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.