An invitation to tropical Alexandrov curvature
File(s)
Accepted version
Author(s)
Améndola, Carlos
Monod, Anthea
Type
Journal Article
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.
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.
Date Issued
2024-05-16
Date Acceptance
2023-02-09
Citation
Algebraic Statistics, 2024, 14 (2), pp.181-214
ISSN
2693-2997
Publisher
Mathematical Sciences Publishers (MSP)
Start Page
181
End Page
214
Journal / Book Title
Algebraic Statistics
Volume
14
Issue
2
Copyright Statement
© 2023 MSP (Mathematical Sciences Publishers).
Identifier
http://arxiv.org/abs/2105.07423v3
Subjects
14T90
math.AG
math.MG
math.MG
Notes
30 pages, 17 figures, 5 tables
Publication Status
Published