论文标题
关于Hölder均衡概率流形的大地测量和动态信息预测
Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities
论文作者
论文摘要
我们在这里考虑转换$ t：m \至m $所描述的离散时间动态，其中$ t $是shift $ t =σ$的动作符号空间$ m = \ {1,2，...，d \}^\ mathbb {n} $，或者，$ t $ d $ $ 1 $ 1 $ t $ s^$ t^$ t^s^$ t^$ t^s^$ t^s^$ t^s^$ t^s^s^$ t^$ t^$ t^$ t^$ t^$ t^$ t^$ t^s^s^s^s^s^s^s^s^s^的时间^$。 $ c^{1+α} $（例如$ x \ to t（x）= d \，x $（mod $ 1）$ \，），其中$ m = s^1 $是单位圆。众所周知，Hölder电位的无限歧管$ \ MATHCAL {N} $的平衡概率$ a：m \ to \ mathbb {r} $是一个分析流形，并且具有与渐近方差相关的天然riemannian指标。我们在这里表明，在假设存在类似于傅立叶的希尔伯特基础的基础上，那里存在地质路径。当存在$ t =σ$和$ m = \ {0,1 \}^\ mathbb {n} $的基础。 在不同的方向上，我们还考虑了一对平衡概率的KL-Divergence $ d_ {kl}（μ_1，μ_2）$。如果$ d_ {kl}（μ_1，μ_2）= 0 $，则$μ_1=μ_2$。尽管$ d_ {kl} $不是$ \ Mathcal {n} $中的度量，但它描述了$μ_1$和$μ_2$之间的接近度。一个自然的问题是：对于固定的概率$μ_1\ in \ Mathcal {n} $中的概率$μ_2$在$ \ MATHCAL {N} $中的一组概率中，该概率最小化$ d_ {kl}（kl}（μ_1，μ_2，μ_2）$。这个最小化问题是信息预测中考虑的主要问题的动态版本。我们认为在$ \ Mathcal {n} $中考虑了这个问题，这种情况是所有概率都是动态不变的，可以为所寻求的解决方案获得明确的方程式。将研究三角形和毕达哥拉斯的不平等现象。
We consider here the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is either the action of shift $T=σ$ on the symbolic space $M=\{1,2,...,d\}^\mathbb{N}$, or, $T$ describes the action of a $d$ to $1$ expanding transformation $T:S^1 \to S^1$ of class $C^{1+α}$ (\,for example $x \to T(x) =d\, x $ (mod $1) $\,), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of equilibrium probabilities for Hölder potentials $A:M \to \mathbb{R}$ is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When $T=σ$ and $M=\{0,1\}^\mathbb{N}$ such basis exists. In a different direction, we also consider the KL-divergence $D_{KL}(μ_1,μ_2)$ for a pair of equilibrium probabilities. If $D_{KL}(μ_1,μ_2)=0$, then $μ_1=μ_2$. Although $D_{KL}$ is not a metric in $\mathcal{N}$, it describes the proximity between $μ_1$ and $μ_2$. A natural problem is: for a fixed probability $μ_1\in \mathcal{N}$ consider the probability $μ_2$ in a convex set of probabilities in $\mathcal{N}$ which minimizes $D_{KL}(μ_1,μ_2)$. This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in $\mathcal{N}$, a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.
