Spacing Test for Fused Lasso

TL;DR

The paper tackles uncertainty quantification after changepoint detection via the one-dimensional fused lasso. It reveals that the fused-lasso path in 1D has a simple, one-sided selection geometry, making the conservative spacing test exact and yielding a closed-form selective p-value identical to the LARS spacing statistic. This leads to exact calibration under the null and high power against alternatives, while maintaining linear-time-like computation and scalability. The work validates the theory numerically and on real data (array-CGH), and discusses broader implications for graph-structured regularization and model-selection-aware inference.

Abstract

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

Figures (10)

• Figure 1: The fused lasso promotes identical values across neighboring coefficients, merging adjacent segments when supported by the data. In the illustration, the merged segments are highlighted in bold red.
• Figure 2: PSI in a nutshell: once a data-driven rule selects a model, the reference distribution becomes a truncated normal, not a full one.
• Figure 3: LARS solution path in the $(\lambda,\beta)$ plane (schematic; not to scale). The red polyline shows the piecewise-linear trajectory of $\beta(\lambda)$ as $\lambda$ decreases from $\lambda_1$ to $\lambda_p=0$. Vertical dashed guides mark the knots $\lambda_1>\lambda_2>\cdots>\lambda_p=0$, and horizontal guides mark the corresponding coordinates in $\beta$. Note that the vertical axis represents the $p$-dimensional coefficient vector $\beta(\lambda)\in\mathbb{R}^p$; for visualization we depict its coordinates stacked as parallel levels (a schematic representation rather than a literal 2D plot).
• Figure 4: Polyhedral lemma (schematic). For fixed $z=(I-c\eta^\top)y$, the selection event $\{Ay\le b\}$ restricts the scalar statistic $\eta^\top y$ to the interval $[\nu^{-}(z),\nu^{+}(z)]$; the gray band indicates this feasible interval for a given $z$.
• Figure 5: Equivalence of selective $p$-values between our lower-envelope formulation and the hit/leave conditioning of Hyun2018.

Theorems & Definitions (10)

• Proposition 1: Leeetal
• Proposition 2: TibshiraniPSI
• Proposition 3: TibshiraniPSI
• Theorem 1: Order preservation and knot identity
• proof
• Corollary 1
• Theorem 2: Lower endpoint equals the next knot
• proof : Proof
• Remark 1: Exactness of the conservative spacing test
• Theorem 3: Knot identity and one-sided selection