Selective inference after convex clustering with $ll_1$ penalization

Résumé: Classical inference methods fail when applied to data-driven test hypotheses. Selective inference is particularly relevant post-clustering, typically when testing a difference in mean between two clusters. Thus, dedicated methodologies are required to obtain statistical guarantees for these selective inference problems. In this work, we address convex clustering with $\ell_1$ 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. Work in collaboration with F. Bachoc (IMT) et P.Neuvial (CNRS/IMT).