  In this paper, we characterize data-time tradeoffs of the proximal-gradient
homotopy method used for solving penalized linear inverse problems under
sub-Gaussian measurements. Our results are sharp up to an absolute constant
factor. We also show that, in the absence of the strong convexity assumption,
the proximal-gradient homotopy update can achieve a linear rate of convergence
when the number of measurements is sufficiently large. Numerical simulations
are provided to verify our theoretical results. All proofs are included in the
online full version of this paper.
