发现一维融合lasso的变点检测可精确计算p值，无需复杂计算。
Spacing Test for Fused Lasso
- 利用LARS路径上唯一单向进入的变点结构，简化推断过程。
- 证明保守间距检验在融合lasso下为精确方法，无需近似。
- 结果简洁高效，适合统计推断与变点检测研究者使用。
一维信号中的变点检测是经典而基础的问题。融合lasso提供了一个优美的凸优化框架，能生成均值的分段常数估计，但对检测到的变点进行不确定性量化仍具挑战。后选择推断（PSI）提供了数据驱动选择后的有效p值计算方法，但其在融合lasso上的应用被认为计算复杂，需追踪正则化路径上的大量“命中”与“离开”事件。本文揭示一维融合lasso具有意外简单的几何结构：每个变点仅以严格单向方式进入，不存在离开事件。这一性质表明Tibshirani等（2016）提出的保守间距检验，此前被视为近似方法，实则为精确结果。选择性分布中的截断区域退化为仅由LARS路径下一个节点给出的单一下界。因此，精确的选择性p值可表示为闭式表达，形式与LARS/lasso设置中的简单间距统计量完全一致，无需额外计算。该发现确立了广义lasso中罕见的可闭式表达的精确后选择推断案例。通过模拟与真实数据验证，结果证实其精确校准与高检验功效。
原文摘要 · Abstract (English)
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
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