Approximate Recovery in Changepoint Problems, from $\ell_2$ Estimation Error Rates

TL;DR

This paper demonstrates that in 1D changepoint detection, a fast $ ext{l}_2$ error rate guarantees approximate localization of true changepoints, and a simple post-processing step improves recovery accuracy, with applications to fused lasso and graph problems.

Contribution

It establishes that $ ext{l}_2$ error rates imply approximate changepoint recovery and introduces a post-processing method to eliminate spurious estimates, extending to graph-based problems.

Findings

Fast $ ext{l}_2$ error rates ensure true changepoint localization.

Post-processing improves the accuracy of changepoint estimates.

Results apply to fused lasso and graph changepoint detection.

Abstract

In the 1-dimensional multiple changepoint detection problem, we prove that any procedure with a fast enough $\ell_2$ error rate, in terms of its estimation of the underlying piecewise constant mean vector, automatically has an (approximate) changepoint screening property---specifically, each true jump in the underlying mean vector has an estimated jump nearby. We also show, again assuming only knowledge of the $\ell_2$ error rate, that a simple post-processing step can be used to eliminate spurious estimated changepoints, and thus delivers an (approximate) changepoint recovery property---specifically, in addition to the screening property described above, we are assured that each estimated jump has a true jump nearby. As a special case, we focus on the application of these results to the 1-dimensional fused lasso, i.e., 1-dimensional total variation denoising, and compare the implications with existing results from the literature. We also study extensions to related problems, such as changepoint detection over graphs.