Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: In this paper, the authors propose to add sparse constraints to the original Schrodinger bridges through optimal control. Specifically, the paper assumes that there exist some intermediate sparse samples during the diffusion process. By modifying the Iterative Proportional Fitting procedure (IPFP) method with spare intermediate constraints, the Iterative Smoothing Bridge (ISB) method is proposed. Experiments show that the ISB method can help the forward and backward drift functions successfully evolve toward the intermediate observations.
strength_and_weaknesses: Strength
- The proposed problem may be important in many different applications with sparse intermediate observations, especially in the medical area.
- It is reasonable to use $L^2$ loss to handle the forward and backward drifts.
- Experimental results are convincing. 

Weakness
- No convergence guarantee of the proposed method.
- For step 2 and 4, since there only exists sparse intermediate observations, to make the algorithm converge, it seems that a large number of samples is needed to make the method converge.
- It is reasonable to assume both $g$ and $\beta$ the same in both equation (5) and (6)?
- I may miss this part in the paper. Empirically, how to define $g$?
- The second paragraph of Step 1 in Sec. 3.1 is not very clear.
clarity,_quality,_novelty_and_reproducibility: Clarity and Quality
- Sec. 3.1 is somewhat ambiguous. The authors need to take more effort on the writing to make the logic much clearer. 
- The results of IPFP should be included in Fig. 3 for comparison.
- There are also some typos:
    - In the fourth-to-last line of page 4, $b_{l,\phi}$ should be $b_{l-1,\phi}$.
    - In the second line under equation (7), it should be $f_{l-1, \theta}$ and $g_{l+1, \phi}$
    - In the experiment of **Single-cell embryo RNA sequences**, why is the PCA used? What happens if the experiment is conducted on the original data?

Novelty
- The paper proposes an interesting problem, and the solution seems work.

Reproducibility
- Without source code, the work is hard to reproduce.
summary_of_the_review: Generally, the paper proposes an interesting problem. But the ambiguity in writing and implementation makes it hard to follow.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper studies the problem of learning a dynamical process given sparse observations of the process at intermediate times using a modification of the Schrodinger bridge process. In particular, the authors propose a method they term Iterative Smoothing Bridge (ISB) which alternates fitting forward and backward drifts parametrized by neural networks with a particle filtering and update step. This latter step is included to incorporate potentially sparse observations of the intermediate time dynamics. The authors discuss connections between their methods and stochastic control and give some theoretical claims. They then give an empirical study of their method.

strength_and_weaknesses: Strengths:
- They study an important problem: that of learning dynamical processes given some sparse observations at intermediate times

- The ISB method is novel

- They achieve some improved empirical results when compared against related methods in the literature

- The work is fairly well written

Weaknesses:
- While their method does offer some improvement compared to related methods, it is not a significant improvement, see for example table 2, figure 6, figure 7

- The authors give little theoretical justification for their method, beyond Prop. 3 which is not rigorously proved and in any case is essentially well-known. 

- The core elements of the method appeared in prior works, namely De Bortoli et al 2021 and Corenflos et al 2021. The novelty of their method is in the combination of these approaches.

Writing feedback:
- Typos in background paragraph on page 3: should be $\mathcal{C} = C([0, T]; \mathbb{R}^d)$. There's an extra "to" in the sentence that begins with $x_t$
- The differentiable re-sampling procedure is hardly explained at all, and only a passing reference to Corenflos et al is given. Comprehension would be greatly aided by giving some discussion of this method, if only in an appendix. Also, it seems that the differentiability of the re-sampling method is not used in your method - is this correct?

De Bortoli, V., Thornton, J., Heng, J., & Doucet, A. (2021). Diffusion Schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34, 17695-17709.

Corenflos, A., Thornton, J., Deligiannidis, G., & Doucet, A. (2021, July). Differentiable particle filtering via entropy-regularized optimal transport. In International Conference on Machine Learning (pp. 2100-2111). PMLR.

clarity,_quality,_novelty_and_reproducibility: As listed above.
summary_of_the_review: While this work proposes a new method -- ISB -- for the important problem of learning dynamical processes with sparse observations, it does not achieve strong empirical performance and has limited conceptual novelty. For these reasons I think it is slightly too weak to merit acceptance.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: Authors present computationally efficient framework for learning data conditional diffusion bridges using Iterative Smoothing Bridge. The proposed framework is assessed by experimental results on both synthetic and real world data.

strength_and_weaknesses: Strength:
1. strong theoretical formulation of the proposed approach
2. well explained algorithm
3. well performed experiments

Weaknesses:
1. lack of rigorous proofs of main propositions (proof sketches of proposition 1 and 2)
clarity,_quality,_novelty_and_reproducibility: Proposed problem is novel, well done experiment section, notation is not standard for stochastic calculus which makes it harder to read
summary_of_the_review: The paper is well written, presents novel framework with experimental assessment of it. Proofs of propositions on which the framework relies are not full but it seems that they are more or less true based on provided sketches.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The authors proposed the iterative smoothing bridge by leveraging ideas from Schr\"{o}dinger bridge and particle filter. Such a method allows us to learn constrained stochastic processes governed by sparse observations at intermediate stages. The algorithm is evaluated on synthetic data, real data, and small-scale high-dimensional data. This is an interesting problem and worth a deeper investigation and the proposed methodology seems to be a reasonable candidate.
strength_and_weaknesses: 
**Pros:** 

1. This paper is proposed to address an interesting problem, a path-constrained Schrodinger bridge, where a particle filter kicks in to tackle the sparse observations; 

**Cons:**

While I am not an expert in the field of particle filters, my biggest concern is the clarity: 

1. more background introduction on particle filters may be needed for non-expert readers in the appendix; 
2. what is the optimal transport map **$\boldsymbol{T}_{\varepsilon}$**;
3. why do we need the H-nearest neighbours?
4. maybe I am wrong, i think simulated annealing only proposed to gradually decrease the noise. Why does decreasing and then increasing the noise scale resemble simulated annealing? 
5. the resampling steps may be detailed.
clarity,_quality,_novelty_and_reproducibility: Clarity: See above

Quality and Novelty: I think leveraging particle filters to address the path-aware kind of Schrodinger bridge is a promising extension. Didn't check the proof.


Reproducibility: Although I like the insights proposed in De Bortoli (2021), the method alone is not that scalable. My question is that does this method extend to the likelihood training framework [1]? 

[1] Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory

summary_of_the_review: An interesting method to solve an important problem, not clear/scalable enough though.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: The paper proposes the  Iterative Smoothing Bridge (ISB) where sparse observations are known on the trajectories for transporting a source to a target distribution. The paper proposes to combine learned diffusion models with particles filtering to achieve this. The paper is promising and the idea of trajectory inference in optimal transport is interesting nevertheless the paper lacks a lot on the clarity. 

After reading the paper multiple times it is not fully clear how the method is implemented and how  the main  building blocks are connected. Authors added few clarifications in the appendix during the rebuttal  but the main paper is still not easy to follow. We suggest the authors to have a more transparent presentation of the proposed method with an algorithm summarizing the steps and the main computations performed. 

 
justification_for_why_not_higher_score: the paper is not yet ready for publishing due to lack of clarity
Decision: Reject