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.

翻译：本文探讨聚类算法输出验证问题。给定数据集$\mathcal{D}$及其被划分为$K$个聚类的分区$\mathcal{C}$，何时能判定所得聚类的正确性或对数据的意义？本文提出一种新范式：当聚类$\mathcal{C}$在损失函数（如K均值失真）意义上表现良好，且具有稳定性（即仅在有微小扰动时仍为最优聚类）时，则认为该聚类具有意义。此外，我们提出一种通用方法，可在后验推断中为聚类$\mathcal{C}$提供近似最优性和稳定性的保证。该方法可应用于存在凸松弛形式的各种聚类准则（亦称损失函数），获得保证仅需求解一个凸优化问题。通过在真实数据集上对K均值与归一化割两种聚类准则进行验证，我们展示了该方法的实际应用价值。我们还证明渐近不稳定性可高概率推导出有限样本不稳定性，从而允许从样本推断总体的聚类倾向性。这些保证无需依赖任何分布假设，但要求数据集$\mathcal{D}$存在稳定聚类。