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