Page - 407 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 parallel transport of thevectorsu1, . . . ,ud of the framealong thedirectionui. A horizontal liftof a differentiablecurvext∈M isacurve inFM tangent toHFM thatprojects toxt.Horizontal liftsare uniqueupto thechoiceof initial frameu0. TFM T∗FM HFMVFM FM×gl(n) FM TMT∗M M h+v → hh+v → v π∗ψ πFM πTM gFM gR Figure3.Relationsbetweenthemanifold, framebundle, thehorizontaldistributionHFM, thevertical bundle VFM, a Riemannian metric gR, and the sub-Riemannian metric gFM, defined below. TheconnectionCprovides thesplittingTFM=HFM⊕VFM. Therestrictionsπ∗|HuM are invertible mapsHuM→Tπ(u)Mwith inversehu, thehorizontal lift. Correspondingly, theverticalbundleVFM is isomorphic to the trivialbundleFM×gl(n). ThemetricgFM :T∗FM→TFMhasanimage inthe subspaceHFM. 2.2.DevelopmentandStochasticDevelopment Let xt be a differentiable curve on M and ut a horizontal lift. If st is a curve inRd with components sit suchthat x˙t=Hi(u)s i t,xt is said tobeadevelopmentof st.Correspondingly, st is the anti-developmentof xt. For each t, thevector st contains framecoordinatesof x˙t asdefinedabove. Similarly, letWtbeanRdvaluedBrownianmotionsothatsamplepathsWt(ω)∈W(Rd).Asolutionto thestochasticdifferentialequationdUt=∑di=1Hi(Ut)◦dWit inFM is calledastochasticdevelopment ofWt inFM. Thesolutionprojects toastochasticdevelopmentXt=π(Ut) inM.Wecall theprocess Wt inRd, that throughϕmapstoXt, thedrivingprocessofXt. Letϕu :W(Rd)→W(M)bethemap that forfixedu sendsapath inRd to itsdevelopmentonM. Its inverseϕ−1u is theanti-development in RdofpathsonMgivenu. Equivalent to the fact thatnormaldistributionsN(0,Σ) inRd canbeobtainedas the transition distributionsofdiffusionprocessesΣ1/2Wt stoppedat time t=1,ageneralclassofdistributionson themanifoldM canbedefinedby stochasticdevelopmentofprocessesWt, resulting inM-valued randomvariablesX=X1. This familyofdistributionsonM introducedin[2] isdenotedanisotropic normaldistributions. Thestochasticdevelopmentbyconstructionensures thatUt ishorizontal, andthe framesare thusparallel transportedalongthestochasticdisplacements. Theeffect is that the frames staycovariantlyconstant, thusresemblingtheEuclideansituationwhereΣ1/2 is spatiallyconstantand thereforedoesnotchangeasWt evolves. Thus,as furtherdiscussedinSection3.2, thecovariance is keptconstantateachof the infinitesimalstochasticdisplacements. Theexistenceofasmoothdensity for the targetprocessXt andsmall timeasymptoticsarediscussed in [3]. Stochastic development gives a map ∫ Diff : FM → Prob(M) to the space of probability distributionsonM. Foreachpointu∈FM, themapsendsaBrownianmotion inRd toadistribution μubystochasticdevelopmentof theprocessUt inFM, startingatuandlettingμubethedistribution of X = π(U1). The pair (x,u), x = π(u) is analogous to the parameters (μ,Σ) for a Euclidean normaldistribution: thepointx∈M represents thestartingpointof thediffusion,andthe frameu representsasquarerootΣ1/2 of thecovarianceΣ. In thegeneralsituationwhereμuhassmoothdensity, theconstructioncanbeusedtofit theparametersu todatabymaximumlikelihood.Asanexample, diffusionPCAfitsdistributionsobtainedthrough ∫ Diff bymaximumlikelihoodtoobservedsamples inM; i.e., itoptimizes for themost likelyparametersu=(x,u1, . . . ,ud) for theanisotropicdiffusion process,givingafit to thedataof themanifoldgeneralizationof theEuclideannormaldistribution. 407