Geometric Algebra-Enhanced Bayesian Flow Network for RNA Inverse Design

Dear reviewer LyRL,

We express our gratitude for your diligent review and the valuable efforts you have made to assist us in enhancing the manuscript.

W1: Complexity: The technical concepts, particularly geometric algebra and Bayesian Flow Networks, may be challenging for some readers.

We understand the reviewer's concerns about the complexity of Geometric Algebra (GA) and Bayesian Flow Networks (BFN). To lower the barrier to understanding, we will take the following measures in the revised manuscript to ensure that the technical details are clearer and more comprehensible:

1. Strengthening and layered explanation of the background section

We will introduce the core concepts of Geometric Algebra in a layered manner, including the basic framework, core operations, practical usage examples, and provide sufficient references for further learning. Additionally, we will further clarify the mechanisms of BFN, including the evolution of continuous-time distributions, differences from traditional diffusion models, handling of discrete data, and visual diagrams of time-step sampling to make it clearer for the readers.

2. Enhancing the writing of the paper

We will clarify the descriptions of symbols and related elements, and unify the writing style. We have also written pseudocode in the appendix for reference, and we will add tables explaining the relevant symbols to make it easier for everyone to understand.

W2 & L2: Data Scarcity: The method depends on the availability of high-quality RNA structural data. & Data Quality Dependence: The method relies on accurate 3D RNA structural data, which may not always be available.

The definition of the RNA inverse design task dictates that sequence generation must be based on existing three-dimensional structures (such as the fixed backbone redesign task described in the paper). This limitation stems from the strong correlation between RNA function and its three-dimensional conformation. Current methods, including RBFN, require generating sequences conditioned on input structures, thus they cannot operate without the support of high-quality structural data. The current RNASolo dataset contains only about 18,955 RNA structures, as of 24-07-2025, which leads to a data scarcity issue in model training. Future improvements can be made in two directions:

The first direction is, as the performance of RNA structure prediction tools improves, predicted three-dimensional structures could supplement real data. For example, similar to how ESM-IF[1] uses AlphaFold2-predicted protein structures for inverse folding training, RNA design could alleviate data bottlenecks by constructing virtual training sets based on predicted structures, thereby expanding sample diversity and improving performance.

The second direction is to overcome data dependency, future work will explore joint modeling of sequences and structures, optimizing both simultaneously through multi-task learning, reducing reliance on single-sided data, much like existing works in protein design[2], aiming at the co-generation of RNA sequences and structures.

[1] Hsu, Chloe, et al. "Learning inverse folding from millions of predicted structures." International conference on machine learning. PMLR, 2022.

[2] Campbell, Andrew, et al. "Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design." International Conference on Machine Learning. PMLR, 2024.

W3 & L3: Limited Wet-Lab Validation: There’s no real-world experimental validation of the method. & Lack of Wet-Lab Testing: Without experimental validation, the real-world applicability is uncertain.

We understand the reviewer's concern regarding wet lab validation. Due to the current limitations of our laboratory conditions, equipment resources, and experimental timelines, we are unable to independently complete functional validation experiments for RNA sequence design (such as in vitro folding validation, structural characterization, or functional activity testing) at this time. We are also very interested in conducting wet lab validations, and we commit to open-sourcing our code upon acceptance of the paper, encouraging researchers worldwide to conduct validations in their own laboratories. For example, we will provide standardized RNA sequence design templates to facilitate rapid experimentation by collaborating labs.

Q1: Does the method work for more dynamic RNA structures?

Theoretically, RBFNs are capable of handling more dynamic RNA structures. The core idea is to treat the dynamic conformations as a collection of multiple static states. However, incorporating more conformations will increase the memory usage due to the expansion of the feature tensor dimensions.

Q2 & L1: What are the computational costs for large-scale RNA design tasks? & Computational Cost: The method could be computationally expensive for large RNA molecules.

Thank you for your attention to the computational cost of RBFN. We re-ran the benchmarks on a GeForce RTX 3090 (24GB VRAM) and have made the test code publicly available. The experimental results show that RBFN requires only 3.05 seconds and 1.84GB of VRAM for long RNA (4269nt), significantly outperforming gRNAde, which takes 66.28 seconds. These results demonstrate that RBFN can efficiently handle ultra-long RNA on consumer-grade GPUs, making it suitable for laboratory-level deployment.

The following is how we add the test code for calculating time and memory usage:

    # Time & Memory Monitoring, Before Running the Model
    torch.cuda.reset_peak_memory_stats(device)
    time_start = time.time()
    
    # Inference (50-step sampling, batch=1)
    with torch.no_grad():
        sequence = model.design(pdb_path, steps=50)

    #After the operation is completed, get the time and GPU memory usage.
    time_end = time.time()
    peak_mem = torch.cuda.max_memory_allocated(device) / 1024**2  # MB
    return time_end - time_start, peak_mem

Here are the test results:

Thank you for reviewing our paper. We noticed that you did not leave detailed comments, and we are eager to address any concerns you may have.

If you have unresolved issues with the paper, we would greatly appreciate it if you could briefly outline them in a follow-up comment. We will carefully revise the manuscript to incorporate your feedback.

Your input is invaluable to improving our work. Thank you again for your time and consideration!

Dear reviewer DZoU,

We express our gratitude for your diligent review and the valuable efforts you have made to assist us in enhancing the manuscript.

W: Motivation for using geometric algebra and a Bayesian Flow Network

We sincerely thank the reviewers for their in-depth questions regarding the motivation behind our method and the limitations of existing RNA modeling approaches. Below, we will systematically elaborate on four aspects: the shortcomings of current methods, the rationale for introducing GA, the design logic of BFN, and the synergistic mechanism by which both address key issues. We will also illustrate the innovativeness of RBFN with experimental results. The relevant content will be updated in the subsequent version of the paper.

1. Core Limitations of Existing RNA Modeling Methods

The current mainstream RNA design methods (such as Rosetta) face the following critical bottlenecks:

(a) Shallow 3D Structure Modeling

Existing tools over-rely on secondary structure predictions (e.g., ViennaRNA), neglecting tertiary interactions. This necessitates error-prone manual feature engineering (e.g., defining distance/torsion constraints in Rosetta), leading to poor generalization to complex conformations and noise sensitivity.

(b) Inefficiency in Discrete Sequence Generation

Autoregressive decoding (e.g., gRNAde) suffers from cumulative errors in long sequences (>500nt), where early prediction errors cascade through attention mechanisms. Additionally, insufficient distribution modeling treats sequence generation as discrete optimization, lacking probabilistic sampling of diverse, stable RNAs, resulting in low diversity/compatibility.

(c) Absence of Multi-State Modeling

Existing tools (e.g., RDesign) mostly focus on designing single states with fixed backbones, whereas biologically active RNAs (e.g., riboswitches) often need to dynamically switch between multiple conformations.

2. Motivation and Advantages of Introducing GA

Geometric algebra (GA) intrinsically encodes geometric inductive biases through its mathematical structure, directly addressing the 3D modeling limitations in existing methods. (See Q3 response for details.)

3. Design Logic and Advantages of BFN

We adopted Bayesian Flow Networks (BFN) based on its demonstrated success in handling discrete tasks such as molecular generation [1], drug design in continuous parameter space [2], and protein sequence design [3]. BFN addresses the efficiency and diversity issues in discrete sequence generation by evolving distributions in a continuous parameter space:

(a) Mechanism of Continuous Time Distribution Evolution

BFN models sequence generation as the evolution of a probability distribution over time ( $t \in [0,1]$ ), directly sampling complete sequences via continuous gradient optimization, thus eliminating cumulative errors in autoregressive decoding.

(b) Low-Variance Sampling and Small Alphabet Adaptation

BFN directly models the categorical distribution of the four bases, minimizing the KL divergence between the model distribution and the true data distribution. By leveraging continuous gradients, it enhances sampling efficiency, resulting in a higher sequence recovery rate (60.12%) for RBFN compared to gRNAde (56.82%).

[1] Song, Yuxuan, et al. "Unified generative modeling of 3d molecules with bayesian flow networks." The Twelfth International Conference on Learning Representations. 2024.

[2] Qu, Yanru, et al. "MolCRAFT: Structure-Based Drug Design in Continuous Parameter Space." Forty-first International Conference on Machine Learning. 2024.

[3] Gong, Jingjing, et al. "Steering Protein Family Design through Profile Bayesian Flow." ICLR. 2025.

4. Synergistic Mechanism of Geometric Algebra and BFN

The 3D structural features encoded by the GA module (e.g., bivector plane relationships) serve as the prior distribution for BFN, guiding sequence generation to ensure spatial compatibility. During BFN’s evolution process, the GA module continuously validates structural feasibility and corrects deviations via gradient backpropagation, enabling dynamic feedback.

In the revised manuscript, we will strengthen the explanation of the method's motivation at Introduction, clearly outlining the limitations of existing methods and the targeted improvements of RBFN in the introduction and related work sections. We believe that RBFN, through the deep integration of Geometric Algebra and Bayesian Flow Networks, provides a new methodological framework for the field of RNA design.

Q1: Metric in Table1

Native Sequence Recover, we will update the version and place it in the table caption. Thank you for pointing out this issue.

Q2: Table2's explain and compare methods

Yes, AR (Autoregressive) in Table 2 refers to Autoregressive Modeling.

In the original submission, RBFN was primarily compared with gRNAde (the current SOTA model) because its publicly available code and evaluation metrics facilitated direct comparison. However, during the Rebuttal phase, we further supplemented the following experiments:

• RhoDesign: An RNA design method based on 3D structure.

• RiboDiffusion: An RNA design method based on a diffusion model, which generates sequences by progressively reversing noise.

Due to space limitations, the results of the additional comparative experiments can be found in the Rebuttal for qKvq.

The methods in Table 1 are mainly based on secondary structure for generating RNA sequences and perform poorly in 3D structure. Therefore, we primarily compared with the SOTA method, gRNAde. Additionally, to demonstrate the improvement of RBFN in 3D structure metrics (such as scTM-score), we retained the comparison with ViennaRNA (a classic 2D tool).

Q3: geometric inductive bias and improve expressivity

We are grateful for the reviewer's in-depth attention to the GA. We provide a targeted response to this issue:

1. Does the Geometric Algebra module provide geometric inductive bias?

Yes, GA intrinsically embeds powerful geometric inductive bias into the model, significantly enhancing its capability to represent complex RNA three-dimensional structures. This advantage stems directly from GA's mathematical structure, which encodes physical-world geometric relationships through:
1) Hierarchical multivectors (scalars, vectors, bivectors, trivectors/pseudoscalars) and
2) Fundamental geometric operations (outer product, inner product, rotors).
Critically, each multivector level captures distinct geometric semantics:

• Vectors (1-vectors) represent directional displacement (e.g., distances between atoms like C4' and P);
• Bivectors (2-vectors) characterize planar orientation (e.g., base pair relationships);
• Trivectors/pseudoscalars (3-vectors) model volumetric properties and global conformational changes.

This hierarchical representation enables the model to directly learn essential spatial patterns—distances, angles, and torsion angles—without relying on error-prone manual feature engineering. The critical role of GA's inductive bias is unequivocally demonstrated in ablation studies: removing GA features ("w/o GA") caused a significant performance drop, reducing sequence recovery rate from 60.1% to 57.0%. These results confirm that GA's geometric priors are fundamental to the model's enhanced ability to capture and exploit RNA structural information.

2. How do multivectors (0/1/2/3...) enhance expressive power?

The hierarchical geometric semantics and compositional capabilities of multivectors fundamentally enhance the expressive power of RNA structure models. Geometric Algebra achieves this through its graded multivector structure, where each level captures distinct physical properties: scalars encode global attributes like energy states; vectors represent directional features (e.g., atomic displacements like C3'→P distances); bivectors define planar orientations critical for base-pairing geometries; and trivectors/pseudoscalars characterize volumetric relationships governing RNA tertiary conformations. These geometric primitives combine compositionally through operations like the geometric product (uv=u⋅v+u∧v), enabling simultaneous modeling of symmetric (scalar) and antisymmetric (bivector) interactions – a capability absent in traditional RNA design frameworks. Further amplifying this expressivity are GA's inherent operational advantages:

• The outer product (u∧v) constructs higher-dimensional subspaces (e.g., generating a bivector plane from two vectors), capturing nonlinear geometric relationships essential for complex motifs;
• Rotors implement efficient spatial transformations (R=e−θB/2) without Euler/matrix decomposition, streamlining conformational modeling;
• Parameter efficiency emerges from GA's compact representation – a single bivector in G3,0,0 space can encode rotational and planar data that would require multiple disconnected parameters in classical GNNs. This unified mathematical framework thus provides both richer structural representation and greater computational economy for RNA design tasks.

3. Is the performance improvement due to an increase in the number of parameters?

The performance improvement is mainly attributed to the structural advantages of Geometric Algebra, rather than a mere increase in the number of parameters. gRNAde (with 1,104,311 parameters) achieved only a 56.82% sequence recovery rate in the single-state task, while RBFN (with 1,106,977 parameters) reached 60.12%, indicating that the structural design of Geometric Algebra is more efficient.

Thank you for your detailed feedback and for acknowledging our clarifications in the Rebuttal regarding the motivation section, as well as for resolving your concerns except for presentation. We clarify that the bullet-point format was adopted in the Rebuttal due to word limit constraints to efficiently address your concerns, but this should not replace the fluid narrative required in the main text.

As you emphasized, the current manuscript fails to sufficiently contextualize Geometric Algebra (GA) and Bayesian Flow Networks (BFNs) for the broader machine learning community, particularly researchers focused on RNA modeling. To address this, we will implement the following precise revisions to ensure content accessibility:

• Strengthen motivation and problem-driven logic in the Introduction:

  • Systematically discuss existing limitations: Explicitly highlight three key shortcomings of current RNA modeling approaches:

    Shallow 3D Structure Modeling, Inefficiency in Discrete Sequence Generation, and Absence of Multi-State Modeling.

  • Elaborate motivation in detail: Introduce the necessity of GA and explain how it resolves these issues. Illustrate BFNs’ advantages using noise sensitivity in RNA sequence generation. Avoid introducing abstract theories first, ensuring ML readers intuitively understand "why these methods are needed". We have added citations to existing methods to validate effectiveness.

• Add a new section in the appendix titled "RNA Structural Characteristics and Modeling Challenges":

  • Provide background knowledge on RNA structure to help the broader machine learning community understand the context.

  • Clarify the rationale and effectiveness of GA by explaining its advantages and logical connections to RNA modeling.

These revisions will establish a dual-layer support system: The Introduction will deliver a problem driven motivation narrative (accessible to all ML researchers), while the appendix will supplement domain background (lowering the RNA knowledge barrier). Together, they will ensure a broader audience can smoothly grasp the innovative value of GA/BFNs.

We sincerely appreciate your invaluable suggestions, which have significantly improved the manuscript's presentation and enhanced its accessibility for readers from the broader machine learning community.

Dear reviewer jqC4,

We express our gratitude for your diligent review and the valuable efforts you have made to assist us in enhancing the manuscript. We denote geometric algebra by GA.

W1: Question about formal presentation

Thank you to the reviewer for pointing out this issue. We have revised the paper and will update it in the updated manuscript. Due to the word limit, we provide a brief summary of the revisions:

• For section 2.1 Problem Definition, to avoid content confusion, we removed the introduction of G and revised the description to re-explain nucleotides so that readers can understand the structure of nucleotides and the content of backbone atoms. The specific symbol content has been moved to section 3.1.

• For section 2.2 GA, we have rephrased the language and added explanations for the symbols, for example:

'''

$\mathbb{G}_{p,q,r}$ denotes a GA over a real vector space with $p$ basis vectors of positive signature ($e_i^2 = +1$), $q$ of negative signature ($e_j^2 = -1$), and $r$ of zero signature ($e_k^2 = 0$), satisfying $p + q + r = n$ for an $n$-dimensional space.

'''

• For section 2.3 Bayesian Flow Networks, we have reorganized the relevant formulas using the symbols and expressions from diffusion models to make it easier for readers to understand. For example:

'''

BFNs introduce a continuous-time parameter $t\in[0,1]$ and define two key distributions: (1) the sender distribution $q_t(\mathbf{x}_t)$ representing the data distribution at time $t$, and (2) the receiver distribution $q_t(\mathbf{x}_t|\mathbf{x}_0)$ describing the forward process from the original data $\mathbf{x}_0$ to the corrupted data $\mathbf{x}_t$. The core of BFNs is the Bayesian inversion mechanism, where the reverse process is given by:

$q_t(\mathbf{x}_0|\mathbf{x}_t) = \frac{q_t(\mathbf{x}_t|\mathbf{x}_0)q_0(\mathbf{x}_0)}{q_t(\mathbf{x}_t)}$

'''

• For section 3.1 GA GNN for RNA Three-Dimensional Structure Modeling, we have provided a more detailed explanation of the symbols and have re-unified some individual symbols. Additionally, we have added symbolic identification for coordinates related to atoms, for example:

'''

For an RNA structure with $m$ conformational states and $n$ nucleotides, we represent it as $m$ graphs $\{ \mathcal{G}^{(1)}, \dots, \mathcal{G}^{(m)}\}$.

...

$\mathfrak{q}(\mathbf{v})$ is obtained based on the three-dimensional coordinate relationships. For a multivector $\mathbf{v} = s + \vec{v} + \mathbf{b} + t\mathbf{I}$ where $s$ is scalar, $\vec{v}$ vector, $\mathbf{b}$ bivector, and $t\mathbf{I}$ pseudoscalar components, the quadratic form is $\mathfrak{q}(\mathbf{v}) = s^2 + \|\vec{v}\|^2 - \|\mathbf{b}\|^2 - t^2$ in $\mathcal{G}_{3,0,0}$.

'''

• For section 3.2 on Bayesian Flow Networks for RNA Generation, we have concretized the relevant symbols and updated the descriptions to make them more focused on the task of RNA design, thereby making it easier for readers to understand. For example:

'''

Before training, we define a uniform prior over the 4 RNA nucleotides: $p_0(\mathbf{y}) = \text{Categorical}(\mathbf{y}|\pi_0)$ with $\pi_0 = [1/4, 1/4, 1/4, 1/4]^\top$.

...

'''

We hope that these detailed revisions and enhancements will make the methods section easier for readers to understand, thereby contributing to scientific research.

W2: Question about Clifford algebra

Thank you for your professional feedback on the use of Clifford algebra in our paper. We will make the necessary revisions in the updated manuscript.

You accurately identified a key conceptual confusion in our paper: in Figure 2, we incorrectly used the interpretative framework of projective GA, misinterpreting 0-vectors, 1-vectors, 2-vectors, and 3-vectors as "points, lines, planes, and solid structures," respectively.

As you correctly pointed out, in standard 3D Euclidean GA ($\mathcal{G_{3,0,0}}$):

• scalars represent quantities without direction

• vectors represent directed line segments

• bivectors represent directed planes/area elements

• trivectors/pseudoscalars represent directed volumes

We acknowledge that this was an oversight in our presentation, stemming from a confusion between different variants of GA. While we used Euclidean GA for calculations in our model design, we inadvertently adopted the common interpretative framework of projective geometry from computer graphics when explaining its biological significance. The intuitive interpretation of nucleotide positions in RNA molecular structures led us to erroneously map Euclidean GA elements onto the projective geometric interpretation.

We will make the following significant revisions in the updated manuscript:

1. Fully correct the interpretation in Figure 2, clearly distinguishing the proper interpretative framework of Euclidean GA.

2. Add a dedicated subsection (Section D.3) to discuss why we chose $\mathcal{G_{3,0,0}}$ over other GAs (such as projective GA $\mathcal{G_{3,0,1}}$ or conformal GA $\mathcal{G_{4,1,0}}$):

• Spatial Representation: RNA structures reside in standard 3D Euclidean space, where $\mathcal{G}_{3,0,0,}$ directly models spatial relationships without extra dimensions. RNA design focuses on nucleotide relative positions, orientations, and local geometry naturally captured by vectors and bivectors in Euclidean GA.
• Task Alignment: While projective GA (PGA) excels at representing points/planes, RNA design prioritizes vector relationships and local geometry over absolute positions. $\mathcal{G}_{3,0,0}$ balances mathematical simplicity with sufficient expressiveness for key RNA structural features.
• Computational Efficiency: $\mathcal{G}_{3,0,0}$ (8 dimensions) offers lower complexity than PGA (16D) or conformal GA (32D). This efficiency is critical for RNA inverse design involving long sequences (thousands of nucleotides).

3. Re-examine all sections of the paper that involve geometric interpretations to ensure conceptual consistency.

We appreciate your identifying this critical issue. Your feedback helped us correct conceptual errors and deepened our understanding of the theoretical foundations of GA in RNA structure modeling. Your professional insights significantly enhanced the rigor and scientific value of our work.

W3&Q3: Performance & inference time of RBFN

Thank you for your attention to performance and the computational cost of RBFN.

While RBFN shows modest but meaningful improvements over gRNAde, the gains align with RNA’s structural complexity:

• 5.8% higher recovery (0.6012 vs. 0.5682),
• 17.0% lower scRMSD error (10.83 vs. 13.04).

Given RNA’s extreme flexibility and shallow energy landscapes—where sub-angstrom precision dictates function—even incremental gains in structural accuracy (e.g., 2.21 Å scRMSD reduction) directly impact biological interpretability.

For the computational cost, we re-ran the benchmarks on a GeForce RTX 3090. The experimental results show that RBFN requires only 3.05 seconds for long RNA (4269nt), 21.7 times faster than gRNAde(AR), which takes 66.28 seconds. These results demonstrate that RBFN can efficiently handle ultra-long RNA on consumer-grade GPUs, making it suitable for laboratory-level deployment.

Here are the test results:

W4&Q1: related work & a plot of the beta distribution

We will supplement the revision with a citation and discussion of this literature to more comprehively present the current progress and challenges in the research, and we will also add a plot of the beta distribution. Using secondary structure is a great idea, which has also given us some inspiration. In our future work, we will further explore how to utilize this information.

Q2: Using 32 nearest neighbors instead of a fixed truncation radius

Thank you for this insightful question. We use fixed neighbor counts (32) instead of distance cutoffs to address fundamental challenges in RNA modeling:

1. Solving Density Heterogeneity:
   RNA regions show extreme density variations:

   • Stems: Tight packing (P-P ~5.9 Å → 6-10 neighbors at 5Å cutoff)
   • Loops: Loose packing (P-P 7-8 Å → 4-7 neighbors)
   • Pseudoknots: Ultra-dense (→15-20+ neighbors)
     Fixed cutoffs cause unstable neighbor counts, risking redundancy in dense areas and missed interactions in sparse regions.

2. Mathematical Necessity for GA-GNN:

   • 32 radial basis functions (RBFs) encode distances.
   • Matching neighbor count (32) ensures each RBF interval is sufficiently sampled.
   • Critical for stable geometric product calculations and multivector normalization.

3. Biophysical Justification:

   • Core functional interactions occur within ~15-20 sequence neighbors (local motif size).
   • 32 neighbors intentionally captures essential long-range tertiary contacts while aligning with RBF dimensions.

In summary, the strategy of using a fixed number of neighbors allows our model to robustly handle the density variability of RNA structures while maintaining the mathematical rigor of GA representations.

We thank the reviewer for their detailed feedback on our manuscript. We appreciate the time and effort taken to evaluate our work and provide constructive suggestions. Here, we systematically elaborate on the core advantages of GA as a structural feature encoder for RNA, its alignment with task-specific characteristics, and clarify the research positioning of this work.

The technical necessity of GA in RNA modeling stems from the intrinsic geometric properties of RNA structures. RNA molecules exhibit hierarchical geometric constraints: helical symmetry of the phosphate backbone, planar rigidity of base pairs, continuous distributions of torsion angles, and topological folding in tertiary structures. Traditional Euclidean representations require complex feature engineering or redundant parameterization to approximate these constraints, whereas GA’s mathematical structure enables models to effectively learn these geometric patterns:

1. Learning rotation invariance: GA’s rotor representation can be efficiently learned by models to automatically encode rigid rotations of base pairs, avoiding information loss during coordinate transformations.

2. Explicit modeling of structural constraints: Geometric features generated by the wedge product (e.g., base plane normal vectors, hydrogen bond directions) can be directly optimized by models, enhancing learning capability for high-dimensional geometric relationships.

3. Unified learning of multiscale features: GA’s multivector hierarchy (scalar-vector-bivector) allows models to simultaneously learn atomic coordinates and secondary/tertiary topological information (e.g., stem-loops, pseudoknots).

These properties establish GA as a task-adaptive learning tool for RNA structure learning. The core contribution of using GA is the first validation of the learnability of GA features in RNA sequence design tasks, the experimental results confirm this. Additionally, the reviewer commented that "other state-of-the-art geometric GNNs would perform equally well." Is there any basis for this statement? Could you provide some references to verify it?

In response to the reviewer’s comments, we have further revised the paper structure to enhance readability, with an emphasis on the representation of GA in RNA modeling. For example:

'''

For an RNA structure with $m$ conformations and $n$ nodes, consider this structure as multiple graphs, i.e., $\{ \mathcal{G}^{(1)}, \dots, \mathcal{G}^{(m)}\}$. For each conformation graph $\mathcal{G}^{(c)} = ( \mathbf{A}^{(c)}, \mathbf{S}^{(c)}, \vec{\mathbf{X}}^{(c)} )$ where $c \in \{1,\dots,m\}$, it contains scalar features, namely the index and sequence representation of nucleotides in RNA $\mathbf{S}\in\mathbb{R}^{n\times m \times f}$, and vector features $\vec{\mathbf{X}} \in \mathbb{R}^{n \times m \times f' \times 3}$: (a) the forward and backward unit vectors along the backbone from the 5' end to the 3' end, ($\vec{\mathbf{x}}\_{i +1}-\vec{\mathbf{x}}\_{i}$ and $\vec{\mathbf{x}}\_{i}-\vec{\mathbf{x}}\_{i-1}$); and (b) the unit vectors, distances, angles, and torsions from each C4' to the corresponding P and N1/N9. The edge features $\{ \mathbf{A}^{(1)}, \dots, \mathbf{A}^{(m)} \}$ from node $j$ to $i$ are initialized as follows: (a) the unit vector from the source node to the target node, $\vec{\mathbf{x}}\_j-\vec{\mathbf{x}}\_i$; (b) the distance in three-dimensional space, $\Vert \vec{\mathbf{x}}\_j-\vec{\mathbf{x}}\_i \Vert\_2$, encoded by 32 radial basis functions; and (c) the distance along the backbone, $j-i$, encoded by 32 sine-position encodings, and finally represented as $\vec{\mathbf{V}}^{(c)} \in \mathbb{R}^{n \times n \times f' \times 3}$. The geometric algebra feature $\mathfrak{q}(\mathbf{v})$ is obtained based on the three-dimensional coordinate relationships. For a multivector $\mathbf{v} = s + \vec{v} + \mathbf{b} + t\mathbf{I}$ where $s$ is scalar, $\vec{v}$ vector, $\mathbf{b}$ bivector, and $t\mathbf{I}$ pseudoscalar components, the quadratic form is $\mathfrak{q}(\mathbf{v}) = s^2 + \|\vec{v}\|^2 - \|\mathbf{b}\|^2 - t^2$ in $\mathcal{G}\_{3,0,0}$.

'''

Dear reviewer qKvq,

We express our gratitude for your diligent review and the valuable efforts you have made to assist us in enhancing the manuscript.

W2 & Q1: Comparison with additional RNA inverse folding baselines are needed, such as RiboDiffusion. & Can you incorporate more RNA inverse folding baselines such as RiboDiffusion or RhoDesign? These models are available on Github (e.g., https://github.com/ml4bio/RhoDesign)

Thank you for the valuable suggestion to incorporate additional RNA inverse folding baselines (RiboDiffusion and RhoDesign). We have extended our experiments and added results for these models in the rebuttal (see table below). While we currently cannot modify the manuscript due to submission system restrictions, we will update the main text with these comparisons in the revised version upon acceptance.

Key Observations and Justifications:

1. RiboDiffusion Performance: While RiboDiffusion achieves high scores (e.g., 1.0000 on 1Q9A and 1XPE), we verified that these instances likely correspond to structures present in its training data (as noted in their repository documentation). Since the training code for RiboDiffusion is not publicly available, we cannot retrain or validate this behavior independently. We will add this clarification to the revised manuscript.

2. RhoDesign Comparison: For RhoDesign (RBFN variant), our model achieves competitive results across most benchmarks. Notably, RBFN outperforms RhoDesign on 10 out of 14 cases (e.g., 0.8588 vs. 0.8148 on 1Q9A, 0.7172 vs. 0.6415 on 2R8S). This suggests that our proposed architecture and training strategy yield robust generalization beyond the training distribution.

W1 & Q2: This paper presented ablation studies to study different modeling choices, but it didn't show why BFN is beneficial. Could the model perform better with a standard diffusion model enhanced with geometric algebra? This part is unclear. & Can you design an experiment to validate the advantage of BFN? I can see the benefit of geometric algebra enhanced network but it's unclear BFN is the right choice. Wouldn't diffusion, GFlowNet, and other methods work as well?

Thank you for your insightful comment regarding the choice of Bayesian Flow Networks (BFN) over other modeling approaches such as standard diffusion models. We appreciate the need for a clearer justification and experimental validation of why BFN was selected for our RNA sequence generation task.

Justification Based on Existing Literature

Our decision to use BFN is grounded in existing research that highlights its advantages, particularly for discrete data tasks [1][2]. Specifically:

• Parameter Space Modeling: As illustrated in Figure 2 of [2], BFN operates in parameter space rather than observation space, which inherently offers lower variance during sampling compared to standard diffusion models. This characteristic is crucial for generating high-quality discrete sequences like RNA.

• Sampling Process Differences: The sampling mechanism in BFN is fundamentally different from diffusion models. While diffusion models gradually denoise data through iterative steps, BFN directly models the flow of probability distributions in parameter space, leading to more efficient and effective sampling processes for discrete tasks.

Given these advantages, we adopted BFN based on its demonstrated success in handling discrete tasks such as molecular structure generation [2], drug design in continuous parameter space [3], and protein family design [4].

Experimental Validation Plan

To further validate the benefits of using BFN over other methods, including standard diffusion models, we are currently implementing a comparative experiment:

1. Baseline Models: We will extend a standard diffusion model or a GFlowNet variant, enhanced with geometric algebra, similar to our BFN approach. This ensures a fair comparison by maintaining consistent enhancements across all models.

2. Results Comparison: By comparing the performance of BFN against these baselines, we aim to demonstrate the specific advantages of BFN in terms of efficiency, quality of generated sequences, and overall robustness.

In summary, while we believe BFN is the right choice for our application due to its unique properties and proven effectiveness based on existing literature and preliminary results, we are committed to providing empirical evidence through controlled experiments to support this claim.

References:

[1] Graves, Alex, et al. "Bayesian flow networks." arXiv preprint arXiv:2308.07037 (2023).

[2] Song, Yuxuan, et al. "Unified generative modeling of 3d molecules with bayesian flow networks." The Twelfth International Conference on Learning Representations. 2024.

[3] Qu, Yanru, et al. "MolCRAFT: Structure-Based Drug Design in Continuous Parameter Space." Forty-first International Conference on Machine Learning. 2024.

[4] Gong, Jingjing, et al. "Steering Protein Family Design through Profile Bayesian Flow." ICLR. 2025.

We hope this clarifies our rationale and addresses your concerns. Thank you for your valuable feedback.