Interplay of minimax estimation and minimax support recovery under sparsity

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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.