We first show that the commonly used dimensionality reduction (DR) methods such as t-SNE and LargeVis poorly capture the global structure of the data in the low dimensional embedding. We show this via a number of tests for the DR methods that can be easily applied by any practitioner to the dataset at hand. Surprisingly enough, t-SNE performs the best w.r.t. the commonly used measures that reward the local neighborhood accuracy such as precision-recall while having the worst performance in our tests for global structure. We then contrast the performance of these two DR method against our new method called TriMap. The main idea behind TriMap is to capture higher orders of structure with triplet information (instead of pairwise information used by t-SNE and LargeVis), and to minimize a robust loss function for satisfying the chosen triplets. We provide compelling experimental evidence on large natural datasets for the clear advantage of the TriMap DR results. As LargeVis, TriMap is fast and provides comparable runtime on large datasets.
In this paper, the authors present a novel dimensionality reduction method named TriMap. TriMap attempts to improve upon the widely-adopted t-SNE algorithm by incorporating global distances through the use of triplets, rather than pairwise comparisons. The authors compare to t-SNE, as well as a newer method called LargeVis which also claims to impose a global distances metric. The authors show that their method is more robust to the addition or removal of clusters and outliers and provides a more meaningful global distance relative to the methods against which they compare.
Technical Quality
The authors’ method is clear and well described and addresses a poignant issue in dimensionality reduction. However, the authors fail to compare their method to a number of relevant dimensionality reduction algorithms which also claim to provide solutions with globally meaningful distances. Such methods include force-directed graph drawing (Fruchterman & Reingold, 1991), diffusion maps (Coifman & Lafon, 2006) and PHATE (Moon et al., 2017). 
Additionally, the handling of outliers is a concern. While the authors claim that the retention of outliers as disconnected from the manifold is a desirable quality of their technique, the presence of many outliers in a dataset (for example, in the Tabula Muris and lyrics datasets) has the potential to mask the interesting portion of the dimensionality reduction. It may be worth commenting on the desirability to identify and remove outliers, and the provision of such a technique in the software upon its release.
Finally, the runtime comparison is of concern. It is common to perform most DR methods on high-dimensional PCA representation of the data, particularly in single-cell genomics (e.g. the Tabula Muris dataset in Part 3.) In this context, both UMAP and PHATE successfully embed the Tabula Muris dataset in less than the reported TriMap time (3.5 and 5 minutes respectively, compared to 15 minutes reported for TriMap.)
Novelty
The authors’ method appears to provide improved results over the compared alternatives, however, it is worth noting that triplet-based embedding is not novel in its own right (van der Maaten & Weinberger, 2012), though one could argue novelty is warranted here due to claimed substantial improvements of results. In this case, the authors should include a comparison to competing triplet-based methods, at least in the appendix. 
Potential impact
The authors’ method has the potential to be used widely across many fields, as a direct replacement for t-SNE. Its adoption is contingent on compelling evidence that it produces results substantially better than UMAP (which is currently heralded as an upcoming replacement for t-SNE in some fields) and other competing methods. The authors may find it worthwhile to provide such comparisons, if not in the main body of the paper at least in the appendix. 
Clarity
The paper is easy to read and makes its point in a reasonably concise manner. Detailed explanation of experiments v) and vi) could be relegated to the appendix. More precise statement of the authors’ tests in Part 2 could be provided by quantifying the results of the tests in a more precise way; it is not clear what the authors seek to achieve by drawing the dotted lines between clusters in Figure 1a, or by providing AUC values in Figure 1.
Detailed Comments
•	In the definition of Equation 2, it is not until one paragraph later than q_{ij}^{~(t’)} is defined – this is confusing and hard to read.
•	The captions for Figures 1 and 3 would be substantially clearer with more detail on the dataset analyzed and in Figure 1, some discussion of the purpose of each subplot.
•	The Figure 3 caption needs a semicolon or period before introducing the bottom panel.
•	The claim that the authors’ heuristic triplet sampling (nearest-neighbor and random sampling) is sufficient to approximate full triplet sampling should be shown in the appendix.
•	The collaboration network analyzed in Part 3 is naturally a graph; it would make sense to cluster and visualize this using a graph-based clustering, rather than coercing it to Euclidean coordinates.
(Note: after reading the revised manuscript I have changed my recommendation from a 6 to a 5)
References
Coifman, R. R., & Lafon, S. (2006). Diffusion maps. Applied and computational harmonic analysis, 21(1), 5-30. https://doi.org/10.1016/j.acha.2006.04.006
Moon, K. R., van Dijk, D., Wang, Z., Burkhardt, D., Chen, W., van den Elzen, A., ... & Krishnaswamy, S. (2017). Visualizing transitions and structure for high dimensional data exploration. bioRxiv, 120378. https://doi.org/10.1101/120378
Fruchterman, T. M., & Reingold, E. M. (1991). Graph drawing by force‐directed placement. Software: Practice and experience, 21(11), 1129-1164. https://doi.org/10.1002/spe.4380211102
L. van der Maaten and K. Weinberger. Stochastic triplet embedding. In 2012 IEEE International Workshop on Machine Learning for Signal Processing, pp. 1–6, Sept 2012. doi: 10.1109/MLSP.2012.6349720.