On the Cardinality of Optimal Representations in the Binary-Source Information Bottleneck

The information bottleneck (IB) seeks a representation $U$ of a source $X$ that retains as much information as possible about a target $Y$, subject to a constraint on $I(U;X)$. A classical argument shows that it suffices to consider representations with at most $|\mathcal{X}|+1$ symbols, and this bo