Title: Multi-Marginal Schrödinger Bridge Matching
Venue: NeurIPS 2025
Accepted: no
Keywords: Optimal Transport

Population Dynamics

Schrödinger Bridge

Trajectory Inference
Abstract: Understanding the continuous evolution of populations from discrete temporal snapshots is a critical research challenge, particularly in fields like developmental biology and systems medicine where longitudinal tracking of individual entities is often impossible. Such trajectory inference is vital for unraveling the mechanisms of dynamic processes. While Schrödinger Bridge (SB) offer a potent framework, their traditional application to pairwise time points can be insufficient for systems defined by multiple intermediate snapshots. This paper introduces Multi-Marginal Schrödinger Bridge Matching (MSBM), a novel algorithm specifically designed for the multi-marginal SB problem. MSBM extends iterative Markovian fitting (IMF) to effectively handle multiple marginal constraints. This technique ensures robust enforcement of all intermediate marginals while preserving the continuity of the learned global dynamics across the entire trajectory. Empirical validations on synthetic data and real-world single-cell RNA sequencing datasets demonstrate the competitive or superior performance of MSBM in capturing complex trajectories and respecting intermediate distributions, all with notable computational efficiency.

Abstract

Understanding the continuous evolution of populations from discrete temporal snapshots is a critical research challenge, particularly in fields like developmental biology and systems medicine where longitudinal tracking of individual entities is often impossible. Such trajectory inference is vital for unraveling the mechanisms of dynamic processes. While Schr¨ odinger Bridge (SB) offer a potent framework, their traditional application to pairwise time points can be insufficient for systems defined by multiple intermediate snapshots. This paper introduces Multi-Marginal Schr¨ odinger Bridge Matching (MSBM), a novel algorithm specifically designed for the multi-marginal SB problem. MSBM extends iterative Markovian fitting (IMF) to effectively handle multiple marginal constraints. This technique ensures robust enforcement of all intermediate marginals while preserving the continuity of the learned global dynamics across the entire trajectory. Empirical validations on synthetic data and real-world single-cell RNA sequencing datasets demonstrate the competitive or superior performance of MSBM in capturing complex trajectories and respecting intermediate distributions, all with notable computational efficiency.

1 Introduction

Understanding the continuous evolution of populations from discrete temporal snapshots represents a significant challenge in various scientific disciplines, particularly in fields like developmental biology [7, 42] and systems medicine [29] where tracking individual entities longitudinally is often unfeasible. The ability to infer trajectories from such snapshot data is crucial for elucidating the underlying mechanisms of dynamic processes. The Schr¨ odinger Bridge (SB) problem, originally rooted in statistical mechanics [43], has garnered substantial interest in machine learning as an entropy-regularized, continuous-time formulation of optimal transport [20, 30]. It seeks to identify the most probable evolutionary path between prescribed initial and terminal distributions, and has been successfully employed in generative modeling [3, 4, 9, 26, 27, 37, 38, 45, 49].

However, many real-world scenarios present observations or constraints at multiple time points, not just at the beginning and end of a process. For instance, in single-cell RNA sequencing (scRNA-seq) experiments, which are pivotal for studying complex biological processes like cell differentiation, cells are typically destroyed upon measurement [6, 17, 28]. This destructive nature makes it impossible to track individual cells over time, thus necessitating the inference of developmental trajectories from population-level snapshots collected at several intermediate stages. Similarly, meteorological systems may have partial observations across various times [11, 32]. Such situations necessitate a multi-marginal generalization of the SB problem (mSBP), where the path measure must align with prescribed marginal distributions at multiple intermediate time points. While the traditional SB framework offers a powerful approach, its standard application to pairwise time points can prove insufficient for systems characterized by multiple intermediate snapshots. Although more specialized methods for mSBP have recently been developed [8, 18, 44], the direct application of some multi-marginal approaches can lead to error accumulation if not carefully managed, particularly

when learned controls are even slightly inaccurate. These challenges highlight the need for robust 39 and scalable solutions for the mSBP that can effectively integrate information across all observed 40 time points. 41

This paper introduces Multi-Marginal Schr¨ odinger Bridge Matching (MSBM), a novel algorithm 42 specifically developed to address the multi-marginal SB problem by building upon and extending the 43 Iterative Markovian Fitting (IMF) algoritmhs [36, 45]. MSBM is designed to effectively manage mul44 tiple marginal constraints by constructing local SBs on each interval and seamlessly integrating them. 45 This local construction strategy, underpinned by a shared global parametrization of control functions, 46 ensures the robust enforcement of all intermediate marginal distributions while crucially preserving 47 the continuity of the learned global dynamics across the entire trajectory. Empirical validations 48 conducted on synthetic datasets as well as real-world single-cell RNA sequencing data demonstrate 49 that MSBM achieves competitive or superior performance in capturing complex trajectories and 50 accurately respecting intermediate distributions, all while exhibiting notable computational efficiency. 51 Our work aims to provide a robust and scalable computational method for these multi-marginal 52 settings, addressing the critical need for consistent and tractable dynamic inference when data is 53 available as snapshots at multiple time points. 54

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We summarize our contributions as follows:

We extend the theoretical and algorithmic foundations of SBs, including the IMF iteration and optimal control perspectives, to the challenging multi-marginal setting.

We introduce an efficient modeling approach for trajectory inference, that constructs and smoothly integrates local SBs across sub-intervals, inherently allows for parallelized training, leading to significant speed-ups.

Through comprehensive experiments on both synthetic and real-world single-cell RNA sequencing data, we demonstrate that MSBM accurately models complex population dynamics and outperforms state-of-the-art methods in both trajectory fidelity and computational speed.

Notation. Let P [0 ,T ] denote the space of continuous functions taking values in R d on the interval [0 , T ] . We use an uppercase letter P ∈ P [0 ,T ] to represent a path measure. For a path measure P ∈ P [0 ,T ] , the marginal distribution at discrete time points T = { t 0 , . . . , t k } , where 0 = t 0 < t 1 < · · · < t k = T is denoted by P T ∈ P T , where we define P T as the set of measures P over R d ×|T | . Additionally, the conditional distribution of P , given T , is denoted by P |T ∈ P [0 ,T ] . Moreover, a path measure P can be defined as mixture. For any Borel measurable set A ∈ B (Ω) , P can be defined by P ( A ) = ∫ R d ×|T | P |T ( A | x T ) d P T ( x T ) , where P ∈ P 0 ,T and P ∈ P T , and we use the shorthand x T := ( x 1 , · · · , x k ) and [0 : k ] := { 0 , 1 , · · · , k } . The Kullback-Leibler (KL) divergence between two probability measures µ and ν on space X is defined as D KL ( µ | ν ) = ∫ X log dµ dν ( X ) dµ ( X ) when µ is absolutely continuous with respect to ν ( µ ≪ ν ), and D KL ( µ | ν ) = + ∞ otherwise. We will often refer to probability measures on R d and their Lebesgue densities interchangeably, under the standard assumption of absolute continuity. Finally, for a function V : [0 , T ] × R d → R , we define the gradient and laplcaian operators with respect to x ∈ R d as ∇V and ∆ V , respectively, and its partial derivative with respect to time t ∈ [0 , T ] as ∂ t V .

2 Preliminaries

2.1 Schr¨ odinger Bridge Matching (SBM)

The Schr¨ odinger Bridge problem (SBP) [16, 43] is a stochastic optimal transport problem [30] that seeks the optimal transport plan for endpoint marginals ρ 0 and ρ T . In this paper, we focus on the dynamical representation, where a reference distribution Q ∈ P [0 ,T ] is induced by the SDEs:

where f t : R d → R d is a drift, σ ∈ R is a diffusion, and W t ∈ R d is a standard Wiener process. 83 With the base reference path measure Q , the dynamic representation of the SB [20, 35, 39] is: 84

Recent advancements in dynamical optimal transport [37, 45] have introduced a novel numerical 85 methodology for solving SBP . This approach reframes SBP by decomposing its dynamical constraints 86 into the time-evolving marginal distributions P t for all t ∈ [0 , T ] and the joint coupling P 0 ,T . This 87 optimization relies on IMF [45], a technique that iteratively refines the path measure P ∈ P [0 ,T ] . 88 IMF alternates between two projection called Markovian and Reciprocal projections to preserve the 89 correct endpoint marginals ( ρ 0 , ρ T ) throughout the optimization. 90

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Reciprocal Projection R . For a given reference measure Q from (1), and a path measure P with marginals specified at end points T = { 0 , T } the reciprocal projection is defined as:

This projection constructs a new path measure by taking the endpoint coupling P 0 ,T from P and forming a mixture of bridge process using Q conditioned on these end points. Sampling from Π := R ( P , T ) involves drawing end points samples ( X 0 , X T ) ∼ P 0 ,T and then generating a path X T t between them using conditional reference measure Q | 0 ,T which induced by following SDEs, for any ( x 0 , x T ) :

If Q | 0 ,T has tractable bridge formulation, for example, when Q is chosen as a Brownian motion i . e ., d X t = σd W t , sampling the path at time t given the endpoints can be performed as:

Markov Projection M . Although the reciprocal projection R in (2) preserves end point marginals ( ρ 0 , ρ T ) , its sampling process in (4) requires both ( X 0 , X T ) , making it non-Markovian and thus ill-suited for generative modeling aimed at sampling from ρ T without knowing X T . The Markov projection M resolves this by projecting Π := R ( P , T ) into a family of Markov process while ensuring P ⋆ = Π t for all t ∈ [0 , T ] . Again, when Q is chosen as a Brownian motion i . e ., d X t = σd W t , the Markov projection of Π , P ⋆ = M (Π , T ) , is induced by following SDEs:

Intuitively, the term E Q T | t [ X T | X t = x ] can be understood as a prediction of the target state X ⋆ t . Flow matching [23] of Bridge matching [37] tackles the approximation X ⋆ T ≈ E Q T | t [ X T | X t = x ] by learning a drift function. This learned drift guides the evolution of X ⋆ t such that its terminal state aligns with the target, often by regressing the drift agains a target drift derived from samples of ( X 0 , X T ) under the reference conditional bridge measure Q | 0 ,T .

Building upon the projections R and M , Schr¨ odinger Bridge Matching (SBM) methods [37, 45] refines the path measure through an alternating iteraive procedure:

Initialized with P (0) = P (0) T Q | 0 ,T , utilizing P (0) T is independent coupling of ρ 0 and ρ T along with the reference conditional bridge measure Q |T . Please refer to [37, 45] for more details.

3 Multi-Marginal Iterative Markovian Fitting

Dynamic SB methods, as discussed in Section 2, have traditionally focused on problems defined 116 by two endpoint marginal distributions, ( ρ 0 , ρ T ) . However, in real-world applications, particularly 117 in fields like developmental biology (e.g., scRNA-seq studies of cellular differentiation), systems 118 are often observed through snapshots at multiple intermediate time points, not just at the beginning 119 and end of a process. This prevalence of multi-stage data highlights a critical limitation of standard 120 SB approaches. While the theoretical extension of SB methods to handle multiple marginals has 121 been explored [1, 31], the development of robust and scalable computational methods for these 122 multi-marginal settings has lagged. Recently, methods with IPF-type objectives have been derived 123 for multi-marginal cases [8, 44]. However, challenges persist in ensuring global dynamic consistency 124 across all intervals, maintaining computational tractability as the number of marginals increases. 125

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In this section, we extends the SBM framework -conventionally applied to problems with two endpoint marginals ( ρ 0 , ρ T ) and foundational to IMF methods -to handle cases involving k + 1 multiple snapshots ( ρ 0 , ρ t 1 , · · · , ρ T ) on discrete time stamps T = { t 0 , t 1 , · · · , t k } where 0 = t 0 < t 1 < · · · < t k = T 1 . Similar to SBP , the dynamic multi-marginal Schr¨ odinger Bridge problem can be formally defined as [10] the entropy minimization problem:

To find a most probable path P mSBP , the solution of mSBP under multiple constraints, we will generalize the principles of SBM in Section 2.1 to the multi-marginal cases in Section 3.1. The extension of dynamic SB optimality [20, 35] and the associated stochastic optimal control problem [39] to multimarginal settings is presented in Appendix A.

3.1 Multi-Marginal Projection operators

To develop multi-marginal extension of SBM, we investigate how the IMF framework can be adapted to scenarios with multiple snapshots ( i . e ., where the set of time points T has cardinality |T | > 2 ). This adaptation necessitates extending the fundamental building blocks of SBM-specifically, the reciprocal projection R and the Markov projection M -to handle multiple marginal constraints.

Multi-Marginal Reciprocal Projection R mm . First, we state and prove a proposition that characterizes the reciprocal structure of conditional path measures. In particular, we focus on a mixture of bridges Π = Π T Q |T ∈ P [0 ,T ] constrained by the marginals at multiple timestamps in T .

Proposition 1 (Reciprocal Property) . For any x T := ( x 0 , x t 1 , · · · , x T ) ∈ R d × ( k +1) and t ∈ [ t i -1 , t i ) , the marginal distribution of Q |T ( ·| x T ) at t satisfies:

Therefore, for any P ∈ P [0 ,T ] the reciprocal projection R mm ( P , T ) admits the following factorization:

A key implication of the reciprocal property, detailed in Proposition 1, is that a mixture of diffusion bridges constrained on T factorizes into independent segments over successive time intervals. This factorization simplifies the analysis and simulation of the overall path measure. Since each segment can then be treated as a standard conditional bridge process as in (3), closed-form sampling, such as in (4), can be applied independently in parallel to each subinterval { t i -1 , t i } i ∈ [1: k ] . This tractability is essential for developing an efficient multi-marginal SBM algorithm.

Multi-Marginal Markov Projection M mm . With the reciprocal property and factorization in (9), we show that the Markov projection on multi-marginal case can be constructed by similar fashion.

Proposition 2 (Multi-Marginal Markovian Projection) . Let Π ∈ P [0 ,T ] admit factorzation in (9) . The multi-marginal Markov projection of Π , P ⋆ := M mm (Π , T ) ∈ P [0 ,T ] , is associated with the SDE:

Moreover, v ⋆ satisfies the Fokker-Planck equation (FPE) [40]:

where p t is marginal density of Π t . In other words, P ⋆ t = Π t for all t ∈ [0 , T ] . d

As established in Proposition 2, constructing a global diffusion process via (10) with the optimal control v ⋆ (11)) yields a multi-marginal Markov projection X ⋆ [0 ,T ] that is continuous over the entire time interval [0 , T ] . The continuity arises because the local Markov projections, X ⋆ [ t i -1 ,t i ] , on each sub-interval are derived from factorized conditional bridge Q | t i -1 ,t i in (9). These bridges are

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anchored by identical marginal distributions at there shared boundaries; for instance, both X ⋆ [ t i -1 ,t i ] and X ⋆ [ t i ,t i +1 ] is guaranteed to match the marginal distribution ρ t i at time t i . Consequently, these local diffusion processes connect seamlessly at adjacent timestamps, resulting in a smooth and well-defined path for X ⋆ [0 ,T ] . The well-defined nature of the global path, in conjunction with the projections R mm and M mm , is fundamental to successfully applying the SBM framework to the mSBP . Finally, the uniquness condition for standard SB [45, Proposition 5] can also be extended to multi-marginal case.

Proposition 3 (Uniqueness) . Let P ⋆ be a Markov measure which is reciprocal class of Q satisfying P ⋆ t = ρ t for all t ∈ T . Then, P ⋆ is unique solution P mSBP of the mSBP .

Building on the projection operators R mm , M mm with the uniquness result of Proposition 3, we can apply the iterative algorithm used in SBM algorithm [45, Algorithm 1] to the multi-marginal setting:

The convergence guarantees proved for the iteration apply equally well to the multi-marginal case.

Proposition 4 (Convergence) . P ( n ) = P mSBP of mSBP as n ↑ ∞ with iterative procedure in (13) .

3.2 Practical Implementation.

In practice, at each iteration n of (13) we approximate the optimal control v ⋆ from (11) by a neural network v θ . By Girsanov theorem, θ are chosen to minimize the following training objective function:

where β T ( t ) = min u { u > t | t ∈ T } ∈ [0 , T ] is the most recent time point in T after time t . With this notation, the SBM can be generalized to the case of multi-marginal constraints. For example, when T = { 0 , T } then (14) reduces to the objective function described in [45].

The learned Markov control v θ ⋆ ( t, x t ) then ensures P θ ⋆ t = Π t for all t ∈ [0 , T ] . Moreover, prior SBM algorithms interleave forward and backward-time Markov projections to re-anchor the terminal distribution and prevent bias between P ( n ) T and Π T accumulate for each n ∈ N . In the multi-marginal setting, we again build the backward-time Markov projection as in Proposition 2 by gluing the local bridge reversals, so that P ⋆ is governed by both SDEs (10) and the corresponding backward dynamics:

where the backward optimal control u ⋆ in (16) can be approximated with neural network u ϕ where ϕ 185 is chosen to minimize the following training objective function with γ T ( t ) = max u { u < t | t ∈ T } : 186

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4 Multi-Marginal Schr¨ odinger Bridge Matching

A na¨ ıve extension of the standard SBM using, multi-marginal projections R mm and M mm in Sec 3, encounters significant limitations not present in the traditional two-endpoint setting. In such an extension, each iteration typically enforces marginal constraints only at the global endpoints ( ρ , ρ ) .

The multi-marginal coupling Π T at each iteration n of (13) is then derived by propagating the projected dynamics in (10) or (15) solely from these end points ρ or ρ , respectively.

0 T ( n ) 0 T

This approach leads to critical issues specific to the multi-marginal context. Firstly, if the learned controls, such as v ⋆ (forward) or u ⋆ (backward), are even slightly inaccurate, significant biases can arise between the inferred intermediate marginals (Π ( n ) t 1 , · · · Π ( n ) t k -1 ) and the target marginals ( ρ t 1 , · · · , ρ t k -1 ) . Secondly, these discrepancies tend to accumulate iteratively. This accumulation is exacerbated because, beyond an initialization Π (0) = P (0) T Q |T with P (0) T , independent joint coupling of { ρ t } t ∈T , where the joint distribution might be informed by all prescribed data distributions, the subsequent self-refinement process for the dynamics often does not directly incorporate the

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Algorithm 1 Training of MSBM

Algorithm 2 Simulation of MSBM (forward)

Input:

Initial ρ 0 , learned control v θ ⋆

Sample X 0 ∼ ρ 0

Simulate forward SDE over

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= [ f t + σv θ ⋆ ( t, X ⋆ t )] dt + σd W t ,

Output:

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intermediate data distributions ( ρ t 1 , · · · , ρ t k -1 ) into its training objective except ρ 0 and ρ T . Without explicit targets for the intermediate marginals guiding each iteration, the inferred paths between ρ 0 and ρ T can 'collapse' or drift away from the desired states. Consequently, precisely satisfying all intermediate constraints becomes increasingly challenging as iterations proceed.

To address this issue of error accumulation and ensure all marginal constraints { ρ t } t ∈T are satisfied, we propose a method that involves constructing local SBs on each interval [ t i -1 , t i ] and then seamlessly gluing them together. Instead of propagating dynamics from the global endpoints ρ 0 and ρ T alone, our approach first establishes local SBs for each segment. The resulting local couplings are then systematically integrated to satisfy all specified marginal distributions { ρ t } t ∈T across the entire time interval [0 , T ] . This local construction strategy helps prevent the compounding of errors at intermediate time points while still aiming to achieve the overall multi-marginal SB solution, P mSBP . The theoretical basis is provided by the following result.

Corollary 5 (Multi-Marginal Schr¨ odinger Bridge) . Assume a sequence of controls { v i , u i } i ∈ [1: k ] , where each v i , u i induced local SBs P i of SBP over local interval [ t i -1 , t i ] with distributions ( ρ t i -1 , ρ t i ) in a forward and backward direction, respectively. If lim t ↑ t i v i ( t, x ) = v i +1 ( t, x ) and lim t ↓ t i -1 u i ( t, x ) = u i -1 ( t, x ) for all i ∈ [1 : k ] , then P mSBP of mSBP induced by following SDEs:

Building upon Corollary 5, we introduce our Multi-Marginal Schr¨ odinger Bridge Matching (MSBM) method to solve the mSBP . A cornerstone of MSBM is divide the global mSBP into local SBP s while maintaining the continuity of the composite drift functions v ⋆ and u ⋆ in (18c) across adjacent intervals, which guarantees a globally continuous diffusion process inducing P mSBP . Furthermore, by explicitly constraining each local SBs, P i , on its corresponding marginals ( ρ t i -1 , ρ t i ) , MSBM is designed to mitigate the accumulation of bias at intermediate marginals, as shown in Figure 1.

̸

A key challenge of the MSBM is rigorously satisfying the continuity conditions at the boundaries of local controls: lim t ↑ t i v i ( t, x ) = v i +1 ( t, x ) and lim t ↓ t i -1 u i ( t, x ) = u i -1 ( t, x ) for all i ∈ [1 : k ] . If these conditions are not met, discontinuities or 'kinks' can arise at the intermediate time steps. Such kinks would imply that the overall path measure P ⋆ = M mm ( P ⋆ , T ) . This would, in turn, hinder the optimlaity for mSBP , because, following Proposition 3, the desired continuous Markov process is a fixed point of both R mm and Markov projections M mm under multiple time points T :

To construct local SBs such that the continuity requirements for forming a valid global solution are 228 met, thereby preventing the aforementioned kinks and ensuring (19), our MSBM introduces a shared 229 global parametrization v θ , u ϕ for its respective local controls { v i , u i } i ∈ [1: k ] for each sub-interval, 230 where each local controls are parallel updated with following aggregate objective function: 231

where T i = { t i -1 , t i } define sub-intervals with local coupling Π T i for end-points marginals in interval [ t i -1 , t i ] and L is defined in (14) and (17) for forward and backward direction, respectively.

The MSBM training procedure, summarized in Algorithm 1, adapts the standard IMF algorithm presented in [45, Algorithm 1]. A key distinction in our MSBM approach is the parallel application of the IMF procedure to each local time interval, utilizing globally shared forward v θ and backward u ϕ across all local intervals. This parallel processing across sub-intervals contributes to a significant reduction in overall training time.

5 Related Work

The solution of SBP often utilize Iterative Proportional Fitting (IPF) [19], with modern adaptations learning SDE drifts for two-marginal settings [4, 9, 13, 49]. A distinct iterative approach, IMF, as featured in [37, 45], offers improved stability by alternating projections onto different classes of path measures. Moreover, emerging research also explores non-iterative algorithm [12, 38]. These methodologies primarily concentrate on the SB problem itself, iteratively refining path measures or directly computing the bridge measure. Moreover, the SB algorithm is studied under the assumption that the optimal coupling is given [27, 46]. While recent studies have extended foundational SB ideas to the multi-marginal setting of mSBP , research in this area remains relatively limited.

In multi-marginal setting, [8] extends the problem to phase space to encourage smoother trajectories and introduces a novel training methodology inspired by the Bregman iteration [5] to handle multiple marginal constraints. Relatedly, [44] presented an approach that, similar to our work, segments the problem across intervals; they learn piecewise SBs and use likelihood-based training to iteratively refine a global reference dynamic. While these methods are often IPF-based or focus on specific reference refinement strategies, our MSBM extends the previous IMF-type algorithm into multimarginal setting and effectively handles multiple constraints. We demonstrate that our MSBM framework offers substantial gains in training efficiency. This enhanced efficiency is primarily attributed to its direct multi-marginal formulation that adeptly manages multiple constraints, thereby circumventing the computationally intensive iterative refinements common in IPF-based methods

Paralleling these SB-centric developments, other significant lines of work model dynamic trajectories by directly learning potential functions or velocity fields, often drawing from optimal transport or continuous normalizing flows. For instance, [18, 24-26] extend SBs to incorporate potentials or mean-field interactions, connecting to stochastic optimal control and earlier mean-field game frameworks [22, 41]. The broader field of trajectory inference from snapshot data, crucial for applications like scRNA-seq, has seen methods like [48] using CNFs with dynamic OT, and [15] employing Neural ODEs on learned data manifolds. More recently, [33, 34] offer variational objectives to learn dynamics from marginal samples.

6 Experiments

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In this section, we empirically demonstrate the effectiveness of our MSBM. Specifically, our goal 267 is to infer a dynamic model from datasets composed of samples from marginal distributions ρ t 268 observed at discrete time points. We evaluate MSBM on both synthetic datasets and real-world single269 cell RNA sequencing datasets, including human embryonic stem cells (hESC) [11] and embryoid 270 body (EB) [32]. To ensure consistency and fair comparison, our experiments follow the respective 271 experimental setups established by baseline methods. In particular, for the petal dataset, we adopt 272 the experimental setup from DMSB [8], and for the hESC dataset, we follow SBIRR [44]. For 273 the EB dataset, we perform evaluations on both 5-dim and 100-dim PCA-reduced data; here, we 274 follow the 100-dim experimental setup of DMSB and the 5-dim setup from NLSB [18]. Accordingly, 275

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we utilize evaluation metrics consistent with previous studies, including the Sliced-Wasserstein Distance (SWD)[2], Maximum Mean Discrepancy (MMD)[14], as well as the 1-Wasserstein ( W 1 ) and 2-Wasserstein ( W 2 ) distances. All experimental results reported are averaged mean value over three independent runs with different random seeds. We highlight the best-performing results in bold and the second-best results in blue. Further experimental details are provided in Appendix C.

6.1 Synthetic Data

Petal The petal dataset [15] serves as a simple yet complex challenge because it mimics the natural dynamics seen in processes such as cellular differentiation, which include phenomena like bifurcations and merges. We compare our MSBM with MIOFlow [15] and DMSB [8] in Figure 2. As shown in Figure 3, we observe that MSBM exhibits the most accurate and clearly defined trajectory, closely resembling the

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ground truth. Furthermore, Figure 2 demonstrates the evaluation results for the trajectories through W 2 and MMD distances, highlighting that MSBM consistently outperforms MIOFlow and DMSB.

6.2 Single-cell Sequencing Data

We evaluated our MSBM on real-world single-cell RNA sequencing data from two sources: 1 ) human embryonic stem cells (hESCs) [11] undergoing differentiation into definitive endoderm over a 4-day period, measured at 6 distinct time points ( t 0 :0 hours, t 1 :12 hours, t 2 :24 hours, t 3 :36 hours, t 4 :72 hours, and t 5 :96 hours); 2 ) embryoid body (EB) cells [32] differentiating into mesoderm, endoderm, neuroectoderm, and neural crest over 27 days, with samples collected at 5 time windows ( t 0 :0-3 days, t 1 :6-9 days, t 2 :12-15 days, t 3 :18-21 days, and t 4 :24-27 days). Following the experimental setup of baselines, we preprocessed these datasets using the pipeline outlined in [48], and the collected cells were projected into a lower-dimensional space using principal component analysis (PCA).

hESC To follow the experimental setup from SBIRR [44], we reduced the data to the first five principal components and excluded the final time point t 6 from our dataset, resulting in three training time points T = { t 0 , t 2 , t 4 } and two intermediate test points T test = { t 1 , t 3 } . Our objective was to train the dynamics based on the available marginals at the training points in T and interpolate the intermediate test marginals at T test , which were not observed during training. Table 1 demonstrates that our proposed MSBM method performs competitively, achieving lower W 2 distances.

Embryoid Body We validate our MSBM on both 5-dim and 100-dim PCA spaces. First, for the 5-dim experiment, we adopt the experimental setup from NLSB. Given 5 observation time points

Table 1: Performance on the 5dim PCA of hESC dataset. W 2 is compute between test ρ t i and generated ˆ ρ t i by simulating the dynamics from test ρ t 0 .

T = { t 0 , t 1 , t 2 , t 3 , t 4 } , we divide the data using train/test splits ρ tr T /ρ te T , with the goal of predicting population-level dynamics from ρ tr t 0 . Similar to NLSB, we train the dynamics based on ρ tr T and

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evaluate the W 1 distance between ρ te t i and the generated ˆ ρ t i from previous test snapshot ρ te t i -1 . In Table 2, we find that MSBM outperforms several SB methods.

For the 100-dim experiment, we borrow the experimental setup from DMSB, where the goal is predict population dynamics given that observations are available for all time points T (denoted as Full in Table 3), or when one of the snapshot is left out (denoted as t i in Table 3, where snapshot ρ tr t i at t i is excluded during training). The high performance in this task represent the robustness of the model to accurately predict population dynamics. In Table 3, MSBM consistently yields performance improvements. Moreover, as shown in Figure 4, the trajectories and generated marginal distributions ˆ ρ T in PCA space further justifies the numerical result and highlights the variety and quality of the samples produced by MSBM.

Computational Efficiency For an fair comparison of training efficiency against recent multi-marginal SB al-

† result from [18], ‡ result from [34]. gorithms, we benchmarked DMSB and SBIRR on the identical hardware configuration employed for MSBM (denoted by ∗ in Table 1). On the hESC dataset, MSBM achieved a runtime over 4× faster than SBIRR. Furthermore, on the petal and 100-dim PCA of EB dataset, MSBM significantly outperformed DSMB in training speed, with detailed results presented in Figure 5.

This enhanced computational efficiency primarily originates from core algorithmic differences. SBIRR, for example, utilizes maximum likelihood training, which requires extensive gradient computations and the storage of all intermediate paths. DMSB employs an IPF-type objective with Bregman Iteration [5]. In contrast, MSBM directly optimizes controls using an IMF-type objective, which not only eliminates the need to store intermediate states but also facilitates parallel computation across sub-intervals. This approach substantially promotes faster convergence of the algorithm.

7 Conclusion and Limitation

This paper revisits previously established frameworks for the SBP , extending them to the mSBP . Specifically, we introduce a computationally efficient framework for mSBP , termed MSBM, which builds upon existing SBM methods [37, 45]. MSBM is tailored for various trajectory inference problems where snapshots of data are available at multi-marginal time steps. Through the successful adaptation of the IMF algorithm to this multi-marginal setting, our approach significantly accelerates training processes while ensuring accurate dynamic modeling when compared to existing methods.

Despite these advantages, the performance degradation of MSBM is more pronounced than that of DMSB when a time point is omitted in Table 3. This may occur because the including velocity term could better accommodate unknown trajectory. Furthermore, the current MSBM framework is restricted to the case involving snapshot data samples, highlighting a need for enhancements to address problems with continuous potentials, such mean-field games [18, 24-26].

Figure 5: Training time

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes **Multi-Marginal Schrödinger Bridge Matching (MSBM)** to address the multi-marginal Schrödinger Bridge problem, with a goal of finding a continuous trajectory for a population given snapshots at multiple intermediate time points. MSBM decomposes the global problem into local bridges on consecutive intervals, then 'glues' them together while sharing a single neural parameterisation. This aims to enforce continuity across the entire trajectory and prevent the error accumulation. The authors demonstrate the method's effectiveness on synthetic datasets and real-world single-cell RNA sequencing data.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n* **Clear presentation**\n* **Important problem** - inferring continuous trajectories from multiple discrete snapshots, which has significant applications in computational biology and other scientific domains\n* **Technically sound** - correctly identifies the primary failure mode of a naive sequential approach (error accumulation and trajectory discontinuities) and proposes a solution through shared, global controls and an aggregate loss function\n* **Consistent results** - the empirical results presented on both synthetic and single-cell datasets are promising\n\n\n**Weaknesses:**\n* **Limited novelty** - the 'global-network + glue' idea is close to multi-marginal flow-matching; prior work [1,2,3] already uses a single shared network across intervals\n* **Missing baselines** - to the best of reviewers knowledge [3] is current SOTA and EB dataset, at least mean number could have been directly taken from the paper for 5 dimensions, authors do not understand why baselines differ across dimensions\n* **Limited number of datasets** - widely-used CITE-seq and MULTI-seq benchmarks from [5] are omitted, while they have been included in most baselines papers, see [1,2,3,4]\n\n\n\n\n[1] Tong, A., Fatras, K., Malkin, N., Huguet, G., Zhang, Y., Rector-Brooks, J., Wolf, G. and Bengio, Y., 2023. Improving and generalizing flow-based generative models with minibatch optimal transport. arXiv preprint arXiv:2302.00482.\n\n[2] Tong A, Malkin N, Fatras K, Atanackovic L, Zhang Y, Huguet G, Wolf G, Bengio Y. Simulation-free schr\" odinger bridges via score and flow matching. arXiv preprint arXiv:2307.03672. 2023 Jul 7.\n\n[3] Kapusniak, K., Potaptchik, P., Reu, T., Zhang, L., Tong, A., Bronstein, M., Bose, J. and Di Giovanni, F., 2024. Metric flow matching for smooth interpolations on the data manifold. Advances in Neural Information Processing Systems, 37, pp.135011-135042.\n\n[4] Neklyudov, K., Brekelmans, R., Tong, A., Atanackovic, L., Liu, Q. and Makhzani, A., 2023. A computational framework for solving wasserstein lagrangian flows. arXiv preprint arXiv:2310.10649.\n\n[5] Lance, C., Luecken, M.D., Burkhardt, D.B., Cannoodt, R., Rautenstrauch, P., Laddach, A., Ubingazhibov, A., Cao, Z.J., Deng, K., Khan, S. and Liu, Q., 2022. Multimodal single cell data integration challenge: results and lessons learned. BioRxiv, pp.2022-04.\n\nQUESTIONS:\n1.  Could the authors further elaborate on the conceptual novelty of the shared global parametrization approach? The use of a shared network across multiple domains is a common pattern in deep learning. Is there a more subtle aspect of extending the Iterative Markovian Fitting (IMF) framework that makes this application particularly non-trivial in the Schrodinger Bridge context, which the reviewer may have missed?\n2. Why omit MFM [3] which is the current SOTA? Could you add them or justify their exclusion?\n3. Can you report results on the CITE-seq and MULTI-seq single-cell benchmarks used in earlier trajectory-inference work?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-06-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces MSBM, a parallelized implementation for solving the multi-marginal Schrödinger Bridge (mSB) problem. MSBM is specifically designed for trajectory inference tasks where data snapshots are available at multiple time points. By adapting the Iterative Markovian Fitting algorithm, the model learns local dynamics from adjacent time steps in a piecewise manner, effectively capturing long-term dynamics in the multi-marginal setting. This approach leads to significant computational acceleration. The method is empirically validated on both synthetic datasets and two real-world developmental single-cell RNA sequencing datasets.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n- The paper addresses a well-motivated and timely problem in the field of trajectory inference using multi-marginal data. \n\n- The proposed method represents a thoughtful and well-justified adaptation of the previous IMF-based model [45], supported by solid theoretical foundations and an efficient algorithmic implementation.\n\n- The manuscript is clearly written and includes a thorough and well-structured literature review that situates the contribution effectively within the existing body of work.\n\n**Weaknesses**\n\nWhile the paper introduces a promising approach, there are several areas where the experimental evaluation could be improved:\n- Inconsistent Baseline Comparisons: The comparison with baseline methods is inconsistent across experiments, making it difficult to draw clear conclusions about the overall performance of the proposed method across different datasets. In particular, SBIRR [44] appears to be the most relevant baseline for this study and should be included in all experimental studies to provide a fair and comprehensive evaluation.\n- Limited Dataset Diversity: The experimental setup primarily focuses on synthetic and scRNA-seq data. To better demonstrate the generalizability and sample-level transport accuracy of the method, it would be valuable to include experiments on image datasets with known ground-truth correspondences. These types of datasets enable direct comparison of sample-to-sample transport, which is not available in the single-cell case.\n- Lack of Evaluation for Generative Capabilities: Despite the relevance of generative modeling in the broader field of dynamic OT and Neural OT, the paper does not explore or evaluate the generative aspects of the proposed method. Including such experiments would strengthen the contribution, especially given the increasing importance of generative OT in practical applications.\n- Claims of Robustness Lack Empirical Support: The authors claim that MSBM enforces all intermediate marginal distributions robustly while preserving global continuity across the trajectory. However, based on the results presented, particularly in Table 1, it appears that the main advantage of MSBM over SBIRR appears to be computational efficiency. There is insufficient empirical evidence to support the claimed robustness compared to other methods, especially SBIRR.\n- Limited Performance in Sparse Time Point Settings: The method may struggle to learn accurate global dynamics when intermediate time points are sparse or missing during training. This limitation is important in practical settings where densely sampled temporal data is not always available and should be discussed more explicitly.\n-Limited Feature Dimensionality in Single-Cell Experiments: For the single-cell data experiments, the rationale behind using a small feature set (e.g., 5 or 100 dimensions) is not clearly justified. One of the core challenges in applying OT methods to single-cell analysis lies in capturing complex cellular trajectories within a high-dimensional feature space. By restricting the dimensionality, the study may not fully reflect the practical difficulties or the expressive power needed for real-world single-cell applications.\n\nQUESTIONS:\n- In Algorithm 2, if the model does not explicitly learn the local dynamics $f_t$ , how is the trajectory generated? Are you using the SDE formulation from Eq (5) or (10)?\n- In Algorithm 1, what happens if intermediate distributions are randomly dropped with a small dropout rate during training? How robust is the method to missing time points in this setting? Could this strategy help reduce overfitting to local dynamics and potentially improve generalization to unseen test time points?\n- MSBM refines the path measure through an alternating iterative procedure from Markov projection to reciprocal projection. How does error accumulation affect the between these projections over iterations? Has this aspect been studied or empirically evaluated?\n- Why is the supplementary material not cited in the main manuscript? In addition to the proofs, I noticed that it includes additional experimental studies that are not referenced or discussed in the main text.\n\nLIMITATIONS:\nI think the main limitation of the proposed solution for the mSBP lies in its inability to effectively balance the optimality of local transport between adjacent time points with the global transport between the initial and terminal distributions.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work tackles the problem of learning population dynamics from discrete temporal observations of samples, a prevalent problem in scientific domains such as cellular and molecular biology. To address this problem, the authors introduce a novel approach, Multi-Marginal Schrödinger Bridge Matching (MSBM). Their proposed framework builds upon advances in Schrödinger Bridge Matching and iterative Markovian Fitting to devise an improved method specifically tailored to better handle the multi-marginal setting. The authors provide a principled and thorough theoretical grounding for their work and further support their claims through a series of empirical experiments on both synthetic and real single-cell datasets. MSBM is shown to achieve competitive performance while exhibiting favourable computational efficiency relative to its counterpart baseline approaches.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n- In general, I think this work presents a novel and valuable contribution to the field with no major drawbacks that significantly reduce the impact of this work. The authors present a computationally efficient and principled framework for extending Schrödinger bridge matching to the multi-marginal setting. Overall, I think this is a well-motivated and well-executed idea.\n- The authors provided thorough and principled theoretical support for their proposed method, presented clearly and easily followed. \n- The proposed method, MSBM, is evaluated over a variety of settings, including synthetic datasets and real single-cell datasets. Through thorough comparison with counterpart methods, the authors demonstrate that MSBM yields SOTA methods for modelling multi-marginal trajectories. \n\n**Weaknesses:**\n\n- No obvious weaknesses that diminish the quality of this work stand out to me. However, I have some clarifying questions and points. Moreover, this work could benefit from a bit of expansion on the side of the empirical experiments. Please see the question section below.\n\nQUESTIONS:\n- It is not entirely clear to me how equation 20a helps address the challenge of MSBM to rigorously satisfy the continuity condition boundaries of the local controls. Could the authors elaborate on this and provide some additional explanation/intuition?\n- Given the advent of flow matching-based frameworks for modelling population dynamics, especially in the multi-marginal setting, I believe it is useful to also include a flow-based model in Figure 5 (results on computational efficiency). For instance, either (or both) OT-CFM and WLF-SB. Also given how closely they perform to MSBM on the 5 PCA EB dataset. \n- In a similar vein, why not include these baselines on the 100-dim PCA dataset and the 5-dim hESC dataset? \n\nMinor comments:\n\n- in Figure 1, labels for _traj_ and _generated_ seem to be reversed.\n- Equation 20a (between lines 231 and 232) may be mislabeled. Is there an equation 20b?\n- Figure 2 is presented before Figure 3. I recommend adjusting this positioning in the presentation.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Multi-Marginal Schrödinger Bridge Matching (MSBM), a new algorithm designed to address trajectory inference when we have access to observations and constraints at multiple time points, not just at the beginning and end of the process. Traditional Schrödinger Bridge (SB) methods, focusing on endpoints, face limitations in scenarios where multiple intermediate stages exist. MSBM extends SB Matching and the Iterative Markovian Fitting (IMF) algorithm to enforce multiple intermediate marginal constraints while preserving global continuity. Empirical evaluations on synthetic data and real-world single-cell RNA sequencing datasets demonstrate that MSBM significantly improves both trajectory inference accuracy and computational efficiency compared to existing state-of-the-art methods.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n**Quality.** The paper is theoretically sound. The notation and propositions in Section 3 appear correct (though I only skimmed the appendix proofs). The structure follows the original SB matching paper quite closely, and the results are intuitive for readers already familiar with that work. That said, the authors also do a good job making the narrative accessible even beyond the original framework they build upon. All claims are well supported by experimental results, which evaluate both the accuracy and speed of the proposed method.\n\n**Clarity.** Overall, the paper is clearly written and accessible. The authors provide a thoughtful problem setup before introducing their solution, and they include helpful intuition to support most of the theoretical results. The exposition is generally clear and well paced.\n\n**Significance.** The problem addressed in this paper is, in my view, very important. Many real-world scientific processes are inherently multimarginal, and standard two-endpoint models often fail to generalize well in such settings. This work generalizes the well-studied SB matching framework and, to the best of my knowledge, is the first to extend the IMF approach to the multimarginal case. I am optimistic that other researchers will adopt this method to analyze diverse scientific datasets and further investigate its theoretical properties.\n\n**Originality.** This paper presents a clear and meaningful extension of SB matching to the multimarginal setting. While the high-level ideas are related to previous work, the multimarginal formulation introduces new theoretical and algorithmic challenges that are addressed with fresh results and experiments. In my view, this makes the contribution original and worthwhile.\n\n---\n\n**Weaknesses**\n\nI elaborate further in the *Questions* section below, but here I summarize a few major weaknesses. Broadly, I found that the paper becomes more difficult to follow in Section 4, and I believe the experimental section could be strengthened.\n\n- I’m not sure I fully understand what is meant by the \"naive extension\" of standard SBM. If this refers to the practical implementation described in Section 3.2, it’s unclear why an entire section is devoted to this approach, given that a different method is ultimately used in practice. If the naive extension is something else, it should be explicitly described.\n- More generally, the connection between Sections 3 and 4 could be made much clearer. In particular, it's not obvious how the theoretical ideas from Section 3 are used to construct Algorithm 1. As it stands, Algorithm 1 appears to perform pairwise estimation using two global neural networks. A paragraph clarifying this connection would significantly improve the clarity of the manuscript.\n- I understand the issue of \"kinks\" arising when the conditions on the $u_i$’s and $v_i$’s are not satisfied. However, it’s unclear to me how using a global parameterization guarantees that such kinks are avoided—couldn’t the $u_i$’s and $v_i$’s still disagree at the endpoints? I think this point would benefit from an additional sentence of intuition, especially near Equation (20a).\n- Corollary 5 is not very clear. What is the main claim being made? It seems that line 215 might be missing either a proposition or a verb (perhaps an \"is\" after \"mSBP\"?). Clarifying the statement would help.\n- I believe Section 5 would benefit from more consistent baselines and experimental setups across tasks. In particular, it would be useful to evaluate the same baselines across all experiments. Also, I found the definition of the “Full” experiment in Table 3 unclear. My understanding of the trajectory inference task is that one splits the time steps into train/test sets—e.g., training on every other step and testing interpolation on the rest. However, it’s not clear what the “Full” experiment entails or why it is needed. This is especially confusing in light of the discussion in Section 7, lines 353–354. Similarly, in Table 2, all time steps are listed—does this imply a different train/test split? Clarifying the experimental setup would be helpful.\n\nQUESTIONS:\nMy main comments and questions are presented in the *Weaknesses* section above. Below, I list some additional minor points I would like the authors to address:\n\n- **Lines 86–87:** The sentence beginning “This approach … and the joint coupling ...” could be rewritten for clarity, as some of the terms used are introduced without sufficient context or prior explanation.\n- **Reference processes:** Do the $Q$ distributions in the new method need to be Brownian motions, or can they be more general reference measures? This does not seem to be discussed in the main text. If more general reference processes are allowed, how are they chosen in practice?\n- **Lines 108–110:** This sentence could benefit from additional intuition or a precise reference to help readers better understand the claim being made.\n- **Notation comment:** Not a question, but I appreciated the use of $\\mathcal{T}$ to denote the two-time-step SB interpolation — it generalizes nicely in the later definition of the multimarginal SB problem after line 130.\n- **Line 164:** What do you mean by “smooth between adjacent timestamps”? If you are referring to continuity, that’s understandable — but smoothness in a Lipschitz or differentiable sense would require stronger conditions. It may also be worth noting that smoothness is not always desirable — for example, in modeling certain biological processes — so this assumption could be context-dependent.\n- **Equations (14) and (17):** Is there a typo in the definitions of $\\beta_{\\mathcal{T}}$ and $\\gamma_{\\mathcal{T}}$? Comparing these to Equation (10) in the original SB matching paper, I would expect $\\beta$ and $\\gamma$ to belong to $\\mathcal{T}$, but with the current definitions, that doesn’t seem to be the case.\n- **Figure 1:** The font size for the legend and time points is quite small — could it be increased for readability?\n- **MMD kernel:** How is the kernel selected for the MMD metric in the experiments? A brief discussion or clarification would be useful.\n- **Layout on page 9:** The layout of tables and figures on page 9 feels a bit cluttered, and the font sizes are small. It might help to aggregate some of the tables or adjust spacing to improve readability.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}