A Class of Nonbinary Symmetric Information Bottleneck Problems

We study two dual settings of information processing. Let $\mathsf{Y} \rightarrow \mathsf{X} \rightarrow \mathsf{W}$ be a Markov chain with fixed joint probability mass function $\mathsf{P}_{\mathsf{X}\mathsf{Y}}$ and a mutual information constraint on the pair $(\mathsf{W},\mathsf{X})$. For the first problem, known as Information Bottleneck, we aim to maximize the mutual information between the random variables $\mathsf{Y}$ and $\mathsf{W}$, while for the second problem, termed as Privacy Funnel, our goal is to minimize it. In particular, we analyze the scenario for which $\mathsf{X}$ is the input, and $\mathsf{Y}$ is the output of modulo-additive noise channel. We provide analytical characterization of the optimal information rates and the achieving distributions.