Selective inference for $k$-means clustering

We consider the problem of testing for a difference in means between clusters of observations identified via $k$-means clustering. In this setting, classical hypothesis tests lead to an inflated Type I error rate. In recent work, Gao et al. (2022) considered a related problem in the context of hierarchical clustering. Unfortunately, their solution is highly-tailored to the context of hierarchical clustering, and thus cannot be applied in the setting of $k$-means clustering. In this paper, we propose a p-value that conditions on all of the intermediate clustering assignments in the $k$-means algorithm. We show that the p-value controls the selective Type I error for a test of the difference in means between a pair of clusters obtained using $k$-means clustering in finite samples, and can be efficiently computed. We apply our proposal on hand-written digits data and on single-cell RNA-sequencing data.