__ Summary and Contributions__: The authors proposed a metric learning algorithm to find a Mahalanobis distance that is robust against adversarial perturbation. They formulated an objective function to learn a Mahalanobis distance, parameterized by a positive semi-definite matrix M, that maximized the minimal adversarial perturbation on each sample.

__ Strengths__: The paper has dealt with the problem of distance metric learning which is a very common and fundamental problem. The idea is simple and interesting. The paper has demonstrated reasonable improvement over several traditional metric learning methods. The theoretical setting upon which the authors build up their approach is well established.

__ Weaknesses__: The experiment is the major problem in the paper.
The authors compared the proposed ARML method with the following baselines: Euclidean, NCA, LMNN, ITML and LFDA. The compared methods are relatively old.
The authors used six public datasets on which metric learning methods perform favorably in terms of clean errors, including four small or medium-sized datasets: Splice, Pendigits, Satimage and USPS, and two image datasets MNIST and Fashion-MNIST, which are wildly used for robust verification for neural networks. The first four datasets are very simple and easy to classify. Thus, the practicability of the proposed method is hard to evaluate. Although the observations are interesting, they are rather limited.

__ Correctness__: The solution is technically sound. The claims made by the authors are somewhat supported by their findings.

__ Clarity__: Paper is understandable and relatively well-written.

__ Relation to Prior Work__: The authors discussed how this work differs from previous contributions to some extent.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The authors present an approach for metric learning that is robust to adversarial perturbations when used in conjunction with K-NN classifier. (Adversarial Robust Metric Learning = ARML). They present an algorithm to learn a Mahalanobis distance (i.e. matrix) M that is robust to additive perturbations to the input. Authors formulate a learning problem to estimate M that approximately maximizes minimal adversarial perturbation. Instead of looking at K neighbors, they consider a triplet loss. For a test point (x), its nearest point of the same class (xp) and its nearest point of a different class (xn), optimize M that decreases the M -distance to the same class point (xp) and increases the distance to other class (xn) normalized by the M -distance between xp and xn. Training is done with SGD over samples of such triplets. AMRL is compared and outperforms a few other approaches. Performance is measured in terms of 1-NN error.

__ Strengths__: Its an interesting paper and a clean and principled formulation of the approximate objective to learn M under adversarial perturbations. Empirical results also show improvement over other algorithms. Since K-NN classifiers are widely used, this is an important area to study and provide mechanism to improve robustness.

__ Weaknesses__: There is no discussion about compuational cost of proposed approach. It would be great to see a comparision table of ARML vs other methods in average training time. Additionally, there is no information about ARML’s effect on classification accuracy. t would be great if there is explaination about their approach vs the existing ones. Some of the missing papers are; “Adversarial Metric Learning” by Chen et al. (2018), “ADVKNN: Adversarial Attacks on K-NN Classifiers with Approximate Gradients” by Li et al. (2019), and “Towards Certified Robustness of Metric Learning” by Yang et al. (2020).

__ Correctness__: Theory and empirical methodology looks correct

__ Clarity__: yes

__ Relation to Prior Work__: Authors discuss related work

__ Reproducibility__: Yes

__ Additional Feedback__: I reviews author's feedback. I appreciate additional comparison to neural network approaches. There is a debate whether deep metric learning is a fair comparison so I am keeping my reviews unchanged.

__ Summary and Contributions__: This paper proposes a metric learning method for robust KNN inference against adversarial examples. A minimal norm of adversarial example for metric learning is derived. Neighbor samples are sampled to optimize the minimal norm. The experimental results demonstrate the effectiveness of the proposed method. The results show that the proposed method does not decrease accuracy in the clean setting.

__ Strengths__: 1. A minimal norm of adversarial example for metric learning is derived. And a corresponding optimization scheme is proposed.
2. The results show that the proposed method does not decrease accuracy in the clean setting.

__ Weaknesses__: 1. No metric learning method against adversarial examples is discussed or compared in the experiment. For example:
Chen, Shuo, et al. "Adversarial metric learning." Proceedings of the 27th International Joint Conference on Artificial Intelligence. 2018.
Mao, Chengzhi, et al. "Metric learning for adversarial robustness." Advances in Neural Information Processing Systems. 2019.
Duan, Yueqi, et al. "Deep adversarial metric learning." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2018.
Zheng, Wenzhao, et al. "Hardness-aware deep metric learning." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2019.
2. The loss function in Eq. (12) is not clear.
3. Sampling instances from the neighborhood of a sample is a important efficiency-related issue, which is not clarified.

__ Correctness__: Yes

__ Clarity__: Yes

__ Relation to Prior Work__: 1. No metric learning method against adversarial examples is discussed or compared in the experiment. For example:
Chen, Shuo, et al. "Adversarial metric learning." Proceedings of the 27th International Joint Conference on Artificial Intelligence. 2018.
Mao, Chengzhi, et al. "Metric learning for adversarial robustness." Advances in Neural Information Processing Systems. 2019.
Duan, Yueqi, et al. "Deep adversarial metric learning." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2018.
Zheng, Wenzhao, et al. "Hardness-aware deep metric learning." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2019.

__ Reproducibility__: No

__ Additional Feedback__: I am satisified with the rebuttal of the authors.

__ Summary and Contributions__: The paper presents a mahalanobis learning algorithm that is certifiable robust to adversarial attacks. The algorithm learns a Mahalabobis matrix which maximizes the minimal adversarial attack on each example. The method is compared against standard learning algorithms on a series of datasets and show that indeed the proposed algorithm has a good robustness to attacks, exhibiting the lowest values of robust error, and often has also the lowest error.
To learn the Mahalanobis matrix it defines an objective it establishes a lower bound for minimal adversarial perturbation of some training instance that is parametrized by the Mahalanobis matrix. The bound is based on the minimal perturbation that given an instance and a negative and a positive instance will change the nearest neighbor relation. Using the triplet result one would need to go over all combinations of negative and positive instances for the given instance to compute the value of the bound, resulting in a quadratic complexity. In the actual algorithm this quadratic complexity is reduced by sampling only the nearest neighbors of any given instance, which adds a second level of approximation.
For the case of one nearest neighbor they propose a variant of the algorithm that instead of relying on a bound computes the exact minimal adversarial perturbation for any given instance, which involves solving a series of quadratic programming problems in order to get the minimum over them.

__ Strengths__: The paper presents what according to the authors is the first certifiable robust metric learning algorithm. The algorithm that it proposes is grounded on a theoretical established lower bound of the minimum adversarial perturbation and is shown to significantly improve the robustness of metric learning, at least when compared to non-robust standard metric learning baseline.

__ Weaknesses__: Since there are no robust-metrix learning algorithms the algorithm proposed here is compared against standard metric learning algorithms, such as LMNN and ITML. It would have been appropriate to see how the proposed algorithm fairs with respect to robust deep learning algorithms.

__ Correctness__: As I stated before, in the apparent absence of robust metric learning algorithms, it would have been useful to provide results from other families of robust algorithms such as the ones the authors themselves cite, e.g. [43, 11, 48].

__ Clarity__: The paper in general is well written. There were a few points that were not clear to me.
I cannot parse equation 6. If I understand well all x_test instances for which |J|=k and |I|=k-1 are missaclassified. The constrain in equation 6 looks for each such instance for the min \delta_{ij}, with j \in i and i in {all same class instances to x_test} - I, that would essentially leave the I and J sets unchanged; in other words we do not want any same class instance of x_test to become a nearest neighbor and kick out a different class nearest neighbor. Shouldn't we be looking instead for all correctly classified instances, i.e. |I|=k, and |J|=k-s, and then among these the minimum perturbation that would reduce the cardinality of I? On the other hand equation 13 which seems to provide the same result for the 1-nn case makes sense. It simply looks for the minimum perturbation over all instances with different class than x that will make one of them the closest instance to x, and thus result in a wrong classification?
In addition in equation 7 what do we know for x? should it be that originally it is closer to x+? which would make sense if we would want to find the minimum perturbation that makes it closest to the x-.
In the experiments how is the \epsilon value determined in tables 1 and 2? I guess the \epsilon value should somehow depend on the average norm of the training instances?

__ Relation to Prior Work__: yes, the discussion with respect to the previous work is satisfactory, as far as I can judge.

__ Reproducibility__: Yes

__ Additional Feedback__: The response of the authors was satisfactory, and in particular the addition of baselines and the addition of a discussion on related work. I will keep my rating.