Over the last years, many variations of the quadratic k-means clustering procedure have been proposed, all aiming to robustify the performance of the algorithm in the presence of outliers. In general terms, two main approaches have been developed: one based on penalized regularization methods, and one based on trimming functions. In this work, we present a theoretical analysis of the robustness and consistency properties of a variant of the classical quadratic k-means algorithm, the robust k-means, which borrows ideas from outlier detection in regression. We show that two outliers in a dataset are enough to breakdown this clustering procedure. However, if we focus on “well-structured” datasets, then robust k-means can recover the underlying cluster structure in spite of the outliers. Finally, we show that, with slight modifications, the most general non-asymptotic results for consistency of quadratic k-means remain valid for this robust variant.

Spectral clustering and co-clustering are well-known techniques in data analysis, and recent work has extended spectral clustering to square, symmetric tensors and hypermatrices derived from a network. We develop a new tensor spectral co-clustering method that simultaneously clusters the rows, columns, and slices of a nonnegative three-mode tensor and generalizes to tensors with any number of modes. The algorithm is based on a new random walk model which we call the super-spacey random surfer. We show that our method out-performs state-of-the-art co-clustering methods on several synthetic datasets with ground truth clusters and then use the algorithm to analyze several real-world datasets.

Clustering graphs under the Stochastic Block Model (SBM) and extensions are well studied. Guarantees of correctness exist under the assumption that the data is sampled from a model. In this paper, we propose a framework, in which we obtain "correctness" guarantees without assuming the data comes from a model. The guarantees we obtain depend instead on the statistics of the data that can be checked. We also show that this framework ties in with the existing model-based framework, and that we can exploit results in model-based recovery, as well as strengthen the results existing in that area of research.

Clustering, in particular $k$-means clustering, is a central topic in data analysis. Clustering with Bregman divergences is a recently proposed generalization of $k$-means clustering which has already been widely used in applications. In this paper we analyze theoretical properties of Bregman clustering when the number of the clusters $k$ is large. We establish quantization rates and describe the limiting distribution of the centers as $k\to \infty$, extending well-known results for $k$-means clustering.

We study the cost function for hierarchical clusterings introduced by [Dasgupta, 2015] where hierarchies are treated as first-class objects rather than deriving their cost from projections into flat clusters. It was also shown in [Dasgupta, 2015] that a top-down algorithm returns a hierarchical clustering of cost at most \(O\left(\alpha_n \log n\right)\) times the cost of the optimal hierarchical clustering, where \(\alpha_n\) is the approximation ratio of the Sparsest Cut subroutine used. Thus using the best known approximation algorithm for Sparsest Cut due to Arora-Rao-Vazirani, the top down algorithm returns a hierarchical clustering of cost at most \(O\left(\log^{3/2} n\right)\) times the cost of the optimal solution. We improve this by giving an \(O(\log{n})\)-approximation algorithm for this problem. Our main technical ingredients are a combinatorial characterization of ultrametrics induced by this cost function, deriving an Integer Linear Programming (ILP) formulation for this family of ultrametrics, and showing how to iteratively round an LP relaxation of this formulation by using the idea of \emph{sphere growing} which has been extensively used in the context of graph partitioning. We also prove that our algorithm returns an \(O(\log{n})\)-approximate hierarchical clustering for a generalization of this cost function also studied in [Dasgupta, 2015]. Experiments show that the hierarchies found by using the ILP formulation as well as our rounding algorithm often have better projections into flat clusters than the standard linkage based algorithms. We conclude with an inapproximability result for this problem, namely that no polynomial sized LP or SDP can be used to obtain a constant factor approximation for this problem.

In this work we develop a theory of hierarchical clustering for graphs. Our modelling 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.

In this paper, we study the mixed linear regression (MLR) problem, where the goal is to recover multiple underlying linear models from their unlabeled linear measurements. We propose a non-convex objective function which we show is {\em locally strongly convex} in the neighborhood of the ground truth. We use a tensor method for initialization so that the initial models are in the local strong convexity region. We then employ general convex optimization algorithms to minimize the objective function. To the best of our knowledge, our approach provides first exact recovery guarantees for the MLR problem with $K \geq 2$ components. Moreover, our method has near-optimal computational complexity $\tilde O (Nd)$ as well as near-optimal sample complexity $\tilde O (d)$ for {\em constant} $K$. Furthermore, we show that our non-convex formulation can be extended to solving the {\em subspace clustering} problem as well. In particular, when initialized within a small constant distance to the true subspaces, our method converges to the global optima (and recovers true subspaces) in time {\em linear} in the number of points. Furthermore, our empirical results indicate that even with random initialization, our approach converges to the global optima in linear time, providing speed-up of up to two orders of magnitude.

A cluster tree provides an intuitive summary of a density function that reveals essential structure about the high-density clusters. The true cluster tree is estimated from a finite sample from an unknown true density. This paper addresses the basic question of quantifying our uncertainty by assessing the statistical significance of different features of an empirical cluster tree. We first study a variety of metrics that can be used to compare different trees, analyzing their properties and assessing their suitability for our inference task. 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 provide a variety of simulations to illustrate our proposed methods and furthermore demonstrate their utility in the analysis of a Graft-versus-Host Disease (GvHD) data set.

Given $iid$ observations from an unknown continuous distribution defined on some domain $\Omega$, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function. Our density estimate is a piecewise constant function defined on a binary partition of $\Omega$. The key ingredient of the algorithm is to use discrepancy, a concept originates from Quasi Monte Carlo analysis, to control the partition process. The resulting algorithm is simple, efficient, and has provable convergence rate. We demonstrate empirically its efficiency as a density estimation method. We also show how it can be utilized to find good initializations for k-means.

We consider the problem of community detection or clustering in the labeled Stochastic Block Model (LSBM) with a finite number $K$ of clusters of sizes linearly growing with the global population of items $n$. Every pair of items is labeled independently at random, and label $\ell$ appears with probability $p(i,j,\ell)$ between two items in clusters indexed by $i$ and $j$, respectively. The objective is to reconstruct the clusters from the observation of these random labels. Clustering under the SBM and their extensions has attracted much attention recently. Most existing work aimed at characterizing the set of parameters such that it is possible to infer clusters either positively correlated with the true clusters, or with a vanishing proportion of misclassified items, or exactly matching the true clusters. We find the set of parameters such that there exists a clustering algorithm with at most $s$ misclassified items in average under the general LSBM and for any $s=o(n)$, which solves one open problem raised in \cite{abbe2015community}. We further develop an algorithm, based on simple spectral methods, that achieves this fundamental performance limit within $O(n \mbox{polylog}(n))$ computations and without the a-priori knowledge of the model parameters.