Page - 381 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 (ii)Notethefollowingeasyinequalities |r1−r2|≤ |r1|+ |r2|≤ √ 2|r|,whichyieldsinh( |r1−r2|2 )≤ 1 2e |r|√ 2 . This last inequalityshowsthatζ2(σ) isfinite forallσ< √ 2. Inorder tocheckζ2( √ 2)=∞, it is necessary toshow: ∫ R2 exp(−|r|√ 2 + |r1−r2| 2 )dr1dr2=∞ (40) The last integral is,uptoaconstant,greater than ∫ Cexp ( −|r|+ |r1−r2|√ 2 ) dr1dr2,where: C={(r1,r2)∈R2 : r1≥−r2,r2≤0}={(r1,r2)∈R2 : r1≥|r2|,r2≤0}. OnC, −|r|+ |r1−r2|√ 2 =−|r|+ r1−r2√ 2 ≥− √ 2r1+ r1−r2√ 2 = −r1−r2√ 2 However, ∫ Cexp (−r1−r2√ 2 ) dr1dr2=∞by integratingwithrespect to r1 andthen r2,whichshows Equation(40). D.TheLawofX inAlgorithm1 As stated in Appendix A, the uniform distribution on O(m) is given by 1 ω′m det(θ), where ω′m= 2 mπm 2/2 Γm(m/2) . LetY(s,V) =V†diag(es1, · · · ,esm)V,with s= (s1, · · · ,sm). SinceX =Y(r,U), for anytest functionϕ :Pm→R, E[ϕ(X)]= 1 ω′m ∫ O(m)×Rm ϕ(Y(s,V))p(s)det(θ) m ∏ i=1 dsi (41) Here, det(θ) = ∧ i<j θij and θij = ∑kVjkdVik. On the other hand, by Proposition 4,∫ Pm ϕ(Y)p(Y| I,σ)dv(Y)canbeexpressedas: (m!2m)−1 × 8m(m−1)4 1 ζm(σ) ∫ O(m) ∫ Rm ϕ(Y(s,V))e− |s| σ det(θ)∏ i<j sinh (|si−sj| 2 ) m ∏ i=1 dsi whichcoincideswithEquation(41). References 1. Pennec,X.;Fillard,P.;Ayache,N.ARiemannianframeworkfor tensorcomputing. Int. J.Comput.Vis. 2006, 66, 41–66. 2. Barachant,A.;Bonnet, S.;Congedo,M.; Jutten,C.MulticlassBrain–Computer InterfaceClassificationby RiemannianGeometry. IEEETrans. Biomed. Eng. 2012,59, 920–928. 3. Jayasumana,S.;Hartley,R.; Salzmann,M.;Li,H.;Harandi,M.KernelMethodsontheRiemannianManifold ofSymmetricPositiveDefiniteMatrices. InProceedingsof the IEEEConferenceonComputerVisionand PatternRecognition(CVPR),Portland,OR,USA,23–28 June2013;pp. 73–80. 4. Zheng, L.; Qiu, G.; Huang, J.; Duan, J. Fast and accurateNearestNeighbor search in themanifolds of symmetricpositivedefinitematrices. InProceedingsof the IEEEInternationalConferenceonAcoustics, SpeechandSignalProcessing(ICASSP),Florence, Italy,4–9May2014;pp. 3804–3808. 5. Dong, G.; Kuang, G. Target recognition in SAR images via classification on Riemannian manifolds. IEEEGeosci. RemoteSens. Lett. 2015,21, 199–203. 6. Tuzel,O.;Porikli,F.;Meer,P.PedestriandetectionviaclassificationonRiemannianmanifolds. IEEETrans. PatternAnal.Mach. Intell. 2008,30, 1713–1727. 7. Caseiro,R.;Henriques, J.F.;Martins,P.;Batista, J.AnonparametricRiemannianframeworkontensorfield withapplicationto foregroundsegmentation.PatternRecognit. 2012,45, 3997–4017. 381