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HRCF: Enhancing Collaborative Filtering via Hyperbolic Geometric Regularization

Menglin Yang, Min Zhou, Jiahong Liu, Defu Lian, Irwin King

arXiv:2204.08176v2cs.IRcs.AI

TL;DR

Euclidean embeddings can poorly represent tree-like, scale-free user-item networks, while hyperbolic space offers roomier geometry for such data. HRCF uses root alignment and an origin-aware penalty to regularize hyperbolic collaborative filtering. The authors report that it outperforms leading Euclidean and hyperbolic baselines and alleviates oversmoothing caused by hyperbolic aggregation.

  • Problem

    Euclidean latent spaces are limited in representing complex, tree-like user-item patterns, while prior hyperbolic recommender approaches seldom explicitly exploit hyperbolic geometry’s inherent advantages.

    Source: §1 · pp. 1–2

  • Method

    HRCF regularizes hyperbolic collaborative filtering by aligning the root with the origin and applying an origin-aware penalty that pushes embeddings away from the origin.

    Source: §1 · p. 2, §4.2 · p. 5

  • Results

    HRCF is reported to outperform leading graph-based hyperbolic and Euclidean models and to alleviate oversmoothing caused by hyperbolic aggregation.

    Source: §1 · p. 2, §6 · p. 8

  • Takeaways & Limitations

    The authors present HRCF as a robust, deeper, and lightweight neural graph collaborative filtering model, with the idea theoretically applicable to other hyperbolic models.

    Source: §6 · p. 8

    The geometric analysis assumes equal-norm points whose common parent is the hyperbolic origin.

    Source: §4.4 · p. 6

Abstract

from arXiv · show

In large-scale recommender systems, the user-item networks are generally scale-free or expand exponentially. The latent features (also known as embeddings) used to describe the user and item are determined by how well the embedding space fits the data distribution. Hyperbolic space offers a spacious room to learn embeddings with its negative curvature and metric properties, which can well fit data with tree-like structures. Recently, several hyperbolic approaches have been proposed to learn high-quality representations for the users and items. However, most of them concentrate on developing the hyperbolic similitude by designing appropriate projection operations, whereas many advantageous and exciting geometric properties of hyperbolic space have not been explicitly explored. For example, one of the most notable properties of hyperbolic space is that its capacity space increases exponentially with the radius, which indicates the area far away from the hyperbolic origin is much more embeddable. Regarding the geometric properties of hyperbolic space, we bring up a Hyperbolic Regularization powered Collaborative Filtering(HRCF) and design a geometric-aware hyperbolic regularizer. Specifically, the proposal boosts optimization procedure via the root alignment and origin-aware penalty, which is simple yet impressively effective. Through theoretical analysis, we further show that our proposal is able to tackle the over-smoothing problem caused by hyperbolic aggregation and also brings the models a better discriminative ability. We conduct extensive empirical analysis, comparing our proposal against a large set of baselines on several public benchmarks. The empirical results show that our approach achieves highly competitive performance and surpasses both the leading Euclidean and hyperbolic baselines by considerable margins.

1 INTRODUCTION

Collaborative filtering uses user-item interactions to predict preferences, but Euclidean embeddings may poorly fit the scale-free, tree-like structure of recommendation networks. HRCF uses hyperbolic geometry and regularization to exploit that structure, with theoretical and empirical benefits reported by the authors.

1 INTRODUCTION

  • Euclidean embeddings are limited in representing complex patterns in low dimensions, and even unbounded dimensions cannot achieve comparably low distortion for tree-like data [31, 24].The passage relates user preferences and item popularity to power-law distributions and tree-like structure.Source: §1 · pp. 1–2
  • Hyperbolic space grows exponentially in volume with radius, offering more room for scale-free data; embeddings near its boundary are reported to be easier to distinguish.The introduction connects this property to inactive users and unpopular items that could be optimized away from the origin.Source: §1 · p. 2
  • Root alignment and an origin-aware penalty push embeddings away from the hyperbolic origin, which the authors say enhances latent hierarchy formation and makes fuller use of embedding space.The theoretical analysis links enlarged node distances to alleviating feature shrinking or oversmoothing from hyperbolic aggregation.Source: §1 · p. 2
  • HRCF’s geometric regularization is reported to outperform leading graph-based hyperbolic and Euclidean models and to address oversmoothing caused by hyperbolic graph aggregation.The authors also report robust performance with smaller embeddings, fewer training epochs, and deeper graph convolutional layers.Source: §1 · p. 2

2 RELATED WORK

The related work moves from graph neural networks for collaborative filtering to hyperbolic graph learning and then hyperbolic recommendation. It identifies limited consideration of hyperbolic geometry’s inherent advantages in existing recommender approaches.

2.1 Graph Neural Networks

  • Graph neural networks encode collaborative signals from user-item interactions through propagation, while existing recommendation GNNs are mainly Euclidean and may underestimate the network’s implicit distribution.The passage distinguishes spectral methods, which use graph-Laplacian eigendecomposition, from spatial methods, which aggregate local neighborhoods.Source: §2.1 · p. 2

2.2 Hyperbolic Graph Neural Networks

  • Hyperbolic GNNs include tangent-space graph convolutions in HGNN, HGCN, and HGAT, and manifold-based aggregation in LGCN; HGCL applies contrastive learning.The cited approaches address tasks or mechanisms including classification, dynamic embeddings, attention, learnable curvature, and Lorentzian-distance centroids.Source: §2.2 · p. 3

2.3 Hyperbolic Recommender Systems

  • Hyperbolic recommender studies cover metric learning, social recommendation, multi-view next-POI recommendation, neighborhood aggregation, and knowledge-aware attention, but usually generalize Euclidean models without fully considering hyperbolic geometry’s inherent advantages.Examples include HyperML, HSCML [50], HyperSoRc, HME, HGCF, and LKGR.Source: §2.3 · p. 3

3 PRELIMINARIES

The preliminaries introduce hyperbolic geometry through the hyperboloid (Lorentz) model and its tangent-space maps. The paper fixes the curvature parameter to 1, giving curvature −1.

3 PRELIMINARIES

  • A tangent space at a hyperboloid point is an n-dimensional vector space, and exponential and logarithmic maps provide projection between tangent and hyperbolic spaces.The logarithmic map is stated to be the inverse of the exponential map.Source: §3 · p. 3
  • The hyperboloid manifold, also called the Lorentz model, is an n-dimensional Riemannian manifold defined by a Lorentz inner product constraint and a positive zeroth coordinate.The paper gives its curvature as −1/K for K>0.Source: §3 · p. 3
  • For simplicity, the paper fixes K=1, so the hyperbolic curvature is −1 throughout the following sections.Source: §3 · p. 4

4 METHODOLOGY

HRCF extends hyperbolic graph collaborative filtering with root alignment and an origin-aware regularizer. Its theoretical analysis addresses aggregation-induced oversmoothing and examines how moving embeddings outward affects geometric distortion.

4.1 (Hyperbolic) Graph Neural Collaborative Filtering

  • Hyperbolic graph collaborative filtering initializes embeddings, aggregates neighborhood messages, pools across layers, and predicts preferences using hyperbolic distance.The initial states are sampled in the tangent space and mapped to hyperbolic space; nearer user-item embeddings indicate greater preference.Source: §4.1 · p. 4

4.2 Hyperbolic Regularized Collaborative Filtering (HRCF)

  • The embedding center, or midpoint, serves as the root, giving a unique task-relevant center that can accommodate networks with single or multiple roots.The midpoint is defined as the centroid minimizing summed squared tangential distances and costs O(n) to compute.Source: §4.2 · p. 5
  • HRCF aligns a root with the hyperbolic origin and regularizes embeddings away from it to retain internal dependencies while using hyperbolic space.The regularizer prevents each item or user embedding norm from becoming too small.Source: §4.2 · p. 5

4.3 Theoretical Analysis

  • The analysis states that tangent-space operations and the exponential map keep embeddings in the tangent space or hyperbolic space, respectively, without violating Lorentz constraints.Aggregation, sum-pooling, root computation, and root alignment are described as weighted sums that preserve the tangent-space condition.Source: §4.3 · p. 5
  • Hyperbolic neighborhood aggregation shrinks connected-node distances, while minimizing Lreg enlarges pairwise distances and prevents aggregation-induced distance decay.Repeated aggregation can bring both connected nodes and nodes sharing neighbors toward similar representations, producing oversmoothing.Source: §4.3 · p. 5

4.4 Geometric Analysis

  • As nodes move farther from the origin, their pairwise hyperbolic distance approaches the sum of their distances from the origin, which can reduce distortion.The corresponding ratio does not change with movement in Euclidean space.Source: §4.4 · p. 6

5 EXPERIMENTS

Across three public recommendation datasets, HRCF outperforms the compared baselines on Recall@K and NDCG@K. Additional experiments report advantages at smaller embedding sizes, across aggregation orders, and with fewer training epochs.

5.1 Experimental Settings

  • Evaluation uses three public datasets, binarizes ratings of at least 4, and compares hyperbolic, Euclidean, factorization, and metric-learning baselines.The experiments use 80/20 train-test splits; settings include 50-dimensional embeddings and 500 epochs. The added regularization is reported to bring negligible computation costs.Source: §5.1 · p. 6

5.2 Overall Performance

  • HRCF comprehensively outperforms the baselines on all datasets across Recall@K and NDCG@K.Table 2 marks the best and second-best results and identifies statistically significant differences between them.Source: §5.2 · p. 6, Table 2 · p. 7
  • Hyperbolic models with ranking loss outperform LightGCN, while HRCF further widens the gap over matrix-factorization and metric-learning methods.The cross-entropy hyperbolic baselines HAE and HVAE are reported as barely satisfactory.Source: §5.2 · pp. 6–7

5.3 Lightweightness w.r.t. Embedding Size

  • HRCF continues to outperform HGCF as embedding dimensions shrink, although smaller embeddings reduce computation burden while degrading performance.The advantage is reported as especially clear on Amazon-CD at dimension 20, supporting the authors’ lightweight-recommender finding.Source: §5.3 · p. 7, Figure 3 · p. 8

5.4 Robustness w.r.t Aggregation Order

  • Across aggregation orders, HRCF outperforms HGCF; HRCF performance rises then stabilizes, whereas HGCF drops suddenly past a threshold.The authors report that HRCF’s increase or saturation with aggregation order supports its use with deeper layers.Source: §5.4 · p. 7, Figure 4 · p. 8

5.5 Convergent Speed w.r.t Training Epoch

  • HRCF repeatedly outperforms HGCF across training epochs and shows larger relative improvement with fewer epochs.The authors interpret the few-epoch results as evidence that HRCF can speed training convergence.Source: §5.5 · pp. 7–8, Table 3 · p. 7

6 CONCLUSION

HRCF uses hyperbolic geometry to place embeddings farther from the origin while maintaining latent structure. The authors conclude that this regularization improves separation, alleviates oversmoothing, and performs competitively against leading Euclidean and hyperbolic models.

6 CONCLUSION

  • HRCF aligns the embedding midpoint with the hyperbolic origin and maximizes the norms of overall embeddings to use the more spacious region while maintaining latent structure.The paper states that hyperbolic volume increases exponentially with radius.Source: §6 · p. 8
  • Theoretical analysis links greater discrepancy between preferred and non-preferred items to alleviating graph-model oversmoothing and supporting deeper hyperbolic graph collaborative filtering.Source: §6 · p. 8
  • Experiments show highly competitive performance against leading graph-based hyperbolic and Euclidean models, and the authors describe HRCF as robust, deeper, and lightweight.The authors say the idea is decoupled from collaborative filtering and could theoretically apply to other hyperbolic models.Source: §6 · p. 8

A PERFORMANCE ON HEAD AND TAIL ITEMS

The section evaluates HRCF separately on popular head items and less-popular tail items. It reports a larger improvement for tail items, while noting that its degree-based split is approximate.

A PERFORMANCE ON HEAD AND TAIL ITEMS

  • Items are ranked by degree: the top 20% are head items, and the remainder are grouped as tail items.The authors caution that this split does not precisely separate popular and unpopular items.Source: A PERFORMANCE ON HEAD AND TAIL ITEMS · p. 10
  • HRCF improves recommendations for both head and tail items, with a substantially larger impact on tail items.The authors attribute the tail gains to placing long-tail nodes farther from the origin, where they have more room to separate.Source: A PERFORMANCE ON HEAD AND TAIL ITEMS · p. 10, Table 4 · p. 10
  • Table 4 reports head- and tail-item performance using Recall@10 and NDCG@10 on Amazon-CD and Amazon-Book.Source: Table 4 · p. 10

B.1 Proof of Proposition 1

The proof invokes Definition 3 to evaluate a point at the hyperbolic origin and identifies the resulting point as lying in the origin’s tangent space.

B.1 Proof of Proposition 1

  • Using Definition 3, the proof considers the inner product of x_E at the hyperbolic origin and concludes that x_E belongs to the tangent space T_oH^n.Source: §B.1 · p. 10

B.2 Proof of Proposition 2

The proof begins by considering the deviation for a point x_T in the tangent space at the origin.

B.2 Proof of Proposition 2

  • For x_T in the tangent space at the origin, the proof sets up a deviation calculation.The supplied passage does not state the resulting conclusion.Source: §B.2 · p. 10

B.3 Proof of Proposition 3

The proofs establish that the exponential map produces points satisfying the Lorentz constraints, describe aggregation at the origin’s tangent space, and relate norm maximization to enlarged pairwise distances.

B.3 Proof of Proposition 3

  • With K = 1, the exponential map at origin o maps the input point into H_K^n, satisfying the Lorentz constraints.This is the conclusion of the proof of Proposition 3.Source: §B.3 · p. 10

B.4 Proof of Theorem 1

  • The proof states that aggregation is applied on the tangent space at the origin according to Proposition 2, then invokes Wang and Leskovec [41]’s Theorem 4.Source: §B.4 · p. 10

B.5 Proof of Proposition 4

  • Proposition 4 proof states that maximizing the average norm enlarges distances across node pairs by 1/N, while the Lreg weight also emphasizes those distances.The passage attributes the enlargement to a coefficient of 1/N and notes the additional emphasis from the hyperparameter before Lreg.Source: §B.5 · p. 10

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