Multiple Clustering Views via Constrained Projections

It is well known that off-the-shelf clustering methods may discover different patterns in a given set of data. This is because each clustering algorithm has its own bias resulting from the optimization of different criteria. Furthermore, there is no ground truth against which the clustering result can be validated. Thus, no crossvalidation technique can be carried out to tune input parameters involved in the process. As a consequence, the user has no guidelines for choosing the proper clustering method for a given data set. The use of clustering ensembles has emerged as a technique for overcoming these problems. A clustering ensemble consists of different clusterings obtained from multiple applications of any single algorithm with different initializations, or from various bootstrap samples of the available data, or from the application of different algorithms to the same data set. Clustering ensembles offer a solution to challenges inherent to clustering arising from its ill-posed nature: they can provide more robust and stable solutions by making use of the consensus across multiple clustering results, while averaging out emergent spurious structures that arise due to the various biases to which each participating algorithm is tuned, or to the variance induced by different data samples. Another issue related to clustering is the so-called curse of dimensionality. Data with thousands of dimensions abound in fields and applications as diverse as bioinformatics, security and intrusion detection, and information and image retrieval. Clustering algorithms can handle data with low dimensionality, but as the dimensionality of the data increases, these algorithms tend to break down. This is because in high dimensional spaces data become extremely sparse and are far apart from each other. A common scenario with high-dimensional data is that several clusters may exist in different subspaces comprised of different combinations of features. In many real-world problems, points in a given region of the input space may cluster along a given set of dimensions, while points located in another region may ∗Department of Computer Science, George Mason University, carlotta@cs.gmu. edu form a tight group with respect to different dimensions. Each dimension could be relevant to at least one of the clusters. Common global dimensionality reduction techniques are unable to capture such local structure of the data. Thus, a proper feature selection procedure should operate locally in the input space. Local feature selection allows one to estimate to which degree features participate in the discovery of clusters. As a result, many different subspace clustering methods have been proposed. Traditionally, clustering ensembles and subspace clustering have been developed independently of one another. Clustering ensembles address the ill-posed nature of clustering, but don’t address in general the curse of dimensionality problem. Subspace clustering avoids the curse of dimensionality in high-dimensional spaces, but typically requires the setting of critical input parameters whose values are unknown. To overcome these limitations we have introduced a unified framework that is capable of handling both issues: the ill-posed nature of clustering and the curse of dimensionality. Addressing these two issues is nontrivial as it involves solving a new problem altogether: the subspace clustering ensemble problem. Our approach takes two different perspectives: in the one case we model the problem as a multiand single-objective optimization one [3, 2, 1]; in the other we take a generative view, and assume that the base clusterings are generated from a hidden consensus clustering of the data [5, 4]. Both directions are promising and lead to interesting challenges. The first can yield general and efficient solutions, but requires as input the number of clusters in the consensus clustering. The second has higher complexity, but provides a principled solution to the “How many clusters? ” question. In this talk, I focus on the first approach. I introduce the formal definition of the problem of subspace clustering ensembles, and heuristics to solve it. The objective is to define methods to exploit the information provided by an ensemble of subspace clustering solutions to compute a robust consensus subspace clustering. The problem is formulated as a multiand single-objective optimization problem where the objective functions embed both sides of the ensemble components: the data clusterings and the assignments of features to clusters.