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Quantization of Ricci Curvature in Information Geometry

Carlos C. Rodriguez

TL;DR

This paper proves the conjecture that the volume-averaged Ricci scalarcomputed with respect to the Fisher information metric is universally quantized to positive half-integers via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples.

Abstract

In 2004, while studying the information geometry of binary Bayesian networks (bitnets), the author conjectured that the volume-averaged Ricci scalar <R> computed with respect to the Fisher information metric is universally quantized to positive half-integers: <R> in (1/2)Z. This paper resolves the conjecture after 20 years. We prove it for tree-structured and complete-graph bitnets via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples. We extend the program to Gaussian DAG networks, where a sign dichotomy holds: discrete bitnets have positive curvature, while Gaussian networks form solvable Lie groups with negative curvature.

Quantization of Ricci Curvature in Information Geometry

TL;DR

This paper proves the conjecture that the volume-averaged Ricci scalarcomputed with respect to the Fisher information metric is universally quantized to positive half-integers via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples.

Abstract

In 2004, while studying the information geometry of binary Bayesian networks (bitnets), the author conjectured that the volume-averaged Ricci scalar <R> computed with respect to the Fisher information metric is universally quantized to positive half-integers: <R> in (1/2)Z. This paper resolves the conjecture after 20 years. We prove it for tree-structured and complete-graph bitnets via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples. We extend the program to Gaussian DAG networks, where a sign dichotomy holds: discrete bitnets have positive curvature, while Gaussian networks form solvable Lie groups with negative curvature.
Paper Structure (34 sections, 16 theorems, 27 equations, 3 tables)

This paper contains 34 sections, 16 theorems, 27 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The 20-year conjecture and the 2004 paper
  3. Corrections to the 2004 results
  4. Summary of new results
  5. Preliminaries: Bitnets and Their Geometry
  6. Bitnets
  7. Volume element and Ricci scalar
  8. Volumes and Curvatures of Classical Topologies
  9. Complete DAGs $\tilde{K}_n$
  10. Directed lines $\tilde{L}_n$ and exploding stars $\tilde{E}_n$
  11. Collapsing stars $\tilde{C}_{n+1}$ (new computations)
  12. Proof of Half-Integer Quantization for Trees
  13. Block-diagonal factorization
  14. The Beta cancellation identity
  15. Loop Topologies
  16. ...and 19 more sections

Key Result

Theorem 1.2

The directed line $\tilde{L}_n$ and the exploding star $\tilde{E}_n$ (one parent, $n-1$ children) have identical Fisher information volumes and identical volume-averaged Ricci scalars:

Theorems & Definitions (39)

  • Conjecture 1.1: Rodríguez, 2004
  • Theorem 1.2: Equal volumes and equal curvatures
  • proof
  • Remark 1.3: Error in 2004
  • Theorem 3.1: Complete DAGs
  • Theorem 3.2: Equal volumes and curvatures
  • Remark 3.3: Interpretation
  • Theorem 3.4: Ricci scalar of $\tilde{C}_{n+1}$
  • Theorem 3.5: Curvature sign inversion
  • proof
  • ...and 29 more