  Teacher-student models provide a framework in which the typical-case
performance of high-dimensional supervised learning can be described in closed
form. The assumptions of Gaussian i.i.d. input data underlying the canonical
teacher-student model may, however, be perceived as too restrictive to capture
the behaviour of realistic data sets. In this paper, we introduce a Gaussian
covariate generalisation of the model where the teacher and student can act on
different spaces, generated with fixed, but generic feature maps. While still
solvable in a closed form, this generalization is able to capture the learning
curves for a broad range of realistic data sets, thus redeeming the potential
of the teacher-student framework. Our contribution is then two-fold: First, we
prove a rigorous formula for the asymptotic training loss and generalisation
error. Second, we present a number of situations where the learning curve of
the model captures the one of a realistic data set learned with kernel
regression and classification, with out-of-the-box feature maps such as random
projections or scattering transforms, or with pre-learned ones - such as the
features learned by training multi-layer neural networks. We discuss both the
power and the limitations of the framework.
