Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design.

Area of Science:

• Machine Learning
• Deep Learning
• Computational Geometry

Background:

• Hyperbolic neural networks excel at representing hierarchical data but face challenges due to the nonlinear nature of hyperbolic space.
• Existing methods often rely on local linearization, limiting their effectiveness in capturing intrinsic hyperbolic geometry.
• The Lorentz group defines natural isometric transformations in hyperbolic spaces, crucial for preserving data structure.

Purpose of the Study:

• To present a novel fully hyperbolic neural network architecture.
• To introduce an isometric and equivariant projection for dimensionality reduction within hyperbolic space.
• To develop a fully hyperbolic graph convolutional neural network leveraging this projection.

Main Methods:

• A novel projection method is proposed to embed data into a lower-dimensional hyperbolic space, creating a nested hyperbolic representation.
• The projection is proven to be isometric and equivariant under Lorentz transformations, enabling computational efficiency and weight sharing.
• A fully hyperbolic graph convolutional neural network architecture is developed to learn projection parameters.

Main Results:

• The proposed nested hyperbolic space representation demonstrates effectiveness in dimensionality reduction, outperforming methods like tangent PCA, principal geodesic analysis (PGA), and HoroPCA.
• The network achieves comparative performance on several publicly available datasets.
• The isometric and equivariant properties of the projection facilitate efficient computation and weight sharing.

Conclusions:

• The novel fully hyperbolic neural network with its unique projection offers an effective solution for hierarchical data representation and dimensionality reduction.
• The isometric and equivariant embedding provides a computationally efficient and theoretically sound foundation for hyperbolic deep learning.
• The developed architecture shows promise for various machine learning tasks involving complex, hierarchical data.