We show that the matrix perspective function, which is jointly convex in the Cartesian product of a standard Euclidean vector space and a conformal space of symmetric matrices, has a proximity operator in an almost closed form. The only implicit part is to solve a semismooth, univariate root finding problem. We uncover the connection between our problem of study and the matrix nearness problem. Through this connection, we propose a quadratically convergent Newton algorithm for the root finding problem.Experiments verify that the evaluation of the proximity operator requires at most 8 Newton steps, taking less than 5s for 2000 by 2000 matrices on a standard laptop. Using this routine as a building block, we demonstrate the usefulness of the studied proximity operator in constrained maximum likelihood estimation of Gaussian mean and covariance, peudolikelihood-based graphical model selection, and a matrix variant of the scaled lasso problem.

Proofs of both Theorems 2 and 4 are based on the following key lemma, Lemma A.1. To prove this lemma, we begin by recalling the definition of directional derivatives. F (x + t h) F (x) t if the limit exists. Now we can prove the lemma: Proof of Lemma A.1. The following lemma shows a representation of an element of this set in terms of M: Lemma A.3.

Summary and Contributions: ### Update ### I have read the rebuttal and the other reviews. I'd like to thank to the authors for their carefully thought-out response. In general, I agree with their position that the organization and exposition are the weakest aspects of the submission. I have increased my score to 7 given the authors' commitment to improve the text and address my and the other reviewers' specific comments. Some point-by-point comments follow: 1) Great, I think readability will be greatly improved when the two sections are merged.

We show that the matrix perspective function, which is jointly convex in the Cartesian product of a standard Euclidean vector space and a conformal space of symmetric matrices, has a proximity operator in an almost closed form. The only implicit part is to solve a semismooth, univariate root finding problem. We uncover the connection between our problem of study and the matrix nearness problem. Through this connection, we propose a quadratically convergent Newton algorithm for the root finding problem.Experiments verify that the evaluation of the proximity operator requires at most 8 Newton steps, taking less than 5s for 2000 by 2000 matrices on a standard laptop. Using this routine as a building block, we demonstrate the usefulness of the studied proximity operator in constrained maximum likelihood estimation of Gaussian mean and covariance, peudolikelihood-based graphical model selection, and a matrix variant of the scaled lasso problem.