Technically, our result is enabled by establishing a link with recent works on optimal denoising of extensive-rank matrices and on the ellipsoid fitting problem.

Gaussian mixtures are a popular model in high-dimensional statistics since, besides being an universal approximator, they often lead to mathematically tractable problems.

This setup contains a vast array of fundamental applications in machine learning, engineering, neuroscience, finance, statisticsandinformation theory [1-10].

Under label corruptions, we prove that this simple estimator achieves minimax near-optimal riskonawiderange ofgeneralized linear models, including Gaussian regression, Poisson regression and Binomial regression.

A recent line of work in high-dimensional statistics working under the Gaussian mixture hypothesis has led to a number of results in the context of empirical risk minimization, Bayesian uncertainty quantification, separation of kernel methods and neural networks, ensembling and fluctuation of random features.