论文标题
信息的无限划分
Infinite Divisibility of Information
论文作者
论文摘要
我们研究了无限划分的概率分布的信息类似物，其中I.I.D.总和由I.I.D的联合分布取代。顺序。随机变量$ x $被称为信息无限分开，如果对于任何$ n \ ge1 $，则存在I.I.D.随机变量的顺序$ z_ {1}，\ ldots，z_ {n} $，其中包含与$ x $相同的信息，即，存在一个注入函数$ f $，因此$ x = f（z__ {1}，\ ldots，\ ldots，z__ {n}）$。虽然没有信息在无限划分的离散随机变量上存在，但我们表明，任何离散的随机变量$ x $都有一个有界的乘法差距到无限的可划分性，也就是说，如果我们在$ f $上删除Indentivity Septionement，则存在I.I.D. $ z_ {1}，\ ldots，z_ {n} $和$ f $满足$ x = f（z__ {1}，\ ldots，z_ {n}）$，并且熵满足$ h（x）/n \ le H（z___ {1}} \ le1.59h（x）/n \ le h（x）/n \ le h（x）/n \ le h（x）/n \ le1.59h（x）我们还研究了一类新的离散概率分布，称为Spectral无限分布分布，我们可以在其中删除乘法差距$ 1.59 $。此外，我们研究$ x =（y_ {1}，\ ldots，y_ {m}）$的情况本身就是一个I.I.D.序列，$ m \ ge2 $，为此，乘法差距$ 1.59 $可以用$ 1+5 \ sqrt {（\ log M）/m} $代替。这意味着，随着$ m $的增加，$（y_ {1}，\ ldots，y_ {m}）$变得更接近光谱以统一的方式无限分组。这可以被视为Kolmogorov统一定理的信息类似物。我们结果的应用包括独立的组件分析，具有保密约束的分布式存储以及分布式随机数生成。
We study an information analogue of infinitely divisible probability distributions, where the i.i.d. sum is replaced by the joint distribution of an i.i.d. sequence. A random variable $X$ is called informationally infinitely divisible if, for any $n\ge1$, there exists an i.i.d. sequence of random variables $Z_{1},\ldots,Z_{n}$ that contains the same information as $X$, i.e., there exists an injective function $f$ such that $X=f(Z_{1},\ldots,Z_{n})$. While there does not exist informationally infinitely divisible discrete random variable, we show that any discrete random variable $X$ has a bounded multiplicative gap to infinite divisibility, that is, if we remove the injectivity requirement on $f$, then there exists i.i.d. $Z_{1},\ldots,Z_{n}$ and $f$ satisfying $X=f(Z_{1},\ldots,Z_{n})$, and the entropy satisfies $H(X)/n\le H(Z_{1})\le1.59H(X)/n+2.43$. We also study a new class of discrete probability distributions, called spectral infinitely divisible distributions, where we can remove the multiplicative gap $1.59$. Furthermore, we study the case where $X=(Y_{1},\ldots,Y_{m})$ is itself an i.i.d. sequence, $m\ge2$, for which the multiplicative gap $1.59$ can be replaced by $1+5\sqrt{(\log m)/m}$. This means that as $m$ increases, $(Y_{1},\ldots,Y_{m})$ becomes closer to being spectral infinitely divisible in a uniform manner. This can be regarded as an information analogue of Kolmogorov's uniform theorem. Applications of our result include independent component analysis, distributed storage with a secrecy constraint, and distributed random number generation.
