Paper: RFold: RNA Secondary Structure Prediction with Decoupled Optimization (ICLR-2024)

Reviewer comment (Experiments):
The experimental results are not sufficiently detailed. The article emphasizes the validity of the proposed method's prediction results but lacks validity results for the baselines.

Author response:
**Q1** The validity results.
 
**A1** Thank you for your kind suggestion. In the ablation study, we compared different post-processing strategies, as shown in Table 7. To eliminate the influence of the backbone model and preprocessing methods, we used RFold as an example. We compared RFold-E, which employs an unrolled algorithm, with RFold-S, which does not use any post-processing. We found that the validity of RFold's post-processing strategies is significantly better than that of others.

Reviewer comment (Theory):
In addition to the specific issues mentioned in the Weaknesses section, the explanation of the method is not very clear. For example, it is not very clear how Eq. 9 is derived. From the reviewer's understanding, $\mathbf{M}$ should represent the ground-truth, and $S_r, S_c$ constitute the decomposition of M. Then, is there any necessity to optimize $S_r$ and $S_c$?

Author response:
**Q2** The explanation of the method.

**A2** Yes, $\boldsymbol{M}$ represents the ground-truth, and $S_r, S_c$ constitute the decomposition of $\boldsymbol{M}$. However, directly optimizing for the final solution is infeasible. Our proposed decoupled optimization approach utilizes the symmetry of the matrix $\widehat{H}$ to decouple the optimization into row-wise and column-wise components. Optimizing $S_r, S_c$​ separately allows for a more manageable and efficient computation process.

Reviewer comment (Novelty):
A key weakness is the technical approaches seem incremental, lacking major deep learning innovations.

Author response:
**The seminal work, E2Efold [1], was accepted by ICLR 2020 as one of the 48 oral presentations.** As shown in the methodology comparison table below, compared to SPOT-RNA, E2Efold modified the backbone model to a Transformer and introduced an unrolling algorithm for post-processing. Although promising, E2Efold did not perfectly address the problem of three constraints.

By employing novel decoupled optimization, **our proposed RFold is the first work to fully satisfy all constraints**. Furthermore, while UFold attempted to enhance performance with more complex pre-processing, our approach utilizes a straightforward seq2map attention to automatically capture pairwise features, thereby improving inference speed. We respect previous works, but believe RFold offers unique contributions.

| Method    | SPOT-RNA      | E2Efold               | UFold                 | RFold                 |
|----------------------------------|---------------|-----------------------|-----------------------|-----------------------|
| pre-processing                   | pairwise concat | pairwise concat       | hand-crafted          | seq2map attention     |
| backbone model                   | ResNet + LSTM | Transformer           | U-Net                 | U-Net                 |
| post-processing                  | ×             | unrolled algorithm    | unrolled algorithm    | decoupled optimization|
| constraint (a)                   | ×             | ✔                     | ✔                     | ✔                     |
| constraint (b)                   | ×             | ✔                     | ✔                     | ✔                     |
| constraint (c)                   | ×             | ×                     | ×                     | ✔                     |
| RNAStralign F1 score | 0.711         | 0.821                 | 0.915                 | **0.977**             |
| Inference time                   | 77.80 s       | 0.40 s                | 0.16 s                | **0.02** s            |

Reviewer comment (Novelty):
The decoupled optimization and Seq2map attention offer straightforward extensions to UFold.

Author response:
**The seminal work, E2Efold [1], was accepted by ICLR 2020 as one of the 48 oral presentations.** As shown in the methodology comparison table below, compared to SPOT-RNA, E2Efold modified the backbone model to a Transformer and introduced an unrolling algorithm for post-processing. Although promising, E2Efold did not perfectly address the problem of three constraints.

By employing novel decoupled optimization, **our proposed RFold is the first work to fully satisfy all constraints**. Furthermore, while UFold attempted to enhance performance with more complex pre-processing, our approach utilizes a straightforward seq2map attention to automatically capture pairwise features, thereby improving inference speed. We respect previous works, but believe RFold offers unique contributions.

| Method    | SPOT-RNA      | E2Efold               | UFold                 | RFold                 |
|----------------------------------|---------------|-----------------------|-----------------------|-----------------------|
| pre-processing                   | pairwise concat | pairwise concat       | hand-crafted          | seq2map attention     |
| backbone model                   | ResNet + LSTM | Transformer           | U-Net                 | U-Net                 |
| post-processing                  | ×             | unrolled algorithm    | unrolled algorithm    | decoupled optimization|
| constraint (a)                   | ×             | ✔                     | ✔                     | ✔                     |
| constraint (b)                   | ×             | ✔                     | ✔                     | ✔                     |
| constraint (c)                   | ×             | ×                     | ×                     | ✔                     |
| RNAStralign F1 score | 0.711         | 0.821                 | 0.915                 | **0.977**             |
| Inference time                   | 77.80 s       | 0.40 s                | 0.16 s                | **0.02** s            |

Reviewer comment (Theory):
How is Equation 10 obtained with the proposed decoupled optimization?

Author response:
Our proposed decoupled optimization approach in RFold simplifies the complex task of RNA secondary structure prediction by dividing it into separate row-wise and column-wise optimizations. This method effectively manages the complexity by handling these aspects independently and then integrating them to form a comprehensive solution. The process relies on the symmetry of the matrix $\widehat{H}$. We utilize Argmax functions for each sub-problem due to their suitability in selecting maximal values while maintaining structure constraints, though they are not the sole method available for such optimizations.

Reviewer comment (Novelty):
It seems that the major point of the proposed method is mainly about the optimization formulation of the assignment matrix learning while the seq2map network architecture and the mapping matrix formulation are mainly inherited from previous works. So probably the novelty of the proposed method is a little bit limited.

Author response:
Thank you for your meticulous review! We would like to claim that both the optimization formulation and the seq2map attention are our main contributions. As stated in the introduction, we decompose deep learning-based RNA secondary structure prediction methods into three key components: preprocessing, the backbone model, and post-processing.
* The pre-processing step means projecting the 1D sequence into 2D matrix. (**1D -> discrete 2D**)
* The backbone model learns from the 2D matrix and then outputs a hidden matrix of continuous values. (**discrete 2D -> continuous 2D**)
* The post-processing step converts the hidden matrix into a contact map, which is a matrix of discrete 0/1 values. (**continuous 2D -> discrete 2D**) 

The methodology comparison is shown in the table below:

| Method                           | SPOT-RNA      | E2Efold               | UFold                 | RFold                 |
|----------------------------------|---------------|-----------------------|-----------------------|-----------------------|
| pre-processing                   | pairwise concat | pairwise concat       | hand-crafted          | seq2map attention     |
| backbone model                   | ResNet + LSTM | Transformer           | U-Net                 | U-Net                 |
| post-processing                  | ×             | unrolled algorithm    | unrolled algorithm    | decoupled optimization|
| constraint (a)                   | ×             | ✔                     | ✔                     | ✔                     |
| constraint (b)                   | ×             | ✔                     | ✔                     | ✔                     |
| constraint (c)                   | ×             | ×                     | ×                     | ✔                     |
| RNAStralign F1 score | 0.711         | 0.821                 | 0.915                 | **0.977**             |
| Inference time                   | 77.80 s       | 0.40 s                | 0.16 s                | **0.02** s            |

It can be observed that the only similarity between RFold and previous methods is the use of U-Net as the backbone model, similar to UFold. Furthermore, the most significant difference of RFold compared to earlier methods is that its predictions for secondary structures are guaranteed to satisfy all three constraints.

Reviewer comment (Theory):
Does the Rol-Column-Softmax operation have a training instability issue? Since this operation is very similar to the Sinkhorn operation and the Sinkhorn algorithm has training stability issues. This reviewer is afraid that the proposed method also has the same drawback.

Author response:
Thank you for your insightful question! We have added the loss curve plot in **Appendix G of the revised manuscript**. It can be observed that the training loss decreased steadily across the training epochs. The possible reasons for this may include: (i) The Softmax function, used in the Row-Column-Softmax, is generally numerically stable; (ii) The objective of the Row-Column-Softmax is not to make the matrix doubly stochastic (as in the Sinkhorn algorithm) but to approximate the Row-Col-Argmax operation in a differentiable manner for training purposes.

Reviewer comment (Experiments):
Suppose Eq (10) is correct, it is unclear how good is the approximation in Eq (14). (1) It'll be great to compare with the approximation (row-softmax \odot col-softmax). (2) argmax can been seen as softmax with temperature T->0. Current approximation uses T=1. It'll be great to add an ablation study to examine how good is the softmax approximation.

Author response:
**Q3** How good is the approximation in Eq (14)?

**A3** Your understanding that row-col softmax is an approximation of row-col argmax is definitely correct. We should use row-column argmax because the predicted contact matrix is meant to be discrete. However, row-column argmax is not differentiable, which is why we use row-column softmax in the training phase for differentiable optimization.

Reviewer comment (Theory):
Eq (1): the order of composition should be swapped. F(x) = G[H(x)]. => F = G\circ H

Author response:
**Q4** Eq (1): the order of composition should be swapped. F(x) = G[H(x)]. => F = G\circ H.

**A4** Thanks for your careful review. We have fixed this issue in the revised manuscript.

Reviewer comment (Writing):
Under Eq (9), what does \otimes refer to? Are S_r and S_c sets of vectors? Maybe better to define S_r and S_c as matrices and use Hadamard product.

Author response:
**Q5** Under Eq (9), what does \otimes refer to? Are $S_r$ and $S_c$ sets of vectors? Maybe better to define $S_r$ and $S_c$ as matrices and use Hadamard product.

**A5** We are sorry for the confusion. In our context, $S_r$ and $S_c$ represent vectors, and the symbol $\otimes$ denotes the Hadamard product. The formulation has been refined in the revised manuscript for greater clarity.

Reviewer comment (Theory):
The soundness of the formulation and approximation slightly make me concern as pointed out in the weakness section. One simple change is to tune down the claim "optimal solution" to "a greedy algorithm solution".

Author response:
**Q6** Tune down the claim "optimal solution" to "a greedy algorithm solution".

**A6** Thank you for your insightful suggestion. We agree that 'a greedy algorithm solution' is suitable because it allows us to decompose the problem into row-wise and column-wise sub-problems. We have refined the description in the revised manuscript.