Distribution free optimality intervals for clustering

• URL: http://arxiv.org/abs/2107.14442v1
• Date: Fri, 30 Jul 2021 06:13:56 GMT
• Title: Distribution free optimality intervals for clustering
• Authors: Marina Meil\u{a}, Hanyu Zhang
• Abstract summary: Given data $mathcalD$ and a partition $mathcalC$ 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 $mathcalC$ 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.
• Score: 1.7513645771137178
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We address the problem of validating the ouput of clustering algorithms. Given data $\mathcal{D}$ and a partition $\mathcal{C}$ 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 $\mathcal{C}$ 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 $\mathcal{C}$. 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 $\mathcal{D}$ admitting a stable clustering.

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