Classification of Heavy-tailed Features in High Dimensions: a Superstatistical Approach

We characterise the learning of a mixture of two clouds of data points with generic centroids via empirical risk minimisation in the high dimensional regime, under the assumptions of generic convex loss and convex regularisation. Each cloud of data points is obtained via a double-stochastic process, where the sample is obtained from a Gaussian distribution whose variance is itself a random parameter sampled from a scalar distribution 𝜚. As a result, our analysis covers a large family of data distributions, including the case of power-law-tailed distributions with no covariance, and allows us to test recent "Gaussian universality" claims. We study the generalisation performance of the obtained estimator, we analyse the role of regularisation, and we analytically characterise the separability transition. * (0, +∞). The family of "elliptic-like" distributions in Eq. (1b) has been extensively studied, for instance, by the physics community, in the context of superstatistics [5, 7] . Mixtures of normals in the form of Eq. 1b are a central tool in Bayesian statistics [65] due to their ability to approximate any distribution given a sufficient number