On geodesic triangles with right angles in a dually flat space (1910.03935v4)

Abstract: The dualistic structure of statistical manifolds in information geometry yields eight types of geodesic triangles passing through three given points, the triangle vertices. The interior angles of geodesic triangles can sum up to $\pi$ like in Euclidean/Mahalanobis flat geometry, or exhibit otherwise angle excesses or angle defects. In this paper, we initiate the study of geodesic triangles in dually flat spaces, termed Bregman manifolds, where a generalized Pythagorean theorem holds. We consider non-self dual Bregman manifolds since Mahalanobis self-dual manifolds amount to Euclidean geometry. First, we show how to construct geodesic triangles with either one, two, or three interior right angles, whenever it is possible. Second, we report a construction of triples of points for which the dual Pythagorean theorems hold simultaneously at a point, yielding two dual pairs of dual-type geodesics with right angles at that point.

• The paper presents novel techniques for constructing geodesic triangles with one or more right angles in dually flat (Bregman) spaces.
• It derives linear system methods and explores dual orthogonality conditions to determine the feasibility of triangulation in non-Euclidean settings.
• The study’s findings offer practical insights for applications in machine learning and data analysis, paving the way for future geometric research.

Geodesic Triangles with Right Angles in Dually Flat Spaces

The paper introduces and analyzes geodesic triangles within the context of dually flat spaces, also known as Bregman manifolds, providing a pivotal step in understanding the geometric implications of such frameworks in information geometry. The dual nature of these spaces allows for exploring unique properties of geodesic triangles, particularly those with right angles.

Overview

Dually flat spaces, a key construct in information geometry, are characterized by a dualistic structure involving a pair of affine connections that interact with a metric tensor. This duality allows for different geodesic connections between points—primal and dual geodesics. The paper systematically examines the formation and properties of geodesic triangles, where the triangles have vertices connected via these geodesics. Specifically, it explores constructing triangles with one, two, and potentially three right angles.

Construction of Right Angle Geodesic Triangles

In Euclidean geometry, the sum of the interior angles in a triangle is constant (π radians), but in dually flat spaces, the curvature can lead to angle excess or defect. The paper extends known geometric principles to these non-Euclidean frameworks, studying when and how such geodesic triangles can be constructed:

• Single Right Angle: The construction involves identifying a third point that satisfies specific orthogonality based on the primal geodesic constraints.
• Double Right Angle: The paper presents a meticulous method for achieving two right interior angles in a Bregman manifold, considerably more complex than the single. This involves solving a linear system of equations derived from the orthogonality conditions between geodesics.
• Triple Right Angles: The endeavor to construct triangles with three right angles is more theoretical as solving the involved non-linear equations often becomes intractable, necessitating further investigation.

Theoretical Implications

The paper further explores situations where dual Pythagorean theorems hold simultaneously at a point, leading to dual orthogonality conditions. This dual orthogonality could theoretically result in Bregman manifolds showcasing unique geometric properties absent in self-dual manifolds like Euclidean geometry.

Numerical Examples and Future Directions

The paper provides numerical examples, especially in the Itakura-Saito manifold, serving as a concrete demonstration of theoretical constructs. This approach highlights potential applications in machine learning and data analysis, where understanding the geometric disposition of data in manifolds could be critical.

Theoretical exploration opens avenues for future research, including extending principles to higher-dimensional geometric constructs (e.g., geodesic quadrangles) and understanding their implications in statistical manifold contexts, especially for spherical-like geometries involved in complex data analysis.

Conclusion

By navigating the intricacies of angle construction in geodesic triangles within Bregman manifolds, this work contributes to a deeper comprehension of the underlying geometry in information spaces. Such insights hold substantial promise for advancing theoretical foundations and practical applications in areas demanding high precision in non-Euclidean spaces. Future work can leverage these foundations to explore higher-dimensional problems, potentially integrating with computational geometry and topology-based data approaches.