  We consider linear regression problems with a varying number of random
projections, where we provably exhibit a double descent curve for a fixed
prediction problem, with a high-dimensional analysis based on random matrix
theory. We first consider the ridge regression estimator and review earlier
results using classical notions from non-parametric statistics, namely degrees
of freedom, also known as effective dimensionality. We then compute asymptotic
equivalents of the generalization performance (in terms of squared bias and
variance) of the minimum norm least-squares fit with random projections,
providing simple expressions for the double descent phenomenon.
