First of all we would like to thank you for the attention you paid to our paper and for all your fruitful remarks. This will be taken into consideration to write the next version of the article. However, we have several comments to clarify some details.

First, you say that because of the thresholding operator, the Shrinkage-Thresholding MALA algorithm does not sample exactly from the target distribution. This is true for the hard thresholding operator (Section 3.2) which avoids the shrinkage of all the active rows (but which cannot draw rows with a norm lower than a given threshold). The two other operators, namely the L2,1 proximal and the soft thresholding function, do not suffer from this flaw and propose new rows with norms as close to zero as needed. This is illustrated in Figure 1. Therefore, in the numerical section, both RJMCMC and STMALA target the right distribution.

In the numerical section, Figure 5 displays the error obtained when the algorithms are used to estimate the activation probabilities (defined in equation (18)). In addition, the ordinate axis of this figure (as well as the one of Figure 10 for example) is in logarithmic scale which explains why there is no plateau in the curves while the algorithms do converge.

Once again, many thanks for your remarks which will help us to write an improved version of the article.

Best regards,

Amandine Schreck and her co-authors.

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