Page - 96 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 vectorspacerelatedby ( R−1(0) .m(0),R−1(0) ( . R(0)+ . m(0)m(0)T )) =(b,B)∈Rn×Symn(R)Thedistance willbe thengivenbythe initial tangentvector: d= √ . m(0)TR−1(0) .m(0)+1 2 Tr [( R−1(0) . R(0) )2] (242) This initial tangentvectorwillbe identifiedby“GeodesicShooting”. LetV= logAB:⎧⎪⎪⎪⎨⎪⎪⎪⎩ dVm dt = 1 2 ( dR dt ) R−1Vm+ 1 2 VRR−1 ( dm dt ) dVR dt = 1 2 (( dR dt ) R−1Vm+VRR−1 ( dR dt )) − 1 2 (( dm dt ) VTm+VTm ( dm dt )) (243) GeodesicShooting is correctedbyusing JacobiField Jandparallel transport: J(t)= ∂χα(t)∂α ∣∣∣ t=0 solutionto d 2J(t) dt2 +R ( J(t), . χ(t) ) . χ(t)=0withR theRiemannCurvarture tensor. Weconsiderageodesicχbetweenθ0 andθ1withan initial tangentvectorV, andwesuppose that V isperturbatedbyW, toV+W. Thevariationof thefinalpointθ1 canbedeterminedthanks to the Jacobifieldwith J(0)=0and . J(0)=W. In termof theexponentialmap, thiscouldbewritten: J(t)= d dα expθ0 (t(V+αW)) ∣∣∣∣ α=0 (244) Thiscouldbe illustrated in theFigure13: Figure13.Geodesicshootingprinciple. Wegivesomeillustration, inFigure14,ofgeodesic shooting tocompute thedistancebetween multivariateGaussiandensity for thecasen=2: 96