Page - 403 - in Differential Geometrical Theory of Statistics

entropy Article AnisotropicallyWeightedandNonholonomically ConstrainedEvolutionsonManifolds † StefanSommer DepartmentofComputerScience,UniversityofCopenhagen,DK-2100CopenhagenE,Denmark; sommer@di.ku.dk † Thispaper isanextendedversionofourpaperpublishedin the2ndConferenceonGeometricScienceof Information,Paris,France,28–30October2015. AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 1September2016;Accepted: 23November2016;Published: 26November2016 Abstract: Wepresent evolution equations for a family ofpaths that results fromanisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises whenperforminginferenceondata thathavenon-trivial covarianceandareanisotropicdistributed. The family canbe interpretedasmostprobablepaths for adrivingsemi-martingale that through stochastic development is mapped to the manifold. We discuss how the paths are projections of geodesics for a sub-Riemannian metric on the frame bundle of the manifold, and how the curvatureof theunderlyingconnectionappears in thesub-RiemannianHamilton–Jacobiequations. Evolution equations for both metric and cometric formulations of the sub-Riemannian metric arederived.We furthermore showhowrank-deficientmetrics canbemixedwith anunderlying Riemannianmetric, andwerelate thepaths togeodesicsandpolynomials inRiemanniangeometry. Examples from the family of paths are visualized on embedded surfaces, and we explore computational representationsonfinitedimensional landmarkmanifoldswithgeometry induced fromright-invariantmetricsondiffeomorphismgroups. Keywords: sub-Riemannian geometry; geodesics;most probable paths; stochastic development; non-lineardataanalysis; statistics 1. Introduction Whenmanifold valueddata have non-trivial covariance (i.e., when anisotropy asserts higher varianceinsomedirectionsthanothers),non-zerocurvaturenecessitatesspecialcarewhengeneralizing Euclideanspacenormaldistributions tomanifoldvalueddistributions: in theEuclideansituation, normal distributions can be seen as transition distributions of diffusion processes, but on the manifold, holonomymakes transport of covariance path-dependent in the presence of curvature, preventingaglobalnotionofaspatiallyconstantcovariancematrix.Tohandle this, in thediffusion principal component analysis (PCA) framework [1], and with the class of anisotropic normal distributions onmanifolds defined in [2,3], data on non-linearmanifolds aremodelled as being distributedaccordingto transitiondistributionsofanisotropicdiffusionprocesses thataremapped from Euclidean space to the manifold by stochastic development (see [4]). The construction is connected to a non-bracket-generating sub-Riemannianmetric on the bundle of linear frames of themanifold, the framebundle,andtherequirement thatcovariancestayscovariantlyconstantgivesa nonholonomicallyconstrainedsystem. Velocityvectorsandgeodesicdistancesareconventionallyusedforestimationandstatistics in Riemannianmanifolds; for example, for estimationof theFrechétmean [5], forPrincipalGeodesic Analysis [6], and for tangent space statistics [7]. In contrast to this, anisotropy asmodelledwith anisotropicnormaldistributionsmakesadistance forasub-Riemannianmetric thenaturalvehicle for Entropy2016,18, 425 403 www.mdpi.com/journal/entropy