论文标题
分层措施的典型性
Typicality for stratified measures
论文作者
论文摘要
此处将欧几里得空间的分层措施定义为可整流措施的凸组合。对于Lebesgue度量，它们可能是单数的，并概括了连续的混合混合物。因此，一个分层的度量$ρ$可以表示为$ \ sum_ {i = 1}^k q_iρ_i$，其中$（q_1，..，..，q_k）$是概率向量，每个$ρ_i$ as $ m_i $ retectiffiffiffiff conterfl $ m_i $ a $ a $ $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a。 $ m_i $ -rectifiable set $ e_i $（例如平滑$ m_i $ -manifold）。我们介绍了一组$ρ^{\ otimes n} $（无内存源）的强烈典型实现，这些实现的可能性很高。典型的实现在层级$ \ {e_ {i_1} \ times \ cdots \ times e_ {i_n} \} $的有限结合中支持其尺寸围绕平均尺寸$ \ sum_ {i = 1}^k q_i m_i $的尺寸。对于每个$ n $，在不同阶层的适当总和均提供了自然的参考“体积”概念；相对于$μ= \ sum_ {i = 1}^kμ_i$，CSISZAR的$ρ$ $ρ$的广义熵量化了指数增长率。此外，我们证明这种广义熵满足链条规则，条件项与每个层中典型实现的体积生长有关。链条规则及其渐近解释在分段连续措施的更一般框架中存在：限制在配备有参考$σ$ finite度量的成对分离集的度量组合。最后，我们确定我们的平均维度概念与Rényi的信息维度相吻合时，将其应用于分层度量，但是此处使用的广义熵与Rényi的维度熵不同。
Stratified measures on Euclidean space are defined here as convex combinations of rectifiable measures. They are possibly singular with respect to the Lebesgue measure and generalize continuous-discrete mixtures. A stratified measure $ρ$ can thus be represented as $\sum_{i=1}^k q_i ρ_i$, where $(q_1,..,q_k)$ is a probability vector and each $ρ_i$ is $m_i$-rectifiable for some integer $m_i$ i.e. absolutely continuous with respect to the $m_i$-Hausdorff measure $μ_i$ on a $m_i$-rectifiable set $E_i$ (e.g. a smooth $m_i$-manifold). We introduce a set of strongly typical realizations of $ρ^{\otimes n}$ (memoryless source) that occur with high probability. The typical realizations are supported on a finite union of strata $\{E_{i_1}\times \cdots \times E_{i_n}\}$ whose dimension concentrates around the mean dimension $\sum_{i=1}^k q_i m_i$. For each $n$, an appropriate sum of Hausdorff measures on the different strata gives a natural notion of reference "volume"; the exponential growth rate of the typical set's volume is quantified by Csiszar's generalized entropy of $ρ$ with respect to $μ=\sum_{i=1}^k μ_i$. Moreover, we prove that this generalized entropy satisfies a chain rule and that the conditional term is related to the volume growth of the typical realizations in each stratum. The chain rule and its asymptotic interpretation hold in the more general framework of piecewise continuous measures: convex combinations of measures restricted to pairwise disjoint sets equipped with reference $σ$-finite measures. Finally, we establish that our notion of mean dimension coincides with Rényi's information dimension when applied to stratified measures, but the generalized entropy used here differs from Rényi's dimensional entropy.
