Low-pass Personalized Subgraph Federated Recommendation
Authors: Wooseok Sim
Hogun Park
Venue: ICLR 2026 Poster (2026)
Keywords: Federated Learning
Recommender Systems
Graph Neural Networks
Primary area: learning on graphs and other geometries & topologies
TL;DR: LPSFed addresses subgraph structural imbalance in federated recommendation via low-pass spectral filtering and a localized bias-aware loss, enabling robust personalized updates that enhance both accuracy and diversity across diverse client subgraphs.

Abstract: Federated Recommender Systems (FRS) preserve privacy by training decentralized models on client-specific user-item subgraphs without sharing raw data. However, FRS faces a unique challenge: subgraph structural imbalance, where drastic variations in subgraph scale (user/item counts) and connectivity (item degree) misalign client representations, making it challenging to train a robust model that respects each client’s unique structural characteristics. 

To address this, we propose a Low-pass Personalized Subgraph Federated recommender system (LPSFed). LPSFed leverages graph Fourier transforms and low-pass spectral filtering to extract low-frequency structural signals that remain stable across subgraphs of varying size and degree, allowing robust personalized parameter updates guided by similarity to a neutral structural anchor. Additionally, we leverage a localized popularity bias-aware margin that captures item-degree imbalance within each subgraph and incorporates it into a personalized bias correction term to mitigate recommendation bias. Supported by theoretical analysis and validated on five real-world datasets, LPSFed achieves superior recommendation accuracy and enhances model robustness.

Published as a conference paper at ICLR 2026
LOW-PASS PERSONALIZED SUBGRAPH FEDERATED RECOMMENDATION
Wooseok Sim, Hogun Park
Department of Artificial Intelligence
Sungkyunkwan University Suwon, Republic of Korea
[dntjr41, hogunpark] ) (akku.ede
ABSTRACT
Federated Recommender Systems (FRS) preserve privacy by training decentralized models on client-specific user-item subgraphs without sharing raw data. However, FRS faces a unique challenge: subgraph structural imbalance, where drastic variations in subgraph scale (user/item counts) and connectivity (item degree) misalign client representations, making challenging to train a robust model that respects each client's 's unique structural characteristics. To address this, we propose a Low-pass Personalized Subgraph Federated recommender system (LPSFed). LPSFed leverages graph Fourier transforms and low-pass spectral filtering extract low-frequency structural signals that remain stable across subgraphs of varying size and degree, allowing robust personalized parameter updates guided by similarity to neutral structural anchor. Additionally, we leverage localized popularity bias-aware margin that captures item-degree imbalance within each subgraph and incorporates it into personalized bias correction term to mitigate recommendation bias. Supported by theoretical analysis and validated on five real- world datasets, LPSFed achieves superior recommendation accuracy and enhances model robustness.
INTRODUCTION
Federated Recommender Systems (FRS) play crucial role in preserving privacy while maintaining recommendation quality. For example, large e-commerce platforms (e.g., Amazon, eBay) can treat user-item interactions via subgraphs, where each client corresponds
specific country region
and accesses only localized data. Under Federated Learning (FL) framework, models exchange parameters instead of raw user data, significantly enhancing data privacy (McMahan al.) 2017). Existing FRS approaches include matrix factorization-
methods, e.g., (Chai et al. 2020).
personalized models (Liu al., [2022, Zhang al.] 2023at [i et al., 22044), and graph-based methods, e.g., FedPerGNNN (Wu et al. [2022] Qu et al.3 that construct subgraphs at the server; however, these methods largely assume comparable client subgraph structures. Meanwhile, decentralized learning naturally introduces substantial heterogeneity across clients from variations in the size and data distributions within local datasets. Heterogeneity has been widely studied in other FL tasks. In image classification FL, it appears through differences in local image quantities and skewed class distributions (Duan et al., 2020 Hsu et al., 2020; Wang et al. 2020a). In graph-based node classification FL., heterogeneity arises as client subgraphs differ in class distributions and class-driven graph topology, with some classes forming dense clusters and others appearing isolated or sparse
al. 2024
Kong al., 2024 Li
2023
Tan al. (025b).
In subgraph-based FRS, the key challenge is subgraph structural imbalance, significant variations in client subgraph size (user/item counts) and connectivity (item degree). This divergence is problematic for spatial Graph Neural Networks (GNNs), such as PinSage, NGCF, and LightGCN (Ying et al.
2018 Wang al. 2019) He et al. 2020), because their multi-hop message passing is highly sensitive to local topology. Consequently, when client subgraphs have vastly different structures, their locally trained models produce misaligned representations, destabilizing federated updates and degrading
"Corresponding author.

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Published as a conference paper at ICLR 2026
Size Degree Client Groups
Avg. Prequue per Sulb
Eigenvalue Distribution by Group
Perform. Comp. Under SubG Imbalance +26.25
Avg. Node Dogroe
30
8K
0.04
Redrus
Ours
Group 55
5..5%
4k
+16 4%
0.02
21
10t
30
50
100
150
200
Number of Nodes (log scale)
Eigenvalue
"Group MB
Overall
Subgraph Size Degree Variation
(b) Structural divergence via Spectrul dist.
(c) Group-wise Performance Gap
Figure I: Empirical observations from the federated recommender systems on the Maazonn-Book (He et al. 2020) dataset. (a) Subgraph size-degree variation: each point is one of 15 client subgraphs, partitioned using spectral clustering, grouped into Large-Dense (LD). M-MMMMM--Balaaaae Balanced (MB), Small-Sparse (SS) by node count and average degree. (b) Structural divergence: Laplacian eigenvalue histograms averaged over each group. highlighting distinct spectral signals. (c) Group-wise Performance Gap: NDCG@ 20 for FedAvg (McMahan) et al. 2017). PFedRec (Zhang ct al.] [2023a), FedPUB (Back et al. (002). and Ours across groups, showing that performance varies significantly depending on subgraph structure. Detailed experimental results in Table 2
recommendation quality. Most existing GNN-based FRS methods adopt such spatial architectures without explicitly accounting for structural imbalance across decentralized subgraphs. Spectral FI. methods TTan et al., 2024; Yu, 2025; Tan et al. 2025a) have been explored as alternative to mitigate client heterogeneity by aligning low-frequency structural components. However, their direct application to FRS is challenging since recommendation graphs are bipartite, lack informative node features, and exhibit severe degree skewness, unlike the homogeneous graph structures typically assumed in spectral FL studies citation social networks.
Beyond representation misalignment, subgraph structural imbalance also amplifies localized popu- larity bias in FRS. Specifically, in dense clients with high average item degree, GNN aggregations dominated by high-degree hub items, overshadowing the long-tail items. Conversely, in sparse clients with low average item degree, the scarcity of connections forces the model rely on a narrow set of relatively higher-degree items, leading to unstable training and poor long-tail learning (Abdollahpouri ct al. 2019 Mansoury et al. 2020 Gao et al. 2022 Zhang al. 2024 Lin al. 2025). As training progresses, this bias fosters a self reif-reinnorcing feedback loop, further consolidating the dominance of popular items (Chaney et al., 2018 Klimashevskaia et al. 2024). Since client isolation in federated
settings prevents the sharing of global context that could mitigate this effect, it is crucial to build adaptive strategies tailored to each subgraph's 's structural characteristics.
Figure 1 highlights three key aspects we examine on Amazon-Book (He et al. 2020) dataset in an FL setting. To simulate this scenario with controlled structural diversity, we partition the global interaction graph into 15 clients using spectral clustering, resulting in structurally distinct groups The figure illustrates that variations in client subgraph size and degree lead to pronounced spectral divergence, which in turn corresponds to a consistent decrease in NDCG@ 20 for existing baseline methods. These observations demonstrate that subgraph structural imbalance directly degrades FRS performance. By leveraging spectral information like Laplacian eigenvalue distributions, our method consistently improves performance across all client groups, demonstrating the effectiveness of spectral signals under structural heterogeneity.
Motivated by these observations, we propose Low-pass Personalized Subgraph Federated Recom- mendation (LPSFed), a robust personalized FRS framework that addresses structural imbalance through two synergistic components. (1) We apply low-pass spectral filtering to each client subgraph to extract its dominant wwwffreqencyy structural signal, which reflects the core connectivity pattern with minimal sensitivity to scale and noise. These signals are used to compute personalized structural similarities between each client and the global model, guiding adaptive parameter updates referenced against neutral structural anchor across heterogencous subgraphs. (2) We incorporate a localized popularity bias-aware margin that captures variations in item-degree distributions across subgraphs and applies personalized correction term during local update.
We evaluate LPSFed on five real-world datasets across diverse FRS scenarios and compare it against representative baselines. The results demonstrate that our method consistently achieves more stable and accurate recommendations. These findings highlight the importance of addressing structural imbalance for improving personalized federated recommendations in subgraph- hhased settings.
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2 PRELIMINARY
2.1 FEDERATED RECOMMENDER SYSTEM
In Federated Recommender System (FRS), user-item interactions are decentralized into C' distinct subgraphs [G1, ..., Gc}. For given client c, we denote its local subgraph as G. Ec), which consists of the node set V. UT and the edge set &.. Here, 2 {u,,.., UM and I = {ir ... ix represent the sets M users and N items within that client c. The recommendation task to predict a preference score Uui for unobserved user-item pairs (u, i) and generate top- K ranked list.
2.2 GRAPH FOURIER TRANSFORM (GFT)
The Graph Fourier Transform (GFT) Ramakrishma et al. 2020 Isufi et al. 2024) extends clas- sical Fourier analysis to graph data by leveraging the graph Laplacian matrix L
I- 1 AD-À, 2, where the adjacency matrix A
0
R with R E RMXN, D is the diagonal 0
R
degree matrix with D,, = Σ, Aij. and I is the identity matrix. We define the node embeddings
Z E (M++))x by stacking user UMx and item V embeddings, where D represents the embedding dimensionality. By eigendecomposition L = PAP", where P is the matrix of eigen- vectors (frequency bases) and diag((A) (Ai, A2, .2................. ... AM+NN) the diagonal matrix of eigenvalues (frequencies). The GFT of an embedding matrix Z is Z FF
Z' while the inverse GFT is: Z = F (Z) PŽ. Through these transforms, graph signals are decomposed into frequency components aligned with the graph's 's topology, enabling selective reconstruction modification for filtering and structural analysis tasks.
2.3 LOW-PASS GRAPH FILTER & CONVOLUTION
Low-pass graph filters (Nt & Maehara 2019 Yu & Qin 2020 Liu al. 2023 preserve meaningful low-frequency structures by suppressing figh-freeeency noise. The filter is defined by a simple
1 gate function, f MM+N- -
where & is the cut-off frequency. The Low-pass Collaborative
Filter (LCF) Yu et al. 2022 is applied as: LCF(Z) = (diag(f). Fz) = PPTZ, where P = P., contains the first & eigenvectors. When & = M + N, the filter becomes all-pass, retaining all frequencies. Adjusting &, LCF selectively preserves low-frequency signals while minimizing the impact of high-frequene noise.
2022)) utilizes these graph filters for efficient convolution operations, leveraging the convolution theorem (Barrett & Wilde 1960). Given an embedding matrix Z and convolution kernel k E IR. M+N graph convolution is defined as:
Low-pass Graph Convolutional Network (LGCN (Yu al.
* k FF
F (diag(k) FZ) Pdiog (i)iog(k)PTZ,
where Ny represents graph convolution, and k is the kernel in the frequency domain. Combining graph convolution with low-pass filtering results in low-pass convolution:
Z Zt,k= F (diag(f) diag(k)) FZ)) = Pd
(2)
where &, denotes low-pass graph convolution, and = k represents the truncated convolution kernel, ensuring computational efficiency by using only the first & eigenvectors, compared to standard graph convolution. Its time complexity is (n) where n denotes the number of non-zen zero elements L. In practice. & < M + N and n & (M + +N),, making the computation efficient for sparse graphs. Instead of performing full eigendecomposition, we compute only the first & eigenvectors using a Lanezos solver (Grimes et al. 1994), which leverages graph sparsity. This computation is performed once during the preprocessing stage, prior to training, and does not affect per-round update or communication costs. LGCN starts with an initial embedding layer ( followed by L graph convolution layers. Each 1-th layer updates feature maps as:
z Pdiag (1) ) TTZ
(3)
After I. iterations, embeddings pooled across all layers to produce the final predictive embeddings.
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Stage Training of Client Models
[On the Client]
Preference Prediction
Biav-Aware Margin
Rus
M
Aº
Subgraph Structural Signals Extraction
R
Stage Computing Similarity
Averaging
K
Buai Pooling MLP
Popularit Hias Information
M
Similarity Score
Stage In Server Aggregation & Distribution
Updated Weights
[On the Server]
Statistical
Pooting MLP
Pool
Personalized Weights Distribution
Random
W Weights
Cliest
Weights
Weights
Predictive Pop. hias Inf orm.
Aggregator
Prod
Graph
M
k
Figure 2: Overview of LPSFed On the Client: Stage 1 applies low-pass GCN and a localized popularity bias-aware loss to train client subgraphs. Stage 2 computes similarities between each client subgraph and server-provided random graph using structural signals. On the Server: Stage aggregates client parameters and distributes them based on personalized similarity scores. Colored arrows indicate stage-wing interactions.
3 METHODOLOGY
In this section, propose the Low-pass Personalized Subgraph Federated Recommendation (LPSFed) which encompasses three key components outlined in Figure 2 and Algorithm
Training of Client Models Applies low-pass GCN and bias-aware loss train subgraph models and extract structural signals, and localized popularity bias information.
[In Client] Stage 1:
[In Client] Stage Computing Structural Similarity Computes the structural similarity by comparing the client's structural signals against the server-provieed neutral structural anchor.
[In Server] Stage 3: Aggregating and Distributing Parameters on the Server Aggregates client parameters the server and distributes them based on personalized similarity scores.
3.1 TRAINING OF CLIENT MODELS
Client Models. (Stage of Figure Each client independently processes its local subgraph using
multi-layer Low-pass Graph Convolutional Network (LGCN) (Yu al. 2022). Within this LGCN, we insert two Multi-Layer Perceptron (MLP) modules: Pooling MLP (Fep.. merges per-layer user and item node embeddings into client-specific single representation per node:
Z Seg.. ((Z ( for Fre==1...,, I,..,,C,
(4)
where C' is the number of clients, and ( (Eq. 3 represents the node embedding matrix at layer I. Ue and V are the pooled user/item embeddings for client c, which is split into user embeddings U ZieM and item embeddings V EM++1M+..
The Predictive MLP (fop..) computes preference score from the user-item pair's pooled embed- dings, which is then converted into the final preference angle R.y:
Rey = arccosstaan(( (LUS, Vi, U O Vi))), for 1,...,C,
(5)
where Sap is a client-specific MLP that outputs an unbounded scalar preference score, which is then converted into the final preference angle Ruy bounded in the range [O, m..
Subgraph Structural Signals Extraction. We extract each client's denoised structural signals by constructing a low-pass convolution kernel distribution k from its subgraph's eigenvalue spectrum:
K
= k ky A fi.e, for I,
(6)
where A is the diagonal eigenvalue matrix of client c, and f is a spectral filter that retains only the first & components. This low-pass kernel effectively suppresses figh-freyyency noise while
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preserving the core structural patterns of each client's subgraph, which are critical for capturing meaningful connectivity and mitigating the impact structural heterogeneity across clients (Shuman et al. 2013). By integrating K into federated learning, model accurately extracts and exploits the denoised structural signals of each client's subgraph.
Localized Popularity Bias-aware Optimization. To address popularity bias, our framework utilizes the bias information in two ways: (1) as an auxiliary contrastive loss to regularize the bias embedding space, and (2) as an adaptive margin in the main recommendation loss (Zhang al.) 2022). First, user and item popularity scores (Puz Pt) are encoded into dddimensional embeddings via Funn (Pu) and foria (pt). The bias score s(-) is the cosine similarity between these embeddings, defined as s(ferar (Pu), feraa (p:)) = cos(Eu:), where the scalar Eus is the angle between the two embedding vectors. This score is first used in an auxiliary bias contrastive loss, Lisoss designed to train the bias encoders (Jpinass J
exp(cos(fu))/T)
Dbios
Σ
log
exp(cos((.o)/T)
Σ exp(cos(Eu:)/T) JEN.
(0+
where is the temperature and No, is the negative set for user Second, we use the same bias angle Éui construct adaptive margin for the main recommendation task:
M min (1 Eus, Rui), for c = l......CC.
where 7 controls the margin strength and R's enforces monotonic decrease. This locally- computed margin then refined by interpolating with a personalized global context (detailed in (Eq. 15). creating the refined margin M that is used in the Bias-aware Contrastive (BC)-loss:
exp(cos( Rusi + MCT/TT)
LBC =
log (u.i)EO+
Σ
exp(cos( + M Σ exp(cos((n..)/7) JEN,
This formulation adaptively penalizes over-recommended items and encourages long-tail exposure, mitigating localized popularity bias.
Localized Popularity Bias Information Computation. Each client aggregates its popularity bias- aware margins into a single representative value to preserve privacy and ensure effective global utilization. The average margin for client e computed
N M for co lece=,..,,C.
1
M
(10)
MN
where C, M, and N are the number of clients, users, and items, respectively. This scalar M represents the overall localized popularity bias information within each client's subgraph and is deliberately aggregated into a single non-invertible value to prevent the exposure of user- or item- level bias characteristics, thereby minimizing privacy risks. The server then utilizes M to construct the global bias context, which guides inter-client bias mitigation. In contrast, sharing the bias encoder parameters (fvarar foru..) is avoided because they encode highly fine-grain fine-grained, client- specific preference patterns. Exchanging such detailed parameters could inadvertently expose sensitive interaction information and lead to instability when aggregated across heterogeneous clients. Additional details on the procedure TRRIICLIENTMODEL in Algorithm 3 line I.
3.2 COMPUTING STRUCTURAL SIMILARITY
(Stage 2 of Figure 2 Since the server cannot access raw subgraph data, it aggregates high-level client statistics (e.g., average node/edge counts) during initialization. Using these statistics, it generates an informed reference graph GR via Erdos Erdos-Renyi (ER) or GNMK models (Erdos et al. 1960 Knuth 1977) at each global epoch, providing neutral structural anchor for similarity comparison. Under strict privacy constraints, this step may be omitted, allowing the server to build G without relying on client-specific statistics. The server performs GCN Gn and generates convolution kernel
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distribution K which is distributed to each client. Each client, without sharing its local kemnel K computes its structural similarity by using the KL-divergence (Hershey & Olsen 2007 between its local kernel K (Eq. 6) and the reference kernel K
K (i) K (i)
Pe (KL(KR K
Σ K (1)
for I...,,C
(11)
To align these scores across clients, the server normalizes the similarities via min-mas max normalization:
(pe (Pe-min(p)) Pe (max(p) min(p)) where p 1 pC].
(12)
The normalized similarity Pe quantifies each client's structural alignment the reference graph. This score guides the personalized FL. updates and mitigates subgraph structural imbalance by reducing the impact of highly divergent clients that would otherwise introduce misaligned representations into the global model. Further details on the procedure COMPUTESIM. in Algorithm
3.3 AGGREGATING AND DISTRIBUTING PARAMETERS ON THE SERVER
Initialization. In the initial communication round, the server initializes the learning environment by distributing global parameters 8 to all clients, ensuring uniform starting point 8 <- 0 for all clients c. Refer to Algorithm line
Aggregation. (Stage of Figure The server aggregates two model parameters and scalar bias predictive MLP's ss Ored (Eq. [5), and averaged margin M
signals: pooling MLP's Ooot (Eq, (Eq. Aggregation involves computing the mean across all clients:
pool C
Σ Poot Oprod
C
Σ c=1
Pred
M = T
1
E.M. M
(13)
c=1
where C is the number of clients. These procedures are run every global training epoch, as described in Global Training Loop part in Algorithm 2 line 5.
Distribution.
The server distributes updated model parameters to each client by adjusting them according to the client's s normalized similarity score Pe (Eq. [12.. This score balances the influence of the global model and the client's 's local model during update:
8 Pool. (updated)
(Opool X Pe) + (f X (1 - pe)),
g
Pred, (updated}
(Opred x Pe) + (f (1 - pe)),
(14)
M (updated)
= (M x Pc) + (M x (1 -.)).
While the updated parameters de ([updated] used directly, the distributed margin M (updated) provides
global bias context the client's next local training round. The client creates a refined margin M by interpolating this received global value with its newly-computed local margin M (Eq 8y:
My = WM UMM
(updated) + (1 w).
(15)
where E [0. 1] balances global guidance and local specificity. It is this refined margin that is then fed into the BC-loss (Eq. 19. ensuring the final loss reflects both global knowledge and structural uniqueness. For implementation details, see the procedure UPDATECLIENT in Algorithm 2 line 11.
3.4 THEORETICAL ANALYSIS
This section presents theoretical analysis of our approach to subgraph structural imbalance. We provide a dual justification: first, we prove that our similarity metric (based on filtered spectrum) is a stable measure of structural similarity, and second, serves as a powerful spectral regularizer. We demonstrate that both our metric and method are fundamentally governed by the underlying graph structure. Detailed proofs in Appendix B
analyze how our low-pass filtering method

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Algorithm 1 LPSFed:: Low-pass Personalized Subgraph Federated Recommendation
Notations.
Client e E (L,.., Server S, local/global epochs ea. eg. parameters θΛu
Opoot, Opves M
I: Server Initialization:
Initialize global model A
Distribute parameters to clients: 0%n f- AAU
3: 3) Aggregate statistics from all clients
Algorithm 2 Global Training Loop
Algorithm 3 Procedures
1:
for epoch to Eg do
# #[Stgee 3]
Proc. TRAINCLIENTMODELLC for epoch = e do Train On. kernel K with GCN return 0 K
#(Stage 1]
2: Generate a statistical random graph GR
3: Train global model with GR + K
3:
4:
for all client (in parallel) do
5:
Vin K TTRAINCLLENTMODEELG 6: Pe t- COMPUTESIM.(C, K*))
Proc. COMPUTESIM. K K #[Stage 2]
7: OAT E.. UNn
7: Compute K L-divergence K & K"
8: Normalize: Pc +-
pe-min(()
8: return
9: for all client do
9:
Oren t OAU +c+ Ogg (1 Pc)
10:
Proc. UPDATECLIENT(C, Onew)
UPDATECLIENT(C, keew)
11:
Apply updated parameters to client
Theorem 3.1 (Structural Comparison via Spectral Distributions). Let G1 (V,, and G2 (V2, E2) be graphs with nı = [V1 and n2 = [22 nodes and l communities each. Er and Eg are sets of edges of G1 and G2, respectively. Moreover, $ (< n) denotes the number of eigenvalues below the cut-off frequency x. Let K' and K² be their respective low-pass filtered eigenvalue distributions:
1
(i) Σ EE G
93 150.j (1.2).
(16)
Under Assumption 1 let Dsirict DEL (Kimee ( denote the KL-divergence between the idealized distributions K and K2 corresponding to the k-commumity structures of G1 and G2, respectively. C, c > are constants. Then, (1) The KL-divergence converges to a limiting value D with probability:
P( - DÅI e) S 44 expep( exp(-cmin(n)) e2 /8))
(17)
(2) The structural similarity is preserved with error:
C
[DI
D Danucl min
(18)
where C is a constant and 0j is the eigengap X A++
( for graph G5-
The theorem above shows that low-pass spectral comparison stabilizes cross-graph similarity. The KL-divergence between filtered eigenvalue distributions converges exponentially, with accuracy bounded by the smallest eigengap, indicating that clearer community separation yields more precise similarity. Therefore, the filtered spectral distribution provides a theoretically grounded basis for using DKL (K Ko) principled proxy for structural divergence.
Theorem 3.2 (Spectral Regularization). Let G (V,E) be client graph with n = | and the eigengap 5 = Ak+1 A > O, where l is the number of communities in the graph G. Denote the low-pass filtered node embedding and the k-community subspace embeddings by z, z" R..
respectively. Assume the raw user/item embeddings are bounded by ||U,ll2, 10ll2,|[V.l] [| r, where r > O is the embedding & norm bound.
Var(z) Var(z") C"
(19)
8
where Var(-) denotes graph smoothness and C' 32 krr..
Our low-pass filtering method serves as a personalized spectral regularizer that bounds representation variance by the eigengap 8, while our similarity metric Pe is also stable measure governed by this same underlying graph property. This shared governance by the eigengap provides the theoretical justification for using Pe (Eq. (2) core personalization weight.
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Table 1: The overall performance comparison Recall 20 and NDCG 20 five datasets, with the best scores highlighted bold and the second-best underlined. The improvement row highlights the gains achieved by our model (LPSFed) compared to the second seennddbest-peeee perffpnero model.
Datasel
Amazon-Book
Gowalla
Movielens- 1M
Yelp2018
Tmall-Buy
Model
Recall
NDCG
Recall
NDCG
Recall
NDCG
Recall
NDCG
Recall
NDCG
FodAvg
0.0642
0.0312
0.1425
0.0660
0.2454
0.1240
0.0721
0.0292
0.0317
0.0164
FedPUB
0.0633
0.0322
0.1433
0.0667
0.2558
0.1209
0.0684
0.0296
0.0333
0.0180
FedMF.
0.0153
0.0072
0.0765
0.0409
0.1679
0.0906
0.0318
0.0150
0.0122
0.0063
F2MF
0,0451
0.0225
0.0961
0.0565
0.1788
0.1120
0.0510
0.0247
0.0207
0.0132
PFedRec
0.0713
0,0242
0.1371
0.0478
0.1508
0.0997
0.0750
0.0225
0.0323
0,0170
FedRAP
0.0082
0.0090
0.0340
0.0458
0.0550
0.0389
0.0129
0.0249
0.0046
0.0052
RedPerGNNN
0.0035
0.0026
0.0958
0.0777
0.1332
0.1124
0.0252
0.0220
0.0032
0.0019
FedHGNN
0.0647
0.0298
0.1230
0.0608
0,2163
0.1031
0.0721
0.0268
0.0362
0.0171
FedSSP
0.0649
0.0356
0.1528
0.0729
0.2564
0.1265
0.0733
0.0315
0.0370
0.0186
LPSTed (BPR)
0.0643
0.0322
0.1529
0.0711
0.2604
0.1281
0.0769
0.0301
0.0362
0.0186
LPSFed
0.0738
0,0442
0.1621
0.0909
0.2646
0.1342
0,0783
0.0379
0.0385
0,0218
Improvement
3.55
1 24.2%
7 6.1%
1 17%
1 3.2%
6.11%
1 4.4%
20.3%
4.11%
1 17.2%
4 EXPERIMENTS
4.1 EXPERIMENTAL SETTINGS
Datasets. We evaluated model on five real-world datasets: Amazon-Book Gowalla (He et al., 2020), Movielens-IM (mov). Yelp2018 (yel), and Tmall-Buy (Tma). Each dataset was split into training, validation, and test sets in 8:1:1 ratio.
Baselines. Nine baselines such FedAvg, FedPUB, FedMF, F2MF PFedRec, FedRAP, Fed- PerGNN, FedHGNN, and FedSSP (MeMahan et al. 2017 Baek et al. 2023 Chai et al. 2020 Liu
al. 2022| Zhang et al.] 2023a Li al. 2024] Wu et al. 2022 Yan al. 2024 Tan et al. 2024)
are used for comparisons. These include standard FL, federated matrix factorization, personalized FRS, calresss-awarr FRS, and Spectral-basee FL methods. We used Recall @ 20 and NDCG@ 20 to measure recommendation accuracy, assessing how well the top 20 recommended items matched user interests (Wang et al., 2019).
Implementation Details. We partitioned each dataset's global interaction graph into four subgraphs using spectral clustering (Damle et al., 2019) to align with the federated recommender systems setting (Table 3, 9, 10, and Figure 3). Results were averaged over six runs across two different partitions. Random graphs were generated using the GNMK (Knuth) 1977) model, which effectively preserves degree distribution compared to the Erdos-Renyi (ER) model (Erdös et al. 1960). Further details dataset statistics, baseline models, clustering methods, random graph generation, and all hyperparameter settings are provided in Appendix D respectively.
4.2 EXPERIMENTAL RESULTS AND ANALYSIS
(RQ1) Overall Performance Comparison. Table I summarizes our comparative analysis against key graph-based FL baselines. We evaluate our framework in two settings: LPSFed(BPR), which applics a standard BPR loss (Rendle et al.) 2209), and our full model LPSFed, which applies a BC loss (Eq. 9). As shown, LPSFed consistently outperforms all nine competing baselines, achieving state- of-the-art performance. LPSFed(BPRR, leveraging our spectral personalization, already demonstrates strong performance by yielding robust personalized similarity measurements. LPSFed further incorporates our proposed bias-aware margin to mitigate local popularity bias. This synergy robustly handles both structural imbalance and localized popularity bias, delivering the significant NDCG gains (e.g.. +24.2% in Amazon Amazon-Book) through more precise top-ranked item recommendations.
(RQ2) Robustness to Subgraph Structural Imbalance. Table 2 Qqaantifies how well each method copes with subgraph structural imbalance on the Amazon-Boon The 15 clients are partitioned into three groups: Large- Dense (LD, # Nodes 40K, Avg. Degree 42.3), < Nodes < 15K, Avg. Degree 20.4), and Small-Sparse (SS, # Nodes < 8K, Avg. Degree 12.8). For each group, report mean Recall @ 20 and NDCG@ 20; the last column averages over all clients.
-aalnnee (MB, 10K
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Table 2: Performance for measuring the impact of subgraph structural imbalance the Amazon- dataset, evaluated using Recall @20 and NDCG@ 20 across a 15-client partition. Clients are grouped into three categories: Dange,Dense, -allnnedd Small-Sparse, Sparse, and Overall (averaged across all clients). The best scores are highlighted in bold and the second-best in underlined. The improvement row shows the gains achieved by our model (L (LPSFed) compared to the second-best pecon--nst-prrfomming model.
Group
Large-Dense
Ballnned
Small-Sparse
Overall
Model
Recall
NDCG
Recall
NDCG
Recall
NDCG
Recall
NDCG
FedAvg
0.0620
0.0280
0.0411
0,0208
0.0275
0.0117
0.0334
0.0152
FedPUB
0.0605
0.0355
0.0528
0.0267
0.0300
0.0123
0.0381
0.0177
FedMF
0.0176
0.0094
0.0127
0.0056
0.0058
0.0027
0.0084
0.0039
F2MF
0.0402
0.0231
0.0341
0.0174
0.0225
0.0085
0.0268
0.0118
PFedRec
0.0619
0.0250
0.0525
0.0185
0.0310
0.0131
0,0388
0.0153
FedRAP
0.0073
0.0082
0.0067
0.0072
0.0048
0.0065
0,0055
0.0068
FedPerGNN
0,0053
0.0037
0.0038
0,0018
0.0039
0.0012
0.0040
0,0015
FedHGNN
0.0630
0.0319
0.0455
0.0237
0.0283
0.0095
0.0352
0,0148
FedSSP
0.0669
0,0380
0,0524
0.0269
0.0317
0,0126
0.0396
0,0181
LPSFed (BPR)
0.0655
0.0375
0.0523
0.0269
0.0306
0.0131
0.0387
0.0184
LPSFed
0.0769
0.0448
0.0550
0.0295
0.0331
0.0146
0.0419
0.0206
Improvement
14.9%
17.9%
1 4.2%
1 9.7%%
1 4.4%
1 11.5%
1 5.8%
13.8%
Table 3: Comparison of the localized popularity bias mitigation on the Amazon Baazon-BBookkao using NDCG@ 20. Best scores are bold. Balanced data includes all data, while the imbalanced dataset excludes less active participants. Arrows show (%) change relative to FedAvg: 1 gain (%),, &. loss (%)
Data Setting
Balanced Dataset
INDCG@20
Imbalanced Dataset INDCG @
Model
Tail
Mid
Head
Overall
Tail
Mid
Head
Overall
FedAvg
0.0017
0.0077
0.0570
0.0264
0.0040
0.0108
0.0924
0.0312
FedPUB
0.0024 1:1
0.008
0.0620 19
0.0279
0.0064 1
0.0130 20
0.0998
0.0322
FedMF
0.0002 1
0,0009 188
0.0130 77
0.0074 172
0.0003 192
0.0013 ss
0.0140 as
0.0072 1
F2MF
0.0037 118
0.0112 1:45
0.0438 23
0.0225 115
0.0050 125
0.0119 10
0.0436 153
0.0225 128
PFedRRe
0.0055 224
0.0125
0.0277 L11
0,0244
0.0088 7020
0.0145
0,0262
0.0242 22
FedRAP
0.0031 1:22
0,0097 725
0.0098 133
0.0110 1.58
0.0014 1.5
0.0055 1.9
0.0107 1x
0,0090 71
FedPerGNN
0.0002 1S8
0.0007 191
0.0042 1.93
0.0015 194
0.0003 192
0.0008 199
0.0046 195
0,0026 192
FedHGNN
0.0019 12
0.0088
0.0555 L3
0.0262
0.0058 145
0.0138
28
0.0770
0.0298 J4
FedSSP
0.0038 124
0.0117
7:52
0.0640 12
0.0317 1.20
0.0067 1.68
0,0167
1:55
0,1019 10
0.0356 14
L.SS (BPR)
0.0031 1:82
0.01
1:47
0.0610 77
0.0301
114
0.0068
7.20
0.0149
3:
0.0974 15
0.0322 13
LPSFed
0.0063 271
0.0143
1.66
0,0752 132
0.0390
0.0078
195
0.0193
1:79
0.1052
0.0442 1:42
LPSFed (BPR), which leverages only spectral personalization, shows competitive performance, although its scores are slightly lower than the FedSSP. However, adding the bias-aware margin (LPSFed) yields significant performance gains across all groups and demonstrates three distinct effects. In group LD, it breaks the feedback loop, improving performance significantly over all baselines (+14, (+14.9% Recall, +17.9% NDCG over the best baseline). In group MB, the spectral-sigee siggall similarity effectively moderates update weights. allowing clients to benefit from the global bias context without overfitting to popular items. group SS, the margin reduces over-dependence on a few popular items, enhancing long-tail recommendation quality.
(RQ3) Localized Popularity Bias Mitigation. Table 3 reports NDCG@20 on Amazon-Boo under two settings: Balanced uses the full dataset, while Imbalanced excludes users and items with fewer than eight interactions. Users are grouped into Tail, Mid, and Head based on interaction proportions (3:2:1 ratio). We compare against FedAvg, and arrows indicate each model's s relative change in NDCG@2 20. Traditional FL models like FedMF and FedPerGNN perform poorly in the Tail group, indicating that they are significantly affected by popularity bias. While PFedRec and FedSSP show notable gains in the Tail, they underperform our model (LPSFed) in the head. Similarly, F2MF, which uses auxiliary features for group-based fairness, performs better than LPSFed (BPR) in the Tail and mid categories on the balanced data, but underperforms in the overall categories on the Imbalanced data. However, L.SSedd consistently achieves the best overall performance on both Balanced and Imbalanced datasets, indicating that model's bias-aware margin removes popularity noise and preserves accuracy across the entire item spectrum.
(RQ4) Ablation Study of Model Components. Table 4 evaluates the contribution of each key component. The w/o LGCN variant, where we replace core low-pass architecture with standard

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Published as a conference paper at ICLR 2026
Table 4: Ablation study of LPSFed on five datasets, reporting Recall @0 and NDCG@D 20 for different model
variants. Bold indicates best performance per column, and the last row shows relative improvements over the second-best.
Dutaset
Amazon-Bood
Gowalla
Moviclens-I1 IM
Yelp2018
Tmall-Buy -uy
Model
Recall
NDCG
Recall
NDCG
Recall
NDCG
Recall
NDCG
Recall
NDCG
wo LGCN
0,0388
0.0216
0.0965
0.0427
0.1622
0,0893
0.0471
0.0207
0.0256
0.0121
w/o bias-aware
0.0643
0.0322
0.1529
0.0711
0.2604
0.1281
0.0769
0.0301
0.0362
0.0186
w/o per
0.0652
0.0368
0.1576
0.0788
0.2625
0,1299
0.0775
0.0315
0.0370
0.0190
woo per & bias-aware
0,0642
0.0312
0.1425
0,0660
0.2454
0.1240
0.0712
0.0292
0.0317
0.0164
wio statisties for
0.0714
0.0403
0.1581
0.0856
0.2625
0.1330
0.0764
0.0372
0.0393
0.0199
LPSSed
0,0738
0.0442
0.1621
0.0909
0.2646
0.1342
0.0783
0.0379
0.0419
0.0206
Improvement
t 3.4%
9.77%
2.3%
1 6.2%
0.8%
1 0.9%
1.0%
1.9%
1 6.6%
3,5%
Effect
Bias Amplification (1)
Avg. Jaccard Index
Effect
sensitivity
NDCG@20
0.045
Magkny
NDCG@20
20
0.044
3.000
0.041
Inbelanced
0.035
0.5
0,918
Balanced
6.000
Balanced Imbalanced
Iterations
16
32
128
256 0.04 0.66 008 0.10 0.12 6.14
(a) Impact o
(b) Bias Amplitication
(c)
Impact of +
(d) Impact of
Figure 3: Hyperparameter sensitivity on the Amazon- Book dataset: (a) Bias-aware margin strength y; (h) Impact
7 on Bias Amplification in Imbalanced set; (c) Low-pass cut-off frequency Φ; (d) Loss Temperature т.
spatial GNN (NGCF [Wang al. 2019)), results a catastrophic performance drop, which highlights the inherent limitations of spatial-based methods in handling structural divergence. Removing the personalization component, denoted w/o per, the bias-aware margin, denoted w/o bias-aware, significantly degrades performance, confirming both modules are essential. Notably, w/o statistics for GR variant enforces stricter privacy by not aggregating any client statistics. Despite this, it performs robustly, achieving the second-best results across most metries. Only the full LPSFed achieves consistently high Recall and precise rankings, confirming that these components enhance both broad coverage and accurate prediction.
(RQ5) Hyperparameter Analysis. Figure shows how key hyperparameters affect NDCG@2I 20 on Amazon-Booe Book under Balanced and Imbalanced settings. (a) Increasing the bias-aware margin strength consistently raises NDCG, as a stronger margin curbs the localized popularity bias and supports long-tail items. This indicates that maximizing the margin strength is the most effective strategy for achieving the intended bias mitigation objective. (b) Higher suppresses the feedback loop by reducing recommendation overlap, thereby limiting popularity amplification. Specifically, 1.0 yields the slowest growth in the Jaccard index, confirming that a stronger margin most effectively disrupts the popularity-driven feedback loop. (c) For the low-pass cut-off frequency &, performance improves more informative structural signals are captured, peaking at & 128, where the cut-off preserves the most spectrally stable frequencies while excluding high-frequency noise that would otherwise reduce robustness. (d) The BC-Loss temperature shows similar pattern: appropriate settings strike a balance between stable optimization and sufficient exploration, while values that too low hinder learning and those that are too high result in overfitting, both leading to lower NDCG.
5 CONCLUSION
In this paper, we introduce Low-pass Personalized Subgraph Federated Recommendation (LPSFed). a robust personalized FRS that simultaneously addresses subgraph structural imbalance and localized popularity bias. Our approach leverages low-pass spectral filtering for stable personalization, while bias-aware margin mitigates feedback loops and improves long- tail recommendations. We provide theoretical justification for this framework, demonstrating that our similarity metric and our spectral filtering method are both governed by the same underlying eigengap. which validates our person- alization strategy. Empirical evaluation five real-world datasets confirms that this synergistic framework achieves of-te-of-the-ar performance and robustness, outperforming existing baselines. Further discussions on the limitations and LLM usage are provided in Appendix and
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ACKNOWLEDGEMENTS
This work was supported by the Institute of Information and Communications Technology Plan- ning and Evaluation (ITTP), the Korea Creative Content Agency (KOCCA), and the National Re- search Foundation of Korea (NRF) through grants funded by the Korean government (RS-2024- 00438686, RS 2S$2022--00336966, STTP-2026RSS R---RRS222--10115 RS- 1119-11190421, IITP-2025- 2024-00360227, RS- ESS-2025-02218777 RS-2025 2025-25442569 24883358...
ETHICS STATEMENT
This research was conducted in accordance with the ICLR Code of Ethics. The focus of our work is the development of robust, privacy-preserving federated recommender systems.
Privacy and Security. A primary ethical consideration in recommender systems is the handling of sensitive user data. Our federated learning framework directly addresses this challenge. By training models decentralized, client-level subgraphs without sharing raw user-item interaction data, our method significantly enhances user privacy and reduces the risk of data exposure inherent in traditional centralized systems. The information shared with the server is limited to model parameters and minimal, non-invertible scalar values (c.g., structural similarity scores), which are designed to prevent the reconstruction of individual user data local graph structures.
Fairness, Bias, and Discrimination. Recommender Systems can perpetuate and amplify existing biases, leading to unfair outcomes. Our research directly confronts this issue by proposing method to mitigate localized popularity bias. By improving the quality of recommendations for diverse and structurally varied client groups, our work aims to promote fairness and provide more equitable exposure for long-tail items. This helps to counteract the feedback loops that often lead to narrow, popularity-dee recommendation landscape, thereby fostering a more diverse and inclusive ecosystem.
Data Usage and Research Integrity. All experiments in this paper were conducted using publicly available and widely adopted benchmark datasets. These datasets consist of anonymized user interactions, and our work did not involve the collection of new data from human subjects. We are committed research integrity and have provided detailed descriptions of our methodology and experimental setup to ensure reproducibility.
REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our work, provide our source code as supplementary material. Our theoretical analysis is presented Section with detailed proofs for all theorems supporting lemmas provided in Appendix B Our code is available at: https: //github. com/ dnt /r LPSFed All experimental settings are described Section 4.1 and Appendix [ which includes dataset statistics, the client subgraph clustering methodology, baseline model details, and complete list hyperparameters.
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An Zhang, Wenchang Ma, Pengbo Wei, Leheng Sheng, and Xiang Wang. General debiasing for traph-based collaborative filtering via adversarial graph dropout. In Proceedings of the ACM Web Conference 2024, pp. 3864-3875, 2024.
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APPENDIX GUIDE
The appendix includes the thcoretical analysis, related work, experimental setup, training efficiency, complexity analysis, additional experimental results, broader impacts, limitations, and LLM Usage. This guide provides a concise overview of its sections for easy navigation.
A NOTATIONS
Table of Notations A List all key notations and symbols used in the paper.
THEORETICAL ANALYSIS
B
Lemma T (1 [Loo-pass Filter Preservation): Shows low-pass filters preserve graph community
structures with an eigengap-basee error bound.
Lemma 2 (Spectral Measure Convergence): Proves eigenvalue distribution convergence in large
graphs.
Lemma (Filter Stability): Establishes low-pass filter stability under Laplacian perturbations.
Assumption (Idealized Distribution) Formalizes the idealized low-pass spectral distributions.
Theorem 3.1 (Structural Comparison): Compares graph structures via KL-divergence of eigen- value distributions.
Theorem 3.2 (Spectral Regularization): Bounds variance of filtered node embeddings for com- munity alignment.
RELATED WORK
Examines collaborative filtering and personalized recom- mendation methods in federated settings.
Federated Recommender Systems C.1
Personalized Federated Learning C.2 Covers approaches for handling data heterogeneity across
clients, particularly in vision and graph domains.
Popularity Bias-aware Recommender Systems
C.3 Discusses methods for mitigating popularity
bias in centralized recommendation settings.
DATASETS AND EXPERIMENTAL SETUP
Datasets D.I 6T
Details Movielens- 1M, Gowalla, Yelp2018, Amazon-Book, Tmall-Buy datasets
Subgraph Clustering D.2
Describes clustering methods for generating client subgraphs in the
federated setting (Table 7)
Random Graph D.3
Describes random graph generation in a federated setting.
Random Graph Sensitivity D.4
Random Graph Sensitivity Analysis (Table 8 Lists baseline models for comparison.
Baselines D.5
Hyperparameter Settings D.6 Summarizes configuration settings.
TRAINING EFFICIENCY & COMPLEXITY
Model Complexity t baselines.
Demonstrates I.PSedd achieves higher efficiency than state-of-the-art
Time Complexity:
LLw-pass Convolution E.1.1
Discusses efficient low-pass filtering and graph convolution
operations.
-LPSFed Framework
Analyzes the overall complexity of the federated learning system. Validates the framework's 's support for large-scale graphs structures.
E.1.2
Scalability E.1.2
Costs E.1.2
Assesses communication and computation overhead (Table
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F ADDITIONAL EXPERIMENTAL RESULTS
Bias Amplification Measurement and Effectiveness of the Bias-Aware Margin F11 Shows bias
mitigation via Jaccard index, with higher margin strengths reducing convergence (Table TO
Correlation Analysis Between Eigengap and Performance F2 Analyzes correlation, confirming
spectral stability benefits within an appropriate filtering range (Table [T].
Effect of Client Variability on Performance F3 Evaluates performance across client counts (4,
10, 20), proving robustness to heterogeneity (Table 12
G BROADER IMPACTS
Broader Impacts G Summarizes societal benefits such as privacy, fairness, and collaborative learning from subgraph-level federated recommender systems approach.
H LIMITATIONS
Limitations
H
Discusses the limitations of this paper.
1 LLM USAGE
LLM Usage |
Details the use an LLM for manusctipt preparation.
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Table Table of Notations.
Symbol
Description
General Notations
C
The total number of clients
An index for specific client, {1, C}
u,i
user an item, respectively The sets of all users and items in a given graph The number of users and items in given graph The Dimensionality of embeddings General model parameters
U.I
M,N
0
Graph Representation
C.C.
A global graph and a client's local subgraph
N,E
The sets of nodes and edges in graph
A, D.
The adjacency. Degree, and Laplacian matrices of graph The user-item interaction matrix
R
A
The i-th eigenvalue and the diagonal matrix of eigenvalues
The matrix of eigenvectors the Laplacian
Low-pass Graph Convolutional Network
Z. V
The node, user, and item embedding matrices
2
The node embedding matrix at the I-th layer The cut-off frequency for the low-pass filter The truncated matrix of the first & eigenvectors
P
k
A convolution kernel in the frequency domain and its truncated version The pooling and Predictive MLPS
formnt t foreed
Bias Mitigation
Pa:Pi
Popularity scores for user and item
Soniast Sésias
Encoders for user and item popularity bias
Evi
The angle between user and item popularity bias embeddings The auxiliary bias loss and the main Bias-aware Contrastive (BC) loss A locally computed adaptive margin for user u, item client
Liass LRC
M
M
The refined margin after interpolation with the global context
7s T.W
Hyperparameters: margin strength, softmax temperature, and interpolation weight Federated Learning (Low-pass Personalized Subgraph Federated Recommendation)
K"
The low-pass convolution kernel distributions for client and the reference graph Gn The structural divergence of client (KL-diivegeenc)
Pc
Pe
The normalized similarity score of client
M
A clicnt c's average margin and the aggregated global average margin Updated (personalized) parameters for client
ge Vipdated
M Mupdated
The updated (personalized) global margin for client
Theoretical Analysis
The eigengap of the graph Laplacian (Ax Az)
Dstract Var(z) z"
The ideal structural divergence between theoretical k-block graphs
The graph smoothness (Dirichlet energy) of an embedding z The theoretical k-community subspace embedding
A NOTATIONS
Table summarrize the key notations used throughout the paper.
B THEORETICAL ANALYSIS
Lemma (Low-pass Filter Preservation). Let G be an undirected graph with normalized Laplacian
L= I D 1/2 111 A12 and k communities. For low-pass filter hxx, (L) with cut-off frequency X between d and At-1:
8v A
h1, (L)x 11 commmnnit x/12 S
Av ||x||2,
(20)
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where Iccommunity is the projection onto community indicator vectors.
Proof. Let L = PAP be the eigendecomposition of L, where A diag(>i,.... An) with 0
A S A2 ........ An < 2. The proof proceeds in four steps:
First, let X E R be the matrix of true community indicators. By the variational characterization of eigenvalues, the first k eigenvectors minimize:
min
txx LX
Σ
A
(21)
X
1=1
Second, let Pj be the matrix of first & eigenvectors. By the Davis KahisnKahannan theorem Davis & Kahan 1970), when the eigengap A++1
X₂ 0:
8vk AR+1 A
P& XRIFF S
(22)
where R is an orthogonal matrix that best aligns PA with Then, the low-pass filter hx (L) with An (Ax, Ab+1) acts as:
1x. L)
Σ P.P. = P.P.
(23)
Lastly, for any signal x, using the projection Iccomunityy
Tx= P XX T x112
||/,, (L)x A.oommnnttyx.l
(24)
(PaP] xRR x )x|12
(25)
S Pk - -XR ||x||2
(26)
8v
|
(27)
Ax+1 Ak
Lemma 2 (Spectral Measure Convergence). Let Gn a graph with vertices and a normalized low-pass filtered eigenvalue distribution with cut-off frequency Ne. Let & (< n) denote the number of eigenvalues below No. For this distribution:
(n)
Kii1))
1
(28)
ΣS1 X
Then there exists a limiting distribution K such that
P( K* K* > e) S 20 20
(29)
where c, > are constants.
Proof. The proof proceeds in steps: By the Matrix Bernstein inequality normalized Laplacians:
Tropp et al., 2015) for
nt?
P[ EL"I| 2 t) < 2n exp(
(30)
Second, let L* LE be the limiting operator. For eigenvalues, Weyl's inequality gives:
p (m)
X < L
L
(31)
For the normalized distribution K.:
(n)
K (1) K (1)l
(32)
Σ Σg=1 Agg
(n)
Σ 1=1
x
1 (n)
X
Φ
+
(2 (m)
x;)-
(33)
1= X (n)
X
(m))²
1=1
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Since eigenvalues of normalized Laplacians lie in [0,2] and & is fixed:
c1
(34)
Therefore, for any >>
K" (i) - kK(()) S CI -
(35)
By setting t = c/C1 in Eq. 30 and applying the union bound over i - 4:
P( K" ac > c) S 24 exp( cncc))
(36)
where c = 1/(4
Lemma 3 (Filter Stability). For low-pass filter hx. (L) and perturbed Laplacian
Laplacian == L E:
E
|/,, - 1xx (E) S
(37)
5x.
where d2g is the minimum gap between eigenvalues separated by Ao
Proof. We prove this in three steps using complex analysis: First, express the filter difference using the Cauchy integral formula:
2Ti /. (wI - L) E(wI - -dd:
h (L) hrs
(38)
where I is contour enclosing eigenvalues below Ac.
For the resolvent norm, when w is on I:
[wI -) dist(w w, o (L))
(39)
dx.
where (L) is the spectrum of L. By taking operator norms:
||, (L) hx. (f)]l2
(40)
f |I(wI
2 [wIl L) [llld//
(41)
E||2
(42)
8x.
The final inequality uses the fact that the contour integral equals 2ri for the characteristic function of (-o0, Ae]-
Assumption 1 (Idealized Low-Pass Spectral Distributions). For each graph {1,2), let
G (Vj, Ej) be undirected (possibly bipartite) graph with normalized Laplacian L and eigenvalues 0 = x < x(6) < ... S (i) < 2. Fix a common low-pass
cut-off d (0,2) (for bipartite graphs choose de < 2), and let Φl3) = max(m x( < Ae}. Define the empirical low-pass eigenvalue distribution:
X()
K E 11-1-1
K()(1) = -00 ()
is Φb) 15)
(43)
Idealized model. We assume G, is generated by k-community structural model that induces an idealized (expected) graph: either
(SBM) k-block (bi)SBM with community proportions ( and block matrix Bli), with expected adjocency A (f), E[A (1) or
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(Graphon) pieceeiise-constnn k-block graphon W,, with associated integral operator's discretization A
Let L (
struct be the normalized Laplacian of A (1)-, with eigenvalues 0 == =>>snas S S Astruct,, and
( strnec max(m: (i)
m
>e
The idcalized low-pass distribution is:
( Struet,
K struee (1)
& (J) strucr
(44)
26
G) strict.
-==1
Structural divergence. The population (idealized) structural divergence between graphs a, b is:
DKu(K [
D struct ==
(45)
where PKL is computed after e-smoothing zero coordinates in the denominator (i.e., replace any zero q: by max[qi, E) for a fixed ε >> 0).
Theorem 3.1
(Structural Comparison via Spectral Distributions). Let G1 &V1 and G2 [V and n2 \22 nodes and & communities each. E₁ and &g are sets
(V2, E2) be graphs with of edges of G₁ and G2, respectively. Moreover, (< n) denotes the number of eigenvalues below the cut-off frequency X. Let K and K2 be their respective low-pass filtered eigenvalue distributions:
1
(1)
9 の E (1,2).
(46)
E ND
Under Assumption let Daruct DKL (K) struct K2 area) denote the KL-divergence between the idealized distributions K sinee and K22
struct corresponding to the k-community structures of Gı and G2, respectively. c, 0 are constants. Then, (1) The KI--divergence converges to limiting value D with probability:
P(IDKZ(K D"I <) S 44 exp(-emin(n)) n2) /8).
(47)
(2) The structural similarity is preserved with error:
C DKL (KL(K* Dstmeel (in 52)
(48)
where C' is a constant and 6y is the eigengap K+1 x for graph Gj.
()
Proof. For each graph Gj. by Lemma 2
P([[K) K2 >
>6) e) 2 exexp(-cnne
(49)
Meanwhile, for the KL-divergence, we decompose:
(KL(K-KK3) (Kz(KK
(50)
< <|KKL IK3) DEL (K! K2
(51)
+ |DKL (K! DrL [K!
(52)
Then, since K' are probability distributions bounded away from (due to low-pass filtering), we can apply the Lipschitz property of KI. KL--dieergence:
DK I - DKL (K! I
(53)
S SCIJK - K + |K2 - K?
(54)
For each graph, Lemma gives the probability bounds for deviations:
P(J K kl E) < 2 exp( exp( =
(55)
P( K2 22
K
c) < 2 exp( 2
(56)
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By the union bound, the probability of either distribution deviating is bounded by:
P K > cor K K > c)
(57)
- 2 exp(-cnn1 + 2 exp( -CnzE?)
(58)
3 20((xp((cmne + exp( (
(59)
s 4 (
(60)
The final bound uses the fact that:
exp(-cm1e) exp(-cnze?) S 2exp((<mmin(tn
(61)
For the structural preservation bound, we proceed in several steps: First, from Lemma | for each graph Gj. the low-pass filter preserves community structure with error:
8v
8V
|1/ 15)x
community x|12 S
|x||
|
(62)
G
A4t1
By Lemma 3 when the Laplacian is perturbed by E, the filter output changes by at most:
E|l2
|/,,, (L') hx. (I S
(63)
85
Similarly, this perturbation affects the filter output by:
Co
||,,, (LP) hv, (IA S
(64)
85*
For the eigenvalues, this implies:
Co 0
(2 - struet <
(65)
Let K be the distribution that perfectly captures the community structure. Then:
K Ka 1o I Ci 05
(66)
where C₁ depends on the number of communities k. For the KL-divergence between distributions:
DKL(K*
K2) DKL DKL(K)nu 3 K K)naa)
(67)
Cz (| K
K Aineeeloo + ||2 K2 Kl
(68)
Cz C1 Ci
+
(69)
81
82
201C₂
(70)
min(01,00) 02)
By defining Damet
DKL (K) J ( and letting C Cg:
C
DeL (K (K) K?) Datueel
(71)
min 82)
This bound shows that the KL-divergence between the filtered distributions approximates the true structural similarity up error controlled by the minimum eigengap of the two graphs.
n
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(Spectral Regularization). Let G EV,E) be client graph with | and the eigengap Ax+1 - x > O, where is the number of communities in the graph G. Denote the low-pass filtered node embedding and the k-community subspace embeddings by z R" respectively. Assume the raw user/item embeddings are bounded by ||Un]|2,) [Vil2 S r, where > 0 is the embedding & norm bound.
Theorem 3.2
Var(z) Var(z") C
(72)
8
where Var(-) denotes graph smoothness and C" = krr
Proof. By Lemma ! (Low-pass Filter Preservation), the raw embedding x E R" represents the initial user/item embeddings with ||x||| Sr. The k-community subspace embedding z" reflects the community structure of the graph and is defined as:
community x,
(73)
where Iggummmmit is the projection onto the community subspace, spanned by the smallest & eigen- vectors of the graph Laplacian L. Since z a projection of x, we have ||z||2 I ||x||| I r. This clarifies that z represents projected into the low-frequency k-community subspace.
The Low-pass filtered embedding is given by z = hx, (L)x, where h,, (L) is low-pass filter that preserves components associated with small eigenvalues of L (low frequencies) and attenuates those the approximation error is bounded as:
with large eigenvalues (high frequencies). By Lemma]] !
8vk ||x|2
8vk 8 r,
2 22
(74)
where 5 A A is the eigengap. This shows that approximates the embedding z".
The Dirichlet energy difference is:
Var(z) - Var(z*)] 2 (z')TIz*) Lz
(75)
Let A = z - zz, with || I SVR Expanding the difference:
8
Lz (z' + A) A L(z" + (x') Lz" + +2((²) LA LA
(76)
80:
Var(z) Var(z*)) = 12(z")
LI LA LAA.
(77)
We bound each term:
Cross term:
LL & |z*||e|LLA|| 1/2 - 12'122 |L||2 ||
(78)
Since ||l|| I 2 (for a normalized Laplacian).
||z || I r. and | - SyEr,
(z') 5r. 8 8VB. r 16VTr2
(79)
32
2((x")TLA LA S 5
(80)
Quadratic term:
8VT
128k7
ATLA
< |
<2
(81)
8
82
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Combining these:
32vkr²
128kr²
[Vr(()) Var(z")1
(82)
8
82
For large 6, the 3 term dominates, yielding:
Var(zz) Var(z") ) C²
C' 32
(83)
C
RELATED WORK
C.I FEDERATED RECOMMENDER SYSTEMS
Federated Recommender Systems (FRS) enhance privacy by processing data locally while sharing only model updates. Methods like FedMF (Chai al. 2020) and LightFR (Zhang al. 2023b use federated matrix factorization for global collaborative filtering without data leakage, while F2MF Liu al. (2022) integrates user/item features for fairness. Additionally, for e-commerce platforms. (Park et al.) 2026) introduces a data valuation framework to quantify the utility of decentralized data during the federation process. To improve recommendation relevance, recent research has shifted toward Personalized Federated Recommendations (PFR), which tailor predictions to individual client preferences. For instance, PFedRee (Zhang et al. 2023a) integrates dual personalization strategies, and FedRAP (Li et al. (024) introduces adjustable personalized layers. Although PFR frameworks address personalization, many treat clients (users) independently. limiting their ability to capture high- order interactions inherent to Graph Neural Networks (GNNs). To alleviate this, existing methods have explored ways to incorporate relational structure. Notable examples include FedPerGNN (Wul al. 2022), which expands subgraphs via third-party server, FedHGNN (Yan al., 2024) utilizes heterogeneous GNN to handle diverse data relationships, and SemiDEFGL (Qu et al. (003)) augments ego-graphs with synthetic common items. However, these approaches mainly focus on augmenting the local neighborhood. Furthermore, F2PGNN (Agrawal et al. (024) addresses the fairness problem; it relies on additional feature information beyond user-item interactions. Importantly, neither leverage graph spectral signals, which capture fundamental structural patterns, nor explicitly address the subgraph structural imbalance caused by variations in client subgraph scales and connectivities.
C.2 PERSONALIZED FEDERATED LEARNING
Personalized Federated Learning (PFL) extends standard FL. by incorporating client-specific adapta- tions to handle heterogeneous data distributions. In vision tasks, heterogeneity driven largely by disparities in local data sizes. For example, FedVC (Hsu et al. 2020) reweights and resamples clients to address data-size gaps, Astraea (Duan et al. 2020) employs augmentation and rescheduling to self-balance imbalanced datasets, and FFL Li et al. (020) uses faimess loss to reweight underrep- resented clients. In graph-based node classification tasks, heterogeneity arises from class imbalance and varying class-driven graph topology. FedPer (Arivazhagan et al., (019) and FedSim (Pali- hawadana et al. (2022) address this by using adaptive layers and similarity-guider aggregation, while G-FML (Yang al. 2023), FedGSL (Zhao al.] (2022), and FedCog (Lei al. 2023) leverage subgraph augmentation and meta-lcarning to align diverse local graphs. Similarly, CUFL (Kang & Park (025) introduces a curriculum guided strategy to adaptively personalize subgraph FL under data heterogeneity. Additionally, FedPUB (Back et al., (223) clusters clients based on random-graph similarity, and FedSSP (Tan et al.) 2024) and S2FGL (Tan et al., (2025a) apply low-pass filters for graph and node classification. However, existing PFL methods for graph data have largely focused on these types of heterogeneity, whereas the core challenge in recommender systems is the structural imbalance of interaction-only subgraphs that vary widely in size and connectivity. As result, they fail to address the subgraph structural imbalance characteristics of FRS, and are ineffective at mitigating structural divergence across clients.
C.3 POPULARITY BIAS-AWARE RECOMMENDER SYSTEMS
GNN-based Recommender Systems, such as NGCF Waag et al. 2019), DGCF (Wang et al. 2020b).
and DHCF (Ji et al. 2020), have significantly improved recommendation accuracy in centralized
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Table 6: The statistics of datasets.
Dataset
Movielens-1M
Gowalla
Yelp2018
Amazon-Boo Booz
Tmall-Buy
Number of Users
6,040
29,858
31,668
52,643
885,759
Number of Items
3,900
40,981
38,048
91,599
1,144,124
Number Interactions
1,000,290
1,027.370
1,561,406
2,984.108
7,592.214
Density
5.431%
0.084%
0.130%
0.062%
0.010%
Table 7: Comparison of different client construction strategies on Movielens- IM.
Method (Aug. + std)
Ego-graph
Random-Come
Interconnected-CCe
Spectral-Clust
Avg. Subgraph Size
141.1 + 16.8
172.8 1: 9.8
2106.6 1 135.7
196.2 1 36.2
Size variance
22276.8 8 + 6905.9
3198.5 +: 1039.8
85083.4 + 45597.3
33370.3 + 17210.8
Avg. Degree
1.97 + 0.0
5,68 + 0.24
7.29 + 0.47
7.57 + 1.07
Degree Variance
1.0 0.0
6.34 + 0.08
2.19 + 1.08
30.22 + 11.45
Avg. Density
0.032 + 0.004
0.035 + 0.001
0.003 + 0.000
0.070 + 0.022
Spectral Entropy
1.0 0.0
6.34 1 0.08
6.42 - 0.15
5.69 1 0.68
Avg. Low-Freq, Energy
0.996 + 0.0
0.365 1 0.010
1.0 + 0.0 8
0.357 1 0.030 4
Avg. Subgraphs/Iter
100
10
systems but often struggle with popularity bias. To address this, recent methods adopt advanced loss functions that mitigate bias and enhance fairness, including DirectAU (Wang et al.) (022), CausE (Bonner Vasile 2018), IPS (Saito et al. 2020), and BC-Loss (Zhang et al. 2022). which rely on global item popularty statistics or direct embedding adjustments. However, in FL, such information is inaccessible, and sharing it would break privacy guarantees, as no client has access to the full interaction graph. Our approach introduces a privacy-preserving bias-aware margin by having each client compress its local popularity distributions skew into a single value, which the server aggregates into a global bias context to regularize local model updates. This process disrupts the popularity-driven feedback loop without compromising client privacy.
D EXPERIMENTAL SETTINGS
D. DATASETS
We use five real-world datasets to evaluate the recommendation performance of our proposed method.
Movielens-Ii (mov) is people's s expressed preferences for movies. These preferences are in the form of tuples, each showing person's rating (0-5 stars) for a specific movie at particular time.
Gowalla (He al. 2020 is location-based social networking website where share their locations by checking in. The friendship network is undirected and was collected using their public API.
Yelp2018 (yel) is derived from the 2018 edition of the Yelp challenge. In this challenge, local businesses such as restaurants and bars are treated as items. Yelp maintains the same 10-core setting to ensure data quality.
Amazon-Book (He et al., 2020 is used in Amazon-Review for product recommendation
purposes.
Tmall-Buy (Tma) is large-scale, real-world dataset from the Tmall e-commerce platform. It is widely used for benchmarking the performance of commercial-scale recommender
systems under conditions of extreme data sparsity.
D.2 SUBGRAPH CLUSTERING
A key challenge in FRS research is the lack of public, partitioned benchmarks with structural imbalance. To create a rigorous and reproducible benchmark to evaluate robustness, we must simulate client partitions. We empirically compared four common partitioning strategies on the Movielens-IM -IM dataset (results Table 7) to find the one that generates the most realistic and challenging structural heterogeneity.
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Table 8: Sensitivity analysis of the anchor graph design in LPSFed. Performance is reported using Recall @0 across five datasets. "w/o statistics GR (GNMK)" constructs the GNMK reference solely from global assumptions, without any client-derived statistics. Bold indicates best performance per column.
Anchor Design
Amazon-Book
Gowalla
ML-1M
Yelp2018
Tmall-Buy
0.0734
0.1599
0.26331
0.0767
0,0410
GNMK
0.0738
0.1621
0.2646
0.0783
0.0419
w/o statistics for GR (GNMK)
0.0714
0.1581
0.2625
0.0764
0.0393
The strategies are: (1) Ego-graph: Each client is I-hop ego network of a single user. (2) Random- Const.: Each client is formed from a random subset of users. (3) Intercomnected-Conss Each client a full bipartite graph induced by a group of users. (4) Spectral-Clust. Our chosen method. As Table [ shows, other methods yield structurally weak or homogeneous subgraphs (e.g., low degree variance). In contrast, Spectral-Cluse
produces subgraphs with high internal connectivity and, crucially, the highest degree variance and diverse structural statistics. Therefore, we adopted spectral clustering (Damle et al.] 2019) for our experiments, as provides the most effective protocol to simulate the subgraph structural imbalance this paper aims to solve. For the main performance evaluation (Table D. we applied spectral clustering to partition the entire user-item interaction graph into four distinct subgraphs, ensuring manageable complexity and clear separations. We also introduced diversity in edge distributions across these subgraphs (approx. +00 variation) to ensure each captures different degrees of user-item engagement. Furthermore, mitigate any bias from a single partitioning outcome, we repeated this spectral clustering process twice, and all reported results are the average of three runs per split (a total of six runs).
D.3 RANDOM GRAPH
To construct random reference graphs, used the average number of user nodes, item nodes, interactions, and mean degree per subgraph to generate two types of bipartite graphs: the Erdos-Renyi Renyi (ER) model (Erdōs et al. 1960) or the GNMK model (Knuth 1977). The ER model creates edges between node pairs with a fixed probability, simulating purely random user-item interactions. In contrast, the GNMK model constructs a bipartite graph with a specified number of edges, resulting in more structured connectivity.
Both ER and GNMK models were evaluated in experiments. The ER model, due its high randomness, struggles to reflect the structural properties of real-world user-item graphs. Its inherent randomness fails to represent community-like structures or degree imbalance, leading to subopti- mal personalization similarity measurements. Moreover, the eigenvalue spectrum of ER graphs asymptotically converges the free convolution of the Gaussian and Wigner semicircular distribu- tions (Ramakrishna et al. 2020), exhibiting unclear spectral gaps. This absence of spectral separation limits the design effective low-pass filters, further reducing the model's utility. In contrast, the GNMK model, which generates graphs with a fixed number of edges, better preserves the structural patterns and degree distribution of clustered subgraphs. This closer alignment with real-world data characteristics resulted in the GNMK model consistently outperforming the ER model, delivering improved performance across all metrics.
D.4 SENSITIVITY TO ANCHOR GRAPH DESIGN
To evaluate the impact of the reference graph used for structural comparison, we conduct an ablation study using two widely adopted random graph models: Sedos-Renyl (ER) and the more structure- aware GNMK model. As shown in Table g the GNMK-based anchor consistently achieves the best performance across all datasets, confirming our motivation that GNMK better captures the structural properties of real-world recommendation graphs. Importantly, we also evaluate a stricter privacy setting "w/o statistics for GR" where the GNMK reference graph is constructed without using any client- statistics. This variant yields performance close to the full model, demonstrating that LPSFed remains robust even without access to spient-specific structural information. These findings validate the effectiveness of the GNMK anchor while demonstrating that our framework maintains stability under stronger privacy constraints.
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D.5 BASELINES
We evaluate our proposed method in comparison to nine baselines:
FedAvg MMMMaha et al. 2017): y is a foundational federated learning approach that transmits
parameter gradients instead of data, enhancing privacy.
2023): adjusts weights within community structures using random graphs and parameter masking to protect data privacy.
FedPUB (Back al.
FedMF (Chai et al.
2020): applies matrix factorization within federated settings to ensure secure and private collaborative filtering.
F2MF (Liu et al.
2022 aims address faimess issues in federated recommender systems
by incorporating faimess constraints into the matrix factorization process, ensuring equitable recommendations across diverse user groups.
PFedRec (Zhang al. 2023ap: offers lightweight, user-specific model for on-device personalization without user embeddings.
FedRAP (Li et al. 2024): : utilizes a dual personalized approach to manage user and item
embeddings separately, optimizing communication efficiency.
FedPerGNN (Wu al. 2022): creates subgraphs for users with shared interactions, utilizing
Local Differential Privacy (LDP) (Qi et al. 2020) for enhanced security during updates.
FedHGNN (Yan et al.,
2024): employs a federated heterogeneous graph neural network explicitly capture the diverse structural relations between users and items for privacy- preserving recommendation.
FedSSP (Tan et al. 2024): leverages spectral knowledge from client graphs to handle hetero- geneity and models personalized preferences, adapted from federated graph classification.
D.6 HYPERPARAMETER SETTINGS
We configure the hyperparameters of the proposed method follows. All user, item, and pop- ularity embeddings are initialized from normal distribution. with an embedding dimension of 64. Optimization performed using the RMSProp optimizer (Ruder) 2016 with a learning rate 0.0005.
Each client performs local epochs of training per communication round. The entire training process consists of 40 global communication rounds, resulting in a total of 200 local training epochs per client. Each client performs local epochs of training per communication round. The entire training process consists of 40 global communication rounds, resulting in a total of 200 local training epochs per client. The network architecture consists of two graph convolution layers, followed by two-layer ML.s used for both pooling and prediction stages (Eq. 5 For the localized popularity bias-aware contrastive loss (C Eq. ) the softmax temperature parameter is set to = 0.1, and the interpolation weight for the refined margin (Eq. 15) is set to 0.25. To ensure training stability, the model freezes parameter updates during the first two global epochs.
E MODEL COMPLEXITY
E.
TIME COMPLEXITY ANALYSIS
E.1.1 LOW-PASS GRAPH CONVOLUTION NETWORK'S TIME COMPLEXITY ANALYSIS
The propagation step in Graph Convolution Network (GCN) can be expressed as - AD - I (I- LI = P(I - A) where it is equivalent to applying a low-pass filter in the frequency domain. The filter, represented thereby emphasizing smooth, global features. Importantly, repeated applications of the Low-pass Collaborative Filter (LCF) are equivalent a single application:
Ai...., AM+N), inherently prioritizes smaller eigenvalues,
(PP') ) (PP' Z LCFFFZ)
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Table 9: Comparison of training efficiency different models. Metrics include time per epoch, number of epochs converge, total training time, and memory usage, with the best scores highlighted in bold.
Model
Time/Epock
#Epoch
Training Time
Memory Usage
FedAvg
20s
90
30m
6.2GB
FedPUB
25s
90
37m
10.2GB
FedMF
23m 48s
150
59h 30m
6.2GB
F2MF
39m 12s
100
65h 20m
6.4GB
PFedRee
38m 47s
135
87h 16m
1,4GB
FedRAP
73m 29s
160
195h 56m
1.6GB
FedPerGNN
6m 38s
200
15h 29m
18.8GB
FedHGNN
368
100
lh
0.6GB
FedSSP
20s
150
50m
6.2GB
LPSFed (BPR)
19s
130
40m
6.2GB
LPSFed
19s
130
39m
6.2GB
ensuring stable feature propagation and avoiding over-smoothing.
Computing the full set of eigenvectors for graph with M + nodes has a time complexity of O((M + ") ). as iigen-decomposition scales cubically. However, most real-world recommendation graphs are sparse, with the number of non-zero elements in the Laplacian L typically scaling linearly with M + N. By leveraging sparsity, the Lanczos algorithm computes subset of eigenvectors P for sparse matrices with time complexity (n) where & represents the cut-off frequency (i.e., the number of retained low-frequency eigenvalues). In practice, where & < M + N and n < (M+NY) this approach ensures computational efficiency.
E. LPSFED FRAMEWORK'S COMPLEXITY ANALYSIS
In federated settings with C' clients, each client independently computes the first & eigenvectors using the Lanczos algorithm. The time complexity per client is On (n))), where n denotes the non-zero elements the client's 's subgraph Laplacian. The server, which generates a global random graph, has comparable computational complexity of On Φ²). Sequential execution would yield a total complexity of O((C+ +)) However, federated learning leverages parallel computation, allowing clients to perform computations simultaneously. As a result, the effective time complexity for the entire system remains (2)
Scalability and Robustness. Our approach ensures scalability and robustness, addressing chal- lenges in large subgraphs and imbalanced data distributions. The use of the Lanczos algorithm leverages the sparsity of real-world recommendation graphs, making eigenvector computations effi- cient even for large-scale datasets. Furthermore, rreeeigendecomposttion ensures that training costs remain low, as this computational step required only once.
Communication and Computational Costs. Communication costs in our framework depend on the size of the exchanged parameters, which occur only at specific aggregation and distribution epochs rather than every training iteration. This reduces the overall communication overhead. On the computational side, the server processes graph representing the average subgraph size of clients, while each client processes its local subgraph, ensuring that both sides operate within their respective resource constraints. By limiting communication frequency and leveraging sparsity, method balances computational and communication overheads, enabling high scalability and practical applicability.
E.2 TRAINING EFFICIENCY ANALYSIS
We compare the training efficiency and resource utilization (computation memory, time) of our model, LPSFed, with several other federated learning models. As shown in Table we evaluate time per epoch, number of epochs to converge (#Epoch), total training time, and memory usage.
It's important to note that LPSFed, FedAvg, FedPUB, FedSSP are all based on the Low-pass Graph Convolution Network (LGGNN (Yu et al. 2022. This architecture allows for pre-computation
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Table 10: Average Jaccard index for paired users iterations, with corresponding to margin strengths and columns to training iterations.
Iteration
20
40
60
80
100
7 0.0
0.0482
0.0497
0.0536
0.0568
0.0600
Y 0.3
0.0478
0.0490
0.0529
0.0554
0.0592
Y 0.5
0.0477
0.0489
0.0522
0.0544
0.0576
Y 0.7
0.0476
0.0486
0.0512
0.0541
0.0570
Y = 1.0
0.0474
0.0478
0.0501
0.0522
0.0553
of cigendecomposition, significantly reducing the computational burden during training. As result, these models achieve faster training times compared to methods that require real-time graph convolutions. IPSFed demonstrates superior training efficiency, converging in 130 epochs with an average of 19 seconds per epoch, resulting in a total training time of 39 minutes and a memory usage of 6.2 GB. This shows an improvement over other models, emphasizing the efficiency of our approach. Leveraging the pre-computed convolution kernel allows our model to operate efficiently in the federated learning framework, capitalizing the advantages of LGCN while optimizing performance through our proposed techniques, even in distributed settings.
F ADDITIONAL EXPERIMENTAL RESULTS
F.1 BIAS AMPLIHICATION MEASUREMENT AND EFFECTIVENESS OF THE BIAS-AWARE MARGIN
Bias Amplification Measurement. To assess how method alleviates the feedback loop caused by popularity bias, we quantify convergence in user behavior using the Jaccard index, follow- ing (Chaney et al. 2018). This metric captures the extent to which recommender systems drive users to interact with increasingyy similar items. For each user u, we identify the most similar user based on the cosine similarity of their preference vectors (0u, 0₂) and compute the Jaccard index over their item interactions:
(t)(t) (t) n (.( (t) (t)
where (t) and (t) denote the item sets interacted by users and up to time t. A higher Jaceard index reflects a stronger feedback loop, indicating behavioral convergence due to repeated exposure popular items. To evaluate the impact of localized popularity bias-aware margin, vary the margin strength y and monitor changes in the Jaccard index over training. The results are presented Table I0
Analysis. Table 10 demonstrates that the Jaccard index tends to increase over training iterations, indicating a typical feedback loop where users increasingly receive similar popular items. However, incorporating a bias-aware margin significantly mitigates this effect. As the margin strength increases. the rate at which the Jaccard index grows progressively reduced. Notably, when y 1.0, the index exhibits the smallest rise, suggesting that stronger margins more effectively suppress convergence driven by popularity.
These results suggest that the bias-aware margin provides regularization signal that limits the reinforcement of frequently recommended items, thereby interrupting the progression the feedback loop. By doing so, our method reduces recommendation redundancy and preserves both person- alization and interaction diversity. This observation consistent with prior findings in (Chaney) et al. 2018), which report that unregulated feedback loops tend to amplify popularity-drive bias and degrade recommendation quality. In contrast, our approach effectively mitigates such dynamics, supporting more equitable and stable recommendation behavior across clients.
F.2 CORRELATION ANALYSIS BETWEEN EIGENGAP AND PERFORMANCE
To examine whether spectral stability contributes to empirical performance, we computed the Pearson correlation between the observed Recall 20 values and the corresponding eigengaps 5 varying
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Table 11: Correlation analysis between eigengap and model performance in Movielens- I.M. Recall @ 20 and corresponding eigengap values are reported for different cut-off frequencies Pearson correlation quantifies linear dependence between eigengap magnitude and model performance.
Cut-off Frequency +
Recall @ @0
0.2556
0.2614
0.2615
0.2622
Eigengap 6
1.17 x 10 10-15 15
4.36 x 10Th 1.61 x 10-2 1.46 x 10-1
Pearson Correlation (Recall ++ 8)
0.5004
Table 12: Effect of client variability on performance using the Amazon- Book dataset. Metrics include Recall and NDCG @20, bold indicates the best results and underlined the second-best results in each setting.
# of Clients
4
10
20
Model
Recall
NDCG
Recall
NDCG
Recall
NDCG
FedAvg
0.0642
0.0312
0.0395
0.0186
0.0220
0.0098
FedPUB
0.0633
0.0322
0.0417
0.0180
0.0232
0.0102
FedMF
0.0153
0.0072
0.0091
0.0058
0,0033
0.0035
F2MF
0.0451
0.0225
0.0325
0.0182
0.0158
0.0078
PFedRcc
0.0713
0,0242
0.0406
0.0169
0,0236
0.0100
FedRAP
0.0082
0.0090
0.0086
0.0085
0.0087
0.0074
FedPerGNN
0.0035
0.0026
0.0034
0.0026
0.0029
0.0025
FedHGNN
0.0647
0.0298
0.0415
0.0188
0.0223
0.0105
FedSSP
0.0649
0.0356
0.0450
0.0219
0.0238
0,0110
I.PSFed (BPR)
0.0643
0.0322
0.0400
0.0195
0.0235
0.0107
LPSFed
0.0738
0.0442
0.0464
0.0259
0.0254
0.0115
Improvement
1 3.5%
1 24.2%
t 3.1%
1 18.3%
1 6.7%
1 4.5%
cut-off frequencies $ As shown in Table 11 the correlation coefficient is 0,5004, indicating a moderate positive relationship. This trend consistent with Figure 33 (c), where performance improves as & increases within moderate range. 'he result supports our theoretical analysis; larger eigengaps 8 enable more stable low-pass filtering by separating meaningful structural components from high-frequency noise, which contributes positively to model performance when applied within an appropriate filtering range.
F.3 EFFECT OF CLIENT VARIABILITY ON PERFORMANCE
Table 12 presents the performance trends of all models under varying numbers of clients (4, 10, and 20) using the Amazon-Book dataset. As the number of clients increases, the overall performance of all baselines degrades. This degradation is attributed the increasing heterogeneity among clients, which results from heightened subgraph structural imbalance and limited user-item interactions per client. These challenges amplify the difficulty of capturing consistent collaborative signals across clients, making recommendations more susceptible to data sparsity and structural diversity. Despite these challenges, LPSFed consistently maintains strong performance across all client settings. This robustness stems from two core components: spectral similarity-guide bias-aware margin. The former helps align model updates with each client's structural uniqueness, while the latter mitigates feedback loops caused by popularity bias. Together, they allow LPSFed to adapt to heterogeneous client environments and preserve recommendation quality, even as the number of clients grows and local data becomes more fragmented.
personalization and
G BROADER IMPACTS
This research on federated recommender systems provides several positive societal impacts. By training models on decentralized subgraph-level interaction data without exchanging raw data, enhances privacy protection, reduces the risk of data exposure, and builds trust in domains where data sensitivity is critical, such as healthcare and finance. Our method also promotes faireess by mitigating localized popularity bias and improving the quality of recommendations for diverse client groups. Furthermore, the federated framework enables silos, allowing clients jointly improve recommendation quality without disclosing proprietary
collaboration across data
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data. These outcomes contribute to improved user experience, sustainable model utility, and the development of a more privacy-preserving recommendation ecosystem.
H LIMITATIONS
Despite these promising advancements, our method inherently relies on spectral computations, such as eigendecomposition, performed during preprocessing stage. Although preprocessing helps avoid computations during training, it still represents a scalability limitation. Future research should therefore prioritize developing scalable spectral approximation techniques and automating hyperparameter selection to further enhance the method's applicability in practical scenarios.
I
LLM USAGE
We utilized a Large Language Model (LLM) as an assistive tool in the preparation of this paper. The LLM'' role was proofread for grammar and spelling and to help refine the text for clarity, conciseness, and word choice. The core concepts, novel methodology, theoretical proofs, and all experimental results were conceived and generated by the human authors.
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Reviews:
[{"review_id": "tzsDaRQ4Fq", "reviewer": "Reviewer_SL1j", "summary": "This paper introduces LPSFed, a federated recommender framework designed to handle subgraph structural imbalance and popularity bias across clients. It applies low-pass spectral filtering to extract stable, low-frequency structural signals and computes a structural similarity between each client and a server-generated reference graph for personalized aggregation. Additionally, it introduces a bias-aware margin loss that mitigates popularity bias by adjusting item ranking based on local degree distributions. Theoretical analysis justifies the stability and regularization of low-pass filtering, and experiments on five real-world datasets demonstrate consistent improvements in recall, NDCG, and robustness over multiple baselines.", "questions": "1 Can similarity computation be done entirely on clients to ensure privacy?\n\n2 How sensitive is performance to the reference graph’s design or randomness?\n\n3 Do empirical eigengaps correlate with observed improvements?\n\n4 Can the bias-aware margin adapt automatically rather than be fixed?\n\n5 Has the model been tested under naturally partitioned (non-synthetic) clients?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Clear and coherent framework integrating personalization and debiasing.\n\n- Sound theoretical motivation for spectral similarity and low-pass regularization.\n\n- Strong empirical results across multiple datasets and settings.\n\n- Effective ablation studies confirming each module’s contribution.", "weaknesses": "- Novelty is incremental; relies heavily on previous spectral FL ideas.\n\n- Theoretical assumptions may not align with real-world bipartite graphs.\n\n- Ambiguity in privacy handling (what information the server receives).\n\n- Sensitivity to anchor graph design and data partitioning not deeply analyzed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "hEdECZiidB", "reviewer": "Reviewer_rJvg", "summary": "The paper targets the challenge of subgraph structural imbalance in FRS. The authors propose LPSFed which uses graph Fourier transform and low-pass spectral filtering to extract stable structural signals, measures structural similarity to guide personalization, and introduce a popularity-bias-aware margin to mitigate feedback loops. Experiments across five datasets show advantages supported by theoretical analysis.", "questions": "Could the authors elaborate on the novelty besides combining existing algorithms and methods?\nCould the authors provide some discussion on how the computation burden caused by spectral computation may be alleviated?\nCan the authors include more fair comparisons with peering baselines such as spectral FRS and fairness-aware FRS?\nCould the authors discuss more about the cut-off frequency and structural anchor determination and their impact and sensitivity?\nCould the authors relax the idealization in math assumptions and give more specifics on value choices for example for C1 and C2 etc. ?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The authors provide a well-motivated spectral view and lots of theoretical analysis. The qualitative tests show advantageous performance with discussions on robustness, completed by the ablation study of model components and hyperparameter analysis. The limitation of reliance on spectral computations is acknowledged.", "weaknesses": "1. Although the motivation of selected algorithms and theories are well motivated, the work seems to be a nice combination of existing proposals applied to this specific challenge, i.e., subgraph structural imbalance. This is not to undermine the effort and importance of such combination, but the novelty by itself is thus unavoidably limited. \n2. As acknowledged by the authors in the appendix, the reliance on spectral computations raises the computation burden, I'd like to see if authors can provide some thoughts on how that could be alleviated. \n3. It seems only one baseline is from the spectral FL family, please include more baselines from this direction, or otherwise explain the reason. The same applies to bias performance tests, which didn't compare with sufficient fairness-aware FRS works. \n4. The cut-off frequency may have a considerable impact on system performance, yet the authors seem skip the discussion on the frequency value choice.  This also applies to the neutral structural anchor.\n5. There seem to be quite some places in the math reasoning that are either vague or too idealized. For example, C1 and C2 only have vague description (actually only C1 has) without specifying how to determine their values; it idealizes the client graphs as having clear k-block community structures, which is quite rare in reality I'm afraid.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "lqbvuZ8x6P", "reviewer": "Reviewer_7AW4", "summary": "The paper introduces LPSFed, a federated recommendation framework designed to handle structural imbalance across user–item subgraphs of different clients. This imbalance arises from variations in subgraph density and connectivity, which often lead to inconsistent representation learning when applying traditional GNN-based federated models. To address this issue, LPSFed employs low-pass spectral filtering to capture smooth, low-frequency graph signals, enhancing representation stability under heterogeneous structures. In addition, a popularity-aware contrastive loss is introduced to balance recommendations and reduce local popularity bias. Experiments on several datasets show consistent improvements over baseline federated models, and theoretical results provide further support for the effectiveness of the proposed spectral regularization strategy.", "questions": "1.  Please provide a detailed explanation of the motivation behind the core method, specifically the similarity comparison based on the low-pass convolution kernel distribution k. The authors should clarify why this kernel distribution is chosen for measuring similarity and discuss its advantages over other possible approaches from both theoretical and empirical perspectives.\n\n2. The model uses spectral domain filtering (Graph Fourier Transform + low-pass filtering) to remove high-frequency noise while preserving low-frequency structural information. Why is it better to retain the low-frequency signal for similarity comparison?\n\n3. While the goal of federated learning is to preserve user privacy, the paper does not discuss whether the client-uploaded information (such as statistical features, spectral kernel distributions, or structural similarity scores) might pose a risk of indirectly revealing the underlying graph structure. A discussion or analysis of potential privacy implications would enhance the completeness of the work and align better with the core principles of federated learning.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The paper offers a novel perspective by applying low-pass spectral filtering to mitigate structural imbalance in federated recommendation. The combination of spectral graph modeling and bias-aware optimization is original and novel.\n\n2. The approach is conceptually clear. The appendix adds useful details on complexity analysis and spectral properties, improving transparency and reproducibility.\n\n3. The study investigates an important but relatively underexplored problem in federated recommendation, namely the structural imbalance among decentralized user–item subgraphs. This imbalance is caused by variations in graph density and connectivity across clients. It can negatively influence the stability and fairness of model training in practical applications. By introducing spectral filtering and bias-aware optimization to address this issue, the paper provides a fresh perspective on improving structural consistency in federated recommender systems. The proposed direction may encourage further research on combining structure-aware learning with privacy-preserving mechanisms to develop more robust and equitable personalized recommendation models.", "weaknesses": "1. The core method of the paper is not sufficiently explained. The authors propose decomposing the embedding layer to generate a low-pass convolution kernel distribution k, which is then used for similarity comparison. However, the paper does not clarify why this specific k is chosen or why it is appropriate for measuring similarity. A more thorough discussion of both the theoretical and empirical reasoning behind this design would help readers understand its advantages over alternative approaches.\n\n2. While the paper claims innovation in \"structural imbalance,\" the uniqueness of the method compared to existing spectral federation methods (such as FedSSP) can be explained more clearly. \n\n3. Although the description of the main method is relatively clear, the workflow diagram(Figure 2) appears somewhat cluttered, which makes it harder to fully understand the overall process.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "CkD5y5Bn0U", "reviewer": "Reviewer_wXeM", "summary": "In this paper, the authors introduce Low-Pass Personalized Subgraph Federated Recommendation (LPSFed) for personalized federated recommendation systems. LPSFed could address the challenge of subgraph structure imbalance and local popularity deviation. LPSFed uses low-pass spectral filtering on heterogeneous subgraphs to extract low-frequency structural signals. These signals are used to calculate the personalized structural similarity between each client and the global model. LPSFed also uses a local popularity bias-aware margin to capture the imbalance of item degrees in each subgraph and incorporates it into the personalized bias correction term to mitigate recommendation bias. Extensive results on multiple datasets demonstrate the effectiveness of the approach.", "questions": "See in weaknesses.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper is easy to follow and generally well presented.\n\n2. It is a timely topic and a very interesting area to explore.\n\n3. The theoretical analysis is solid.", "weaknesses": "1. The authors acknowledge in the Appendix and Limitations that spectral calculations are performed in the preprocessing stage. If the eigenvectors of the entire graph (or subgraph) are directly calculated on a very large graph (such as Tmall), it may not be feasible in terms of memory/time.\n\n2. The author averages the local margins of all (u,i) of each client into a scalar $M^c$ and uploads it to the server. A single average value cannot reflect the skewness (such as long tail) or multimodal distribution within the client; the average value is sensitive to outliers and will mask important information.\n\n\n3. Figure 1 shows a 15-client partition for Amazon-Book; however, D.2 in the main experiment states, \"In the main experiment, we partitioned the full graph into four subgraphs.\" The paper does not clearly explain the correspondence between the different experimental settings and the reasons behind them.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": {}}]

Author rebuttal:

Decision: Accept (Poster)