Spacing Test for Fused Lasso

This study addresses the unresolved problem of selecting the regularization parameter in the fused lasso. In particular, we extend the framework of the Spacing Test proposed by Tibshirani et al. to the fused lasso, providing a theoretical foundation for post-selection inference by characterizing the selection event as a polyhedral constraint. Based on the analysis of the solution path of the fused lasso using a LARS-type algorithm, we derive exact conditional $p$-values for the selected change-points. Our method broadens the applicability of the Spacing Test from the standard lasso to fused penalty structures. Furthermore, through numerical experiments comparing the proposed method with sequential versions of AIC and BIC as well as cross-validation, we demonstrate that the proposed approach properly controls the type I error while achieving high detection power. This work offers a theoretically sound and computationally practical solution for parameter selection and post-selection inference in structured signal estimation problems. Keywords: Fused Lasso, Regularization parameter selection, Spacing Test for Lasso, Selective inference, Change-point detection