Page - 367 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 Section5 isanapplicationof thepreviousmaterial to theclassificationofdatawithvalues inPm, whichcontainoutliers(abnormaldatapoints).AssumetobegivenatrainingsequenceY1, · · · ,Yn∈Pm. UsingtheEMalgorithmdevelopedinSection4, it ispossible tosubdivide this sequence intodisjoint classes. Toclassifynewdatapoints,aclassificationrule isproposed. Therobustnessof this rule lies in the fact that it is basedon thedistances betweennewobservations and the respectivemedians of classes insteadof themeans [15]. This rulewill be illustratedbyanapplication to theproblem of textureclassification incomputervision. Theobtainedresultsshowimprovedperformancewith respect torecentapproacheswhichuse theRiemannianGaussiandistribution[15]andtheWishart distribution[17]. 2.RiemannianGeometryofPm ThegeometryofSiegelhomogeneousboundeddomains, suchasKählerhomogeneousmanifolds, havebeenstudiedbyFelixA.Berezin[18]andP.Malliavin[19]. ThestructureofKählerhomogeneous manifolds has been used in [20,21] to parameterize (Toeplitz–) Block–Toeplitzmatrices. This led to aHessianmetric from information geometry theorywith aKähler potential given by entropy andtoanalgorithmtocomputemediansof (Toeplitz–)Block–ToeplitzmatricesbyKarcherflowon Mostow/BergerfibrationofaSiegeldisk. Optimalnumerical schemesof thisalgorithminaSiegel diskhavebeenstudied,developedandvalidated in [22–24]. Thissection introduces thenecessarybackgroundontheRiemanniangeometryofPm , thespace ofsymmetricpositivedefinitematricesofsizem×m. Precisely,Pm is equippedwith theRiemannian metric known as the affine-invariantmetric. First, analytic expressions are recalled for geodesic curves andRiemanniandistance. Then, twoproperties are stated,which are fundamental to the following. Theseareaffine-invarianceof theRiemanniandistanceandtheexistenceanduniquenessof Riemannianmedians. The affine-invariant metric, called the Rao–Fisher metric in information geometry, has the followingexpression: gY(A,B)= tr(Y−1AY−1B) (5) whereY∈PmandA,B∈TYPm, thetangentspacetoPmatY,whichis identifiedwiththevectorspace ofm×m symmetricmatrices. TheRiemannianmetricEquation(5) inducesaRiemanniandistanceon Pm as follows. The lengthofasmoothcurve c : [0,1]→Pm isgivenby: L(c) = ∫ 1 0 √ gc(t)(c˙(t), c˙(t))dt (6) where c˙(t)= dcdt . ForY,Z∈Pm, theRiemanniandistanced(Y,Z), calledRao’sdistance in information geometry, isdefinedtobe: d(Y,Z)= inf{L(c),c : [0,1]→Pm isasmoothcurvewith c(0)=Y,c(1)=Z} . This infimumisachievedbyauniquecurve c=γ, calledthegeodesicconnectingYandZ,which has the followingequation[10,25]: γ(t)=Y1/2(Y−1/2ZY−1/2)tY1/2 (7) Here,andthroughoutthefollowing,allmatrixfunctions(forexample,squareroot, logarithmorpower) areunderstoodassymmetricmatrix functions [26]. Bydefinition,d(Y,Z)coincideswithL(γ),which turnsout tobe: d2(Y,Z)= tr [log(Y−1/2ZY−1/2)]2 (8) Equipped with the affine-invariant metric Equation (5), the space Pm enjoys two useful properties, which are the following. The first property is invariance under affine 367