Classification of Heavy-tailed Features in High Dimensions: a Superstatistical Approach
This work addresses the challenge of classification in high-dimensional settings with non-Gaussian data, which is incremental as it extends existing analyses to more general distributions.
The paper tackles the problem of learning from high-dimensional data with heavy-tailed features by analyzing a mixture model with random variances, covering power-law distributions and testing Gaussian universality claims. It provides analytical results on generalization performance, regularization effects, and the separability transition.
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.