Graph Trend Filtering Networks for Recommendations

1 INTRODUCTION

• The introduction motivates collaborative filtering as modeling historical user–item interactions to predict likely interactions, with matrix factorization [21] and neural collaborative filtering as examples.Matrix factorization uses user–item embedding inner products; neural collaborative filtering replaces the inner product with neural networks.Source: §1 · p. 1
• Graph collaborative filtering methods such as NGCF [46] and LightGCN [20] propagate user and item embeddings over interaction neighborhoods to exploit high-order connectivity.This propagation, motivated by GNN feature aggregation, is presented as a key to their advances in recommendation.Source: §1 · p. 1
• Existing systems often treat interactions uniformly, although their reliability varies, leading to suboptimal representations and motivating adaptive propagation.The paper identifies non-adaptive propagation and non-robustness in GNN-based recommendation methods from the perspective of Laplacian smoothness.Source: §1 · pp. 1–2
• Unreliable implicit feedback—including random or bait clicks and purchases followed by negative reviews—can dominate training data and obscure users’ actual preferences.Such interactions can be difficult to identify or prune, making recommenders vulnerable to them.Source: §1 · p. 2
• GTN applies graph trend filtering to learn the adaptive reliability of user–item interactions during recommendation.The method is motivated by trend filtering and graph trend filtering.Source: §1 · p. 2
• The paper reports comprehensive experiments and ablation studies on real-world datasets to demonstrate GTN’s effectiveness.Source: §1 · p. 2

2.1 Notations and Definitions

• The paper represents user–item histories as a bipartite graph, with adjacency and Laplacian matrices, an oriented incidence matrix, and user/item embeddings.It also establishes matrix/vector notation and defines matrix norms and inner products.Source: §2.1 · p. 2

2.2 Embedding Propagation and Smoothness

• LightGCN’s performance drops significantly as injected random user–item interaction noise increases, confirming its sensitivity to unreliable interactions.The preliminary study evaluates LightGCN under different rates of random interaction perturbation.Source: Figure 2 · p. 3, §2.2 · p. 3
• Standard propagation pulls connected user and item embeddings together equally, so random clicks can make items appear highly relevant and undermine robustness.The paper frames this as inappropriate smoothness caused by unreliable interactions.Source: §2.2 · p. 3

3.1 Design Motivation

• The embedding-smoothness penalty downweights interactions whose user and item embeddings differ substantially, preserving those differences while smoothing more similar embeddings for robust, adaptive propagation.The L1 graph trend filtering prior promotes piecewise-smooth embeddings.Source: §3.1 · pp. 3–4

3.2 Graph Trend Collaborative Filtering

• The convex smoothness objective is solved with a primal-dual PAPC iteration [33], which the stated parameter settings guarantee will converge to its optimum.The authors state that each iteration has complexity O(|E|d), the same order as embedding propagation.Source: §3.2 · p. 4

3.3 Graph Trend Filtering Networks

• GTN initializes user and item embeddings randomly when features are unavailable, filters them with K GTCF iterations, and scores recommendations by the inner product of final representations.When features are available, they can be used directly or concatenated with randomly initialized embeddings.Source: §3.3 · p. 5
• The model learns its initial embeddings with pairwise BPR loss, ranking observed interactions above unobserved ones and applying L2 regularization to reduce overfitting.Training examples pair an observed user-item interaction with an unobserved interaction for that user.Source: §3.3 · p. 5

4.1 Experimental Settings

• Experiments use Gowalla, Yelp2018, Amazon-Book, and LastFM as recommendation benchmarks.The first three datasets are from NGCF [46] and LightGCN [20], while LastFM is from KGAT [45].Source: §4.1.1 · p. 5

4.1.1 Datasets.

• Observed user–item interactions are positive instances, paired with randomly sampled unobserved items as negatives; dataset statistics appear in Table 1.The experiments closely follow LightGCN settings for fair comparison.Source: §4.1.1 · p. 5, Table 1 · p. 5
• Comparisons include MF, NeuCF [21], GC-MC [2], Mult-VAE, NGCF [46], DGCF [47], and LightGCN [20].The baselines span classic and neural collaborative filtering, graph auto-encoder, VAE, and GNN-based methods.Source: §4.1.2 · p. 5

4.1.2 Baselines.

• The study excludes SGL [49], stating that self-supervised learning is orthogonal to its collaborative-filtering contribution and that combining them is future work.Source: §4.1.2 · p. 5

4.1.3 Evaluation Metrics.

• Recommendation ranking is evaluated with Recall@K and NDCG@K at K=20, averaged across test-set users; higher values indicate better performance.Source: §4.1.3 · p. 5
• The PyTorch implementation uses Adam and searches embedding sizes {16, 32, 64, 128, 256, 512}, batch sizes {128, 512, 1024, 2048}, and learning rates {0.0005, 0.001, 0.005, 0.01, 0.05, 0.1}.Baseline hyperparameters follow their papers; NGCF and LightGCN settings are followed for fair comparison.Source: §4.1.4 · p. 6

4.2 Performance Comparison of Recommender Systems

• GTN achieves the best Recall@20 and NDCG@20 across four datasets, improving on the strongest LastFM baselines by 12.77% on NDCG@20 and 9.61% on Recall@20.The paper attributes this improvement to local adaptive smoothness over user-item interactions.Source: §4.2 · p. 6
• MF performs relatively poorly, while NeuCF and GNN-based methods are stronger; NGCF and DGCF perform similarly and surpass MF and NeuCF, reflecting the value of local graph structure and high-order connectivity.Stacked propagation layers let GNN-based methods capture collaborative signals through high-order connectivity.Source: §4.2 · p. 6
• GTN routinely outperforms LightGCN on Recall@20 and NDCG@20 across four datasets, using adaptive embedding smoothness to mitigate unreliable interactions that can cause inappropriate smoothness.LightGCN is a strong baseline, but its propagation can pull user and item embeddings together even when interactions are unreliable.Source: §4.2 · p. 6

4.3.1 Robustness against random perturbation.

• The robustness test injects random edges into test data and evaluates models trained on clean data across perturbation rates from 5% to 50%.The tested rates are 5%, 6%, 8%, 10%, 15%, 20%, 30%, and 50%.Source: §4.3.1 · p. 6
• Across tested perturbation rates, GTN consistently outperforms NGCF and LightGCN, maintaining high performance as noise rises and widening its margin at higher noise.LightGCN drops significantly above a 20% perturbation rate, especially on Gowalla and Yelp2018; NGCF is almost the worst-performing baseline but relatively stable.Source: §4.3.1 · pp. 6–7

4.3.2 Sparsity pattern.

• Across all datasets, GTN yields a much larger share of embedding-difference elements with absolute value below 0.2 than other methods, supporting the trend captured by GTCF.The sparsity ratio is the portion of elements whose absolute value is less than 0.2.Source: §4.3.2 · p. 7, Table 4 · p. 7

4.4.1 Effect of 𝜆

• Increasing λ from 0 to 1 significantly improves GTN, while performance remains stable for λ values above 2.GTN is relatively insensitive to λ over the tested range of 0 to 9, and λ controls embedding smoothness.Source: §4.4 · p. 7

4.4.2 Effect of

• Across GNN-based methods, performance generally rises and then falls with depth, peaking at three layers in most cases; additional layers can degrade results.The study varies layer count from 1 to 5 and compares NGCF, LightGCN, and GTN.Source: §4.4.2 · p. 7
• With five layers on LastFM, GTN improves over the strongest baselines by 19% on Recall@20 and 23% on NDCG@20.GTN achieves the best prediction performance on four datasets for Recall@20 and NDCG@20 in most cases.Source: §4.4.2 · p. 7
• The objective loss decreases during the filtering process, with curves across four datasets verifying convergence of graph trend collaborative filtering.Source: §4.4.3 · p. 7

4.4.3 Convergence of GTCF.

• At the same training epoch, GTN outperforms NGCF and LightGCN and converges faster in most cases.Source: §4.4.4 · p. 7

5.1 Collaborative Filtering

• CF models similar-behaving users as sharing preferences and commonly embed users and items; MF predicts by inner product, while NeuCF and DSCF use neural methods [21] [18].DSCF uses a Bi-LSTM-based language model to capture information from high-order neighbors and enhance user representations.Source: §5.1 · pp. 7–8

5.2 GNNs-based Recommender Systems

• GNN recommenders propagate over user-item graphs; PinSage, GraphRec [15], and NGCF [46] lead to simplified LightGCN [20], reported as state-of-the-art.Earlier CF methods treat interactions independently and can miss deep collaborative signals [46]; GNNs learn node embeddings by aggregating and transforming local-neighbor messages [22] [23] [24] [51].Source: §5.2 · pp. 8–9

5.3 Trend Filtering

• GTN presents the first use of graph trend filtering as a collaborative-filtering technique for recommender systems.Trend filtering progressed from time-series estimation to regression [32] [39] [40], tensor denoising [36], and graph signals; Elastic GNNs [32] used it for robustness against adversarial graph attacks.Source: §5.3 · p. 9

• Existing GNN-based collaborative filtering methods exhibit non-adaptive propagation and non-robustness when viewed through Laplacian smoothing on user-item graphs.Source: §6 · p. 9
• GTN introduces a graph trend collaborative filtering framework that captures adaptive reliability in user-item interactions.Source: §6 · p. 9
• Experiments on four real-world datasets demonstrate GTN’s effectiveness and the benefits of modeling adaptive interaction reliability for recommendations.Source: §6 · p. 9