  The goal of lossy data compression is to reduce the storage cost of a data
set $X$ while retaining as much information as possible about something ($Y$)
that you care about. For example, what aspects of an image $X$ contain the most
information about whether it depicts a cat? Mathematically, this corresponds to
finding a mapping $X\to Z\equiv f(X)$ that maximizes the mutual information
$I(Z,Y)$ while the entropy $H(Z)$ is kept below some fixed threshold. We
present a method for mapping out the Pareto frontier for classification tasks,
reflecting the tradeoff between retained entropy and class information. We
first show how a random variable $X$ (an image, say) drawn from a class
$Y\in\{1,...,n\}$ can be distilled into a vector $W=f(X)\in \mathbb{R}^{n-1}$
losslessly, so that $I(W,Y)=I(X,Y)$; for example, for a binary classification
task of cats and dogs, each image $X$ is mapped into a single real number $W$
retaining all information that helps distinguish cats from dogs. For the $n=2$
case of binary classification, we then show how $W$ can be further compressed
into a discrete variable $Z=g_\beta(W)\in\{1,...,m_\beta\}$ by binning $W$ into
$m_\beta$ bins, in such a way that varying the parameter $\beta$ sweeps out the
full Pareto frontier, solving a generalization of the Discrete Information
Bottleneck (DIB) problem. We argue that the most interesting points on this
frontier are "corners" maximizing $I(Z,Y)$ for a fixed number of bins
$m=2,3...$ which can be conveniently be found without multiobjective
optimization. We apply this method to the CIFAR-10, MNIST and Fashion-MNIST
datasets, illustrating how it can be interpreted as an
information-theoretically optimal image clustering algorithm.
