Selective inference after convex clustering with $\ell_1$ penalization

TL;DR

This paper develops a selective inference framework for convex clustering with  penalization, providing statistical guarantees and efficient algorithms for testing differences in cluster means.

Contribution

It introduces a polyhedral characterization for clustering with  penalization, enabling valid statistical tests and efficient computation in both one-dimensional and multi-dimensional settings.

Findings

Validated statistical guarantees through numerical experiments

Demonstrated power to detect mean differences between clusters

Implemented methods in the R package poclin

Abstract

Classical inference methods notoriously fail when applied to data-driven test hypotheses or inference targets. Instead, dedicated methodologies are required to obtain statistical guarantees for these selective inference problems. Selective inference is particularly relevant post-clustering, typically when testing a difference in mean between two clusters. In this paper, 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, than 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 multi-dimensional clustering with $\ell_1$ penalization, and also to more general multi-dimensional clusterings that aggregate one-dimensional ones. With various numerical experiments, we validate our statistical guarantees and we demonstrate the power of our methods to detect differences in mean between clusters. Our methods are implemented in the R package poclin.