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In this manuscript we consider the problem of generalized linear estimationon Gaussian mixture data with labels given by a single-index model. Our firstresult is a sharp asymptotic expression for the test and training errors in thehigh-dimensional regime. Motivated by the recent stream of results on theGaussian universality of the test and training errors in generalized linearestimation, we ask ourselves the question: "when is a single Gaussian enough tocharacterize the error?". Our formula allow us to give sharp answers to thisquestion, both in the positive and negative directions. More precisely, we showthat the sufficient conditions for Gaussian universality (or lack of thereof)crucially depend on the alignment between the target weights and the means andcovariances of the mixture clusters, which we precisely quantify. In theparticular case of least-squares interpolation, we prove a strong universalityproperty of the training error, and show it follows a simple, closed-formexpression. Finally, we apply our results to real datasets, clarifying somerecent discussion in the literature about Gaussian universality of the errorsin this context.

Are Gaussian data all you need? Extents and limits of universality in high-dimensional generalized linear estimation