SOLVING SCHRODINGER BRIDGE PROBLEM VIA STOCHASTIC ACTION MINIMIZATION

This paper appears to be a solid paper but not ready.

1. Experiments are only evaluated on baby experiments.

2. Comparison between the proposed formulation (including imaginary part) and the original one (Tianrong's version.) is not studied enough.

3. It is a little bit ad-hoc to use the combination of Yasue Lagrangian, GM Lagrangian (19) and Lagrangian associated with cross entropy。 More principled approach is suggested.

Some of the theorems are phrased quite awkwardly (e.g., Theorem 4. "Stochastic action minimization of GM Lagrangian (19) is associated Fisher information production")

It's not really clear that the "Toy model" example is reproducible.

When discussing the algorithmic approaches in Section 5.1, the authors say "forward and backward loss functions (35) and (36) were iteratively minimized until convergence", but do not explain what sense of (weak) convergence they consider.

There are many typos (ßaction).

"Our approach is the most similar to the work of (Chen et al. (2021a)) on likelihood training of the SBP."

An important and highly relevant reference was presented by Neklyudov et al. (2023). A direct comparison and discussion on how your proposal differs from that approach is necessary for a successful publication of this work.

• Neklyudov, Kirill, et al. "Action Matching: Learning Stochastic Dynamics from Samples." International Conference on Machine Learning (ICML)(2023).

"Our approach offers great simplicity since it does not require transformation into the Euler-Lagrange equations."

There are several approaches that propose alternatives to simulation via Euler-Lagrange equation, i.e., Tong et al. (2023) and Shi et al. (2023) but also proposals by Liu et al. (2023) and Somnath et al. (2023). I would highly recommend comparing to them apart from standard SB solvers such as De Bortoli et al. (2021) and Chen et al. (2022).

• Liu, Guan-Horng, et al. "I$^ 2$SB: Image-to-Image Schrödinger Bridge." International Conference on Machine Learnint (ICML) (2023).
• Somnath, Vignesh Ram, et al. "Aligned Diffusion Schrödinger Bridges." Conference on Uncertainty in Artificial Intelligence (UAI) (2023).

"We demonstrate the utility of our approach on the toy model dataset in 2D."

The authors propose a new training method for solving the Schrödinger bridge problem. However, the current analysis is reduced to a single experiment on a toy model dataset in 2D. And looking at the results, for anyone who ever trained a simple SB, the performance is underwhelming. In order to fully assess the utility and potential of this method, it would be crucial to test it on more complex tasks and provide a direct comparison to existing methods that are all published alongside with code. This would provide a more comprehensive evaluation and help validate the usefulness of the proposed approach.

"In section (2) we review the background on the SBP problem. In the following section (3) we introduce the stochastic ßaction."

The paper contains a lot of spelling errors and requires another round of proofreading, e.g., ßaction -> action, SBP problem -> SB problem, there are a lot of "the"s missing, equations (See appendix -> equations (see appendix, amsmath, etc.

Besides the language errors, the paper is citing very sparsely, e.g., for statements such as "system of equations is usually solved using two neural networks" it is always a good idea to credit the authors and provide those readers not familiar with the work with useful references to study. Concretely, citations are missing at important positions throughout the paper, and important literature is not cited in the first place.

Major

• Theorems 1, 3, and 4 are known results in the field, and the authors provide links to the corresponding articles, so only theorem 2 may be considered novel. It would be better to present the results of theorems 1,3 and 4 in the distinct background section. Otherwise, - it is unclear whether the authors claim these results as their own.
• It is not clear how theorem 2 actually relates to the Schrödinger Bridge problem.
• The proposed IPF-based algorithm and objectives for optimization are already known [1, Eq 18, 19].
• The authors evaluate the proposed approach only on the one toy 2D setup, and their approach shows poor performance even in matching the marginal distributions at initial and final times.
• The authors do not provide any experimental evidence of whether their algorithm is capable of restoring true trajectories of the Schrödinger Bridge.
• There is no discussion of previously developed methods (e.g. [1, 2, 3, 4, 5]) for the Schrödinger Bridge problem and how the proposed method relates to them.

Minor

• There is an error in the title (Schrödinge instead of Schrödinger).

In the end, although there are some intesting ideas, the overall paper seems to be clearly rushed and unfinished. Hence, I put “reject”.

Links

[1] Chen T., Liu G. H., Theodorou E. Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory //International Conference on Learning Representations. – 2021.

[2] Vargas, Francisco, et al. "Solving schrödinger bridges via maximum likelihood." Entropy 23.9 (2021): 1134.

[3] De Bortoli, Valentin, et al. "Diffusion Schrödinger bridge with applications to score-based generative modeling." Advances in Neural Information Processing Systems 34 (2021): 17695-17709.

[4] Gushchin, Nikita, et al. "Entropic neural optimal transport via diffusion processes." In Advances in Neural Information Processing Systems, 2023

[5] Shi, Yuyang, et al. "Diffusion Schr" odinger Bridge Matching." In Advances in Neural Information Processing Systems, 2023

• Unsatisfactory presentation and writing style for the conference.
• The authors do not comment on the advantages of their approach compared to existing mentioned approaches for providing numerical solutions for the same problem.
• Limited demonstration on numerical experiments (the authors show only. a single toy example)
• No comparison with competing methods.
• They provide a lot of unecessary historical information and in the Appendix they repeat widely known and available calculations, thereby obscuring their contribution.