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 by sampling from a possibly uncountable superposition of Gaussian distributions, whose variance has a generic probability density $\varrho$. Our analysis covers therefore a large family of data distributions, including the case of power-law-tailed distributions with no covariance. We study the generalisation performance of the obtained estimator, we analyse the role of regularisation, and the dependence of the separability transition on the distribution scale parameters.

翻译：我们通过高维机制下的经验风险最小化，刻画了两类具有一般质心的数据点云混合的学习过程，该过程假设采用一般凸损失函数与凸正则化。每类数据点云通过从可能不可计数的多个高斯分布的叠加中采样得到，其方差具有一般概率密度 $\varrho$。因此，我们的分析覆盖了大量数据分布族，包括具有无协方差特性的幂律尾分布情形。我们研究了所得估计器的泛化性能，分析了正则化作用，以及可分性相变对分布尺度参数的依赖性。