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.

In this work we develop a theory of hierarchical clustering for graphs. Our modeling assumption is that graphs are sampled from a graphon, which is a powerful and general model for generating graphs and analyzing large networks. Graphons are a far richer class of graph models than stochastic blockmodels, the primary setting for recent progress in the statistical theory of graph clustering. We define what it means for an algorithm to produce the "correct" clustering, give sufficient conditions in which a method is statistically consistent, and provide an explicit algorithm satisfying these properties.

Quick Shift is a popular mode-seeking and clustering algorithm.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The authors propose a novel approach for hierarchical clustering of multivariate data. They construct cluster trees by estimating minimum volume sets using the q-One-Class SVM, and evaluate their method on a synthetic data set and two real word applications. While their new method seems to perform better than other approaches based on density estimation, I am not convinced by the benefits in practical applicability as the authors did not compare their method to the most commonly used hierarchical clustering techniques (agglomerative clustering with average linkage/ward). Minor comment: Rather than splitting their data once in a training and test set, the authors should perform 10-fold/5-fold cross-validation for a more reliable estimation of the generalizability of their method.

We investigate the problem of estimating the cluster tree for a density $f$ supported on or near a smooth $d$-dimensional manifold $M$ isometrically embedded in $\mathbb{R}^D$. We study a $k$-nearest neighbor based algorithm recently proposed by Chaudhuri and Dasgupta. Under mild assumptions on $f$ and $M$, we obtain rates of convergence that depend on $d$ only but not on the ambient dimension $D$. We also provide a sample complexity lower bound for a natural class of clustering algorithms that use $D$-dimensional neighborhoods.

The goal of hierarchical clustering is to construct a cluster tree, which can be viewed as the modal structure of a density. For this purpose, we use a convex optimization program that can efficiently estimate a family of hierarchical dense sets in high-dimensional distributions. We further extend existing graph-based methods to approximate the cluster tree of a distribution. By avoiding direct density estimation, our method is able to handle high-dimensional data more efficiently than existing density-based approaches. We present empirical results that demonstrate the superiority of our method over existing ones.

A comparison between neural network clustering (NNC), hierarchical clustering (HC) and K-means clustering (KMC) is performed to evaluate the computational superiority of these three machine learning (ML) techniques for organizing large datasets into clusters. For NNC, a self-organizing map (SOM) training was applied to a collection of wavefront sensor reconstructions, decomposed in terms of 15 Zernike coefficients, characterizing the optical aberrations of the phase front transmitted by fluidic lenses. In order to understand the distribution and structure of the 15 Zernike variables within an input space, SOM-neighboring weight distances, SOM-sample hits, SOM-weight positions and SOM-weight planes were analyzed to form a visual interpretation of the system's structural properties. In the case of HC, the data was partitioned using a combined dissimilarity-linkage matrix computation. The effectiveness of this method was confirmed by a high cophenetic correlation coefficient value (c=0.9651). Additionally, a maximum number of clusters was established by setting an inconsistency cutoff of 0.8, yielding a total of 7 clusters for system segmentation. In addition, a KMC approach was employed to establish a quantitative measure of clustering segmentation efficiency, obtaining a sillhoute average value of 0.905 for data segmentation into K=5 non-overlapping clusters. On the other hand, the NNC analysis revealed that the 15 variables could be characterized through the collective influence of 8 clusters. It was established that the formation of clusters through the combined linkage and dissimilarity algorithms of HC alongside KMC is a more dependable clustering solution than separate assessment via NNC or HC, where altering the SOM size or inconsistency cutoff can lead to completely new clustering configurations.

The goal of hierarchical clustering is to construct a cluster tree, which can be viewed as the modal structure of a density. For this purpose, we use a convex optimization program that can efficiently estimate a family of hierarchical dense sets in high-dimensional distributions. We further extend existing graph-based methods to approximate the cluster tree of a distribution. By avoiding direct density estimation, our method is able to handle high-dimensional data more efficiently than existing density-based approaches. We present empirical results that demonstrate the superiority of our method over existing ones.

This paper derives complexity results for active learning queries to hierarchical clustering. The result is a partition or "cut", c, of the cluster tree, where the "flat" clustering is defined by the clusters at the leaves of a subtree of nodes AB(c) that have the same root as the original cluster tree. Learning occurs by making pairwise judgments on items (leaf nodes). All pairwise judgments form a "ground truth" matrix \Sigma. Given consistency conditions, this is an equivalent way to represent a clustering.

Hierarchical clustering is commonly applied to two types of objects: 1. sets of points 2. graphs (in which case it is usually called hierarchical graph partitioning) What can be said about the statistical properties of hierarchical clustering? In case (1), we can look at the underlying density from which the points are sampled, define a suitable (infinite) "cluster tree" for this density and then assert that a particular hierarchical clustering procedure returns finite trees that converge to this cluster tree in some suitable sense. Recent work by Eldridge et al has used a criterion called "merge distortion" to assess the discrepancy between the target (infinite) tree and the tree estimated from a finite sample. Specific algorithms have been found to be consistent in the sense of having the right limit, and their rates of convergence have been determined. The present paper is interested in extending this methodology to case (2).