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Random geometric graphs in high dimension.

Vittorio Erba1,2, Sebastiano Ariosto1, Marco Gherardi1,2

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This study analyzes random geometric graphs in high-dimensional spaces, crucial for machine learning. It reveals that soft graphs approximate Erdös-Rényi models, while hard graphs show deviations, offering insights into data analysis.

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Area of Science:

  • Machine Learning
  • Graph Theory
  • High-Dimensional Data Analysis

Background:

  • Machine learning algorithms often rely on nearest neighbor computations to define geometric graphs.
  • Existing research primarily focuses on low-dimensional random geometric graphs.
  • High-dimensional data is prevalent in modern machine learning applications.

Purpose of the Study:

  • To analyze the infinite-dimensional limit of hard and soft random geometric graphs.
  • To compute the average number of k-cliques in these graphs.
  • To understand the behavior of local observables in high-dimensional spaces.

Main Methods:

  • Consideration of the infinite dimensions limit for random geometric graphs.
  • Analytical computation of the average number of k-cliques.
  • Comparison of soft and hard random geometric graph ensembles.
  • Numerical validation for dimensions d≳10.

Main Results:

  • Soft random geometric graphs with continuous activation functions converge to Erdös-Rényi graph limits.
  • Hard random geometric graphs can exhibit systematic deviations from these limits.
  • Analytical results derived for infinite dimensions provide good approximations for high dimensions.

Conclusions:

  • The choice of ensemble (soft vs. hard) significantly impacts local observables in high-dimensional random geometric graphs.
  • Soft graphs offer a more predictable behavior, aligning with established models.
  • The study provides a theoretical framework and practical approximations for analyzing high-dimensional graph structures in machine learning.