This paper proposes a new parametrization of the multivariate linear regression problem. It shows that under this new parametrization, it is easier to employ sparsity inducing penalty terms on the inverse covariance matrix. The paper suggests a sequential relaxation algorithm. The reviewers noted the novelty of the approach and numerous strengths. The simulation experiments (in the supplementary material) explore the method in the context of several connectivity scenarios. However, one weakness is the exploration of the performance of the model on real data scenarios. It would also be helpful to explore the sensitivity of the approach to violations of assumptions. There were other minor weaknesses around initialization, and clarity. Despite these minor weaknesses. The author's response addressed many of the concerns the reviewers raised and the discussion around the manuscript and thought it was overall a positive contribution to the community. I recommend this paper for acceptance.