AMBIENTES DE APRENDIZAJE LÚDICOS

he canonical divergence D induces the metric g and the connections∇and∇∗. The same holds for the mean canonical divergence D∇mcd

if∇is integrable, then it is notgenerally true that X(q,p) =−gradqD∇mcd(p‖·)

mean canonical divergenceD∇mcd(p‖q):=12(D(p‖q) +D∗(q‖p))(64)which obviously satisfiesD(∇∗)mcd(p‖q) =D∇mcd(q‖p)

he energy of the geodesicγp,qas the symmetrized version of the canonical divergence:12(D(p‖q) +D(q‖p))=12∫10∥∥ ̇γp,q(t)∥∥2dt

D(p‖q) =∫10t∥∥ ̇γp,q(t)∥∥2dt(61)whereγp,qdenotes the geodesic from p to q.

D(p‖q) =∫10(1−t)∥∥ ̇γq,p(t)∥∥2dt

n-dimensional dual manifold(M,g,∇,∇∗). Consider a∇-geodesicγq,p:[0, 1]→Mconnectingqandp. We define a tangent vector fieldXt(p,q)along this geodesic:Xt(q,p):=X(γq,p(t),p)(52)Obviously,X0=X(q,p)(53)X1(q,p) =0(54)Definition 3.A canonical divergence from p to q is defined by the path integralD(p‖q) =∫10〈Xt(q,p), ̇γq,p(t)〉dt

∫10〈X(γ(t),p), ̇γ(t)〉dt=−∫10〈gradγ(t)Dp, ̇γ(t)〉dt=−∫10(dγ(t)Dp)( ̇γ(t))dt=−∫10d Dp◦γd t(t)dt=Dp(γ(0))−Dp(γ(1))=Dp(q)−Dp(p) =Dp(q) =D(p‖q)(13)In particular, we can apply this derivation to the geodesic connectingqandpeven when theintegrability ofXis not guaranteed and obtain the definition of a general canonical divergence

functionsDpsatisfying the condition of Equation (12) then they are uniqueup to a constant that can vary withp, and we can therefore assumeDp(p) =0

manifold is dually flat, a canonical divergence was introduced by Amari and Nagaoka [2], which isa Bregman divergence

a divergence exists for any such manifold. However, it isnot unique and there are infinitely many divergences that give the same geometrical structure

When a coordinate systemξ:p7→ξp= (ξ1p, . . . ,ξnp)∈Rnis given inM, we pose one condition that, for two nearby pointsξpandξq=ξp+∆ξ,Dis expanded asD(p‖q) =12Dgij(p)∆ξi∆ξj+O(‖∆ξ‖3)(2)and(Dgij(p))ijis a positive definite matrix.

A divergence functionD(p‖q)is a differentiable real-valued function of two pointspandqin amanifoldM. It satisfies the non-negativity conditionD(p‖q)≥0(1)with equality if and only ifp=q.