Paper:
Matrix Completion Under Monotonic Single Index Models
Most recent results in matrix completion assume that the matrix under consideration is low-rank or that the columns are in a union of low-rank subspaces. In real-world settings, however, the linear structure underlying these models is distorted by a (typically unknown) nonlinear transformation. This paper addresses the challenge of matrix completion in the face of such nonlinearities. Given a few observations of a matrix that are obtained by applying a Lipschitz, monotonic function to a low rank matrix, our task is to estimate the remaining unobserved entries. We propose a novel matrix completion method that alternates between lowrank matrix estimation and monotonic function estimation to estimate the missing matrix elements. Mean squared error bounds provide insight into how well the matrix can be estimated based on the size, rank of the matrix and properties of the nonlinear transformation. Empirical results on synthetic and real-world datasets demonstrate the competitiveness of the proposed approach.

Review:
This paper proposed extends the single index model in statistics to low-rank matrix estimation. Particularly, the authors proposed to recover the underlying low-rank matrix given the observations after nonlinear transformation. An algorithm based on alternative minimization is proposed to estimate both the transformation function and the underlying low-rank matrix. Theoretical analysis is given for the proposed algorithm. Finally, experiments on both synthetic data and benchmark datasets are conducted to demonstrate the effectiveness of the proposed algorithm in recovering the matrix.  Major comments   1. A quick tip is that the bound in (18) can be greatly simplified. For example, the forth term is dominated by the second term in the setting of this paper.  2. Based on the proof in the appendix, the error bound obtained in Theorem 1 should hold with high probability. The authors should point this out explicitly. 3. As the authors also mention in the paper that the problem being solved is a non-convex problem, a natural question is whether the proposed algorithm will converge to a stationary point. However, this question is not answered in this paper. 4. Only the error bound of the one-step estimator ($T = 1$) is given. However, the error bound of the proposed algorithm is much more important and essential when T is larger than one. Also, as indicated by the experimental part, the one-step estimator is worse than the SVD estimator. In this sense, the analysis is not complete.  5. Line 333, the statement of "For such matrices $\mu_1 = O(), \mu_2 = O()$" is NOT obvious to me.Further justifications are required for rigorous analysis.    6. Line 347, the statement of "If we are given ...." contradicts with the definition of $\tilde{O}$ in Line 306 that there should be a logarithmic term, let alone that a constant is missing.  7. In Line 085, the setup of the problem is that the entries in the matrix are observed with noise; however, in the experimental part, the settings correspond to noiseless observations. The authors should at least add more experiments corresponding to the settings being studied in this paper.    8. The writing and organization of the proof (Appendix) in this paper should be improved significantly. Based on the current writing, it is almost impossible to check the correctness of the theoretical analysis.i) Line 104, it is not clear what A1-A8 refer to.  ii) Line 226 (Appendix), it is not evident what I'_4 is, which leads to confusion over statement in line 233.  iii) Line 191, "Hence, A - Z" ---> "Hence, A - \hat{Z}".  iv) (31) is redundant.