What Should an Information Bottleneck Encoder Learn? Separating Representation Learning from Rate Control

arXiv:2610.36255v1 Announce Type: new Abstract: The information bottleneck (IB) trades compression of an input $X$ against prediction of a target $Y$ through a multiplier $\beta=1/\lambda$, yet each operating point is usually served by training, or fully conditioning, a new encoder. For jointly Gaussian data, we show that the IB solution factorizes as $A(\beta)=D(\beta)W$: the information-bearing directions $W$ do not depend on $\beta$, and $\beta$ acts only through scalar gains. We ask whether this separation of \emph{representation} from \emph{rate control} survives beyond Gaussian models, treating the budget as a given operating parameter rather than a quantity to be optimized.