Hyperbolic Neural Networks (1805.09112v2)

Abstract: Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural network layers. This makes it hard to use hyperbolic embeddings in downstream tasks. Here, we bridge this gap in a principled manner by combining the formalism of M\"obius gyrovector spaces with the Riemannian geometry of the Poincar\'e model of hyperbolic spaces. As a result, we derive hyperbolic versions of important deep learning tools: multinomial logistic regression, feed-forward and recurrent neural networks such as gated recurrent units. This allows to embed sequential data and perform classification in the hyperbolic space. Empirically, we show that, even if hyperbolic optimization tools are limited, hyperbolic sentence embeddings either outperform or are on par with their Euclidean variants on textual entailment and noisy-prefix recognition tasks.

• The paper’s main contribution is the principled extension of Euclidean neural networks into hyperbolic space using M"obius gyrovector formalism.
• It develops smooth adaptations of core components like FFNNs and GRUs by incorporating operations such as M"obius addition and scalar multiplication.
• Empirical evaluations demonstrate that hyperbolic models outperform Euclidean counterparts in hierarchical tasks, such as textual entailment.

Overview of "Hyperbolic Neural Networks"

This paper, authored by Ganea, Becigneul, and Hofmann, focuses on extending neural network architectures into hyperbolic space to leverage their inherent tree-like properties. The authors address the existing gap in the applicability of hyperbolic embeddings in downstream tasks by introducing hyperbolic versions of core deep learning components, such as multinomial logistic regression (MLR), feed-forward neural networks (FFNN), and recurrent neural networks (RNNs), including gated recurrent units (GRUs).

Contributions

The paper's primary contribution lies in the principled adaptation of deep learning tools from Euclidean to hyperbolic spaces. By employing the formalism of M\"obius gyrovector spaces alongside the Riemannian geometry of the Poincaré model, the authors propose a smooth framework for operations within hyperbolic space. This framework is capable of continuous deformation between Euclidean and hyperbolic geometries dependent on curvature.

Theoretical Framework

The theoretical basis involves the Poincaré ball model and gyrovector spaces. This allows geometric operations analogous to vector operations in Euclidean space but adapted to the constraints of hyperbolic geometry. The authors rigorously develop mathematical operations—such as M\"obius addition and scalar multiplication—that maintain the structure of hyperbolic space and extend into neural network computations.

Empirical Evaluation

The authors empirically demonstrate that hyperbolic sentence embeddings perform competitively, often surpassing Euclidean embeddings in tasks with an implicit hierarchical structure. Notable tasks include textual entailment and synthetic datasets modeling hierarchical noisy-prefix recognition. The experiments confirmed that hyperbolic models, particularly for GRUs and FFNNs, exhibit advantages when the underlying data geometry is more tree-like—a natural fit for the hyperbolic space's geometric properties.

Implications and Future Directions

The move to hyperbolic neural networks addresses challenges in embedding complex structures more naturally than Euclidean spaces. The potential for hyperbolic embeddings lies in areas where hierarchical and taxonomic data representation is crucial, opening pathways for more effective natural language processing and network analysis.

Future work could explore the development of optimization methods specific to hyperbolic spaces to improve the training of such networks. Non-convexity in hyperbolic spaces offers distinct challenges that warrant deeper investigation. Broader implications include the potential application in fields like biology and complex network analysis where inherent data structures align with hyperbolic geometry.

By bridging the gap between hyperbolic geometry's theoretical advantages and practical neural network applications, the paper provides a solid groundwork for future advancements in geometric deep learning.