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. From the computational viewpoint, we show that the problem of finding the equilibria of our clustering game is equivalent to locally optimizing a polynomial function over the standard simplex, and we provide a discrete-time dynamics to perform this optimization.

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.

We present a novel feature selection algorithm for the $k$-means clustering problem. Our algorithm is randomized and, assuming an accuracy parameter $\epsilon \in (0,1)$, selects and appropriately rescales in an unsupervised manner $\Theta(k \log(k / \epsilon) / \epsilon 2)$ features from a dataset of arbitrary dimensions. We prove that, if we run any $\gamma$-approximate $k$-means algorithm ($\gamma \geq 1$) on the features selected using our method, we can find a $(1 (1 \epsilon)\gamma)$-approximate partition with high probability. Papers published at the Neural Information Processing Systems Conference.

We prove that any set of $n$ points in $d$ dimensions (rows in a matrix $A \in \RR {n \times d}$) can be projected into $t \Omega(k / \eps 2)$ dimensions, for any $\eps \in (0,1/3)$, in $O(n d \lceil \eps {-2} k/ \log(d) \rceil)$ time, such that with constant probability the optimal $k$-partition of the point set is preserved within a factor of $2 \eps$. The projection is done by post-multiplying $A$ with a $d \times t$ random matrix $R$ having entries $ 1/\sqrt{t}$ or $-1/\sqrt{t}$ with equal probability. A numerical implementation of our technique and experiments on a large face images dataset verify the speed and the accuracy of our theoretical results. Papers published at the Neural Information Processing Systems Conference.

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.

Links between probabilistic and non-probabilistic learning algorithms can arise by performing small-variance asymptotics, i.e., letting the variance of particular distributions in a graphical model go to zero. For instance, in the context of clustering, such an approach yields precise connections between the k-means and EM algorithms. In this paper, we explore small-variance asymptotics for exponential family Dirichlet process (DP) and hierarchical Dirichlet process (HDP) mixture models. Utilizing connections between exponential family distributions and Bregman divergences, we derive novel clustering algorithms from the asymptotic limit of the DP and HDP mixtures that feature the scalability of existing hard clustering methods as well as the flexibility of Bayesian nonparametric models. We focus on special cases of our analysis for discrete-data problems, including topic modeling, and we demonstrate the utility of our results by applying variants of our algorithms to problems arising in vision and document analysis.

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. Papers published at the Neural Information Processing Systems Conference.

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. Papers published at the Neural Information Processing Systems Conference.

We formulate clustering aggregation as a special instance of Maximum-Weight Independent Set (MWIS) problem. For a given dataset, an attributed graph is constructed from the union of the input clusterings generated by different underlying clustering algorithms with different parameters. The vertices, which represent the distinct clusters, are weighted by an internal index measuring both cohesion and separation. The edges connect the vertices whose corresponding clusters overlap. Intuitively, an optimal aggregated clustering can be obtained by selecting an optimal subset of non-overlapping clusters partitioning the dataset together.