The information bottleneck method
<|reference_start|>The information bottleneck method: We define the relevant information in a signal $x\in X$ as being the information that this signal provides about another signal $y\in \Y$. Examples include the information that face images provide about the names of the people portrayed, or the information that speech sounds provide about the words spoken. Understanding the signal $x$ requires more than just predicting $y$, it also requires specifying which features of $\X$ play a role in the prediction. We formalize this problem as that of finding a short code for $\X$ that preserves the maximum information about $\Y$. That is, we squeeze the information that $\X$ provides about $\Y$ through a `bottleneck' formed by a limited set of codewords $\tX$. This constrained optimization problem can be seen as a generalization of rate distortion theory in which the distortion measure $d(x,\x)$ emerges from the joint statistics of $\X$ and $\Y$. This approach yields an exact set of self consistent equations for the coding rules $X \to \tX$ and $\tX \to \Y$. Solutions to these equations can be found by a convergent re-estimation method that generalizes the Blahut-Arimoto algorithm. Our variational principle provides a surprisingly rich framework for discussing a variety of problems in signal processing and learning, as will be described in detail elsewhere.<|reference_end|>
@article{tishby2000the,
    title={The information bottleneck method},
    author={Naftali Tishby (Hebrew University and NEC Research Institute),
  Fernando C. Pereira (ATT Shannon Laboratory), and William Bialek (NEC
  Research Institute)},
    journal={arXiv preprint arXiv:physics/0004057},
    year={2000},
    archivePrefix={arXiv},
    eprint={physics/0004057},
    primaryClass={physics.data-an cond-mat.dis-nn cs.LG nlin.AO}
}