he canonical divergence D induces the metric g and the connections∇and∇∗. The same holds for the mean canonical divergence D∇mcd
10 Matching Annotations
- Jul 2017
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if∇is integrable, then it is notgenerally true that X(q,p) =−gradqD∇mcd(p‖·)
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mean canonical divergenceD∇mcd(p‖q):=12(D(p‖q) +D∗(q‖p))(64)which obviously satisfiesD(∇∗)mcd(p‖q) =D∇mcd(q‖p)
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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
Fazendo a mudança de variável $$ s \mapsto t(s) = 1- s $$ e usando o fato de que \( \gamma_{q,p}(s) = \gamma_{p,q}(1 - s) \), temos: $$ \begin{aligned} D(q||p) & := \int0^1 s ||\dot{\gamma}{q,p}(s)||^2 ds \ & = - \int1^0 (1 - t) ||\dot{\gamma}{p,q}(t)||^2 dt \ & = \int0^1 ||\dot{\gamma}{q,p}(t)||^2 dt - D(p||q) \end{aligned} $$
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D(p‖q) =∫10t∥∥ ̇γp,q(t)∥∥2dt(61)whereγp,qdenotes the geodesic from p to q.
Até o momento, a conexão dual parece não desempenhar nenhum papel.
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D(p‖q) =∫10(1−t)∥∥ ̇γq,p(t)∥∥2dt
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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
Qual o papel da conexão dual?
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∫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
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functionsDpsatisfying the condition of Equation (12) then they are uniqueup to a constant that can vary withp, and we can therefore assumeDp(p) =0
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manifold is dually flat, a canonical divergence was introduced by Amari and Nagaoka [2], which isa Bregman divergence
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