Spacing Test for Fused Lasso

Detecting changepoints in a one-dimensional signal is a classical yet fundamental problem. The fused lasso provides an elegant convex formulation that produces a stepwise estimate of the mean, but quantifying the uncertainty of the detected changepoints remains difficult. Post-selection inference (PSI) offers a principled way to compute valid $p$-values after a data-driven selection, but its application to the fused lasso has been considered computationally cumbersome, requiring the tracking of many hit'' and leave’’ events along the regularization path. In this paper, we show that the one-dimensional fused lasso has a surprisingly simple geometry: each changepoint enters in a strictly one-sided fashion, and there are no leave events. This structure implies that the so-called \emph{conservative spacing test} of Tibshirani et al.\ (2016), previously regarded as an approximation, is in fact \emph{exact}. The truncation region in the selective law reduces to a single lower bound given by the next knot on the LARS path. As a result, the exact selective $p$-value takes a closed form identical to the simple spacing statistic used in the LARS/lasso setting, with no additional computation. This finding establishes one of the rare cases in which an exact PSI procedure for the generalized lasso admits a closed-form pivot. We further validate the result by simulations and real data, confirming both exact calibration and high power. Keywords: fused lasso; changepoint detection; post-selection inference; spacing test; monotone LASSO