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Where are we in embedding spaces? A Comprehensive Analysis on Network Embedding Approaches for Recommender Systems

Sixiao Zhang, Hongxu Chen, Xiao Ming, Lizhen Cui, Hongzhi Yin, Guandong Xu

arXiv:2105.08908v1cs.IRcs.LG

TL;DR

The paper addresses uncertainty about when hyperbolic space should be considered for recommender systems. It analyzes and compares hyperbolic and Euclidean embeddings across models, datasets, and latent sizes, reporting that hyperbolic space favors particular settings and that HSCML performs comparably or better than existing Euclidean social recommendation methods.

  • Problem

    It is unclear which recommender models and datasets suit hyperbolic space, and how its performance compares fairly with Euclidean space across latent sizes.

    Source: §1 · p. 2

  • Method

    The paper theoretically analyzes and empirically compares hyperbolic and Euclidean spaces across recommender models, datasets, and latent sizes, and proposes SCML and HSCML.

    Source: §1 · p. 2

  • Results

    Hyperbolic space favors distance models, low-density datasets, and small latent sizes, while Euclidean performance becomes comparable as dimensions grow; HSCML matches or exceeds existing Euclidean social methods.

    Source: §1 · p. 2, §4.3.2 · p. 7, §8 · p. 9

  • Takeaways & Limitations

    The paper’s guidance is to consider model type, dataset density, and latent size when choosing between hyperbolic and Euclidean spaces.

    Source: §1 · p. 2

    Hyperbolic operations require exponential and logarithmic maps for simple operations such as matrix multiplication and bias addition, adding computational complexity.

    Source: §4.3.3 · p. 7

Abstract

from arXiv · show

Hyperbolic space and hyperbolic embeddings are becoming a popular research field for recommender systems. However, it is not clear under what circumstances the hyperbolic space should be considered. To fill this gap, This paper provides theoretical analysis and empirical results on when and where to use hyperbolic space and hyperbolic embeddings in recommender systems. Specifically, we answer the questions that which type of models and datasets are more suited for hyperbolic space, as well as which latent size to choose. We evaluate our answers by comparing the performance of Euclidean space and hyperbolic space on different latent space models in both general item recommendation domain and social recommendation domain, with 6 widely used datasets and different latent sizes. Additionally, we propose a new metric learning based recommendation method called SCML and its hyperbolic version HSCML. We evaluate our conclusions regarding hyperbolic space on SCML and show the state-of-the-art performance of hyperbolic space by comparing HSCML with other baseline methods.

1 INTRODUCTION

Euclidean embeddings can distort hierarchical and scale-free interaction data, while increasing their dimensionality raises training and storage costs. The paper analyzes when hyperbolic spaces suit recommender models and datasets, and reports hypotheses about model type, dataset density, and latent size.

1 INTRODUCTION

  • Euclidean embeddings can distort scale-free and hierarchical data, so reducing distortion may require larger latent sizes and greater training and storage resources.Source: §1 · p. 1
  • Hyperbolic space expands exponentially and can preserve hierarchies, properties the paper connects to real-world data with tree-like structure and power-law distributions.The paper contrasts this with Euclidean space’s polynomial expansion.Source: §1 · pp. 1–2
  • The paper addresses missing guidance on when hyperbolic embeddings suit recommendation models and datasets through theoretical analysis and comparisons across six benchmark datasets and six models.Source: §1 · p. 2
  • Hyperbolic space is hypothesized to favor distance models, low-density datasets, and small latent sizes, with Euclidean performance becoming comparable as dimensions increase.The paper also proposes SCML and HSCML and reports that HSCML achieves state-of-the-art performance against baselines on two benchmark datasets.Source: §1 · p. 2

2 PRELIMINARIES

The preliminaries introduce the Poincaré ball as a learnable hyperbolic model, distinguish projection and distance latent-space models, and describe geometric operations used for optimization and neural layers.

2.1 The Poincaré Ball Model

  • The Poincaré ball is presented as suitable for gradient-based learning; its geodesics are circles orthogonal to the sphere, and its distance follows those geodesics.Its origin can serve as a hierarchy’s root, embeddings remain within the unit ball, and conformality supports inner products for matrix-factorization methods.Source: §2.1 · p. 2

2.2 Latent Space Model

  • Latent-space models represent nodes as embeddings and compare their relations; projection models use inner products, whereas distance models use distances between embeddings.The text identifies matrix factorization as a projection-model example and metric learning as a well-known distance-model family.Source: §2.2 · pp. 2–3

2.3 Poincare Ball Optimization and Geometry

  • Hyperbolic optimization maps embeddings to a tangent space for Euclidean operations, then uses exponential and logarithmic maps to move between the tangent space and hyperbolic space.Source: §2.3 · p. 3
  • A hyperbolic linear layer takes a hyperbolic input but uses Euclidean weights and bias defined on the tangent space, with weight multiplication and bias addition defined there.The preliminaries also define the Poincaré ball’s Riemannian metric, norm, and inner product.Source: §2.3 · p. 3

3 HYPOTHESES

The paper motivates three hypotheses about when hyperbolic embeddings may suit recommender systems better than Euclidean embeddings: by model type, dataset density, and latent size.

3 HYPOTHESES

  • Distance models are hypothesized to suit hyperbolic embeddings better because they can use the space beyond the center, unlike projection models whose norm regularization limits that use.The paper also argues that distance-based placement can represent hierarchical structure through layers around a root.Source: §3 · pp. 3–4
  • Hyperbolic space is hypothesized to perform better on low-density datasets, while Euclidean space may become comparable when dataset density is high.The proposed rationale is that hyperbolic space can use hierarchy, whereas Euclidean space needs more user-item pairs to reach comparable performance.Source: §3 · p. 4
  • Hyperbolic space is hypothesized to perform better at small latent sizes, while the performance gap is expected to shrink as latent size increases.The paper attributes this expectation to hyperbolic space’s ability to represent hierarchy with fewer dimensions and to both spaces sharing an upper bound at sufficiently large sizes.Source: §3 · p. 4

4 EVALUATING HYPERBOLIC SPACE

The evaluation compares Euclidean and hyperbolic versions of latent-space recommendation models across general-item and social recommendation, using six datasets to test hypotheses about model type, density, and latent size.

4.1 Baselines

  • Five standard projection and distance baselines span general-item and social recommendation; top-n social recommendation lacks distance models, motivating a new method.The baselines were chosen to compare spaces with little noise from auxiliary model components.Source: §4.1 · p. 4, Table 2 · p. 5

4.2 Datasets

  • The six datasets are MovieLens 1M/100K, Amazon Cellphones/Automotive, Epinions, and Ciao; MovieLens densities are about 4% and 6%, versus below 0.4% for the other four.Source: §4.2 · p. 5

4.3 Evaluation Metrics

  • Top-n tasks report HR/NDCG and rating tasks MAE/RMSE; evaluation uses leave-one-out or 60/20/20 splits, ranks all negative items, and averages five trials.Source: §4.3 · p. 5, §4.3.1 · p. 5
  • The experiment compares spaces rather than model rankings: Euclidean and hyperbolic settings share basic parameters, while space-specific parameters are tuned separately.Source: §4.3.1 · p. 5

4.3.2 Results.

  • Hyperbolic space favors distance over projection models, benefits lower-density datasets, and leads especially at small latent sizes before Euclidean space becomes comparable.These patterns are reported across the model, dataset-density, and latent-size comparisons.Source: §4.3.2 · pp. 6–7, Figure 2 · p. 6, Figure 3 · p. 6

4.3.3 Drawbacks of Hyperbolic Space.

  • Hyperbolic operations require exponential and logarithmic maps, increasing computation—especially for neural DMF versus simpler CML/SoRec—and inverse-hyperbolic calculations require clipping invalid values.The added computation is described as much greater for DMF, while it causes little trouble for CML and SoRec.Source: §4.3.3 · p. 7

4.3.4 Comments on Using Hyperbolic Space.

  • Use hyperbolic embeddings for distance models and low-density datasets; prefer Euclidean space for dense data, and choose relatively small hyperbolic latent sizes to save resources.The authors recommend Euclidean space for dense data because its performance is comparable and its computational complexity is lower.Source: §4.3.4 · p. 7

5 SOCIAL COLLABORATIVE METRIC LEARNING

SCML extends metric learning to top-n social recommendation by jointly modeling user–item interactions and social connections. Its hyperbolic version, HSCML, substitutes hyperbolic distance for Euclidean distance.

5 SOCIAL COLLABORATIVE METRIC LEARNING

  • SCML generalizes Collaborative Metric Learning for top-n social recommendation, addressing the stated lack of distance models for this task.The method is applied in both Euclidean and hyperbolic spaces; the hyperbolic version is called HSCML.Source: §5 · p. 8
  • The method samples positive and negative items and neighbors for each user to form separate item and social-connection losses.Positive pairs belong to R or S, respectively, while the corresponding negative pairs do not.Source: §5 · p. 8
  • SCML uses Euclidean distance, whereas HSCML uses hyperbolic distance, with safety margins and hinge loss defining the pairwise comparisons.The safety margins apply to the item and social losses.Source: §5 · p. 8
  • The item loss draws interacted users and items closer while pushing negative user–item pairs apart; the social loss draws socially connected users closer.The combined objective weights the social loss with λ.Source: §5 · p. 8

6 EVALUATING SCML AND HSCML

The evaluation compares SCML with HSCML and benchmarks HSCML against existing methods. HSCML is comparable to state-of-the-art embedding methods and outperforms other baselines; on the low-density datasets, hyperbolic performance leads SCML, while Euclidean performance catches up as latent size grows.

6 EVALUATING SCML AND HSCML

  • The section benchmarks SCML and HSCML against Euclidean baselines on Ciao and Epinions, aiming to assess whether HSCML can match leading methods with a simple distance model.Most baseline results are taken from RML-DGATs; the authors note that those datasets differ slightly from theirs and that the RML-DGATs implementation was unavailable.Source: §6 · p. 8, §6.1 · p. 8

6.1 Experiment Settings

  • The evaluation compares SCML and HSCML using HR and NDCG at cutoffs 1, 5, 10, 15, and 20, with λ=0.1 and tuned item and social margins.For the HSCML baseline comparison, latent size is 128; the experiments use 999 sampled negatives per user and average results over 10 runs.Source: §6.1 · p. 8

6.2 Experimental Results

  • On low-density Epinions and Ciao, HSCML clearly outperforms SCML, while Euclidean performance gradually catches up as latent size increases.The authors relate the comparison to SCML being a distance model and the datasets' low densities.Source: §6.2.1 · p. 8

6.2.1 SCML vs. HSCML.

  • HSCML is comparable to SAMN and RML-DGATs and outperforms other baseline approaches in top-n recommendation.The paper attributes this result to a simple hyperbolic distance model performing comparably or better than baselines that use memory modules and GAT units.Source: §6.2.2 · p. 9, Table 3 · p. 9

7 RELATED WORK

Prior work connects hyperbolic geometry with treelike, scale-free, and hierarchical structures, motivating its use in recommendation and related tasks. Recommender research has applied hyperbolic space to metric learning, neural models, scalable systems, heterogeneous networks, next-POI, and path-based recommendation.

7 RELATED WORK

  • Work on treelike graphs and hierarchical data relates hyperbolic space to real-world scale-free and hierarchical structures, providing a theoretical basis for applying it to link prediction, node classification, and recommendation.The cited research also studies the relationship between hyperbolic space and languages and complex networks.Source: §7 · p. 9
  • Recommendation-related applications include hyperbolic metric learning, neural and graph neural networks, scalable recommendation, heterogeneous-network link prediction, next-POI, and path-based recommendation.Examples named include HyperBPR [26], HyperML, hyperbolic graph neural networks [19], and a scalable hyperbolic recommender [3].Source: §7 · p. 9

8 CONCLUSION

The paper compares Euclidean and hyperbolic recommendation across methods, datasets, and latent sizes, and introduces SCML and HSCML for social recommendation. Experiments show HSCML can match or outperform existing Euclidean social recommendation methods.

8 CONCLUSION

  • The analysis compares hyperbolic and Euclidean spaces across recommendation methods, datasets, and latent sizes.The authors describe it as the first work to analyze hyperbolic space’s advantages and disadvantages and offer guidance on its use.Source: §8 · p. 9
  • SCML and its hyperbolic version HSCML are distance models proposed for social recommendation.Source: §8 · p. 9
  • HSCML can reach comparable or better performance than existing Euclidean social recommendation methods.Source: §8 · p. 9
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