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Adaptive indefinite kernels in hyperbolic spaces.

Pengfei Fang1

  • 1School of Computer Science and Engineering, Southeast University, Nanjing, 210096, China; Key Laboratory of New Generation Artificial Intelligence Technology and Its Interdisciplinary Applications (Southeast University), Ministry of Education, China.

Neural Networks : the Official Journal of the International Neural Network Society
|October 24, 2024
PubMed
Summary
This summary is machine-generated.

This study introduces adaptive indefinite kernels for hyperbolic embeddings, enhancing data hierarchy representation. These novel kernels outperform traditional positive definite kernels across various machine learning tasks.

Keywords:
Data hierarchyHyperbolic spaceIndefinite Lorentz kernelsIndefinite Poincaré kernels

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Area of Science:

  • Machine Learning
  • Geometric Deep Learning
  • Kernel Methods

Background:

  • Hyperbolic embeddings leverage negative curvature for data hierarchy.
  • Kernelization aims to boost hyperbolic embedding representation power.
  • Existing methods use positive definite kernels, potentially limiting hyperbolic space properties.

Purpose of the Study:

  • Develop adaptive indefinite kernels for hyperbolic embeddings.
  • Address limitations of positive definite kernels in indefinite spaces like Kreĭn space.
  • Enhance representation power by utilizing Kreĭn space structures.

Main Methods:

  • Propose an adaptive embedding function in the Lorentz model.
  • Define indefinite Lorentz kernels (iLks) based on the embedding function.
  • Extend iLks to the Poincaré ball as indefinite Poincaré kernels (iPKs).

Main Results:

  • Demonstrate significant performance gains over baseline methods.
  • Showcase improved representation power compared to positive definite kernels.
  • Validate effectiveness across diverse learning scenarios like image classification and few-shot learning.

Conclusions:

  • Adaptive indefinite kernels effectively utilize Kreĭn space structures.
  • The proposed indefinite kernels offer superior representation power in hyperbolic learning.
  • This work advances kernel methods for hyperbolic representation learning.