RFold: RNA Secondary Structure Prediction with Decoupled Optimization

Dear Reviewer G4jZ,

Thank you for your professional comments. We apologize for the issues in the presentation. We have addressed these issues in the revised manuscript, highlighting the changes in blue font.

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.

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.

Q3 Why the Row-Col Softmax function is used to approximate the Row-Col Argmax function for training?

A3 Thank you for your insightful comment. The use of the Row-Col Softmax function to approximate the Row-Col Argmax function in the RFold model for RNA secondary structure prediction is primarily due to the differentiability requirements of the training process.

The Argmax function, which is used in the inference phase of RFold, is non-differentiable. On the other hand, the Softmax function is a differentiable approximation of the Argmax function. It provides a "soft" version of the maximum, assigning probabilities to each element in such a way that larger values get higher probabilities。The Row-Col Softmax function serves as a differentiable proxy during training. It allows the model to learn the appropriate parameters by providing gradient information which is not possible with the Row-Col Argmax function.

Once the model is trained, during the inference phase, the Row-Col Softmax function can be replaced with the Row-Col Argmax function. This is because inference does not involve backpropagation or gradient computations, and the Argmax function provides a clear, decisive prediction that is necessary for the final RNA structure prediction.

Dear reviewer G4jZ,

We would like to kindly remind you that ICLR allows for revisions, and the textual issues you mentioned have been considered minors in other reviews, as we are also reviewers of mutiple papers. In the initial round of review, you graciously expressed that:

"If the author can effectively address my concerns, especially those related to presentation, I will reconsider this article and possibly revise my evaluation."

We have diligently revised the manuscript based on your suggestions. As the rebuttal deadline is approaching, we would greatly appreciate hearing your response. Thanks for your valuable review and comments again. We have addressed all your concerns and questions. If you have any other questions, please feel free to discuss with us.

Best,

Authors

Dear Reviewer BXWa,

Thank you for your constructive comments!

Q1 A key weakness is the technical approaches seem incremental, lacking major deep learning innovations. The decoupled optimization and Seq2map attention offer straightforward extensions to UFold. This suggests the work may be better suited for a venue focused on the specific domain.

A1 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.

Q2 How is Equation 10 obtained with the proposed decoupled optimization?

A2 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.

[1] RNA Secondary Structure Prediction By Learning Unrolled Algorithms, ICLR 2020.

Dear Reviewer Y2Rp,

Thank you for your thoughtful and inspiring comment!

Q1 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.

A1 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:

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.

Q2 Does the Rol-Column-Softmax operation have a training instability issue?

A2 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.

Dear Reviewer fGRe,

We sincerely appreciate your careful and insightful comments!

Q1 $\boldsymbol{\widehat{H}}$ can have negative values?

A1 Thank you again for your careful review! We have defined $\boldsymbol{\widehat{H}} = (\boldsymbol{H} \odot \boldsymbol{H}^T) \odot \boldsymbol{\bar{M}}$ below Eq.6. The element-wise multiplication $\boldsymbol{H} \odot \boldsymbol{H}^T$ results in a matrix where each element is the square of the corresponding element in $\boldsymbol{H}$. Since squaring any real number results in a non-negative value, this operation yields a matrix with non-negative elements. $\boldsymbol{\bar{M}} \in \{0,1\}$ contains only non-negative values, typical for a constraint matrix, often containing 0s and 1s to indicate the presence or absence of constraints. Thus, the final element-wise multiplication in $\boldsymbol{\widehat{H}}$ does not introduce any negative values.

Q2 The row-col-argmax($\boldsymbol{\widehat{H}}$) may be not optimal.

A2 The example is interesting. However, considering $\boldsymbol{\widehat{H}} = (\boldsymbol{H} \odot \boldsymbol{H}^T) \odot \boldsymbol{\bar{M}}$ and $\boldsymbol{\bar{M}}$ is defined as $\boldsymbol{\bar{M}}_{ij}:=1$ if $x_i x_j \in \mathcal{B}$ and $|i-j| \geq 4$. This example is not suitable for our case because it excludes the diagonal.

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.

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.

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.

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.