The paper studies the minimum number of linear measurements requires to recover a sparse, low rank, or otherwise structured signal when using convex relaxations. The key contribution is the extension of the class of linear measurements designs by requiring only the first and second moments of the measurement ensemble. This is an important extension. The reviewers and also I recommend a modification in the title since "learning" in all generality is not proved in the manuscript but rather the convex formulations are. As a result, it would be important to highlight this distinction.