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.