Distribution free optimality intervals for clustering
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We address the problem of validating the ouput of clustering algorithms. Given data 𝒟 and a partition 𝒞 of these data into K clusters, when can we say that the clusters obtained are correct or meaningful for the data? This paper introduces a paradigm in which a clustering 𝒞 is considered meaningful if it is good with respect to a loss function such as the K-means distortion, and stable, i.e. the only good clustering up to small perturbations. Furthermore, we present a generic method to obtain post-inference guarantees of near-optimality and stability for a clustering 𝒞. The method can be instantiated for a variety of clustering criteria (also called loss functions) for which convex relaxations exist. Obtaining the guarantees amounts to solving a convex optimization problem. We demonstrate the practical relevance of this method by obtaining guarantees for the K-means and the Normalized Cut clustering criteria on realistic data sets. We also prove that asymptotic instability implies finite sample instability w.h.p., allowing inferences about the population clusterability from a sample. The guarantees do not depend on any distributional assumptions, but they depend on the data set 𝒟 admitting a stable clustering.
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