Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: The authors consider the problem of finding the dynamics of a population from snapshots of its temporal marginals. They introducing an estimator which which is the sum of the goodness of fit to data term and entropy-based term (a.k.a. prior/regularizer). Instead of minimization of the entropy-regularized functional over the space of all paths, they introduce the “reduced” functional for optimization that includes additional entropy-regularized Optimal Transport problems between intermediate positions. The reformulated problem turns out to be equivalent to the initial problem. This follows from Theorem 3.1 which the authors prove (they call it the Representer theorem as its idea is analogous the idea of the original representer theorem). The resulting “reduced” problem is practically more tractable as it considers only a finite amount of time moments. The authors use Mean-Field Langevin (MFL) dynamics to approach the functional and recover the trajectory of the stochastic process, finding the optimal corresponding marginal distributions simultaneously. In the experimental session, they compare proposed approach with Global Waddington-OT (gWOT) in the simulated and real data experiments in small dimensions.
strengths_and_weaknesses: **Strengths**

(1) While most other methods are based on recovering the Waddington potential directly, the current paper uses point of clouds (particles) to recover the desired dynamics.

(2) The theoretical part of the paper is interesting, specifically I would like to highlight the Theorem 3.1 in the spirit of Representer which reduces optimization over paths to optimization over finite products of measure spaces at observation times. Additionally, the paper proves the result on the exponential convergence of the functional which they optimize (Theorem 3.3).
 
**Weaknesses**
 
(1) The method is point-cloud-based (particle-based) and it is presumably not very scalable for high dimensions as it would require exponential amount of particles.

(2) The authors claim that existing approaches can be difficult to establish rigorous guarantees and establish guarantees for their method. However, when it comes to practice, it seems like in their method a large gap anyway appears between the theory and practice. For example, Thm. 3.3 is proved for measures, but in practice they are finitly approximated by particles (eq. (12)); entropy term is estimated by the nearest neighbor estimator (line 259); sinkhorn terms are also computed between discrete distributions – all these approximations introduce minor biases here-and-there which raises questions to which extent the proposed theory (Theorem 2.1, 3.3.) is still applicable and advantageous w.r.t. The other methods.

(3) The comparisons are a little bit limited – only the gWOT method is included. What about (some) of the other related methods? Specifically, seems like [6] or [9] might be relevant.

(4) The method has some hyperparameters which are not very clear how to choose.

(5) There is not clear explanation of the final algorithm with description of all the inputs, outputs, parameters, etc. Seems like these details can be collected from the main text and appendix, but this is a negative factor for reproducibility and readability.

questions: (1) Consistency Theorem 2.1 seems to be formulated in [1] for a different than (3) functional (lines 110-111). Does the consistency hold for (3) which the authors use?

(2) How do you solve numerically PDE (11)?  

(3) What is the computational complexity of the proposed method?

(4) How well does the method scale to higher dimensions? It would be great to see the error dynamics on some simulated dataset with the increase of the dimension.

(5) The unbalanced OT appears rapidly and out of nowhere. Is it true that all the theoretical results of the previous sections apply to formulation (14) instead of the original entropic balanced OT?

limitations: The authors sufficiently fully describe the limitations.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper tackles the challenge of recovering the law of a stochastic process from sample observations of its marginals. The authors propose to approximate an existing, continuou-time, consistent estimator minimising a given regularised functional by another reduced, discrete-time estimator defined as the sum of a functional G and an entropy term H. The functional G is built from solutions to Schrodinger bridge problems between consecutive marginals. To obtain the reduced estimator, the authors consider a Langevin SDE of Mc-Kean-Vlasov type and show that its dynamics converge to the unique minimiser of the reduced optimisation problem. Numerical experiments are performed on simulated data from a bifurcating SDE as well as on real-world RNA-sequencing data. 
strengths_and_weaknesses: Recovering the law of a stochastic process from sporadic observations of its marginals is an important and difficult task, both at a theoretical and practical level, and this work makes good contributions in this direction. I enjoyed reading the paper and found its material original, well-written and the mathematical claims thoroughly justified in the appendix.

One possible weakness is the scalability of the proposed algorithm to high-dimensional and long trajectories, as noted by the authors themselves in their conclusion.

questions: Is it possible to provide a more concrete interpretation of the integral in L. 156?

Can you provide additional details or comment on the time- and space-complexities of the proposed algorithm? In particular, how does it scale in the length and dimensionality of the paths?

How are the regularising constants chosen in the experiments?

In Figure 1, for large values of N, gWOT seems to start outperforming MFL. Does this trend continue if N is chosen in the order of 10^3, 10^4...?

Would the proposed methodology apply if one considered a different reference measure than Brownian motion (BM)? In particular, what would happen if one considered instead a non-Markovian reference process, for example fractional BM, or other processes with memory?

Regarding benchmarking as well as assessment of the success for the proposed method, have you considered divergences for probability measures supported on pathspace, such as the families of MMD distances studied in [1]?

[1] Salvi, C., Lemercier, M., Liu, C., Horvath, B., Damoulas, T., & Lyons, T. (2021). Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems, 34, 16635-16647.
limitations: As the authors highlight in their conclusion, the proposed algorithm suffers from the typical drawbacks of min-entropy estimators which makes it not scalable to high-dimensional problems.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: In this work authors study the problem of fitting a law on the path space induced by gradient SDE with additive and reflective Brownian motion to the set of prescribed time marginals. This is similar to multi-marginal OT with a difference the transport plan is restricted the be the law induced by an SDE.  The problem has been recently studied in Lavenant et al where, essentially, an estimator used in the current paper has been developed and analysed. Authors observed (which is not new) that one can translate entropic optimisation problem on the Wiener space (as in Lavenant et al.) to the optimisation problem on the product space of probability measures supported on the compact set \mathcal X (the compactness being a technical condition). This reformulation then open the avenue for mean-filed langevin dynamics (MFLD) (or gradient flow on W_2 space)  to be used as an optimisation ‘algorithm’ for which convergence along with the convergence speed can be deduced from recent works in the literature. Indeed mean-filed langevin dynamics has been recently studied by Mei et al (2018) and Hu et al (2019) where convergence has been proved rigorously. These result have been recently extended by Chizat 2022 and Nitanda et al 2022 that showed that under appropriate conditions convergence of MFLD is exponential. In the current paper authors showed, by assuming compactness of the space \mathcal X, that results of Chizat and Nitanda et al apply.  

It’s worth remarking that conceptually this work is related to ‘Mean-Field Neural ODEs via Relaxed Optimal Control’ by Jabir at al 2019 and 
‘Gradient Flows for Regularized Stochastic Control Problems’ by Siska et all 2020, where detailed analysis of coupled MFLD has been studied. In these works the cost function was slightly simpler and significant amount of regularisation was required to establish convergence.  

strengths_and_weaknesses: Strengths:
- Very interesting contribution tackling difficult problem of proving convergence rates for gradient flow algorithms for the inference on the path space. 
-  The paper is (mainly) well written making highly technical material accessible to broad audience 
- cell-RNA sequencing is a very promising area of applications for the ML models and I anticipate significant impact of this work on it

Weaknesses:
- The compactness of \mathcal X is not a natural assumption but seem necessary for the analysis to go through 
- Some parts of the proofs are written in haste and I would appreciate more details (see comments bellow)
- When citing other works, especially in the proofs, please give exact detail of the results used rather than just the name of the paper… 
questions: - It wasn’t clear where in the proof you used the fact that only SDE with drift of grade it type are needed? I would appreciate more discussion on that in the appendix 
- When stating theorems please make sure assumptions are clearly stated 
- In eqn 10 reflected BM is used. How do I know X lives in \mathcal X ? Don’t I need to consider Skorohod problem?
- Similarly to derive 11 from 10 Ito formula is used. Please Don’t you need Ito-Tanaka formula here? 
- Proof of Lemma B.2 : for completes please provide more details rather then just citing [21] and [23]
- In deriving LSI compactness of \mathcal X is critical. Can you comment more about the constants how the diameter of D affects convergence ?
- please provide more details in the proof of Proposition D.1 rather than just citing other’s people work. That will make the paper easier to read. 

limitations: Mathematically main limitation is the compactness of \mathcal X, but the result is still very interesting.
review_title: null

Meta-review: metareview: This paper studies the challenging problem of inferring the trajectory of a stochastic process from sample observations of its marginals. Earlier work of Lavenant et al. introduced a consistent estimator based on an optimization problem over continuous time. The main contributions of this paper are in (1) introducing a discrete time variant, based on Schrodinger bridges and minimizing an entropy regularized optimal transport problem over the marginals rather than over path space. In Theorem 3.1 they show an interesting "Representer theorem" which shows that the two formulations are equivalent. And (2) they prove consistency of their estimator. In Theorem 3.3 they show exponential convergence. The main weakness is that the bounds are asymptotic in nature, and they do not get, for example quantitative bounds on how many particles are needed as the dimension grows. Overall this is still a nice contribution and seems like an accept. 
confidence: Certain
Decision: Accept