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.

翻译：我们通过经验风险最小化，在一般凸损失和凸正则化的假设下，描述了高维场景中两个具有一般质心的数据点云混合的学习过程。每个数据点云通过双重随机过程生成，其中样本来自高斯分布，而该分布的方差本身是从标量分布$\varrho$中采样的随机参数。因此，我们的分析涵盖了包括无协方差的幂律尾分布在内的一大类数据分布，并使我们能够检验近期提出的“高斯普适性”论断。我们研究了所得估计量的泛化性能，分析了正则化的作用，并解析地表征了可分性转变。