Flow Matching for Denoised Social Recommendation

1. Differences between redundancy and errors Thanks! Redundancy refers to similar suggestions that do not provide additional value. Errors, on the other hand, represent inaccurate or flawed information, which could arise from noisy data sources, incorrect predictions, or misclassifications. While redundancy can reduce model efficiency by introducing unnecessary information, errors can degrade model accuracy and lead to incorrect recommendations.

2. More detailed explanation about using generative models rather than traditional denosing models. Thanks! We have already make a comparison to some tranditional denoising models like DSL [1], and the results show that generative models should ideally provide more accurate and diverse predictions because it can better capture the patterns in the noisy data.

[1] Denoised Self-Augmented Learning for Social Recommendation

3.ODE sampling method. Thanks! We agree, and we present it as follows, which will also be added in our revised paper. Euler method:
$y_{n+1} = y_n + h f(t_n, y_n)$

Runge-Kutta method:
$k_1 = h f(t_n, y_n), \quad k_2 = h f(t_n + \frac{h}{2}, y_n + \frac{k_1}{2})$
$k_3 = h f(t_n + \frac{h}{2}, y_n + \frac{k_2}{2}), \quad k_4 = h f(t_n + h, y_n + k_3)$
$y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)$

4.Condition and supervisory signal The conditions mentioned refer to the labels. These labels guide the interpretation and application of the noise model, defining the structure and nature of the noise at each step. They play a crucial role in conditioning the system, ensuring that the model effectively accounts for variations in the data. In the revised version, we will further elaborate on their role, explaining how these labels influence the model's behavior under different scenarios and how they help in refining the noise removal process for more accurate predictions.

5. Time complexity Thanks! The theoretical time complexity has been discussed in the model module of the paper. In the revised version, we will also include a comparison of the training time to provide a clearer understanding of the model's efficiency.

6. Social homogeneity. Thanks! We agree. Following most works [1][2], we also define the isotropy as noise obeys the standard normal distribution N(0,1), which leads to the uniform properties in all directions. While, anisotropy refers to noise distributions that deviate from such uniformity. For example, Gaussian noise with a non-diagonal covariance matrix introduces variability across different directions in the data space. While social homogeneity is widely accepted in the social graph domain, we will explicitly address and clarify this concept within the revised version, relating it to the behavior of isotropic and anisotropic noise in social graphs.  [1]A Directional Diffusion Graph Transformer for Recommendation [1]Denoising Diffusion Probabilistic Models

1.Consistent with the denoising objective ? Thanks! Our denoising process aligns with the general denoising objective in diffusion models. While typical diffusion models use stochastic noise processes, our method is based on ODEs, achieving a more direct reconstruction objective.

2.Types of noise? Thanks！While we acknowledge that other types of noise (e.g., label noise, varying intensities) exist, this paper primarily emphasizes modeling anisotropic edge noise. However, addressing various noise types systematically remains a valuable direction for future work. Generally, traditional diffusion models using isotropic Gaussian noise treat all noise uniformly, which limits their ability to explicitly differentiate between noise types. In contrast, our flow-matching approach supports customized anisotropic noise modeling, implicitly differentiating between noise types by adapting noise patterns to relational data structures.

3.After denoising, can the representation align with the collaborative signals? Yes, the representations obtained after our flow-matching denoising procedure explicitly capture relational (collaborative) structures. By modeling anisotropic noise, our approach preserves critical collaborative signals, resulting in meaningful and effective representations that align closely with recommendation tasks.

4.A more detailed explanation of the flow-matching process is needed. Thanks！We will include a more detailed explanation of the flow-matching process in the revised version, particularly focusing on the sampling process.

5.Advantages of the flow-matching method itself Thanks! While the advantages of the flow-matching method are indeed emphasized, they are directly relevant to social recommendation contexts. Specifically, flow-matching is particularly effective in handling the anisotropic noise structures commonly found in social graphs, which is central to our work. By better modeling these complex relationships, flow-matching improves the accuracy of recommendations in social networks. 

6.How does flow matching specifically enhance performance within social recommendation scenarios? Flow matching enhances performance in social recommendation by addressing two key challenges: structural noise and representation degradation caused by isotropic assumptions. Unlike diffusion-based models that rely on isotropic Gaussian noise, flow matching captures the anisotropic nature of social graphs, where user preferences vary across communities. It learns a deterministic velocity field to directly map noisy representations to clean ones, avoiding stochastic sampling and preserving fine-grained patterns.

7. Time complexities in the experimental section. Thanks! We will include this comparison in the revised version regarding training time.

8.Recall rate metrics. Thanks！ Recall rate metrics are already included in the primary experimental results of the paper. These metrics provide a comprehensive evaluation of the model's performance in terms of its ability to baselines social recommendations. 

9. Figures, color scheme, coordinate system distorted Thanks! In the revised version, we will make an effort to modify the figure for better clarity.

10. Different sampling methods Thanks! We agree that the derivation of the ODE equations requires more detailed mathematical treatment. Moreover, within the experimental section or the ablation study, we plan to visually illustrate the essential steps of the ODE derivation and explicitly demonstrate their impacts on model performance (primarily focusing on the Euler and Runge-Kutta methods).

11.Whether applying generative models to both the user and bipartite graph Thanks！it's important to note that the bipartite graph and the social graph have different characteristics. In particular, the bipartite graph, which typically models user-item interactions, does not necessarily exhibit the same anisotropic noise distribution as the social graph, which captures more complex relationships between users or between items with varying strengths. Thus, the assumption of anisotropic noise may not always hold in bipartite graph scenarios. We have not yet explored this direction of a unified generative framework specifically for bipartite graphs, but we recognize the potential value of doing so. Our current focus is on modeling anisotropic noise within social graphs, and we plan to investigate the applicability of generative models to bipartite graphs in future work.

1. More theoretical foundation and mathematical descriptions. Thanks, and we agree. Following most works [1][2], we also define the isotropy as noise obeys the standard normal distribution N(0,1), which leads to the uniform properties in all directions. While, anisotropy refers to noise distributions that deviate from such uniformity. For example, Gaussian noise with a non-diagonal covariance matrix introduces variability across different directions in the data space. In the revised version, we will provide rigorous mathematical characterizations of these concepts.

[1] A directional diffusion graph transformer for recommendation [2] Denoising Diffusion Probabilistic Models

2. More detailed derivation of the ODE equation, and correlated experiments should be added. Thanks! We agree, and we present it as follows, which will also be added in our revised paper. Euler method:
$y_{n+1} = y_n + h f(t_n, y_n)$

Runge-Kutta method:
$k_1 = h f(t_n, y_n), \quad k_2 = h f(t_n + \frac{h}{2}, y_n + \frac{k_1}{2})$
$k_3 = h f(t_n + \frac{h}{2}, y_n + \frac{k_2}{2}), \quad k_4 = h f(t_n + h, y_n + k_3)$
$y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)$

Moreover, we plan to visually illustrate the essential steps of the ODE derivation and explicitly demonstrate their impacts on model performance (primarily focusing on the Euler and Runge-Kutta methods).

3.[1] suggests that diffusion models do not necessarily require the noise addition phase. Thanks! What [1] suggests is indeed highly relevant to our RecFlow. In essence, RecFlow aligns closely with the perspective: rather than relying solely on the standardized noise addition typical of traditional diffusion models. Concretely, RecFlow employs a flexible flow-matching mechanism, avoiding the masking of relational structures caused by standard isotropic noise. This allows RecFlow to preserve relational information more accurately. Thus, our research provides complementary evidence supporting the assertion of [1] that diffusion models need not strictly rely on traditional isotropic noise. We will explicitly discuss this alignment in the next revision.

[1]Is Noise Conditioning Necessary for Denoising Generative Models?

4. Color Scheme Thanks! We will consider adjusting the figure in the revised version to make it clearer.

5. Comparison with other heuristic denoising methods. Thanks! We have already made a comparison with some traditional denoising models like DSL [1], and the results show that the generative model should ideally provide more accurate and diverse predictions because it can better capture the patterns in the noisy data. Additionally, we agree that comparing with heuristic denoising methods would be beneficial, and we plan to include such a comparison in future work. As for the impact of different types of noise, we will incorporate a case study to illustrate how isotropic and anisotropic noise affect model performance, which will further highlight the advantages of our flow-matching approach[2]

[1]Denoised Self-Augmented Learning for Social Recommendation [2]Denoising Diffusion Probabilistic Models

6. More discussion on stochastic differential equations (SDEs) based methods. Thanks! There are existing works [1] using SDE approaches for social recommendation. Additionally, SDE methods predominantly rely on isotropic noise, making them inadequate for effectively capturing the anisotropic characteristics inherent in the social networks emphasized in our study. We will add these discussion in the final version. [1] Score-based Generative Diffusion Models for Social Recommendations

1. It is unclear whether it addresses label errors, outdated connections, or other types of noise. The noise we primarily address is graph-based noise, where social edges may represent outdated or misleading connections, potentially degrading the quality of recommendations. Regarding the type of noise mentioned, the paper focuses on anisotropic noise in social graphs. This refers to the varying structure of relationships across different parts of the network, where some edges might be more meaningful or relevant than others. The anisotropic nature of this noise distinguishes it from the isotropic Gaussian noise commonly assumed in other models.

2. Why denoising is limited to the social network, ignoring noise on user-item bipartite graph. The characteristics of bipartite graphs and social graphs differ significantly; while bipartite graphs typically model user-item interactions, social graphs capture more complex relationships between users or items with varying strengths. It is important to note that bipartite graphs may not necessarily exhibit the same anisotropic noise distribution as social graphs. Although we have not yet explored a unified generative framework specifically for bipartite graphs, we recognize its potential value. Our current focus is on modeling anisotropic noise within social graphs, with plans to investigate the applicability of generative models to bipartite graphs in future research.

3. More discussion on stochastic differential equations (SDEs) based methods. Thanks! There are existing works [1] using SDE approaches for social recommendation. Additionally, SDE methods predominantly rely on isotropic noise, making them inadequate for effectively capturing the anisotropic characteristics inherent in the social networks emphasized in our study. We will add these discussion in the final version. [1] Score-based Generative Diffusion Models for Social Recommendations

4. Supervisory signals? The supervisory signal mentioned refers to the label, which acts as a condition in the model. These labels guide the interpretation and application of the noise model, specifying the characteristics of the noise to be modeled in each scenario. By conditioning the system on these labels, we ensure that the model adapts its behavior according to the specific type of noise or variation present in the data. The labels allow the model to differentiate between different data distributions or contexts, improving its ability to make accurate predictions. In the revised version, we will further elaborate on the role of these labels in conditioning the system.

5. Robustness? We have already conducted a robustness analysis, where we examine the impact of the noise scale factor (τ) on the noising process. As the noise scale decreases from 1 to 0.1, model performance improves, with higher Recall@20 and NDCG@20 values for both Yelp and Epinions, demonstrating the effectiveness of RecDiff's denoising mechanism. However, when the noise scale reaches a certain threshold (τ = 10−2 and 10−3), excessive noise causes performance degradation, particularly in NDCG@20, as too much noise interferes with the model's ability to retain important user-item information.

6. Lacks discussion of existing score-based social recommendation methods Existing works like [1] apply SDEs to recommendation, akin to Q2, but use isotropic noise, limiting their capture of social networks' anisotropic traits, highlighted in our study. Anisotropic noise, reflecting community patterns interactions, is key to understanding social graphs. We'll detail these SDE methods in the final version's Related Work section for better context. As for the comparison, we primarily focus on Ciao and Epinions, as SGSR does not include a comparison with Yelp.

[1]Score-based Generative Diffusion Models for Social Recommendations

7. Lacks a clear explanation of the specific types of noise in social graphs

Thanks! We agree that the explanation of the specific types of noise in social graphs could be more detailed. The noise we primarily address is graph-based noise, where social edges may represent outdated or misleading connections, potentially degrading the quality of recommendations. Regarding the type of noise mentioned, the paper focuses on anisotropic noise in social graphs. This refers to the varying structure of relationships across different parts of the network, where some edges might be more meaningful or relevant than others. The anisotropic nature of this noise distinguishes it from the isotropic Gaussian noise commonly assumed in other models.