  In this paper, we present a sharp analysis for a class of alternating
projected gradient descent algorithms which are used to solve the covariate
adjusted precision matrix estimation problem in the high-dimensional setting.
We demonstrate that these algorithms not only enjoy a linear rate of
convergence in the absence of convexity, but also attain the optimal
statistical rate (i.e., minimax rate). By introducing the generic chaining, our
analysis removes the impractical resampling assumption used in the previous
work. Moreover, our results also reveal a time-data tradeoff in this covariate
adjusted precision matrix estimation problem. Numerical experiments are
provided to verify our theoretical results.
