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A Note on Divergences.

Xiao Liang1

  • 1Institute of Technology, University of Washington Tacoma, Tacoma, WA 98402, U.S.A. xlianguw@uw.edu.

Neural Computation
|August 25, 2016

View abstract on PubMed

Summary
This summary is machine-generated.

A counterexample disproves Amari's conjecture on divergences in neural computation. The study identifies conditions under which weighted f-divergences that are also Bregman divergences become weighted [Formula: see text]-divergences.

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Area of Science:

  • Neural computation
  • Information theory
  • Machine learning

Background:

  • Divergences are crucial for neural computation tasks like learning and inference.
  • Amari (2009) proposed a conjecture regarding these divergences.

Purpose of the Study:

  • To investigate Amari's conjecture.
  • To explore weighted f-divergences and weighted [Formula: see text]-divergences.
  • To establish relationships between different types of divergences.

Main Methods:

  • Mathematical analysis of divergence properties.
  • Development of a counterexample to Amari's conjecture.
  • Proof of a specific relationship between weighted f-divergences, Bregman divergences, and weighted [Formula: see text]-divergences.

Main Results:

  • A counterexample is provided, demonstrating that Amari's conjecture does not hold universally.
  • It is proven that any divergence that is both a weighted f-divergence and a Bregman divergence must be a weighted [Formula: see text]-divergence.
  • This finding simplifies to Amari's (2009) main theorem under specific conditions ([Formula: see text] = [Formula: see text]).

Conclusions:

  • Amari's conjecture requires refinement as it is not generally true.
  • The study clarifies the relationship between weighted f-divergences, Bregman divergences, and weighted [Formula: see text]-divergences.
  • The results contribute to a deeper understanding of divergence measures in neural computation.