Kernel functions based on triplet comparisons
Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a lowdimensional Euclidean embedding of the data set that reflects the given similarity triplets, we aim at defining kernel functions that correspond to high-dimensional embeddings. These kernel functions can subsequently be used to apply any kernel method to the data set.
The authors propose two kernel functions based on comparing triplets of data examples. The first kernel function is based on ranking of all the other objects with respect to the objects being compared. The second function is based on comparing whether the compared objects rank similarly with respect to the other objects. The idea is interesting and it is potentially impactful to create such “ordinal” kernels, where the data are not embedded in metric spaces. 
Comments:
1. The authors should provide their main claims somewhere and demonstrate it well in the paper. Why should anybody use a triplet comparison kernel? From what I understand, with triplet comparison kernel, less number of triplets are sufficient to capture the same information that a full pairwise similarity matrix can capture (sec. 2.2). It is also faster. Is that correct?
2. How is the first kernel (k_1) and the one proposed in (Jiao & Vert 2015) different?
3. In Figure 3, can you quantify how well the proposed kernel performs with respect to computing all pairwise similarities? The synthetic datasets are not hard, and it may also be useful to demonstrate some harder case?
4. The kernels seem to have time advantages compared to other embedding approaches such as GNMDS (from the experiments in Sec. 4.2). However in terms of clustering performance they are outperformed by embedding approaches (the authors claim that the performance is close for all algorithms). How many landmark points were chosen for computing the kernels? Can you comment on the size of landmark set vs. running times? 
5. Also, in terms of running times, can you comment whether the reason for the slow performance of the embedding algorithms is the inherent computational complexity? Could it be an implementation issue?
6. The authors should attempt to provide results that are legible and easy to read. If there is lack of space, supplementary material can be provided with the rest of results.
Update
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The authors have answered several of my comments satisfactorily.