Seite - 421 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 7.2. PriorsandLow-RankEstimation The low-rank cometric formulationpursued in Section 5gives a natural restriction of (21) to u∈ FkM, 1≤ k≤ d. As forEuclideanPCA,mostvariance isoftencaptured in thespanof thefirst k eigenvectorswith k d. Estimates of the remaining eigenvectors are generally ignored, as the varianceof theeigenvectorestimates increasesasthenoisecapturedinthespanof the lasteigenvectors becomes increasingly uniform. The low-rank cometric restricts the estimation to only the first k eigenvectors, andthusbuilds theconstructiondirectly into themodel. Inaddition, itmakesnumerical implementationfeasible,becauseanumerical representationneedonlystoreandevolved×kmatrices. Asadifferentapproachforregularizingtheestimator (21), thenormalizingterm−N log(detgRu)can beextendedwithotherpriors (e.g., anL1-typepenalizing term). Suchpriors canpotentiallypartly removeexistenceanduniqueness issues,andresult inadditional sparsityproperties thatcanbenefit numerical implementations. Theeffectsof suchpriorshaveyet tobe investigated. In the k= d case, thenumberofdegreesof freedomfor theMPPsgrowsquadratically in the dimensiond. Thisnaturally increases thevarianceofanyMPPestimategivenonlyonesample fromits trajectory. Thelow-rankcometric formulationreducesthegrowthtolinear ind. Thenumberofdegrees of freedomishoweverstillk times larger thanforRiemanniangeodesics.With longitudinaldata,more samplesper trajectorycanbeobtained, reducing thevarianceandallowingabetterestimateof the MPP.However, for theestimators (20)and(21)above,estimatesof theactualoptimalMPPsarenot needed—only their squared length. It canbehypothesizedthat thevarianceof the lengthestimates is lower thanthevarianceof theestimatesof thecorrespondingMPPs. Further investigationregarding thiswillbe thesubjectof futurework. 7.3. Conclusions Theunderlyingmodelofanisotropyusedinthispaperoriginates fromtheanisotropicnormal distributions formulated in [2]andthediffusionPCAframework[1]. Becausemanystatisticalmodels aredefinedusingnormaldistributions, thisapproachto incorporatinganisotropyextends tomodels suchaslinearregression.Weexpect thatfindingmostprobablepaths inotherstatisticalmodelssuchas regressionsmodelscanbecarriedoutwithaprogramsimilar to theprogrampresented in thispaper. ThedifferencebetweenMPPsandgeodesicsshowsthat thegeometricandmetricpropertiesof geodesics, zero acceleration, and local distanceminimization are not directly related to statistical properties such as maximizing path probability. Whereas the concrete application and model determines ifmetricorstatisticalpropertiesare fundamental,moststatisticalmodelsare formulated withoutreferringtometricpropertiesof theunderlyingspace. It canthereforebearguedthat thedirect incorporationofanisotropyandtheresultingMPPsarenatural in thecontextofmanymodelsofdata variation innon-linerspaces. Acknowledgments: The authorwishes to thank PeterW.Michor and Sarang Joshi for suggestions for the geometric interpretation of the sub-Riemannian metric on FM and discussions on diffusion processes on manifolds. The work was supported by the Danish Council for Independent Research, the CSGB Centre for StochasticGeometry andAdvancedBioimaging fundedby agrant from theVillum foundation, and the ErwinSchrödinger Institute inVienna. Conflictsof Interest:Theauthordeclaresnoconflictof interest. References 1. Sommer,S.DiffusionProcessesandPCAonManifolds.Availableonline: https://www.mfo.de/document/ 1440a/OWR_2014_44.pdf (accessedon24November2016). 2. Sommer, S. Anisotropic distributions on manifolds: Template estimation and most probable paths. In InformationProcessing inMedical Imaging;LectureNotes inComputerScience;Springer: Berlin/Heidelberg, Germany,2015;Volume9123,pp. 193–204. 3. Sommer,S.;Svane,A.M.Modellinganisotropiccovarianceusingstochasticdevelopmentandsub-riemannian framebundlegeometry. J.Geom.Mech. 2016, inpress. 421