The Double-Sided Information-Bottleneck Function

We consider a two-terminal variant (double-sided) of the information bottleneck problem, which is related to biclus-tering. In our setup, $x$ and Y are dependent random variables and the problem is to find two independent channels $\mathrm{P}_{\cup 1\times}$ and $\mathrm{p}_{\vee 1!}$ (setting the Markovian structure $\cup\rightarrow\times\rightarrow \mathrm{Y}\rightarrow$ V) that maximize $I(\cup;\mathrm{V})$ subject to constraints on the relevant mutual information expressions: $I(\cup;\mathrm{X})$ and $I(\mathrm{V};\mathrm{Y})$ . For jointly Gaussian X and Y, we show that Gaussian channels are optimal in the low-SNR regime, but not for general SNR. Similarly, it is shown that for a doubly symmetric binary source, binary symmetric channels are optimal when the correlation is low, and are suboptimal for high correlation. We conjecture that Z and S channels are optimal when the correlation is 1 (i.e., $\mathrm{X}=\mathrm{Y})$ , and provide supporting numerical evidence.