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Entropy2016,18, 396
computationalcomplexity.Unfortunately, eigenfunctionsof theLaplacianoperatorareknownonDn
butnotoncompactsub-domains. This iswhythestudyof thismethodhasnotbeendeepened.
3.3. Kernels
LetK :R+→R+beamapwhichverifies the followingproperties:
(i) ∫ RdK(||x||)dx=1;
(ii) ∫
Rd xK(||x||)dx=0;
(iii) K(x>1)=0;
(iv) sup(K(x))=K(0).
Let p ∈ Dn. Generally, given apoint p on aRiemannianmanifold, expp defines an injective
applicationonlyonaneighborhoodof0. On theSiegel space, expp is injectiveon thewhole space.
Whenthe tangentspaceTpDn isendowedwith the local scalarproduct,
||u||= d(p,expp(u)),
where ||.|| is theEuclideandistanceassociatedwiththelocalscalarproductandd(., .) is theRiemannian
distance. The correspondingLebesguemeasureonTpDn is noted Lebp. Let exp∗p(Lebp)denote the
push-forwardmeasureofLeppby expp. The functionθpdefinedby:
θp : q → θp(q)= dvoldexp∗p(Lebp) (q) (6)
is thedensityof theRiemannianmeasureonDnwithrespect to theLebesguemeasureLebp after the
identificationofDn andTpDn inducedby expp (seeFigure1).
Figure1.M isaRiemannianmanifold,andTxM is its tangentspaceatx. Theexponentialapplication
inducesavolumechangeθxbetweenTxMandM.
GivenKandapositiveradius r, theestimatorof f proposedby[1] isdefinedby:
fˆk= 1
k∑i 1
rn 1
θxi(x) K (
d(x,xi)
r )
. (7)
Thecorrective factor θxi(x) −1 isnecessary since thekernelKoriginally integrates toonewith
respect to theLebesguemeasureandnotwithrespect to theRiemannianmeasure. It canbenoticed
that thisestimator is theusualkernelestimator in thecaseofEuclideanspace.Whenthecurvatureof
thespace isnegative,which is thecaseof theSiegel space, thedistributionplacedovereachsample
xihasxi as intrinsicmean. Thefollowingtheoremprovidesconvergencerateof theestimator. It isa
directadaptationofTheorem3.1of [1].
Theorem1. Let (M,g) be aRiemannianmanifold of dimensionn andμ itsRiemannianvolumemeasure.
LetXbearandomvariable taking itsvalues inacompact subsetCof (M,g). Let0< r≤ rinj,where rinj is the
infimumof the injectivity radiusonC.Assumethe lawofXhasa twicedifferentiabledensity f with respect to
352
Differential Geometrical Theory of Statistics
- Titel
- Differential Geometrical Theory of Statistics
- Autoren
- Frédéric Barbaresco
- Frank Nielsen
- Herausgeber
- MDPI
- Ort
- Basel
- Datum
- 2017
- Sprache
- englisch
- Lizenz
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Abmessungen
- 17.0 x 24.4 cm
- Seiten
- 476
- Schlagwörter
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Kategorien
- Naturwissenschaften Physik