Interplay of minimax estimation and minimax support recovery under\n sparsity

In this paper, we study a new notion of scaled minimaxity for sparse\nestimation in high-dimensional linear regression model. We present more\noptimistic lower bounds than the one given by the classical minimax theory and\nhence improve on existing results. We recover sharp results for the global\nminimaxity as a consequence of our study. Fixing the scale of the\nsignal-to-noise ratio, we prove that the estimation error can be much smaller\nthan the global minimax error. We construct a new optimal estimator for the\nscaled minimax sparse estimation. An optimal adaptive procedure is also\ndescribed.\n