HAKG: Hierarchy-Aware Knowledge Gated Network for Recommendation

1 INTRODUCTION

• Knowledge graphs provide item-related real-world facts that can alleviate cold-start and improve recommendation explainability.Traditional collaborative-filtering systems are described as suffering from data sparsity and cold-start problems.Source: §1 · p. 1
• Existing propagation-based methods are described as failing to model hierarchical structures and relations and to capture high-order collaborative item signals.The paper says interaction data and KGs exhibit hierarchy, while item signals can also arise from long-range user-item connectivity.Source: §1 · pp. 1–2
• Hyperbolic space expands exponentially and is presented as a suitable geometry for preserving hierarchical data.The paper describes it as providing more room to preserve distances between hierarchical data than Euclidean space.Source: §1 · p. 2, Figure 2 · p. 2
• HAKG embeds users, items, entities, and relations in hyperbolic space and uses relation-aware aggregation and an angle constraint for hierarchical relations.The angle constraint is intended to preserve item attributes in the embedding space.Source: §1 · p. 2
• HAKG uses dual item embeddings to separately propagate collaborative signals and knowledge associations, then gates their fusion to identify user behavior patterns.The framework treats user-item interactions and the KG as two information channels.Source: §1 · p. 2
• HAKG significantly outperforms state-of-the-art methods, including CKAN, Hyper-Know, and KGIN [47], on three real-world datasets.The paper also reports that its learned hyperbolic embeddings reveal underlying data hierarchies and relations.Source: §1 · p. 2
• The paper frames HAKG as knowledge-aware recommendation that accounts for hierarchy and high-order collaborative item signals.These are the two factors the introduction identifies as missing from existing propagation-based methods.Source: §1 · pp. 1–2
• The contributions also include experiments on three public benchmark datasets to evaluate HAKG.The introduction states that these experiments demonstrate the model's superiority.Source: §1 · p. 3

2 PROBLEM FORMULATION

• User-item behavior, such as clicks or reviews, is represented as a bipartite graph whose edges indicate user-item interactions.The formulation concerns implicit feedback.Source: §2 · p. 3
• The KG is a heterogeneous graph of head-relation-tail facts, and the formulation assumes every item also appears as a KG entity.That assumption allows items in the interaction graph to be linked with KG entities for auxiliary semantics.Source: §2 · p. 3
• The task is to predict how likely a user is to adopt an item with which they have never engaged.The prediction uses a user-item graph and a knowledge graph as inputs.Source: §2 · p. 3

3 METHODOLOGY

• Hierarchy-aware modeling embeds users, items, entities, and relations in hyperbolic space, aggregates relation-aware KG context, and applies an angle constraint to learn item attribute semantics.Gated aggregation uses dual item embeddings to separately encode high-order collaborative signals and KG associations, then fuses useful semantics for user representations.Source: §3 · p. 3

3.1 Preliminaries

• Hyperbolic space has constant negative curvature and expands exponentially with radius, a property used to capture hierarchy in interactions and KGs.HAKG uses the Poincaré ball model with curvature −1 for gradient optimization.Source: §3.1 · p. 3
• The Poincaré ball domain is defined within the unit ball in R^n.The displayed domain condition is an inequality on the coordinates.Source: §3.1 · p. 3
• A tangent space at a point on the ball is a Euclidean approximation used for aggregation; exponential and logarithmic maps connect it with hyperbolic space.The passage also identifies Möbius addition as an operation used in the model.Source: §3.1 · p. 3

3.2 Hierarchy-Aware Modeling

• Hyperbolic embeddings represent users, items, entities, and relations, reflecting the hierarchy in interaction data and knowledge graphs.The paper motivates this choice by the high distortion of Euclidean representations for hierarchical data.Source: §3.2 · p. 3

3.2.1 Hyperbolic Relation-Transitive Aggregation.

• The KG aggregator integrates first-order relational contexts in hyperbolic space, preserving relation dependencies by mapping contexts through tangent space and back.It avoids hyperbolic mean pooling, and stacked layers capture higher-order knowledge associations.Source: §3.2.1 · pp. 3–4

3.2.2 Angle Constraint of Hierarchical triplets.

• When relation types are unavailable, HAKG can infer hierarchicalness with Krackhardt scores or treat item-connected entities as attributes; the constraint is relaxed to avoid possible embedding collapse.Randomly masked dimensions restrict the angle constraint to a subset of hyperbolic planes, leaving room for other item semantics.Source: §3.2.2 · pp. 4–5
• An angle constraint uses nested entailment cones to encode hierarchical entities as item attributes in the embedding space.The constraint encourages an item embedding to lie inside its connected entity’s cone.Source: §3.2.2 · p. 4

3.3 Gated Aggregation with Dual embeddings

• Dual item embeddings keep KG knowledge associations separate from high-order collaborative signals, addressing methods that mix or omit the latter.Collaborative representations encode multi-hop user-item connectivity, including co-occurrence relationships.Source: §3.3 · p. 5

3.3.1 Collaborative Aggregation for Items.

• Collaborative item representations are recursively updated from neighboring users in the user-item graph to capture high-order connectivity.The collaborative representation is distinct from the item’s KG-derived knowledge representation.Source: §3.3.1 · p. 5

3.3.2 Information Gated Aggregation for Users.

• A learnable gate adaptively balances knowledge and collaborative item signals when distilling information for user aggregation.A high gate value indicates behavior is attributed mainly to item attributes rather than similarities among items.Source: §3.3.2 · p. 5

3.4 Model Prediction

• HAKG sums user and item representations across layers, then adds cosine-similarity predictions from item behavior and attributes.The two information types remain separately encoded in the final representations.Source: §3.4 · p. 5

3.5 Model Optimization

• The contrastive loss increases similarity for observed user-item pairs and decreases it for sampled unobserved pairs, with the objective combining prediction and angle losses.The paper states this loss alleviates convergence problems by introducing more negative samples and penalizing uninformative ones.Source: §3.5 · pp. 5–6

3.6.1 Model Size.

• HAKG omits nonlinear activations and learnable transformation matrices, which prior work associated with better performance and fewer redundant parameters.Its listed parameters are ID embeddings and W1/W2 for gated aggregation.Source: §3.6.1 · p. 6

3.6.2 Time Complexity.

• Each training epoch has complexity O(|G_k|dL + |G_b|dL), with angle-constraint and space-mapping costs also described.The KG and user-item aggregation costs are O(|G_k|dL) and O(|G_b|dL), respectively; angle constraint costs O(|I|), and mappings cost O(|G_b| + |G_k|).Source: §3.6.2 · p. 6
• With the same embedding size and propagation depth, HAKG has complexity comparable to KGAT, CKAN, and KGIN.Source: §3.6.2 · p. 6

4 EXPERIMENTS

• HAKG experiments compare state-of-the-art knowledge-aware recommendation performance, test effects of hierarchical modeling, gated propagation, dual embeddings, and propagation depth, and assess hierarchy insights, using all-ranking recall@20 and ndcg@20.Source: §4 · p. 6, §4.1.2 · p. 6

4.2 Performance Comparison (RQ1)

• HAKG performs best on all three datasets, exceeding the strongest baselines in ndcg@20 by 15.43% on Alibaba-iFashion, 8.21% on Yelp2018, and 9.79% on Last-FM.Source: §4.2 · p. 7, Table 2 · p. 8

4.3.1 Impact of Angle Loss & Gated Aggregation.

• Removing the angle constraint and gated aggregation degrades performance; angle loss models hierarchical relations, while gating supplies discriminative user-behavior signals.The compared variants remove both components, remove angle loss alone, or replace gating with point-wise addition.Source: §4.3.1 · p. 8, Table 3 · p. 8

4.3.2 Impact of Hierarchical Modeling.

• Replacing hyperbolic operations with Euclidean ones lowers performance, while predicted hierarchical relation types perform similarly to ground-truth types.The alternatives compare Euclidean operations, Krackhardt-score-predicted relations, and given hierarchical relation types.Source: §4.3.2 · p. 8, Table 4 · p. 8

4.3.3 Impact of Dual Item Embeddings.

• Removing collaborative item embeddings consistently degrades performance, with a larger decline on Alibaba-iFashion, where auxiliary KG information is smallest.The study compares dual embeddings against single knowledge embeddings used for user aggregation.Source: §4.3.3 · p. 8, Table 5 · p. 8

4.3.4 Impact of Model Depth.

• More aggregation layers generally improve performance, especially on Last-FM, but HAKG remains competitive at L=1 and is less depth-sensitive than other propagation methods [45] [47].The authors attribute its lower depth sensitivity to separate aggregation schemes and a gate that capture patterns from interactions and KG.Source: §4.3.4 · pp. 8–9, Table 6 · p. 8

4.4 Hierarchies Visualization (RQ3)

• The interaction visualization shows distance to origin increasing exponentially for less popular items, while the KG visualization displays hierarchical entity relations.Examples include bottoms categorized into pants, jeans, and skirt, and T-shirt as a type of tops; the authors report that these relations preserve item attributes in the embedding space.Source: §4.4 · p. 9, Figure 4 · p. 9

• Embedding methods use KG embeddings as item representations, path methods seek semantic user-item connectivity, and propagation methods aggregate neighborhood information end to end.The cited examples include CKE, KPRN [48], KGAT [45], CKAN, and KGIN [47].Source: §5 · p. 9
• Embedding approaches may miss high-order user-item dependence, while path methods require domain-specific patterns that can be time-consuming and generalize poorly [23].Source: §5 · p. 9
• Prior hyperbolic recommendation methods do not model KG relations at finer-grained hierarchical levels or preserve crucial high-order collaborative item signals [54].HAKG instead targets hierarchical relations for item profiling and uses dual item embeddings to propagate collaborative and knowledge information separately.Source: §5 · pp. 9–10

• HAKG captures underlying data hierarchy in hyperbolic space and characterizes items using hierarchical relations in the KG.Source: §6 · p. 10
• Experiments on three real-world datasets demonstrate HAKG's superiority.Source: §6 · p. 10
• Future work will investigate numerical errors in hyperbolic space and examine operations such as addition and multiplication there.Source: §6 · p. 10