Page - 410 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 Letnowutbeapath inFM, andchoosea local trivializationut=(xt,u1,t, . . . ,ud,t) suchthat the matrix [uiα,t] represents thesquarerootcovariancematrixΣ 1/2 atxt. Sinceutbeingaframedefinesan invertiblemapRd→TxtM, thenorm‖·‖Σ abovehasadirectanalogue in thenorm‖·‖ut definedby the innerproduct 〈v,w〉ut = 〈 u−1t v,u −1 t w 〉 Rd (5) forvectorsv,w∈TxtM. The transportof the framealongpaths ineffectdefinesa transportof inner productalongsamplepaths: thepathscarrywith themthe innerproductweightedbytheprecision matrix,which in turn isa transportof thesquarerootcovarianceu0 atx0. The innerproductcanequivalentlybedefinedasametricgu :T∗xM→TxM. Againusing that u can be considered amapRd → TxM, gu is defined by ξ → u(ξ◦u) , where is the standard identification (Rd)∗→Rd. Thesequenceofmappingsdefininggu is illustratedbelow: T∗xM → (Rd)∗ → Rd → TxM ξ → ξ◦u → (ξ◦u) → u(ξ◦u) . (6) Thisdefinitionuses theRd innerproduct in thedefinitionof . Its inversegives the cometric g−1u :TxM→T∗xM; i.e.,v → (u−1v) ◦u−1. TxM → Rd → (Rd)∗ → T∗xM v → u−1v → (u−1) → (u−1) ◦u−1. (7) 3.1. Sub-RiemannianMetric on theHorizontalDistribution Wenow lift the path-dependentmetric defined above to a sub-Riemannianmetric on HFM. Foranyw,w˜∈HuFM, the liftof (5)byπ∗ is the innerproduct 〈w,w˜〉= 〈 u−1π∗w,u−1π∗w˜ 〉 Rd . The innerproduct inducesasub-RiemannianmetricgFM :TFM∗→HFM⊂TFMby 〈w,gFM(ξ)〉=(ξ|w) , ∀w∈HuFM (8) with (ξ|w)denoting the evaluation ξ(w) for the covector ξ ∈ T∗FM. The metric gFM gives FM a non-bracket-generating sub-Riemannian structure [23] on FM (see also Figure 3). It is equivalent to the lift ξ → hu(gu(ξ◦hu)) , ξ∈TuFM (9) of themetricgu above. In framecoordinates, themetric takes the form u−1π∗gFM(ξ)= ⎛⎜⎝ξ(H1(u))... ξ(Hd(u)) ⎞⎟⎠ . (10) In terms of the adapted coordinates for TFM described in Section 2.3, with w = wjDj and w˜= w˜jDj,wehave 〈w,w˜〉= 〈 wiDi,w˜jDj 〉 = 〈 u−1wi∂xi,u −1w˜j∂xj 〉 = 〈 wiuαi ,w˜ juαj 〉 Rd = δαβwiuαi w˜ juβj =Wijw iw˜j 410