Intuitive dissection of the Gaussian information bottleneck method with an application to optimal prediction

Efficient signal representation is essential for the functioning of living and artificial systems operating under resource constraints. A widely recognized framework for deriving such representations is the information bottleneck method, which yields the optimal strategy for encoding a random variable, such as the signal, in a way that preserves maximal information about a functionally relevant variable, subject to an explicit constraint on the amount of information encoded. While in its general formulation the information bottleneck method is a numerical scheme, it admits an analytical solution in an important special case where the variables involved are jointly Gaussian. In this setting, the solution predicts discrete transitions in the dimensionality of the optimal representation as the encoding capacity is increased. Although these signature transitions, along with other features of the optimal strategy, can be derived from a constrained optimization problem, a clear and intuitive understanding of their emergence is still lacking. In our work, we advance our understanding of the Gaussian information bottleneck method through multiple mutually enriching perspectives, including geometric and information-theoretic ones. These perspectives offer intuition about the set of optimal encoding directions, the nature of the critical points where the optimal number of encoding components changes, and the way the optimal strategy navigates between these critical points. We then apply our treatment of the information bottleneck to a previously studied signal prediction problem, obtaining insights into how different features of the signal are encoded across multiple components to enable optimal prediction of future signals. Altogether, our work deepens the foundational understanding of the information bottleneck method in the Gaussian setting, motivating the exploration of analogous perspectives in broader, non-Gaussian contexts.