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HyperML: A Boosting Metric Learning Approach in Hyperbolic Space for Recommender Systems
Lucas Vinh Tran, Yi Tay, Shuai Zhang, Gao Cong, Xiaoli Li
TL;DR
Recommender-system metric-learning approaches had been confined to Euclidean space, motivating exploration of hyperbolic user-item representations. The paper proposes HyperML, using Möbius gyrovector spaces and a distortion term, and reports that it outperforms Euclidean metric-learning baselines across ten benchmarks, with gains up to 32.32%.
Problem
Metric-learning approaches for personalized ranking had covered only Euclidean space, leaving hyperbolic user-item representations unexplored in recommender systems.
Source: §1 · p. 1Method
HyperML performs user-item metric learning in hyperbolic space using Möbius gyrovector operations and a distortion term to balance accuracy with distance preservation.
Source: §1 · pp. 1–2Results
Across ten public benchmarks, HyperML outperforms CML and LRML on all benchmarks, with gains up to 32.32% on standard ranking metrics.
Source: §1 · p. 2Takeaways & Limitations
The reported results demonstrate the effectiveness of hyperbolic geometry for personalized recommendation across the evaluated datasets.
Source: §1 · p. 2, §5 · p. 8The mapping f from Euclidean space to hyperbolic space is taken as an identity function.
Source: §2.3 · p. 3
Abstract
from arXiv · showhide
This paper investigates the notion of learning user and item representations in non-Euclidean space. Specifically, we study the connection between metric learning in hyperbolic space and collaborative filtering by exploring Mobius gyrovector spaces where the formalism of the spaces could be utilized to generalize the most common Euclidean vector operations. Overall, this work aims to bridge the gap between Euclidean and hyperbolic geometry in recommender systems through metric learning approach. We propose HyperML (Hyperbolic Metric Learning), a conceptually simple but highly effective model for boosting the performance. Via a series of extensive experiments, we show that our proposed HyperML not only outperforms their Euclidean counterparts, but also achieves state-of-the-art performance on multiple benchmark datasets, demonstrating the effectiveness of personalized recommendation in hyperbolic geometry.
1 INTRODUCTION
The paper introduces hyperbolic metric learning for recommender systems, extending metric-learning approaches that had been limited to Euclidean space. It proposes HyperML with a distortion term and reports gains over Euclidean metric-learning baselines across benchmark datasets.
1 INTRODUCTION
- Prior metric-learning recommenders, including CML and LRML, designed distance functions over users and items but covered only Euclidean space.The paper positions its work as the first exploration of metric learning in hyperbolic space for recommender systems.Source: §1 · p. 1
- HyperML applies metric learning in hyperbolic space and uses Möbius gyrovector spaces to formalize common Euclidean operations.These operations include addition, multiplication, and exponential and logarithmic maps.Source: §1 · pp. 1–2
- Hyperbolic space's exponential expansion means moving a point a given distance requires less force than in Euclidean space, motivating the pull-push mechanism.The authors state that this mechanism could boost performance significantly.Source: §1 · p. 1
- A distortion term helps preserve local and global distance relationships while multi-task learning controls performance.The authors describe this as balancing accuracy with preservation of distances.Source: §1 · pp. 1–2
- Across ten public benchmarks, HyperML outperforms CML and LRML on all benchmarks, with gains up to 32.32% on standard ranking metrics.The authors also report that HyperML outperforms its Euclidean counterparts and a suite of competitive baselines.Source: §1 · p. 2
2 HYPERBOLIC METRIC LEARNING
HyperML adapts metric learning to hyperbolic space, using its exponential expansion to model user-item relationships. It combines local pull-and-push learning with a distortion objective intended to preserve global structure.
2.1 Hyperbolic Geometry & Poincaré Embeddings
- Hyperbolic space expands exponentially, and HyperML places users near the origin with positive and negative items around them to represent a triplet as a tree.The Poincaré ball is chosen for its conformality and convenient parameterization.Source: §2.1 · p. 2
2.2 Gyrovector spaces
- Möbius gyrovector operations generalize distance and vector operations across curvature, recovering Euclidean distance as c approaches 0.The model uses exponential and logarithmic maps to connect the ball and tangent space; these maps also recover their Euclidean counterparts as c approaches 0.Source: §2.2 · p. 3
2.3 Model Formulation
- The pull-and-push hinge loss brings observed user-item pairs closer than unobserved pairs, subject to a safety margin.The loss uses observed implicit feedback and does not adopt the ranking-loss weight used in.Source: §2.3 · p. 3
- A distortion objective addresses global structure by minimizing differences between Euclidean pair distances and their mapped hyperbolic distances.The mapping f is the identity in this paper; the distortion objective is jointly trained with pull-and-push loss, balancing representation distortion against precision.Source: §2.3 · pp. 3–4
- HyperML learns parameters with projected Riemannian stochastic gradient descent, rescaling Euclidean gradients according to the Poincaré-ball metric.Updates use a retraction onto the ball; the gradient-scaling factor depends on the point's Euclidean distance from the origin.Source: §2.3 · p. 4
3 EXPERIMENTS
Across ten benchmark datasets, HyperML outperforms all baselines on HR@10 and nDCG@10. Further analyses show that performance depends on the scaling variable and distortion weight, while the learned embeddings display clustering structure.
3.1 Experimental Setup
- Evaluation used one-class collaborative filtering with nDCG@10 and HR@10 against 100 sampled negatives, comparing inner-product, neural-network, and metric-learning baselines.The last interacted item was held out for testing and the penultimate item for validation.Source: §3.1 · pp. 4–5
3.2 Experimental Results
- HyperML consistently outperforms all baselines on HR@10 and nDCG@10 across ten benchmark datasets.Table 2 compares the results on both ranking metrics.Source: Table 2 · p. 6, §3.2 · p. 6
3.3 Model Convergence Analysis
- After convergence, HyperML and CML form item-embedding spheres around users, whereas LRML’s item embeddings move in the opposite direction.HyperML’s user embeddings tend to lie closer to the origin than CML’s, while item embeddings show similar spread.Source: Figure 3 · p. 7, §3.3 · p. 7
3.4 Comparison of Hyperbolic Variants
- HyperML and HyperTS both achieve highly competitive results over Euclidean metric-learning models across Meetup, Clothing, Sports, and Cell phones.HyperTS applies optimization after mapping embeddings to the tangent space at the origin using the log0 map.Source: Table 3 · p. 7, §3.4 · p. 7
3.5 Effect of Scaling Variable
- The best scaling variable is c=0.5 on Goodbooks and c=2.0 on Games; HyperML’s performance oscillates at other tested values.The authors note that c should be set carefully for each dataset.Source: Table 4 · p. 7, §3.5 · p. 7
3.7 Metric Learning Visualization
- Despite training only on implicit interactions, HyperML’s MovieLens1M item embeddings cluster by genre, and nearby items tend to be liked by users with similar interests.The visualization also shows sphere-structured embeddings.Source: Figure 5 · p. 8, §3.7 · p. 8
3.6 Accuracy Trade-off with Different Multi-task Learning Weight
- The best γ differs by dataset—0.75 on Meetup and 1.0 on Automotive—and removing distortion (γ=0) lowers performance versus CML.The authors describe γ as trading off distortion minimization against model accuracy.Source: §3.6 · p. 8, Figure 4 · p. 8
4 RELATED WORK
Recommender systems use several kinds of matching functions, while prior work had not explored metric learning in hyperbolic space. Related work includes Euclidean distance-based models and hyperbolic representations in other applications.
4 RELATED WORK
- Recommender systems have used inner products, distance or similarity metrics, and neural networks as matching functions.Inner-product models factorize interactions, while neural models use stacked nonlinear transformations.Source: §4 · p. 8
- CML was proposed as a Euclidean-distance recommendation baseline in response to concerns that inner products lack expressiveness because they violate the triangle inequality.Recent approaches also use neural models to approximate interaction functions.Source: §4 · p. 8
- Hyperbolic representation learning has been used in other applications, but its use for metric learning in recommender systems was identified as unexplored.A hyperbolic user-item graph embedding existed, but it did not use metric learning.Source: §4 · p. 8
5 CONCLUSION
The paper introduces HyperML, a hyperbolic metric-learning recommendation model with a distortion term, and reports results on ten datasets that outperform Euclidean baselines including CML and LRML.
5 CONCLUSION
- HyperML is presented as the first recommender model to explore metric learning in hyperbolic space and adds a distortion term to control representations.The authors report experiments on ten datasets and results over Euclidean baselines, including CML and LRML.Source: §5 · p. 8
- Experiments on ten datasets show HyperML outperforming Euclidean baselines, including state-of-the-art metric-learning models CML and LRML.The authors suggest these results may inspire future recommendation research in hyperbolic space.Source: §5 · p. 8
- The authors suggest HyperML's results may inspire future work exploring hyperbolic space for recommendation.Source: §5 · p. 8