Flow Matching for Denoised Social Recommendation
Authors: Yinxuan Huang
KE LIANG
Zhuofan Dong
Xiaodong Qu
Wang Tianxiang
Yue Han
Jingao Xu
Bin Zhou
Ye Wang
Venue: ICML 2025 poster (2025)
Keywords: Graph Machine Learning; Social Recommendation; Flow matching
Primary area: applications->social_sciences

Abstract: Graph-based social recommendation (SR) models suffer from various noises of the social graphs, hindering their recommendation performances. Either graph-level redundancy or graph-level missing will indeed influence the social graph structures, further influencing the message propagation procedure of graph neural networks (GNNs). Generative models, especially diffusion-based models, are usually used to reconstruct and recover the data in better quality from original data with noises. Motivated by it, a few works take attempts on it for social recommendation. However, they can only handle isotropic Gaussian noises but fail to leverage the anisotropic ones. Meanwhile the anisotropic relational structures in social graphs are commonly seen, so that existing models cannot sufficiently utilize the graph structures, which constraints the capacity of noise removal and recommendation performances. Compared to the diffusion strategy, the flow matching strategy shows better ability to handle the data with anisotropic noises since they can better preserve the data structures during the learning procedure. Inspired by it, we propose RecFlow which is the first flow-matching based SR model. Concretely, RecFlow performs flow matching on the structure representations of social graphs. Then, a conditional learning procedure is designed for optimization. Extensive performances prove the promising performances of our RecFlow from six aspects, including superiority, effectiveness, robustnesses, sensitivity, convergence and visualization.

Flow Matching for Denoised Social Recommendation
Yinxuan Huang 1*
Ke Liang"
Zhuofan Dong² Xiaodong Qu Tianxiang Wang'
Yue Han
Jingao Xu'
Bin Zhou'
Ye Wang'
Abstract
Graph-based social recommendation (SR) models suffer from various noises social graphs, hin- dering their recommendation performances, Both graph-level redundancy and graph-level missing will indeed influence the social graph structures, further influencing the message propagation pro- cedure of graph neural networks (GNNs). Gen- erative models, especially diffusion models, are usually used to reconstruct and recover the data in better quality from noisy input. Motivated by it, few works take attempts on for social recommendation. However, they can only han- dle isotropic Gaussian noise and fail address anisotropic noise. Moreover, anisotropic rela- tional structures in social graphs are commonly seen, which existing models cannot sufficiently utilize the graph structures, which constraints the capacity of noise removal and recommendation performances. Compared the diffusion strat- egy, the flow matching strategy better handles anisotropic noise, preserves data structures more effectively during the learning process. In- spired by this, we propose RecFlow, the first flow- based SR model. Concretely, RecFlow performs flow-based method on the structure representa- tions of social graphs. Then, conditional learn- ing procedure is designed for optimization. Ex- tensive performances prove the promising per- formances of our RecFlow from six aspects, in- cluding superiority, effectiveness. robustnesses, sensitivity, convergence and visualization. Code are available at ?
DDPM
Matching
User User Graph
Velocily Field Direction
Convergence
Figure I. Illustration of denoising diffusion probabilistic models (DDPM) and flow matching based models where blue and red nodes represent normal and noisy data. Compared DDPM, flow- based models can get better discriminative capacity, further leading to better denoise performance.
1. Introduction
As online content continues grow exponentially, the chal- lenges of managing information sensitivity and urgency have become increasingly pressing (Lin et al., 2025; Wang et al., 2024), driving the rise of recommendation systems (Liu et al., 2023b). Despite advancements, recommendation systems still face challenges such as collaborative infor- mation sparsity. The rise of social media has shifted their focus from user-item interactions integrating social net- works to enhance recommendations, By leveraging social relationships auxiliary information, social recommenda- tion (SR) systems can mitigate these issues, making SR a key research area (Liang et al., 2023). Early models of SR relied on matrix factorization (Salakhutdinov & Mnih, 2007: Yang et al., 2013), which drew upon social theories to exploit the influence of nearby or connected users on individual preferences. Additionally, social relationships naturally form graph-structured data, making graph neural networks (GNNs) an effective tool for graph representa- tion learning ( (Wang et al., 2023a; Dai et al., 2023). GNNs have demonstrated exceptional performance in aggregating neighborhood information of nodes and have been widely
'Equal contribution 'School of Computer Science, National University of Defense Technology, Changsha, Hunan, China UUniversity of Chicago, Chicago. Illinois. United States Harbin Institute of TeehnologySheenzhenn,, Shenzhen, Guangdong, China. Correspondence to: Ye Wang <Yewang nudt.edu.en.>.
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Flow Matching for Denoised Social Recommendation
applied in the social recommendation domain, enabling the deep exploration and effective utilization of more valuable information (Huang et al., 2021a; Liang et al., 2023).
In summary, we make the following contributions:
We propose RecFlow, a novel social recommendation model that explicitly captures anisotropy in social net- works through velocity fields, addressing the limita- tions of isotropic noise assumptions in conventional diffusion models.
Despite advancements in social recommendation systems, current approaches often struggle to effectively mitigate two critical graph structural issues: redundancy and incomplete- ness (Lin et al.. 2023). These limitations primarily stem from noisy social connections in real-world data, where low- quality relationships inject interference that significantly degrades recommendation performance (Lin et al., 2024c;a). Graph neural networks (GNNs) exhibit particular vulnera- bility to such noise due to their inherent reliance on message propagation mechanisms across social edges, a character- istic that amplifies error transmission through the network (Wang et al., 2023a; Lin et al., 2024b). Recent studies have extended diffusion models to graph- based recommendation systems. DiffRec (Wang et al., 2023b) applies continu- ous diffusion by adding Gaussian noise to user/item em- beddings and optimizes them through a denoising process. RecDiff (Li et al., 2024b) introduces a multi-step diffusion and denoising framework for modeling complex social con- nections. These efforts build upon the growing success of diffusion models in various domains (Liu et al., 2024; Lee et al., 2024; Jiang et al., 2024; Deng et al., 2024).
By integrating flow matching with social recommenda- tion, RecFlow models directional data dynamics more effectively, leading to improved user preference repre- sentations.
Extensive experiments on benchmark datasets demon- strate the effectiveness of RecFlow, with significant performance gains. A visualization experiment further visualizes the evolution of velocity fields over time, highlighting the impact of our approach.
2. Related Work
In this section, we provide a comprehensive review of re- lated studies in the areas of social recommendation and generative models, and clarify how our work aligns with and builds upon the existing research.
Social data often exhibit strong anisotropy, as shown by the directional distribution of vector fields in our prelimi- nary analysis (Figure 3). This conflicts with the isotropic Gaussian noise (where noise is modeled as c NNO, r))) assumption in conventional denoising diffusion probabilistic models (DDPM), leading to two key issues: representation degradation, as isotropic noise blurs user embedding dis- tinctiveness, and unstable training, as DDPM's iterative de- noising undermines convergence consistency (Kingma et al., 2023). To address these limitations, we adopt Flow Match- ing, a generative approach that learns continuous velocity fields to construct direct sampling paths and model an ODE flow, enabling it to handle non-isotropic noise(modeled by <~NN N(,,)) Σ) with μ + 0 or non-diagonal Σ) (Zhao et al., 2024; Lipman et al., 2022). Figure 1 illustrates the differ- ences between these generative approaches: DDPM injects isotropic noise without directional awareness, while Flow Matching guides data towards clean distributions in a more discriminative manner, improving denoising performance on graph-structured data (Lipman al., 2022). Building on this, we propose RecFlow, baweb social recom- mendation model that leverages Flow Matching to capture the directional dynamics of user interactions, enabling more stable training. better denoising, and improved recommen- dation accuracy. Building on this, we propose RecFlow a flow-based social recommendation model that leverages Flow Matching to capture the directional dynamics of user interactions, enabling more stable training, better denoising, and improved recommendation accuracy.
2.1. Graph-based Social Recommendation
Graph-based Social Recommendation has gained signifi- cant attention for incorporating social relationships. Early works like DWf et al., 2019b) used Graph Con- volutional Networks (GCNs) to model social influence, while later models like GraphRec et al., 2019) and DANSER(WL et al., 2019c) added attention mechanisms to account for varying influence levels. More recent ap- proaches, such as MHCN(Yu et al., 2021b) and HOSR(Liu et al., 2020), capture higher-order relationships and dis- tant influences. Models like RecoGCN(XU et al., 2019), DGRec(Song et al., 2019b), TGRec(Bai et al., 2020), and KCGN(Huang et al., 2021b) integrate diverse data sources, including agent, temporal, and knowledge graph informa- tion. To address noisy social relations, recent methods like DSL(Wang et al., 2023a) and GDMSR(Quan et al., 2023) focus on denoising by identifying and removing irrelevant or redundant social connections, thus improving recommen- dation quality and efficiency. These methods struggle with noisy or irrelevant social connections. Flow matching mod- els, by modeling influence spread and denoising, provide a promising solution, refining user representations to enhance the robustness and accuracy of recommendations.
2.2. Diffussin Model-basee Recommendation
Generative recommenders have attracted interest in recent studies. Some studies focused on leveraging diffusion mod-
2

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Flow Matching for Denoised Social Recommendation
els to enhance data representation and mitigate noise inher- ent in social connections. For instance, RecDiff(Li al., 2024a) employed a latent diffusion paradigm to denoise user representations derived from social networks, demonstrating improved robustness in handling the diverse noisy effects of user social contexts. Similarly, DiffuASR(Liu et al., 2023a) proposed a diffusion-based pseudo sequence genera- tion framework, and fills in the gap between the generations of continuous images and discrete sequences. CGSoReccHe et al., 2024) proposed a condition-guided social recommen- dation model, leveraging a conditional constraint in the diffusion process to incorporate social connections. This allows the model to refine user preferences based on their social connections. Similarly, DIEX et al., 2023) uses difftssion framework to model user preferences, and then trains textual decoder generate explanations based on the denoised user representation, enhancing the inter- pretability of diffusion recommenders. In contrast to these established approaches, our proposed Reeflow introduces a novel methodology by leveraging flow matching models (Liu et al., 2022), making it more effective in capturing intricate patterns.
x), the model solves least squares regression problem, optimizing the velocity field vo to closely match the ideal flow between and 2.
The training of the neural network involves minimizing the loss function I,, defined as:
C2 - f Es, [1(===))=
c
- co f)1l]
(4)
This loss quantifies the discrepancy between the ideal and predicted velocities over the time interval [o, 1]. enabling the flow follow the desired trajectory by accurately pre- dicting the velocity at any point The parameterized neural network vy is thus trained to minimize L, facilitating an efficient and accurate modeling of transitions from x: to z.
4. Proposed Model
As illustrated in Figure 2, we integrated collaborative and social signals within unified flow-based generative frame- work, the overall architecture of RecFlow consists of three key components: Graph-based Collaborative Pattern Encod- ing, the RecFlow Module, and a Joint Optimization Module.
3. Preliminary
4.1. Problem Statement
In this section, we briefly introduce the preliminaries of flow-matching models. A flow-matching model is a type of generative model that bridges the gap between a source dis- tribution Pa and a target distribution Ps by learning a neural network to parameterize the velocity field of an Ordinary Differential Equation (ODB). The ODE is defined as:
Users and items are defined the sets U (w1, U2,..., Un) and V = {11, ......, ). Interactions between users and items are represented by the matrix R E
respectively.
[ IVI,, where the element Ta.v = indicates that user u interacts with item v, and Ta, = 0 otherwise.
I
Social relationships between users are described by the ma- trix S E | where Su,u² = 1 signifies a social inter- action between user u and user u', and su,u' 0 otherwise. Based on these interaction matrices, the following graph structures are constructed:
dr. = up (a+,, t)dl,
where vn denotes the learnable velocity field parameterized by a neural network. The ODE ensures that the intermediate distributions 2a remain consistent with the learned proba- bility path for all E [0, 1]. and the velocity field Uo directs the flow from the initial state the target state 2, effec- tively transforming the source distribution into the target distribution over time.
Collaborative Graph G,- = (U. V. Er), where the edge set Eyp = { (v, v) Tu,o = 1 represents interac- tions between users and items.
This process converts samples from I ~ Pr to align with ps. The interpolation between æ and is defined through the linear blend 2, = tz + (1 t)x, satisfying the ODE:
Forward Process:
Social Graph G. = (U, E.), where the edge set E, { {u, u/) Su, = 1 } represents social relationships between users.
Our RecFlow leverages both collaborative and social graphs specifically, the collaborative graph G, generates node em- beddings denoted as F,, and the social graph G, generates node embeddings denoted as E.. The predicted user-item interaction value fu is computed as:
dx. = (= 15-xx)a dt
(2)
Reverse Process: Conversely, this process generates sam- ples starting from 2 ~ P2 and reverses the flow dynamics. The reverse ODE, mirroring the forward process, is defined as:
dx. (xr z)dt
(3)
uu. = Pred(eu: Eo),
(5)
The effectiveness of the rectified flow depends on the precise estimation of velocity v. To align u with the direction (z
where en and e, are the embeddings of the user u and item v. These e, and e, are derived from both the collaborative
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Flow Matching for Denoised Social Recommendation
Algorithm 1 RecFlow Training
The initial embeddings Ej are randomly generated learnable parameters. And D, is the corresponding diagonal degree matrix, defined as:
Input: Users' social interaction embedding E
Output: The reconstructed embedding which denoted as 60
0 R
R
1: Set E # Initialize the social interaction embedding while not converged do
A
0
2:
3:
~~ (,,((011) 1)
# Sample time
where Ly is the normalized Laplacian matrix of Gr, and is the identity matrix.
4:
x ~ Pr
# Sample data
5: p~p;
# Sample noise
6:
Zt VV (z(z))
# Conditional flow
For the user's social graph G., embedding propagation fol- lows similar process:
7:
Gradient step: (ollu, 2,
8: end while
F (T.g T) F. F.(-1).
(8)
Algorithm 2 RecFlow Inference
L, = D2 * SD.
Input: Users' interaction vectors Tu. u 1,2...., [U;; optimized parameter
S being the adjacency matrix of the social graph G,. T. is the normalized Laplacian matrix. After propagating through L layers, the final embedding for each user-item pair is obtained by aggregating embeddings across all layers as eu, Σt u where e represents the I-th layer embedding between user and u'. Similarly, for user-item pair (u, v), the predicted embedding is êu,v El Cu..-
Output: Predicted user embeddings or interaction out- comes
1: for E U do
2: Let Tu <- Zu
# Initialize with user data
3: Sample ~ pt.
Time step sampling
4: Calculate Ia + fr
# Run learned model
5: if nreessoosiroccessing then
6:
(2))ess(21)
# Final prediction
4.3. RecFlow Module
7: end if
8: end for
In the forward process, the RecFlow module begins by sam- pling time step t from a uniform distribution (1 1). This time step is then encoded into time embedding vector E,, which is concatenated with the user representation E, ob- tained from the social encoder. The perturbed input æ, is defined as linear interpolation between Gaussian noise z and the socially-informed embedding E,;
graph G, and the social graph G., and are learned jointly during the model training, respectively.
4.2. Graph-based Collaborative Pattern Encoding
Drawing inspiration from the effectiveness of simpli- fied Graph Neural Networks (GNNs), we incorporated a lightweight Graph Convolutional Network (lightGCN) as the graph encoder our RecFlow architecture (Jiang et al., 2023). LightGCN is widely recognized as a robust graph recommender for modeling implicit interactions in top-k recommendations, construct collaborative graph to encode user-item interactions using the Rec Encoder, and user-user graph to capture social relationships among users using the Social Encoder. On the user-item graph G, the embeddings are propagated across layers using the follow- ing equation:
= tz (1- F)E,
(9)
Unlike conventional flow fow-matching methods that typically treat z as the origin and interpolate toward data samples, our formulation conditions the diffusion trajectory on the social embedding E.. enabling the model to incorporate social context into the forward process.
In the reverse process, we train a vector field estimator vg (xe, 1) to approximate the velocity field. Here, 0 denotes the set of learnable parameters. The estimator is defined as:
Ug (a t)
FC(x) = o b) (10)
E = (L, + EQ-
(6)
where E, is the time embedding step || denotes vector concatenation, and FC represents consecutive fully connected layers. The function ((-) denotes non-linear activation (e.g., ReLU or GELUU, and W, b are the weight matrix and bias vector of each linear transformation.
4 = D. A.D
where A, HVM is the adjacency matrix of the bipartite graph G,, the embedding matrix EX!) E R ((B+++)) captured the embeddings in the I-th iteration of the GCNs. And I,p is the normalized Laplacian matrix.
Specifically, starting from Te. the model integrates the reverse-time flow using an ODE solver to obtain the clean

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Flow Matching for Denoised Social Recommendation
Graph-based Collaboration Pattern Encoding
Jointly Optimization
Rec Encoder
Dpr
ODE Solver
LCF
Collaborative Graph
V
RecFlow Module
Social Encoder
Vector Field Estimator
vo(2e, t)
tz
Time Embedding
User-User Graph
Forward Process
Reverse Process
Figure Our RecFlow consists three main components: Graph-based Collaborative Pattern Encoding responsible for obtaining embeddings of the collaborative graph and the social graph; RecFlow Module learns continuous velocity fields on the social graph to effectively denoise anisotropic structures; and Joint Optimization integrates the objectives of collaborative encoding and flow-based social representation learning.
embedding zo, which approximates the original user pref- erence vector in the latent space. The reverse integration is performed as:
with positive and nega- tive item in pairwise training setup following the Bayesian Personalized Ranking (BPR) paradigm (Rendle et al., 2012). The conditional flow-matching loss Letm- computed over sampled diffusion steps f, guides the learning of the reverse- time vector field. Additionally, L. regularization (weight decay) with coefficient λι is applied to all trainable param- eters @ to prevent overfitting. The form of LCrM follows the definition in Eq. (3) and the flow estimation process illustrated in Figure 2.
Here, (u, v+, x-) denotes
20 = Iz 1. Vo(x,, dr
(11)
During inference, the model fixes the optimized parameters 0, and no further training performed, then we sample latent representation 2 ~ (O, I) and timestep and then applies the learned reconstruct the user embedding e..
4.4. Joint Optimization
4.5. Disucssion and Analysis
To integrate social relationships with encoded user-item interaction patterns, RecFlow employs a hidden-space re- flow mechanism to generate the final user embeddings for prediction. This process is defined as:
In this section, present detailed analysis of the time and space complexity of our RecFlow model. Besides, we fur- ther provide the theoretical analysis between our RecFlow and original DDPM methods to better illustrate the effi- ciency the flow matching strategy.
Au, 2 c, == =' éo(e
(12)
where denotes sampled diffusion time step for user u, e', and e', represent the initial embeddings of the user and item obtained from the respective graph encoders, and ( denotes the learned reflow adjustment from the vector field estimator.
Time Complexity, Initially, RecFlow performs graph-level information propagation both the holistic collaborative graph G, and the social graph G.. this process requires O([E, + E.1) x d) calculations for message passing. The overall time complexity of RecFlow during training is dom- inated by the graph-level propagation and the gradient up- dates, resulting in:
The model is optimized by minimizing joint loss function that combines recommendation and diffusion objectives:
O((E, (E,1)
(14)
L
Σ
- log of - -aa A Eleim
Letm (13)
for each iteration of training. And complexity of flow match-
(u,
5

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Flow Matching for Denoised Social Recommendation
ing is O(N), eliminating multi-step iterations and requiring only one global optimization.
Visualisation Dataset
Visualisation Yelp Dataset
Space Complexity. The space complexity is primarily de- termined by the storage required for the graphs and em-
beddings. The collaborative graph G, requires storing the
adjacency matrix of the user-item interactions, which has space complexity of O(UU) x [V]), the social graph Gs requires storing the adjacency matrix of the user-user inter- actions, which has space complexity of ((U)??) assuming a dense representation. Thus, the overall space complexity of RecFlow is:
Compaienc
Complnere
Figure 3. Illustration of the anisotropic attributes of two typical datasets, ie., Ciao and Epinions. Each point in the 2D visualization represents user-user interaction embedding projected onto two principal components. The uneven and elongated distributions of points along certain directions highlight the anisotropy of the data, indicating that user user relationships are not uniformly distributed but instead exhibit directional concentration.
O(UU) +UP (IU) + [V]) d)
(15)
This accounts for the storage of the graph structures and the user and item embeddings. And space complexity is O(D), dependent solely on data dimensionality and independent of timesteps.
Q3: Is RecFlow robust enough to effectively handle noisy and sparse data in social recommendation (SR)?
Theoretical Analysis. In the context of generative mod- eling, Flow Matching and DDPM both aim to generate data through controlled transformations of noise. FM con- structs linear interpolation between Xo and X1, leading to continuous and deterministic path. Mathematically, this is deseribed as:
Q4: How do different settings impact the performance of RecFlow?
Q5: How does the complexity of our method compare that of alternative approaches?
X; = (1- X)x0 + tX1
(16)
Before showing and analyzing the experimental results, we first present the experimental settings below.
where X₂ follows a simple ODE-driven trajectory. In con- trast, DDPM follows a stochastic diffusion process:
5.1. Experiment Settings
X, / + BBE
(17)
Datasets and Evaluation Metries. We conducted exper- iments on three publicly available social recommendation datasets: Ciao, Yelp and Epinions. Detailed statistics for these datasets are provided in Table 2. Our datasets are obtained from RecDiff (Li et al., 2024a)²
where is sampled from a Gaussian prior. The nonlinear and stochastic nature of DDPM results in higher variance in sam- pling paths, leading to inefficiencies. And FM is governed by an Ordinary Differential Equation (ODE), which enables continuous time evaluation and faster sampling via adaptive solvers. In contrast, DDPM relies on discrete Stochastic Differential Equations (SDEs), requiring a large number of steps for accurate generation. Empirically, FM achieves comparable quality with fewer function evaluations, reduc- ing computational overhead.
We conducted a preliminary analysis of the Ciao and Epin- ions dataset. Figure.3 shows the distribution of social data in the Ciao and Epinions dataset. After reducing the data from high- dithe-iimniionall space to two dimensions using Principal Component Analysis (PCA), it can be observed that the data points are more spread out along Component 1, while the distribution is more concentrated and less variable along Component 2. This aligns with the characteristies of social network data, where there are often dominant relational pat- terns, while others secondary or sparse. In the social graph, certain user groups with strong internal connections form tightly-knit social clusters, reflecting prominent rela- tional patterns in the data. Additionally, the presence of a dominant direction in the data causes the vector field to display a clear anisotropic distribution, indicating that user interactions are more concentrated along specific directions.
5. Experiment
In this section, we present a series of experiments conducted to evaluate the performance of our RecFlow method, focus- ing on the following five questions:
Q1: How does RecFlow perform in comparison to other state-of-the-art social recommendation methods?
Q2: What are the key contributions of RecFlow's 's main modules?
Datasets are available at butps//gttuhh.oom//KKDDDSSse
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Flow Matching for Denoised Social Recommendation
Table 1. Statistics of experimental datasets, our datasets
Table Overall performance analysis.
Data
Ciao
Yelp
Epinions
Ciae
Yelp
Epinions
# Users
1,925
99,262
14,680
Method
Recall
NDCG
Recall
NDCG
Recall
NDCG
TrustMF
0.539
0.343
0.371
0.193
0.265
0.195
# Items
15,053
105,142
233,261
SAMN
0.604
0.384
0.403
0,208
0.329
0,226
# Interactions
23,223
672,513
447,312
DifINet
0.528
0.328
0.557
0.292
0.384
0.273
GraphRec
0.540
0.335
0.419
0.201
0.334
0.246
Social Ties
65,084
1,298,522
632,144
DGRec
0.517
0.319
0.410
0.209
0.326
0,236
NGCF
0.559
0.363
0.450
0.230
0.353
0.243
MHCN
0.621
0.378
0.567
0.292
0.438
0.321
Evaluation Protocols. In our experiments, we utilized two commonly used metrics.: Hit Ratio HR N and Normalized Discounted Cumulative Gain (NDCG) @ N as metries, where N represents the number of items recommended to the user, widely ultilized in Top-N recommendations. We applied 7:1:2 ratio to split each dataset into training, validation, and test sets, adhering to common data partitioning practices in graph-based recommendation systems.
KCGN
0.602
0.350
0.460
0.234
0.2201
0.1456
SMIN
0.588
0.354
0.485
0.251
0.333
0.228
GDMSR
0.560
0.355
0.513
0,246
0.368
0.241
DSL
0.606
0.389
0.504
0.259
0.365
0.267
RecDiff
0.712
0.419
0.597
0.308
0.460
0.336
RecFlow
0.725
0.438
0.618
0,341
0.486
0,341
(+4.5%) over RecDiff. For Yelp and Epinions, it maintains robust gains: 0.618 vs. 0.597 Recall(+3.5% and 0.341 vs. 0.308 NDCG (+10.7%) on Yelp; 0.486 vs. 0,460 Recall (+ 5.6%) on Epinions, demonstrating adaptability to vary- ing data densities. Notably, methods with self-supervised learning (SSL)-MHCN (local-global contrast), KCGN, SMIN (hierarchical relations), and DSL (predictive con- sistency) consiteet-y outperform traditional approaches. SSL mitigates noise propagation and interaction sparsity by extracting latent relational patterns, enabling stable repre- sentation learning. RecFlow's denoising process, guided by a velocity field, directly optimizes trajectories toward clean data distributions, suppressing noise during refinement. This explains its 5.6-10.7% improvements over diffusion-based RecDiff sparse datasets.
Compared Baselines. We compared RecFlow with 12 base- line models, representing the latest advancements in social recommendation research. These encompass conventional and attention-basee methods, graph-based recommendation models using collaborative filtering, as well as other GNN- based social recommendation systems, as outlined below: PMF (Salakhutdinov & Mnih, 2007), TrustMF (Yang et al., 2013), GraphRec (Fan et al., 2019), DifINet (Wu et al., 2019a), DGRec (Song et al., 2019a), NGCF (Wang al.,
2019), MHCN (Yu al., 2021a), KCGN (Huang et al.,
2021a), SMIN (Long et al., 2021), GDMSR(Quan et al.,
2023), DSL (Wang et al., 2023a), RecDiff (Li et al., 2024a).
Implementation Details. All experiments are conducted on a machine with an RTX A800 for a fair comparison. The experimental settings and hyper-parameteed details of our RecFlow framework are elaborated in. The learning rate was tuned within [5e-1, 1e-3, 5] with a 0.96 decay factor per epoch. Batch sizes were selected from [1024, 2048, 4096, 8 8192], and hidden dimensions from [64, 128, 256, 512]. The parameter y was set according to the Ypet- percentile of node embedding distances for each dataset. The optimal number of GNN layers was chosen from [1, 2, 3, 4]. The Timestep embedding size is selected from 4,8.1 And the batch size for Ciao is 2048, while for Yelp and Epinions is 4096. Regularization weights A were selected from [1e-3,11 3, 12-2,122 le le",, ].
5.3. Ablation Study (RQ2)
To evaluate the impact of different components in the RecFlow framework, performed ablation study using three benchmark datasets: Ciao, Yelp, and Epinions. The results are presented in Table 4.
w/o Flow: This configuration excludes the holistic flow-matching module, leaving only the GNN for learn- ing user-item and social relations. As shown in Table 4, the absence of the flow module results in a signif- icant decrease in both Recall and NDCG across all datasets. Specifically, Recall drops by approximately 13% (Ciao), 7% (Yelp), and 12% (Epinions), while NDCG decreases by about 13% (Ciao), 16% (Yelp), and 31% (Epinions). This emphasizes the importance of the denoising mechanism in our model.
5.2. Performance Comparison (RQ1)
Table 2 summarizes the experimental results across three datasets, with RecFlow's metrics bolded and top base- lines underlined. RecFlow demonstrates marked improve- ments over existing approaches, more specifically, on Ciao, RecFlow achieves 0.725 Recall (+2.0%) and 0.438 NDCG
w/o CL: In this configuration, we remove the condi- tional learning (CL. guidance for flow matching. The
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Flow Matching for Denoised Social Recommendation
Recail@20 Noise Scale
NDCG@20 vs. Noise Scale
Table Comparison of different sampling methods
0,070
0.065
Yelp Epinions
0.035
Clao
Yelp
Epinions
0.060
Epinions
0.055
Method
NDCG Recall NDCG Recall NDCG
8 recalla
0.050
0.081
0.045
0.029
ODE-Salver
0.725
0.438
0.618
0.341 0.486
0.341
0.040
0.035
0.027
RK4
Multistep Heun
0.699 0.403 0.590 0.317
0.710 0.411 0.594
0.322
0.453
0.460
0.320
0.325
0.030
10'
Noise Scale (log scale)
10a1
10-
10"1
0.025
10P
Noise Scale (log scalle)
10
10
10
Heun
0.683 0.399
0.581 0.319
0.449
0.318
Euler
0.670
0.383
0.570
0.311
0.438
0.305
Figure 4. Robustness Analysis, the blue bars represent the Yelp dataset, while the green bars represent the Epinions dataset.
Table Ablation Analysis: "wo Flow" denotes the removal of the RecFlow module, "CL" denotes the removal of conditional learning, and "both" indicates the removal both components simultaneously.
Convergence Comparison: Flow Matching vs DDPM
Flo Matching
1.0
0.8
Ciao
Yelp
Epinions
Residual Emor (Log
Method
Recall NDCG Recall NDCG Recall NDCG
0.6
ReeFlow RecFlow w/o Flow
0.725 0.438
0.618
0.341
0.486
0.341
0.633
0.380
0.573
0.301
0.429
0.297
RecFlow w/o CL
0.692
0.401
0.597
0.320
0.443
0,312
RecFlow w/o CL of both 0.621
0.407
0.589
0.312
0.417 0,302
0.0
0.0
1.0
Time Sneps
results show a notable performance decline, particu- larly in NDCG across all datasets, with decrease of around 8% (Ciao), 11% (Yelp), and 12% (Epinions). This demonstrates the critical role of Conditional Learning in improving model accuracy.
Figure 5. The convergence comparison of the loss curves is shown in the legend. The blue curve represents the convergence of RecFlow, while the yellow curve corresponds to the convergence after applying DDPM.
5.4. Robustness Analysis (RQ3)
w/o both: In this case, both the flow module and the CL label guidance are removed. The performance ex- periences a significant drop, with Recall decreasing by about 14% (Ciao), 5% (Yelp), and 7% (Epinions), and NDCG dropping by approximately 7% (Ciao), 9% (Yelp). and 15% (Epinionss) This highlights the essen- tial contributions of both the now-matching process and label guidance in enhancing the model's 's ability to learn effective user-item and social relationships.
This section examines the effect of the noise scale factor (r) on the noising process. By scaling the minimum and maximum noise in the scheduler to T mmin and T 5 respectively, we test the model's performance at different noise scales (1, 0.1, 0.01, 0.001). The results, shown in Figure 4, reveal the following:
Increasing the noise scale improves model perfor- mance, with higher Recall and NDCG@20 values for both Yelp and Epinions the noise scale decreases from to 0.1. This demonstrates the effectiveness of the RecDiff framework's 's denoising mechanism.
The choice of sampling method creates clear trade-off between computational cost and model accuracy (see Table 3): the basic Euler method, with its single first-order step, yields the lowest Recall and NDCG, while Heun's 's two- stage predictor-correst corrector boosts both metrics modestly at only twice the cost. Leveraging history, the multistep Heun scheme further raises performance, and the four-stage RK4 delivers similar gains by reducing local error through fourth- order updates. Finally, an adaptive ODE solver such as Dormand-Prinee which dynamically adjusts its step size to satisfy error tolerances, consistently achieves the highest Recall and NDCG on Ciao (and likewise leads on Yelp and Epinions), demonstrating that, when resources allow, more precise integration yields the strongest quality.
Excessive noise beyond threshold leads to perfor- mance degradation, especially for Yelp and Epinions. As noise scales reach 10 and 10 10-3, a noticeable decline in NDCG@ @ 20 suggests that too much noise interferes with the model's ability to retain important user-item data.
5.5. Convergence Analysis (RQ4)
In Figure.5, RecFlow exhibits a faster convergence speed compared to DDPM, as the horizontal axis represents the
8

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Flow Matching for Denoised Social Recommendation
Velecty Feed Dimection Froch
Weldity (Epoch
leacity Drrction Eppon
Vilictt Directonn
Welocity (Epech
Velecity Fold (Epech
0.6
64
0.4
044
0.4
02
0.2
0.2
0.0
-0.2
-02
-02
-02
-0.4
-14
-0.4
-0.4
-0.6
-0.90-0.255
0.25 9.50
0.50
0.00
-08
-02
80
04
0.
04
00
-04
02
-0.52-0222 0.2 6.51
24
-02
06
02
-04 -02
00
64
Figure The top row shows the velocity field direction of our RecFlow, while the bottom row displays that of DDPM. is evident that applying the flow-matching velocity field results in clear directional pattern, aligning with the anisotropy observed in the previous dataset visualizations. In contrast, the application DDPM does not exhibit any noticeable directionality, with directions appearing random throughout.
diffusion model's time steps, ranging from 0 1. The verti- cal axis indicates the residual error each time step. It is evident that the residual error of Flow Matching decreases rapidly in the early time steps, demonstrating a significantly faster convergence trend. In contrast, DDPM's error de- creases at slower pace, highlights the advantage Flow Matching in modeling vector fields and achieving efficient convergence, making it more suitable for handling complex distributions and data scenarios.
RecFlow effectively captures and aligns with the anisotropy of the data, DDPM fails to exhibit clear directional conver- gence, further emphasizing the advantage of RecFlow in learning meaningful patterns from the social graph structure.
6. Conclusion
In this paper, we proposed RecFlow, novel generative so- cial recommendation framework based now-matching. Diffusion-based method primarily target the removal of isotropic noise, therefore damage the structural representa- tion, and the aim of Recflow is to fill the gap. By learning continuous velocity fields on social graphs, Recflow con- structs a direct sampling path from noisy input the target distribution, thereby enabling faithful representation learn- ing. To validate the effectiveness, experiments several strong baselines, and the results con- sistently demonstrate the superior performance and robust- ness of our proposed RecFlow. In the future, we will explore more scalable extensions of RecFlow to further assess its practicality and effectiveness real-world recommendation datasets.
5.6. Visualization of Flow matching velocity field direction over different epochs (RQ5)
To better understand how the velocity field evolves during training, we design experiment that visualizes the direc- tion of the vector field across different epochs. In Figure 6, the data points are unevenly distributed, showing clear direc- tionality and concentration, which highlights the anisotropy of social data. The model captures this anisotropy within the velocity field, represented by arrows that dynamically adjust direction the epoch evolves, demonstrating the convergence dynamics of velocity field directions.
conduct extensive
In the early stages (Epoch 0-38), directions exhibit high dispersion (-0.50 to +0.50), reflecting unstable parameter ad- justments. As training progresses (Epoch 59-99), directions progressively cluster near the origin (range: -0.25 to +0.25), indicating stabilized optimization trajectories. Notably, after Epoch 78, data density intensifies with directional similarity, signifying unified parameter updates.
Impact Statement
This paper introduces RecFlow, a flo-matchhnn social rec- ommendation model that captures anisotropic in user inter- actions. By leveraging flow matching, RecFlow enhances representation learning and denoising efficiency, emphasizes the practical benefits for personalized recommendation. We also acknowledge potential societal risks, such as bias ampli- fication, and highlight the need for faimess and robustness in future recommendation systems.
In comparison, as shown in the bottom row of Figure 6, the velocity field direction of DDPM lacks any noticeable directionality across epochs, with the directions appearing random throughout. This contrast highlights that while

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Flow Matching for Denoised Social Recommendation
Acknowledgments
Jiang, Y., Xia, L., Wei, W., Luo, D., Lin, K., and Huang, C. Diffmm: Multi-modal diffusion model for recommenda- tion, 2024. https:/ /aaxivi.v... wwgg/bbs./. aba/2/bs//26 2466. 11781.
This work is supported by National Natural Science Founda- tion of China (No 62302507) and Hunan Provincial Natural Science Foundation (No. 2023JJ40684).
Kingma, D. P., Salimans, T., Poole, B., and Ho, J. Varia- tional diffusion models, 2023. https:/ //atxis://rrvv...
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12


Reviews:
[{"review_id": "Jr6ZbTIW4S", "reviewer": "Reviewer_Ec7S", "summary": "This paper introduces RecFlow, a social recommendation system using flow matching to handle noise in social graphs. Unlike traditional diffusion models that assume isotropic noise, RecFlow better captures the anisotropic nature of social data by learning velocity fields that preserve structural relationships. Experiments across three datasets show RecFlow outperforms state-of-the-art methods, with 2-5.6% improvements in Recall and 4.5-10.7% in NDCG metrics. The approach requires fewer computational steps than traditional diffusion models while maintaining better data structure preservation.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. It is unclear whether it addresses label errors, outdated connections, or other types of noise.**\nThe 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.\n\n\n**2. Why denoising is limited to the social network, ignoring noise on user-item bipartite graph.**\nThe 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.\n\n**3. More discussion on stochastic differential equations (SDEs) based methods.**\nThanks! 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.\n[1] Score-based Generative Diffusion Models for Social Recommendations\n\n\n**4. Supervisory signals?**\nThe 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.\n\n**5. Robustness?**\nWe 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.\n\n**6. Lacks discussion of existing score-based social recommendation methods**\nExisting 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.\nAs for the comparison, we primarily focus on Ciao and Epinions, as SGSR does not include a comparison with Yelp.\n| Model     | Recall (Epinions) | NDCG (Epinions) | Recall (SGSR) | NDCG (SGSR) |\n|-----------|-------------------|-----------------|---------------|-------------|\n| SGSR      | 0.645             | 0.425           | 0.470         | 0.315       |\n| RecFlow   | 0.725             | 0.438           | 0.486         | 0.341       |\n\n[1]Score-based Generative Diffusion Models for Social Recommendations\n\n**7. Lacks a clear explanation of the specific types of noise in social graphs**\n\nThanks! 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.", "claims_and_evidence": "a) As far as I know, the errors in social networks may include label errors (i.e., annotation mistakes) and some outdated or irrelevant connections. \nb) The paper mentions denoising, but it's unclear which type of noise this refers to, or if it addresses other kinds of noise. \nc) There is no detailed explanation provided in the paper. Additionally, why is the denoising process limited to the social network, while noise can also exist on the user-item edges? Why isn't denoising applied to the user-item bipartite graph as well?", "methods_and_evaluation_criteria": "There is a score-based work from 2024: Liu, C., Zhang, J., Wang, S., Fan, W., & Li, Q. (2024). Score-based Generative Diffusion Models for Social Recommendations. It also applies generative models to social recommendations, but no comparison is made in the paper.", "theoretical_claims": "How is the supervisory signal introduced in the generative model in the paper? \nThe paper doesn't provide a detailed explanation, or at least I didn't fully understand it.", "experimental_designs_or_analyses": "The paper includes extensive experiments covering performance, ablation, visualization, convergence, and sensitivity analysis\n\nHowever, one question remains: how is the method demonstrated to be robust against newly introduced noise? Furthermore, how exactly is the noise removed? This aspect should also be addressed in the experiments.", "relation_to_broader_scientific_literature": "Previous work has already explored score-based methods for social diffusion, and I’ve provided the paper title in the Evaluation Criteria section. \n\nGiven methods like DDPM and score-based models, which are based on SDEs, what advantages does flow matching, which is based on ODEs, offer in comparison?", "essential_references_not_discussed": "Liu, C., Zhang, J., Wang, S., Fan, W., & Li, Q. (2024). Score-based Generative Diffusion Models for Social Recommendations.", "other_strengths_and_weaknesses": "Strengths:\n1. The proposed RecFlow method introduces a novel flow matching approach to social recommendation that effectively handles anisotropic noise in social graphs. This represents a significant advancement over traditional diffusion models that assume isotropic Gaussian noise.\n2. The experimental evaluation is comprehensive, covering three datasets and comparing against 12 baselines. The results show consistent improvements (2-5.6% in Recall and 4.5-10.7% in NDCG) over state-of-the-art methods, with thorough ablation studies and analyses of robustness, sensitivity, and convergence.\n\nWeaknesses:\n1. The paper lacks discussion of existing score-based social recommendation methods, creating a gap in the literature review. This omission makes it difficult to fully contextualize the contribution within the broader landscape of generative approaches to social recommendation.\n2. Despite focusing on denoising, the paper provides insufficient explanation of the specific types of noise being addressed in social graphs. While it mentions \"graph-level redundancy\" and \"graph-level missing,\" it doesn't clearly define these concepts or explain how they manifest in real-world data, limiting understanding of the method's practical applications.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Please refer to my comments above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nj0gpHFdMn", "reviewer": "Reviewer_jYLr", "summary": "The study presents Recflow, a flow-matching model for social recommendation systems, which addresses challenges in traditional recommendation methods, especially in noisy social networks. The key issue with many graph-based approaches is their inability to handle noisy edges in social graphs, which can degrade performance. The study highlights the effectiveness of using flow-matching models to capture anisotropic characteristics of social data, unlike traditional isotropic Gaussian noise used in diffusion models, which can obscure relational structures. Moreover, the study offers a detailed comparison of RecFlow with other generative models like DDPM mathmatically, demonstrating its superior ability to handle the anisotropy inherent in social data. The computational efficiency is also highlighted, with RecFlow requiring fewer steps for convergence compared to traditional diffusion models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. More theoretical foundation and mathematical descriptions.** \nThanks, 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.\n\n[1] A directional diffusion graph transformer for recommendation\n[2] Denoising Diffusion Probabilistic Models\n\n**2. More detailed derivation of the ODE equation, and correlated experiments should be added.**\nThanks! We agree, and we present it as follows, which will also be added in our revised paper.\nEuler method:  \n$$ y_{n+1} = y_n + h f(t_n, y_n) $$\n\nRunge-Kutta method:  \n$$ k_1 = h f(t_n, y_n), \\quad k_2 = h f(t_n + \\frac{h}{2}, y_n + \\frac{k_1}{2}) $$  \n$$ k_3 = h f(t_n + \\frac{h}{2}, y_n + \\frac{k_2}{2}), \\quad k_4 = h f(t_n + h, y_n + k_3) $$  \n$$ y_{n+1} = y_n + \\frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) $$\n\n\nMoreover, 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).\n| Model        | Ciao Recall | Ciao NDCG | Epinions Recall | Epinions NDCG |\n|--------------|-------------|-----------|-----------------|---------------|\n| Euler        | 0.725       | 0.438     | 0.486           | 0.341         |\n| Runge-Kutta  | 0.720       | 0.435     | 0.483           | 0.339         |\n\n\n**3.[1] suggests that diffusion models do not necessarily require the noise addition phase**. \nThanks! 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.\n\n[1]Is Noise Conditioning Necessary for Denoising Generative Models?\n\n**4. Color Scheme**\nThanks! We will consider adjusting the figure in the revised version to make it clearer.\n\n**5. Comparison with other heuristic denoising methods.**\nThanks! 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]\n| Model    | Ciao Recall | Ciao NDCG | Epinions Recall | Epinions NDCG |\n|----------|-------------|-----------|-----------------|---------------|\n| GDMSR    | 0.560       | 0.355     | 0.368           | 0.241         |\n| DSL      | 0.606       | 0.389     | 0.365           | 0.267         |\n| RecFlow  | 0.725       | 0.438     | 0.486           | 0.341         |\n\n\n[1]Denoised Self-Augmented Learning for Social Recommendation\n[2]Denoising Diffusion Probabilistic Models\n\n**6. More discussion on stochastic differential equations (SDEs) based methods.**\nThanks! 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.\n[1] Score-based Generative Diffusion Models for Social Recommendations", "claims_and_evidence": "Although the paper provides detailed experiments, first analyzing the dataset to demonstrate the inherent anisotropy of social data, and later using visualization methods to show that flow matching can partially match this anisotropy, I believe that a more theoretical definition and expression of isotropy and anisotropy, as discussed in the paper, are needed. Unfortunately, the theoretical foundation and mathematical descriptions in the paper are not sufficiently developed.", "methods_and_evaluation_criteria": "I think it is meaningful to evaluate the method proposed in the paper using graph-based recommendation datasets and commonly used metrics in recommendation systems.", "theoretical_claims": "As mentioned in the claims, I believe that the core argument of the paper, the so-called \"isotropic\" and \"anisotropic\" noise, requires a more rigorous mathematical definition. Additionally, I think the derivation process of the ODE equation should be more detailed, and this should also be fully reflected in the experimental section.", "experimental_designs_or_analyses": "As mentioned earlier, the derivation process of the ODE equation must include a mathematical procedure, which needs to be explained in detail in the both methd and experiment section. If there is no visualized process, this part should at least be presented in the ablation study.", "relation_to_broader_scientific_literature": "The latest work by [1] suggests that diffusion models do not necessarily require the noise addition phase.\n\n\n[1] Sun, Qiao, et al. “Is Noise Conditioning Necessary for Denoising Generative Models?” arXiv, 2025.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "The study introduces a flow-matching diffusion model for social recommendation systems, highlighting the potential of generative models in capturing user interactions and social graph structures. Through extensive experiments, it demonstrates RecFlow's significant advantages in improving recommendation accuracy and diversity compared to existing models.\nIn addition, I believe the color scheme of the figures in the paper is inconsistent, and it doesn't align well with the main content. Some explanatory notes should be added to enhance the readability of the article.", "other_comments_or_suggestions": "I believe it would be beneficial to compare this approach with other denoising techniques, such as some heuristic denoising methods, to explain why flow matching performs better. Additionally, it would be even more helpful if a case study could be included to illustrate how different types of noise (isotropic and anisotropic) affect performance.", "questions_for_authors": "Why was the method based on stochastic differential equations (SDEs) not adopted in this context?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Hx27exx7es", "reviewer": "Reviewer_avbL", "summary": "The paper introduces a generative model for social recommendation systems, using flow-matching to efficiently handle noise in social networks while preserving relational structures. It provides a thorough theoretical analysis and experiments demonstrating its superiority, fast convergence, and better fitting performance across various data types.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1.Consistent with the denoising objective ?**\nThanks! 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.\n\n**2.Types of noise?**\nThanks！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.\n\n**3.After denoising, can the representation align with the collaborative signals?**\nYes, 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.\n\n**4.A more detailed explanation of the flow-matching process is needed.**\nThanks！We will include a more detailed explanation of the flow-matching process in the revised version, particularly focusing on the sampling process. \n\n**5.Advantages of the flow-matching method itself**\nThanks! 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. \n\n**6.How does flow matching specifically enhance performance within social recommendation scenarios?**\nFlow 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.\n\n**7. Time complexities in the experimental section.**\nThanks! We will include this comparison in the revised version regarding training time.\n| Model    | Ciao | Epinions |\n|----------|------|----------|\n| RecDiff  | 4.5s | 9.1s     |\n| RecFlow  | 4.3s | 8.7s     |\n\n\n**8.Recall rate metrics.**\nThanks！ 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. \n\n**9. Figures, color scheme, coordinate system distorted**\nThanks! In the revised version, we will make an effort to modify the figure for better clarity.\n\n**10. Different sampling methods**\nThanks! 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).\n| Model        | Ciao Recall | Ciao NDCG | Epinions Recall | Epinions NDCG |\n|--------------|-------------|-----------|-----------------|---------------|\n| Euler        | 0.725       | 0.438     | 0.486           | 0.341         |\n| Runge-Kutta  | 0.720       | 0.435     | 0.483           | 0.339         |\n\n\n**11.Whether applying generative models to both the user and bipartite graph**\nThanks！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.\nWe 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.", "claims_and_evidence": "1. Is the denoising process described in the paper consistent with the denoising objective of generative models? \n2. Given that noise can vary in type and intensity, how does the paper address these different objectives? \n3. Do diffusion models have the ability to differentiate between different types of noise? \n4. After denoising, can the representation align with the collaborative signals?", "methods_and_evaluation_criteria": "NA", "theoretical_claims": "1. A more detailed explanation of the flow-matching process is needed. \n2. The advantages mentioned in the paper seem to be primarily focused on the flow-matching method itself, rather than specifically in the context of social recommendation. \n3. How does flow matching specifically enhance performance within social recommendation scenarios? More clarification is required to better understand its application and benefits in this domain.", "experimental_designs_or_analyses": "1. Since the theoretical section analyzes how flow-matching methods have lower time complexity compared to DDPM-based methods, it would be useful to compare their time complexities in the experimental section. \n2. Regarding convergence analysis, for recommendation tasks, in addition to the convergence curves, it would be helpful to compare recall rate metrics as well.", "relation_to_broader_scientific_literature": "No.", "essential_references_not_discussed": "All discussed yet.", "other_strengths_and_weaknesses": "1. The figure in Introduction only reveals the direction of the arrows when zoomed in, which adds an extra reading burden for the reader.\n2. The color scheme of the figures in the paper is inconsistent, and it doesn't align well with the main content. Some explanatory notes should be added to enhance the readability of the article.", "other_comments_or_suggestions": "1. If different sampling methods are utilized, they should be compared in the experimental section. \n2. Also, is the coordinate system in the introduction distorted?", "questions_for_authors": "Applying generative models on the user graph and on the product graph, as in Jiang, Y., Yang, Y., Xia, L., & Huang, C. (2023). *DiffKG: Knowledge Graph Diffusion Model for Recommendation*. arXiv preprint arXiv:2312.16890. https://arxiv.org/abs/2312.16890, as well as applying generative models on the bipartite graph—could these approaches form a unified generative framework for graph-based recommendations? Have you made any attempts in this direction?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UqCRpQKl8S", "reviewer": "Reviewer_ZxUy", "summary": "1. While there have been many prior works on generative recommendation systems, few have explored the direction of noise. This paper addresses the challenges posed by noisy social networks. \n2. It provides a detailed theoretical explanation of the advantages of the flow-matching model, particularly in comparison to DDPM. \n3. The paper also includes comprehensive experiments that demonstrate the various strengths of the model.\n In a word, RecFlow effectively addresses the anisotropic noise problem in social recommendation through flow-matching, outperforming existing methods in terms of various perspectives.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Differences between redundancy and errors**\nThanks! 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.\n\n**2. More detailed explanation about using generative models rather than traditional denosing models.**\nThanks! 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.\n| Model    | Ciao Recall | Ciao NDCG | Epinions Recall | Epinions NDCG |\n|----------|-------------|-----------|-----------------|---------------|\n| GDMSR    | 0.560       | 0.355     | 0.368           | 0.241         |\n| DSL      | 0.606       | 0.389     | 0.365           | 0.267         |\n| RecFlow  | 0.725       | 0.438     | 0.486           | 0.341         |\n\n[1] Denoised Self-Augmented Learning for Social Recommendation\n\n\n**3.ODE sampling method.**\nThanks! We agree, and we present it as follows, which will also be added in our revised paper.\nEuler method:  \n$$ y_{n+1} = y_n + h f(t_n, y_n) $$\n\nRunge-Kutta method:  \n$$ k_1 = h f(t_n, y_n), \\quad k_2 = h f(t_n + \\frac{h}{2}, y_n + \\frac{k_1}{2}) $$  \n$$ k_3 = h f(t_n + \\frac{h}{2}, y_n + \\frac{k_2}{2}), \\quad k_4 = h f(t_n + h, y_n + k_3) $$  \n$$ y_{n+1} = y_n + \\frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) $$\n\n**4.Condition and supervisory signal**\nThe 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.\n\n\n**5. Time complexity**\nThanks! 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.\n| Model    | Ciao | Epinions |\n|----------|------|----------|\n| RecDiff  | 4.5s | 9.1s     |\n| RecFlow  | 4.3s | 8.7s     |\n\n\n**6. Social homogeneity.**\nThanks! 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. \n[1]A Directional Diffusion Graph Transformer for Recommendation\n[1]Denoising Diffusion Probabilistic Models", "claims_and_evidence": "1. Could you explain how redundancy differs from errors in this context? \n2. I would appreciate a more detailed explanation of how generative models outperform traditional denoising models, if possible, could you provide a small experiment that demonstrates this in action?", "methods_and_evaluation_criteria": "NA", "theoretical_claims": "I would appreciate a more detailed explanation of the ODE sampling method and the conditions mentioned in the model section.", "experimental_designs_or_analyses": "The theoretical section is already presented quite thoroughly, but the experimental section needs to align with the time complexity discussed in the theory. It would be helpful to directly compare the theoretical time complexity with the experimental results to validate the claims.", "relation_to_broader_scientific_literature": "NA", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "1. The paper is well-structured, with each section clearly explained, and the theoretical content is presented rigorously. \n2. However, the discussion of the problem itself could be further developed. The Introduction section should provide an explanation of the theory behind social homogeneity. While social homogeneity is widely accepted in the social graph domain, it is still important to explicitly address and clarify this concept within the paper.", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_flag": "Flag this paper for an ethics review.", "ethics_expertise_needed": ["Discrimination / Bias / Fairness Concerns"], "ethical_review_concerns": "NA"}}]

Author rebuttal:

Meta-review: All reviewers give positive scores. The paper is well-structured and the experimental evaluation is comprehensive. Extensive performances prove the promising performances of our RecFlow from six aspects, including superiority, effectiveness, robustnesses, sensitivity, convergence and visualization.
Decision: Accept (poster)