Page - 353 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 theRiemannianvolumemeasure. Let fˆk be the estimatordefined inEquation (7). There exists a constantCf such that ∫ x∈M Ex1,...,xk[(f(x)− fˆk(x))2]dμ≤Cf( 1 krn +r4). (8) If r∼ k −1n+4 , ∫ x∈M Ex1,...,xk[(f(x)− fˆk(x))2]dμ=O(k− 4 n+4). (9) Proof. SeeAppendixA. It canbecheckedthatontheSiegel space rinj=+∞andthat, foran isometryα,wehave: fˆk(x,x1,...,xk)= fˆk(α(x),α(x1), ...,α(xk)). Each locationanddirectionareprocessedassimilarlyaspossible. Thisdensityestimatorcanbeused fordataclassificationonRiemannianmanifolds, see [26]. Inorder toobtain theexplicit expressionof theestimator,onemusthavetheexplicit expression of the Riemannian exponential, of its inverse, and of the function θp (see Equations (6) and (7)). Theseexpressionsaredifficultandsometimes impossible toobtain forgeneralRiemannianmanifolds. In thecaseof theSiegel space, the symmetric structuremakes thecomputationpossible. Since the space ishomogeneous, thecomputationcanbemadeat theorigin iI∈Hnor0∈Dn andtransported to thewholespace. In thepresentwork, therandomvariable lays inDn.However, in the literature, theCartanandIwasawadecompositionsareusuallygivenfor the isometrygroupofHn. Thus,our computationstarts inHnbeforemovingtoDn. TheKilling formontheLiealgebrasp(n,R)of the isometrygroupofHn inducesascalarproduct onp. This scalarproduct canbe transportedon exp(p)by leftmultiplication. Thisoperationgives exp(p)aRiemannianstructure. It canbeshownthatonthisRiemannianmanifold, theRiemannian exponentialat the identitycoincideswith thegroupexponential. Furthermore, φ : exp(p) → Hn g → g.iI (10) isabijective isometry,uptoascalingfactor. Since thevolumechangeθp is invariantunderrescaling of themetric, thisscalingfactorhasnoimpact. Thus,Hn canbe identifiedwith exp(p)and expiI∈Hn with exp|p. Theexpressionof theRiemannianexponential isdifficult toobtain ingeneral;however, itboilsdownto thegroupexponential in thecaseof symmetric spaces. This is themainelementof the computationofθp. TheRiemannianvolumemeasureon exp(p) isnotedvol′. Let ψ : K×a → p (k,H) → Adk(H). Leta+be thediagonalmatriceswithstrictlydecreasingpositiveeigenvalues. LetΛ+be thesetof positiverootsofsp(n,R)asdescribed inp. 282 in [20] , Λ+={ei+ej, i≤ j}∪{ei−ej, i< j}, where ei(H) is the i-thdiagonal termof thediagonalmatrixH. LetCc(E)be the set of continuous compactly supported functionson the spaceE. In [27], atpage73, it is given that for all t∈Cc(p), thereexists c1>0suchthat∫ p t(Y)dY= c1 ∫ K ∫ a+ t(ψ(k,H))∏ λ∈Λ+ λ(H)dkdH, (11) 353