Page - 415 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 In (16), thecurvature termsappear in thecovariantacceleration fornormalMPPs,whilecubic splines leads to thecurvature termappearing in the thirdorderderivative. 5.CometricFormulationandLow-RankGenerator Wenowinvestigateacometric gFkM+λgR,where gR isRiemannian, gFkM is a rank kpositive semi-definite innerproductarisingfromk linearly independent tangentvectors,andλ>0aweight. WeassumethatgFkM is chosenso thatgFkM+λgR is invertible, eventhoughgFkM is rank-deficient. The situation corresponds to extracting the first k eigenvectors in Euclidean space PCA. If the eigenvectorsareestimatedstatistically fromobserveddata, thisallowstheestimationtoberestricted toonly thefirstkeigenvectors. Inaddition,an importantpractical implicationof theconstruction is thatanumerical implementationneednot transporta fulld×dmatrix for the frame,butapotentially much lowerdimensionald×kmatrix. Thispoint is essentialwhendealingwithhigh-dimensional data, examplesofwhichare landmarkmanifoldsasdiscussed inSection6. Whenusingthe framebundle tomodelcovariances, thesumformulation isnatural toexpressas acometric comparedtoametricbecause,with thecometric formulation,gFkM+λgR representsasum of covariancematrices insteadofa sumofprecisionmatrices. Thus, gFkM+λgR canbe intuitively thoughtofasadding isotropicnoiseofvarianceλ to thecovariancerepresentedbygFkM. To pursue this, let FkM denote the bundle of rank k linear mapsRk → TxM. We define acometricby 〈 ξ, ξ˜ 〉 = δαβ(ξ|hu(uα))(ξ˜|hu(uβ))+λ 〈 ξ, ξ˜ 〉 gR forξ, ξ˜∈T∗uFkM. Thesumoverα,β is forα,β=1,. . . ,k. Thefirst termisequivalent to the lift (9)of thecometric 〈 ξ, ξ˜ 〉 = ( ξ|gu(ξˆ) ) givenu :Rk→TxM.Note that in thedefinition(6)ofgu, themapu is not inverted; thus, thedefinitionof themetric immediatelycarriesover to therank-deficientcase. Let (xi,uiα),α=1,. . . ,kbeacoordinatesystemonFkM. Theverticaldistribution is in thiscase spannedby the dkvectorfieldsDjβ = ∂ujβ . Except for index sumsbeingover k insteadof d terms, thesituation is thussimilar to the full-rankcase.Note that (ξ|π−1∗ w)=(ξ|wjDj)=wiξi. Thecometric incoordinates is 〈 ξ, ξ˜ 〉 = δαβuiαξiu j βξ˜j+λg ij Rξiξ˜j= ξi ( δαβuiαu j β+λg ij R ) ξ˜j= ξiWijξ˜j withWij = δαβuiαu j β+λg ij R. We can thenwrite the corresponding sub-Riemannianmetric gFkM in termsof theadaptedframeD gFkM(ξhD h+ξhγD hγ)=WihξhDi (17) because (ξ|gFkM(ξ˜))= 〈 ξ, ξ˜ 〉 = ξiWijξ˜j. That is, thesituation isanalogousto(11), except the termλg ij R isaddedtoWij. ThegeodesicsystemisagaingivenbytheHamilton–Jacobiequations.As in the full-rankcase, thesystemisspecifiedbythederivativesofgFkM: ∂ylg ij FkM=W ij ,l , ∂ylg ijβ FkM=−Wih,lΓ jβ h −WihΓ jβ h,l , ∂ylg iαj FkM=−Γ iα h,lW hj−ΓiαhW hj ,l , ∂ylg iαjβ FkM=Γ iα k,lW khΓ jβ h +Γ iα kW kh ,lΓ jβ h +Γ iα kW khΓ jβ h,l , ∂ ylζ gijFkM=W ij ,lζ , ∂ ylζ g ijβ FkM=−Wih,lζΓ jβ h −WihΓ jβ h,lζ , ∂ ylζ giαjFkM=−Γ iα hW hj ,lζ −Γiαh,lζW hj , ∂ ylζ g iαjβ FkM=Γ iα k,lζ WkhΓ jβ h +Γ iα kW kh ,lζΓ jβ h +Γ iα kW khΓ jβ h,lζ , Γiαh,lζ = ∂ylζ ( Γihku k α ) = δζαΓihl , W ij ,l =λg ij R ,l , W ij ,lζ = δilujζ+δ jluiζ . 415