Classical inference methods fail when applied to data-driven test hypotheses. Selective inference is particularly relevant post-clustering, typically when testing a difference in the mean between two clusters. Thus, dedicated methodologies are required to obtain statistical quarantees for these selective inference problems. In this work, we address convex clustering with l1 penalization, by leveraging related selective inference tools for regression, based on Gaussian vectors conditioned to polyhedral sets.
In the one-dimensional case, we prove a polyhedral characterization of obtaining given clusters, then enables us to suggest a test procedure with statistical guarantees. This characterization also allows us to provide a computationally efficient regularization path algorithm. Then, we extend the above test procedure and guarantees to some multi-dimensional clusterings. Our methods are implemented in the R package poclin.