7 On Geodesic Triangles with Right Angles in a Dually Flat Space
183
Multiplying Eq. 7.123 by sqr(θ
x
p )sqr(θ
y
p ), Eq. 7.124 by sqr(θ
x
q )sqr(θ
y
q ), and Eq. 7.125
by sqr(θ
x
r )sqr(θ
y
r ), we get a system of polynomial equations to solve [30]:
(θ
x
q − θ
x
p )sqr(θ
y
p )(θ
x
r − θ
x
p ) + (θ
y
q − θ
y
p )sqr(θ
x
p )(θ
y
r − θ
y
p ) = 0, (7.126)
(θ
x
r − θ
x
q )sqr(θ
y
q )(θ
x
p − θ
x
q ) + (θ
y
r − θ
y
q )sqr(θ
x
q )(θ
y
p − θ
y
q ) = 0, (7.127)
(θ
x
p − θ
x
r )sqr(θ
y
r )(θ
x
q − θ
x
r ) + (θ
y
p − θ
y
r )sqr(θ
x
r )(θ
y
q − θ
y
r ) = 0, (7.128)
where sqr(x) = x
2 . The system of polynomial equations has 3 equations with 6
positive variables defining 3 pairs of distinct points (θ
x
p , θ
y
p ), (θ
x
q , θ
y
q ), and (θ
x
r , θ
y
r ).
When considering the extended Kullback–Leibler manifold, since the Hessian
matrix at p is diag
1
θ x
p
,
1
θ
y
p
, we get the following system of polynomial equations
(after multiplying the first equation by θ
x
p θ
y
p , the second equation by θ
x
q θ
y
q and the
third equation by θ
x
r θ
y
r ):
⎧
⎨
⎩
(θ
x
q − θ
x
p )θ
y
p (θ
x
r − θ
x
p ) + (θ
y
q − θ
y
p )θ
x
p (θ
y
r − θ
y
p ) = 0,
(θ
x
r − θ
x
q )θ
y
q (θ
x
p − θ
x
q ) + (θ
y
r − θ
y
q )θ
x
q (θ
y
p − θ
y
q ) = 0,
(θ
x
p − θ
x
r )θ
y
r (θ
x
q − θ
x
r ) + (θ
y
p − θ
y
r )θ
x
r (θ
y
q − θ
y
r ) = 0.
(7.129)
Fixing three variables, we get a cubic system of three equations in three unknowns.
We used Wolfram alpha™ to check for potential real solution(s) of the system:
a>0, b>0, c>a, x>0, y>c,z>0,
(c-a)*b*(y-a)+(x-b)*a*(z-b)=0,
(y-c)*x*(a-c)+(z-x)*c*(b-x)=0,
(a-y)*z*(c-y)+(b-z)*y*(x-z)=0
The system does not admit solution in the positive orthant. We also checked the
feasibility of the system for the Itakura–Saito manifold:
a>0, b>0, c>a, x>0, y>c,z>0,
(c-a)*b*b*(y-a)+(x-b)*a*a*(z-b)=0,
(y-c)*x*x*(a-c)+(z-x)*c*c*(b-x)=0,
(a-y)*z*z*(c-y)+(b-z)*y*y*(x-z)=0.
The system does not admit solution in the positive orthant.
Thus it is an ongoing task to report an example of such a ∇-triangle with three
right angles for an asymmetric Bregman divergence, or to prove that such a triangle
can never exist.
183
Multiplying Eq. 7.123 by sqr(θ
x
p )sqr(θ
y
p ), Eq. 7.124 by sqr(θ
x
q )sqr(θ
y
q ), and Eq. 7.125
by sqr(θ
x
r )sqr(θ
y
r ), we get a system of polynomial equations to solve [30]:
(θ
x
q − θ
x
p )sqr(θ
y
p )(θ
x
r − θ
x
p ) + (θ
y
q − θ
y
p )sqr(θ
x
p )(θ
y
r − θ
y
p ) = 0, (7.126)
(θ
x
r − θ
x
q )sqr(θ
y
q )(θ
x
p − θ
x
q ) + (θ
y
r − θ
y
q )sqr(θ
x
q )(θ
y
p − θ
y
q ) = 0, (7.127)
(θ
x
p − θ
x
r )sqr(θ
y
r )(θ
x
q − θ
x
r ) + (θ
y
p − θ
y
r )sqr(θ
x
r )(θ
y
q − θ
y
r ) = 0, (7.128)
where sqr(x) = x
2 . The system of polynomial equations has 3 equations with 6
positive variables defining 3 pairs of distinct points (θ
x
p , θ
y
p ), (θ
x
q , θ
y
q ), and (θ
x
r , θ
y
r ).
When considering the extended Kullback–Leibler manifold, since the Hessian
matrix at p is diag
1
θ x
p
,
1
θ
y
p
, we get the following system of polynomial equations
(after multiplying the first equation by θ
x
p θ
y
p , the second equation by θ
x
q θ
y
q and the
third equation by θ
x
r θ
y
r ):
⎧
⎨
⎩
(θ
x
q − θ
x
p )θ
y
p (θ
x
r − θ
x
p ) + (θ
y
q − θ
y
p )θ
x
p (θ
y
r − θ
y
p ) = 0,
(θ
x
r − θ
x
q )θ
y
q (θ
x
p − θ
x
q ) + (θ
y
r − θ
y
q )θ
x
q (θ
y
p − θ
y
q ) = 0,
(θ
x
p − θ
x
r )θ
y
r (θ
x
q − θ
x
r ) + (θ
y
p − θ
y
r )θ
x
r (θ
y
q − θ
y
r ) = 0.
(7.129)
Fixing three variables, we get a cubic system of three equations in three unknowns.
We used Wolfram alpha™ to check for potential real solution(s) of the system:
a>0, b>0, c>a, x>0, y>c,z>0,
(c-a)*b*(y-a)+(x-b)*a*(z-b)=0,
(y-c)*x*(a-c)+(z-x)*c*(b-x)=0,
(a-y)*z*(c-y)+(b-z)*y*(x-z)=0
The system does not admit solution in the positive orthant. We also checked the
feasibility of the system for the Itakura–Saito manifold:
a>0, b>0, c>a, x>0, y>c,z>0,
(c-a)*b*b*(y-a)+(x-b)*a*a*(z-b)=0,
(y-c)*x*x*(a-c)+(z-x)*c*c*(b-x)=0,
(a-y)*z*z*(c-y)+(b-z)*y*y*(x-z)=0.
The system does not admit solution in the positive orthant.
Thus it is an ongoing task to report an example of such a ∇-triangle with three
right angles for an asymmetric Bregman divergence, or to prove that such a triangle
can never exist.