Recent work has looked at extending the k-Means algorithm to incorporate background information in the form of instance level must-link and cannot-link constraints. We introduce two ways of specifying additional background information in the form of δ and constraints that operate on all instances but which can be interpreted as conjunctions or disjunctions of instance level constraints and hence are easy to implement. We present complexity results for the feasibility of clustering under each type of constraint individually and several types together. A key finding is that determining whether there is a feasible solution satisfying all constraints is, in general, NP-complete. Thus, an iterative algorithm such as k-Means should not try to find a feasible partitioning at each iteration. This motivates our derivation of a new version of the k-Means algorithm that minimizes the constrained vector quantization error but at each iteration does not attempt to satisfy all constraints. Using standard UCI datasets, we find that using constraints improves accuracy as others have reported, but we also show that our algorithm reduces the number of iterations until convergence. Finally, we illustrate these benefits and our new constraint types on a complex real world object identification problem using the infra-red detector on an Aibo robot.
Constrained clustering has been well-studied for algorithms like K-means and hierarchical agglomerative clustering. However, how to encode constraints into spectral clustering remains a developing area. In this paper, we propose a flexible and generalized framework for constrained spectral clustering. In contrast to some previous efforts that implicitly encode Must-Link and Cannot-Link constraints by modifying the graph Laplacian or the resultant eigenspace, we present a more natural and principled formulation, which preserves the original graph Laplacian and explicitly encodes the constraints. Our method offers several practical advantages: it can encode the degree of belief (weight) in MustLink and Cannot-Link constraints; it guarantees to lowerbound how well the given constraints are satisfied using a user-specified threshold; and it can be solved deterministically in polynomial time through generalized eigendecomposition. Furthermore, by inheriting the objective function from spectral clustering and explicitly encoding the constraints, much of the existing analysis of spectral clustering techniques is still valid. Consequently our work can be posed as a natural extension to unconstrained spectral clustering and be interpreted as finding the normalized min-cut of a labeled graph. We validate the effectiveness of our approach by empirical results on real-world data sets, with applications to constrained image segmentation and clustering benchmark data sets with both binary and degree-of-belief constraints.
Abstract. We explore the use of instance and cluster-level constraints with agglomerative hierarchical clustering. Though previous work has illustrated the benefits of using constraints for non-hierarchical clustering, their application to hierarchical clustering is not straight-forward for two primary reasons. First, some constraint combinations make the feasibility problem (Does there exist a single feasible solution?) NP-complete. Second, some constraint combinations when used with traditional agglomerative algorithms can cause the dendrogram to stop prematurely in a dead-end solution even though there exist other feasible solutions with a significantly smaller number of clusters. When constraints lead to efficiently solvable feasibility problems and standard agglomerative algorithms do not give rise to dead-end solutions, we empirically illustrate the benefits of using constraints to improve cluster purity and average distortion. Furthermore, we introduce the new γ constraint and use it in conjunction with the triangle inequality to considerably improve the efficiency of agglomerative clustering.
Abstract.Clustering with constraints is an active area of machine learning and data mining research. Previous empirical work has convincingly shown that adding constraints to clustering improves performance, with respect to the true data labels. However, in most of these experiments, results are averaged over different randomly chosen constraint sets, thereby masking interesting properties of individual sets. We demonstrate that constraint sets vary significantly in how useful they are for constrained clustering; some constraint sets can actually decrease algorithm performance. We create two quantitative measures, informativeness and coherence, that can be used to identify useful constraint sets. We show that these measures can also help explain differences in performance for four particular constrained clustering algorithms.
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