The problem of clustering with constraints is receiving increasing attention. Many existing algorithms assume the specified constraints are correct and consistent. We take a new approach and model the uncertainty of constraints in a principled manner by treating the constraints as random variables. The effect of specified constraints on a subset of points is propagated to other data points by biasing the search for cluster boundaries. By combining the a posteriori enforcement of constraints with the log-likelihood, we obtain a new objective function. An EM-type algorithm derived by variational method is used for efficient parameter estimation. Experimental results demonstrate the usefulness of the proposed algorithm. In particular, our approach can identify the desired clusters even when only a small portion of data participates in constraints.

Clustering ensembles have emerged as a powerful method for improving both the robustness as well as the stability of unsupervised classification solutions. However, finding a consensus clustering from multiple partitions is a difficult problem that can be approached from graph-based, combinatorial, or statistical perspectives. This study extends previous research on clustering ensembles in several respects. First, we introduce a unified representation for multiple clusterings and formulate the corresponding categorical clustering problem. Second, we propose a probabilistic model of consensus using a finite mixture of multinomial distributions in a space of clusterings. A combined partition is found as a solution to the corresponding maximum-likelihood problem using the EM algorithm. Third, we define a new consensus function that is related to the classical intraclass variance criterion using the generalized mutual information definition. Finally, we demonstrate the efficacy of combining partitions generated by weak clustering algorithms that use data projections and random data splits. A simple explanatory model is offered for the behavior of combinations of such weak clustering components. Combination accuracy is analyzed as a function of several parameters that control the power and resolution of component partitions as well as the number of partitions. We also analyze clustering ensembles with incomplete information and the effect of missing cluster labels on the quality of overall consensus. Experimental results demonstrate the effectiveness of the proposed methods on several real-world data sets.

In combination of multiple partitions, one is usually interested in deriving a consensus solution with a quality better than that of given partitions. Several recent studies have empirically demonstrated improved accuracy of clustering ensembles on a number of artificial and real-world data sets. Unlike certain multiple supervised classifier systems, convergence properties of unsupervised clustering ensembles remain unknown for conventional combination schemes. In this paper, we present formal arguments on the effectiveness of cluster ensemble from two perspectives. The first is based on a stochastic partition generation model related to re-labeling and consensus function with plurality voting. The second is to study the property of the "mean" partition of an ensemble with respect to a metric on the space of all possible partitions. In both the cases, the consensus solution can be shown to converge to a true underlying clustering solution as the number of partitions in the ensemble increases. This paper provides a rigorous justification for the use of cluster ensemble.

Clustering ensembles have emerged as a powerful method for improving both the robustness and the stability of unsupervised classification solutions. However, finding a consensus clustering from multiple partitions is a difficult problem that can be approached from graph-based, combinatorial or statistical perspectives. We offer a probabilistic model of consensus using a finite mixture of multinomial distributions in a space of clusterings. A combined partition is found as a solution to the corresponding maximum likelihood problem using the EM algorithm. The excellent scalability of this algorithm and comprehensible underlying model are particularly important for clustering of large datasets. This study compares the performance of the EM consensus algorithm with other fusion approaches for clustering ensembles. We also analyze clustering ensembles with incomplete information and the effect of missing cluster labels on the quality of overall consensus. Experimental results demonstrate the effectiveness of the proposed method on large real-world datasets.

A data set can be clustered in many ways depending on the clustering algorithm employed, parameter settings used and other factors. Can multiple clusterings be combined so that the final partitioning of data provides better clustering? The answer depends on the quality of clusterings to be combined as well as the properties of the fusion method. First, we introduce a unified representation for multiple clusterings and formulate the corresponding categorical clustering problem. As a result, we show that the consensus function is related to the classical intra-class variance criterion using the generalized mutual information definition. Second, we show the efficacy of combining partitions generated by weak clustering algorithms that use data projections and random data splits. A simple explanatory model is offered for the behavior of combinations of such weak clustering components. We analyze the combination accuracy as a function of parameters controlling the power and resolution of component partitions as well as the learning dynamics vs. the number of clusterings involved. Finally, some empirical studies compare the effectiveness of several consensus functions.