Paper:
Learning Mahalanobis metric spaces is an important problem that has found numerous applications. Several algorithms have been designed for this problem, including Information Theoretic Metric Learning (ITML) [Davis et al. 2007] and Large Margin Nearest Neighbor (LMNN) classification [Weinberger and Saul 2009]. We consider a formulation of Mahalanobis metric learning as an optimization problem, where the objective is to minimize the number of violated similarity/dissimilarity constraints. We show that for any fixed ambient dimension, there exists a fully polynomial-time approximation scheme (FPTAS) with nearly-linear running time. This result is obtained using tools from the theory of linear programming in low dimensions. We also discuss improvements of the algorithm in practice, and present experimental results on synthetic and real-world data sets. Our algorithm is fully parallelizable and performs favorably in the presence of adversarial noise.

Review:
The paper proposes a method to handle Mahalanobis metric learning thorough linear programming. The authors consider the specific setup where examples are labelled as similar or dissimilar and the task is to find a mapping such that the feature-space distance between examples is i) smaller than a certain value if the examples are labelled as 'similar' and ii) greater than a possibly different value if the examples are labelled as 'dissimilar'. Arguments from the theory of linear programming are leveraged to define exact and approximated algorithms.
I would tend to reject the paper because I do not fully understand where is where is the main novelty.  Transforming the problem into a linear programming does not look a very complicated step given the specific setup considered in the paper. Moreover, it is not clear enough if there are computational or theoretical gains in following the proposed approach instead of applying other existing methods. Especially because the provided experiments seem to show that there is no improvement  in the accuracy, the authors should have spent some more words  to motivate their strategy.
Questions:
- Are there any theoretical guarantees for the proposed approximation? Is the proposed approximation strategy completely new or similar approaches have been already applied to slightly different setups?
- What are the key differences between the proposed method and with other convex approximations for learning Mahalanobis metrics? As the experimental performance of the proposed approach and other existing methods, what are the net advantages to be associated with the geometric approximation?
- Why is the approximation needed?
- In the proof, why is it true that, for a given solution, adding a constraint implies this constraint is not satisfied?