Title: Deep Momentum Multi-Marginal Schrödinger Bridge
Venue: NeurIPS 2023
Accepted: yes
Keywords: Schrödinger Bridge

Trajectory Inference

Optimal Transport
Abstract: It is a crucial challenge to reconstruct population dynamics using unlabeled samples from distributions at coarse time intervals. Recent approaches such as flow-based models or Schrödinger Bridge (SB) models have demonstrated appealing performance, yet the inferred sample trajectories either fail to account for the underlying stochasticity or are unnecessarily rigid. In this article, we extend SB into phase space and propose $\underline{D}$eep $\underline{M}$omentum Multi-Marginal $\underline{S}$chrödinger $\underline{B}$ridge (DMSB), a novel computational framework that learns the smooth measure-valued spline for stochastic systems that satisfy position marginal constraints across time. By tailoring the celebrated Bregman Iteration and extending the Iteration Proportional Fitting to phase space, we manage to handle high-dimensional multi-marginal trajectory inference tasks efficiently. Our algorithm outperforms baselines significantly, as evidenced by experiments for synthetic datasets and a real-world single-cell RNA sequence dataset. Additionally, the proposed approach can reasonably reconstruct the evolution of velocity distribution, from position snapshots only, when there is a ground truth velocity that is nevertheless inaccessible.

Deep Multi-Marginal Momentum Schrödinger Bridge

Abstract

It is a crucial challenge to reconstruct population dynamics using unlabeled samples from distributions at coarse time intervals. Recent approaches such as flowbased models or Schrödinger Bridge (SB) models have demonstrated appealing performance, yet the inferred sample trajectories either fail to account for the underlying stochasticity or are unnecessarily rigid. In this article, We extend the approach in [1] to operate in continuous space and propose Deep Momentum Multi-Marginal Schrödinger Bridge (DMSB), a novel computational framework that learns the smooth measure-valued spline for stochastic systems that satisfy position marginal constraints across time. By tailoring the celebrated Bregman Iteration and extending the Iteration Proportional Fitting to phase space, we manage to handle high-dimensional multi-marginal trajectory inference tasks efficiently. Our algorithm outperforms baselines significantly, as evidenced by experiments for synthetic datasets and a real-world single-cell RNA sequence dataset. Additionally, the proposed approach can reasonably reconstruct the evolution of velocity distribution, from position snapshots only, when there is a ground truth velocity that is nevertheless inaccessible.

1 Introduction

We consider the multi-marginal trajectory inference problem, which pertains to elucidating the dynamics and reactions of indiscernible individuals, given static snapshots of them taken at sporadic time points. Due to the inability of tracking each individual, one considers the evolution of the statistical distribution of the population instead. This problem received considerable attention, and associated applications appear in various scientific areas such as estimating cell dynamics [2, 3], predicting meteorological evolution [4], and medical healthcare statistics tracking [5]. [6, 7] constructed an energy landscape that best aligned with empirical observations using neural network. [8, 9] learn regularized Neural ODE [10] to encode such potential landscape. Notably, in the aforementioned work, the trajectory of samples is represented in a deterministic way. In contrast, [11, 12] employ Schrödinger Bridge (SB) to determine the most likely evolution of samples between marginal distributions when individual sample trajectories are also affected by environmental stochasticity. Yet, these approaches scale poorly w.r.t. the state dimension due to specialized neural network architectures and computational frameworks.

SB can be viewed as a solution to the entropy-regularized optimal transport problem. SB seeks a nonlinear SDE that yields a straight path measure between two arbitary distributions. The straightness is implied by achieving optimality of minimizing transportation costs (i.e. 2-Wasserstein distance ( W 2 )). We note SB is often related to Score-based Generated Model (SGM), both of which can be used for generative modeling by constructing certain Stochastic Differential Equation (SDE) that links data distribution and a tractable prior distribution (i.e. 2 marginals). SGM accomplishes the generative task by first diffusing data to prior through a pre-specified linear SDE, during which a neural network is also learned to approximate the score function. Then this score approximator is used to reverse this diffusion process, and consequently establish the generation. Critically-damped

Langevin Diffusion (CLD) [13] extends the SGM SDE to the phase space by introducing an auxiliary velocity variable with a tractable Gaussian distribution at both the initial and terminal time. The resulting trajectory in the position space becomes smoother, as stochasticity is only injected into the velocity space, and the empirical performance and sample efficiency are enhanced due to the structure of the critical damped SDE. The connection between SGM and SB has been elaborated in [14, 15] and scalable mean matching Iterative Proportional Fitting algorithm (IPF) is proposed to estimate SB efficiently in high dimensional cases. Applications of SB, such as image-to-image transformation [16, 17], RNA trajectory inference [11], solving Mean Field Game[18], Riemannian interpolation [19], demonstrate the effectiveness of SB in various domains.

In this work, we start with SB in phase space (termed momentum SB, mSB in short), and then further investigate mSB with multiple empirical marginal constraints present in the position space, which was formulated as multi-marginal mSB (mmmSB) in [1]. This circumvents the need for expensive space discretization which does not scale well to high dimensions. We also address the challenge of intricate geometric averaging in continuous space setup by strategically partitioning and reorganizing the constraint sets. Furthermore, we enhance the algorithm's computational efficiency by incorporating the method of half-bridge IPF. The optimality of transportation cost in SB leads to straight trajectories, and if one solves N 2-marginal SB problems and connect the resulting trajectories to match N+1 marginals, the connected trajectories will have kinks at all connection points. On the contrary, in mmmSB, the optimality of transportation cost leads to a smooth measure-spline over the state space that also interpolates the empirical marginals. Therefore, this approach is highly suitable for problems originated from physical systems and/or those that should have smooth trajectories, such as trajectory inference in single-cell RNA sequencing. Our research will emphasize on solving mmmSB efficiently in high-dimensions (thus the approach will differ from that in the seminal work [1]; see Sec.4). The differences between our algorithm and prior work are demonstrated in Table.1, and the main contributions of our work are fourfold:

We extend the mean matching IPF to phase space allowing for scalable mSB computing.

We introduce and tailor the Bregman Iteration [21] for mmmSB which makes it compatible with the phase space mean matching objective, thus the efficient computation is activated for high dimensional mmmSB.

We show how to overcome the challenge of sampling the velocity variable when it is not available in training data, which enhances the applicability of our model.

We show the performance of proposed algorithm DMSB on toy datasets which contains intricate bifurcations and merge. On realistic high-dimension (100-D) single-cell RNA-seq (scRNA-seq) datasets, DMSB outperforms baselines by a significant margin in terms of the quality of the generated trajectory both visually and quantitatively. We show that DMSB is able to capture reasonable velocity distribution compared with ground truth while other baselines fail.

2 Preliminary

2.1 Dynamical Schrödinger Bridge problem

Dynamical Schrödinger Bridge problem has been extensively studied in the past few decades. The objective of the SB problem is to solve the following optimization problem:

where π ∈ Π( ρ 0 , ρ T ) belongs to a set of path measures with its marginal densities at t = 0 and T being ρ 0 and ρ T . ξ is the reference path measure (i.e., [14] sets ξ as Wiener process from ρ 0 ). The optimality of the problem (1) is characterized by a set of PDEs (3).

Theorem 2.1 ([22]) . The optimal path measure π in the problem (1) is represented by forward and backward stochastic processes

in which Ψ , ̂ Ψ ∈ C 1 , 2 are the solutions to the following coupled PDEs,

The stochastic processes of SB in (2a) and (2b) are equivalent in the sense of ∀ t ∈ [0 , T ] , p (2 a ) t ≡ p (2 b ) t ≡ p SB t . Here p SB t stands for the marginal distribution of SB at time t , which also represents the marginal density of stochastic process induced by either of Eq.2. The potentials Ψ t and ̂ Ψ t explicitly represent the solution of Fokker-Plank Equation (FPE) and Hamilton-Jacobi-Bellman equation (HJB) after exponential transform [14] where FPE describes the evolution of samples density and HJB represents for the optimality of Eq.1. Furthermore, the marginal density also obeys a factorization of p SB t = Ψ t ̂ Ψ t . Such rich structures of SB will later on be used to construct the log-likelihood objective (Thm.B.1) and Langevin sampler for velocity (§4.4).

To solve SB, prior work have primarily used the half-bridge optimization technique, also known as Iterative Proportional Fitting (IPF), in which one iteratively solves the optimization problem with one of the two boundary conditions [14, 15, 23],

with initial path measure π (0) := ξ . By repeatedly iterating over aforementioned optimizations until the algorithm converges, the SB solution will be attained as π SB ≡ lim d →∞ π ( d ) [24]. In addition, [25] shows that the drift term in SB problem can also be interpreted as the solution Stochastic Optimal Control (SOC) problem by having optimal control policy z ∗ = 2 ∇ x log Ψ( t, x t ) :

This formulation will be used later on for constructing phase space likelihood objective function in §3. Regarding solving the half-bridge problem, abundant results exist in the literature for the vanilla SB described above [14, 15, 23], but we will be solving a different SB problem; see Prop.4.1 for formulation and §.4 for a solution.

2.2 Bregman Iterations for Multiple Constraints

Bregman iteration [21] can be viewed as a multiple marginal generalization of IPF, and it is widely used to solve entropy regularized optimal transport problem [1] with multiple constraints. The algorithm can efficiently solve problems in the form of,

SBProblem

Multi Marginal

MomentumSBProblem

MomentumSBProblem

Problem

min

1/112dxdt

min

1z1 l2dxdt

min

1/z1 12dxdt

Formulations

u

D

D

dx, = v,dt

dx = v,dt

mp8+1p²z=p

x~Po，x~PT

[xovo] ~Poo，[x,V] ~pT

x~Pi∈ [0,1...T]

Baseline:

Prop 3.1

PhaseSpaceLikelihood

Decoupleconstraints

SolutionsLikelihood Objective IPF

Objective IPF (ours)

and Optimize(Sec4)

DMSB (ouralgorithm)

VanillaSB(Chenetal.)

DMSB (ours)

Snapshot po

Toy

Snapshot p1

Example

Snapshotp2

Snapshot p3

Generated Samples

Individual Traj

where K is the intersection of multiple closed convex constraint sets K l : K = ∩ L l =1 K l . Bregman Projection (BP) is defined as optimization w.r.t one of the constraint K l ,

and d -th Bregman Iteration (BI) is recursively computing BP over all the constraints in K :

The initial condition for ( d +1 )-th BI is π ( d +1 , 0) = π ( d,L ) . Under certain conditions (see e.g., [24]), one has that π ( d,L ) converges to the unique solution:

3 Momentum Schrödinger Bridge

We first describe how to conduct half-bridge IPF training in the phase space, which can be used to solve momentum SB (mSB) problem with two marginals constraints. This scalable phase space half-bridge technique will then be applied to multi-marginal cases (Sec.4). Fig.1 demonstrates how we develop an algorithm based on [14]. Notations used in following sections are listed in Table.2. mSB extends SB problem to phase space, which consists of both position and velocity.

We will first consider boundary distributions that depend on both x and v , although eventually we will use this as a module to find transport maps between two distributions that only depend on position x , as velocity v is an auxiliary variable artificially introduced for obtaining smooth transport. Conceptually, as an entropy regularized optimal transport problem, SB tries to obtain the straightest path between empirical marginals of positions x with additive noise, but mSB aims at finding the smooth interpolation between empirical marginals of x [26] conditioned on boundary velocity distributions (see Fig.1). Such smooth measure-valued splines in the position space are obtained by the optimization problem in the phase space [1]:

Similar to Theorem 2.1, one can derive a set of PDEs using the potential functions Ψ( t, x , v ) and ̂ Ψ( t, x , v ) , and subsequently apply IPF procedure to solve the problem. The formulation of the phase space PDE can be found in Appendix.B.2. Such PDE representation of mSB results in a straightforward yet innovative log-likelihood training that enables efficient optimization of the IPF.

Proposition 3.1 (likelihood bound) . The half-bridge IPF in phase space

represents the bound of the likelihood and gives approximate likelihood training:

̂ Z t ∆ = ( 0 ̂ z t ) and ̂ p t is the density of path measure induced by eq.5 at time t . A similar result for log( m T , T ) can be obtained in a similar derivation.

Proof.

Remark 3.2. After optimizing ̂ Z t , the reference path measure becomes eq.5, which implies π ∈ Π( · , µ T ) , i.e., the constraint in half-bridge IPF is satisfied. A path measure π is induced by either Z t or ̂ Z t . As being mentioned in Remark.2.2. One half-bridge IPF is basically one BP and one IPF is one BI. Prop.3.1 provides a convenient way to perform one BP in the form of π := arg min π ∈K l D KL ( π || ¯ π ) by maximizing log-likelihood given constraint K and reference path measure ¯ π .

Prop.3.1 provides an alternative way to conduct the BI which will be heavily used in mmmSB §3, and it is computationally efficient after parameterizing and discretization (§4.4).

4 Deep Momentum Multi-Marginal Schrödinger Bridge

We first state the problem formulation of momentum multi-marginal Schrödinger Bridge (mmmSB). Different from previous two marginals case, we consider the scenario where N + 1 probability measures µ t i are lying at time t i . In addition, velocity distributions are not necessarily known.

Proposition 4.1 ([1]) . The dynamical mmmSB with multiple marginal constraints reads:

For i e [O...N]

For j ∈ [N,..0]

min J(π)

(d)-th BI

min J(π)

minJ(π)

π

s.t

π

EKbridge

Solved bySec.4.3

For j ∈ [N,..0]

For ie[O...N]

(d + 1)-th BI

min J(π)

min J(π)

minJ(π)

Kbridge

:Solved by Sec.4.4

S.t  n e K!

π中

S.t π ∈ K0

π

s.t

π+

EKbridge

and K is the intersection of close convex set of K t i

The problem described in Prop.4.1 can be solved by classical BI algorithm integrated with Sinkhorn method [1]. However, due to the curse of dimensionality and unfavorable geometric explicit solution, the BP cannot be applied in high-dimensional and continuous state space directly. To tackles these difficulties, we parameterize the forward and backward policies z t and ̂ z t by a pair of neural networks. We further decouple and resemble the constraints by which it enables the scalable likelihood IPF and avoids the geometric averaging issue under mmmSB context.

4.1 Decoupling and Reassembling Constraints

We decompose the constraint set (7) by

One can notice that the K 0 t i and K 1 t i share similar structure as simpler boundary marginal conditions K t 0 and K t N , hence we can get rid of the notorious geometric averaging (see §4 in [1]). Notably, this type of constraint provides an opportunity to utilize Proposition 3.1 for optimization, but the joint distribution of x and v is still absent. We classify the constraints into two categories:

By following BI (§2.2), we execute optimization w.r.t. (6) while projecting the solution to subset of K boundary or K bridge iteratively. The sketch can be found in Fig.2. The next sections will provide more details on obtaining the joint distribution µ and optimizing within each constraint set.

Hereafter, we only demonstrate the optimization for forward policy z t given reference path measure ¯ π driven by fixed backward policy ̂ z t . The procedure can be applied for the ̂ z t and vice versa.

4.2 Optimization in set K boundary

We first show how to optimize forward policy z t w.r.t. objective function (6) given the reference path measure ¯ π driven by fixed backward policy ̂ z t under one subset of K boundary .

Proposition 4.2 (Optimality w.r.t. K boundary ) . Given the reference path measure ¯ π driven by the backward policy ̂ z t from boundary µ t i +1 in the reverse time direction, the optimal path measure in the forward time direction of the following problem

Proof. See appendix.B.6

Conceptually, the above optimization objective with K bridge constraint aims at finding a continuous path measure close to reference path measure ¯ π while any intermediate marginals constraints will not be considered. The boundary condition of reference path measure in the next iteration p t 0 ( v t 0 , x t 0 ) is determined by eq.10b. Fortunately, the empirical samples from this distribution are available, though the analytic representation of the distribution ˆ q t 0 ( v t 0 , x t 0 ) is unknown. Hence we can utilize these samples as empirical sources from boundary distribution ˆ q t 0 ( v t 0 , x t 0 ) for the next BP. For further explanation and intuition, one can find it in Appendix.G

4.4 Parameterization and Training Objective Function

Inspired by the success of prior work [14], we parameterize path measure π by forward policy z θ t or backward policy ̂ z ϕ t combined with one of constraints in K boundary or K bridge (see Fig.8 in Appendix for visualization). We adopt Euler-Maruyama discretization and denote the timestep as δ t . Notably, eq.9b and eq.10b can be implied by minimizing phase space NLL in Prop.3.1. This leads to the following objective function, termed as phase space mean matching objective, which will be used to train neural networks that represent z θ t and ̂ z ϕ t after time discretization:

The velocity boundary condition for the reference path measure in the succeeding BP is encoded in eq.9b or eq.10b, but the representation of conditional distribution eq.9b is not clear. We leverage the favorable property of SB to parameterize and sample from such distribution.

When π t i : t i +1 ≡ π ∗ t i : t i +1 , the following equations need to hold ∀ t ∈ [ t i , t i +1 ] :

where ˆ p t and ˆ q t denote the marginal density and conditional velocity distribution of the reference path measure at time t , respectively.

Proof. See appendix.B.5

Remark 4.3. When the ground truth distributions of velocity γ t i are available, one can simply sample from γ t i since the joint distribution µ t is available in this case. In order to matching the reference path measure in KL divergence sense, one needs to match both the intermediate path measure eq.9a and the boundary condition eq.9b. In the traditional two-boundary SB case, matching the boundary condition is often disregarded due to either having a predefined data distribution or a tractable prior. However, in our specific case, as the velocity is not predefined, it becomes imperative to address this issue and optimize it through the application of Langevin dynamics.

4.3 Optimization in set K bridge

The formulation of optimization under K bridge is similar to the previous section but differs by the boundary condition (eq.10b):

Proposition 4.4 (Optimality w.r.t. K bridge ) . Given the reference path measure ¯ π driven by the backward policy ̂ z t from boundary µ t N in the reverse time direction, the optimal path measure in the forward time direction of the following problem

when π t 0 : t N ≡ π ∗ t 0 : t N , the following equations need to hold ∀ t ∈ [ t 0 , t N ] :

position

t = 0.00(constraint

ρ

0

)

t = 0.66

t = 1.32(constraint

ρ

1

)

t = 1.99

t = 2.66(constraint

ρ

2

)

t = 3.33

t = 4.00(constraint

ρ

3

)

velocity

Proposition 4.5 ([27, 28]) . If π θ and ¯ π ϕ shares same path measure, then

Prop.4.5 suggests that one can use p t i ( v t i | x t i ) := ˜ p θ,ϕ t i to imply condition (9b) and obtain samples from such distribution by simulating Langevin dynamics. Namely, we first sample position from ground truth x t i ∼ ρ t i , and then sample v t i ∼ ˜ p θ,ϕ t using eq.11. One can further adopt the same regularization [29] to enforce the condition of Prop.4.5.

4.5 Training Scheme

Here we introduce the scheme to traverse BI (see Fig.2). In one BI, all constraints must be iterated once. For the sake of L MM, the reference path measure should be induced by opposite direction. A single BI cannot be recursively repeated due to the conflict of reference path measure direction. For example (see Fig.2), at the end of d -th BI, ¯ π is yielded by forward policy while the first BP of d -th BI is also optimizing forward policy which violates L MM. Instead, we reschedule the optimization order. Specifically, in ( d +1) -th BI, we optimize backward policy at the first BP and the last BP.

5 Experiments

Setups: We test DMSB on 2D synthetic datasets and real-world scRNA-seq dataset [30]. We choose state of the art algorithms MIOFlow [9] and NLSB [11] as our baselines. We tune both models to the best of our hardware capacity. We choose Sliced-Wasserstein Distance (SWD)[31] and Maximum Mean Discrepancy (MMD)[32] together with visualization as our criterion. The detailed setup of training and evaluation can be found in Appendix.C.

Synthetic Datasets: The Petal [9] and Gaussian Mixture Model (GMM) dataset are simple yet challenging, as they mimic natural dynamics arising in cellular differentiation, including bifurcations and merges. We compare our algorithm with MIOFlow in Fig.3. DMSB can infer trajectories aligned with ground truth distribution more faithfully at timesteps when snapshots are taken.

In GMM experiments (see Fig.4), we choose standard Gaussian at initial and terminal time steps while fourmodal GMM and eight-modal GMM are placed at intermediate time steps. Besides good position trajectory, it is almost serendipity that DMSB can also learn the reasonable velocity trajectory without any access to ground truth velocity information. This paves the way for our later velocity estimation for the RNAsc dataset. scRNAseq Dataset: The emergence of single-cell profiling technologies has facilitated the acquisition of high-resolution single-cell data, enabling the characterization of individual cells at distinct developmental states [7]. However, because the cell population is eliminated after the measurement, one may only gather statistical data for single samples at particular timesteps, which neither preserves

DMSB(ours)

MIOFlow[8]

Ground Truth

t

0

t

1

t

2

t

3

t

4

any correlations over time nor provides access to the ground truth trajectory. The diversity of embryonic stem cells after development from embryoid bodies, which comprises mesoderm, endoderm,

neuroectoderm, and neural crest in 27 days, is demonstrated by the scRNA-seq dataset. The snapshot of cells are collected between ( t 0 : day 0 to 3, t 1 : day 6 to 9, t 2 : day 12 to 15, t 3 : day 18 to 21, t 4 : day 24 to 27). Snapshot data are prepossessed by the quality control [30] and then projected to feature space by principal component analysis (PCA). We inherit processed data from [8]. We validate DMSB on 5-dim and 100-dim PCA space to show superior performance on high-dimension problems compared with baselines. We further show that DMSB can estimate better velocity distribution compared with baselines when the ground truth is absent during training and testing.

We testify the performance of our model by computing MMD and SWD with full snapshots and when one of snapshots is left out (LO). We postpone the comparison of all the models on 5-d RNA space to the appendix (see Fig.9 and Table.6) because the problem is relatively simple and all models can infer accurate trajectory. Table.3 summarizes the average MMD and SWD between estimated marginal and ground truth over different snapshot timesteps. DMSB outperforms prior work by a large margin in high (100) dimensional scenarios. The visualization (Fig.5) in PCA space further justifies the numerical result and highlights the variety and quality of the samples produced by DMSB.

PC1

PC2

NLSB[10]

PC1

MIOFlow[8]

PC1

DMSB(ours)

PC1

Ground Truth

PC3

PC4

PC3

PC3

PC3

PC5

PC6

PC5

PC5

PC5

t

0

t

1

t

2

t

3

t

4

PC1

PC2

t

3

PC1

PC2

t

4

PC3

PC4

PC3

PC4

PC5

PC6

PC5

PC6

DMSB Predicted Velocity

Ground Truth Velocity

Interestingly, Fig.4 demonstrates that DMSB can reconstruct reasonable evolution of the velocity distribution which was not accessible to the algorithm. We further validate such property in 100-D RNAsc dataset. During the training and testing, all the models do not have access to the ground truth velocity. We run the experiments of 100-D and 5-D RNAsc datasets and average the discrepancy between ground truth velocity and estimated velocity over snapshot time. The numerical values are listed in the Table.7 and Table.6. The plot of velocity and position can be found in Fig.9 and Fig.10. The plot illustrates that while all models are capable of learning reasonable trajectories, only DMSB has the ability to estimate a plausible velocity distribution. This property holds even for

100-D RNA dataset (see Fig.5,11,12). This is notable, despite the velocity estimated by DMSB does not perfectly match the ground truth, because it should be noted that the proposed phase space SDE and the optimality of OT are artificial and may not necessarily represent the actual RNA evolution. Moreover, as individual evolutions cannot be tracked, possibilities such as {A → A, B → B} versus {A → B, B → A} can not be discerned, which renders exact velocity recovering almost impossible.

6 Conclusion and Limitations

In this paper, we propose DMSB, a scalable algorithm that learns the trajectory which fits the different marginal distributions over time. We extend the mean matching objective to phase space which enables efficient mSB computing. We propose a novel training scheme to fit the mean matching objective without violating BI which is the root of solving mmmSB problem. We demonstrate the superior result of DMSB compared with the existing algorithms.

Amain limitation of this work is, the rate of convergence to the actual mmmSB has not been quantified after neural network approximations are introduced. Even though [15] theoretically analyzed the convergence of mean matching iteration, supporting its outstanding performance [14], the iteration still fails to converge to the actual SB [34] precisely due to practical neural network estimation errors accumulating over BI. However, recent work [35] shows the convergence of SB when training error exists. In addition, DMSB cannot simulate the process with death and birth of cells which can be potentially described as unbalanced optimal transport [36].

References

Yongxin Chen, Giovanni Conforti, Tryphon T Georgiou, and Luigia Ripani. Multi-marginal schrödinger bridges. In International Conference on Geometric Science of Information , pages 725-732. Springer, 2019.

Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, et al. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Cell , 176 (4):928-943, 2019.

Karren D Yang and Caroline Uhler. Scalable unbalanced optimal transport using generative adversarial networks. arXiv preprint arXiv:1810.11447 , 2018.

Mike Fisher, Jorge Nocedal, Yannick Trémolet, and Stephen J Wright. Data assimilation in weather forecasting: a case study in pde-constrained optimization. Optimization and Engineering , 10(3):409-426, 2009.

Kenneth G Manton, XiLiang Gu, and Gene R Lowrimore. Cohort changes in active life expectancy in the us elderly population: Experience from the 1982-2004 national long-term care survey. The Journals of Gerontology Series B: Psychological Sciences and Social Sciences , 63(5):S269-S281, 2008.

Tatsunori Hashimoto, David Gifford, and Tommi Jaakkola. Learning population-level diffusions with generative rnns. In International Conference on Machine Learning , pages 2417-2426. PMLR, 2016.

Charlotte Bunne, Laetitia Papaxanthos, Andreas Krause, and Marco Cuturi. Proximal optimal transport modeling of population dynamics. In International Conference on Artificial Intelligence and Statistics , pages 6511-6528. PMLR, 2022.

Alexander Tong, Jessie Huang, Guy Wolf, David Van Dijk, and Smita Krishnaswamy. Trajectorynet: A dynamic optimal transport network for modeling cellular dynamics. In International conference on machine learning , pages 9526-9536. PMLR, 2020.

Guillaume Huguet, Daniel Sumner Magruder, Oluwadamilola Fasina, Alexander Tong, Manik Kuchroo, Guy Wolf, and Smita Krishnaswamy. Manifold interpolating optimal-transport flows for trajectory inference. arXiv preprint arXiv:2206.14928 , 2022.

Tian Qi Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. In Advances in Neural Information Processing Systems , pages 6572-6583, 2018.

Takeshi Koshizuka and Issei Sato. Neural lagrangian schr \ " odinger bridge. arXiv preprint arXiv:2204.04853 , 2022.

Lénaïc Chizat, Stephen Zhang, Matthieu Heitz, and Geoffrey Schiebinger. Trajectory inference via mean-field langevin in path space. arXiv preprint arXiv:2205.07146 , 2022.

Tim Dockhorn, Arash Vahdat, and Karsten Kreis. Score-based generative modeling with critically-damped langevin diffusion. arXiv preprint arXiv:2112.07068 , 2021.

Tianrong Chen*, Guan-Horng Liu*, and Evangelos A Theodorou. Likelihood training of schrödinger bridge using forward-backward sdes theory. arXiv preprint arXiv:2110.11291 , 2021.

Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion schrödinger bridge with applications to score-based generative modeling. arXiv preprint arXiv:2106.01357 , 2021.

Yuyang Shi, Valentin De Bortoli, George Deligiannidis, and Arnaud Doucet. Conditional simulation using diffusion schrödinger bridges. In Uncertainty in Artificial Intelligence , pages 1792-1802. PMLR, 2022.

Guan-Horng Liu, Arash Vahdat, De-An Huang, Evangelos A Theodorou, Weili Nie, and Anima Anandkumar. I 2 sb: Image-to-image schr \ " odinger bridge. arXiv preprint arXiv:2302.05872 , 2023.

Guan-Horng Liu, Tianrong Chen, Oswin So, and Evangelos A Theodorou. Deep generalized schr \ " odinger bridge. arXiv preprint arXiv:2209.09893 , 2022.

James Thornton, Michael Hutchinson, Emile Mathieu, Valentin De Bortoli, Yee Whye Teh, and Arnaud Doucet. Riemannian diffusion schr \ " odinger bridge. arXiv preprint arXiv:2207.03024 , 2022.

Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456 , 2020.

Lev M Bregman. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR computational mathematics and mathematical physics , 7(3):200-217, 1967.

Michele Pavon and Anton Wakolbinger. On free energy, stochastic control, and schrödinger processes. In Modeling, Estimation and Control of Systems with Uncertainty , pages 334-348. Springer, 1991.

Francisco Vargas. Machine-learning approaches for the empirical schrödinger bridge problem. Technical report, University of Cambridge, Computer Laboratory, 2021.

Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, and Gabriel Peyré. Iterative bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing , 37(2):A1111-A1138, 2015.

Paolo Dai Pra. A stochastic control approach to reciprocal diffusion processes. Applied mathematics and Optimization , 23(1):313-329, 1991.

Jean-David Benamou, Thomas O Gallouët, and François-Xavier Vialard. Second-order models for optimal transport and cubic splines on the wasserstein space. Foundations of Computational Mathematics , 19(5):1113-1143, 2019.

Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications , 12(3):313-326, 1982.

Edward Nelson. Dynamical theories of Brownian motion , volume 106. Princeton university press, 2020.

Hung-Yu Tseng, Lu Jiang, Ce Liu, Ming-Hsuan Yang, and Weilong Yang. Regularizing generative adversarial networks under limited data. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 7921-7931, 2021.

Kevin R Moon, David van Dijk, Zheng Wang, Scott Gigante, Daniel B Burkhardt, William S Chen, Kristina Yim, Antonia van den Elzen, Matthew J Hirn, Ronald R Coifman, et al. Visualizing structure and transitions in high-dimensional biological data. Nature biotechnology , 37 (12):1482-1492, 2019.

Nicolas Bonneel, Julien Rabin, Gabriel Peyré, and Hanspeter Pfister. Sliced and radon wasserstein barycenters of measures. Journal of Mathematical Imaging and Vision , 51:22-45, 2015.

Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test. The Journal of Machine Learning Research , 13(1):723-773, 2012.

Murray Rosenblatt. Remarks on some nonparametric estimates of a density function. The annals of mathematical statistics , pages 832-837, 1956.

David Lopes Fernandes, Francisco Vargas, Carl Henrik Ek, and Neill DF Campbell. Shooting schrödinger's cat. In Fourth Symposium on Advances in Approximate Bayesian Inference , 2021.

Yu Chen, Wei Deng, Shikai Fang, Fengpei Li, Nicole Tianjiao Yang, Yikai Zhang, Kashif Rasul, Shandian Zhe, Anderson Schneider, and Yuriy Nevmyvaka. Provably convergent schr \ " odinger bridge with applications to probabilistic time series imputation. arXiv preprint arXiv:2305.07247 , 2023.

Yongxin Chen, Tryphon T Georgiou, and Michele Pavon. The most likely evolution of diffusing and vanishing particles: Schrodinger bridges with unbalanced marginals. SIAM Journal on Control and Optimization , 60(4):2016-2039, 2022.

Yang Song, Conor Durkan, Iain Murray, and Stefano Ermon. Maximum likelihood training of score-based diffusion models. arXiv e-prints , pages arXiv-2101, 2021.

Jiongmin Yong and Xun Yu Zhou. Stochastic controls: Hamiltonian systems and HJB equations , volume 43. Springer Science & Business Media, 1999.

Kenneth Caluya and Abhishek Halder. Wasserstein proximal algorithms for the schrödinger bridge problem: Density control with nonlinear drift. IEEE Transactions on Automatic Control , 2021.

Ioannis Exarchos and Evangelos A Theodorou. Stochastic optimal control via forward and backward stochastic differential equations and importance sampling. Automatica , 87:159-165, 2018.

Tianrong Chen, Ziyi O Wang, Ioannis Exarchos, and Evangelos Theodorou. Large-scale multiagent deep fbsdes. In International Conference on Machine Learning , pages 1740-1748. PMLR, 2021.

Tianrong Chen, Ziyi Wang, and Evangelos A Theodorou. Deep graphic fbsdes for opinion dynamics stochastic control. In 2022 IEEE 61st Conference on Decision and Control (CDC) , pages 4652-4659. IEEE, 2022.

Hopf Eberhard. The partial differential equation ut+ uux= µ xx. Communications on Pure and Applied Mathematics , 3(3):201-230, 1950.

Julian D Cole. On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of applied mathematics , 9(3):225-236, 1951.

Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101 , 2017.

Francisco Vargas, Pierre Thodoroff, Neil D Lawrence, and Austen Lamacraft. Solving schrödinger bridges via maximum likelihood. arXiv preprint arXiv:2106.02081 , 2021.

A Appendix

B Proof in §3 and §4

Before stating our proofs, we provide the assumptions used throughout the paper. These assumptions are adopted from stochastic analysis for SGM [27, 37, 38], SB [39], and FBSDE [40-42].

(i) µ t i with finite second-order moment for all t i .

(ii) f and g are continuous functions, and | g ( t ) | 2 > 0 is uniformly lower-bounded w.r.t. t .

(iii) ∀ t ∈ [0 , T ] , we have ∇ v log p t ( m t , t ) , ∇ v log Ψ( · , · , · ) , ∇ v log ̂ Ψ( · , · , · ) , Z ( · , · , · ; θ ) , and ̂ Z ( · , · , · ; ϕ ) Lipschitz and at most linear growth w.r.t. x and v .

(iv) Ψ , ̂ Ψ ∈ C 1 , 2 .

Assumptions (i) (ii) (iii) are standard conditions in stochastic analysis to ensure the existenceuniqueness of the SDEs; hence also appear in SGM analysis [37]. Assumption (iv) allows applications of Itô formula and properly defines the backward SDE in FBSDE theory. Finally, assumption (v) assures the exponential limiting behavior when performing integration by parts. w.o.l.g, we denote f = [ v , 0 ] T .

B.1 Proof of Proposition.3.1

The results of the Prop.3.1 is part of results of Prop.B.4 which gives the results for both forward and backward likelihood objective.

Theorem B.1. The optimization problem

will induce the coupled PDEs,

and the optimal control of the problem is

Proof. One can write the Lagrange by introducing lagrangian multiplier ϕ :

By taking the minimization within the bracket, The optimal control is,

By Plugging it back, the optimality of the aforementioned problem is presented as:

Theorem B.2. The optimal forward and backward processes are represented as:

in which f = [ v , 0 ] T . Optimal control is expressed as,

where Ψ and ̂ Ψ are the solution of following PDEs,

Proof. By Lemma.B.1, we notice that the optimal control is:

By leveraging Hopf-Cole [43, 44] transformation, here we define

Then we can have the following expressions:

Thus, we can have.

Then we can represent the optimal control as:

Then the solution of such mSB is characterized by the forward SDE:

Due to the structure of Hopf-Cole transform, one can have

According to [27, 28], the reverse drift of such SDE (eq.23) u b ∗ t should admits,

which yields The backward optimal control

Thus, the optimal forward and backward process is

And Ψ and ̂ Ψ satisfy following PDEs,

Lemma B.3. By specifying f = [ v , 0 ] T , The PDE shown in 20 can be represented by following SDEs

Where:

Proof. One can write

By applying Itô's lemma,

Similarly, one can have,

Noticing:

Following the above derivation, one can have,

By defining

One can conclude the results.

Proposition B.4. The log-likelihood at data point m 0 can be expressed as

By maximizing the log-likelihood at time t = 0 then t = T iteratively, ( z t , ̂ z t ) will converge to the solution of phase space SB.

t

Proof. from Lemma.B.3, one can have:

A similar result can be obtained for log p ( m T , T ) .

One can notice that the likelihood objective is a continuous time analog of the mean matching objective proposed in [15], and iterative optimization between logp ( m 0 , 0) and log p ( m T , T ) are the continuous analog of IPF. Hence, the convergence proof will keep valid (see Proposition 4 in [15]).

The equivalence of KL divergence optimization in IPF and likelihood optimization is widely analyzed in [14, 15, 18]. The objective function will eventually boil down to the mean matching objective shown in the above proposition.B.4.

Proposition B.5 (Optimality w.r.t. K boundary ) . . Given the reference path measure ¯ π driven by the policy ̂ z t from boundary µ t i +1 in the reverse time direction, the optimal path measure in the forward time direction of the following problem

When π t i : t i +1 ≡ π ∗ t i : t i +1 , the following equations need to hold ∀ t ∈ [ t i , t i +1 ] :

where ˆ p t and ˆ q t denote the marginal density and conditional velocity distribution of the reference path measure at time t and t i , respectively.

Proof. Due to the similarity of optimization for K boundary, the close form solution of the next path measure is (see §4 in [1] for detail):

By denoting the transition kernel of parameterized SDE driven by backward policy ̂ z t as q ( ·|· ) , and the time range between t i and t i +1 is discretized into S interval by EM discretization. Then one can

get

Doing eq.36 revursively

According to [15], given the policy ̂ z t , the transition kernel q s ( m t i +( s +1) · δ t | m t i + s · δ t ) can be estimated by ̂ z t (see Proposition 3 in [15])and it can be treated as the label for the forward policy z t for all s . Thus, if π t i t i +1 is aligned with π ∗ t i t i +1 , then one can construct following objective function for policy z t :

due to the special structure of Z t and ̂ Z t

(40)

Where 1 , 2 , 3 corresponds to F k , B k , and B k +1 in [15] respectively. Furthermore, we need to find a density function p t i ( v t i | x t i ) which satisfies

to be the new boundary condition.

Proposition B.6 (Optimality w.r.t. K bridge ) . Given the reference path measure ¯ π driven by the policy ̂ z t from boundary µ t N in the reverse time direction, the optimal path measure in the forward time direction of the following problem

when π t 0 t N ≡ π ∗ t 0 t N , the following equations need to hold ∀ t ∈ [ t 0 , t N ] :

Proof. Same proof as B.5.

Remark B.7. The optimizer of such a problem can be represented as

Where the notation R represents for the time reversal. The Proposition.4.4 is basically using neural network Z θ t to approximate eq.44.

C Experiment Details

We test DMSB on 2D synthetic datasets and realworld scRNA-seq dataset [30]. We parameterize z ( t, m ; θ ) and ̂ z ( t, m ; ϕ ) with residual-based networks for all datasets (see.fig.6). The network adopts position encoding and is trained with AdamW[45] on one Nvidia 3090 Ti GPU. We use constant g(t) for simplicity though the framework can adopt time varying function g(t). We set the time horizon T = t N = 1 · N and interval δ t = 0 . 01 . We use EM discretization throughout the whole paper. For scRNA-seq dataset, we split data into train and test subsets(85% and 15%).All the experiment results are simulated by all-step push forward from initial data points at time t = t 0 .

MIOFlow and NLSB setup: We use the official implementation of NLSB and MIOFlow.For MIOFlow, we report the best performance for all experiments w/GAE(or AE) and w/o GAE(or AE) embedding. For NLSB, we enlarge the size of the neural network to the best of our GPU capacity for a 100-dimensional scRNA-seq dataset and report the best performance during the training.

We evaluate the velocity of NLSB, as an SDE model, by its estimated drift term at time steps t = { 1 , 2 , 3 , 4 , 5 } . Because MIOflow w/ GAE simulates trajectories in the latent space, we estimate the velocity by using the forward finite difference technique with discretization 1 E -3 sec after mapping from the latent code to the original space. We run the experiments of 100-D and 5-D RNAsc datasets and average the discrepancy between ground truth velocity and estimated velocity over snapshot time. The numerical values are listed in the Table.7 and Table.6. The plot of velocity and position can be found in Fig.9 and Fig.10. We do not want to underestimate any prior work and tried out best to tune the prior work. Feel free to communicate with the first author if one can reproduce better results in the experiment section, and we are willing to update it.

Metrics and Evaluations The 1-Wasserstein Distance suffers from the curse of dimensionality seriously. In the main paper, we are using Sliced-Wasserstein Distance (SWD) and Maximum Mean Distance as our criterion for 100-dim RNA dataset. An example is listed in the following toy code. One can notice that W 1 suffers from the curse of dimensionality seriously, the distance between two gaussian samples is even larger than the distance between gaussian and zeros (See following code snapshot). Hence such a metric is not suitable for high dimension ( ≥ 100 ) dataset evaluation even though some papers report W 1 . In order to better evaluate our model compared with baselines, we are using W 1 , Energy Distance, Max-sliced Wasserstein distance, Sliced-Wasserstein Distance and MMD. Our metric is adapted from Geoloss ( W 1 and Energy ), POT (Sliced Wassersetein and Maximum-Sliced Wasserstein) and this repo (MMD).

Trajectories Cache Similar to prior work [14, 15], we also need to cache the trajectories for training purposes. We cache 4096 trajectories for each Bregman Projection.

which can also be represented as,

Special Clarification for NLSB

We evaluate the velocity of NLSB, as an SDE model, by its estimated drift term at time steps t = { 1 , 2 , 3 , 4 , 5 } . It may not be reasonable to consider the drift term as the real velocity, but the drift term can certainly depict a trend of SDE, so we still provide the result here.

Training: We use Exponential Moving Average (EMA) with a decay rate of 0.999. Table.7 details the hyperparameters used for each dataset.The learning rate for all the datasets is set to be 2e-4 and the training batching size is 256 . For computation efficiency, we cache large batch size of empirical samples from reference trajectory and sample training batch size from the cache data. The hyperparameters can be found in Table.7.

mt

ResBlock

ResBlock

FClayers

Output

Position

L

encoding

FC layers

Langevin sampling: The Langevin sampling procedure for the velocity is summarized in 2. Given some pre-defined signal-to-noise ratio r (we set snr =0.15 for all experiments), the Langevin noise scale σ at each time step t and each corrector step i is computed by

D Algorithms

Algorithm 1 Sampling Procedure of DMSB

Algorithm 2 Langevin Sampler at t i marginal constraint

Algorithm 3 DMSB Training

Algorithm 4 Function OptSubSet (Optimization for subsets)

E Additional Diagram

Reference path 元tt

Ht

ut

μt2

μto

μt

Optimized backward path π

tist;

ut

μt2

μto

ut

Current boundary condition

Boundary condition of Next BP

μto

ut2

Optimization w.r.t Kboundary

d

utz

μto

Optimization w.r.t Kbridge

F Additional Experiment

↓

±

±

±

±

±

-

4

-

2

0

2

4

PC1

-

4

-

2

0

2

4

PC2

NLSB

-

4

-

2

0

2

4

PC1

-

4

-

2

0

2

4

MIOFlow

-

4

-

2

0

2

4

PC1

-

4

-

2

0

2

4

DMSB

-

4

-

2

0

2

4

PC1

-

4

-

2

0

2

4

Ground Truth

-

4

-

2

0

2

4

PC3

-

4

-

2

0

2

4

PC4

-

4

-

2

0

2

4

PC3

-

4

-

2

0

2

4

-

4

-

2

0

2

4

PC3

-

4

-

2

0

2

4

-

4

-

2

0

2

4

PC3

-

4

-

2

0

2

4

t

0

t

1

t

2

t

3

t

4

-

4

-

2

0

2

4

PC1

-

4

-

2

0

2

4

PC2

NLSB

-

4

-

2

0

2

4

PC1

-

4

-

2

0

2

4

MIOFlow

-

4

-

2

0

2

4

PC1

-

4

-

2

0

2

4

DMSB

-

4

-

2

0

2

4

PC1

-

4

-

2

0

2

4

Ground Truth

-

4

-

2

0

2

4

PC3

-

4

-

2

0

2

4

PC4

-

4

-

2

0

2

4

PC3

-

4

-

2

0

2

4

-

4

-

2

0

2

4

PC3

-

4

-

2

0

2

4

-

4

-

2

0

2

4

PC3

-

4

-

2

0

2

4

t

0

t

1

t

2

t

3

t

4

PC1

PC2

NLSB

PC1

MIOFlow

PC1

DMSB

PC1

Ground Truth

PC3

PC4

PC3

PC3

PC3

PC5

PC6

PC5

PC5

PC5

t

0

t

1

t

2

t

3

t

4

PC0

PC1

t

0

PC0

PC1

t

1

PC0

PC1

t

2

PC0

PC1

t

3

PC0

PC1

t

4

PC2

PC3

PC2

PC3

PC2

PC3

PC2

PC3

PC2

PC3

PC4

PC5

PC4

PC5

PC4

PC5

PC4

PC5

PC4

PC5

ground truth

predict

G Intuitions of Propositions and Theorems

Remark for Proposition 3.1: Within each half-bridge IPF, the variable Z t (or ̂ Z t ) is essentially learning the reverse-time stochastic process induced by ̂ Z t . This process can also be viewed as minimizing the approximated parameterized negative log-likelihood.

Remark for Proposition 4.2: In order to matching the reference path measure in KL divergence sense, one need to match both the intermediate path measure eq.(9a) and the boundary condition eq.(9b). In the traditional two boundary SB case, matching the boundary condition is often disregarded due to either having a predefined data distribution or a tractable prior. However, in our specific case, as the velocity is not predefined, it becomes imperative to address this issue and optimize it through the application of Langevin dynamics.

Remark for Proposition 4.3: Following the same argument as the remark of Proposition 4.2, it becomes evident that, in this particular scenario, there is no need to account for the data distribution since there are no position constraints when optimizing with K . Consequently, the optimal solution will inherently align faithfully with the reverse diffusion and adapt to the boundary conditions imposed by the reference path measure.

Remark for Proposition 4.5: We indeed underexplained an important nontrivial fact (thank you so much for catching it): the unique structure of SB leads to a beautiful fact that the score is proportional to the sum of the forward and backward drift terms. This facilitates the sampling of velocity. Specifically, the score function can be obtained using eq(11), as supported by the findings in eq(24). It can also be understood as the one realization of Nelson duality (see Lemma 1 in [46] and [28]).

H Complexity

Here we provide the complexity of our algorithm.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper tackles the multi-marginal Schrodinger bridge space in phase space, by proposing a computationally tractable solver (DMSB). It leverages alternating Bregman projections to adapt the iterative proportional fitting algorithm in order to deal with multiple convex constraint sets.\n\nThe DMSB framework constrains the stochasticity of particle dynamics to their velocity component and therefore ensures smooth trajectories. The authors show how the proposed algorithm better describes scRNA-seq data by reconstructing lower-variance trends in the dynamics and (approximately) recovers the velocity field of particles.\n\nSTRENGTHS:\nThe paper builds on well-known results involving Schrödinger bridges (SBs) and convex optimization to practically solve the multi-marginal SB problem while ensuring smooth trajectories. It extends SBs [13] by considering random velocities, rather than random infinitesimal displacements. Furthermore, it generalizes the popular critically-damped Langevin diffusion (CLD) [12] framework to non-gaussian velocity marginals at the extremes.\n\nA key contribution consists in performing 2 alternate optimization stages over the two sets $k_{\\text{boundary}}$ and $k_{\\text{bridge}}$, containing appropriately-partitioned constraints. The extension of the mean-matching objective to the phase space is also novel but relatively straightforward.\n\nThe content is well-motivated and appropriately described. Both mathematical results and algorithmic solutions are accompanied by remarks and informal insight, which help to understand their purpose and significance.\nSeveral tables and diagrams further clarify the presentation by (i) precisely stating the notation used (Table 2), (ii) the problem description (Figure 1), (iii) the comparison against previous work (table 1), and the structure of the algorithm (Figure 2).\n\nWEAKNESSES:\nThe rationale behind the decoupling of constraints, i.e. the requirement that marginalizing joint densities over the previous/next timesteps yield the same distribution, could be made more explicit to the reader. In particular, it would be advisable to explain why the solution proposed by the authors avoids the “geometric averaging issue” (line 156) which affects instead the solution in [20].\n\nI think that Proposition 4.5 should be followed by a more extensive discussion on the differences between constraints in Eqs. 9b and 10b: Why a conditional distribution appears in the former and a joint one in the latter?\n\n**Miscellaneous**\n\nIn addition, I point out minor inaccuracies found while reviewing this draft:\n- The quantities optimized in the formulas in the first row of Figure 1 are incorrect ($u$ and $a$ do not appear anywhere in the optimization objective).\n- Proposition 4.1 looks more like a definition (and it is not even stated as a proposition in the work cited as a source).\n- Wrong sections are cited on the right of Figure 2.\n- There are spelling mistakes in lines 34, 66, 89, and 246.\n- I would suggest refining the references to the Appendix, by clearly specifying if the link points to a proposition or a section (e.g., in line 128).\n\nQUESTIONS:\n**Experiments**\n\nIt is unclear why the colors used in Figure 3 vary across the rows. Could it be possible to identify the position and velocity of each group of points by using the same colors in both graphs?\n\nContrary to what was stated in the text (“it is almost serendipity that DMSB can also learn the reasonable velocity trajectory without any access to ground truth”, line 230), I don’t find the approximate recovery of velocities surprising. Given the formulation of momentum Schrödinger bridges (lines 125-126), the algorithm must find a reasonable velocity field, in order to produce meaningful trajectories.\n\nLIMITATIONS:\nAs stated by the authors, the benefits brought by DMSB are not detectable when learning simple trajectories in relatively low-dimensional spaces. It is therefore unclear whether many practical problems would benefit from it since this comes at the expense of doing without convergence guarantees.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper aims at solving efficiently in high dimensions the multi marginal momentum Schrödinger Bridge, that is Schrödinger Bridge in phase space with multiple marginal constraints. They also tackle the issue of marginals constraints where only the positions are enforced. They reach this objective by proposing a new algorithm: DMSB.\n\nThe paper first focuses on adapting the iterative proportional fitting (IPF) algorithm to the momentum Schrödinger Bridge with fully specified marginals in phase space. They show that the IPF can be seen as an alternating maximization of log-likelihood.\n\nNext they focus on the multi-marginal momentum Schrödinger Bridge. They propose a new formulation of the marginals constraints in order to apply their version of IPF as log-likelihood maximization via policy specification. Moreover they prove that the optimality conditions for the optimal bridge (under their new set of constraints) allows to sample the velocities when they aren't initially specified.\n\nIn order to train efficiently in high dimension they adapt a neural network parametrization of the policies known in the classical SB setting to the momentum SB setting.\n\nFinally they test the the performance of DMSB on synthetic data. The performance on high dimensional data is higher than the baselines algorithm. Moreover they are able to recover velocity distributions which is a novelty.\n\nThe contributions can be summarized as follows:\n- Extend the IPF to the momentum SB\n- Introduce a new set of constraints for the Bregman iterations which allows for a neural net parametrization of the policies and sampling of velocities when they are not specified.\n- Proposal and benchmark of a new algorithm DMSB made to solve momentum multi-marginal SB\n\nSTRENGTHS:\nThe presented use case of their algorithm is cell profiling over time. In that regard the proposed algorithm performs well above baselines and adresses the following technical challenges: sampling of the velocities, curse of dimensionality, partial informations on the marginals.\n\nThe sampling of velocities and curse of dimensionality are both tackled by a combination of proposition 3.1 and proposition 4.2/4.3. Proposition 3.1 proposes a novel and interesting formulation of the half bridge using log-likelihood. This is combined with a novel way to deal with the marginal constraints which is a key advance towards efficient high dimensional computing with respect to prior works.\n\nGlobally the article puts swiftly together multiple well known ideas and adds key ingredient (decoupling of marginal constraints, log-likelihood IPF) in order to produce an algorithm which removes limitations from prior algorithms. Those limitations being :\n- lack of scalability\n- need for fully specified velocities\n- robustness to missing marginals\n\nFinally the benchmark contains multiple metrics which is helpful in the high dimensional context since the curse of dimensionality renders some metrics less meaningful as they point out.\n\nThe explicit description of the algorithm in the appendix is appreciated because it makes the article whole: it contains the theoretical and practical aspects.\n\nWEAKNESSES:\nThough inspired by a prior work the section on the neural nets parametrization lacks clarity. How does the log-likelihood minimization is tied to the mean matching objective and thus the loss Lmm? This link is succintly pointed out in the appendix B.5 equation (38).\n\nThe training scheme section goes rapidly over how the neural networks are trained. The figure 8 in appendix clarifies a lot the training. The explanation on the discretization method is unclear in section 4.5.\n\nFinally the main advantage of the algorithm DSMB is the scalability however there are no complexity analysis of the algorithm with respect to dimensions and number of marginals.\n\nQUESTIONS:\nHow long does training take for scRNA-seq using DMSB compared to MIOFlow and NLSB ?\n\nLIMITATIONS:\nThe author address the limitations of the works which are mainly the proofs of the theoretical convergence of the algorithm in the neural nets parametrization case of the momentum multi marginal SB. Proofs which are key to back up the interesting practical results.\n\nThey also sharply point out the limitation which is tied to their use case which is the impossibility for mmmSB to model for the death and birth of cells.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors present an algorithm, DMSB (Deep Multi-Marginal Momentum Schrödinger Bridge), to approximate solutions to an extension of the Schrödinger Bridge (SB) problem into phase space (mmmSB), where (i) marginal constraints on the position are given across time and (ii) stochasticity is only introduced on the velocity variable (which makes trajectories smoother). This framework aims at solving multi-marginal trajectory inference problems, i.e., inferring likely stochastic dynamics of particles on a time interval, given snapshots of them at certain time steps. This work proposes several contributions: (i) it extends the approach proposed by [1] (which solves a single-variable formulation of SB) to phase space (Proposition 3.1), (ii) it presents an efficient numerical scheme for the Iterative Proportional Fitting (IPF)-type algorithm presented by [2] and restated in Proposition 4.1, to solve mmmSB in practice, (iii) it shows great performance on realistic high-dimensional single-cell RNA sequencing ($d=100$) compared to previous baselines [3,4].\n\n[1] Likelihood training of schrödinger bridge using forward-backward sdes theory, Chen et al., 2021.\n\n[2] Multi-marginal Schrödinger bridges, Chen et al., 2019\n\n[3] Manifold interpolating optimal-transport flows for trajectory inference, Huguet et al., 2022.\n\n[4] Neural Lagrangian Schr$\\backslash$\" odinger bridge, Koshizuka et al., 2022.\n\nSTRENGTHS:\nThis work provides a well-motivated and theoretically well grounded adaptation of the SB problem to the multi-marginal setting with smooth trajectories. The authors derive a solid framework and present convincing numerical experiments on real-world data.\n\nWEAKNESSES:\n- In my opinion, this paper is not easy to follow and some statements are provided without context or comment, which consists in the main limitation for me at this stage. Since I find the content pretty good, I will increase my score if the authors accept to be clearer in the main paper, by answering my questions and following my general comments given below.\n\n- This paper lacks a theoretical result of convergence (at least non-quantitative, in the ideal setting where the neural networks would perfectly fit the drift terms of the forward and backward SDEs). It is not clear to me how the convergence result stated in [5] extends to the current setting.\n\n[5] Iterative Bregman projections for regularized transportation problems, Benamou et al., 2015.\n\nQUESTIONS:\n- What is the reference measure $\\xi$ used in the momentum-like setting (see Line 125 and Eq (6)) ? As far as I understand, it is the path given by the Brownian motion propagated along the velocity variable with the corresponding ODE $d x_t= v_t dt$. The authors should state it explicitly.\n- In my opinion, $\\bar{\\pi}$ is the solution of the problem given in Proposition 4.4 without any other constraint. Does the solution given in the main paper actually include the constraint on the boundary ? What is $q_{t_0}$ in Line 187 ?\n- I recommend the authors to specify the set of measures on which the optimization is performed in Propositions 4.1, 4.2 and 4.3: is it a collection of couplings or joint distributions over all the N states ?\n- As far as I understand, the constraints given by $K_{bridge}$ enable to smooth out the trajectories of the particles along the whole time interval. To prove the efficiency of their method, could the authors compare their method with the procedure where the only constraints are given by $K_{boundary}$ (which should work too if I am not wrong) ?\n- What is the definition of $\\mathcal{L}_{reg}$ (Algorithm 4) ?\n- How many Bregman iterations did you perform in your experiments ?\n\nMajor comments:\n- I recommend the authors to give the proof of Proposition B.6 in the appendix, since it is not straightforward.\n- For sake of clarity, the authors should introduce in the main paper a consistent notation to refer to couplings (eg, $\\pi_{t_1, t_2}$) and joint distributions (eg, $\\pi_{t_1:t_2}$) in order to avoid any confusion.\n\nMinor comments:\n- The authors should insist more on the (stochastic optimal control) formulation of the SB problem that they are solving in this paper. I don't think it is well highlighted in Figure 1.\n- There is inconsistency of the control variable ($a$ or $u$) in Appendix B.1.\n- I think there is a typo between $\\hat{m}$ and $\\bar{m}$ in Algorithms 3 and 4.\n- For sake of clarity, I think that the system of PDEs in Eq (20) should be displayed in the main paper.\n\nLIMITATIONS:\nUnlike [1] or [5] where experiments on Celeba dataset ($d\\geq3072$) are performed, the dimension does not exceed 100 in the experiments of this paper. In my opinion, it is crucial to present experiments with such order of dimensionality to study the scalability of DMSB.\n\n[1] Likelihood training of schrödinger bridge using forward-backward sdes theory, Chen et al., 2021.\n\n[5] Score-based generative modeling with critically-damped langevin diffusion, Dockhorn et al., 2021.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses the topic of multi-marginal trajectory inference in high dimensions using Schrödinger Bridge (SB). In particularly, the authors focus on the so-called momentum SB in phase space where the resulting trajectories in position space are smooth interpolations between the intermediate marginals. The motivation for this is that in real physical systems smooth trajectories are often more likely than abrupted changes in drift direction at intermediate marginals. After an introduction of the preliminaries of Schrödinger Bridge and Bregman iterations, the phase space formulation of SB is introduced, which this work then builds upon. By restructuring the problem formulations and introducing an efficient training scheme for the involved function approximators, the authors show how to solve the multi-marginal momentum SB problem in a computationally efficient way. This is then utilized in the experiments which comprise both a synthetic as well as a 100-dimensional real-world use-case.\n\nSTRENGTHS:\n- Originality: The authors make several important technical contributions that allow for a computationally efficient solution of the (multi-marginal) momentum SB problem in high-dimensional phase spaces. Related work and the foundations the authors build upon are discussed and cited in the manuscript. \n- Quality: All original propositions are supported with detailed proofs in the appendix (which I did not check). The experimental results show a convincing improvement over the state of the art. In the conclusion, limitations of the work are addressed.\n- Clarity: The paper has a good introduction of the work it builds on. In addition, the novel contributions are clearly stated. The submission is well written and quite accessible on pages 1-4 where the relevant prior work is explained. \n- Significance: The presented contributions are likely to enable machine learning practitioners to apply the momentum (multi-marginal) SB method to many interesting high-dimensional trajectory-inference problems. The advances over the state of the art are demonstrated convincingly on both an artificial and a real-world use-case. Further, the approach is also applicable if the marginal velocity distributions are not available, which is probably relevant in many real world use-cases where velocity information can be much harder to obtain the positional information.\n\nWEAKNESSES:\n- Clarity: The novel contributions are likely difficult to understand for a non-expert reader. While proofs are presented in great detail in the appendix, a more intuitive explanation or interpretation of propositions 3.1, 4.2, 4.4, 4.5 is lacking. In addition, reproduction and/or extension of the results is probably a significant effort because the source-code is not provided.\n\nQUESTIONS:\n- My main suggestion to the authors would be to make the ideas behind their original propositions accessible to a wider audience by providing some intuition and explanation beyond the mathematical proofs.\n- In addition, open-sourcing the code would reduce the entry barrier for practitioners for applying the method.\n- In the abstract it reads \"In this article, we extend SB into phase space\". Is that claim really justified? I had the impression that SB in phase space was introduced in reference 20 and that the novel contribution is in the derivation of a computationally efficient way for half-bridge iterative proportional fitting via negative log-likelihood minimization.\n\nLIMITATIONS:\nLimitations of the presented approach are discussed in the \"conclusion and limitations\" section.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors extend the diffusion Schrodinger bridge methodology to mult-marginal setting whereby each marginal is ordered sequentially, in addition, introducing  momentum into the diffusion Schrodinger bridge framework. The method shows excellent performance on trajectory inference tasks.\n\nSTRENGTHS:\n- The proposed work fills a gap in the literature, and the authors demonstrate how diffusion model based Schrodinger bridges [1] and implementation of [2] can feasibly be plugged into the multi-marginal setting [3]. \n- The method shows excellent performance on trajectory inference tasks, outperforming other methods. The approach feels quite natural.\n\n[1] Bortoli et al https://arxiv.org/abs/2106.01357 \\\n[2] Chen et al https://arxiv.org/abs/2110.11291 \\\n[3] Chen, Multi-marginal Schrodinger bridges, 2019\n\nWEAKNESSES:\n- The computational complexity of the algorithm is not discussed. It appears multiple (2N reverse diffusions?) must be computed per IPF step. This seems quite costly. I would appreciate discussion and more transparency on this.\n\n- It is very difficult to clearly see what is the main algorithm and training procedure from the main text. I see algorithms are provided in the appendices but these are also not very well detailed in my opinion. The paper would improve in clarity and reproducibility if this was given more attention. This is especially true as no code is provided. Indeed, some training details like caching trajectories are mentioned in passing (as was introduced in [1], which should be cited) but it is not clear from this paper what the authors mean and without a reference it is difficult to know.\n\n- Possible error in likelihood proof / explanations\n\nIn line 454, $\\mathbb{E}[y_0]$ (and for $\\tilde{y}_0$) is decomposed into $\\mathbb{E}[y_T]$ minus the path integral of the SDE for $y$. This seems fine. However, my understanding is that unless the diffusion has fully converged to the prior (i.e. after convergence of the IPF, or with infinite regularization in SB), the term  $\\mathbb{E}[y_T]=\\mathbb{E}[\\log p(m_T)]$ is from the terminal point in the diffusion (which is not fixed until after convergence) hence must depend on the network parameters used to define the diffusion. This is commented on in Theorem 2 of [3]. This term is dropped in [2] and in this paper, which makes the likelihood interpretation incorrect for training each step of the IPF.\n\nInstead further work is required, as in [3], to show that time-reversal is a bound of the likelihood and gives **approximate** likelihood training.\n\nThe training procedure will still work as it coincides with the time-reversal of a diffusion and the same training procedure as given in [1] but the likelihood explanation does not seem justified as it is currently written. \n\nIt appears other reviewers had similar concerns with [2], and also that the overall procedure really is not likelihood training but IPF with time-reversal where each time-reversal can be viewed as approximate likelihood. https://openreview.net/forum?id=nioAdKCEdXB\n\n\n-------- \n\n\nI am conflicted by this paper. Whilst overall I think it is a good contribution methodologically and believe it can be justified with extra work. I have concerns about clarity and more importantly that this likelihood interpretation is misleading and possibly incorrect.\n\nI would be happy to increase my score if the authors address this. I am happy to be corrected if I misunderstand the likelihood interpretation.\n\n- [1] Bortoli et al https://arxiv.org/abs/2106.01357\n- [2] Chen et al https://arxiv.org/abs/2110.11291\n- [3] Song et al, Maximum likelihood training of score-based diffusion models, 2021\n\nQUESTIONS:\n- Why use mean matching of [1] rather than divergence based training advocated in [2]? I agree they are theoretically equivalent up to an integration by parts but practically different. Similarly, the authors state the method is based on [2] (e.g. table 1 and line 116) but then if using the training objective given by [1] is it not more accurate to say the method is based on [1]?\n- What is the complexity and training time for this method? It appears that one must perform multiple reverse time diffusions per IPF step.\n\n- [1] Bortoli et al https://arxiv.org/abs/2106.01357\n- [2] Chen et al https://arxiv.org/abs/2110.11291\n\nLIMITATIONS:\n- The authors discuss lack of theoretical convergence results\n- Computational complexity and time to train is not discussed", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-07"}