Page - 368 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 transformations [10,25]. Recall thatanaffinetransformationofPm isamappingY →Y ·A,whereA is an invertible realmatrixofsizem×m, Y ·A=A†YA (9) and † denotes the transpose. DenotebyGL(m) thegroupofm×m invertible realmatricesonPm. Then, the action of GL(m) onPm is transitive. Thismeans that for any Y,Z ∈ Pm, there exists A∈GL(m), such thatY.A=Z.Moreover, theRiemanniandistanceEquation(8) is invariantbyaffine transformations in thesense that forallY,Z∈Pm: d(Y,Z)= d(Y ·A,Z ·A) (10) whereY ·AandZ ·AaredefinedbyEquation(9). Thetransitivityof theactionEquation(9)andthe isometrypropertyEquation(10)makePm aRiemannianhomogeneousspace. Theaffine-invariantmetricEquation(5)turnsPm intoaRiemannianmanifoldofnegativesectional curvature [10,27].Asaresult,Pm enjoys thepropertyof theexistenceanduniquenessofRiemannian medians. TheRiemannianmedianofNpointsY1, · · · ,YN∈Pm isdefinedtobe: YˆN=argminY N ∑ n=1 d(Y,Yn) (11) where d(Y,Yn) is theRiemanniandistance Equation (8). IfY1, · · · ,YN donot belong to the same geodesic, then YˆN existsandisunique[28].Moregenerally, foranyprobabilitymeasureπonPm , the medianofπ isdefinedtobe: Yˆπ =argminY ∫ Pm d(Y,Z)dπ(Z) (12) Note thatEquation(12) reduces toEquation(11) forπ= 1N∑ N n=1δYn. If thesupportofπ isnot carriedbyasinglegeodesic, thenagain, Yˆπ existsandisuniquebythemainresultof [28]. To end this section, consider the Riemannian volume associated with the affine-invariant Riemannianmetric [10]: dv(Y)=det(Y)− m+1 2 ∏ i≤j dYij (13) where the indicesdenotematrixelements. TheRiemannianvolumeisusedtodefinethe integralofa function f :Pm→Ras: ∫ Pm f(Y)dv(Y)= ∫ . . . ∫ f(Y)det(Y)− m+1 2 ∏ i≤j dYij (14) where the integralon theright-handside isamultiple integralover them(m+1)/2variables,Yijwith i≤ j. The integralEquation(14) is invariantunderaffinetransformations. Precisely: ∫ Pm f(Y ·A)dv(Y)= ∫ Pm f(Y)dv(Y) (15) whereY ·A is theaffine transformationgivenbyEquation (9). It takesona simplified formwhen f(Y)onlydependsontheeigenvaluesofY. Precisely, let thespectraldecompositionofYbegivenby Y=U†diag(er1, · · · ,erm)U,whereU isanorthogonalmatrixand er1, · · · ,erm are theeigenvaluesof Y. Assumethat f(Y)= f(r1, . . . ,rm), thenthe invariant integralEquation(14) reduces to: ∫ Pm f(Y)dv(Y)= cm× ∫ Rm f(r1, · · · ,rm)∏ i<j sinh (|ri−rj| 2 ) dr1 · · ·drm (16) 368