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First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. Summary: The authors propose a new Newton-like method to optimize the sum of a smooth (convex) cost function and multiple decomposable norms. Their contributions are (1) an active subspace selection procedure that allows to speed up the solution of the quadratic approximation problem (2) a proof that solving the quadratic approximation problem over the (changing) active subspace still leads to convergence. The authors also provide numerical results showing that, for two important problems, their methods gives 10x speed up over state-of-the-art methods and, in the appendix, give numerical results that illustrate which fraction of the speed up is due to the quadratic approximation technique and which fraction of the speed up is due to the active subspace selection method. Quality: The amount of critical information in the appendix makes this paper more suited for a journal than a conference.