# Spacing Test for Fused Lasso

arXiv: https://arxiv.org/abs/2509.14229 · HTML: https://mlsanity.com/paper/2509.14229/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 13 references that are in the map.
- First posted on arXiv 17 Sep 2025.

## Builds on (most cited first, 3 of 3)

- [Exact post-selection inference, with application to the lasso](https://mlsanity.com/paper/1311.6238/) (2013)
- [Exact Post-Selection Inference for Sequential Regression Procedures](https://mlsanity.com/paper/1401.3889/) (2014)
- [A significance test for the lasso](https://mlsanity.com/paper/1301.7161/) (2013)

## Abstract (arXiv)

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 conservative spacing test of Tibshirani et al. (2016), previously regarded as an approximation, is in fact 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