Seite - 94 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 This is theEuler-Poincaréequationofgeodesic.Wecanobservethatwehaveobtainedareduction of the followingEuler-Lagrangeequation[27,156,187]: { .. R+ . m . mT− .RR−1 .R=0 .. m− .RR−1 .m=0 associatedto the informationgeometrymetricds2= dmTR−1dm+ 12Tr (( R−1dR )2). The Fisher information defines a metric turning Nn = {(m,R)∈Rn×Sym+(n)} into a Riemannianmanifold. The innerproductof twotangentvectors (m1,R1)∈Tn and (m2,R2)∈Tn at thepoint (μ,Σ)∈Nn isgivenby: g(μ,Σ) ((m1,R1) ,(m2,R2))=m T 1Σ −1m2+ 1 2 tr ( Σ−1R1Σ−1R2 ) (227) andthegeodesic isgivenby: l(χ)= t1 t0 √ gχ(t) ( . χ(t), . χ(t) ) dt (228) Wecanalsoobserve that themanifoldofmultivariateGaussian ishomogeneouswithrespect to positiveaffinegroupGA+(n): ds2Y= ds 2 X forY=Σ 1/2X+μwithGA+(n)={(μ,Σ)∈R×GL(R)/det(Σ)>0} (229) characterizedbytheactionofthegroup (m,R) → ρ.(m,R)= ( Σ1/2m+μ,Σ1/2RΣ1/2T ) ,ρ∈GA+(n) with [ Y 1 ] = [ Σ1/2 μ 0 1 ][ X 1 ] (230) ds2Y= d ( Σ1/2m+μ )T( Σ1/2RΣ1/2T )−1 d ( Σ1/2m+μ ) + 1 2 Tr ((( Σ1/2RΣ1/2T )−1 d ( Σ1/2RΣ1/2T ))2) ds2Y= dm TR−1dm+ 1 2 Tr (( R−1dR )2) = ds2X (231) Since thespecialorthogonalgroupSO(n)={δ∈GL(R)/det(δ)=1} is thestabilizersubgroup of (0, In),wehavethe following isomorphism: GA+(n)/SO(n)→Nn={(m,R)∈Rn×Sym+(n)} ρ=(μ,Σ) → ρ.(0, In)= ( μ,Σ1/2Σ1/2T ) =(μ,Σ) (232) We can then restrict the computation of the geodesic from (0, In) and thenwe can partially integrate thesystemofequations: ⎧⎨⎩ . m=Rb . R=R ( B−bmT) (233) where ( R−1(0) .m(0),R−1(0) ( . R(0)+ . m(0)m(0)T )) =(b,B)∈Rn×Symn(R)are the integrationconstants. FromthisEuler-Poincaréequation,wecancomputegeodesicsbygeodesic shooting [188–191] usingclassicalEriksenequations [192–195],by the followingchangeofparameters: { Δ(t)=R−1(t) δ(t)=R−1(t)m(t) ⇒ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ . Δ=−BΔ+bmT . δ=−Bδ+(1+δTΔ−1δ)b Δ(0)= Ip,δ(0)=0 with ⎧⎨⎩ . Δ(0)=−B . δ(0)=b (234) 94