Page - 405 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 is thenotionof Frechét orKarchermeans [5,10], definedasminimizers of the squareRiemannian distance.Generalizationsof theEuclideanprincipalcomponentanalysisprocedure tomanifoldsare particularlyrelevant fordataexhibitinganisotropy.Approaches includeprincipalgeodesicanalysis (PGA, [6]), geodesicPCA(GPCA, [11]), principalnested spheres (PNS, [12]), barycentric subspace analysis (BSA, [13]), andhorizontal componentanalysis (HCA, [14]). Commonto theseconstructions areexplicit representationsofapproximating low-dimensional subspaces. The fundamental challenge here is that thenotionofEuclidean linear subspaceonwhichPCArelieshasnodirect analogue in non-linearspaces. AdifferentapproachtakenbydiffusionPCA(DPCA, [1,2])andprobabilisticPGA[15] is tobase thePCAproblemonamaximumlikelihoodfitofnormaldistributions todata. InEuclideanspace, this approachwasfirst introducedwithprobabilisticPCA[16]. InDPCA, theprocessof stochastic development [4] isused todefineaclassof anisotropicdistributions thatgeneralizes the familyof Euclidean space normal distributions to themanifold context. DPCA is then a simplemaximum likelihoodfit in this familyofdistributionsmimickingtheEuclideanprobabilisticPCA.Theapproach transfers thegeometriccomplexitiesofdefiningsubspacescommonintheapproaches listedaboveto theproblemofdefiningageometricallynaturalnotionofnormaldistributions. InEuclidean space, squareddistances‖x−x0‖2 betweenobservations x and themean x0 are affinelyrelatedto thenegative log-likelihoodofanormaldistributionN(x0,Id). ThismakesanML fitof themeansuchasperformedinprobabilisticPCAequivalent tominimizingsquareddistances. Onamanifold, distances dg(x,x0)2 coming fromaRiemannianmetric g are equivalent to tangent spacedistances‖Logx0x‖2whenmappingdata fromM toTx0Musingthe inverseexponentialmap Logx0.AssumingLogx0xaredistributedaccordingtoanormaldistribution in the linearspaceTx0M, this restores the equivalencewithamaximumlikelihoodfit. Let{e1, . . . ,ed}be the standardbasis forRd. Ifu :Rd→ Tx0M is a linear invertiblemapwithue1, . . . ,ued orthonormalwith respect to g, thenormaldistribution inTx0M canbedefinedasuN(0,Id) (seeFigure2). M x0 Expx0v M x0 ϕu(wt) (a) (b) Figure 2. (a) Normal distributions uN(0,Id) in the tangent space Tx0M with covariance uuT (blue ellipsis) canbemapped to themanifold by applying the exponentialmapExpx0 to sampled vectorsv∈Tx0M (redvectors). Thiseffectively linearises thegeometryaroundx0; (b)Thestochastic developmentmap ϕu mapsRd valuedpathswt to M by transporting the covariance in each step (blueellipses)givingacovarianceut alongtheentire samplepath. Theapproachdoesnot linearise around a single point. Holonomyof the connection implies that the covariance “rotates” around closed loops—aneffectwhichcanbe illustratedbycontinuing the transportalong the loopcreated bythedashedpath. TheanisotropicmetricgFMweightsstep lengthsbythetransportedcovariance ateachtime t. Themapucanberepresentedasapoint in theframebundleFMofM.Whentheorthonormal requirement on u is relaxed so that uN(0,Id) is a normal distribution in Tx0M with anisotropic covariance, thenegative log-likelihood inTx0M is related to (u −1Logx0x) T(u−1Logx0x) in thesame way as the precisionmatrixΣ−1 is related to the negative log-likelihood (x−x0)TΣ−1(x−x0) in 405