Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper proposes a new diffusion method incorporating physical information by carefully designing the prior for the downstream task. The suggested diffusion approach incorporates physical prior into bridges, in contrast to other diffusion methods that learned diffusions as a combination of forward-time diffusion bridges. The method was evaluated on molecules and 3D point cloud generation tasks, results of which show advantages in terms of generating quality, efficiency, and energy calculation convenience. They demonstrate the superiority of the method in terms of both quantity and quality.

strengths_and_weaknesses: ### Strength
* The paper is generally clear, well structured, and easy to follow.
* The theory and accompanying research are explained clearly.
* They properly motivate the need for the method and describe the proposed model thoroughly.
* Injecting task-dependent priors into a diffusion method is non-trivial and novel
* They reasonably select the benchmarks emphasizing the strength of the suggested model.


### Weakness
* Typos
  * Line 42: repeated citation [14] 
  * Line 109: ofdenoised -> of denoised
  * Line 196: bound(x) -> bond(x)
  * Line 240: [26] seems to be an incorrect citation because it's nothing to do with a molecule application.
* Table 1 omits uniqueness comparisons that are important to determine the superiority of the model. Including uniqueness comparisons is important because there may be models that generate molecules with low uniqueness and high novelty only. (I'd be happy to increase my score if this issue is resolved)

questions: * Taking fewer time steps sometimes could mislead because actual wall clock time spent for each time step could be different. Wondering if a wall clock comparison can be presented.

limitations: * The scientific significance of merely producing molecular conformation is limited. The usefulness of this work would be higher if this model could be applied to real-world drug discovery applications, as the authors emphasized.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper proposes a framework for incorporating physics driven prior bridges in diffusion models for improved molecule and point cloud generation. In order to do so, a Lyapunov function based method is developed to construct and determine bridges. The work seems to improve on existing methods for molecule and point cloud generation.
strengths_and_weaknesses: ***Strengths***
1) The proposed contribution of Lyapunov function to construct prior bridges is novel in my opinion and is a sensible approach for incorporating informative priors.

2) The proposed method seems to improve molecule generation and point cloud generation benchmarks.

***Weaknesses***

1) For GeomDrug datasets, although the atom stability seems to improve, the overall molecule stability seems to be 0%, i.e., none of the predicted molecules are correct. Although this is the case with other existing models as well, it is surprising to see that the proposed method using informative physics informed priors does not improve this aspect of generation. Will inferring the edges during the diffusion process (instead of doing so in a post processing manner with RDKit) improve this aspect?

2) I did not find any mention of computational complexity of the proposed method. It would be nice to see how feasible is the proposed approach from a practical perspective. In drug discovery, often a deserted outcome is the de novo generation of molecules at a reasonable speed. Hence, knowing about the computational requirements and the time taken on average for generating realistic molecules will be a good information to have.
questions: See questions in weaknesses sections above.
limitations: The limitations are mentioned in the weaknesses section above.
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 introduces a novel diffusion-based generative model that leverages the prior information to learn the diffusion process satisfying the bridge condition. Specifically, the paper proposes a  Brownian bridge with extra drift term that incorporates the prior information, and aims to learn the diffusion process by fitting the trajectories drawn from the mixture of these bridges. The paper utilizes Lyapunov functions to guarantee the bridge condition, and further introduces energy functions to guide the training process for each downstream task. The proposed method, Bridge with Priors, is able to generate samples with better quality compared to the baselines in both molecule generation and point cloud generation tasks.
strengths_and_weaknesses: **Strength**

1. The paper proposes a novel approach of leveraging the prior information in the form of diffusion bridges to learn the diffusion process, which is clearly different from injecting inductive bias into the model architectures. Moreover, instead of using the time-reversal technique which most of the previous works are based on, the paper learns the generation process as a mixture of diffusion bridges which was introduced in [Peluchetti, 2022], which is also a new direction for the diffusion model.

1. The paper has a clear motivation of exploiting problem-dependent prior information to generate high-quality samples. Especially, for the point cloud generation, the paper addresses the problem of unevenly distributed points and provides a solution by using the Risze energy function. 

1. Although the paper is based on mathematically heavy theory, the paper is well-written and understandable with clear notations and formulations, except for the part explaining the mixture of diffusion bridges and the loss function (lines 102-112). 

1. The paper shows promising results for both molecule generation and point cloud generation tasks

1. I believe the experiments are well-conducted with clear data/training/evaluation setups and compared with the state-of-the-art baselines (EDM). Especially, comparing previous diffusion models, the Bridge without Prior, and the Bridge with Prior clearly shows the advantage of exploiting prior information for the generation. 

**Weaknesses**

1. Lines 102-111 are too concise with necessary details omitted. Without reading [Peluchetti, 2022], it would have been impossible to understand the concept of "mixture of diffusion bridges" denoted as $\mathbb{Q}^{\Pi^{\ast}}$. 

1. If I understood correctly, the term $b_t(Z_t|Z_1)$ in the loss function of Eq.(2) should be the drift term of the mixture $\mathbb{Q}^{\Pi^{\ast}}$, not the drift term of a specific $x$-bridge $\mathbb{Q}^{x}$. Is this term analytically accessible? 

1. The computation of the loss function of equation (2) needs clarification. I assume the expectation in Eq.(2) was computed by a Monte Carlo estimate with a sample $(t,z_1,z_0,z_t)$ where $z_1\sim \text{Data}$, $x_0\sim\Pi_{0|1}(dx_0|x_1)$ and $z_t\sim p_{t|0,1}(z_t|z_0,z_1)$. Although [Peluchetti, 2022] shows that $p_{t|0,1}(z_t|z_0,z_1)$ has explicit form, how is $x_0\sim\Pi_{0|1}(dx_0|x_1)$ sampled? 

1. The condition of line 157: $\mathbb{E}_{\mathbb{Q}^{x,bb}}\|f_t(Z_t)\|^2$ does not seem trivial which could be satisfied for most practical functions, although the provided intuition in lines 158-161 is convincing. It would be clearer if the authors provide proof that the condition is satisfied for $f$ induced from the proposed energy functions.

1. The reason for incorporating the energy function (prior information) into the Brownian bridge by $f_t(\cdot)=-\nabla E(\cdot)$ is not clear. Is the energy function minimized when $t$ approaches 1?

1. I would like to see more experimental results to the authors' claim for "less sampling time", either for GEOM-DRUG or the point cloud generation tasks. I can see that in Table 2, Bridge+Force with 500 steps outperforms EGM for 1000 steps in the QM9 dataset which is impressive, but as the molecules of QM9 are small, it would be more convincing if the claim is proven for larger datasets. It would be great if there is an ablation study section in the paper to emphasize the advantage of less sampling time.

(*) I would like to raise my score if the presentation is improved and the addressed concerns/questions are clarified.

**Minor correction**
- The explanation for the abbreviation SMLD is missing. 
- The SDE of $\mathbb{P}^{\theta}$ and Eq.(1) use t-subscript notation for drift and diffusion coefficients, while Eq.(4) does not use such t-subscript. I recommend unifying the notation to prevent confusion.
- No bold in row Airplane, column 100 Steps-COV-EMD in Table 4.
- Condition #3 of Theorem A.1. in the supplementary file is missing.
- In my opinion, Algorithm 1 and Figure 1 do not give much information. It would be great to add more details about "mixture of diffusion bridges" and the explicit form of $b_t(Z_t|Z_1)$.

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**References**
- Peluchetti, Non-denoising forward-time diffusions, 2022
questions: 1. Is Bridge with Priors designed for a more complex process such as Ornstein–Uhlenbeck process? Or is it applicable only to the Brownian bridge which is possible due to the singular characteristic of the drift term $\sigma^2_t\frac{x-Z_t}{\beta_1-\beta_t}$?

1. How to determine the learnable parameter $\alpha$ as in $s^{\theta}_t(z) = \alpha f_t(z)+\tilde{s}^{\theta}_t(z)$?

1. What is the meaning of "to ensure that $U$ is minimized at time $t=1$" in line 147?

1. Is the condition in line 157: $\mathbb{E}_{\mathbb{Q}^{x,bb}} [||f_t(Z_t)||^2] <\infty$  correct? 
In the supplementary file, the condition of Proposition A.4. is $\mathbb{E} [\int^1_0||f_t(Z_t)||^2]  < \infty$.

1. Why is the molecular stability of GEOM-DRUG 0.0 for all models? Is it because the molecules are large?

1. Is the training time of Bridge with Priors larger than previous diffusion models? Additionally, although the comparison with the Schr&ouml;dinger Bridge methods [De Bortoli et al., 2021, Chen et al., 2022] is out-of-the-scope of this paper, I believe that the comparison would be interesting as Bridge with Priors may have less training/sampling time and competitive or better performance.

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**References**
- De Bortoli et al., Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling, NeurIPS 2021, 
- Chen et al., Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory, ICLR 2022

limitations: Training the model takes a long time, which is a common limitation of diffusion models. The paper also discusses the future works in section 6, which seems to be a promising research direction.
review_title: null

Meta-review: metareview: 
All reviewers agreed that this work has many positive aspects, such as originality of the idea, technical soundness and practical relevance. In the initial reviews, some concerns about the experimental evaluation have been raised. In particular, one reviewer mentioned potential problems regarding the uniqueness of generated molecules. This issue, however could be addressed reasonably well in the rebuttal. 
I do share the generally positive perception of this paper. Therefore, I recommend to accept to paper.
confidence: Certain
Decision: Accept