Graph clustering methods such as spectral clustering are defined for general weighted graphs. In machine learning, however, data often is not given in form of a graph, but in terms of similarity (or distance) values between points. In this case, first a neighborhood graph is constructed using the similarities between the points and then a graph clustering algorithm is applied to this graph. In this paper we investigate the influence of the construction of the similarity graph on the clustering results. We first study the convergence of graph clustering criteria such as the normalized cut (Ncut) as the sample size tends to infinity.

We are often interested in casting classification and clustering problems in a regression framework, because it is feasible to achieve some statistical properties in this framework by imposing some penalty criteria. In this paper we illustrate optimal scoring, which was originally proposed for performing Fisher linear discriminant analysis by regression, in the application of unsupervised learning. In particular, we devise a novel clustering algorithm that we call optimal discriminant clustering (ODC). Thus, our work shows that optimal scoring provides a new approach to the implementation of unsupervised learning.

Hypergraph clustering refers to the process of extracting maximally coherent groups from a set of objects using high-order (rather than pairwise) similarities. Traditional approaches to this problem are based on the idea of partitioning the input data into a user-defined number of classes, thereby obtaining the clusters as a by-product of the partitioning process. In this paper, we provide a radically different perspective to the problem. In contrast to the classical approach, we attempt to provide a meaningful formalization of the very notion of a cluster and we show that game theory offers an attractive and unexplored perspective that serves well our purpose. Specifically, we show that the hypergraph clustering problem can be naturally cast into a non-cooperative multi-player clustering game, whereby the notion of a cluster is equivalent to a classical game-theoretic equilibrium concept.

Clustering is a basic data mining task with a wide variety of applications. Not surprisingly, there exist many clustering algorithms. However, clustering is an ill defined problem - given a data set, it is not clear what a "correct" clustering for that set is. Indeed, different algorithms may yield dramatically different outputs for the same input sets. Faced with a concrete clustering task, a user needs to choose an appropriate clustering algorithm.

A number of objective functions in clustering problems can be described with submodular functions. In this paper, we introduce the minimum average cost criterion, and show that the theory of intersecting submodular functions can be used for clustering with submodular objective functions. The proposed algorithm does not require the number of clusters in advance, and it will be determined by the property of a given set of data points. The minimum average cost clustering problem is parameterized with a real variable, and surprisingly, we show that all information about optimal clusterings for all parameters can be computed in polynomial time in total. Additionally, we evaluate the performance of the proposed algorithm through computational experiments.

Dimensionality reduction is commonly used in the setting of multi-label supervised classification to control the learning capacity and to provide a meaningful representation of the data. We introduce a simple forward probabilistic model which is a multinomial extension of reduced rank regression; we show that this model provides a probabilistic interpretation of discriminative clustering methods with added benefits in terms of number of hyperparameters and optimization. While expectation-maximization (EM) algorithm is commonly used to learn these models, its optimization usually leads to local minimum because it relies on a non-convex cost function with many such local minima. To avoid this problem, we introduce a local approximation of this cost function, which leads to a quadratic non-convex optimization problem over a product of simplices. In order to minimize such functions, we propose an efficient algorithm based on convex relaxation and low-rank representation of our data, which allows to deal with large instances.

Although spectral clustering has enjoyed considerable empirical success in machine learning, its theoretical properties are not yet fully developed. We analyze the performance of a spectral algorithm for hierarchical clustering and show that on a class of hierarchically structured similarity matrices, this algorithm can tolerate noise that grows with the number of data points while still perfectly recovering the hierarchical clusters with high probability. We additionally improve upon previous results for k-way spectral clustering to derive conditions under which spectral clustering makes no mistakes. Further, using minimax analysis, we derive tight upper and lower bounds for the clustering problem and compare the performance of spectral clustering to these information theoretic limits.

Many clustering techniques aim at optimizing empirical criteria that are of the form of a U-statistic of degree two. Given a measure of dissimilarity between pairs of observations, the goal is to minimize the within cluster point scatter over a class of partitions of the feature space. It is the purpose of this paper to define a general statistical framework, relying on the theory of U-processes, for studying the performance of such clustering methods. In this setup, under adequate assumptions on the complexity of the subsets forming the partition candidates, the excess of clustering risk is proved to be of the order O(1/\sqrt{n}). Based on recent results related to the tail behavior of degenerate U-processes, it is also shown how to establish tighter rate bounds.

We introduce a new prior for use in Nonparametric Bayesian Hierarchical Clustering. The prior is constructed by marginalizing out the time information of Kingman's coalescent, providing a prior over tree structures which we call the Time-Marginalized Coalescent (TMC). This allows for models which factorize the tree structure and times, providing two benefits: more flexible priors may be constructed and more efficient Gibbs type inference can be used. We demonstrate this on an example model for density estimation and show the TMC achieves competitive experimental results.

Learning the number of clusters is a key problem in data clustering. We present dip-means, a novel robust incremental method to learn the number of data clusters that may be used as a wrapper around any iterative clustering algorithm of the k-means family. In contrast to many popular methods which make assumptions about the underlying cluster distributions, dip-means only assumes a fundamental cluster property: each cluster to admit a unimodal distribution. The proposed algorithm considers each cluster member as a ''viewer'' and applies a univariate statistic hypothesis test for unimodality (dip-test) on the distribution of the distances between the viewer and the cluster members. Two important advantages are: i) the unimodality test is applied on univariate distance vectors, ii) it can be directly applied with kernel-based methods, since only the pairwise distances are involved in the computations.