Page - 393 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 Proof. Letusconsider theplan[11]givenbythedensity: d: (x,y) → ∫ 1 0 K(‖x−γ1(η)‖)K(‖y−γ2(η)‖)dη whereeachcurve isparametrizedbythescaledarclength. Theassociatedtransportcost isgivenby: C= ∫ Rq×Rq ‖x−y‖2 ∫ 1 0 K(‖x−γ1(η)‖)K(‖y−γ2(η)‖)dηdxdy lettingu= y−xandusingFubinigives: C= ∫ 1 0 ∫ Rq K(‖x−γ1(η)‖) ∫ Rq ‖u‖2K(‖u+x−γ2(η)‖)dudxdη. The inner termcanbewrittenas:∫ Rq ‖u‖2K(‖u+x−γ2(η)‖)du= ∫ Rq ‖u+γ2(η)−x‖2K(‖u‖)du = ∫ Rq ‖u‖2K(‖u‖)du+2〈γ2(η)−x, ∫ Rq uK(‖u‖)du〉+‖γ2(η)−x‖2. (11) The integral: ∫ Rq uK(‖u‖)du iszeroand,usingspherical coordinates:∫ Rq ‖u‖2K(‖u‖)du= ∫ R+ rq+1K(r) ∫ Sq−1 dσdr=Vol(Sq−1)M withM= ∫ R+ rq+1K(r). Puttingbackthisvalue in theexpressionof thecostgives: C=Vol(Sq−1)M ∫ 1 0 ∫ Rq K(‖x−γ1(η)‖)dxdη+ ∫ 1 0 ∫ Rq K(‖x−γ1(η)‖)‖γ2(η)−x‖2dxdη =Vol(Sq−1)M+ ∫ 1 0 ∫ Rq K(‖x−γ1(η)‖)‖γ2(η)−x‖2dxdη =Vol(Sq−1)M+ ∫ 1 0 ∫ Rq K(‖x‖)‖γ2(η)−γ1(η)+x‖2dxdη. (12) Finally: ∫ Rq K(‖x‖)‖γ2(η)−γ1(η)+x‖2dx= ∫ Rq K(‖x‖)‖γ2(η)−γ1(η)‖2dxdη +2〈γ2(η)−γ1(η), ∫ Rq xK(‖x‖)dx +Vol(Sq−1)M. (13) Asbefore, themiddle termvanishes,andthefirstone integrates to: ∫ 1 0 ‖γ2(η)−γ1(η)‖2dη so that: C=2Vol(Sq−1)M+ ∫ 1 0 ‖γ2(η)−γ1(η)‖2dη. This result indicates that thedensitiesassociatedwithcurvesγ1,γ2 usingthesmoothingprocess describedabovecannotbe too far (with respect to theWassersteindistance) fromeachother if the geometricL2 distanceD(γ1,γ2) is small. In fact, theupperboundinProposition2canbe interpreted 393