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

Understand

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model.

• Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime.
• Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?".
• Our formula allow us to give sharp answers to this question, both in the positive and negative directions.