Paper:
Statistical Inference for Cluster Trees
A cluster tree provides a highly-interpretable summary of a density function by representing the hierarchy of its high-density clusters. It is estimated using the empirical tree, which is the cluster tree constructed from a density estimator. This paper addresses the basic question of quantifying our uncertainty by assessing the statistical significance of topological features of an empirical cluster tree. We first study a variety of metrics that can be used to compare different trees, analyze their properties and assess their suitability for inference. We then propose methods to construct and summarize confidence sets for the unknown true cluster tree. We introduce a partial ordering on cluster trees which we use to prune some of the statistically insignificant features of the empirical tree, yielding interpretable and parsimonious cluster trees. Finally, we illustrate the proposed methods on a variety of synthetic examples and furthermore demonstrate their utility in the analysis of a Graft-versus-Host Disease (GvHD) data set.

Review:
The article discusses an inference problem for cluster trees arising from density estimation and proposes a method of constructing confidence sets by a bootstrap. In addition, to aid visualisation, additional pruning method is explored. The authors approach the problem by topological consideration of comparing trees, and propose an $\ell_\infty$ metric-based pruning. The main contribution seems to develop the framework for its theoretical justification, though some numerical examples are also included.   Overall, the problem is interesting and important, and the careful consideration of the theory is impressive, the solution provided is reasonable and practical. 
1. Although I appreciate its importance of theoretical development, the numerical example does not seems to demonstrate well the potential of the proposed method. In particular, since the main objective is the confidence set, I wonder if there is a way to show the richness of the confidence sets so that the readers better appreciate the visualisation by pruning.   2. Related to the first point, I originally thought that the partial ordering was considered for comparing different trees, but the way the partial ordering is defined seems to require the main split should occur at exactly the same location (Fig.2), which would never happen for two different estimates, although the variation of the branching points would be small enough to define an approximation. So the partial ordering is well defined for pruning of the given estimate, but it wouldn't help for comparing the various trees within the confidence sets. Or do I miss something here?   3. The definition of the metrics in Section 3 doesn't necessarily assume that $p$ and $q$ are the density functions representing the same underlying population (like KDE estimates from two different realisations of the population), but any functions that are defined on the same domain so that they don't have to be close at all. Is it true?   4. Similarly, for $p_{inf}$ and the definition of $a$, if we consider density functions, unless they have compact support, $p_{inf}$ and $a$ would be zero. Then in Lemma 1, "more generally" means whether it has compact support or not?  5. Considering the distance with respect to the truth, intuitively as KDE has bias proportional to the second derivatives, the main error would occur at peaks and valleys, which are directly linked to the merging height. Then the equivalence of $d_\infty$ and $d_M$ is expected. However, this doesn't seem the case when we compare two different functions, the relative difference between these functions don't have to be amplified at those landmarks, yet I feel that the same principle should be relevant. Could you give some insights into the source of my confusion?  6. In figure 4, the second example (mickey mouse data), I would expect that heights would be different for the main part and two small clusters as the probabilities would be different. Why is it not the case?