Pareto-Optimal Data Compression for Binary Classification Tasks
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→Z≡f(X) that maximizes the mutual information I(Z,Y) while the entropy H(Z) is kept below some fixed threshold. We present a new 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∈{1,…,n} can be distilled into a vector W=f(X)∈Rn−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=gβ(W)∈{1,…,mβ} by binning W into mβ bins, in such a way that varying the parameter β 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 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. We find that these Pareto frontiers are not concave, and that recently reported DIB phase transitions correspond to transitions between these corners, changing the number of clusters.