  We consider a least squares regression problem where the data has been
generated from a linear model, and we are interested to learn the unknown
regression parameters. We consider "sketch-and-solve" methods that randomly
project the data first, and do regression after. Previous works have analyzed
the statistical and computational performance of such methods. However, the
existing analysis is not fine-grained enough to show the fundamental
differences between various methods, such as the Subsampled Randomized Hadamard
Transform (SRHT) and Gaussian projections. In this paper, we make progress on
this problem, working in an asymptotic framework where the number of datapoints
and dimension of features goes to infinity. We find the limits of the accuracy
loss (for estimation and test error) incurred by popular sketching methods. We
show separation between different methods, so that SRHT is better than Gaussian
projections. Our theoretical results are verified on both real and synthetic
data. The analysis of SRHT relies on novel methods from random matrix theory
that may be of independent interest.
