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 bound is known to be tight whenever $|\mathcal{X}| \geq 3$. We show that the binary case behaves differently: if $X$ is binary and $Y$ is finite, then for every joint distribution of $(X,Y)$ and every rate constraint, the IB optimum is attained by a binary $U$. Hence the bound $|\mathcal{U}| \leq |\mathcal{X}|+1$ sharpens to $|\mathcal{U}| \leq |\mathcal{X}|$ for binary sources. The proof combines a separating hyperplane argument with the observation that, for a binary source, the ratio of the second derivatives of the two entropy functions involved is concave.