158
F. Nielsen
θ
η
θ
η
Fig. 7.4 The 2 3 = 8 types of geodesic triangles visualized both in the primal θ-coordinate system
and the dual η-coordinate system. Primal geodesic edges are shown in red and are straight in the
θ-coordinate system. Dual geodesic edges are shown in blue and are straight in the η-coordinate
system. Refer to Fig. 7.3 for point coordinates
In this paper, we raise and (partially) answer the following questions on a statistical
manifold:
Q1. When and how can one build geodesic triangles with one, two and three right
angles?
Q2. When and how can one build dual geodesic triangles T and T
∗ such that at a
triangle vertex, we have two pairs of dual geodesics emanating from that vertex
that are simultaneously orthogonal?
Q3. When is there a relationship or inequality between α(T ) and α(T
∗
)?
In this work, we are interested in studying these questions and unraveling some
properties of geodesic triangles in the particular case of dually flat spaces, Bregman
manifolds. In Bregman manifolds, the Pythagorean theorem allows one to check that
a pair of dual geodesics is orthogonal at a given point by checking some divergence
identity. In general, the Pythagorean theorem is useful to prove uniquess of projections in particular settings, interpret geometrically projections based on the divergence, and design derivative-free projection algorithms [1]. In information geometry,
F. Nielsen
θ
η
θ
η
Fig. 7.4 The 2 3 = 8 types of geodesic triangles visualized both in the primal θ-coordinate system
and the dual η-coordinate system. Primal geodesic edges are shown in red and are straight in the
θ-coordinate system. Dual geodesic edges are shown in blue and are straight in the η-coordinate
system. Refer to Fig. 7.3 for point coordinates
In this paper, we raise and (partially) answer the following questions on a statistical
manifold:
Q1. When and how can one build geodesic triangles with one, two and three right
angles?
Q2. When and how can one build dual geodesic triangles T and T
∗ such that at a
triangle vertex, we have two pairs of dual geodesics emanating from that vertex
that are simultaneously orthogonal?
Q3. When is there a relationship or inequality between α(T ) and α(T
∗
)?
In this work, we are interested in studying these questions and unraveling some
properties of geodesic triangles in the particular case of dually flat spaces, Bregman
manifolds. In Bregman manifolds, the Pythagorean theorem allows one to check that
a pair of dual geodesics is orthogonal at a given point by checking some divergence
identity. In general, the Pythagorean theorem is useful to prove uniquess of projections in particular settings, interpret geometrically projections based on the divergence, and design derivative-free projection algorithms [1]. In information geometry,