High-dimensional logistic entropy clustering
Minimization of the (regularized) entropy of classification probabilities is a versatile class of discriminative clustering methods. The classification probabilities are usually defined through the use of some classical losses from supervised classification and the point is to avoid modelisation of the full data distribution by just optimizing the law of the labels conditioned on the observations. We give the first theoretical study of such methods, by specializing to logistic classification probabilities. We prove that if the observations are generated from a two-component isotropic Gaussian mixture, then minimizing the entropy risk over a Euclidean ball indeed allows to identify the separation vector of the mixture. Furthermore, if this separation vector is sparse, then penalizing the empirical risk by a $\ell_{1}$-regularization term allows to infer the separation in a high-dimensional space and to recover its support, at standard rates of sparsity problems. Our approach is based on the local convexity of the logistic entropy risk, that occurs if the separation vector is large enough, with a condition on its norm that is independent from the space dimension. This local convexity property also guarantees fast rates in a classical, low-dimensional setting.