Spectral clustering is useful for a wide-ranging set of applications in areas such as biological data analysis, image processing and data mining. However, the computational and/or communication resources required by the method in processing large-scale data sets are often prohibitively high, and practitioners are often required to perturb the original data in various ways (quantization, downsampling, etc) before invoking a spectral algorithm. In this paper, we use stochastic perturbation theory to study the effects of data perturbation on the performance of spectral clustering. We show that the error under perturbation of spectral clustering is closely related to the perturbation of the eigenvectors of the Laplacian matrix. From this result we derive approximate upper bounds on the clustering error. We show that this bound is tight empirically across a wide range of problems, suggesting that it can be used in practical settings to determine the amount of data reduction allowed in order to meet a specification of permitted loss in clustering performance.

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). We associate our algorithm with the existing unsupervised learning algorithms such as spectral clustering, discriminative clustering and sparse principal component analysis. Thus, our work shows that optimal scoring provides a new approach to the implementation of unsupervised learning. This approach facilitates the development of new unsupervised learning algorithms.

Learning distance functions with side information plays a key role in many machine learning and data mining applications. Conventional approaches often assume a Mahalanobis distance function. These approaches are limited in two aspects: (i) they are computationally expensive (even infeasible) for high dimensional data because the size of the metric is in the square of dimensionality; (ii) they assume a fixed metric for the entire input space and therefore are unable to handle heterogeneous data. In this paper, we propose a novel scheme that learns nonlinear Bregman distance functions from side information using a non-parametric approach that is similar to support vector machines. The proposed scheme avoids the assumption of fixed metric because its local distance metric is implicitly derived from the Hessian matrix of a convex function that is used to generate the Bregman distance function. We present an efficient learning algorithm for the proposed scheme for distance function learning. The extensive experiments with semi-supervised clustering show the proposed technique (i) outperforms the state-of-the-art approaches for distance function learning, and (ii) is computationally efficient for high dimensional data.

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. Experiments are presented which show the superiority of our approach over state-of-the-art hypergraph clustering techniques.

We investigate methods for selecting sets of labeled vertices for use in predicting the labels of vertices on a graph. We specifically study methods which choose a single batch of labeled vertices (i.e. offline, non sequential methods). In this setting, we find common graph smoothness assumptions directly motivate simple label selection methods with interesting theoretical guarantees. These methods bound prediction error in terms of the smoothness of the true labels with respect to the graph. Some of these bounds give new motivations for previously proposed algorithms, and some suggest new algorithms which we evaluate. We show improved performance over baseline methods on several real world data sets.

Little work has been done to directly combine the outputs of multiple supervised and unsupervised models. However, it can increase the accuracy and applicability of ensemble methods. First, we can boost the diversity of classification ensemble by incorporating multiple clustering outputs, each of which provides grouping constraints for the joint label predictions of a set of related objects. Secondly, ensemble of supervised models is limited in applications which have no access to raw data but to the meta-level model outputs. In this paper, we aim at calculating a consolidated classification solution for a set of objects by maximizing the consensus among both supervised predictions and unsupervised grouping constraints. We seek a global optimal label assignment for the target objects, which is different from the result of traditional majority voting and model combination approaches. We cast the problem into an optimization problem on a bipartite graph, where the objective function favors smoothness in the conditional probability estimates over the graph, as well as penalizes deviation from initial labeling of supervised models. We solve the problem through iterative propagation of conditional probability estimates among neighboring nodes, and interpret the method as conducting a constrained embedding in a transformed space, as well as a ranking on the graph. Experimental results on three real applications demonstrate the benefits of the proposed method over existing alternatives.

We present a new method for clustering based on compression. The method doesn't use subject-specific features or background knowledge, and works as follows: First, we determine a universal similarity distance, the normalized compression distance or NCD, computed from the lengths of compressed data files (singly and in pairwise concatenation). Second, we apply a hierarchical clustering method. The NCD is universal in that it is not restricted to a specific application area, and works across application area boundaries. A theoretical precursor, the normalized information distance, co-developed by one of the authors, is provably optimal but uses the non-computable notion of Kolmogorov complexity. We propose precise notions of similarity metric, normal compressor, and show that the NCD based on a normal compressor is a similarity metric that approximates universality. To extract a hierarchy of clusters from the distance matrix, we determine a dendrogram (binary tree) by a new quartet method and a fast heuristic to implement it. The method is implemented and available as public software, and is robust under choice of different compressors. To substantiate our claims of universality and robustness, we report evidence of successful application in areas as diverse as genomics, virology, languages, literature, music, handwritten digits, astronomy, and combinations of objects from completely different domains, using statistical, dictionary, and block sorting compressors. In genomics we presented new evidence for major questions in Mammalian evolution, based on whole-mitochondrial genomic analysis: the Eutherian orders and the Marsupionta hypothesis against the Theria hypothesis.

The k -modes algorithm [1] extends the k -means paradigm to cluster categorical data by using (1) a simple matching dissimilarity measure for categorical objects, (2) modes instead of means for clusters, and (3) a frequency-based method to update modes in the k -means fashion to minimize the cost function of clustering. Because the k -modes algorithm uses the same clustering process as k -means, it preserves the efficiency of the k -means algorithm. Although the k -modes algorithm is very efficient, it suffers the problem that the clustering results are sensitive to the selection of the initial points. Hence, a better initial points selection procedure would improve the reliability and accuracy of clustering results. To that end, an iterative initial-points refinement algorithm for k -modes clustering has been presented in [2]. As shown in [2], the new initialization pr ocedure greatly improves the reliability and accuracy of final clustering results. Despite the su ccess of Ref. [2], the following observations motivate us to further pursue other alternative initialization methods.

We present a tool that, from automatically recognised names, tries to infer inter-person relations in order to present associated people on maps. Based on an in-house Named Entity Recognition tool, applied on clusters of an average of 15,000 news articles per day, in 15 different languages, we build a knowledge base that allows extracting statistical co-occurrences of persons and visualising them on a per-person page or in various graphs.

We are presenting a text analysis tool set that allows analysts in various fields to sieve through large collections of multilingual news items quickly and to find information that is of relevance to them. For a given document collection, the tool set automatically clusters the texts into groups of similar articles, extracts names of places, people and organisations, lists the user-defined specialist terms found, links clusters and entities, and generates hyperlinks. Through its daily news analysis operating on thousands of articles per day, the tool also learns relationships between people and other entities. The fully functional prototype system allows users to explore and navigate multilingual document collections across languages and time.