复合信息瓶颈展望
The Compound Information Bottleneck Outlook
摘要：
建立了复合信息瓶颈规划模型并进行了分析。在这个问题中，假设马尔可夫链$ \mathsf{X} \rightarrow \mathsf{Y} \rightarrow \mathsf{Z} $具有固定的边际分布$\mathsf{P}_{\mathsf{X}}$和$\mathsf{P}_{\mathsf{Y}}$，并且在给定来自给定类的$\mathsf{Y}$的$\mathsf{Z}$的条件概率的选择上，在来自不同类的对$(\mathsf{X},\mathsf{Y})$的联合概率的\textit{worst choice}下，寻求$ \mathsf{X} $和$ \mathsf{Z} $之间的互信息最大化。我们考虑几个类的基础上的极端：互信息;最小相关性;总变差;和相对熵类。我们提供的值，界限，和各种表征的具体情况下，这个问题：二进制对称的情况下，标量高斯的情况下，矢量高斯的情况下，对称模加性的情况下。最后，对于一般情况下，我们提出了一个Blahut-Arimoto型交替迭代算法，找到一个一致的解决方案，这个问题。
英文摘要：
We formulate and analyze the compound information bottleneck programming. In this problem, a Markov chain $ \mathsf{X} \rightarrow \mathsf{Y} \rightarrow \mathsf{Z} $ is assumed with fixed marginal distributions $\mathsf{P}_{\mathsf{X}}$ and $\mathsf{P}_{\mathsf{Y}}$, and the mutual information between $ \mathsf{X} $ and $ \mathsf{Z} $ is sought to be maximized over the choice of conditional probability of $\mathsf{Z}$ given $\mathsf{Y}$ from a given class, under the \textit{worst choice} of the joint probability of the pair $(\mathsf{X},\mathsf{Y})$ from a different class. We consider several classes based on extremes of: mutual information; minimal correlation; total variation; and the relative entropy class. We provide values, bounds, and various characterizations for specific instances of this problem: the binary symmetric case, the scalar Gaussian case, the vector Gaussian case and the symmetric modulo-additive case. Finally, for the general case, we propose a Blahut-Arimoto type of alternating iterations algorithm to find a consistent solution to this problem.
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