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The 1-norm was proven to be a good convex regularizer for the recovery ofsparse vectors from under-determined linear measurements. It has been shownthat with an appropriate measurement operator, a number of measurements of theorder of the sparsity of the signal (up to log factors) is sufficient forstable and robust recovery. More recently, it has been shown that such recoveryresults can be generalized to more general low-dimensional model sets and(convex) regularizers. These results lead to the following question: to recovera given low-dimensional model set from linear measurements, what is the "best"convex regularizer? To approach this problem, we propose a general framework todefine several notions of "best regularizer" with respect to a low-dimensionalmodel. We show in the minimal case of sparse recovery in dimension 3 that the1-norm is optimal for these notions. However, generalization of such results tothe n-dimensional case seems out of reach. To tackle this problem, we proposelooser notions of best regularizer and show that the 1-norm is optimal amongweighted 1-norms for sparse recovery within this framework.

Optimality of 1-norm regularization among weighted 1-norms for sparse recovery: a case study on how to find optimal regularizations