On Geodesic Triangles with Right Angles in a Dually Flat Space

Abstract

The dualistic structure of statistical manifolds in information geometry yields eight types of (possibly mixed type) 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 work, 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.

A. Notations

F :

Strictly convex and $C^3$ real-valued function

$F^*$ :

Dual Legendre-Fenchel convex conjugate

$\theta (p)=(\theta ^1(p),\ldots ,\theta ^D(p))$ :

primal coordinates of point p

$\eta (p)=(\eta _1(p),\ldots ,\eta _D(p))$ :

dual coordinates of point p

$\theta _{ab}=\theta _a-\theta _b$ :

notational shortcut

$\eta _{ab}=\eta _a-\eta _b$ :

notational shortcut

$D_F(p:q)$ :

Divergence between points

$B_F(\theta (p):\theta (q))$ :

Bregman divergence

$A_F(\theta (p):\eta (q))$ :

Fenchel-Young divergence

$(M,g,\nabla ,\nabla ^*)$ :

Dually flat space (Bregman manifold)

$T_p$ :

Tangent plane at p

$g_p(u,v)$ :

inner product between two vectors u and v of $T_p$

[$g_{ij}$:

=[$g(e_i,e_j)$]$_{ij}=\nabla ^{2F}(\theta )$] Riemannian metric

[${g^*}^{ij}$:

=[$g^*({e^*}^i,{e^*}^j)$]$_{ij}=\nabla ^2F^*(\eta )$] dual Riemannian metric

$\prod _{p,q}(v)$ :

primal parallel transport of $v\in T_p$ to $T_q$

$\prod _{p,q}^*(v)$ :

dual parallel transport of $v\in T_p$ to $T_q$

$\gamma _{ab}(t)$ :

Primal geodesic: $\theta (\gamma _{ab}(t))=(1-t)\theta (a)+t\theta (b)$

$\gamma _{ab}(t)^*$ :

Dual geodesic: $\eta (\gamma _{ab}^*(t))=(1-t)\eta (a)+t\eta (b)$

$(v)_B$ :

vector components in basis B, arranged in a D-tuple

[v:

$_B$] vector components in basis B, arranged in a D-dimensional column vector

$B_p=\{e_i=\partial _i=\frac{\partial }{\partial \theta ^i}\}$ :

natural basis at $T_p$

$B_p^*=\{{e^*}^i=\partial ^i=\frac{\partial }{\partial \eta _i}\}_i$ :

reciprocal basis at $T_p$ so that $g(e_i,{e^*}^j)=\delta _{i}^j$

$v_{ab}=\frac{d}{\mathrm {d}t}\gamma _{ab}(0)=\dot{\gamma }_{ab}(0)$ :

tangent vector of $\gamma _{ab}(t)$ at a with contravariant components $\theta (b)-\theta (a)$

$v_{ab}^*=\frac{d}{\mathrm {d}t}\gamma _{ab}^*(0)=\dot{\gamma }_{ab}^*(0)$ :

tangent vector of $\gamma _{ab}^*(t)$ at a with covariant components $\eta (b)-\eta (a)$

[$v^i$:

$_B$] contravariant components of vector v, $v^i=g(v,{e^*}^i)$

[$v_i$:

$_B$] covariant components of vector v (meaning $[v]_{B^*}$), $v_i=g(v,e_i)$

$g_p(u,v)$ :

inner product at $T_p$ of two vectors: $g_p(u,v)=u_i v^i=u^i v_i$.