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We study the supervised clustering problem under the two-componentanisotropic Gaussian mixture model in high dimensions and in the non-asymptoticsetting. We first derive a lower and a matching upper bound for the minimaxrisk of clustering in this framework. We also show that in the high-dimensionalregime, the linear discriminant analysis (LDA) classifier turns out to besub-optimal in the minimax sense. Next, we characterize precisely the risk of$\ell_2$-regularized supervised least squares classifiers. We deduce the factthat the interpolating solution may outperform the regularized classifier,under mild assumptions on the covariance structure of the noise. Our analysisalso shows that interpolation can be robust to corruption in the covariance ofthe noise when the signal is aligned with the "clean" part of the covariance,for the properly defined notion of alignment. To the best of our knowledge,this peculiar phenomenon has not yet been investigated in the rapidly growingliterature related to interpolation. We conclude that interpolation is not onlybenign but can also be optimal, and in some cases robust.

Minimax Supervised Clustering in the Anisotropic Gaussian Mixture Model: A new take on Robust Interpolation