Page - 95 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 The initial speedof thegeodesic isgivenby (. δ(0), . Δ(0) ) . Thegeodesicshooting isgivenbythe exponentialmap: Λ(t)=exp(tA)= ∞ ∑ n=0 (tA)n n! = ⎛⎜⎝ Δ δ ΦδT ε γT ΦT γ Γ ⎞⎟⎠withA= ⎛⎜⎝−B b 0bT 0 −bT 0 −b B ⎞⎟⎠ (235) This equation can be interpreted by group theory. A could be considered as an element of Lie algebra so(n+1,n) of the special Lorentz group SOO(n+1,n) andmore specifically as the elementpofCartanDecomposition l+pwhere l is theLiealgebraofamaximalcompactsub-group K=S(O(n+1)×O(n))of thegroupG=SOO(n+1,n).Weknowthat its exponentialmapdefinesa geodesiconRiemannianSymetric spaceG/K. Thisequationcanbeestablishedbythe followingdevelopments: . Λ(t)=A.Λ(t)⇒ ⎛⎜⎜⎝ . Δ . δ . Φ . δ T . ε . γ T . Φ T . γ . Γ ⎞⎟⎟⎠ = ⎛⎜⎝−B b 0bT 0 −bT 0 −b B ⎞⎟⎠. ⎛⎜⎝ Δ δ ΦδT ε γT ΦT γ Γ ⎞⎟⎠ (236) Wecanthendeduce that: ⎧⎨⎩ . Δ=−BΔ+bδT . δ=−Bδ+εb (237) Ifε=1+δTΔ−1δ, then(Δ,δ) issolutiontothegeodesicequationpreviouslydefined. Sinceε(0)=1, it suffices todemonstrate that . ε= . τwhereτ=δTΔ−1δ. From . Λ(t)=Λ(t).A,usingthat . δ T =bTΔ−bTΦT,wecandeduce:{ . ε=bTδ−bTγ . τ=bTδ−bT((τ−ε)Δ−1δ+ΦTΔ−1δ) (238) Then . ε = . τ, ifγ = (τ−ε)Δ−1δ+ΦΔ−1δ, that could be verifiedusing relationΛ.Λ−1 = I, by observingthat: Λ−1=exp(−tA)=Λ(−t)= ⎡⎢⎣ Γ γ ΦTγT ε δT Φ δ Δ ⎤⎥⎦ (239) Λ.Λ−1= I⇒ { Δγ+εδ+Φδ=0 ΔΦT+δδT+ΦΔ=0 ⇒ { γ=−εΔ−1δ−Δ−1Φδ ΦTΔ−1+Δ−1δδTΔ−1+Δ−1Φ=0 ⇒ { γ=−εΔ−1δ−Δ−1Φδ ΦTΔ−1δ+τΔ−1δ+Δ−1Φδ=0 (240) Wecanthencomputeγ fromtwolastequations: γ=(τ−ε)Δ−1δ+ΦTΔ−1δ (241) As . τ=bTδ−bT((τ−ε)Δ−1δ+ΦTΔ−1δ) thenwecandeduce that .τ=bTδ−bTγandthen .τ= .ε. To interpret elements ofΛ, (Γ(t),γ(t)) = (Δ(−t),δ(−t)), opposite points to (Δ(t),δ(t)), and ε= 1+δTΔ−1δ=1+γTΓ−1γ. Then thegeodesic that goes through theorigin (0,In)with initial tangent vector (b,−B) is the curvegivenby (δ(t),Δ(t)). Then thedistancecomputation is reducedtoestimate the initial tangent 95