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 $\varrho$. 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.

Multiple-input and multiple-output processing techniques in fiber optical communications have been proposed as a promising approach to meet increasing demand for information throughput. In this context, the multiple channels correspond to the multiple modes or multiple cores or both in the fiber. In this paper, we characterize the distribution of the mutual information with Gaussian input in a simple channel model for this system. Assuming significant crosstalk between cores, negligible backscattering and near-lossless propagation in the fiber, we model the transmission channel as a random complex unitary matrix. The loss in the transmission may be parameterized by a number of unutilized channels in the fiber. We analyze the system in a dual fashion. First, we evaluate a closed-form expression for the outage probability, which is handy for small matrices. We also apply the asymptotic approach, in particular the Coulomb gas method from statistical mechanics, to obtain closed-form results for the ergodic mutual information, its variance as well as the outage probability for Gaussian input in the limit of large number of cores/modes. By comparing our analytic results to simulations, we see that, despite the fact that this method is nominally valid for large number of modes, our method is quite accurate even for small to modest number of channels.