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An Invitation to Tropical Alexandrov Curvature

Carlos Améndola, Anthea Monod

Abstract

We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric, which has been useful in various statistical analyses, particularly in phylogenomics. Alexandrov curvature is a generalization of classical Riemannian sectional curvature to more general metric spaces; it is determined by a comparison of triangles in an arbitrary metric space to corresponding triangles in Euclidean space. In the polyhedral setting of tropical geometry, triangles are a combinatorial object, which adds a combinatorial dimension to our analysis. We study the effect that the triangle types have on curvature, and what can be revealed about these types from the curvature. We find that positive, negative, zero, and undefined Alexandrov curvature can exist concurrently in tropical settings and that there is a tight connection between triangle combinatorial type and curvature. Our results are established both by proof and computational experiments, and shed light on the intricate geometry of the tropical projective torus. In this context, we discuss implications for statistical methodologies which admit inherent geometric interpretations. This paper is dedicated to Bernd Sturmfels on the occasion of his 60th birthday.

An Invitation to Tropical Alexandrov Curvature

Abstract

We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric, which has been useful in various statistical analyses, particularly in phylogenomics. Alexandrov curvature is a generalization of classical Riemannian sectional curvature to more general metric spaces; it is determined by a comparison of triangles in an arbitrary metric space to corresponding triangles in Euclidean space. In the polyhedral setting of tropical geometry, triangles are a combinatorial object, which adds a combinatorial dimension to our analysis. We study the effect that the triangle types have on curvature, and what can be revealed about these types from the curvature. We find that positive, negative, zero, and undefined Alexandrov curvature can exist concurrently in tropical settings and that there is a tight connection between triangle combinatorial type and curvature. Our results are established both by proof and computational experiments, and shed light on the intricate geometry of the tropical projective torus. In this context, we discuss implications for statistical methodologies which admit inherent geometric interpretations. This paper is dedicated to Bernd Sturmfels on the occasion of his 60th birthday.

Paper Structure

This paper contains 21 sections, 9 theorems, 74 equations, 17 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Motivation: Tropical Statistics and Metric Geometry
  3. The Tropical Projective Torus, Tropical Metric, and Tropical Line Segments
  4. Curvature: Classical and Alexandrov
  5. Curvature for Curves
  6. Alexandrov Curvature
  7. Triangle Sizes.
  8. Related Notions: Combinatorial and Discrete Curvature
  9. Negative and Positive Alexandrov Curvature in ${\mathbb R}^3/{\mathbb R}{\bf{1}}$: A Sampling Procedure and a Theorem
  10. Computing tropical and Euclidean distances between the vertex $a$ and the $bc$ edge.
  11. Computing tropical and Euclidean distances between the vertex $a$ and the $bc$ edge.
  12. Combinatorial Types of Tropical Triangles in the Plane
  13. Regions of Skinniness and Fatness: Sampling Tropical Triangles
  14. A Procedure for Sampling Triangles.
  15. Fatness Everywhere: Triangle type T3
  16. ...and 6 more sections

Key Result

Proposition \oldthetheorem

For $n \geq 3$, $({\mathbb R}^{n-1}, \| \cdot \|_{\mathrm{tr}})$ is not a Hilbert space.

Figures (17)

  • Figure 1: Types of Tropical Line Segment: (a) L1; (b) L2; (c) L3.
  • Figure 2: An example of a skinny triangle in the sense of Alexandrov ollivier2011visual.
  • Figure 3: (a) Example of a skinny tropical triangle; (b) Corresponding Euclidean comparison triangle.
  • Figure 4: Tropical distance function vs. Euclidean distance function from Example \ref{['ex:eqskinny']}
  • Figure 5: (a) Example of a fat tropical triangle; (b) Corresponding Euclidean comparison triangle.
  • ...and 12 more figures

Theorems & Definitions (33)

  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Example \oldthetheorem
  • ...and 23 more