 — Figure E.7: Some qualitative figures for reconstruction of intermediate cell populations using 3-day intervals.

Case	Method	$L$	$G$	Updates	Time (s)
Anisotropic Gaussian	Decoder	50	3	298,500	907.6
Anisotropic Gaussian	Encoder	50	3	298,500	902.5
Anisotropic Gaussian	Symmetric	50	6	597,000	1799.7
MNIST: 10 images	Decoder	95	32	1,216,000	128.2 $\pm$ 6.1
Ancestor, serum	Decoder	75	4	217,800	176.4 $\pm$ 1.7
1 day, serum	Decoder	60	2	55,680	194.4 $\pm$ 2.7

Appendix

Table F.1: Selected training budgets and recorded times for

L

outer iterations,

G

directed bridges per iteration

Wasserstein Barycenters

Wasserstein barycenters are probability distributions that minimize weighted optimal-transport distance to a collection of distributions.

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Karlsruhe Institute of Technology, Karlsruhe, Germany

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Open paper AskPDF arXiv

Learning generative models in settings where the source and target distributions are only specified through unpaired samples is gaining in importance. Here, one frequently-used model are Schrödinger bridges (SB), which represent the most likely evolution between both endpoint distributions. To accelerate training, simulation-free SBs avoid the path simulation of the original SB models. However, learning simulation-free SBs requires paired data; a coupling of the source and target samples is obtained as the solution of the entropic optimal transport (OT) problem. As obtaining the optimal global coupling is infeasible in many practical cases, the entropic OT problem is iteratively solved on minibatches instead. Still, the repeated cost remains substantial and the locality can distort the global transport geometry. We propose quantized diffusion Schrödinger bridges (QDSB), which compute the endpoint coupling on anchor-quantized endpoint distributions and lift the resulting plan back to original data points through cell-wise sampling. We show that the regularized optimal coupling is stable w.r.t. anchor quantization, with an error controlled by the quality of the anchor approximation. In real-world experiments, QDSB matches the sample quality of existing baselines, requiring substantially less time. Code and data are available at github.com/mathefuchs/qdsb.

Abstract

Wasserstein Barycenters

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