On the Consistency of Quick Shift
Quick Shift is a popular mode-seeking and clustering algorithm. We present finite sample statistical consistency guarantees for Quick Shift on mode and cluster recovery under mild distributional assumptions. We then apply our results to construct a consistent modal regression algorithm.
In this paper the authors submit proofs regarding various notions of consistency for the Quick Shift algorithm. They also introduce adaptions of the Quick Shift algorithm for other tasks paired with relevant consistency results. 
Quick Shift is a mode finding algorithm where one chooses a segmentation parameter \tau, constructs a kernel density estimator, and assigns a mode, m, at samples where the KDE magnitude at m is larger than the KDE magnitude evaluated at all samples within a \tau-ball of m. In this setting a true mode are those points in the sample space where the density is maximized within a \tau ball. Quick shift can also be used to assign samples to a mode by ascending a sample to the nearest sample where the KDE magnitude is larger and repeating until a mode is reached. The authors present proofs of consistency for both of these tasks: mode recovery and mode assignment.
The authors also propose an adaptation of Quick Shift for cluster tree recovery and modal regression for which they also prove consistency.
The proofs presented in this paper are dense and technical; I cannot claim to have to have totally validated their correctness. That being said I found no glaring errors in the authors' reasoning. My largest concern with this paper is reference [1]. [1] plays a central role in the proofs for individual mode recovery rates. [1] is an "anonymous  "unpublished manuscript" on Dropbox. I spent a fair amount of time searching for any other reference to this manuscript and I could find none. The authors give no explanation for this manuscript. I'm inclined to think that I cannot reasonably accept a paper whose results rely upon such a dubious source.
Additionally there are many other smaller issues I found in the paper. Some of them are listed here by line:
99: period after "we" should be removed
101-103: The sentence "For example, if given an input X the response y have two polarizing tendancies, modal regression would be able to adapt to these tendancies," is very strange and unclear. Also "tendencies" is misspelled.
103: "indepth" is not a word. Replace with "in-depth" or "indepth"
106: "require" should be "requires" 
112: Why is the "distribution \script{F}" introduced? This is never mentioned elsewhere. We can simply proceed from the density. 
113: I'm guessing that the "uniform measure" is the same thing as the "Lebesgue measure." "Lebesgue measure" is more standard terminology in my opinion.
115: Why is the norm notation replaced with absolute value notation? This occurs later as well elements which are clearly vectors, line 137 and Lemma 2 for instance.
139: Remark 3 should have a citation.
145: C is never defined.
150: Your style changes here. Assumption 3 uses parentheses and here Assumption 4 uses square brackets.
158: The final "x" in the set definition should be "x'"
242: "f(x)" should be to the right of "inf"
280: "conditiona" should be "conditional"
286: What exactly is meant by "valid bandwidth choices?"
Update: I've bumped the paper by two points. I still have concerns with the numerous small technical errors I found