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Researchers introduced an extended geometric Jensen-Shannon divergence (G-JSD) for positive densities, offering a new tool for machine learning and information sciences. This extended G-JSD provides a more general approach than the standard G-JSD, with applications in Gaussian distribution analysis.

Keywords:
Bhattacharyya distanceChernoff informationJeffreys divergenceJensen–Shannon divergenceTaneja divergenceexponential familiesf-divergencegeometric mixturesinformation monotonicityprojective γ-divergencesquasi-arithmetic meansseparable divergencetotal variation distance

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Area of Science:

  • Information Theory
  • Machine Learning
  • Mathematical Statistics

Background:

  • The geometric Jensen-Shannon divergence (G-JSD) is widely used due to its closed-form solution for Gaussian distributions.
  • Existing G-JSD definitions have limitations, particularly with positive densities and normalization of geometric mixtures.

Purpose of the Study:

  • To introduce a novel, extended geometric Jensen-Shannon divergence (extended G-JSD) applicable to positive densities and measures.
  • To analyze the relationship between the extended G-JSD, the standard G-JSD, and other divergence measures like Jeffreys divergence and Bhattacharyya distance.
  • To explore the properties and applications of the extended G-JSD, including its behavior with Gaussian distributions and its potential as a regularization technique.

Main Methods:

  • Developed a new definition for the geometric Jensen-Shannon divergence, termed extended G-JSD, for positive densities.
  • Derived closed-form formulas for both G-JSD and extended G-JSD for multivariate Gaussian distributions.
  • Investigated the properties of the extended G-JSD, including its classification as an f-divergence and its information geometry characteristics.
  • Explored Monte Carlo estimations and approximations using projective γ-divergences.

Main Results:

  • The extended G-JSD is defined for positive densities without normalizing geometric mixtures, generalizing the standard G-JSD.
  • Explicit expressions for the gap between extended G-JSD and G-JSD are provided for probability densities.
  • Both G-JSD and extended G-JSD can be expressed using Jeffreys divergence and Bhattacharyya distance/coefficient.
  • The extended G-JSD is an f-divergence, satisfying information monotonicity and invariance properties.
  • Closed-form formulas for multivariate Gaussian distributions were derived for both divergences.
  • Unlike the square root of the standard Jensen-Shannon divergence, the square roots of G-JSD and extended G-JSD do not form metric distances.

Conclusions:

  • The extended G-JSD offers a more versatile and generalizable divergence measure for positive densities and measures.
  • The derived formulas and properties facilitate practical applications, especially with Gaussian distributions in machine learning.
  • The study highlights the interpretability of G-JSDs as regularizations of the ordinary JSD.