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：我们通过经验风险最小化在高维情形下刻画了两类具有任意质心的数据点云的混合学习过程，其中假设采用一般凸损失函数与凸正则化方法。每个数据点云均通过对可能不可数多个高斯分布的叠加进行采样获得，其方差具有一般概率密度函数$\varrho$。因此，我们的分析涵盖了包含无协方差幂律尾分布在内的大类数据分布族。我们研究了所得估计器的泛化性能，分析了正则化的作用，以及可分离性转变对分布尺度参数的依赖关系。